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ACCRETION DISC INSTABILITIES IN CATACLYSMIC

VARIABLE STARS

A thesis submitted

for the degree of

Doctor of Philosophy

by

R ebecca W ynn

Astronomy Group

Department of Physics and Astronomy

University of Leicester

November 2000

UMI Number: U133235

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Acknowledgments

I would like to thank Professor Andrew King for his supervision during the course

of this work. Many thanks also go to Dr Rudolf Stehle, Dr James Murray and Dr

Graham Wynn, for allowing me use of their computer codes, and to Norma, for all

her help during my time at Leicester University. I acknowledge receipt of a PPARC

studentship, and use of STARLINK facilities.

A large amount of gratitude goes to my parents for their unlimited guidance and

support throughout my time as a student. Thanks to everyone for livening up life

in the underpass (Fraser, Louise, Mike, Chris, Anastasia, James, Klaus, Rudi, Paul

and Unal), and cheers to everyone who has helped me unwind after work over a beer

or two.

Finally, the biggest thanks go to Graham, for keeping me sane, and helping me

believe in myself.

Accretion Disc Instabilities in Cataclysmic Variable Stars

Rebecca Wynn

November 2000

A ID hydrodynamical code is used to model the viscous evolution of VY Scl stars, which are a subclass of Cataclysmic Variable. Low states arise as a result of occasional drops in the mass transfer rate, which probably result from the passage of starspots across the inner Lagrangian point on the secondary star. The model includes the heating of the accretion disc by irradiation from the white dwarf and shows that outbursts from the low state can be suppressed if the temperature of the white dwarf is sufficiently high (Twd ~ 40 000 K).

A magnetic propeller model is used to show that the quiescent value of the viscosity parameter of the accretion disc within WZ Sge is likely to be c oid ~ 0.02, in agreement with estimates of Q:Cold f°r other dwarf novae. Assuming the white dwarf in WZ Sge to be weakly magnetic it is shown that, in quiescence, material close to the white dwarf can be propelled to larger radii, depleting the inner accretion disc. This has the effect of stabilizing the inner disc and allowing the outer disc to accumulate mass. Numerical models yield an estimated recurrence time of tree ~ 30 ± 10 yr, in agreement with the observed recurrence time of trec ~ 33 yr. The model is also used to follow WZ Sge through outburst, producing lightcurves that are in good agreement with observation.

Finally, high-speed K-band photometry of WZ Sge is presented. Analysis of the data reveals a strong oscillation at 27.88 ± 0.01 s, along with weaker oscillations at slightly longer periods. The principal oscillation is attributed to the presence of a rapidly rotating weakly magnetic white dwarf, and possible explanations for the weaker oscillations are discussed. The long term brightness variation in the K-band lightcurves is analysed, providing tentative evidence of a precessing, elliptical disc. The observational properties can be explained if the white dwarf possesses a weak magnetic field.

Contents

Acknowledgments ii

1 Introduction 2

1.1 Accretion in Binary System s................................................................... 2

1.2 Mass transfer in C V s ............................................................................... 4

1.3 Non-magnetic Cataclysmic V ariab les .................... 8

1.3.1 Accretion disc form ation ................................ 8

1.3.2 Observational evidence for accretion discs . ........................... 13

1.4 CV classification ..................................................................................... 16

1.5 Thermal-viscous disc in s tab ilitie s .......................... 19

1.6 Accretion in magnetic C V s...................................................................... 23

1.6.1 The role of the magnetic f i e ld ...................................................... 25

1.7 Magnetic Cataclysmic Variable Classification ........................... 25

1.7.1 P o la r s .............................................................................................. 25

1.7.2 Intermediate P o lars ............................................ 26

1.8 CV e v o lu tio n ........................................................................................... 27

1.9 Thesis sum m ary........................................................................................ 32

iii

2 Numerical techniques 33

2.1 Introduction............................................................................................... 33

2.2 Eulerian Grid C o d e . 33

2.2.1 Radial Disc Structure.............................................................. 34

2.2.2 Vertical Disc Structure - the / — £ re la tio n s ............................ 37

2.2.3 Approximations for the / — £ r e la t io n ............................... 38

2.2.4 The hot b ra n c h ....................................................................... 38

2.2.5 The unstable b ra n c h ..................................................................... 39

2.2.6 The cool, optically thin branch .................................................. 40

2.2.7 The cool, optically thick b ra n c h .................................................. 40

2.2.8 Solving the partial differential equations on an Eulerian grid . 41

2.2.9 Van Leer Upwind differencing Scheme......................................... 43

2.2.10 Boundary Conditions..................................................................... 45

2.3 Smoothed Particle Hydrodynamics ....................................................... 46

2.3.1 Kernel In te rp o la tio n ................ 47

2.3.2 Equations of m o t io n .................................................................... 48

2.3.3 SPH artificial v iscosity ................................................................. 49

2.3.4 Time stepping................................................................................. 49

2.3.5 Boundary Conditions ..................................................... 50

2.4 H y d isc ..................................................................................................... . 51

2.4.1 Equations of M o tio n ................................................................ . 52

2.4.2 Particle-particle Interactions ..................................................... 54

2.4.3 Boundary conditions and time-stepping............................... 55

2.5 Chapter Summary ................................................................................... 56

iv

3 The Low States of VY Scl stars 57

3.1 Introduction............................................................... 57

3.2 The disc m o d e l ............................................................ 60

3.2.1 Radial disc s tru c tu re .................................................................... 60

3.2.2 Vertical disc s t ru c tu re .................................. 63

3.2.3 Numerical M e th o d .................................................... 63

3.2.4 Model Parameters ........................................... 63

3.3 Results............................................................................ 64

3.3.1 Cool white dwarf m o d e l .............................................................. 65

3.3.2 Hot white dwarf m odel................................................................. 67

3.3.3 Discussion....................................................................................... 69

3.4 Conclusions............................................................................................... 71

4 M agnetically-Driven Outbursts of WZ Sagittae 73

4.1 Introduction............................................................................................... 73

4.2 The quiescent value of a in WZ S g e ...................................................... 77

4.2.1 The quiescent disc mass as a limit on Ofcoi d ................................. 77

4.2.2 Extracting a Coid from the outburst lightcurve ........................... 79

4.3 A magnetic solution?................................................ 82

4.3.1 WZ Sge as a magnetic propeller................... 82

4.3.2 Calibration of the numerical viscosity......................................... 83

4.3.3 The magnetic m o d e l ............................................... 84

4.3.4 Numerical re su lts ............................................... 88

4.4 Observational Consequences................................................................... 92

v

4.4.1 Truncated D iscs....................................................................... 92

4.4.2 Superhumps................................................................................ . 95

4.5 The spin evolution of the white dw arf..................................................... 98

4.6 Conclusions.................................................................................................. 100

5 WZ Sge in outburst 103

5.1 Introduction............................................................................................. . 103

5.2 Modelling DN outbursts using S P H ...........................................................104

5.2.1 Magnetic m odel........................................................................ 107

5.2.2 System Param eters.......................................................................... 109

5.2.3 Numerical S calings................................................................. 109

5.3 R esults............................................................................................................I l l

5.3.1 Disc A na ly sis .............................................................................. . 116

5.4 Conclusion............................................................................................. ... . 119

6 Photom etric Techniques 120

6.1 Introduction.......................................................................................... ... . 120

6.2 The Photometric S c a le ......................................................................... ... . 120

6.3 M agnitudes...................................................................................................121

6.4 Photometric C alib ration ............................................................................. 122

6.5 Infrared C am eras......................................................................................... 124

6.6 Image reduction............................................................................................ 124

6.7 Aperture Photom etry............................ 126

6.8 E r r o r s ................................................................................................... ... . 126

6.9 Conclusions.................................................................................................. 127

vi

7 K-band observations of WZ Sagittae 128

7.1 Introduction................................................................. 128

7.2 Observations and data reduction................................................................ 131

7.3 K-band Lightcurves......................................................................................132

7.4 Short period oscillations.............................................................................134

7.4.1 Lomb-Scargle periodogram s............................... 134

7.4.2 Discussion of the oscillations in WZ S g e ................ 135

7.4.3 WZ Sge as a magnetic propeller.............................. 139

7.5 The search for superhumps in quiescence................................................. 142

7.6 A comparison with SPH lightcurve simulations ............................144

7.6.1 Numerical tech n iq u e .......................................................................144

7.6.2 Generating lightcurve p ro file s ....................................................... 145

7.6.3 Results.............................................................. 147

7.7 Conclusions.................................................................................................. 152

7.8 REFEREN CES........................................................... 153

1

Chapter 1

Introduction

1.1 Accretion in Binary System s

Accretion discs appear in many important astrophysical situations. They are re

sponsible for the dramatic outbursts of binary systems containing an accreting black

hole, neutron star or white dwarf (WD). The extreme luminosities of the nuclei of

active galaxies are thought to arise from an accretion disc around a supermassive

black hole. Accretion discs also play a pivotal role in the early stages of star and

planet formation in young stellar objects (YSOs), with direct evidence for accre

tion discs in Hubble Space Telescope observations of YSOs, such as (3 Pictoris. The

accretion processes in these systems share a lot of common traits with the binary

systems which are the focus of this thesis.

Binaries, by their nature, reveal more about themselves than other astronomical

objects, and are therefore an extremely important class of accreting system. In

particular, we are able to gain information about the masses and dimensions of

these systems. For example, the binary separation a, is given in terms of the orbital

Introduction 3

period, P, a fundamental observable quantity, via Kepler’s law:

4t t2o 3 = G (M, + M2) M0P 2 (1.1)

where Mi and M2 are the masses of the two stars, expressed in solar mass units.

Here we have assumed that the stars execute circular orbits. Tidal forces make this

a good approximation for binary systems (see eg. Warner 1995 for a full review of

this topic). The importance of accretion in binary systems is further emphasized by

the fact that a majority of stars are likely to be members of binaries, which, at some

stage in their evolution will undergo mass transfer.

In this thesis I shall focus on accretion discs in Cataclysmic Variable Stars (CVs),

which are binary systems consisting of an accreting white dwarf and a lower main

sequence (LMS) companion. CVs are of particular interest since the accretion pro

cesses taking place lead to spectacular increases in brightness (hence the name Cat

aclysmic Variable). Understanding the physical processes responsible for these dra

matic changes in brightness (known as outbursts) is crucial to our knowledge of

accretion discs in CVs, as well as in other astrophysical systems.

CVs provide an ideal astrophysical laboratory for accretion disc studies. They

are plentiful, bright systems and are therefore easy to observe. In addition, the

time-scales that are of interest are observationally favourable. Generally, the orbital

periods of CVs lie in the range 80 min to several hours, making it possible to cover

several orbital periods in a single night of observations. Disc outbursts typically last

a few days and occur at intervals of weeks to months. In astrophysical terms, these

are short time-scales.

Finally, CVs probably provide one of the best opportunities to study the accretion

Introduction 4

cmCO

Test ptle.

CO

Figure 1.1: Schematic picture of a test particle placed in the gravitational field of a binary system with a white dwarf of mass Mi and a LMS star of mass M2. The centre of mass of the system is labelled as cm.

process in isolation since the accretion discs tend to be bright, and other sources of

luminosity in the system (in particular the LMS star) are relatively unimportant.

1.2 Mass transfer in CVs

Given that the orbital periods of CVs lie in the range 80 min to several days, it is

more convenient to express equation (1.1) in the form

a = 3.5 x 1010M y3(l + q)1 P ^ c m (1 .2)

where Phr is the orbital period in hours, and q is the mass ratio

M2Q = Mi (1.3)

where Mi and M2 are the primary (WD) and secondary (LMS) masses, expressed

in solar mass units. For typical CVs, with a WD (typically Mi ~ M0 , R ~ 109

cm), and a dwarf secondary, equation (1.2) gives a value typically less than a solar

diameter.

Introduction 5

The small separation between the stars in CVs means that the gravitational

field of the WD is able to disrupt the outer layers of the LMS star, resulting in

mass flowing towards the WD. The conditions for mass transfer can be found by

considering the motion of a test particle in the gravitational field of the system (see

fig 1.2). This problem was first studied in the 19th century by Edouard Roche in

connection with the destruction or survival of planetary satellites, and is commonly

known as the Roche Problem. In the Roche approach, it is assumed that the two

massive stars execute circular orbits around their common centre of mass. This is a

good approximation for CVs since any eccentricities in the orbit are eliminated by

tidal effects on time-scales far shorter than the mass transfer time-scale. In addition,

it is assumed that both stars are centrally condensed so that they can be treated as

point masses (again this is a good approximation), and are massive enough to be

unaffected by the gravitational field of the test particle.

Motion of gas between the stars is determined by the the Euler equation (set

up in the frame of reference rotating with the binary orbit such that both stars

remained fixed) which is given by

dw/dt + (v.V)v = — V<1> — 2 ( w A v ) - -V P (1.4)P

where u is the angular velocity of the WD and LMS star around their common centre

of mass. The term — 2(u A v) is the Coriolis force per unit mass (this is required

since we are in the co-rotating frame), and <E> is the effective potential seen by the

test particle,

j r . / \ G M \ G M 2 1 / \ 0 / \

= “ 9 (w A r ) L 5jr — ri | [r — F2 1 2,

and is the sum of the gravitational potentials as well as the centrifugal potential, ri

Introduction 6

and r 2 are the position vectors of the centres of the two stars, the centre of mass

being the origin. The potential 3> is known as the Roche potential, and results from

the Roche problem described above.

It is very useful to consider the equipotential surfaces of <F, which also determine

the shape of both stars. Figure 1.2 shows the equipotential surfaces in the orbital

plane. It should be noted that some of the forces acting on the accreting gas, in

particular the Coriolis force are not represented by <F. The shape of the equipo-

tentials are governed by the mass ratio q (although the general features remain the

same), whilst the linear scale is determined by the binary separation a. The most

interesting surface is that at which the equipotential surfaces of both stars merge.

The two areas, known as Roche Lobes, join at the inner Lagrangian point L\. If a

star is close to filling its Roche Lobe, material close to L\ finds it much easier to

pass through to the Roche Lobe of the other star than to escape the system, and

mass transfer occurs. In CVs, the LMS star fills its Roche Lobe, and transfers mass

to the WD via a stream centered on the Li point.

It can be seen from figure 1.2 that the secondary star is distorted from a sphere.

For convenience, the radius of the LMS star in a CV is defined as the radius of a

sphere having the same volume as the Roche Lobe. This has to be calculated numer

ically since $ is a complicated function. For 0.1 < q < 0.8, a good approximation is

given by Paczynski (1971),

Combining equations (1.2) and (1.6), we find that the mean density p of the lobe

(1.6)

Introduction 7

Figure 1.2: Equipotential surfaces of a binary system (taken from Pringle and Wade, 1985). The surface which defines the Roche Lobe passes through Li, the inner Lagrangain point.

filling star is determined solely by the binary period P, i.e,

^ = H " i i 5p- gcm_3 ™

In the case of an LMS star, the radius is directly proportional to the mass, and we

find

M2 ~ 0.11Phr. (1.8)

Hence, by simply measuring the orbital period, we are able to obtain an estimate of

the mass of the secondary star.

The Roche problem has illustrated that if a secondary star fills its Roche Lobe,

any small perturbations in the vicinity of Li (these are always present, caused by e.g.

pressure forces) will cause gas to flow through to the Roche Lobe of the primary.

Mass transfer will continue via a stream centered on the Li point so long as the

secondary remains in contact with its Roche Lobe. In the following sections of

Introduction 8

this chapter I shall consider in more detail the fate of the transferred mass in both

non-magnetic and magnetic CVs.

1.3 Non-m agnetic Cataclysmic Variables

1.3.1 Accretion disc formation

In this section I shall consider the fate of the accretion flow in the case where the

white dwarf does not possess a magnetic field strong enough to influence the motion

of the gas. A consequence of Roche Lobe overflow is that the accretion stream has

a high specific angular momentum and the WD sees the accretion flow as a stream

being squirted through a nozzle which rotates around in the binary plane. In the

case of CVs (which have fairly short orbital periods), the orbital angular velocity u

causes the stream (as seen from the WD) to move almost orthogonally to the line

joining the centre of the two stars. In a non-rotating frame of reference the velocity

components along and perpendicular to the instantaneous line of centres (i>|| and v±

respectively) are,

V||^cs (1.9)

and

v±_ ~ b\U (1-10)

where b\ is the distance from the WD to the L\ point, and cs is the sound speed in the

envelope of the secondary star. The accretion stream is presumably pushed through

Li by pressure forces, and v\\ is therefore unlikely to be greater than about lOkms-1

for normal stellar envelope temperatures (< 105 K). An order of magnitude estimate

Introduction 9

file : stream .002 property : dens file : stream .004 property : dens file : stream .006 property : dens

o

o

o

1- 0 .4 -0 .2 0 0.2 0.4

o

o

o

?oI

■0.4 0.2 0.40.2 0

o

o

o

?■4*?

-0 .4 -0 .2 0.2 0.40N = 3451 T = 0.2 N = 6586 T = 0.4 N = 9696 T = 0.6

file : stream .008 property : dens

-0 .4 -0 .2 0 0.2 0.4

N = 12769 T = 0.8

file : stream .010 property : dens file : stream .020 property : dens

-0 .4 -0 .2 0 0.2 0.4

N = 15810 T = 1

-0 .4 -0 .2 0 0.2

N = 30999 T = 2

Figure 1.3: Numerical simulation (using HyDisc, as described in chapter 2) showing the formation of a ring of material at Rcirc-

for v± can be obtained by using the formula of Plavec and Kratochvil (1964),

b\ « (0.5 — 0.2271ogg) a. (1.11)

Combining equations (1.11) and (1.2) we obtain

v± ~ lOOM^3 (1 + q)1 Pd"ay/3 km s-1 (1-12)

Hence the accretion stream passing through L\ is highly supersonic. The stream

of material will be further accelerated by the gravitational field of the WD, and

therefore the flow will remain supersonic as it travels further into the primary’s

Roche Lobe. This allows pressure forces to be neglected (since they are only able

to travel through the gas at the sound speed), and the stream therefore follows a

Introduction 10

ballistic trajectory that is governed by Mi alone. The mass flow can be treated, to

a good approximation, as a stream of test particles which will adopt an elliptical

orbit around the WD. The orbit precesses slowly due to the gravitational field of the

LMS star, and hence the continuous accretion stream will intersect itself, resulting in

shocks (due to the supersonic nature of the stream) which dissipate energy. Hence the

material adopts the minimum energy configuration for a fixed angular momentum,

which is a circular Keplerian orbit (this process is depicted in figure 1.3). The

Keplerian orbit at Rdrc has the same specific angular momentum as the stream had

on passing through L\ (since the gas has had little opportunity to get rid of angular

momentum), allowing Rdrc to be written as

Utilising equations (1.11), and (1.1), equations 1.13 and 1.14 can be combined to

give

The circularization radius Rcjrc is therefore a function of both q and a. In the case

of CVs, Rcirc is always larger than the WD radius, and lies well within the primary’s

Roche Lobe, allowing a ring of material to accumulate.

Collisions of gas elements, shocks and viscous dissipation will cause energy dis

sipation, which will result in energy being radiated away from the ring. This loss of

energy forces the gas deeper down into the the potential well of the WD, which in

Rcirc^(f> (-^circ) — b- UJ (1.13)

where ity, the circular velocity, is given by

( d \ — (V(l> {R c irc ) — ( p ) ' -K'circ '-circ

(1.14)

R c i r c / a = (1 + q) [0.5 - 0.2271og9]4. (1.15)

Introduction 11

Cool star

White DwarfMass stream

Accretion disc

Bright Spot

Figure 1.4: Disc formation in a binary system. Material passing through the L\ point hits the outer edge of the accretion disc, forming a bright spot.

turn requires it to lose angular momentum. Therefore, the ring of gas redistributes

its angular momentum (by means of viscous torques), with a net outward flow of

angular momentum. The time-scale for angular momentum distribution is longer

than both the dynamical time-scale of the material, as well as the radiative cooling

time-scale. The inner region of the ring will therefore spiral slowly towards the white

dwarf via a series of perturbed Keplerian orbits, whilst the outer parts of the ring

will gain angular momentum, and spiral outwards. Thus the gas spreads gradually

into a viscously driven accretion disc (this is depicted in figure 1.4). As the accretion

disc expands, angular momentum will eventually be removed from the outer edge

of the accretion disc and transmitted back to the binary orbit via tidal interaction

with the secondary, limiting the outer disc radius. Once the accretion disc is estab

lished, material in the accretion stream will hit the outer edge of the accretion disc,

resulting in a bright spot in the disc (see fig 1.4).

The nature of the viscosity in accretion discs is still unknown. Recent results

Introduction 12

by Hawley and Balbus (1995) have led to an encouraging model in which the vis

cosity arises from a magneto-hydrodynamical instability: seed fields are amplified

by Kepler rotation of the disc. Stresses in the field lines then transport angular

momentum between disc annuli. Whilst this model looks promising, it is still to be

fully developed on the global scale of an accretion disc.

In the absence of any fully developed, global model for accretion disc viscosity,

theories currently rely on a parameterization. In this approach (commonly known

as the a parameterization) the kinematic viscosity of the disc is approximated as

v ~ aCsH, where H is the scale height in the disc, and cs is the local sound speed

(a full review is presented in Frank, King and Raine 1992). This approach has led

to many successful developments in the study of accretion discs.

Accretion discs are a powerful source of energy. Consider a gas element of mass

m that is just grazing the WD surface. The binding energy of the element is given

by

E - r i r 11161

Assuming that the element starts with negligible binding energy far from the star,

the total disc luminosity (assuming the disc to be in a steady state), is

GM iM 1 ,sc _ 2R\ ~ 2 ° (1-17)

where M is the accretion rate and Lacc is the accretion luminosity. The rest of the

Lacc is released very close to the star, in a thin boundary layer. Hence the accretion

disc is an extremely efficient way of extracting the available energy.

Introduction 13

1.3.2 Observational evidence for accretion discs

CVs contain small faint secondary stars, and are therefore ideal systems in which to

observe the accretion disc. CVs provide the best observational evidence of accretion

discs, and provide a good comparison for accretion disc theory.

The most basic observational requirement is to find evidence for circular motion

of material around the WD. This can be seen in the strong H and He emission lines in

CVs. If the inclination is sufficiently high that we are observing the system almost

edge on, the emission lines are found to be double peaked with extensive wings.

This is exactly what is expected for a rotating disc of optically thin gas. Velocity-

broadening of the line gives the projected circular velocity v sim of the outer edge

of the disc, whilst the broad wings are a good indication of Keplerian motion, with

the velocity increasing as material gets closer to the WD (for a detailed review see

Warner 1995).

Eclipsing systems show even stronger evidence for accretion discs. Spectra taken

during eclipse ingress show no blueward peaks, but they gradually return, and at

eclipse egress, the redward peaks disappear. The only explanation for these observa

tions is that the gas is in roughly circular motion (and orbiting in the same sense as

the secondary) about the white dwarf. Eclipsing systems where the inclination and

masses can be obtained show that the emitting gas extends from close to the white

dwarf to near the WD Roche Lobe radius (Warner 1995), exactly what is predicted

from accretion theory.

More recently, Doppler tomography techniques have been used to obtain velocity

distribution maps of CVs (eg Marsh and Horne 1988). At each orbital phase we are

Introduction 14

able to obtain a ID velocity distribution from the observed line profile. Therefore,

if we measure the line profiles at successive orbital phases, we can combine the ID

velocity profiles and produce a single 2D velocity map. The resulting velocity map,

known as a Doppler tomogram, is plotted in the frame co-rotating with the binary,

with the centre of mass usually placed at the origin (see figure 1.5 for a schematic

diagram). The equations required to transform the observed velocities (vx,i,vy,i)

from the inertial frame to the co-rotating frame (vX)Co5 %co) are

Vx,co = cos (ut)vxj + sin {ut)vy,i (1.18)

vy,co = —sin (ut)vxj + cos (ut)vyj (1-19)

Inversion to an image in spatial co-ordinates is complicated, and is best performed

by the construction of Doppler tomograms from theoretical models. However, evi

dence for accretion discs have been seen in the Doppler tomograms of many CVs, in

the form of a diffuse ring centered on the WD (see eg. Kaitchuck et al 1994).

Doppler tomography is able to go much further than merely predicting the pres

ence of accretion discs. It is an extremely useful technique for understanding the

dynamics of accretion discs, and their interaction with the accretion stream. Re

cently, the method has led to the discovery of spiral shock waves in outbursting CVs,

(Steeghs k Stehle 1999) a feature that has been firmly predicted in accretion theory.

Eclipse mapping techniques can be used to gain insight about the spatial distri

bution of the accretion discs in CVs (eg. Horne 1985). Since the secondary star fills

its Roche Lobe, the system geometry can be specified given i (the system inclination)

and q. By assuming an arbitrary brightness distribution for the disc (in fitting with

Introduction 15

2

1

02000 1000 1000 20000

Velocity (km /s)

Inner edge of disc

Position coordinatesVelocity coordinates

Figure 1.5: The top plot illustrates the double-peaked line emission expected from a Keplerian accretion disc. The bottom left plot shows a schematic diagram of a typical Doppler tomogram for a CV (taken from Marsh and Horne 1988); the main features are the Roche Lobes of the two stars and the disc emission, which can be seen as a diffuse ring centered on the origin. The bottom right hand plot demonstrates how points on the Doppler tomogram transform into spatial coordinates.

Introduction 16

the system geometry), the system lightcurve can be predicted. The most likely mass

distribution is determined by varying the assumed distribution until a closest match

to the observed lightcurve is discovered. If a x 2 fit is used, there are an extremely

large number of distributions that will give an equally good fit. The maximum en

tropy method (Horne, 1985) adds the additional constraint that the disc has to be

as smooth as possible, which is measured in terms of the image entropy.

If a system is observed simultaneously at many different wavelengths, it is possible

to obtain a spectrum at any point in the map. An important result of this technique

is the observed temperature distribution of the accretion discs. The temperature

distribution for the Dwarf Novae Z Cha in outburst (where the disc is believed to

be in a steady state) is shown to be Te oc R ~3/4 (eg Horne & Cook, 1985), in good

agreement with accretion theory.

1.4 CV classification

The long term lightcurves of Cataclysmic variables vary widely from one system to

another, and are classified accordingly into the following sub-types (for a detailed

review see Warner 1995):

(a) Classical Novae (CN). These systems have undergone only one observed

eruption. CNs undergo the largest brightness changes found in CVs; the range

from pre-nova brightness to maximum brightness ranges from 6 to greater than

19 magnitudes. The amplitude of the eruptions is correlated to the rate at which

the nova declines after maximum; the largest eruptions last the shortest duration

Introduction 17

(termed fast novae), whilst the lower amplitude eruptions sometimes last for many

years (slow novae).

CN eruptions are understood to be thermonuclear runaways of hydrogen accreted

onto the WD surface. The accreting hydrogen forms a hot atmosphere on the surface

of the secondary star, which eventually undergoes unstable hydrogen burning (re

sulting in fusion). This leads to a thermonuclear explosion in which the gas, which

contributes only a tiny fraction of the WD mass (~ 10-4 — 10-5Mo) is carried away

in a shell rich in fusion products.

(b) Dwarf Novae (DN). DN undergo outbursts of typically 2 — 5 magnitudes,

and are more common and outburst more frequently than CN. The interval between

outbursts is typically weeks to months, whilst the outbursts themselves last between

2 — 20 days. As an example, the longterm lightcurve of the DN SS Cyg is shown in

figure 1.6. There are three distinct subtypes of DN:

(i) Z Cam systems show occasional standstills in their lightcurves, with the mag

nitude around 0.7 mag lower than the maximum brightness. The standstills last

from tens of days to years, during which time no outbursts are observed.

(ii) SU UMa stars are identified as DN whose lightcurves undergo superoutburst

behaviour. Superoutbursts are typically 1 magnitude brighter and have a dura

tion several times longer than normal outbursts. Superoutbursts typically occur at

intervals of several months, interspersed with normal outbursts.

(iii) U Gem stars include all DN that are neither Z Cam or SU UMa stars.

DN outbursts are understood to be a release of gravitational energy, caused by

a temporary enhancement in the rate of mass transfer through the accretion disc.

The reasons for this are discussed in section 1.5.

Introduction 18

e ' i I * M ! ! ! >. !• t I » * * » * • i • ; • I i: ♦< \ ' . . 1 1 *, " *

12 i t W v f f W ' ^ v i»vi *K V t - ihiv'vi W u W i* y W '^ ^8 : ------------------------------- mw .a r A , . * i i . a < . i . i * \

W' u *

!4

12

• M h o i ‘ i i n i < ; • 1 1>< t i » ; i ' \ i V . ; .1; •, *, > * i p , . i». >

I ' ' <) '> 1:1 ■''L| \' 1"' ii'l •'!.’• r! 1 i i '; :* !i; i MW' M it V w {i v? W w i / *< V m w i »'>>'*'» i i W * 'a w W * wv

12 8

m :i \ - •. ! 1 U 1 f • > ! SI V s '• ?, • ! ) I '• ' 1 ! ‘i ! * : \ ' > ! S, ! Ji < ; i . 1 ! ! •' : J •; : J ! : ? :'■ £ • i 1 : • : : ■ • ; '• ."

: ! •! '1 P J . i l ^ U ' .vjW U ^L V ' I 1" ! ' b , ’i* :!A ' !<^Wni Wk UW«| mU wk *<W • V» ^ * * *S/'A >»UW Ww W i Vw W*.

I X'I

12A* Ww ww m '^w w'y ****

' 1 »•I t ' M M M, * ', 1 I n 114 > "’ • • > 1 I ' * '» ' » ' • ! 'I < •

i n

i !’H ‘1 ,■ i:i .***** y y V ix w W **»w wW vWw

12

1 ’ * », * i «•.;ii

i ' m * i ) AAVSOV»« «kV \*N WW W Smmi

Figure 1.6: The long term light curve of the dwarf nova SS Cyg. Outbursts last many days and occur at intervals of weeks to months.

(c) Recurrent Novae (RN) are previously identified CV that are found to re

peat their eruptions. Outbursts repeat after several decades, and vary from 7 — 9

magnitudes. As in the case of CN, a shell of gas is ejected at high velocities during

an eruption.

(d) Nova Like (NL) systems are the remaining sub-class of CN. There are no

significant outburst seen in systems belonging to this group, which includes UX

UMa stars (identified by the broad absorption lines in their spectra), AM Her stars

Introduction 19

(see section 1.7.1), and VY Scl stars. VY Scl stars (also called ’anti-dwarf novae’)

show occasional reductions in brightness from an otherwise constant brightness level.

These brightness reductions are believed to be due to a temporary drop in mass

transfer rate.

1.5 Thermal-viscous disc instabilities

Observations reveal that disc outbursts result from a sudden increase in mass flow

through the accretion disc (Warner 1995). Originally, the enhanced mass flow was

assumed to be due to an increase in mass flow from the LMS star, —m2 (Bath

1969), but was later rejected because the luminosity of the bright spot did not rise

significantly during outburst. In addition, the periodic nature of DN outbursts is

a problem; if enhanced mass flow from the secondary is responsible for triggering

outbursts, we would expect the resulting lightcurves to be irregular, not periodic in

nature.

This led to an investigation of the stability of the accretion disc itself after

Smak(1971) and Osaki(1970) independently proposed that disc outbursts could be

achieved with a fixed value of —m2 if there was some mechanism in the disk which

stores up mass and then dumps it onto the white dwarf on a short time-scale.

Calculations of the vertical structure of the accretion disc show that the disc will

be unstable when hydrogen becomes partially ionised (see Cannizzo and Kaitchuck

1992 for a review). To understand this more fully consider a disc annulus at a

radial distance R from the WD (figure 1.7). The annulus has a thermal scale height

H = flkCs (Frank King & Raine, 1992), (where Qk is the Keplerian angular velocity

Introduction 20

T eff

R + d RR - dR RFigure 1.7: The vertical cross-section of a disc annulus at radius R. The annulus has a surface density E = pH, where H is the thermal scale height.

of the material and cs is the midplane isothermal sound velocity), and a surface

density E = pH, where p is the volume density of the material. The annulus is

fed at R + dR by a constant accretion rate ra0, but loses mass through R — dR at

a variable accretion rate m\. It is assumed that viscous heating is balanced by

radiation losses through the annulus surfaces, such that the annulus is in thermal

equilibrium. The disk surface has an effective temperature Teff, which is related to

the midplane temperature Tc via the opacity in the vertical direction, (Frank King

& Raine, 1992).A/t— Tc 4 « aT e 4 (1.20)

The thermal stability of the annulus can be considered by slowly decreasing rhi below

rho- Initially, rhi = ra0, and E remains unchanged since there is no net change in

mass. If rhi is decreased slightly the mass in the annulus, and therefore E increases.

Introduction 21

XB

Figure 1.8: The thermal instability curve, showing the limit cycle behaviour of the disc annulus.

This also results in an increase in the midplane temperature Tc. Thus, if we plot Tc

vs X we find a positive slope (A-B in figure 1.8).

This behaviour continues until Tc « 10000K when hydrogen becomes partially

ionised. The opacity now increases rapidly with a small increase in Tc, and the

annulus is no longer in thermal equilibrium since dissipation at the surface is much

lower than viscous heating. Tc rises rapidly on a thermal time-scale (B-C in figure

1 .8 ) during which time hydrogen becomes fully ionised, and thermal equilibrium is

restored. Viscosity is now more efficient due to the higher temperature, and rhi

becomes higher than rh0. The annulus slowly gets depleted, and X and Tc gradually

fall (C-D figure 1.8). When Tc falls sufficiently hydrogen begins to recombine, and

once again the annulus falls out of thermal equilibrium since radiative cooling exceeds

Introduction 22

viscous heating. Tc now falls on a thermal time-scale (D-A, figure 1.8), until hydrogen

is neutral and thermal equilibrium is restored.

Thus a limit cycle is established, with the annulus alternating between the hot

and cool branches of the S shaped curve. The two branches are linked by a third neg

atively sloped branch, which is unstable. To illustrate this, consider a disc annulus

(as described above) situated somewhere along the unstable branch, with mj = rh0.

If a small negative perturbation is applied to E, rhi will increase accordingly (as the

annulus moves along the negatively sloped branch). E will continue to decrease in

response to the increase in rhi, forcing rhi to increase further. The disc annulus is

therefore unstable, since any perturbation (positive or negative) in rhi or £ results

in a growing instability.

The end points of the unstable branch are governed by two surface densities, Eb

(or rhmax) being the highest value possible in the cold state before an instability sets

in, and Ea (or rhcrjt) being the lowest value which allows the disk to remain fully

ionised. Limit cycle behaviour will only occur if rhmax < rho < rhcrjt . If rh0 > rhcrjt

the annulus will remain in the hot state with hydrogen fully ionised and if rho < mmax

the annulus remains fully neutral in the cold state.

We can extend the thermal instability model to the whole accretion disc by

treating it as a series of neighbouring annuli. Each radius of the accretion disc has

a different S curve, and we find that Ea and Eb vary with radius as (Ludwig and

Meyer, 1998):

Ea oc Eb oc r 1' 10 ( 1 - 2 1 )

If Ea < E(r) < Eb at any point in the disc the disc will become unstable at that

Introduction 23

radius. The annulus makes a thermal transition to the hot (or cold) state, and

the steep radial surface density and temperature gradients cause mass and heat to

diffuse rapidly into adjacent annuli, such that they too become unstable and switch

to the hot (or cold) state. Therefore heating (cooling) waves travel in both radial

directions from the point where the instability first occurs.

This model now allows us to explain the various subtypes of CVs in terms of

the mean mass transfer rate from the secondary i.e. — ra2. If mmax < m2 < rhcrjt

the disc will behave as a DN; outbursts correspond to a hot fully ionised disk while

during quiescence the disk is cool and neutral. If —m2 > mcrit the disc will be in the

hot steady state, with hydrogen fully ionised, and the system is nova-like. Finally, if

—rh2 ~ mcrit the system behaves as a DN but may occasionally slip into the nova-like

state producing long standstills in the light curve, i.e Z Cam systems.

1.6 Accretion in magnetic CVs

The presence of a white dwarf with a large magnetic field has a significant effect

on the accretion disc formation outlined in section 1.3.1. The accretion flow, which

is ionized by collisions as well as irradiation from the white dwarf (in the form of

X-rays) experiences a magnetic force per unit volume of:

V B 2 (B.V)B+ (1 '22)

where B is the local magnetic field. Fm consists of a pressure term exerted by the

magnetic field (f^), and a tension which acts along the magnetic field lines ( ^ ) .

This magnetic force acts as a barrier to the accretion flow, and prevents plasma

crossing the field lines.

Introduction 24

It is expected, if the magnetic field is sufficiently high, for the accretion flow

to become threaded at some point by the field lines and channelled down onto

small regions of the white dwarf surface, known as pole caps. The magnetized flow

is ballistic, and experiences a strong shock before being accreted on to the WD

surface, which releases X-ray emission. The X-ray luminosity will only be visible

for a fraction of the white dwarf spin cycle, hence a periodic modulation is seen

corresponding to the white dwarf spin period. This is a fundamental observational

signature of magnetic CVs.

The precise details of the accretion flow in magnetic CVs are rather complicated.

However, three time-scales can be used to describe the nature of the accretion flow;

3 1 /2the local dynamical time-scale tdyn = , the magnetic time-scale tmag =

k~1 |vj-vf|x (see c^aP^er 4), and the viscous time-scale tvjSC.

The magnetic field plays an important role wherever tmag < tdyn- A rough esti

mate of the radius where this begins to hold can be gained by considering the ram

pressure of the accretion flow (pv2) and the local magnetic pressure. In the case

of spherically symmetric accretion, and a dipolar magnetic field, the ram and gas

pressures are equal when (Frank King and Raine 1992):

rM = 5.1 x 108 V r62/ 7M r 1/7/4o7cm (1.23)

where M i6 is the accretion rate in units of 1 0 16gcm“ 3 and / / 30 is the magnetic moment

/i in units of 1 0 3 0Gcm3. rM is known as the Alven radius.

Introduction 25

1.6.1 The role of the magnetic field

In non-magnetic disc formation (section 1.3), the accretion disc forms a Keplerian

ring of gas at R c jrc that viscously spreads into an accretion disc. Let us now consider

a similar process for magnetic systems in two extreme cases. Firstly consider a

magnetic system in which tm > R c \ r c , equivalent to tmag ( R C[ r c ) < tv-lsc (R d r c ). In this

case magnetic stresses will quickly dissipate the angular momentum stored in the

accretion ring, preventing an accretion disc from forming.

In the other extreme case < R d T C , and accretion disc formation is unaffected

by the magnetic field. However, as the accretion disc spreads inward to the point

where £mag < £dyn, the magnetic field will remove angular momentum from the disc

more efficiently than viscous torques, and the accretion disc will be truncated at

n n - rM-

1.7 M agnetic Cataclysmic Variable Classification

1.7.1 Polars

Polars (otherwise known as AM Her stars, after the prototype) are discless systems

in which RM > a. In all Polars, tmag < tdyn < £visc for R > R c i r c , and therefore field

aligned flow occurs before material is able to circularize and form an accretion disc.

An important property of polars is the synchronous rotation of the white dwarf

with the orbital period. Since material (and AM) is being accreted on to the WD,

there must be an opposing torque acting to keep the WD spin period constant. The

Introduction 26

most likely cause of this torque is the interaction between the magnetic field of the

white dwarf and the secondary stars (Campbell 1985).

1.7.2 Intermediate Polars

The second class of magnetic CV are intermediate polars (or DQ Hers), in which the

observed WD spin period is shorter than the orbital period. In many intermediate

polars (IPs) we find that Rm — Rdrc, which makes the accretion scenario more

complicated. In this thesis I follow King (1993) and Wynn and King (1995), and

assume that material moving through the magnetosphere interacts with the local

magnetic field via a velocity-dependent acceleration of the general form:

fmag = ~k[v - Vf]± (1.24)

where v and Vf are the velocities of the material and field lines, and the suffix _L

refers to the velocity components perpendicular to the magnetic field lines. This

equation is used to represent the dominant term of the magnetic force with k ~

playing the role of a magnetic alpha (this is discussed in more depth in chapter 5 ).

Using this model, plasma will exchange orbital energy and angular momentum

with the magnetic field on the magnetic time-scale £mag. Material at radii greater

than the corotation radius

fic<>= (G M 1Ps2pin/ 4 7r2 ) 1 /3 (1.25)

where the magnetic field lines rotate at the local Keplerian velocity, will experience

a net gain of angular momentum, and will be re-captured by the secondary star, or

ejected from the system completely. Conversely, material inside Rco will lose angular

momentum, and be accreted onto the WD.

Introduction 27

The spin rate of a magnetic white dwarf accreting via an accretion disc reaches

an equilibrium when the rate at which angular momentum is accreted by the white

dwarf is balanced by the braking effect of the magnetic torque. It is believed that

the majority of IPs are in spin equilibrium, with PSpin/^orb — 0.1 (i.e the white dwarf

spins faster than the binary orbit), and Rm ~ Rc0 > R c\rc (Wynn and King 1995).

There are however definite exceptions to this rule, most noticeably AE Aqr.

Wynn, King and Horne (1997) show that the rapid spin down of the WD in this

system, and its unusual Doppler tomogram can be explained if the rapidly rotating

WD acts as a magnetic propeller, in which most (> 99%) of the accretion flow is

centrifugally ejected from the binary on a time-scale < tayn. The required condition

for this is tmag(RCirc) < £dyn(#circ) and Rcirc > Rco (King and Wynn 1999).

Another possible exception is WZ Sge, which is the main focus of the work in

this thesis. In chapter 4 I show that the strange properties of WZ Sge (in particular

its long recurrence time-scale) can be explained if the system is a magnetic pro

peller. However, unlike AE Aqr, mass is not ejected from the system entirely, but

accumulates in a ring close to the tidal radius.

1.8 CV evolution

In order for Roche Lobe overflow to occur, the donor star must either expand to fill

its Roche Lobe, or the binary separation must shrink, such that the Roche Lobe is

reduced onto the star’s surface. In the case of CVs evolutionary expansion can gener

ally be discarded, since the secondary star is low in mass (M2 < 1). The evolutionary

Introduction 28

time-scale for stars of this mass is far longer than the age of the galaxy. Therefore,

the binary separation must be shrinking, resulting in Roche Lobe overflow.

The angular momentum of the CV is given by

/ Cn \ 1 /2J = Mi M2 ( w “ ) • (1.26)2 \ M l + M2J v '

Assuming that the total mass of the system is conserved, i.e all the mass lost from

the secondary is accreted onto the WD, logarithmic differentiation of equation (1.26)

givesa 2 j 2 ( - M 2 )~a = T + - V i ( 1 - « ) (L27)

where Mi = —M2. It can be seen from 1.27 that in the case of CVs (q < 1 ), conserva

tive mass transfer (J = 0) will cause the binary to expand (a > 0). Logarithmically

differentiating equation (1.6), allows equation (1.27) to be written in terms of the

secondary star’s Roche Lobe radius, i.e:

2 j 2 (—M2 )|1 2 8 )

Therefore in CVs, where q < 5/6 typically, conservative mass transfer will cause the

Roche Lobe to expand away from the secondary star’s surface, and mass transfer

would cease. Therefore, the only way in which long term mass transfer can be

sustained in CVs, is for the system to lose orbital angular momentum, such that

j < 0 .

When identifying the angular momentum loss process that drives mass transfer,

it is useful to consider the orbital period of the binary. At short orbital periods

(Phr < 2 ), the high orbital velocities in the system make gravitational radiation

a strong candidate. Indeed, in the case of short period CVs, general relativity

Introduction 29

theory provides acceptable models that are able to predict mass transfer rates and

evolutionary time-scales.

However, at longer periods gravitational radiation is insufficient to account for

the observed mass transfer. For periods longer than Phr > 3, the angular momentum

loss is believed to be a combination of gravitational radiation and magnetic stellar

wind braking. Magnetic braking arises because the stellar wind of the secondary

star is forced to co-rotate with the magnetic field of the star. Co-rotation occurs out

to large radii, and therefore even an extremely low mass loss in the form of a stellar

wind can provide a strong braking torque, which ultimately passes to the binary

orbit via tides.

Since the angular momentum loss process depends on orbital period, it is useful

to consider the period distribution of CVs (figure 1.9). The histogram contains three

important features:

• There is a long period cut-off at Phr — 14 hours.

• There is a period gap in the range 2 < Phr < 3

• A short period cut-off can be observed at Porb — 80 min.

The maximum period cut-off can be explained by the conditions for stable mass

transfer, for which we require M2 < M\ (Frank King and Raine 1992). Since the

white dwarf mass cannot exceed the Chandrasekhar limit we require M2 < Mi <

1.44. From equation (1.2) we see that the period required for stable mass transfer

is Phr < 14, in agreement with the observed distribution.

The minimum period is more difficult. The most common explanation was first

introduced by Paczynski & Sienkiewicz (1981), where it is assumed that as Porb

Introduction 30

23 22 21 20 19 18 17 16 15 14 13 12

Z 11 10 9 8 7 6 5 4 3 2 1

0o o m ro co ^ cn 05 -o oococo °

P er io d h r

Figure 1.9: Period distribution histogram for CVs (John Barker, private communication).

Introduction 31

approaches 80 min, the secondary star deviates from the main sequence, such that it

becomes oversized for its mass. As a result R2 cannot decrease rapidly at the same

rate as M2, and at some point the star will begin to expand adiabatically in response

to mass loss. This causes the orbital period to increase, lowering the efficiency of

gravitational radiation, and producing a drop in the mass transfer rate. It is thought

that the absolute minimum period of CVs is just below 80 min, at the point where

the secondary leaves the main-sequence completely, and becomes degenerate.

As in the case of the minimum period, the explanation of the period gap is less

clear than the maximum period. The most common view point is that the secondary

star loses its radiative core and becomes fully convective at Phr — 3. This structure

change will almost certainly have an effect on the internal dynamo mechanism that is

responsible for the secondary’s magnetic field. If the magnetic field falls as a result of

the structure change, magnetic braking will become less efficient, and mass transfer

will cease as the Roche Lobe loses contact with the secondary. Alternatively the

structure change may reduce the rate at which stellar wind leaves the secondary star,

again reducing the effects of magnetic braking. Gravitational radiation continues

throughout the period gap, and shrinks the binary orbit until mass transfer resumes

at P^ — 2 .

To summarize, a CV is presumed to form by a common envelope stage, which

occurs when the white dwarf progenitor expands to a red giant phase. Angular

momentum loss then shrinks the binary, resulting in accretion on to the white dwarf.

At Phr ~ 3 magnetic braking is reduced, and mass transfer ceases. Gravitational

radiation continues to shrink the binary separation until the Roche Lobe makes

contact with the secondary Phr — 2, and mass transfer resumes. The minimum

Introduction 32

period cut-off at PGrb — 80 min is thought to be a result of the departure of the

secondary star from the main sequence.

1.9 Thesis summary

In this chapter I have summarized the basic accretion processes which occur in both

magnetic and non-magnetic CVs, and introduced the thermal-viscous disc instability

model (DIM), which will be the main focus of this thesis.

The work in this thesis is concentrated on two CV objects, namely VY Scl and

WZ Sge. Chapter 2 contains details of the theoretical techniques used throughout

the work included in the thesis, and in chapter 3 I show that the long term lightcurves

of VY Scl stars allow constraints to be placed on the WD temperature.

Chapters 4 and 5 provide numerical models of WZ Sge in quiescence and out

burst, and show that the long recurrence time can be explained using standard DIM

viscosities so long as the WD has a weak intrinsic magnetic field. Finally, chapter 6

contains details of the observational techniques used in chapter 7, in which I present

K-band lightcurves of WZ Sge, and argue that the short period oscillations and long

term brightness can be attributed to a magnetic WD.

Chapter 2

Numerical techniques

2.1 Introduction

This chapter describes the physics and numerical techniques used in chapters 3, 4,

and 5 of the thesis. I give details of three numerical codes, an Eulerian grid code,

and two different particle methods, SPH and HyDisc. I discuss the strengths and

weaknesses of the numerical methods, and describe astrophysical calculations for

which they are suited.

2.2 Eulerian Grid Code

The Eulerian Grid Code used is a 1-D version of the Stehle and Spruit (1999) hy

drodynamic code. Calculations are made on a radial grid of N fixed, linearly spaced

mesh points. The disc is assumed to be geometrically thin and axisymmetric, allow

ing a separate treatment of the vertical and radial disc structure. The code includes a

33

Numerical techniques 34

thermodynamic treatment of the vertical disc structure, and is therefore extremely

useful for modelling DN outbursts. The code does have certain limitations, and

using the same method. First, the fixed grid does not allow the radial disc bound

ary to vary, and cannot be used to model outer disc radius changes. In addition,

since the code is 1-D, it cannot be used to model non-axisymmetric features such as

superhumps in SU Uma stars.

2.2.1 Radial Disc Structure

The basic equations for radial disc structure are the continuity, Navier Stokes and

energy conservation equations, which are written in terms of the vertically integrated

surface density E, the angular momentum p#, the radial momentum pr, and the mid

plane temperature Tc.

The equation of continuity is

with pj = rityE, the angular momentum per unit area. The Navier Stokes equation

for the radial flow velocity is

whilst it is extremely useful for studying DNs, it is not possible to model all CVs

(2 .1)

where r is the radial distance from the white dwarf. The Navier Stokes equation for

the azimuthal velocity is

(2 .2)

(2.3)


where pr = urE is the momentum per unit area in the radial direction, and gr is

the gravitational acceleration in the radial direction. The pressure term in equation

(2.3) consists of radiation pressure as well as gas pressure, and can be written as

E kT 4a aP = + — T 4 2.4

fimu 3c

where k is the Boltzman constant, and a is the Stefan-Boltzman constant. H is the

isothermal scale height, given by

kT \ , F h

1/2

t f = ( — J / ^ k (2.5)

where is the Keplerian angular velocity.

In equations (2.2) and (2.3), the vertically integrated components of the stress

tensor are given by

M = - r f ™ (2.6)

Ta = - f { 2 ^ ~ i ^ rVr)) (2-7)

Tm, = - ! ( 2^ - ~ (rvr) j (2.8)

where / is the viscosity integral (see below). For a thorough treatment of viscous

stress tensors in cylindrical co-ordinates, see Landau and Lifshitz (1987).

The energy equation is given by

Cy( d P \ 1 d

= ~ 2HT Y f f f ) ~r~dr K ) " 2aT‘s {T' S ’r’ Mulx)

dd dvr vT 1 d ( nrAacT%d T \MT dr Trr dr T* * r + r dr ( r2H ZkP )

( r X - E - T l 12 91

where Cy is the specific heat at constant volume, and k, is the Rosseland mean

opacity. The first term on the rhs represents compressional heating, whilst the


second term is radiative cooling from the disc surface. Viscous heating is included,

as well as radial energy fluxes carried by radiative and viscous processes (these are

the last terms in the energy equation).

In addition to the disc structure equation, a description for the viscosity is re

quired. The standard alpha viscosity prescription introduced in chapter 1 (Shakura

and Sunyaev 1973) is modified in order to account for non-Keplerian rotation. The

viscosity parameterisation used is that of Godon (1995)

2 TI\dQ/dR\ ( 1 VU~ 3 aCs d n K/d R \ l + ( f l / t f ) | l - f i 2 / f i | | J ( ^

where f2 and Qk are the local and Keplerian angular velocities of the disc material,

Cg (Tc) is the midplane isothermal sound velocity, and H is the isothermal scale

height. For D = r2K, this reduces to the standard Shakura Sunyaev description. For

the viscosity integral / we use

f Hf = I updz ~ z/cE (2-11)Jo

where vc is the mid plane value of v as described in equation (2 .1 0 ).

Previous calculations (eg Ludwig and Meyer 1998) have shown, that in order

to produce DN lightcurves where the ratio of time spent in outburst to quiescence

is similar to observation, different values for a need to be used in quiescence and

outburst. As in the case of Ludwig and Meyer (1998), we take ahot = 0.2 and

cooi = 0.05 for the hot and cold states respectively.


2.2.2 Vertical Disc Structure - the / — £ relations

In order to solve the radial disc equations, the vertically integrated variables need

to be expressed in terms of Tc and £ at each radius. Detailed results of the vertical

disc calculations are given by Ludwig et al (1994). In this section I discuss the prin

ciples that govern the / — £ relation, before presenting the analytic approximations

employed in the code (along with their sources).

The / — E relation expresses the assumption that the disc is in local vertical thermal

equilibrium, i.e., at any given radius

Q+ (Tc) = Q~ (Teff) (2 .1 2 )

where Q+ is the rate at which heat is being generated in the disc (by viscous pro

cesses), and Q~ is the rate at which heat is being radiated away from the disc

surface.

Since the disc is thin, the temperature gradient will be effectively in the z-

direction. As in stellar structure theory, the flux in the vertical direction through a

surface z=constant is given by (eg. Kippenhahn and Weigert 1990)

_ ^ 4gT(z)a dT« (213>

where t ( z ) is the optical depth (here it assumed that all viscous heat generation

occurs at the mid plane). It is assumed when writing equation (2.13) that the disc

is optically thick, with optical depth

r = [ Kpdz = ac £ (2.14)Jo

where K(p T) is the opacity corresponding to the mean density and temperature in


the vertical direction. The mean density is calculated using

p(z) = pc exp2p_J p

(2.15)

whilst the mean temperature is approximated using T = 0.5(TC + Teff). To complete

the relation, opacity relations are required, which give n(p, T, p), where p ~ 1 is the

mean molecular weight. At high temperatures, opacities from Alexander (1975) are

used, whilst low-temperature opacities are taken from Williams (1980).

2.2.3 Approximations for the / — E relation

In the previous section I outlined how a relation for crTe4ff (T, £, r, Mwd,m) can be

obtained assuming the disc to be in local thermal equilibruim. Detailed results for

the vertical structure are given in Ludwig et al (1994). The resulting S-shaped

thermal equilibrium curve consists of the hot branch (where the disc material is

fully ionised), an intermediate branch (where partial ionisation occurs), and a cool

branch (un-ionised), which can be either optically thin or thick, depending on the

temperature. A schematic diagram of the S-curve is shown in fig 2 .1 . In the code,

the S-curve uses a combination of results from Canizzo (1993), Ludwig et al (1994),

and Ichikawa and Osaki(1992). The relations are given below.

2.2.4 The hot branch

The relations used are taken from Canizzo (1993)

T = 8039 K M^ 7d>1 rr„3 /7 < 7 2 £ 3/ 7 ^ (2.16)


- 8

hot b r a n c h- 9

u n s t a b l e b r a n c h- 1 0

cool opt ical ly thick b r a n c h

cool opt ical ly thin b r a n c h

- 1 31.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

log £ / g e m 2

Figure 2 .1 : Schematic drawing of the / — E relation, consisting of a hot branch (where the disc material is fully ionised), an intermediate branch (where partial ionisation occurs), and a cool branch (unionised), which can be either optically thin or thick, depending on the temperature.

and

Teff = 2414 K rr02 7 /5 6 < „ 7 2 £ 5 /1 4 (2.17)

where Mwd,i = AfWD/M 0 , r i0 = r / 1 0 locm, ahot/0.2 and /z0 .6 = ///O.6 . a can be

eliminated by combining equations (2.16) and (2.17), giving

Teff = 37346 K r%8 T? E2“ 1 /2 ^ (2.18)

where T5 = T/105 K and — tf/lO2 g cm- 2

2.2.5 The unstable branch

Calculations of the vertical structure (Ludwig et al 1994) give the transition values

for the hot branch to the unstable branch (denoted by a suffix A), and that from


the unstable branch to the cool optically thick branch (denoted by a suffix B),

Ea = 32 g cm" 2 rtf" a j j ft™ (2.19)

Eb = 185 g cm' 2 (2.20)

TeffiA = 8322KM^°D31rr001 (2.21)

Teff,B = 6317 K M™h rfcT (2.22)

where a:c,o.o5 — a;Cooi/0.05. The corresponding mid-plane temperatures at these

points are

Ta = 35503 K 2 r ? 0043 f t £ (2-23)

Tb = 13588 K o;C)o‘o5 ^wd!i r io° 3 Milo- (2.24)

The effective temperature on the unstable branch itself is given by interpolation:

T r r l ° g PV ^a) ! ( Teff B ^ , r rlog Teff = -— log ----- + log Teff,A (2.25)log (Tb/Ta ) \T eff,a /

2.2.6 The cool, optically thin branch

The cool branch includes optically thick structure for higher temperatures and op

tically thin structure for low temperatures. For the cool, optically thin branch, the

relation of Ichikawa and Osaki (1992) is used,

Te)f = 3416 K ( 3 ^ ) 2 37 ( | ) 0 41 K g , i rr„0 23 (2.26)

2.2.7 The cool, optically thick branch

The transition from the optically thin to optically thick state occurs at temperature

T, at which point

Et = 78 g cm" 2 r }007 a ^ 53 (2.27)


5.0

4.5

ocn 4.0 o

3.5

3.00.5 1.0 1.5 2.0 2.5

log H / g e m 2

Figure 2.2: The / — £ relation used in the code for different radii, assuming ahot — 0.2, and a coid = 0.05 (with fi = 1.0 and M\ = 1.0)

Teff/r = Tt = 3416 K < 01605 (2.28)

The effective temperature on the cool optically thick branch is given by interpolation

between points B and T, using a similar equation to (2.25). With the above approx

imations, the / — £ relations are fully defined in the code, allowing the radial disc

structure equations to be solved. Figure 2.2 shows the / — E relation for different

radii, using ahot = 0.2, and a coid = 0.05 (with (l — 1.0 and Mi = 1.0).

2.2.8 Solving the partial differential equations on an Eule

rian grid

The partial differential equations are solved using the method of finite differences

with a multi step solution procedure. In order to ensure numerical stability the


centre of cell cell boundary

v r (N) v r

Pr (N) Pr

r (N) r (N+l)

P<h(N> I (N) P^N+l) X (N+l)

v > > Tc (N) V«,<N+1> Tc (N+l)

grid cell N grid cell N+l

(N+l)(N+l)

Teff(N) T f/N+l)

Figure 2.3: Schematic drawing of grid cells N and N +l. Scalar quantities are assigned to the centre of the grid cell, whilst the velocity and momentum in the radial direction are assigned at the interface between grid cells

time-step is limited by the Courant-Fredrichs-Lewy condition, which is chosen to

ensure that during any given time-step, material moving at the sound speed does

not travel further than the distance from one grid cell boundary to the next. The

calculations are performed on a grid of concentric rings numbered from 1 to N, where

1 is the innermost ring and N is the outermost ring. For the boundary conditions,

two ’ghost rings’ are added with numbers 0 and N+l.

The grid used in the code is staggered, such that scalar quantities (as well as the

azimuthal velocity and angular momentum) are assigned at the centre of the rings,

whilst the radial flow velocity is assigned at the interface between grid cells. This is

shown schematically in figure 2.3.

Staggered grid codes are particularly useful for calculating the advection terms


in the disc structure equations, since the radial velocity at the boundary is imme

diately known. As a result, it is straight forward to calculate fluxes at the grid cell

boundaries.

2.2.9 Van Leer Upwind differencing Scheme

The advection from one grid cell to the next is done using the Van Leer upwind

differencing scheme (Van Leer 1977). As an example, it is described here in the case

of the continuity equation,

The second term on the left hand side equation (2.29) requires calculation of the

flux passing through the boundaries of the grid cell. The staggered grid means that

the velocities at the cell boundaries are already known. However, the surface density

needs to be interpolated in order to obtain the value at the cell boundary.

The simplest way to interpolate E is to assume

This method is not particularly useful, since it does not take into consideration the

direction of the gas flow. One way of considering the flow direction is to take the

where the flow is moving into. This is represented by equations (2.31) and (2.32).

(2.29)

E(i + l/2 ) = l[E(i) + E(i + l)] (2.30)

surface density of the cell where the flow is coming from, and ignore the grid cell

E (i + 1 / 2 ) = E (i) if vr > 0 (2.31)

E (i+ 1 /2 ) = E (1 + 1 ) if vr < 0 (2.32)


This method is an improvement on simply averaging neighbouring cells.

The Van Leer scheme goes one step further and interpolates E between three

neighbouring grid cells, 2 grid cells on the side where the flow is coming from and

one cell where the flow is advancing to. This scheme is known to be numerically

robust, and has a low numerical diffusivity.

The scheme works as follows: If vr (i) > 0, we interpolate between the grid cells

(i — 1 ), i and (i + 1 ) only if the surface density increases or decreases monotonically

with radius, i.e. only if E(i — 1) < E(i) < E(i — 1) or E(i — 1) > E(i) > E(i — 1). This

condition breaks down close to discontinuities in the fluid (e.g. in the neighbourhood

of shocks). Near discontinuities, the Van Leer method is a poor approximation and

a lower order method is required. If this is the case equations (2.31) and (2.32) are

If there are no discontinuities in the fluid the Van Leer interpolation is used,

which reads (Van Leer 1977)

The surface density at the grid cell boundaries are then simply found by extrapo

lating from the centre of the grid cell to the required radii using the local gradient

given by equation (2.33).

Here, I have outlined the Van Leer scheme in the case of the continuity equa

tion. The same method can be used in the case of all advection terms, or wherever

it is necessary to interpolate variables assigned at the centre of grid cells to the

boundaries.

employed.

(2.33)


2.2.10 Boundary Conditions

The inner disc boundary (r(0)) is set equal to the radius of the white dwarf. At the

inner boundary it is assumed that T — 0 and E = 0, and that the azimuthal velocity

of the grid cells 0 and 1 is equal to the Keplerian flow velocity.

The outer disc radius (r(N)) is fixed at 0.7jRi (R\ is the primary Roche Lobe

radius), where accretion theory predicts the outer disc radius is limited by tidal

is assumed to be in Keplerian rotation, such that angular momentum is supplied to

gradient at the inner and outer disc edges) are unlikely to hold in realistic systems.

However, it is expected that any differences will become negligible within a few grid

cells, and will not affect the global behaviour of the accretion disc.

As mentioned in section 2 .2 .1 , a 2 alpha viscosity description is used in order to

simulate disc outbursts. The alpha value for each ring is defined in the following

way

forces. Mass is supplied to the outer edge of the disc at the rate — M2, and once again

the disc at the rate — A/^K-^out- The temperature gradient at the inner and outer

edges of the disc (i.e. between grid cells N and N +l, and cells 0 and 1) is assumed

to be zero.

Some of the chosen boundary conditions (in particular the zero temperature

& — ^hot -^eff ^ (2.34)

log (Teff/T efF,B)

a — » c o ld T gff < T eff5B (2.36)


2.3 Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) treats fluids as a set of points, each having

a velocity, thermal energy, and mass (a thorough introduction to SPH is given by

Monaghan 1988 and Monaghan 1992). In order to evolve the set of points (which are

referred to as particles), it is necessary to construct the forces which each particle

experiences. Unlike the Eulerian code described in the previous section, SPH does

not need a grid to calculate spatial derivatives. Instead, they are found by analytical

differentiation of interpolation formulae.

SPH has a wide variety of applications, ranging from fluid dynamics to highly

supersonic rock collisions. Here, SPH is used to model the 3-D accretion flow in

CVs. The code has been specifically designed for thin disc problems (test results are

given in Murray 1996), and has previously been used to model tidally unstable discs

(Murray 1998, 2000), tilted discs (Murray & Armitage 1998), and disc outbursts

(Truss et al 2000).

Unlike the Eulerian grid code, in which the grid cells and boundaries are fixed,

SPH is a fully Lagrangian particle method, allowing greater flexibility and far higher

spatial resolution. The 3-D calculations allow the development of spiral arm struc

tures and non-axisymmetric discs, features that the 1 -D Eulerian grid code obviously

cannot cope with. One of the main limitations of the code as it stands is that it

does not contain a full thermodynamic treatment (a version of the code that in

cludes thermodynamical calculations similar to the Eulerian grid code does exist,

but is extremely slow on existing machines). However, a recent paper (Truss et al,

2000), shows how DN and SU UMa outbursts can be simulated using a simple, E —


related viscous switching technique. Whilst a full thermodynamic treatment would

obviously be highly superior, the viscous switching technique is extremely useful for

modelling general disc outburst properties. In chapter 6 , the SPH code is used to

follow WZ Sge during outburst.

2.3.1 Kernel Interpolation

In order to follow the motion of the set of particles, it is necessary to construct the

forces which a fluid element would experience. This is done by means of integral

interpolants. The integral interpolant of any function A(r) is defined as

< i4(r) > = J i4(r) w (r — r \h ) dr (2.37)

where w(u, h) is an interpolating kernel with the properties

J w(u, h) du = 1 (2.38)

and

lim tu(u, h) = <5(u) (2.39)

h is the characteristic length-scale of the kernel, commonly referred to as the smooth

ing length.

For numerical purposes, the integral is evaluated by dividing the fluid into N small

volume elements (particles) with masses mi ,m 2 ,m3 ra^. This allows equation

(2.37) to be approximated as

N A< i4(r) > = 2 mk— w(r ~ rk , h ) (2-40)

k=i P*


where k is a particle label. In this way, the local fluid density is can be written as

N

< p(r) >= mkw(r - rk, h) (2.41)k=l

An important feature of SPH is that if a differentiable kernel is used, a differen

tiable interpolant of a function can be constructed from its values at the particles. In

this way, derivatives can be obtained by ordinary differentiation, without the need

for finite differences or a grid. For example

N AvVA(r) = mk— Vw(r - r k, h) (2.42)

k=i P*

2.3.2 Equations of motion

In order to obtain a momentum equation for the fluid, an estimate of V P /p is needed,

where P is the pressure. In order to ensure that linear and angular momentum are

conserved, rather than calculate the gradient directly the pressure gradient is written

in a symmetrized form (see eg. Monaghan 1992), i.e.

-V P = V ( — | + -^Vp (2.43)P \ P ) P

The momentum equation for particle i can therefore be written

d v / p p . \

—i = _ ^ mk _T + _ i + £ + r Vjwfc - rk, h) (2 .4 4 )at k \ Pk Pi J

where g is the gravitational force of the binary per unit mass, and T is the viscous

force per unit mass (see below for a discussion of the artificial viscosity used in the

code). Similarly (see Monaghan 1992 for details), the SPH energy equation is


2.3.3 SPH artificial viscosity

In addition to the energy and momentum equations, a term for the viscous force per

Murray 1996) which in the continuum limit (i.e. linpv^oo, lim/^o) is a combination

of shear and bulk viscosities. The viscous force per unit mass is given by

k, is a constant that depends on the form of the kernel used. In 3 dimensions, with

a cubic spline kernel (Monaghan 1992), A: = 1/10. c is the sound speed, v is the

fluid velocity, £ is the artificial viscosity parameter, and L is a viscous length scale.

In the case of approximately Keplerian discs, the divergence term in equation (2.46)

can be neglected, and the viscous force per unit mass reduces to the following

Note that this equation is of a very similar form to the Shakura-Sunyaev viscosity

parameterization outlined in chapter 1 .

2.3.4 Time stepping

The dynamical time-scale of particles in an accretion disc (which is short compared

to viscous or pressure forces) increases as tdyn & r3/2 (Frank, King and Raine 1992).

Since a large fraction of the disc mass lies near the outer radius, it is extremely inef

ficient to evolve the entire particle set using time steps appropriate to the particles

located at the inner edge of the accretion disc. A useful way of increasing com

putational efficiency where such a range of time-scales exists is operator splitting,

unit mass is required. The code uses an artificial viscosity term (for more detail see

(2.46)

r = kQc l . (2.47)


which allows the gravitational force to be separated from the pressure and viscous

forces (which are computationally more demanding). In this scheme all particles

are integrated forward (using a fourth order Runge-Kutta method (for a thorough

explaination see Press et al. (1988)) to a common time-step, tc, with each parti

cle taking a different (integer) number of time-steps (depending on its dynamical

time-scale) to reach tc. Once all particles have been evolved to tc, the new particle

properties are then used for the pressure/viscosity calculations, which are all cal

culated using a single common time step. The common time-step used has to be

lower than the pressure and viscous time-scales, and is calculated using the Courant-

Friedrichs-Lewy condition.

2.3.5 Boundary Conditions

In SPH we do not encounter many of the problems encountered when using grid

based codes, in particular the need for ghost cells, and zero temperature and surface

density gradients at disc boundaries. In SPH, spatial boundaries for the accretion

disc are simply defined as follows:

• The inner disc radius is set as r = Rm. Any particle crossing this radius is

assumed to be accreted by the white dwarf, and is removed from the particle

set.

• The boundary of the secondary is defined as r = R l, where R l defines the

Roche lobe of the secondary star. Any particle crossing this boundary is as

sumed to be re-accreted by the secondary star.


• If a particle reaches the escape radius, defined as r = Resc, the particle is

assumed to have escaped the system altogether.

Particles are injected into the system at the LI point with a specified velocity (val

ues similar to the sound speed in the atmosphere of the secondary star are most

appropriate) and angle from the line of centre of the stars (usually set so that the

particles are injected just ahead of the line joining the two stars). The injected

accretion stream has a Gaussian surface density profile across its cross section, and

the particles carry a specific angular momentum equivalent to that of the L\ point.

2.4 Hydisc

HyDisc treats the accretion flow as a set of discrete particles, each of which carry

mass and angular momentum. In this way, the code is fairly similar to the SPH

code described in the previous section, and has the same strengths and weaknesses

compared to grid based codes. Full details of the design and implementation of the

code is given in Whitehurst (1988). One of the main differences between Hydisc

and SPH is the way in which particles interact with each other. In Hydisc, each

particle only interacts with its nearest neighbour (unlike SPH, where all particles

within a certain radius (2H) are considered). Whilst this method is less sophisticated

than kernel interpolation, it is more intuitive, and computationally more efficient.

In addition, rather than using analytical forms for the artificial viscosity and pres

sure, particle-particle interactions simulate artificial viscosity and pressure forces via

inelastic collisions (see below).

HyDisc has been used extensively to model accretion discs in CVs, in particular


systems containing a magnetic white dwarf (see eg King & Wynn 1999, Pearson,

Wynn & King 1997, Wynn, King &; Horne 1997, and Wynn & King 1995). Here,

HyDisc is used to model WZ Sge in quiescence (see chapter 5).

2.4.1 Equations of M otion

In order to follow the motion of individual particles, the equations of motion need to

be set up in the co-ordinate system used in HyDisc. This system is a set of Cartesian

co-ordinates centred on the white dwarf star which co-rotates with the binary (such

that the x-axis lies along the line of centres of the two stars at any time). As in the

case of SPH, the particle masses are assumed to be negligible compared to the two

stars, such that the equations of motion are those of the three body problem (see

chapter 1). In addition to gravity, a magnetic dissipation term (due to the dipolar

magnetic field of the primary) is introduced, which is described in detail in chapter

Lagrange’s equations can be written (for an arbitrary co-ordinate system) as

where are the coordinate set, L is the Lagrangian function (i.e. kinetic energy -

where k = l/tdrag is the magnetic drag co-efficient, m is the particle mass, and vm

is the velocity of the particle with respect to the magnetic field.

5.

(2.48)

potential energy), and F is the dissipation function. In Hydisc, the dissipation is

due to a magnetic drag force (see chapter 5), and is given by

(2.49)


Now let us consider these equations in the HyDisc coordinate system. For a

system of separation a with the primary and secondary stars situated at a distance

of d\ and d2 respectively, the inertial velocity of any particle can be written in terms

of the Hydisc coordinate system as

Vi = [x - 2/fi]x + [y + + cfi)]y (2.50)

Hence the Lagrangian becomes

1 / • / - » \ 2 ! / • r \ t \ \ 2 G M j M qL = ~(x — yQ) + - (y + fl(:r + di)) H---------------1------------------------ (2-51)2 2 ri r 2

where is the orbital angular velocity, rq — (x2+y2)1/2 is the distance of the particle

from the primary star, and r2 = ((x + a)2 + y2)1 2 is the distance of the particle from

the secondary star.

In order to find the magnetic dissipation term in HyDisc coordinates it is first

necessary to write the velocity of a particle with respect to the magnetic field in

circular polar coordinates (centred on the white dwarf), and in the co-rotating frame,

i.e

V m = t\T + riOeo - rx(u - Q ) e 0 (2.52)

where e# is the unit vector normal to r, in the direction of the white dwarf spin.

Taking the dot product of equation (2.52) with itself, and converting to HyDisc

coordinates, we obtain

vrn = X 2 + y2 - 2(yx - yx)(uo - Q) + (x2 + y2)(oJ - £1)2. (2.53)

Substituting 2.53 into 2.49, and applying the result along with (2.51) to Lagrange’s

equations (2.48), the equations of motion can be written as

X = 2yQ + x Q 2 + d ift2 - G M ^ X _ G M 2U &(x + A ) _ k i _ ^ ^ 54jrf r 2


and

y = -2x0. + y O 2 - GA/i-NL _ G M ^ 0 V _ ^ + k x ^ _ ^ ^r

2.4.2 Particle-particle Interactions

In addition to the equations of motion described above, particles are able to interact

with each other simulating viscous and pressure forces. In such interactions the

particles are assumed to be incompressible, and neighbouring particles separated

by a distance r are regarded as exerting a pressure force upon each other through

their common surface area Ap. Assuming that the particles form an ideal gas, the

pressure force over this area is given by P — ppc2, where pp is the mean density, and

Cg is the isothermal sound speed. Using the approximation Ap ~ r 2, the pressure

acceleration corresponding to this force is given by

ap ~ c2/ r (2.56)

In addition to this pressure, ’viscous’ forces are generated from the collisions

between particles. For two colliding particles separated by a distance r, and closing

with a supersonic velocity —u, a shock front will develop between the particles,

decelerating the particles in a time tcoii — \r/u\. This produces an average repulsive

acceleration

av ~ Qu2/r. (2.57)

Here Q is the coefficient of restitution of the collision, and lies in the range 0.5 <

Q < 1.0 for inelastic, non-penetrative collisions. Hence, using (2.56) and (2.57), the


total acceleration between particles can be written as

Cs2/ r + Qu2/r u < 0a = <

cs2/ r u > 0

2.4.3 Boundary conditions and tim e-stepping

The boundaries of the computational zone are identical to that of the SPH code,

and are defined in section 2.3.5 (i.e if particles cross certain radii they are assumed

to be accreted on to the WD, passed back to the secondary, or escape the system

altogether). Similarly, the particles are injected through the Li point with velocities

typical of the secondary’s atmosphere.

As in the case of SPH, a multi time step procedure is used in order to increase

computational efficiency. In HyDisc, each particle is allotted a time-step from the

allowed set

At[ = 2_1At where i = 0,1,2.... (2.58)

where At is the largest allowed time-step, which is governed by the particle collision

time tcoii. Each particle is moved through its time-step At[ according to the equations

of motion, allowing particles with the same time step to interact with each other

simulating pressure and viscous forces. This scheme is far more efficient than using

the shortest time-step for all particles.


2.5 Chapter Summary

In this chapter I have outlined three different numerical approaches that can be used

to simulate accretion discs in CVs. Chapters 3,4 and 5 contain results achieved using

the codes.

Chapter 3

The Low States of VY Scl stars

3.1 Introduction

Cataclysmic variables (CVs) are short-period binaries in which a low-mass main

sequence star (the secondary) transfers mass to a white dwarf primary via Roche lobe

overflow. The long-term photometric behaviour of these systems is of considerable

interest, as it is strongly influenced by the stability of the accretion discs which

form in all systems where the white dwarf is not strongly magnetic. For example,

the outbursts of the dwarf nova subclass are now thought to result from a thermal-

viscous disc instability driven by partial hydrogen ionization (see Chapter 1 ). This

instability is only present for average mass transfer rates below a critical value Mcrjt

necessary to keep most of the disc hydrogen ionized.

King & Cannizzo (1997; hereafter KC) suggested that the long-term photometric

behaviour of all classes of non-magnetic CVs could be explained as the reaction of the

disc to intrinsic but irregular variations of the mass transfer rate from the secondary.

57

The Low States of V Y Scl stars

The latter are probably caused by the passage of starspots across the Li region of

the secondary stars. In the strongly magnetic AM Her systems, where matter can be

directly accreted onto the white dwarf, variations in the mass transfer rate —M2 are

directly reflected in the light curves. Thus the extended AM Her low states really

are epochs of very low mass transfer (Gansicke, Hessman & Mattei 1998). The light

curves of dwarf novae, in constrast, show no obvious signs of such variations in the

mass transfer rate. KC explained this fact by noting that the outbursts can continue

for a long time even in the absence of mass transfer, since each outburst consumes

only a few percent of the disc mass. The mass transfer rate is likely to return to its

usual value well before the disc mass is exhausted, thus explaining the rarity of low

states among dwarf novae.

Non-magnetic CVs which do not show normal dwarf nova eruptions are called

novalike variables. Most of the time, these systems have a fairly constant brightness,

but at irregular intervals of months to years they may enter low states. The systems

named as UX UMa or VY Scl stars, depending on whether the drop in optical

brightness is smaller or greater than one magnitude. The usual high states of the

novalike variables evidently result from the fact that the average mass transfer rate

< —M2 > exceeds Mcrjt . The discs are therefore hot, bright, and relatively stable.

The low states of VY Scl systems, however, do not straightforwardly fit the

scheme proposed by KC: while a drop of —M2 below the value Mm\n allowing the

disc to remain stably in the cool state does cause the disc to become cool and faint,

the observed low states can last for hundreds of days, sometimes with rather irregular

light variations (cf. Warner 1995). Two examples of this behaviour can be seen in


EdQDE-Zo<s-J<coM>

Figure 3.1: The light curve of the VY Scl system MV Lyr, derived from 5810 visual observations made by members of the AAVSO, AFOEV, and VSOLJ.

the long-term visual light curves of the systems MV Lyr (fig. 3.1) and TT Ari (fig.

3.2). In this state one might expect the system to behave as a dwarf nova, having

ample time to produce several disc outbursts. Indeed, the simulations by KC show

that prominent outbursts from the low state should occur, quite contrary to what is

observed.

Following a private communication by Hessman & King (hereafter HK), KC

noted that the unwanted outbursts are inevitable if the inner disc is cool in the

low state. Although the disc surface density must initially lie below the critical

value at which the outbursts are triggered, this density will evolve on the cool-state

MV Lyr

2441000 2441500 2442000 2442500 2443000

2

4

6

8

2443500 2444000 2444500 2445000 2445500

2446000 2446500 2447000 2447500 2448000

1970 June - 1997 August

2448500 2449000 2449500 2450000 2450500J.D.


viscous timescale and eventually produce an outburst. To prevent this, the inner

unstable regions of the disc have to be removed. KC/HK proposed that this might

occur because the white dwarf in VY Scl systems is known to be quite hot (up to

~ 50 000 K; cf. Warner 1995, Table 2.8), probably as a result of heating by the

relatively high average accretion rate. Radiation from this star is therefore able to

keep a substantial fraction of the inner disc in the hot state even when the mass

transfer rate has fallen well below Mmin (a more modest version of this effect, with a

cooler white dwarf, can account for the UV delay observed in dwarf nova outbursts:

King 1997; Stehle & King 1997). If this fraction is sufficiently large, the cool outer

part of the disc will remain stably in the low state. In this chapter full numerical

calculations of the thermal and viscous evolution of the accretion discs in VY Scl

systems are presented, including the effects of irradiation by the white dwarf, in

order to test the suggestion by HK.

3.2 The disc model

3.2.1 Radial disc structure

The evolution of a geometrically thin, axisymmetric accretion disc is computed using

1 -D hydrodynamics in the radial direction. The disc equations (presented in detail

in section 2.2.1) are written in terms of the vertically integrated surface density E,

the angular momentum the radial momentum pr, and the mid-plane temperature

Tc. These quantities are evolved in time using the continuity, Navier-Stokes, and

energy conservation equations. Included in the Navier-Stokes equations are the full


wQPE-MZO<SJ<Pm

1 1 - 1 - 1 1— I

TT A r i■ i i 1 i i

— 1------1------1 1 1 1 1 | 1 1

• ’ v —• . jfc 1 m

■ i 1 i i i i 1 i i

. i 1 i

•

i i 1 i

1 1 1 | 1 1 1

0 oo —

1 1 1 1 1 1 12443500 2444000 2444500 2445000 2445500

i i i i | i i

V

i “i j i i i i | i i 1

*

i i i | i i i

•atim.

g U f t i •i i i i i i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i i i 1 i i i

2446000 2446500 2447000 2447500 2448000i i- i i | i i

ttnmin -a***i i | i i i i I ' i i , i , i i i i | i i i

AAVSO Light Curvei i i i 1 i i i i 1 i i i i 1 i i , , 1 , i i i 1 i i i

2448500 2449000 2449500 J.D.

2450000 2450500

Figure 3.2: The light curve of the VY Scl system TT Ari, derived from 5052 visual observations made by members of the AAVSO, AFOEV, and VSOLJ.

viscous stress tensors and the local radial pressure gradients. The pressure is given

bynKT. An .

(3-1)p _ pKTc + 4o _ T l

(imp 3c

where o is the Stefan-Boltzmann constant.

The energy equation described in section 2.2.1 includes heating by viscosity and

compression as well as lateral heat diffusion and cooling via radiation from the disc

surface (equation (2.9)). For VY Scl stars, an additional term needs to be added

to equation (2.9) to include irradiation of the disc by the central white dwarf. The

central white dwarf has a fixed luminosity given by

Tw d = 47tctjRL d Tw d 4, (3.2)

with Twd the effective temperature. The radius Rwd of the white dwarf is taken


from Nauenberg (1972). Accretion on to the white dwarf produces a boundary layer

luminosity

G M w dM wdL b l — <*bi d W -3 )

itW D

where cmbl ~ 0.5 (even for a rapidly rotating white dwarf: Stehle & King 1997).

Eqs. (3.2) and (3.3) can be combined into an expression for the irradiation flux for

both sides of the disc:

f . = Lirr 2tt2RS ( L w d + L b l )

W Darcsin p — p ( l — p2) | (3.4)

(Adams et al., 1988; King, 1997), where p~l = R/Rwo is the relative cylindrical

disc radius. This expression assumes that the boundary layer luminosity is radiated

by the entire surface of the white dwarf rather than by some narrow ring in the disc.

The kinematic viscosity used is a version of the Shakura-Sunyaev alpha param

eterization given by Godon (1995):

i \ 22 W / d R \ / ____________________3 \dSlK/dR\ l l + ( f l / f f ) | l - f i 2/ ^ |

(3.5)

where Q(R) and Qk are the local and Keplerian angular velocities of the disc ma

terial, Cs(Tc) is the midplane isothermal sound velocity, and H = cs/f2K is the

isothermal vertical scale height. As in the calculations of Ludwig & Meyer (1998;

Stehle &; King 1997), we take Qfhot = 0-2 and a cooi = 0.05 for the hot and cool states

respectively.


3.2.2 Vertical disc structure

The vertical disc structure is that of Ludwig & Meyer (1998), which relates the

surface temperature Teff to the mid-plane temperature Xc. Details of the vertical

disc calculations are given in section 2 .2 .2 .

The resulting S shaped thermal equilibrium curve consists of a hot branch (with

matter fully ionized), an intermediate branch (partial ionization), and a cool branch

(unionized matter) which can be optically thin or thick. Analytic approximations

for the different branches are used in the code, as described in section 2.2.3.

3.2.3 Num erical M ethod

Numerical calculations are performed using a 1-D Eulerian grid code (see section

2.2 for details). The equations are solved using finite-differences in time on a radial

grid of 200 linearly spaced grid cells, numbered 1 to N (1 being the innermost cell).

Radial advection is solved using the upwind differencing scheme of Van Leer (1977),

and is implimented in the same way as Stehle and Spruit (1997) (see also Stone and

Norman, 1992). The time step is limited by the Courant-Friedrich-Levy condition,

which is chosen to ensure that the Kepler frequency is resolved at the inner edge

of the disc. Full details of solving the equations using an Eulerian grid code, and

setting up boundary conditions are given in secitons 2 .2 . 8 and 2 .2 .1 0 .

3.2.4 M odel Parameters

Unfortunately, the system parameters of the most prominent VY Scl stars (TT Ari,

MV Lyr, VY Scl) are not well known. We therefore performed our calculations


using the system parameters of the eclipsing VY Scl star, LX Ser: orbital period

Porh = 3.79 hr, white dwarf mass Mwd = 0.41M®, and secondary mass M2 = 0.36Mo

(Ritter &; Kolb 1998). These result in an outer disc boundary of 2.1 x 10locm and

a disc inner radius (= Rwd) of 1.07 x 109 cm.

We began our calculations with a stationary disc in the hot state, as given by

the Shakura-Sunyaev solution (Frank, King & Raine 1992). The accretion rate at

the outer edge must be greater than the critical rate Mcrit(Rdisc) at the outer disc

boundary below which hydrogen is no longer fully ionized and instabilities occur. We

find V/crit (R-disc) ~ 5 X 1017gs *, so setting —M2 = 7 x 1017gs 1 ensures this condition

is met. We initiate a low state by completely and instantaneously switching off mass

transfer from the secondary. In reality the reduction in mass transfer is likely to be

gradual, over a timescale £var ^ 0 (see below).

3.3 Results

If the white dwarf is relatively cool, we recover essentially the results of KC. Thus

we first present results for Twd = 20 000 K (to which we shall refer as the “cool”

white dwarf model), and show that further outbursts are observed after the initial

drop into the low state. We then present results for Twd = 40 0 0 0 K (the “hot”

white dwarf model). We shall see that this temperature is high enough to prevent

further outbursts.


3.3.1 Cool white dwarf model

The solid lines in Fig. 3.3 show the results for the cool model: the top panel (a)

shows the visual light curve (the flux of the system at A5500A); the middle panel

(b) is the UV light curve (the flux at A1500A); and the bottom panel (c) shows

the evolution of the disc mass with time. In all light curves the disc is assumed to

radiate locally as a black body, while the white dwarf and secondary are taken as

uniform black bodies.

Mass transfer is switched off at t = 8 d, leading to an exponential decay in lumi

nosity. The disc light decays quasi-statically until t = 12d, when the surface density

at the outer disc edge £(i?djSC) falls below the critical value £ min required to keep

the disc in the hot state. This results in a cooling wave propagating inwards from

i?disc to a radius R tr « 2.8 x 109cm ^0.13i?disc, which is reached at t « 20d. The

cooling wave cannot travel beyond Rtr since the inner region of the disc is kept hot

by irradiation from the central white dwarf and boundary layer.

In quiescence, the disc consists of a small hot inner region which contains only

a small fraction of the total disc mass but contributes significantly to the disc lumi

nosity, and a cool outer disc with a low luminosity. The high viscous flow in the hot

inner disc rapidly removes any material passing through R = R tr, and thus behaves

as a small steady state accretion disc with an accretion rate M (Rtr).

The outer disc evolves on a cool viscous time scale, during which M (R tr) and

£(i?tr) slowly rise. This produces a gentle exponential growth in the disc luminosity,

which can only be seen in the visual because the primary and inner disc completely

dominate the UV light curve in quiescence. This continues until t « 32d, when


M ( R t r ) > Merit ( R t r ) , initiating a disc outburst. A heating wave starts at R = R t r

and travels outwards, forcing the outer part of the disc into the hot state. The

heating wave reaches R = 0.33i?diSc, where it is reflected as a cooling wave, returning

the disc to a cool state. The outburst lasts for ~ 2d, during which the luminosity

rises by a factor ~ 2 in the visual and ~ 1 . 8 in the UV.

Less than 0.1% of the total disc mass is accreted during the outburst, allowing

further outbursts. Successive outbursts get brighter because the heating wave travels

further out towards the disc edge during the outburst. The inter-outburst timescale

also gets longer, because the mass in the disc is reduced with each outburst. Out

bursts will continue until the disc mass falls enough to ensure that M CTi t ( R t T ) can no

longer be reached.

At t = 90d we arbitrarily returned the mass transfer rate at the outer disc edge

instantaneously to —M2 = 7 x 1017gs_1. Mass quickly accumulates in the outer

part of the disc, causing the visual luminosity to rise immediately. This continues

until t « 95d, when £(i?diSC) > £ Crit(-Rdisc)> producing a heating wave which quickly

propagates inwards and returns the whole disc to the hot state.

It should be noted that there is a delay of ~ 5d before the UV light increases after

mass transfer has been switched on. This is because the relevant density information

initially travels to the hot inner region on the cool viscous timescale, which is very

long. The UV only responds significantly after the heating wave has propagated

inwards to R = R t r, i.e. after t = 90d.

A peak can be seen in both light curves (Figs. 3.3a & 3.3b), because there is

more mass in the disc than a hot steady state disc with the same value of —M2 and

The Low States o f V Y Scl stars

RdiSC. This occurs because to return the disc to the hot state requires E = Emax

somewhere, whereas the steady-state disc will have E < Emax everywhere. However,

in practice —M2 will return more gradually than to its usual value than assumed

here, and this peak would be much less noticeable.

3.3.2 H ot w hite dwarf model

The dashed lines in fig. (3.3) show the corresponding evolution of a disc with the same

parameters, but this time a white dwarf temperature Twd = 40 000 K. Mass transfer

is switched off at t « l l d and the disc decays quasistatically until t = 15d before it

falls into quiescence as in the case of the cool model. This time, however, the hot

white dwarf keeps a larger region of the disc fully ionized, and the cooling wave only

reaches Rtr « 8.9 x 109 « 0.4TdiSC- The critical density required for an outburst

varies as E a R n / 10 (Ludwig & Meyer 1998), and E (Rtr) is therefore significantly

higher in the hot white dwarf case.

Initially, the outer disc continues to evolve on a cool viscous timescale, and the

luminosity begins to rise exponentially as before, but E(i?tr) never exceeds Ecrjt (-Rtr)-

Instead, E(i?tr) reaches a maximum value of ~ 50gem-2 at t « 60d after which it

decreases once more because there is no longer enough mass in the cool outer disc

to increase it further.

For t > 60d, the disc light slowly decays away, and the luminosity decreases very

slowly until t « 80d when mass transfer is switched on again, and the system returns

to the high state.


- 4

- 6

GO

- 2lO

- 4

- 6

- 9.6o

^ - 9.8

10.0O jO

£ - 10.2

- 10.40 20 40 60 80 100

TIME [days]Figure 3.3: Evolution of (a) the flux at A= 5500A, (b) the flux at A= 1500A and (c) the disc mass, assuming a white dwarf temperature of 20 000 K (solid lines) or 40000K (dashed lines).


3.3.3 Discussion

As mentioned earlier, the kink in the theoretical light curves which marks the initial

high-state/low-state transition (figs. 3.3a & 3.3b) results from the quasi-static evo

lution of the disc after the immediate cessation of mass transfer. Similar features

are also seen in observed light curves: a drop of ~0.7 magnitudes over 30 — 50d can

be seen in the light curve of MV Lyr (Fig. 3.4) around JD 2449720; Honeycutt,

Cannizzo & Robertson (1994; hereafter HCR) present RoboScope observations of

the VY Scl star V794 Aql showing ~0.4 magnitude drops over periods of ~ 30d;

and P G 1000+667 even shows ~ 0 . 8 magnitude drops over ~ 100d. All of these shal

low declines are followed by steeper ones, just like those seen in fig. 3.3: see the

(somewhat arbitrary) superposition of the theoretical light curve on that of MV Lyr

shown in fig. 3.4.

HCR attempted to explain the initial drops as the timescales for the passage of

cooling waves, but were forced to adopt very small values of Q h ot and a :Cooi in order to

reproduce the light curves: the cooling and heating waves are factors of a faster than

purely viscous diffusion. However, there are obviously two timescales involved: a

longer one associated with the initial drop (~ 0 . 0 1 magd-1), and a very much shorter

one corresponding to the final drop down to the low state (~ 0.02 — 0.03 mag d-1).

To be detectable in the visual, any reduction in the mass transfer rate from the

secondary must be processed by the disc either on the viscous timescale or on the

(shorter) heating/cooling front timescale. Since the high-state disc is everywhere

above the critical surface density at which a cooling front can propagate, it must

generally evolve viscously (i.e. reduce E) before it can evolve thermally in response


to a drop in mass transfer rate. Thus, it is natural to assume that the observed

initial high-state/low-state transitions reflect surface density evolution on the high-

state viscous timescale, while the shortest observed timescale - the rapid drop down

to the low state (~ 1.3magd_1) - corresponds to that of a cooling front. If mass

transfer stops suddenly and (*hot is of order 0 . 2 (as in our simulations), this timescale

is actually too short compared with observation (Figs. 3.3 & 3.4: ~ 0.16 mag d-1).

The most obvious explanation is that the presumed starspots responsible for the

low state do not appear suddenly, i.e. £var / 0 : the shortest timescales for high-low

transitions observed in AM Her systems are a few days, about the shortest time one

would expect such large-scale structures to move into place or appear. Thus the

secondary can take days or longer to shut off the disc’s supply of matter completely,

i.e. tvar ~ days. The disc responds by relaxing viscously until a sufficiently low

surface density is reached to trigger a cooling front.

Similarly, the timescale to return to the high state is set by the longer of two

timescales: that for resuming the high-state mass transfer rate from the secondary,

and that for viscously rebuilding a massive disc. The rise in the simulated light

curves - where mass transfer is assumed to resume instantaneously - occurs initially

on the slow viscous timescale of the cool outer disc and then on the rapid heating

front timescale (~ 1.5magd_1). The rise times of HCR are much longer because

they assumed a low value of ahot- Had we included the effects of stream overflow

(Schreiber & Hessman 1998), the delay between the resumption of mass transfer and

the start of the heating front could have been made even shorter.

The thermal-viscous instability model, modified by the inclusion of important


effects like the heating of the inner disc by the white dwarf, succeeds very well in

describing the rapid transitions to and from the high states and the extended low

states with little photometric activity seen in VY Scl stars, given a simple on-off

behaviour of the mass transfer rate. However the detailed photometric behaviour

of these systems is more complex than this simple picture, as they also show excur

sions to intermediate brightness states from the low state (cf the ‘outbursts’ around

JD 2449900 and 2450330 in Figs. 3.1 & 3.4). Since intermediate mass transfer states

are observed in AM Her stars, we must evidently look to the disc response to more

complicated mass transfer variations for a possible explanation of these intermedi

ate states. The intermediate-state variations do not resemble disc outbursts, so it

is likely again that the presence of a hot central region in the disc prevents these.

3.4 Conclusions

The viscous evolution of the accretion discs in VY Scl stars has been studied using a

1-D version of Stehle &; Spruit’s (1997) hydrodynamic code. The models include the

heating of the accretion disc by irradiation from the white dwarf, which prevents the

inner region of the disc from returning to the low state. This hot inner disc behaves

as a small, steady state accretion disc, and rapidly accretes any material passing

through the transition radius.

VY Scl low states are initiated by switching off the mass transfer completely, and

results show that the outbursts from the low state seen in the simulations by King

& Cannizzo (1998) can be suppressed if the temperature of the white dwarf is high

enough ( T w d ~ 40 0 0 0 K).


11

12

w§ 13E -I—H

55O< 14 S

OT 151— 4>

16

17

Figure 3.4: Comparison of observed MV Lyr (from Fig. 3.1) and theoretical LX Ser (Fig. 3.3) light curves (the theoretical curve has been shifted in both axes by an arbitrary amount in order to make a clean comparison between the curves possible). The resumption of mass transfer in the model was not chosen in order to reproduce the secondary rise to the intermediate state: it corresponds to an assumed return to the high-state mass transfer rate at an arbitrary time.

In conclusion, the light curves of the VY Scl systems result from the response

of their accretion discs, which are irradiated by relatively hot white dwarfs, to oc

casional drops in the mass transfer rate. These probably result from the passage of

starspots across the inner Lagrangian point on the secondary star.

+# MM** * *

** #

* #t>

***

* *

L--- -*

** * * ***#

*** ** ♦ ** *# * * *** *• * * ** **** **m * * * ** *** *t* ** m * *m* * #**** **

2449800 2449900JD

Chapter 4

Magnetically-Driven Outbursts of

WZ Sagittae

4.1 Introduction

The SU UMa stars are a subclass of Dwarf Nova (DN) whose light curves exhibit

superoutburst behaviour. Superoutbursts typically occur at intervals of several

months, interspersed with normal outbursts every few weeks. In general super

outbursts are around one magnitude brighter than normal outbursts and last a few

weeks rather than 2-3 days. During superoutburst a ‘superhump’ appears in the

optical lightcurve at a period slightly longer than the orbital period.

WZ Sge is an unusual SU UMa star because as well as only producing superout

bursts the recurrence time is extremely long: superoutbursts have been observed in

1913, 1946 and 1978, making the recurrence time-scale tvec ~ 33 years.

73

Magnetically-Driven Outbursts of WZ Sagittae

Normal outbursts in DN can be explained by a thermal-viscous driven disc in

stability due to the partial ionisation of hydrogen (see section 1.5). Osaki (1995)

proposed that superoutbursts are due to a thermal-tidal instability. During each

normal outburst, only a small fraction of the total disc mass is deposited onto the

white dwarf (WD), leading to an accumulation in disc mass and angular momentum.

The outer disc radius therefore expands until the tidal radius is reached. At this

point enhanced tidal interaction with the secondary star is assumed to increase the

mass accretion rate through the disc, triggering a superoutburst. This model is able

to explain the superoutburst behaviour of SU Uma stars using the same viscosity

parameters as the standard disc instability model (DIM), namely a coid ~ 0.01 and

a hot ~ 0 .1 .

Smak (1993) argued that the viscosity in WZ Sge had to be far lower than the

standard values (aCoid < 5 x 10-5), for the following reasons:

• SI In the standard DIM, the inter-outburst recurrence timescale is governed

by the viscous timescale of the inner accretion disc, which is far shorter than

the observed value of 33y. Lowering o;coid slows down the viscous evolution of

the accretion disc, increasing the inter-outburst time-scale.

• S2 Integrating the critical surface density using the standard DIM a coid gives

a maximum disc mass that is far lower than the observed estimate of the mass

accreted during superoutburst, AM ^c ~ 102 4g. Lowering the viscosity allows

more mass to be accumulated in the outer regions of the accretion disc to fuel

the outburst accretion rate.

There is however no obvious reason why the quiescent accretion disc in WZ Sge


should have a much lower viscosity than the other SU UMa stars. This has led a

number of authors to suggest other reasons to account for the long recurrence time

and high disc mass prior to superoutburst. In order to produce a long recurrence time

without requiring a low a coid the inner, most unstable regions of the disk must be

stabilized. Hameury, Lasota and Hure (1997) and Warner, Livio and Tout (1996) find

that the disk can be stabilized if the inner regions are removed, either by evaporation

into a coronal layer (see also Meyer-Hofmeister, Meyer & Liu, 1999), or by the

presence of a magnetosphere. Hameury et al. (1997) produce a marginally stable

disk that requires an episode of enhanced mass transfer to trigger a disk outburst.

The recurrence time in this model is governed by the mass transfer fluctuation cycle,

which has no obvious physical connection with the regular 33 year outburst cycle

observed in WZ Sge. Warner et al. (1996) find that under certain conditions the

disk will be marginally unstable and produce outside-in outbursts with the required

recurrence time. However, neither of the above models address the problem of how

to accumulate enough disc mass during quiescence in order to explain the mass

accreted in outburst ( S 2 ) . Since only ~ 1021 g is available in the disc just prior

to outburst both of the above models require mass to be added to the disc during

the outburst. The authors appeal to irradiation of the secondary star to increase

the transfer rate during outburst and supply the missing mass. In addition, Warner

et al. only produce normal outbursts, not superoutbursts. The reason for this is

that their disc radius RdiSC ^ 1 . 1 x 1 0 10cm is far smaller than the tidal radius, so

enhanced tidal interaction with the secondary (required to produce a superoutburst)

is not possible.


In this chapter I argue that the value of a coid in WZ Sge is consistent with stan

dard DIM values. In section 4.2 a similar calculation to Smak (1993) is performed

adopting the system parameters suggested by Spruit and Rutten (1998). For a fixed

disc mass I find that a coid oc R ^ t (where R0ut is the outer disc radius), and show that

a value of a coid ~ 0.02, and a moderate increase in Rout results in a value for AMacc

consistent with observation. A similar value is obtained for a coid by estimating the

cool viscous timescale from the observed superoutburst profile.

X-ray observations of WZ Sge by Patterson et al (1998) have revealed a coherent

27.86 s oscillation in the ASCA 2-6 keV energy band (see also Patterson, 1980).

The most compelling explanation for these data is that WZ Sge contains a magnetic

white dwarf. A recent paper by Lasota, Kuulkers and Charles (1999) interprets WZ

Sge as a DQ Her star and the 27.86 s oscillation as the spin period of the white

dwarf (PSpin)- The authors then suggest that WZ Sge is an ejector system: i.e. most

of the material transferred from the secondary is ejected from the system, similar to

the case of AE Aqr (Wynn, King and Horne 1997). In this model an accretion disc

is recreated in outburst, which again is triggered by a mass transfer event.

In section 4.3 I show that if the magnetic torque is not sufficient to eject mass

completely from the system, but merely propels it further out into the Roche Lobe

of the WD (we refer to the system as a weak magnetic propeller), SI and S2 can be

explained with a standard DIM value for a Coid- The injection of angular momentum

the disc receives from the propeller radically alters the surface density profile (com

pared with that of a conventional DIM disc). Numerical calculations of the magnetic

interaction show that the outer disc evolves to outburst on a timescale ~ 30 years,

Magnetically-Driven Outbursts of WZ Sagittae 77

avoiding SI. The model simultaneously provides a solution to S2 : the disc is forced

in the outer regions of the disc. The resulting increase in f?out is sufficient to allow

the disc to be massive enough, prior to outburst, to be consistent with observation.

The outburst is initiated as in the standard DIM, and does not require any episode

of enhanced mass transfer. Some of the observational consequences of the weak pro-

WD is considered.

4.2 The quiescent value of a in WZ Sge

4.2.1 The quiescent disc mass as a limit on a coid

Following Smak (1993) a value for c^id is estimated by considering the disc mass

immediately prior to outburst. An outburst is triggered when the surface density at

some radius E (R) becomes greater than the maximum value (Ecrjt) allowed on the

cool branch of the E — Te relation (eg. Cannizzo 1993). Therefore, the maximum

possible disc mass prior to outburst is given by:

to expand to the tidal radius (and a little beyond), allowing material to accumulate

peller model are discussed in section 4.4, and in section 4.5 the spin evolution of the

max (4.1)

where R m and Rout are the inner and outer disc radii respectively. Calculations of

the vertical disc structure yield (eg. Ludwig et al 1994)

(4.2)


where Mwd,i — Afi/Af0 , R\o = R / 1 0 10 cm, a C)o.o5 = «coid/0-05, and is the mean

molecular weight. Assuming that /x ~ 1 and assuming Rm <C R0ut, (4.1) and (4.2)

can be combined to give

A /m a x — 3.75 X 102 2MWd i »C)0.05 ^out,1 0- (4-3)

Although WZ Sge has a long observational history the values of some of its fun

damental parameters are still uncertain. For instance, Smak (1993) deduces that

A/wd,i = 0-45 and q = M2 /M 1 = 0.13, whereas the spectroscopic observations

of Spruit and Rutten (1998; SR following) lead to values of Mwd,i = 1-2 and

q = 0.075. Lasota, Kuulkers and Charles (1999) point out that a white dwarf mass

of Mwd,i = 0.45 is too small if it is assumed to be rotating with a period of 27.87 s,

so for the means of this paper the system masses suggested by SR are adopted.

Smak (1993) estimated the total mass accreted during outburst to be around

AMacc ~ 1-2 x 1024 g. Setting AMacc < Mmax, and using the system parame

ters Mwd,i = 1.2Mq q = 0.075, Rout = 0.37a = 1.7 x 1 0 10cm (SR), (4.3) gives

coid ~ 0.006. While this value is around a factor of 2 lower than the standard DIM

value of a coid ~ 0 .0 1 , it is 2 orders of magnitude higher than the value obtained by

Smak. However, setting Mmax « A M ^, we find a coici cx Rout3'9, i-e. a Coid is very

sensitive to disc radius (Smak fixed Rout = 1010 cm, and hence the sensitivity of

a to Rout was not noticed previously). Therefore, any error in the observationally

inferred value will produce a large discrepancy in a coid- The observational estimate

in SR (-Rout = 0.37a) was determined from models of the accretion stream - disc

impact region, which was found to be rather extended in the case of WZ Sge. A

theoretical estimate of the tidal radius can be found by utilizing the particle code of


VisualMagnitude

Outburst maximumHot viscous decay

Thermal transition to cold state

Cool viscous decay

Time

Figure 4.1: Schematic diagram of a superoutburst profile.

Whitehurst (1988) assuming the system parameters detailed above. This approach

yields a tidal radius of Rout ~ 0.5a. Assuming this value for Rout and repeating

the above calculation we find that a:coid < 0 .0 2 , in good agreement with the standard

DIM value. This outer disc radius agrees well with the magnetic model presented in

section 4.3.3.

4.2.2 Extracting a coid from the outburst lightcurve

A second, independent estimate of a coid can be obtained using the observed 1946

and 1978 outburst profiles (Patterson et al. 1981). Superoutbursts have similar

observational profiles to the outbursts of X-ray transients (King and Ritter 1998),

i.e they have a rapid rise to outburst maximum, followed by an exponential decay


on the hot viscous time-scale until the disc is no longer hot enough to remain fully

ionized and falls into quiescence. This produces a steep decline in brightness on

a thermal time-scale followed by a slower decline as the disc readjusts on the cool

viscous time-scale (see figure 4.1 for a schematic diagram).

Figure 4.2 (Kuulkers 1999, private communication) presents the visual lightcurves

for the 1913, 1946 and 1978 outbursts. It is possible to determine ocoid by considering

the visual luminosity as the disc returns to quiescence, i.e. ~ 30 — 90 days after the

outburst began. The disc brightness decays exponentially on the cool viscous time

scale tviSC,c (this is clearly seen in the 1946 outburst, but is less obvious in 1978),

such that the visual luminosity is given by:

F(t) = F0 exp f r — ) > (4-4)v^visc,c '

where

i ? 2tvisc,c ( 4 ‘^ )

and vc ~ a coidc%H (eg. Frank, King and Raine 1992). Here cs is the adiabatic sound

speed, and H is the disc semi-thickness. During quiescence the disc temperature is

T<5000 K. Assuming that (i ~ 1 , the sound speed is cs ~ ^ 2 ~ ~ 6 . 4 x 105 cm s_1.

If we consider R ~ 1010cm, and assume that H ~ O.li? for a geometrically thin disc

(4.5) becomes

Ocoid^l.5 x 105 t~ ^ c (4.6)

where tvjSC,c is in seconds. From the 1946 outburst lightcurve it can be seen that,

as the disc falls into quiescence, the visual flux drops by one magnitude in ~ 130 ±

40 days. (4.6) gives a value of a coid ^0.012 ± 0.004, in reasonable agreement with

the previous estimate.

mag

nitu

de


WZ Sge

□ 1 9 1 3 (pg) o 1946 ( V / p g ) . 1978 (V/v i s )

oo

o

CN

OOo*» •

160 2000 40 80 1 2 0

Time (d ay s a f t e r m a x i m u m )

Figure 4.2: Visual lightcurves for WZ Sge during the 1913, 1946, and 1978 superoutbursts (Kuulkers. E, private communication. See also Patterson et al (1981), and references therein).


4.3 A magnetic solution?

4 .3 .1 W Z S g e a s a m a g n e t ic p r o p e lle r

In sections 4.1 and 4.2 it has been argued that the value of a coid in WZ Sge is

unlikely to be different from other DN. This is consistent with the observed AMacc

if .Rout ~ 0.5a. However the recurrence time problem ( S I ) still remains unresolved.

The observation of a 27.87 sec. oscillation in the quiescent X-ray lightcurve of

WZ Sge (Patterson et al., 1998) suggests that the white dwarf is magnetic. The

introduction of a magnetic field to a rapidly rotating WD raises the possibility of

increasing the recurrence time by removing the inner, most unstable regions of the

accretion disc. This is a result of the magnetic torque which, rather than accreting

the inner disc, forces mass close to the white dwarf to larger radii (the magnetic

propeller effect). The inner disc is depleted and the surface density profile of the

disc is radically altered as material is forced to orbit at radii close to the tidal limit.

Hence, the outbursts of the magnetic disc are regulated by the long X evolution time

scale (Xcrit/X ) of the outer disc. The magnetic propeller effect not only increases

trec but allows the disc to accumulate enough mass prior to outburst to supply the

observed AM^c, with the stable outer disc acting as a large reservoir of mass. In this

way both S I and S2 can be explained simultaneously with a coid ~ 0 .0 2 , and with no

appeal to episodes of enhanced mass transfer. In this section the theoretical basis for

the magnetic propeller model is outlined, and the results of numerical experiments

are presented.


4 .3 .2 C a lib r a t io n o f t h e n u m e r ic a l v is c o s ity

Numerical simulations of the magnetic model are presented in section 4.3.4, and

utilize a modified version of the particle code of Whitehurst (1988), described in

detail in section 2.4. In order to calibrate the internal viscosity of the code in

terms of a coid, simulations were first carried out with a non-magnetic WD. These

experiments also serve to highlight the extent of problem SI.

Observations indicate that the mass transfer rate during quiescence is —M2 ~

1015 g s_ 1 (Smak 1993). To reduce the run time of the code the mass transfer rate is

increased to —M2 ~ 1016 g s-1, the estimated recurrence time is scaled accordingly.

The recurrence time in the simulation is a factor of 10 lower than if the correct

mass transfer rate had been used, since the higher mass transfer rate causes the

disc mass to build up more quickly. The evolution of the peak surface density of

the numerical disc is shown in figure 4.3 as a fraction of £ crjt , and can be seen to

grow exponentially with time. An outburst occurs when hydrogen becomes partially

ionized ( £ / £ crit > 1). Extrapolating the surface density shows that £ / £ Crit ~ 1 after

tree ~ 36±6 days (scaled to the observed mass transfer rate), which is typical of most

DN but much shorter than that of WZ Sge. Throughout the simulation the peak

value of £ / £ Crit remains close to the inner edge of the disc. The outburst would be

expected to commence at this point and be inside-out in nature. Eventually, after

several of these ‘normal’ outbursts, the mass of the outer disc should have increased

sufficiently to initiate a superoutburst (as outlined in section 4.1).

The viscosity of the numerical disc may be calibrated in terms of Q;coid via the

cool viscous time-scale at the outburst initiation radius. Setting R ~ 0.04a ~


(JI

w

o - 2 . 0

5 10 15 20 25 30 35Scaled t ime (days)

Figure 4.3: Evolution of the peak surface density as a fraction of £ crit, with a nonmagnetic white dwarf.

2 x 109 cm and using the estimate of trec above, equation (4.5) gives a coid~0.01, in

good agreement with the values obtained in section 4.2. The numerical calculations

presented below are thus consistent with the usual DIM viscosity prescription, and

would produce normal SU UMa type outbursts if the magnetic effects were removed.

4 .3 .3 T h e m a g n e t ic m o d e l

A description for the magnetic interaction is used which assumes that as material

moves through the magnetosphere it interacts with the local magnetic field via a

velocity dependent acceleration of the general form:

flmag = ~k[v ~ Vf]_L (4.7)


where v and V/ are the velocities of the material and magnetic field respectively,

A version of this description has been successfully applied to a number of other

intermediate polars by King (1992); Wynn & King (1995); Wynn, King & Horne

(1997) and King & Wynn (1999). The magnetic acceleration (4.7) is intended to

represent the dominant term of the magnetic interaction, with k playing the role of

a “magnetic a ” . The magnetic time-scale can be written in terms of k as

I shall refer to a system as a weak magnetic propeller when, in quiescence, the

magnetic field is strong enough for the system to behave as a magnetic propeller,

but not strong enough to eject a significant fraction of the transferred mass from

the Roche lobe of the primary (as is the case in AE Aqr). During outburst the

field should not be strong enough to prevent accretion. This requires the following

hierarchy of time-scales:

where t visc,h and tvisc,c are the hot and cold viscous time-scales respectively. Assuming

that the temperature of the quiescent disc to be T < 5000 K and ac= 0.02, and that

in outburst we have T> 104 K, a^= 0.1, (4.5) gives

A further condition for a system to act as a weak propeller is that the magnetic inter-

and the suffix _L refers to the velocity components perpendicular to the field lines.

(4.8)

visc,h mag ~ tvisc,c (4.9)

1 0 days < £mag < 1 0 0 days. (4.10)

action radius takes place at radii exceeding the corotation radius Rco = (GMiP2pin/ i n 2

where the fieldlines rotate with the local (circular) Kepler velocity. A spin period


of 27.87 s for the WD in WZ Sge gives Rc0 ~ i?wD ~ 109 cm, easily fulfilling this

requirement.

It now remains to identify tmag. There are a number of different models for the

inner disc - magnetosphere interaction which have been applied to magnetic CVs.

Here two extreme cases are examined, one in which the accretion disc is completely

magnetized, and conversely, that of a perfectly diamagnetic disc. In both cases it

is assumed that the stellar field has a dipolar geometry, and that any local field

distortions may be treated as perturbations on this structure.

When the magnetic field lines permeate the inner disc any vertical velocity shear

tends to amplify the toroidal field component B$ (eg. Yi and Kenyon, 1997). Diffu

sive losses counteract this amplification and in steady state, these effects balance to

give

Here 7 is a parameter that accounts for the uncertainty in the vertical velocity

shear and is of the order unity, r < 1 represents an uncertainty in the diffusive loss

timescale, whilst £1* and fix are the angular velocities of the WD and disc respec

tively. The tension in the field lines results in a magnetic acceleration which may be

approximated by

rc is the radius of curvature of the field line. It is possible to parameterize rc as

B# _ 7 (ft* - flK) (4.11)

(4.12)

where pd is the disc density, n is the unit vector perpendicular to the field line and


where (5 is a constant of order unity. Combining (4.13) and (4.12), with H ~ 0.1 R,

the magnetic acceleration is given by

Qmag = 2 7r.Rr/?/)dVK ~ VK ( 4 '14)

and tmag by

A7rRT/3pd v£tr

2 n RVK ~ p—■* snin(4.15)mag ' 1 D 2

VK ~±

At RdTC ~ 1 0 locm, the disc density is pd < 1 0 _6 gcm~ 3 (from (4.2), with H ~ 0.1 R).

Assuming 7 ~ l i t i s possible to constrain the magnetic moment p ~ BZR 3

using (4.10), giving

5 x 1 0 31G cm3 < p < 1 0 32G cm3 (4.16)

For the case of a diamagnetic flow it is assumed that the inner regions of the disc

are broken up into a series of individual filaments, as predicated by the analysis of

Aly & Kuijpers (1990). In order to follow the evolution of the diamagnetic gas blobs

(which are formed well outside the corotation radius) the approach of Wynn, King

and Horne (1997) is followed. This approach assumes that the diamagnetic blobs

interact with the field via a surface drag force, characterized by the time-scale:

Imag ~ CApblbB~2 ^ (4.17)Iv - V / I j.

where ca is the Alfven speed in the medium surrounding the plasma, B is the local

magnetic field, p\> is the plasma density, and Ib is the typical length-scale over which

field lines are distorted. Making the approximations that pbh ~ Ecrit (1010cm), ca — c

and v ~ vk we find that

1029G c m 3 < / i < 1 0 3°G cm 3. (4.18)


4.3.4 Numerical results

Numerical calculations were performed in 3D with a dipolar stellar magnetic field

structure, by implementing the above prescription for the magnetic acceleration in

a version of the particle code of Whitehurst (1988; see also Wynn, King k, Horne,

1997 and King & Wynn, 1999). The inclination of the dipole axis to the spin axis

was assumed to be 30°, as the observation of an X-ray periodicity indicates a non

zero magnetic inclination (the inclination of the magnetic dipole was not found to

affect the main conclusions significantly). A typical magnetic disc and its associated

surface density profile is shown in figure 4.4. Figure 4.5 shows the evolution of the

peak surface density of the magnetic disc. In this case the surface density grows

approximately linearly with time. The difference in the form of the surface density

profile over the non-magnetic case comes about because viscous readjustment of

the disc material is dominated by magnetic stresses. The magnetic torques inject

angular momentum into the disc material and result in the steady state X profile

shown in figure 4.4. Material piles up close to the tidal radius (~ 0.5a) where the

tidal and magnetic torques balance. The evolution of the disc to (super)outburst,

when X/Ecri t~ l , takes far longer than in the non-magnetic case. Therefore two

methods are adopted for determination of the recurrence time:

(a) Extrapolation of the curve in figure 4.5 yields a recurrence time of trec ~

30 ± 10 years, in good agreement with observation. Note: the periodic variation

superimposed on the linear relationship is caused by the accretion disc becoming

eccentric (see section 4.4).


file : wzsge_evol1.050 property : dens

- 0 .5 0 0.5

N = 23016 T = 50

CM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

radius (binary separations)

Figure 4.4: Grey-scale density plot of the simulated accretion disc in WZ Sge (upper panel), and distribution of £ /£ crit with disc radius (lower panel).


(b) Since the £ profile presented in figure 4.4 is the steady state profile, the evo

lution of the surface density of an annulus of the magnetic disc can be approximated

by

14191

where Mf = —fM 2, and / is the fraction of the total disc mass stored in the annulus,

i.e.

/ ~ (4.20)

where b = Rout — R\n (see figure 4.6 for a schematic diagram). Note: The value

of AR is unimportant since it cancels out when (4.19) and (4.20) are combined.

Combining (4.19), (4.20), and (4.2), with —M2 ~ 1015gs-1, the recurrence time is

given by

tree ~ 406i?a years (4.21)

where R a = R /a (the radius where the outburst begins) and b& = b/a. During the

simulation the peak value of varies between 0.4 < Ra < 0.5, so it is reasonablecrit

to assume that an outburst will begin in this region of the disc. The disc is truncated

at R\n ~ 0.1a due to the magnetic field, while the tidal radius is iodise ~ 0.5a. Setting

0.4 and 0.4 < Ra < 0.5, the recurrence time is

23 < trec ^ 47years, (4.22)

in agreement with the value obtained by extrapolation.

In the magnetic model the peak value of £ / £ crjt is forced to larger radii as a

result of the propeller effect. Therefore the outer regions of the disc will become

unstable first, producing an outside-in superoutburst. During the superoutburst a


•c 0.015oI

0.010w

40 60 80 100 120Scaled t ime (days)

Figure 4.5: Evolution of the peak surface density as a fraction of £ crjt , assuming a weakly magnetic white dwarf.

large fraction of the disc mass will be accreted. Immediately prior to outburst the

surface density profile of the disc will resemble that of figure 4.4 with the peak value

given by £ crjt (the requirement for outburst). Approximating the profile as linear

between the inner disc edge (R[n ~ 0 .1 a) and the radius at which the surface density

peaks (J?peak ~ 0.45a), the surface density gradient can be written as

fly y>0.45a y0.45a_ _ ~ crit crit ( 4 .23)OR Rpeak ~ Rin 0.35a

where 12 ~ 877 g cm- 2 is the value of £ crjt at R = 0.45a, the mean position of the

peak in the surface density profile. Hence the surface density can be approximated


as

E (R) ~^ 0 .4 5 0 yi0.4a

crit R — 0.1 rit (4.24)0.35a 0.35a

where the latter term in equation (4.24) is the intercept obtained when E is extrap

olated back to R = 0. The total mass of the disc is then

fR out 2i7TMd ~ / 2ttRE(R) dR = —

** R \ n vJ •

2 t t E ^ 35a

IP~3

IP 1 2 0 J

out

Rin(4.25)

Setting R\n = 0.1 a and R0ut = 0.45a (cf. figure 4.4) we obtain Md ~ 9 x 1023 g.

This is in excellent agreement with the observation of Smak (1993) that AMacc ~

—M^rec ~ 1024 g. The magnetic disc is able to store this amount of mass stably

because the magnetic propeller forces material to orbit close to the tidal limit at

~ 0.45a. This outer reservoir of mass can be seen in the surface density profile of

the magnetic disc presented in figure 4.4.

4.4 Observational Consequences

4.4.1 Truncated Discs

The weak magnetic propeller model for the outbursts of WZ Sagittae results in

a number of observationally testable predications. The major consequence of the

model, the existence of a substantial hole in the inner disc, has already been sug

gested by Mennickent and Arenas (1998). The authors estimate the ratio of the

inner and outer disc radii (R = it4n/#out) from observations of the HQ emission line

widths. WZ Sge was found to show an extremely large value of R ~ 0.3. This value

agrees remarkably well with figure 4.4 where Rout ~ 0.5a and R\n ~ 0.15a, and is


J / L .crit

Disc annulus

Radius"discm

<— »AR

Figure 4.6: Schematic diagram showing the radial surface density profile of the accretion disc in WZ Sge. The recurrence time can be estimated by considering the evolution of one annulus of the disc.

much larger than the value expected if the accretion disc extended all the way to

the WD surface (R ~ 0.03).

The existence of a ring-like disc in WZ Sge should have a discernible effect on

the expected disc spectrum. Assuming the magnetic disc to be optically thick, the

local viscous and magnetic dissipation rates allow simple, synthetic disc spectra

to be constructed: the disc is divided in to a number of cells with the effective

temperature for each being determined by the local dissipation. Figure 4.7 shows

the predicted spectrum between 0 . 1 and 1 0 microns (solid line). Superimposed on


- v 1 0

E 10 3 .

CMIE

1 0

10

10

3

7

wavelength (jum)

Figure 4.7: Theoretical prediction of the disc spectrum between 0.1 and 10 microns (solid line), along with the disc spectrum extracted from observations (dashed line).

this (dashed line) is the disc spectrum presented by Ciardi et al (1998). (Note: the

latter was obtained by modelling the spectra of both the WD and secondary stars,

and subtracting them from the observed spectrum.) It is encouraging that these

spectra are in reasonable agreement, with the peak emission occurring at around 0.4

microns in both cases.

A useful diagnostic of the accretion dynamics in WZ Sge comes in the form of

the Ha Doppler tomograms presented by SR. The numerical model allows the re

construction of theoretical tomograms from the simulated accretion flow. Figure 4.8

shows such a theoretical tomogram with an integration time of 4 Porb (the integra

tion time used by SR), weighted by the local mass density in the disc. The velocity


resolution is 50 km s-1. The theoretical tomogram is symmetric about the origin

(apart from the accretion stream), with velocity limits ~ 1300 km s- 1 and ~ 400

km s-1. The density structure and velocity field of the magnetic flow compares

favourably with the observations presented by SR.

4.4.2 Superhumps

Numerical simulation of the magnetic flow suggests that it may be possible for the

quiescent disc to become eccentric. This would raise the possibility that the quiescent

lightcurve of WZ Sge may display superhumps (which are normally associated with

superoutbursts of SU UMas). The magnetic disc allows this possibility as it reaches

the 3:1 resonance radius (see Frank, King and Raine, 1992) after ~ 100POrb? with the

subsequent evolution of an elliptical, precessing disc. Figure 4.9 shows the evolution

of the eccentric, magnetic disc. The superhumps associated with this eccentric disc

would be difficult to detect in quiescence, but it is predicted that the superhump

period should appear at around the 0.1 magnitude level in the K-band lighcurve, and

at the 0.04 magnitude level in the V-band. However, it should be emphasized that

the quiescent superhumps may be purely numerical. The high disc mass used in the

simulations requires a large particle mass. However —M2 is very low, resulting in a

low particle density in the accretion stream. For this reason the interaction between

the stream and the outer disc is poorly resolved. Increased mass resolution may well

suppress the development of superhumps in quiescence (Murray, 1998). That being

said, it is interesting to calculate the theoretical, quiescent superhump period. Figure

4.10 shows the phase angle of the semi-major disc axis (in the co-rotating frame)


1500

1000

500

0

-500

-1000

-1500-1500 -1000 -500 0 500 1000 1500

vx

Figure 4.8: Mass density weighted dopplertomogram of WZ Sge, integrated over four orbital periods, and with a velocity resolution if 50 km/s.


file : wzsge_evol 1.050 property : dens file : wzsge_evol 1.100 property : dens

in6

o

mdI

file : wzsge_evo!1.150 property : dens

ino

o

inoI

-0 .5 0 0.5

N = 46357 T = 100

-0 .5 0 0.5

N = 23016 T = 50

-0 .5 0 0.5

N = 62594 T = 150

Figure 4.9: Grey-scale density plot showing the evolution of the eccentric, magnetic disc. Here, N is the number of particles in the disc, and T is the time (in units of Porb) since the start of the calculation

versus time. The phase of the semi-major axis was determined via integration of the

disc mass in the radial direction over phase bins of width 0.17T radians. From figure

4.10 a superhump period of Psh ~ 1.030 ± 0 . 0 0 1 Porb is obtained with q = 0.075,

longer than the value of ~ 1.008Porb observed during outburst (Patterson et al,

1981). Smoothed particle hydrodynamics simulations display a positive correlation

between superhump period and mass ratio (Murray, 2000). Setting q ~ 0 . 0 2 would

lead to a quiescent superhump period in agreement with the outburst observations.


V 4 0 0CDCD

o' 300 (1)

"O

3? 200CDCo0 100&o

Cl

2 4 6t / p —orb

Figure 4.10: Plot showing the phase angle of the semi-major disc axis (in the co- rot at ing frame) versus time.

4.5 The spin evolution of the white dwarf

The spin evolution of the white dwarf in WZ Sge is determined by angular momen

tum transfer in the quiescent and outburst phases. In quiescence the WD acts as

a magnetic propeller, ie. material approaching the WD is forced to larger radii.

We can approximate this interaction by assuming the transferred gas, with specific

angular momentum (GMiRCirc)1/2, is forced to orbit at a mean outer radius P out-

Here, Rcirc is the circularization radius defined by 27r62/P 0rb — (GMiRCirc)5, where b

is the distance between the WD and the LI point. The associated spin down torque


on the WD is then

j d = I), (4.26)

where —M2 ~ 1015gs_1, _R0ut ~ 0.5a, Rc\rc ~ 0.34a, and $ > 1 if material at

Rout loses angular momentum to the secondary star via tides (this will be expected

to occur once the outer edge of the disc reaches the tidal radius). Conversely, in

outburst, the WD accretes angular momentum at the rate

3u = (4.27)

From the results of Smak (1993) the average accretion rate during outburst can be

estimated as M^c ~ 1024g/14d ~ 1018 g s_1.

The evolution of the white dwarf spin period can be written in terms of J as

P O7r T^ = __£?[£_ (4.28)P spin P s v m J

where I is the moment of inertia of the WD. Averaging over a duty cycle the torque

on the WD is given by

< j > ~ (4.29){tree + tQ)

where tQ is the outburst duration.

From the above estimates (and using the assumption that Macc o ~ M2trec), we

find that the WD will be in spin equilibrium (i.e. Jnt0/Jytrec ~ 1-0) when $ ~ 1.14.

It is assumed that WZ Sge is in spin equilibrium (or spinning up or down on a

very long timescale), since if this was not the case, it is highly unlikely that we

would observe WZ Sge during its propeller phase. Therefore, during quiescence,

approximately 14 percent of the angular momentum stored in the accretion disc


must be passed back to the secondary, and hence the binary orbit. This is expected,

since the disc expands beyond the 3:1 resonance radius during quiescence as a result

of the magnetic propeller, allowing enhanced tidal interaction with the secondary

star. This effectively lowers the gravitational radiation angular momentum loss from

the system, leading to a lower mass transfer rate than other DN with similar binary

parameters (this could account for the extremely low mass transfer rate observed in

WZ Sge (Smak, 1993)).

During quiescence, the spin down torque on the WD leads to the following spin

down rate

Pspin ~ --Pepin (4-30)

where I is the WD’s moment of inertia. During the 33 year recurrence time in WZ

Sge, equation (4.32) predicts a decrease in the WD spin period (from 27.87s) of

AP ~ 1.456 x 10- 7 s. This change in the WD spin period is obviously too weak to

be detected in observations.

4.6 Conclusions

This chapter has shown that the outbursts of WZ Sge can be explained using the

standard DIM value for a coid if the white dwarf in the system acts as a magnetic

propeller. Numerical models of the magnetic disc can explain the the long inter

outburst time trec ~ 30=1=10 years, and produce a disc massive enough (A/diSC ~ 102 4g)

to fuel the outburst accretion rate. No episode of enhanced mass transfer is required

to trigger outbursts, or to supply mass during outburst.


The weak-propeller models assume that the magnetic tension force is the domi

nant cause of angular momentum transfer in the quiescent disc, and that the force

is proportional to the local shear between the disc plasma and the magnetic field. In

outburst angular momentum transfer is dominated by viscosity, and mass accretion

on to the white dwarf is enabled. This results in an estimated WD magnetic moment

in the range

5 x 1 0 31 G cm3 < n< 1 0 32 G cm3 (4-31)

if the disc is assumed to be fully magnetized, or a value in the range

1029 G cm3 < n < 1 0 3° G cm3 (4.32)

if the disc is assumed to be diamagnetic. These values for the WD magnetic moment

would suggest that WZ Sge is a short period equivalent of the intermediate polars

(IPs). Most IPs lie above the period gap (P0rb ~ 3 hours) and have magnetic moments

in the range 1031 G cm3 1034 G cm3. King and Wynn (1999) pointed out the

only two confirmed IPs with P0rb~2 hours (EX Hya and RX1238-38) have n> 1033

G cm3 and long spin periods (67 min and 36 min respectively). WZ Sge would

therefore be the first weakly magnetic //< 1 0 32 G cm3 CV to be found below the

period gap.

In differentiating between the models of the disc-magnetosphere interaction it is

useful to consider the case of AE Aquarii. This system is thought to contain a WD

with a magnetic moment ~ 1032 G cm3, which ejects ~ 99% of the mass transfer

stream from the system (Wynn, King and Horne, 1997). If the magnetic moment

of the WD in WZ Sge followed //> 1031, it would satisfy the criteria neccessary to

become an ejector system. Hence in the case of a fully magnetized disc, it would


seem that the presence of the disc itself is the only protection the system has from

ejecting virtually all of the transferred mass. If the disc was completely accreted or

destroyed for any reason WZ Sge would resemble a (very faint) short period version

of AE Aqr. This would not apply to the case where the WD has a magnetic moment

in the range predicted by the diamagnetic model.

Mennickent and Arenas (1998) find evidence for ring-like accretion discs in long

supercyle length SU UMa stars from a radial velocity study of the Ha emission lines

of these systems. WZ Sge is the most extreme of these objects with a recurrence

time of 12000 days and R = i?in/-R0ut — 0-3, which is in very good agreement with

the magnetic disc presented in figure 4.4. Other SU UMas show a strong positive

correlation between supercyle times and the R parameter. This result is in excellent

agreement with the model presented in this chapter for WZ Sge. The WDs in the

other, less extreme, SU UMa stars would simply act as less efficient propellers: that

is the WDs would have weaker magnetic moments, or lower spin rates (which may

reflect various stages of the spin up/down cycle postulated above). Strong candidates

for WZ Sge-like systems are RZ Leo (tvec < 4259 days, R ~ 0.16), CU Vel (trec ~

700 - 900 days, R ~ 0.15) and WX Cet (trec ~ 1000 days, R ~ 0 .1 2 ), (taken from

Mennickent and Arenas, 1998, table 5).

Chapter 5

WZ Sge in outburst

5.1 Introduction

In chapter 4, the long recurrence time of WZ Sge was explained (with standard

DIM viscosity values) using a magnetic propeller model in which the white dwarf

possesses a weak intrinsic magnetic field. In this chapter, the model is developed

further, and used to follow WZ Sge through outburst. This is achieved using a

smoothed particle hydrodynamics code (see chapter 2), that uses Viscous switching’

to simulate outbursts (Truss et al., 2000). The numerical method is described in

detail in sections 5.2 and 5.3.

Results show that, so long as the magnetic time-scale satisfies tvisc,h ~ tmag < tvjSC)C,

the WD acts as a magnetic propeller during quiescence, and an outburst is initiated

when £ ~ Ecrit at some point in the accretion disc. The hole at the centre of the

accretion disc closes up during outburst, due to the increased ram pressure of the

disc material. As well as allowing the WD to accrete in outburst (this is essential

103

WZ Sge in outburst

for the disc to return to quiescence), this result is able to account for the observed

disappearance of the 27.87 sec. WD spin period (Patterson et al 1998). The visual

outburst profile has a rise time of ~ 1 day, and a peak brightness of ~ 9 magnitudes,

in agreement with observation (Patterson et al. 1981).

5.2 M odelling DN outbursts using SPH

The smoothed particle hydrodynamics code described in chapter 2 needs to be de

veloped in order to model the outburst cycle of WZ Sge. Early attempts (Murray

1998, Armitage & Murray 1999) to model DN outbursts using SPH were made by in

stantaneously increasing the artificial viscosity throughout the entire accretion disc.

Whilst DN outbursts can be achieved in this way, the approach is rather simplistic,

and does not produce the gradual rise to outburst as observed in DN lightcurves.

A more realistic model can be achieved by allowing the artificial viscosity to

vary locally in response to disc conditions. As seen in chapter 2, the SPH artificial

viscosity is given by

r = k(cL (5.1)

where k = 1/10 for a 3D Cubic Spline Kernel, ( is the artificial viscosity parameter,

c is the sound speed, and L is the viscous length scale (usually set equal to the

smoothing length). The SPH code assumes that the accretion disc to be isothermal

(i.e. c is constant), and therefore the artificial viscosity of any particle can be

modified by varying (. When considering the interaction between two particles, the

mean value of ( for the two particles is used. (Cleary and Monoghan (1999) found

WZ Sge in outburst

that this form of averaging was suitable for heat conduction between particles of

vastly different temperatures).

It now has to be decided how viscosity changes are triggered. The simplest

approach (Truss et al 2000) is to allow the shear viscosity (or £) of each particle

to depend on the local surface density, E. In this approach, when E in a quiescent

region of the accretion disc increases to a value greater than Ec, then that region of

the disc is assumed to have been triggered into the hot state. If this occurs, (, the

viscosity parameter (for all particles involved) is increased to the value suitable for

a hot accretion disc. Conversely, a hot region of the disc will remain in such a state

as long as E stays above a second critical value Eh, where Eh < Ec. If the local

value of E falls below Eh, C returns to its quiescent value. The dependence of ( (and

hence the artificial viscosity) on E is shown schematically in figure 5.1. Note that

this dependence is a simplified version of the / — E relation described in chapter

1 (the main difference is due to the fact that the SPH code assumes an isothermal

accretion disc, and therefore the viscosity does not depend on the disc temperature).

Vertical structure calculations (Cannizzo et al., 1988) give critical surface densi

ties of

Eh = 8.5 g cm- 2 Mwd“ ^ i )05 cth.ol (5-2)

Ec = 11.4 g cm- 2 r j005 aJ f 5 (5.3)

In the SPH model, the radial dependence of Eh and Ec is retained, but the magnitude

of the gradients is reduced in order to lower the run time of the code (numerical

scalings are discussed in detail in section 4.4).

In previous calculations (Murray 1998, Armitage & Murray 1999), the artificial

WZ Sge in outburst 106

hot

t,cold

Figure 5.1: The simplified surface density trigger conditions used in the code. Ec and Eh are functions of disc radius.

viscosity was increased instantaneously to its outburst value. The form and means

of the viscous state of unstable accretion discs is unknown. Here, it is assumed

that the viscosity changes on the thermal time-scale, and is included in the code

by using an appropriate functional form for the transition. An hyperbolic tangent

function is used, since it models both the initial exponential change, followed by

a smooth, asymptotic, approach to the outburst value. In terms of the Shakura-

Sunyaev viscosity parameter, the viscosity during a transition from quiescence into

outburst is given by


where £th is the total time-scale for the transition (Truss et al 2000 show that a value

of a few thousand seconds is appropriate for DN). The resulting function is shown

is figure 5.2. Conversely, for the transition between the hot and cold states we find

This modified version of the SPH code has recently been used by Truss et al

(2000) to model DN outbursts. Following Truss et al, we increase computational

speed by dividing the disc into a number of concentric annuli (typically 1 0 0 ), and

use the azimuthally smoothed surface density of each ring to determine the viscosity

of all particles in that annulus. Truss et al compare this method with a version of

the code in which single particles can change state, and find that the two different

approaches produce similar results.

5.2.1 M agnetic model

In addition to outburst triggers, a model for the magnetic interaction has to included

in the SPH code. The model used is identical to that described in section 4.3.3, i.e.,

as material moves through the magnetosphere it interacts with the local magnetic

field via a velocity dependent acceleration of the general form

where v and Vf are the velocities of the material and magnetic field respectively, and

the suffix ± refers to the velocity components perpendicular to the field lines. The

magnetic time-scale can be written in terms of k as

(5.5)

fc[v - Vf]± (5.6)

WZ Sge in outburst

'hot

co ld

t

Figure 5.2: Functional form of the outburst trigger. The viscosity is switched on a local time-scale tth-

In order to model outbursts successfully, k has to be calibrated in order to satisfy

the following hierarchy of time-scales:

^visc,h ~ ^mag ~ ^visc,c (5.8)

The easiest method of calibrating k is to examine the orbits of single test particles.

When a single particle enters the primary’s Eoche Lobe, it adopts a ballistic tra

jectory around the WD (see section 1.3.1). If the WD possesses a weak magnetic

field, the magnetic torque gives a small amount of angular momentum to the parti

cle each time it passes close to the WD (as opposed to an ejector model, where the

test particle receives sufficient angular momentum to eject it from the system on its


first orbit). Hence, k should be calibrated such that the orbit of the test particle

gradually gets larger, but does not gain enough angular momentum to escape from

the potential well of the WD. Calculations show that a value of k « 3 x 10- 5s- 1

satisfies this condition.

5.2.2 System Parameters

WZ Sge has a well observed orbital period of 4898 s (eg. Patterson et al, 1981), and

a mass transfer rate during quiescence of —M2 ~ 1015 g s- 1 (Smak, 1993). ASCA

observations (Patterson et al, 1998) reveal a 27.87s oscillations in the 2-6 keV band,

which is taken to be the WD spin period. Numerical calculations are carried out

using these system parameters, along with the system masses quoted by Spruit and

Rutten, namely q = 0.075 and Mi = 1.2M©. The exact choice of q and M\ is not

vitally important as the model is being used to test the plausibility of the propeller

model (in chapter 7 I fit theoretical lightcurves to observed eclipse profiles, and find

that q = 0.05 provides the closest fit).

5.2.3 Numerical Scalings

In order to reduce the run time of the code (SPH is computationally demanding since

calculations involve around 40 000 particles), artificially high values for the viscosity

parameter are used. I take ac = 1.0, and = 10.0, and scale all time-scales

accordingly (Frank King and Raine, 1992, show that at a given radius tviSC oc a -4/5).

In addition, the magnitude of the critical surface densities are reduced from the

values given in equations (5.2) and (5.3). When doing this, it is important to consider

WZ Sge in outburst

the ratio of £ c to Eh, since this effects the ratio of time spent in quiescence compared

to outburst.

The accretion disc emission depends on both the viscosity and disc mass, and

therefore the above scalings affect the amplitude of disc outbursts. The change

in outburst amplitude can be estimated by considering the dissipation rate of the

accretion disc, which can be written as (eg. Frank, King & Raine. 1992)

M )

where v is the viscosity, and Q, is the angular velocity of the gas at radius R. The

dissipation rate is therefore proportional to vE.

The SPH code uses a simple 1:10 ratio of a in the hot and cold states, and is

independent of surface density. The surface density triggers are set at

£ c = 1.07— gem- 2 (5.10)

Eh = 0.33— gcm~ 2 (5-11)a

where a is the binary separation. The surface density just before outburst will be

close to £ c, and close to Eh during outburst. Therefore, the ratio of dissipation in

outburst to quiescence is approximately

A , = <*hot £ h ^ g o g ^ ^

T 'c C^coldEc

Now consider a non-isothermal accretion disc. The artificial viscosity for a thin,

Shakura-Sunyaev accretion disc can be written as


where Cg is the sound speed and H is the scale height. Assuming radiation pressure

can be neglected

cl = ^ oc Tc (5.14)

where Tc is the mid-plane temperature. Using equations (5.2) and (5.3) for the

critical surface densities, the ratio of dissipation (for a fixed radius) is given by

Dh _ 8.25ahot^c,hotQho°t8Dc ~ U A a coldT c,colda ; ™ 6 1 ' j

Setting Q;coid = 0.01 and ahot = 0.1, and adopting Cannizzo’s (1993,b) values of

^ c ,h o t ~ 60 0 0 0 K and T C)COid — 4 0 0 0 K, the ratio of dissipation in outburst to quies

cence is around 13.1. The compression factor introduced in outburst amplitude by

SPH is approximately 4.25.

5.3 Results

Fig. 5.3 shows simulated lightcurves for WZ Sge in the U,V and K bands. These

are calculated by assuming that the accretion disc is optically thick, and that each

annulus in the disc radiates as a black body. The Planck function is then simply

integrated over the chosen waveband, and summed over all annuli (cf. section 7.6.2).

A 1.2M0 WD of 16 000K (Ciardi et al.), also assumed to be a black body, is added

to the disc emission, giving the lightcurves shown in fig 5.3. Note that the numeri

cal scalings introduced in section 2.2.3 lead to a compression in outburst amplitude

of A M ~ 1.57 magnitudes. The outburst amplitude seen in the theoretical visual

lightcurve is 3.5 magnitudes (fig 5.3). Accounting for the compression factor this


leads to a predicted visual flux increase of 5.07 magnitudes, with a maximum bright

ness of ~ 9 magnitudes. The rise-time in the visual is ~ 1 day, followed by a much

slower decline in brightness (taking ~ 3 days to drop by 1 magnitude) after the

outburst peak. These values are in reasonable agreement with the observed visual

outburst profiles of WZ Sge (fig. 4.2). Fig. 5.4 shows the lightcurves superimposed

on the same axis, with the quiescent brightness levels normalized to that in the

V-band. The results show an outburst amplitude (accounting for the compression

factor) of 4.27 magnitudes in the U-band, and 2.77 magnitudes in the K-Band. A

time delay of ~ 10 hours can be seen between the U-band and visual lightcurves,

a feature that has frequently been observed in DN (King 1997, Livio and Pringle

1992).

At ~ 39.5 days the system falls into quiescence, resulting in a sharp decay on

the thermal time-scale. This is followed by a slow exponential decline to the pre

outburst brightness level, making the total outburst duration ~ 6 days. However,

in the simulation the ratio of Emax to Emjn is a factor of 3 lower than that of Can-

nizzo (1993). Therefore the time spent in outburst needs to be scaled accordingly,

increasing the outburst duration to around 20 days. This is in reasonable agreement

with observation.

The X-ray luminosity of the system can be estimated by considering the gravita

tional energy released as particles are accreted onto the WD. The particles used in

the simulation have a mass Mp, and therefore the X-ray luminosity is approximated


38 39 40 41 4236 3735

1438 39 40 41 4236 3735

12.0

12.5

13.0

13.5 -

14.04136 37 38 39 40 4235

Tim e(days)

Figure 5.3: U-band, K-band and Visual lightcurves of WZ Sge.

where dN/dt is the rate at which particles are accreted, and rj is an efficiency fac

tor, expected to be <C 1. The X-ray lightcurve is shown in figure 5.5. The X-ray

luminosity does not begin to rise until t ~ 36.9 days, 17 hours after the outburst is

initiated. This delay corresponds to the time it takes for the hole at the centre of

the accretion disc to close up, allowing accretion on to the WD (see section 5.3.1).

Mag

nitu

de

WZ Sge in outburst

10

1

12

13

14

1535 36 37 38 39 40 41 42

timpfDavc^

Figure 5.4: Superimposed theoretical U-V-K lightcurve profiles for WZ Sge, showing a UV delay of ~ 10 hours. Times marked 1-4 correspond to the surface density profiles shown in fig 5.7.

X-ray

di

ssip

atio

n (e

rg/s

)


2.010

.5*10

. 0 * 1 0

36 37 38 39 40 41 42time (Days)

Figure 5.5: Theoretical X-ray lightcurve for WZ Sge. The X-ray luminosity rises ~ 17 hours after the onset of outburst, corresponding to the time taken for the central hole to fill.


5.3.1 Disc Analysis

The lightcurves in fig. 5.3 show that the magnetic propeller model is able to produce

disc outbursts in WZ Sge. In this section I provide a clearer picture of the disc

evolution by following the dynamic behaviour of the accretion disc itself through

outburst. Figure 5.6 shows a series of grey scale density maps of the accretion disc

as the system goes into outburst. This clearly shows the hole at the centre of the

accretion disc closing up, allowing the WD to accrete material. The propeller is

quenched during outburst, which accounts for the disappearance of the 27.87s 2-6

keV oscillation (Patterson et al. 1998) during outburst, since this is believed to be

due to magnetically channelled accretion.

Figure 5.6 shows doppler tomograms of WZ Sge, calculated at the same time

intervals as the grey scale density maps. Spiral features can clearly be seen in the

Doppler tomograms. The spiral arms are tightly bound in quiescence, but as the

system moves into outburst, the arms appear to unwind slowly, as expected for a

disc in the high viscosity state (Steeghs and Stehle 1999).

The viscous evolution of the disc can be clearly seen in fig. 5.7, which shows

the surface density profile of the disc at times indicated in fig. 5.4. The outburst

begins at the peak of the surface density distribution, which occurs at R ~ 0.4a in

fig 5.7, panel 1. This corresponds to the start of the visual and K-band outbursts

(see fig 5.4). Panels 2 and 3 of fig. 5.7 show a ’density wave’ (analogous to a heating

wave in a full thermodynamic treatment) moving inwards on the hot viscous time-

scale TvjSC)h ~ R2/aCsH ~ 5d. Panel 2 coincides with the initiation of the U-band

outburst, and accounts for the delay between the U-band and visual lightcurves,

WZ Sge in outburst

-1.0 -0.5 0.0 0.5

0.5

0.0

-0.5

- 1.0

, ...........5

: /Jr

, . —... ,.„■

, , i . .. -1-.-. 1 * . I-

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 1 5 ,1,000,

-1.0 -0.5 0.0 0.51.0

0.5

0.0

-0.5

- 1.0

/ r/

i I w

J .y J

* j-1-*_-.-.-«-1 -----.-»_

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 1 5 .............

-1.0 -0.5 0.0 0.5 1.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5x 1000 km/s

Figure 5.6: Grey scale density plots (lhs), and doppler tomograms of WZ Sge during rise-time. The top panel corresponds to the start of the outburst, the middle panel ~ 10 hours after the start of the outburst, and the bottom panel ~ 17 hours after the start of the outburst.

(sjiun qds)

3 (squn

qds) 3

WZ Sge in outburst

4 .0 * 1 0 1

0.2 0.40.0 0.6 C.8

.0x10'

R /a R /a

13

13

13

^ 5 .0*10 M

olZ0.0 0.2 0.60.4 0.8

2.0x10*

0.2 0.40 .0 0.80.6

R /a R /a

Figure 5.7: Evolution of the surface density profile through outburst. The straight lines show the radial dependence of the upper and lower surface density trigger. The times corresponding to plots 1-4 are marked on fig. 5.4

WZ Sge in outburst

since hot material (~ 20 OOOif) is required in inner regions of the disc (i.e. radii

0.25a) to cause a rise in the U-band luminosity. Eventually (panel 3), mass

reaches the inner regions of the disc, and accretion occurs (the inner disc boundary

is set at Rm ~ 0.02a). This gives rise to the increase in X-ray flux seen in fig 5.5.

Once in this state, the disc slowly drains until the disc enters quiescence once again,

at which point the propeller begins to act, and a hole is gradually swept out in the

centre of the disc once more. The 27.87s oscillation is not expected to return until

the inner regions of the disc have viscously adjusted on the cool viscous time-scale.

5.4 Conclusion

In this chapter I have followed WZ Sge through outburst, and shown that the mag

netic propeller model is plausible. I have produced theoretical visual lightcurves,

with an outburst amplitude of around 5 magnitudes and a rise time of ~ 1 day, in

good agreement with observation. The outburst profile shows a steady, exponential

decay after peak brightness, again in close agreement with the observed lightcurve

profile. I have shown that the UV delay in WZ Sge is ~ 10 hours, and predicted that

the rise in X-ray luminosity will occur ~ 17 hours after the outburst is initiated,

corresponding to the time it takes to fill the hole at the centre of the accretion disc.

Doppler Tomograms show that as the system goes into outburst, high velocity

regions become more densely populated, and the spiral arm features slowly unwind.

Grey-scale density maps clearly show the hole at the centre of the accretion disc

closing up during outburst, and is able to explain the disappearance of the 27.87s

(2-6 keV) oscillation (Patterson et al. 1998) during outburst.

Chapter 6

Photometric Techniques

6.1 Introduction

In chapters 4 and 5, a magnetic propeller model was used to explain the long recur

rence time in WZ Sge. Chapter 7 presents the first high-speed K-band photometry

of WZ Sge, which is used to test some of the observational predictions of the pro

peller model (see section 4.4). In this chapter I introduce the photometric tequniques

which are used in chapter 7.

6.2 The Photom etric Scale

The apparent magnitude scale was first introduced by Hipparchus (2 nd century BC).

Stars (viewed with the naked eye) were divided into six classes from the brightest (1st

magnitude) to the faintest (6 th magnitude). In 1856, Pogson defined a photometric

scale that approximately agreed with Hipparcus’s brightness classes, given by

mi - m2 = —2.51ogio ( h / h ) (6.1)

120

Photometric Techniques 121

where mi and m 2 are the magnitudes of stars 1 and 2 , and I\ and / 2 are their

apparent brightnesses (or fluxes) respectively. The scale is such that brighter stars

have lower magnitudes than fainter stars.

6.3 M agnitudes

Photometric measurements are taken with the use of filters, which block all radiation

bar that in a specified wavelength range. The apparent magnitude zero for all

filters is defined using the star Vega (ct Lyr). Vega is always defined as having

apparent magnitude zero, regardless of the filter that is in use. The zero point for

an observation of a star depends on the intrument used to observe it, and is defined

by rearranging equation (6 .1 ), giving

mi + 2.5 log Ji = m2 + 2.5 log I2 = ZPi$ (6.2)

where Z P is the instrumental zero point.

In order to compare the magnitudes of two stars at different distances, it is neccessary

to normalize the apparent magnitude scales to the same distance and interstellar

extinction. The normalized magnitudes, termed absolute magnitudes (denoted by

an upper case M), are defined at a distance of 1 0 parsecs, and are related to the

apparent magnitudes via equation (6.3).

M\ = m \ — (5 log d - 5) — Ax (6.3)

Here, d is the distance of the star in parsecs, and A\ is the interstellar extinction to

the star.


6.4 Photom etric Calibration

As radiation passes through the Earth’s atmosphere the intensity decreases as pho

tons are scattered out of the beam. The degree of scattering is proportional to the

size of the column of atmosphere it has passed through, and also varies with wave

length. The size of the column of atmosphere is called the airmass, and its units are

defined such that one airmass corresponds to the height of the column directly over

head. The airmass (AM) increases as observations are taken closer to the horizon,

and can be approximated by,

A M = sec z (6.4)

where z is the angular distance of the star from the zenith. Hardy (1962) obtained a

more accurate relationship for the airmass by considering the curvature of the Earth,

but found that for z < 2, equation (6.4) is accurate to two decimal places.

The intensity I \ of the radiation as it passes through the Earth’s atmosphere can be

written as

h = h (o) e*-4 (6.5)

where I \ (0 ) is the intensity before entering the atmosphere, and T\ is the optical

depth. If a given star is observed at two different airmasses, (6.5) can be rearranged

to obtain the optical depth, i.e.,

_ In / A>1 - In I x ,2

TA_ AM 2 - AMi

which can be rearranged to give


CD

□lTMm00

C N

00

Airmass

Figure 6.1: Plot showing a typical airmass curve (F.Kenyon, private communication).

Utilising equation (6.2), the intensities can be expressed in terms of zero point giving

Z P x - Z P 2k = T\ (2.5 log e) = (6 .8)A M 2 - A M i

If measurements are taken at several different airmasses, and zero point is plotted

as a function of airmass (these plots are referred to as airmass curves), a straight

line is obtained, of gradient k (where k is known as the air mass coefficient).

Therefore, photometric calibration can be achieved by observing a number of stan

dard stars (these have a known magnitude) at different airmasses, and for different

filters. An example of a typical airmass curve is shown in figure 6.1.


6.5 Infrared Cameras

The observations presented in chapter 6 were taken using UFTI, the infra-red camera

situated at the United Kingdom Infrared Telescope on Mauna Kea, Hawaii. UFTI

operates in a similar way to optical charged couple devices (CCD).

CCDs are semiconductor devices, cooled to very low temperatures (~ 150K) in or

der to reduce thermal noise. Light falling on the camera during an exposure creates

hole-electron pairs. The electrons are stored using a positively charged electrode,

and a series of confining electrode ’gates’. At the end of the exposure, the gate

electrodes are opened in turn, and the charge stored in each gate is transferred to

a read-out point, where it is converted into a digital signal (see Kitchen 1997 for a

more detailed description of the process). The number of electons represented by

one analogue to digital count (referred to as an ADU) depends on the gain of the

CCD.

UFTI is also a semiconductor device, but needs to be cooled to a much lower temper

ature than optical CCDs, in order to reduce the dark current. The detection precess

is very similar to that for optical CCDs, but the array is considerably smaller, and

hence sky coverage is far lower.

6.6 Image reduction

The images produced by UFTI (and CCDs in general) are in the form of digital

ized pixels, and are therefore ideal for computer reduction. Raw CCD images have


an instrumental signature which needs to be removed in order to obtain accurate

photometry. The steps required are as follows:

• An artificial offset is added to each pixel charge in order to prevent the ana

logue digital converter receiving a negative signal (due to noise on a low level

background). The offset, known as the bias level, has an intrinsic noise known

as the read-out noise. A ’bias frame’ is obtained by taking an image of zero ex

posure time, which is then subtracted from the raw image in a process known

as de-biasing. The de-biased signal is directly proportional to the number of

photons detected.

• Dark current in the signal, produced by thermal noise (this is particularly

important in infra-red arrays) needs to be removed. This is achieved by taking

an exposure for the same length as the image, with the telescope shutter closed.

This ’dark frame’ can then be subtraced off the image, such that the image

consists only of photon detections from the observed sources.

• The amount of charge stored per photon varies slightly from one array element

to the next. This non-uniformity can be corrected by ’flat fielding’ the image.

A flat field can be created by imaging a uniformly bright source (e.g. a blank

region of night sky), and normalizing the resulting image to a mean value of

1 . In infrared imaging the sky is relatively bright, and it is more usual to

use median filtering of the actual observations, Median filtering removes the

individual sources, leaving the uniformly bright background sky only. This can

be normalized to a mean of 1.0 as before. Each image is divided through by


the normalized flat field, such that the ADU for each array element represents

the same flux.

6.7 Aperture Photom etry

Photometry is obtained by counting the total number of ADUs (for the reduced im

age) in each source. Firstly a value for the background sky for each source is found

by taking a median average of pixel values in an annulus centred on the source. The

annulus radius should have a radius large enough for the signal to be uncontami

nated by the central source, yet be small enough to be representative of the local

background level.

The ADUs in the source can then be found by adding the signal above the back

ground level in a circular aperture of sufficient radius centred on the source. The

gain of the CCD allows the number of ADUs to be converted into the number of

electrons and hence electrons detected, which, when divided by the exposure time

gives a flux. The magnitude of the source can then be obtained from equation (6.2).

6.8 Errors

There are many different sources of error that occur in photometric observations,

including errors in the estimation of the dark sky background as well as the error

resulting from the read out noise. In order to obtain an accurate reading the signal

coming the source must be far higher than the error associated with it. A quantity

that is required to calculate photometric errors is the ’signal to noise ratio’.


Assuming that the detections come from a Poissonian distribution, asky, the

standard deviation of the sky background photon flux per pixel is given by

where Ss is the sky background photon flux per pixel. The signal to noise ratio is

then given by

R O N is the read-out noise.

6.9 Conclusions

In this chapter I have described the photometric scale, and shown how accurate

photometry can be obtained from flux measurements. The methods outlined here

were used for the K-band observations of WZ Sge, presented in chapter 7.

(6.9)

v/>, + ( N x Ss) + RO N(6 .10)

where Fs is the flux from the source, N is the number of pixels in the aperture and

Chapter 7

K-band observations of WZ

Sagittae

7.1 Introduction

As explained in chapter 4, WZ Sge is an unusual SU UMa star, because as well as only

producing superoutbursts, the recurrence time is extremely long: superoutbursts

have been observed in 1913, 1946, and 1978, making the inter-outburst time-scale

« 33 years. WZ Sge is the prototype of a class of systems with similar outburst

properties (often referred to as TOADS, or Tremendous Outburst Amplitude Dwarf

Novae).

The orbital parameters of WZ Sge place it at the extreme end of the CV class:

WZ Sge has a very short orbital period of Porb ~ 82 minutes (Krzeminski 1962),

very mass low secondary (M2 < O.O7M0 , Spruit and Rutten 1998), and extreme mass

ratio (g< 0.075, Spruit and Rutten 1998). The mass transfer rate is also very low

128

K-band observations of WZ Sagittae 129

(—M2 ~ 1015 g s-1, Smak 1993). This, along with the fact that the orbital period is

extremely close to the minimum period, suggests that WZ Sge may be an extremely

old CV. The accretion process in WZ Sge is therefore worth close study since it may

have important consequences for CV evolution.

WZ Sge has been observed to show rapid coherent oscillations during quiescence.

The oscillations are seen at either 27.87s, or 28.98s, or at both periods simultane

ously, and were first discovered by Robinson, Nather and Patterson in 1978 (RNP

hereafter). Similar oscillations have been detected in many CVs, and are usually

associated with DQ Her stars, in which the observed oscillations are due to mag

netically channelled accretion onto a rapidly spinning white dwarf. However, the

oscillations seen in WZ Sge are more difficult to explain than other DQ Hers. In

particular, the simultaneous presence of 2 coherent oscillations, the large amplitude

variation, and the disappearance of the oscillations altogether during outburst (and

possibly for many years after the system has returned to quiescence), are difficult

to explain using a standard DQ Her model. This has led some authors to suggest

that the oscillations are due to non-radial g-mode pulsations of the white dwarf (eg

RNP, Skidmore et al 1997).

In this chapter I present the first infrared oscillations in WZ Sge, which show

a strong feature at 27.88s, and weaker oscillations at slightly longer periods, and

attempt to explain the oscillations features (in all wave-bands) using a magnetic

WD model. Such a model has a number of favourable aspects, in particular it is

able to account for the long interoutburst time-scale without invoking a low value for

the viscosity. In chapter 4 (see also Wynn, Leach & King, 2000), I showed that the


recurrence time of 33 years can be explained using standard DIM (disc instability

model) values for a, if the WD has a magnetic moment of /x<1032 G cm3. The

system acts as a magnetic propeller, since as material approaches the WD, it is

propelled to larger radii by the magnetic torque. In this chapter I argue that the

magnetic propeller model is able to explain some important observational features

(in particular the transient, varying amplitude nature of the oscillations).

One of the possible features of the model presented in chapter 4 is the presence

of a cool precessing elliptical disc in quiescence. Unlike in outburst, the superhumps

will be weak, and therefore difficult to detect in the lightcurves (in chapter 4 I predict

that the variation will be too weak to detect in the visual, and will have a amplitude

of <0.05 magnitudes in the infra-red). In section 7.5 an attempt is made to find

the preccessional period of the accretion disc by searching for evidence of long term

brightness variation in the lightcurves of WZ Sge. I find a brightness variation of

amplitude ±0.05 magnitudes, which would be consistent with a period several times

longer than P0rb, and can be interpreted as tentative evidence of an eccentric disk in

quiescence. Finally, simulated lightcurves are produced using a smoothed particle

hydrodynamics code. The resulting lightcurve profiles produced using a magnetic

propeller model are very similar to those observed. I conclude that the WD in

WZ Sge possesses a weak magnetic field, which is able to explain the observational

properties of the system, as well as the long recurrence time.

K-band observations of WZ Sagittae

.V .V .* / -jf / V . 5V

A ‘ V % 4 *

^ S s « .. y? ^ ,<r -•v **• * • ** -

Orbital p h a se

Figure 7.1: K-band lightcurves for WZ Sge, using an integration time of 7 seconds. Observation start times (given in JD0 ) are 2451000+: (a) 388.841473, (b) 388.984579, (c) 389.760877, (d) 389.890591. Error bars are not shown on the plot, but are typically ± 0 . 0 2 magnitudes.

7.2 Observations and data reduction

Observations were taken on July 28th and 29th 1999, using the 3.8m United Kingdom

Infrared Telescope (UKIRT) situated at Mauna Kea, Hawaii. WZ Sge was observed

in the near-infrared K broadband filter, using a 256 x 256 sub array of the 1024 x

1024 UFTI camera. The array has a spatial resolution of 0.091” per pixel, a total

field of view of 92” x 92” (for the full array), a read noise of ~ 26e_ , and a system

gain of 7e_/DN.

In order to reduce the effects of residual heat in the array, observations were


taken using a 5 point jitter pattern, with a 10” offset. We used an integration time

of 7 seconds per image, and a sampling time of « 22s (this varied by « 2s due to the

write time of the array). In order to obtain lightcurves, differential photometry was

performed using a bright companion (Mk « 10.5 magnitudes) 8 ” to the west of WZ

Sge for comparison (differential photometry was used since it is able to remove the

effects of changing weather conditions). Absolute photometry (required to calculate

the mean magnitude of WZ Sge) was obtained by observing a standard star at

the beginning, middle and end of each night. The images were reduced using the

Starlink Software package IRAF DAOPHOT. A flat field was constructed by taking

a median filter of 50 consecutive frames. Aperture photometry was performed using

a circular aperture of radius 10 pixels for both WZ Sge and its companion star, and

a sky aperture of inner radius 30 pixels and width 10 pixels (centered on each star

in turn).

7.3 K-band Lightcurves

The K-band lightcurves for the two nights are shown in fig. 7.1. The phasing was

accomplished using the ephemeris of Patterson et al (1998). The lightcurve profiles

look very similar to those presented by Ciardi et al (1978): there is a sharp eclipse at

phase 0.0, and a double humped W Ursae Majoris-type variation, with the humps

occurring at phases « 0.2 and « 0.7. Double humped modulations are usually

attributed to ellipsoidal variations of the secondary star. However, Ciardi et al

(1998) find that the secondary star only contributes ~ 5% of the total IR flux (since

it has an extremely weak spectrum), and it is therefore extremely unlikely that the


secondary star causes the double humped variation in WZ Sge, as this amounts to

~ 20% of the total system flux. A likely origin for this modulation is the varying

aspect of an optically thick hotspot (Ciardi et al 1998), and I will return to this in

section 7.6.3.

As in the case of visual observations (eg. RNP) the primary eclipse in the Ri

band lightcurves is asymmetric (this could not be confirmed in previous infra-red

observations due to insufficient time resolution), with the ingress time longer than

the egress time (the ratio of ingress and egress times is consistent with RNP, at ap

proximately 10:7), whilst the eclipse depth is approximately 0.2 magnitudes. Unlike

the optical lightcurves (eg RNP), we find no dips in the hump occurring after the

primary eclipse, and the lightcurve profiles are far less variable from one orbit to the

next.

Figure 7.2 shows the whole set of data obtained over both nights, folded on the

orbital period (4897.83 s) into 200 phase bins. Folding the data reduces the scatter,

allowing the lightcurve structure to be more closely examined. Fig. 7.2 shows that

in addition to the sharp primary eclipse, ’knees’ in the lightcurve (resembling a

broader, shallower eclipse) can be seen at orbital phase 0.90 and 0.105 (the effect

is particularly noticeable during ingress). In section 7.6, I attempt to fit simulated

lightcurves to the folded lightcurve, and suggest that the deep eclipse at phase 0 . 0

is the primary eclipse, whilst the broad shallow eclipse is due to occultation of the

accretion disc itself.

mag

nitu

ded

diff

eren

ceK-band observations of WZ Sagittae 134

13.70

13.80

13.90 • • •

14.00 •

V%V#*14.10

14.20

14.30

1.200.80 1.00 1.100.90orbital phase

Figure 7.2: Folded lightcurve showing the eclipse region of WZ Sge. The two nights of data have been folded on the orbital period into 2 0 0 phase bins.

7.4 Short period oscillations

7.4.1 Lomb-Scargle periodograms

The Lomb-Scargle formalism (Scargle 1982) was used to search for short periods in

the data, since the method takes into account both unevenly sampled data, and the

noise of the observations. At low frequencies we find power only at Porb5 and its

harmonics. Fig. 7.3 shows the average power spectrum for data over both nights for

periods ranging between 27.5 and 29.0 seconds (periods less than 27.5 seconds were

considered to be too close to the sampling period of the observations, and there were


no significant features in the spectrum for periods greater than 29.0 s). Note: the

power spectra were oversampled in frequency by a factor of 2 .

Figure 7.3 shows that there is a significant signal at 27.88 ± 0.01 s, as well as

weaker peaks at 27.99 ± 0.02, and 28.14 =t 0.01. The Lomb Scargle formalism allows

us to predict the ’false alarm’ probability, p f , that the signal is due to noise. We

find that the probability that the 27.88 and 27.99 s signals are real is 1 — pf >99%,

whilst the probability for the 28.14 s signal is 1 — pf « 77%. In order to test that

the features in the periodogram were not merely an artifact of the sampling time,

a ’dummy’ data set was produced, in which the magnitudes of all the data points

were set to zero. The resulting periodogram did not show any features at periods

27 - 30 seconds, just a strong peak close to the sampling period (~ 22 seconds).

The dominant period has frequently been seen in the B-band (eg RNP, Patter

son, 1980), the U-Band (Skidmore et al, 1997), as well as in HST UV observations

(Skidmore et al, 1999), and ASCA X-ray observations (Patterson et al, 1998). Peri

ods close to 28.14 and 38.33 s have also been reported in some of the fore-mentioned

papers (a detailed listing of all previously detected observations can be found in

Skidmore et al, 1999).

7.4.2 Discussion of the oscillations in WZ Sge

Since the 27.87s oscillation was first discovered in 1978 (RNP), there have been two

competing models regarding the origin of these oscillations. The first of these models

assumes that the oscillations are produced by non-radial g-mode pulsations of the

white dwarf. This model was first introduced (RNP) in order to explain the presence

K-band observations o f WZ Sagittae 136

of simultaneous signals, and has continued to be a plausible model. However, since

little is known about the effects of accretion on to a pulsating white dwarf, it is

extremely difficult to make firm predictions about the observable properties. How

ever, Skidmore et al (1997), find that the oscillation colours are very blue, requiring

unplausibly high WD temperatures, and conclude that the oscillations cannot be

caused by direct observations of a pulsating white dwarf.

The second model suggests that WZ Sge contains a rapidly rotating magnetic

WD, making the system a member of the DQ Her family. Patterson (1980) found

that the 27.87s oscillation is a stable clock, favouring magnetically channelled accre

tion on to a small fraction of the white dwarf surface. Here, the satellite periods were

identified as beat phenomena between the fundamental oscillation period and the

longer orbital periods found in the accretion disc (see also Lasota et al 1999). The

discovery of the principal oscillation in the 2-6 keV band (Patterson et al 1998) pro

vides fairly strong evidence of magnetically channelled accretion. However difficulties

with the DQ Her model continue to exist. Firstly, the 27.87s signal was unobserved

for 16 years before returning. In addition, the X-ray pulses are extremely weak,

and along with the 27.87s principal period, the 28.09s UV oscillation (Welsh et al

1997) also contains a Lya absorption feature, suggesting that that the oscillation

originates on the WD surface, and is not due to reprocessing off the accretion disc.

K-band observations o f WZ Sagittae

40

27.88 s

k -0£oQ .a) 20 >

27.99 s

28.14 s

27.6 27.8 28.0 28.2 28.4 28.6 28.8 29.0period (s)

Figure 7.3: Lomb Scargle periodogram showing relative power versus period between 27.5 and 29.0 s. Indicated on the plot are the principal signal occurring at 27.88±0.01 s, as well as weaker signals at 27.99 ± 0.01 s and 28.14 ± 0.01 s.


0.5

>> 0.0

-0.5

-1 .0 L- 1.0 -0.5 0.0 0.5 1.0

x

0.5

-0.5

- 1.0-1.0 -0.5 0.0 0.5 1.0

xFigure 7.4: Grey-scale mass weighted density map of WZ Sge just before (top) and during (bottom) outburst.


7.4.3 WZ Sge as a magnetic propeller

The magnetic model presented in chapters 4 and 5 naturally accounts for the long

recurrence time (33 years) without invoking an extremely low viscosity. These chap

ters showed that the recurrence time (and high disc mass prior to outburst) can be

explained if the WD is weakly magnetic, producing a truncated accretion disc. The

magnetic torque prevents accretion during quiescence, propelling material further

out into the Roche Lobe of the WD (i.e. the magnetic field is lower than in typical

DQ Hers, yet insufficient to eject material completely from the system, as in the

case of AE Aqr (see Wynn,King & Horne, 1997)).

This model predicts that the accretion rate should be extremely low during qui

escence, which accounts for the weak/transient nature of the X-ray oscillations. In

addition, we expect many magnetohydrodynamic instabilities to exist near the in

ner edge of the accretion disc which could produce short bursts of accretion on to

the white dwarf. This could explain why the oscillations are extremely variable in

amplitude (an observational feature that is difficult to explain using the g-mode os

cillation model). Other DQ Her stars show oscillations in X-Ray, IR and Optical

wavebands (Warner, 1995), and therefore the discovery of IR oscillations in WZ Sge

supports existing evidence of a magnetic WD.

The magnetic propeller model produces very different disc dimensions to a non

magnetic system with similar binary parameters. Numerical modelling in chapter 4

predicts an inner radius of Rin « 0.2a, and an outer disc radius of R0XJLt « 0.55a.

Assuming that oscillations longer than the 27.87s WD spin period are beats resulting

from reprocessing of X-ray emission from the WD on the surface of the disc, we find


that the Keplerian orbits associated with these beat periods can be written as

Here PK is the Keplerian period of material at the inner disc edge, and P0bs is the

observed beat period. Assuming Mi = 1.2MQ we find the 28.96s (i beat — 0.28a)

to the inner disc edge, whilst the 28.14s period corresponds to the outer disc/hot

spot (i^beat — 0.65a). The 27.99s oscillation leads to a radius of 1.19 a, which could

possibly result from reprocessing on the secondary surface. A tighter constraint on

q is required to match these radii precisely.

The magnetic propeller model also requires the oscillations to disappear during

outburst. Numerical modelling through outburst shows that when the accretion disc

switches into the hot state, the ram pressure of the material is able to compress the

magnetosphere, making the propeller extremely inefficient (see chapter 5). This can

be seen in fig. 7.4, which shows a grey-scale density map of the accretion disc just

before, and during outburst (the sph code used to produce the result is described

in more detail in chapters 2 and 5). During outburst the hole at the centre of the

accretion disc closes up, and the system resembles a non-magnetic CV (although the

outer disc radius is larger due to the transfer of angular momentum from the WD to

the disc during quiescence). We therefore predict that there will be no observational

evidence pointing towards a magnetic WD during outburst.

The propeller model also predicts that the 27.87s will be extremely weak for

PlGMi(7.1)

where

1 1 1(7.2)

and 29.69s (i^beat — 0.20a) oscillations correlate closely to material orbiting close


- 0 . 1 o

-0 .0 5

0.00

0.05

0 . 1 O(

- 0 . 1 o

-0 .0 5

0.00

0.05

0 . 1 OO 5 10 15 20 25

T i m e ( / P o r b )

Figure 7.5: Figure showing the long term brightness trend in WZ Sge for N=2 (top plot) and 7V=10 (bottom plot). AM is the deviation from the mean lightcurve (marked with a dashed line).

some time after outburst. Chapter 5 shows that during outburst the WD accretes

a large fraction of the disc mass, so when the disc returns to quiescence the surface

density is very low, and we expect virtually no accretion until the disc has had time

to build up mass near the inner disc edge (this is a difficult time-scale to predict

since it depends on the mass transfer rate, the magnetic interaction time-scale and

the cold viscous time-scale). Hence we expect the 27.87s oscillation to disappear

during outburst and be very low in amplitude (too weak to be detected) for some

time afterwards.

n = 2

5 1 O 1 5 20 25

n = 1 0


7.5 The search for superhumps in quiescence

WZ Sge’s last outburst was 1978, hence the disc has had a long time to evolve

while in quiescence. If WZ Sge is a magnetic propeller, we expect the disc to have

had sufficient time to expand beyond the 3:1 resonance radius (chapter 4), and

there should be evidence of a precessing elliptical disc in the observational data. A

signature of such a disc are superhumps in the lightcurve (Lubow, 1994), but these

are extremely difficult to observe in quiescence because the disc is relatively cool. In

chapter 4 I showed that during quiescence, the superhump variation is unlikely to

exceed ±0.05 magnitudes (this is also likely to be an overestimate since the accretion

stream in the model was poorly resolved.) As mentioned in section 7.1, searching

for periods slightly longer than Porb proved fruitless due to insufficient data (as well

as the small amplitude of the superhump variation).

Therefore, rather than search for the superhump period PSh, I will attempt to

find Pprec? the precession period of the elliptical accretion disc where

— = —------------------------------------------------- (7.3)sh orb 1 prec

As the semi major-axis of the elliptical accretion disc precesses, the effective area

(and hence brightness) observed, will vary, and this should manifest itself in the

observed lightcurves. This would result in a long term brightness variation over the

two nights of observations,with a small amplitude (< 0.05) magnitudes, and a period

several times longer than the orbital period.

In order to search for long term brightness variations, it is necessary to remove the

effects of short period oscillations. This is achieved by binning each orbital period


into N phase bins, such that all short period variations are smoothed out. Bins with

the same phase are then averaged over successive orbital periods to obtain a mean

lightcurve, which is then subtracted from each binned lightcurve. Confirmation that

the trend was due to WZ Sge and not the companion star used for differential pho

tometry was obtained by performing similar analysis using absolute photometry for

WZ Sge and the companion star. On the first night, when conditions were photo

metric, it could be clearly seen that the companion showed no signs of variation,

whilst WZ Sge showed a long term variation identical to that obtained using dif

ferential photometry. Differential photometry was used, since this method ensures

that brightness fluctuations due to changing weather conditions (more prevalent on

the second night) are removed.

Fig 7.5 shows the resulting long term brightness trend for N=2 (top plot), and

7V=10 (bottom plot), where AM is the deviation from the mean lightcurve. There

is clearly a long term trend in the data, with an amplitude of around ± 0.07 mag

nitudes.

Whilst there is tentative evidence for superhumps in the data, it is extremely

difficult to fit a sinusoid to the trend in the data due to insufficient coverage, and

further observations are required in order to obtain a definite period. However, the

brightness appears to be increasing on both nights, and the brightness at the start

of the second night is lower than that at the end of the first night, implying that a

minimum might occur somewhere in between. If we assume that this is the case, we

can place crude limits on Pprec of 15±5P0rb- Using equation (2 ), this corresponds to

a superhump period of 1.053 < Psh ^ 1.11 P0rb- Alternatively, there could be a much


0.5

-0.5

- 1.0-1.0 -0.5 0.0 0.5 1.0

x

Figure 7.6: Numerically simulated grey-scale dissipation map of WZ Sge, weighted to the local K-band dissipation.

longer trend in the data (resulting in a lower value for Psh), in which case we are

only observing a fraction of the precession period. Further observations are required

to confirm whether this is the case.

7.6 A com parison w ith SPH lightcurve sim ula

tions

7.6.1 N um erica l technique

In this section we show that the observed K-band lightcurves have very similar

profiles to numerically simulated lightcurves, in which it is assumed that WZ Sge

is a magnetic propeller. Numerical calculations were performed using smoothed

particle hydrodynamics, a Lagrangian method for modelling the dynamics of fluids.


For a detailed explanation of the code see chapter 2. In order to model the effects of

a magnetic field, a magnetic interaction term was introduced to the code. We adopt

a prescription for the magnetic interaction which assumes that as material moves

thorough the magnetosphere it interacts with the local magnetic field via a velocity

dependent acceleration of the general form

flmag = “ &[V “ V f]± (7.4)

where v and Vf are the velocities of the material and magnetic field respectively,

and the suffix _L refers to the velocity components perpendicular to the magnetic

field lines. This description is described in more detail in chapter 4. In order to

produce a magnetic propeller, it was necessary to calibrate k. This was achieved

by varying k , and following the trajectory of a single particle until it slowly gained

angular momentum as a result of the magnetic torque (but not enough to eject the

particle from the Roche Lobe of the WD). This required k & 3 x 10- 5 s-1.

7 .6 .2 G e n e r a t in g l ig h tc u r v e p r o file s

Lightcurves were generated from the numerical simulations using a simple eclipse

mapping technique. Assuming that the accretion disc is covered by a Cartesian grid

in the x and y directions then the total observed flux at a given phase can be written

as

^ = (7-5)Z= 1

where A is the area of each pixel, n is the number of grid cells in each direction, / is

the flux of the grid cell, and R is the fraction of the grid cell that is visible at phase <j>.

For each phase, the co-ordinate system is transformed from the inertial frame such


that the x and y axis lie along and perpendicular to the line of sight respectively.

Assuming the secondary Roche lobe is spherical (this is a good approximation at

high inclination) the observability of each cell is determined by the following limits:

R x ,c (7.6)

(y\ - R y ,c )2 + (z\ - Rz,c)2 ^ R 2 (7.7)

where Xi and y* are the grid cell coordinates, i?X)C Ry c and RZ)C mark the centre of

the Roche lobe, and R r is the radial extent of the Roche lobe. If (7.6),and (7.7) are

satisfied, then the grid cell is hidden from view and makes no contribution to the

total brightness. Note that this is a very simple calculation, and I do not consider

whether grid cells are hidden by other cells along the line of sight.

In order to produce a lightcurve using this technique the local k-band dissipation

in the accretion disc needs to be calculated. This was achieved by dividing the

accretion disc into 800 x 800 grid cells (running from -LI to +L1), and assuming

that each grid cell behaved as a blackbody. Since the total bolometric dissipation

D for each cell is known (calculated by summing the viscous force acting on each

particle), the K-band dissipation of each grid cell, K cen, is found by integrating the

Planck function between 2.0 and 2.44 microns, i.e.

r 2 A 4

ATceu = A / BA(T)dA (7.8)J 2.0

where

<tT4 = D /A (7.9)

and A is the area of the grid cell. The total K-band dissipation is then found by

summing over all grid cells. Fig. 7.6 shows a typical K-band dissipation map of

-bon

d m

og

k-bo

nd

mag

k-

bond

m

ag

k-bo

nd

mag


4.5

5.0

5.5

6.02.0 2.2

3.6 F 3.8 =- 4 .0 =- 4.2 i- 4.4 =-

Orbital p h a se

2.0 2.23.63.84.04.24.4

Orbital p h a se

1.6 3.6 F 3.8 =- 4.0 £ 4.2 =- 4.4 5-

1.8 2.0 2.2Orbital p h a se

2.0 2.21.6 1.8Orbital p h a se

Figure 7.7: Simulated eclipse profiles for (a) the WD, (b) the accretion disc, (c) the disc and a varying brightspot, and (d) a modulated accretion disc.

the accretion disc in WZ Sge (during quiescence), assuming that the disc is being

observed face on. The accretion disc is circular, with strong spiral arm features.

In addition, an extended hot spot region can be seen where the accretion stream

impacts the disc (this is consistent with the UV observations of Spruit and Rutten,

1998).

7.6.3 R esults

WZ Sge was modelled using an orbital period of 4898s, a WD spin period of 27.87s,

and a mass transfer rate of - M 2 ~ 1015 g s- 1 (Smak 1993). Results are presented

for three different sets of system parameters:

-ban

d M

ag.

K-b

and

Mag

. K

-ban

d M

agK-band observations of WZ Sagittae 148

13.4

13.814.014.214.4

IP* — —!

0.8

13.814.014.214.4

0.8

13.814.014.2

* 14.4 14.6

1.21.00.8 1.40.6Orbital P h a se

Figure 7.8: Plot showing a simulated lightcurve (solid line) consisting of a WD, varying hot spot and accretion disc, superimposed on top of the observed eclipse profile. Results are shown for q = 0.07, q = 0.05, and q = 0.03.

(1) Mi = 1.2M© q = 0.07 * = 80°.

(2) Mi = 1.0M© q = 0.05 * = 81°.

(3) Mi = 1.0Mo q = 0.03 i = 83°. The parameters in set (1 ) are very close

to the values given in Spruit and Rutten (1998). However, observations show that

the mass ratio could be as low as q = 0.03 (Patterson, private communication).

Parameter set (2) has an intermediate mass ratio. Ciardi et al (1998) show that the

WD contributes around 25% of the total K-band system brightness. Therefore, a

WD temperature of 16 0 0 0 K (Ciardi et al.) is used in all of the following theoretical

eclipse profiles, and is assumed to be a perfect black body.

-bon

d M

ag.

K-b

and

Mag

. K

-ban

d M

ag.


3.4 3.6

3.8 4.0 4.24.4

0.8

3.84.04.24.4

z ' n3.8 =- ■*«4.0 E-4.2 E-4.4 E- 4.6 L

0.6

0.8 t A

0.8 1.0 1.2 1.4Orbital P h a se

Figure 7.9: Plot showing a simulated lightcurve (solid line) consisting of a WD and modulated accretion disc emission, superimposed on top of the observed eclipse profile. Results are shown for q = 0.07, q = 0.05, and q = 0.03.

Eclipse profiles were generated using the method outlined in section 7.6.2. The

contribution of the various components is shown for q — 0.05 in figure 7.7. (a) shows

the white dwarf contribution, with a sharp eclipse at phase 1 .0 , whilst plot (b) shows

the accretion disc emission. A broad shallow eclipse can be seen between phases 0.9

and 1 .1 , which occurs as the secondary star passes in front of different regions of the

accretion disc. The disc profile shows no indication of the double humped variation.

This is not surprising, since the effect is thought to be due to the changing orientation

of the hot spot, or possibly due to shielding of different regions of the accretion disc.


These effects have not been considered in the lightcurve simulation, but can be

included as follows:

(i) Since the hot spot is elongated (Spruit and Rutten 1998), we expect the

effective area projected by the hot spot to vary sinusoidally on half the orbital period

(depending whether we are observing the hot spot face on or edge on). This will

produce an additional sinusoidal variation in the resulting lightcurve (this feature

is strong in WZ Sge since the bright spot contributes a high fraction of the total

system luminosity). The luminosity of the accretion disc will therefore consist of the

following components,

L to t = £d isc + A sin(47r(t - <j>)) (7.10)

where A is the amplitude of the hot spot variation, t is the time (expressed in terms

of the orbital period), and ( j ) is a phase offset. Plot 7.7(c) shows the resulting K-band

magnitude of the accretion disc using A = 1 . 6 x 1029 ergs s-1, and 0 = 0.175 (i.e.

minimum emission occurs at orbital phase 0.05, with a maximum at orbital phase

0.3).

(ii) The SPH simulation shows strong spiral shock arms in the accretion disc.

The shock arms are expected to have a larger scale height than their surroundings

due to the increased local temperature. The raised spiral arms will shadow certain

areas of the accretion disc, and will vary as the accretion disc changes orientation.

This would modulate the total disc emission, and can be modelled using

Ltot = Tdisc x [1.0 + Bsin(4n(t - (/>))] (7.11)

where B — 0.1 and (f) = 0.175. Plot 7.7(d) shows the accretion disc emission including


this effect. The phases of maximum and minimum emission are identical to that in

case (i).

Figure 7.8 shows simulated lightcurves including a varying hotspot (solid line)

superimposed on top of the observed folded eclipse profile for the three sets of system

parameters. Figure 7.9 shows similar results, using disc shielding in order to produce

the SU UMa like variation. The simulated eclipse profiles look extremely similar in

both cases, making it very difficult to distinguish between the models. The results

show that the primary eclipse is too wide using q = 0.07, and too narrow for q = 0.03.

The results for q = 0.05 are very encouraging, since the primary eclipse and broad

shallow eclipse of the accretion disc closely resemble that of the observed lightcurve.

An interesting feature of our lightcurves is the primary eclipse itself. This is pro

duced by the white dwarf passing behind the secondary star, and not the occultation

of the brightspot, as is currently believed. However, since the simulated lightcurve

is a good fit to the observations, perhaps it is worth considering whether the white

dwarf is responsible for the eclipse. The main reason that the brightspot was believed

to be responsible was the asymmetric nature of the eclipse, with the ingress time

exceeding the egress time. In addition, the depth, and width of the eclipse varies

in time in the optical, something not associated with white dwarf eclipse profiles

in DN. However, in a magnetic propeller model, magnetohydrodynamic instabilities

close to the inner disc edge will lead to bursts of accretion on to the white dwarf.

Therefore the emission of the WD and surrounding magnetosphere is likely to be

extremely variable in the optical wave-band.


7.7 Conclusions

I have analysed oscillations in high-speed K-band photometry of WZ Sge in quies

cence, and found a strong oscillation at 27.88 ±0.01 s, along with weaker oscillations

at 27.99 ± 0 . 0 2 s and 28.14 ± 0.01 s. I have argued that the properties of these,

and all other short period oscillations in WZ Sge can be explained using a magnetic

propeller model. In particular the model predicts that the oscillations should be seen

in many different wave-bands, and should disappear during, and for some time after

outburst. In addition, the oscillation amplitudes are expected to be highly variable

(due to the fluctuating accretion rate), which is extremely difficult to explain using

the g-mode oscillation model.

Tentative evidence of superhumps has been found in the K-band lightcurves by

examining the underlying long term brightness variation. However, further observa

tions are required in order to obtain a definite period. An SPH code has been used to

model the magnetic propeller model, producing theoretical lightcurve profiles. The

model is able to account for the main features seen in the observed lightcurve pro

files, and the model lightcurves closely resemble those observed without the need for

a full parameter search. The best results were achieved using the system parameters

Mi = l.OM0, q = 0.05, and i = 81°.

I conclude that a magnetic propeller model for WZ Sge succeeds in explaining

many of the observational properties of the system.

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