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Oxford Physics

M. Phys. Major Option C1

Astrophysics

Problem Sets 2016-2017

DrG.Cotter, Dr J.Devriendt, & Prof. Ph. Podsiadlowski

Denys Wilkinson Building, Keble Road,

Oxford OX1 3RH1

[email protected]

JD—7th October 2016

C1 Astrophysics problems 2016-2017

2

Chapter 1

Introductory problems

For Tutorial 1

You will need to refer to courses from previous years including the 1st and3rd year astrophysics short options.

1.1 Blackbody spectra and magnitude sys-

tems

Explain what is meant by a black-body and describe applications of thisconcept in astrophysics.The intensity of radiation emitted per unit wavelength interval by a black-body of temperature T is given as a function of wavelength by

B(λ) =2hc2

λ5

1

exp(hc/λkT)− 1.

Obtain an approximation for this formula appropriate for wavelengths suchthat hc/λkT << 1. In what region of the electromagnetic spectrum wouldyou expect this approximation to apply to the radiant emission from stars?The apparent magnitude m of a hot star of surface temperature 40 000 Kis measured at the two wavelengths 440 nm and 550 nm, corresponding tothe colour filters known as B and V. The colour index mB − mV is foundto be -0.35. Compare this with the theoretical colour index expected for ablack-body at this temperature.[The apparent magnitude is related to the observed flux f bym = −2.5log10f+k where k is a constant for a given wavelength. The colour index for a star

3


of surface temperature 11 700 K is zero.]

1.2 Optics and spectroscopy

Explain what is meant by the angular resolution and magnification of atelescope. Show that the limit of resolution is given approximately by λ/Dwhere λ is the wavelength of the incident radiation and D is the diameterof the telescope aperture. What is the typical angular resolution (in secondsof arc) achieved with a large ground-based telescope operating at visiblewavelengths from a good site?Give expressions for the resolving power and angular dispersion of a diffrac-tion grating, defining all quantities.In order to study the internal motion of galaxies astronomers often need tomeasure velocity differences differences of order 10 km s−1. Consider a gratingspectrograph equipped with a camera of focal length 2 m and a CCD detectorof pixel size 20µm. The spectrograph is used to study galaxies in the [Oiii]emission line whose wavelength is 500.7 nm. Given that the light is incidentnormally on the grating and the first-order spectrum falls on the detector,estimate the minimum grating ruling, in lines per mm, required to give thedesired velocity resolution.

1.3 Stellar Structure

Explain what is meant by the term hydrostatic equilibrium in stellar structureand discuss its importance. Derive an expression for the timescale on whichchanges occur if equilibrium conditions are disturbed.A star is completely supported by radiation pressure, and transport of energyis by radiation only. Use the equation of state p = aT 4/3 and the radiativetransport equation

L =16πr2acT 3

3κρ

dT

dr

(where L is the luminosity, κ the opacity, ρ the density, r the radius, cthe speed of light and a the radiation constant) to show that in hydrostaticequilibrium the luminosity is given by

L =4πGMc

κ

where G is the gravitational constant and M is the total mass of the star.What would happen if L were suddenly increased beyond this value?

4

Tutorial 1: Introduction

1.4 Accretion

Explain what is meant by escape velocity and device an expression for theescape velocity of a particle at distance R from a compact object of mass M .

By assuming that the maximum escape velocity is the speed of light, c, obtainan expression for the radius, RS of a black hole of mass M . Evaluate thisradius for the cases M = 10M and M = 106M.

What rate of matter infall (in units of M per year) would be needed to powera quasar of luminosity 1039 W if a black hole of mass 106M lay at the quasarcore? Assume conversion of gravitational potential energy into luminositywith 100% efficiency. Does your answer support accretion of matter by ablack hole as a likely model for quasar energy generation?

1.5 Cosmological models

State the cosmological principle and define the terms homogeneity and iso-tropy. Give an example showing that a homogeneous universe need not beisotropic.

The Friedmann-Robertson-Walker metric for a homogeneous and isotropicuniverse is given by

ds2 = −c2dt2 + a(t)2

[dr2

1− kr2+ r2(dθ2 + sin2θdφ2)

]

where ds is the proper time interval between two events, t is the cosmic time,k measures the spatial curvature, and r thea and φ are the radial, polarand azimuthal co-ordinates respectively. Discuss the physical significance ofa(t), the scale factor, sketching its form for the three cases of a matter-onlyuniverse with positive, zero, and negative spatial curvature.

Define the term luminosity distance. By considering the amount of radiationwhich is received in a unit area located a co-moving distance away from asource, show that the luminosity distance dlum is given by the formula

dlum = a0r0(1 + z)

where r0 is the co-moving distance and z is the red-shift.

Describe how the luminosity distance of Type Ia supernova might be used toconstrain cosmological parameters, and discuss the observations which arerequired and any key assumptions of the method.

5


1.6 High-redshift galaxies

Discuss the significance of the observation that special lines of distant galaxiesare redshifted compared to their rest wavelengths.For what range of redshifts would the hydrogen Lyman-α line (rest wavelength121.6 nm) be redshifted in to the visible part (400–700 nm) of the spectrum?What particular problems are encountered in attempting to detect Lyman-αfrom galaxies with redshifts greater than about 7?Assuming the Hubble constant H0 to be 70 km s−1 Mpc−1, determine therange of distances (in Mpc) corresponding to the range of redshifts you havecalculated. You may take the relationship between distance d and redshift zto be

d =cz

H0

(1 + z2)

(1 + z)2.

What sort of celestial objects are observed with redshifts z > 2? Outlinefurther evidence which favours these objects being located at cosmologicaldistances.

1.7 Early Universe

Give an account of the observational evidence for the hot Big Bang model ofthe Universe.The Friedmann and fluid equations respectively are given by

(a

a

)2

=8πG

3ρ− kc2

a2

and

ρ+ 3a

a

(ρ+

p

c2

)= 0

where a is the scale factor, ρ is the density and p is the pressure (a and ρ arethe derivatives of these quantities with respect to time.) Use these equationsto derive the acceleration of the Universe.Hence demonstrate that it the Universe is homogeneous and the strong energycondition ρc2 + 3p > 0 holds, the Universe must have undergone a Big Bang.

6

Chapter 2

Radiative Processes I

For Tutorial 2

2.1 Emission lines in planetary nebulae

Explain why most of the bright emission lines in the spectrum of a planetarynebula arise from elements whose abundances are small compared to hydro-gen. What determines the relative brightness of lines from different elementsand from different stages of ionization of a given element, and what quantit-ies need to be estimated or measured in order to estimate the abundance ofthe elements that produce the emission lines?

How do observations made in different spectral regions contribute to ourknowledge of the processes that produce the spectrum and the conditions inthe nebula?

2.2 Critical density of a planetary nebula

An element of total number density nE exists mainly in two stages of ion-ization, i and i + 1, with number densities ni and ni+1. Express ni/nE andni+1/nE in terms of the ionization rate βi (from i) and the recombination rateγi (to i). In a planetary nebula what processes are the main contributors toβi and γi? How do ni/nE and ni+1/nE vary if the electron number densityne is increased?

Consider a forbidden transition in the optical spectrum of a planetary neb-ula. The transition occurs between an excited level (2), with number densityn2, and the ground level (1), with number density n1. Write down the ap-

7


propriate form of n2/n1. Hence explain what is meant by the critical density.

Two such lines occur in an ion, from levels 3 and 2 to level 1. Assuming thatthe wavelengths λ21 ' λ31, show that their flux ratio is given by

F31

F21

' g3

g2

A31

A21

(n∗e(2) + ne)

(n∗e(3) + ne),

where Aji is the spontaneous transition probability, gj is the statistical weightof level j and n∗e(j) is the critical density for a transition from level j.

Using the data for singly ionized sulphur (S II) given in the table below, find

(a) the values of F31/F21 when (i) ne n∗e(2) and (ii) when ne n∗e(3)

(b) the value of ne at which F31/F21 has its greatest dependence on ln(ne),and the corresponding value of F31/F21.

Show that if this value of F31/F21 can be measured to within ±10% thenlog10(ne) can be determined to within ±0.18.

Table: Data for S II lines.λji/nm Transition (ji) Aji/s−1 Critical density n∗e(j)/m−3

673.1 2D3/2 − 4S3/2 (21) 1.7× 10−3 3× 1010

671.6 2D5/2 − 4S3/2 (31) 6.3× 10−4 1× 1010

[At a fixed electron temperature Te the collisional excitation rate is

Cij ∝exp(−hc/λijkBTe)

gi]

2.3 Detailed balance and coronal approxima-

tion

Discuss what is meant by detailed balance and the coronal approximation inthe context of processes that determine the number density N2 of an excitedstate in a two level atom. In each case give the expression for N2/N1, whereN1 is the ground state number density.

Emission lines arising from permitted and spin-forbidden electric dipole trans-itions to a common atomic energy level are observed in stellar transition re-gions. Explain how their relative intensities can be used to determine the

8

Tutorial 2: Radiative Processes I

electron number density Ne. Explain why there is an electron density belowwhich this method cannot be used.

The diagram shows some transitions observed in Fe XIV.

Transition λ /nm Cij (relative) Aji (relative)

2D5/2 → 2P3/2 21.9 9 62D3/2 → 2P1/2 21.1 10 52D3/2 → 2P3/2 22.0 1 1

Using the relative collisional excitation rate coefficients Cij and the relativetransitionprobabilities Aji given in the table, derive an expression for the relativeintensities of the lines at wavelengths λ = 21.9 nm and λ = 21.1 nm in termsof the population ratio N(2P3/2)/N(2P1/2). What relative intensity would beexpected if the 2P3/2 and 2P1/2 level number densities were determined bydetailed balance at a temperature Te = 2× 106 K?

In a solar active region the observed intensity ratio of the above lines is 0.15.Use the collisional excitation rate coefficient Cij = 7× 10−15 m3 s−1 and thetransition probability Aji = 60 s−1 for the 2P3/2 → 2P1/2 transition to showthat N(2P3/2)/N(2P1/2) depends on Ne. Find the value of Ne.

[The relation between the collisional de-excitation and excitation rate coef-ficients is Cji = (gi/gj)Cij exp (Wij/kBTe) where gi and gj are the statisticalweights of the lower and upper levels respectively and Wij is the excitationenergy of level j above level i.]

2.4 Intersystem lines of cool stars

Using a model for a two-level ion (ground state plus one excited state), discussthe processes which should be included when considering the formation of anintersystem (semi-forbidden) line in a cool-star transition region. Hence showthat the rate at which energy is emitted in an intersystem line of wavelengthλ21 is given by

E21 =hc

λ21

A21NE

NH

∫ (Nion

NE

)f1(Ne, Te)NH

A21 + f1(Ne, Te) + f2(Ne, Te)dV ,

where A21 is the spontaneous transition probability, Te is the electron tem-perature and NE, NH, Nion and Ne are the number densities of the elementunder consideration, hydrogen, ions and electron s, respectively.

9


Show that

f1(Ne, Te) = C12Ne and f2(Ne, Te) = C21Ne,

where C12 and C21 are rate coefficients for collision and excitation.

Intersystem lines of Si III and C III are observed in spectra of cool starswith a range of surface gravities. Assuming that both lines are formed atTe = 4.5× 104K, use the data in the table below to calculate the maximumand minimum values of the ratio E(Si III)/E(C III).

In the spectrum of a planetary nebula, the ratio E(Si III)/E(C III) is ob-served to be less than 0.1. Discuss the differences between the physicalconditions in planetary nebulae and cool star transition regions and suggestthe main cause of this small energy ratio.

Data for Si III and C III

Ion Transition λ (nm) Ω A21(s−1) NE/NH Nion/NE

Si III 3s2 1S0−3s3p 3P1 189.2 2.8 1.5× 104 3.5× 10−5 0.79

C III 2s2 1S0−2s2p 3P1 190.9 0.32 1.0× 102 3.5× 10−4 0.46

The ionization potentials of Si III and C III are 33.5eV and 47.9eV, respect-ively.

[ The rate coefficient for collisional excitation is

C12 =8.63× 10−12 Ω 10

−[6.25×106

λ21Te

]g1T

1/2e

m3 s−1 ,

where Ω is the collision strength given in the table, g1 is the statistical weightof the lower level, λ21 is in nm and Te is in K. ]

10

Tutorial 2: Radiative Processes I

J

3d D

3p P

5/23/2

3/21/2

2

2

11


12

Chapter 3

Radiative Processes II

For Tutorial 3

1. A hot star is embedded in a dusty region. Silicate dust grains have aradius of 0.1µm and have an efficiency Qν ∝ ν, such that the efficiency ratio<QIR > / <QUV > = 2×10−2a Tgr, while carbon grains have a radius of0.01µm and have an efficiency Qν ∝ ν2, such that the efficiency ratio <QIR >/ <QUV > = 4×10−4a2T2

gr, where a is in µm . Calculate the temperaturesthat these grains will attain at a distance of 20 AU from the central starof a planetary nebulae with a temperature of 40 000 K and a luminosity of104 L. Comment on these temperatures and the possib le behaviour as afunction of distance from the star in this object.Observations in the mid-infrared at λ = 10µm show emission extending todistances of 0.01pc from the central star. What is the likely explanation forthis emission.

2. Describe the main sources of opacity at visible wavelengths in the pho-tospheres of hot (B-type) stars, solar-type stars and cool stars. Explain thephysical conditions that give rise to the different opacity sources. [8]Estimate the temperature at which the number of hydrogen atoms in the firstexcited state is equal to the number in the ground state for a thermal distribu-tion. Is this level population likely to be realised in practice? [7]Without detailed derivation, justify the following expression for a greyatmosphere, defining all terms used and stating any assumptions made:

S(τ) = 34π

(τ + 2

3

)F

13


Adopting a grey atmosphere approximation, estimate the temperature rangeswhich may be investigated by measurements at the limb and at the centre ofthe solar disk.

3. Observations of a HII region in a galaxy at a distance of 3Mpc give thefollowing values for the intensities of Hydrogen recombination lines:

Line Wavelength Observed Intensity Relative Intrinsic Intensity(µm) Iν (10−18 Wm−2)

H(Gamma) 0.434 0.0119 47H(Beta) 0.486 0.0455 100H(alpha) 0.656 0.702 285

Theoretical line intensities relative to Hβ = 100 calculated by Hummer &Storey are also listed.The extinction expressed in magnitudes relative to an extinction of A(V)=1 magnitude in the optical can be approximated by:

A(x) = 1.0 + 0.826(x-1.83) - 0.320(x-1.83)2

where x is the inverse wavelength 1/λ in units of µm−1.

What is the extinction, expressed as magnitudes of visual extinction A(V)towards the emitting region?What information can be obtained from the extinction-corrected hydrogenline fluxes, and how might the calculated value of the extinction be checked?Give an estimate of the total number of ionizing photons emitted by thestars, and estimate the size of the resulting HII region, assuming an electrondensity ne ∼ 1010m−3.

The hydrogen recombination coefficient α = 2.6× 10−19m3s−1

4. Rest-frame ultraviolet absorption lines from the ground state of C II havebeen detected towards the QSO 0347 - 3819 at a redshift of 3.025. Theground state of the C+ ion consists of two levels 2s22p 2P0

1/2,3/2 with an e

nergy separation of ∆E = 63.42 cm−1.

14

Tutorial 3: Radiative Processes

Assuming thermal equilibrium, obtain an expression for the level populationof the ground and excited fine structure levels within the ground state, andestimate the excitation temperature using the column densities in the twolevels and the oth er information below. Comment on the value of Tex found.

The column densities estimated from the transitions from the ground andexcited levels within the ground state are 5.05 ±0.28 × 1015 cm−2 and 1.92±0.1×1013 cm−2 respectively. For the ground level J=1/2 and for the excitedlevel J=3/2.

Appendix: Useful Constants and Unit Conver-

sions

• G = 6.673 10−8 cm3 g−1 s−2

• mH = 1.673 10−24 g

• 1 pc = 3.086 1016 m

• 1 yr = 3.160 107 s

15


16

Chapter 4

High-Energy Astrophysics I

For Tutorial 4

4.1 Flux density, brightness and temperature

In the lectures we discussed the core of the quasar 3C273, an extremely brightradiosource. Consider now a less-extreme example, where a source has a fluxdensity of 10 Jy at 178 MHz and subtends a solid angle of one arcmin2.

(a) What is the brightness temperature of the radiosource at 178 MHz?

(b) You may be surprised that the temperature is not dramatically higherthan that of very hot stars. Why then is it so difficult to observe starsat radio wavelengths?

(c) For enthusiasts only, and strictly non-examinable! If you’ve grasped theessential point in part (b), consider this question: What magnitude isthe faintest astronomical object that can be seen by the human eye indaylight? Google for quantities that you think are relevant.

4.2 Shocks

Using the ideal gas law and the strong shock jump conditions, show that thetemperature of gas downstream of a shock is given by

Td =3

16

mv2u

kB

17


Figure 4.1: Brightness temperature.

18

Tutorial 4: High-Energy I

where m is the mean particle mass in the gas and v2u is the bulk speed of the

gas upstream of the shock.Why do the temperature and density of the upstream gas not appear in thisrelation?In the lectures we considered the blast wave of SN1993J, which had an ini-tital expansion speed of 20 000 km s−1. Assuming the interstellar mediumaround the supernova to be hydrogen plasma, estimate the initial temperat-ure behind the blast wave.Hydrogen plasma falls radially onto a white dwarf, passing through a shockvery close to the surface. Show that the temperature behind the shock isgiven by

kBT

mec2=

3

32

mp

me

RS

R∗

where R∗ is the radius of the star and RS is its Schwarzschild radius. Whatis kBT if R∗ = 6000 km and M = M?

4.3 Particle acceleration

Outline the physical mechanism by which it is thought that electrons areaccelerated to ultra-relativistic energies in strong non-relativistic shocks suchas are found in supernova remnants.Suppose that an electron is involved in a collision which increases the elec-tron’s total energy by a factor β. Suppose further that there is a probabilityp that this electron remains within the region where further collisions mayoccur. Show that the expected distribution of electron energies is

N(E)dE ∝ EkdE

Where k = −1 + (ln p/ ln β).Assuming that k = −2 and that the electrons are accelerated to energies atwhich they emit synchrotron radiation, show that the power-law region ofthe synchrotron spectrum will have a form

Sν ∝ ν−0.5

You may assume that a synchrotron electron emits almost all its radiationat a characteristic frequency νcrit ∼ γ2eB

2πmeand that the power radiated is

P = 4γ2β2cσTB2

3µ0

19


4.4 Synchrotron spectrum

Describe, with the aid of an annotated sketch, the shape of the spectrumof continuum radio emission from the lobes of a powerful extragalactic ra-diosource. Include in your discussion the optically thick and optically thinregions of the spectrum, along with an explanation of these terms, and de-scribe how the shape of the spectrum is modified by radiation losses as theradiosource ages.Show that a relativistic electron with Lorentz factor γ, passing through aregion containing a uniform magnetic flux density B, has a gyrofrequency

ν =eB

2πγme

Hence show that such an electron emits radiation whose spectrum is stronglypeaked at a characteristic frequency νcrit which is given by

νcrit ∼γ2eB

2πme

The power radiated by the electron is given by

P =4γ2β2cσTB

2

3µ0

where β is the speed of the electron relative to the speed of light and σT isthe Thomson cross section. Using these formulae for P and νcrit, calculate acharacteristic timescale for the synchrotron lifetime of the electron.The powerful giant radiosource 3C236 is 6 Mpc across, with the host galaxyat the centre. The radio spectrum of the lobes close to the host galaxy showsa cut-off in emission, assumed to be due to radiative losses, at frequenciesabove about 1 GHz. Taking the magnetic flux density in this region to be0.3 nT, estimate the age of the synchrotron plasma near the host galaxy.On the assumption that this plasma was left behind near the host galaxy bythe radiosource jets just as they began to expand into intergalactic space,estimate the expansion speed of the radiosource. What factors may causethese estimates of age and expansion speed to be unreliable?

4.5 Equipartition/Minimum Energy

Describe the evidence that the radio emission from powerful extragalacticradio sources is produced by the synchrotron mechanism.

20

Tutorial 4: High-Energy I

A radio source contains a population of relativistic electrons with Lorentzfactors γ 1 and a uniform magnetic flux density of magnitude B. Thenumber density of electrons with Lorentz factors in the range γ to γ + dγ isgiven by

n(γ)dγ = n1

(γ

γ1

)−kdγ

for γ greater than some limit γ1, where n1 is a constant and k is a constantgreater than 2. Show that Jν , the power emitted per unit volume per unitfrequency interval, is given by

Jν ∝ n1γk1γ

1−kB.

You may assume that each electron emits all its synchrotron radiation at afrequency ν = γ2eB/2πme and that the power radiated by each electron isP = 4

3cσTumγ

2 where um is the energy density of the magnetic field and σT

is the Thomson scattering cross-section.If the energy density stored in relativistic electrons, ue, is

ue =n1γ

21mec

2

k − 2

show that, for a given observed value of Jν , the total energy density in thesource in the form of relativistic electrons and magnetic field has a minimumvalue which occurs when

ue =4

3um.

The radio source Cygnus A is about 100 kpc across. The flux density of themagnetic field in the radio source is thought to be about 6 nT. Estimate alower limit to the total energy content of the radio source and explain whatimplications this has for how Cygnus A is powered.

21


22

Chapter 5

High-Energy Astrophysics II

For Tutorial 5

5.1 Accretion discs

Describe four methods used to determine the masses of black holes in galacticnuclei.

What is the physical significance of the Eddington luminosity, LEdd? Derivethe express ion

LEdd =4πGmpMc

σT

for pure hydrogen plasma accreting onto an object of mass M , where σT isthe Thomson cross-section, clearly stating any assumptions made.

Assume that the emission from a geometrically-thin, optically-thick accretiondisc is dominated by a component from its innermost part. Take this to bea uniform circular region of radius r ∼ 6GM/c2. By equating the luminosityemitted from this region to the Eddington luminosity, derive an approximateexpression for the black-body temperature of radiation from the inner disc.Estimate the temperatures for black holes of 10 M and 108 M and commenton your results in the context of the observed spectra of X-ra y binaries andactive galaxies.

The galaxy NGC 4258 contains a black hole of mass 3.5× 107 M. Calculatethe Eddington luminosity and compare it with the observed X-ray luminosityof 4× 1033 W.

Contrast the properties of accretion discs with M ∼ MEdd and M MEdd.

[ σT = 6.6× 10−29 m2. ]

23


5.2 Doppler beaming and boosting

In the lectures we derived the formula for the apparent projected velocityβappc of material in a jet with velocity βc at an angle θ to the line of sight,

βapp =β sin θ

1− β cos θ

Use this formula to show that for any particular jet, the maximum apparentspeed is

βmaxapp = γβ,

which is seen when the jet is viewed from a direction such that

β = cos θ.

Now a demonstration of the relativistic abberation of solid angle.Suppose a source is moving along the θ = 0 axis (spherical polar coordinates).If the source emits photons isotropically in its rest frame (the S ′ frame) thenthe angular distribution of photons in an annulus of solid angle between θ′

and θ′ + dθ′ will be given by

P (θ′) dθ′ =1

2sin θ′ dθ′.

(Recall this from you old kinetic theory notes if you have forgotten it!). Usethe relativistic abberation formulae to transform θ′ to the observer’s (S)frame θ and hence show that the distribution of photons in the S frame isgiven by

P (θ) dθ = D2 1

2sin θ dθ,

where D is the relativistic Doppler factor, [γ(1− βcosθ)]−1.Figure 5.1 shows how the Doppler factor varies as a function of γ and θ.There are couple of points to note.First, D varies with cos(θ), so for a receding source you can calculate it as[γ(1 − βcosθ)]−1 with 90 < θ < 180, or you can say [γ(1 + βcosθ)]−1 with0 < θ < 90. Be alert!Second, D becomes less than one while the jet is still pointing towards you!The peak of the boosting is occuring off to the side; you’re looking at thefainter “shoulder” of the boosted beam.Now a calculation. Detailed study of the single-sided jet in the quasar 3C273suggests that the jet is pointing towards us, at about 10 from the line-of-sight, has a bulk Lorentz factor γ = 11, and a synchrotron spectral index of0.5. The deepest radio maps of 3C273 have a dynamic range of about 5000and show no sign whatsoever of a counterjet. Is this surprising?

24

Tutorial 5: High-Energy II

Figure 5.1: Doppler boosting of a relativistic jet.

5.3 Inverse-Compton scattering

Explain what is meant by Thomson scattering and inverse-Compton scatter-ing. Give two examples where inverse-Compton scattering is important inastrophysics.The power P in scattered radiation due to Thomson scattering is given by

P = cσTU

where σT is the Thomson scattering cross-section and U is the energy dens-ity of incident radiation. Use this relation to derive an expression for theaverage power of radiation inverse Compton scattered by a population ofrelativistic electrons with Lorentz factors γ ∼ 1000. Assume that the distri-bution of electrons is isotropic and that the radiation being scattered is atradio frequencies. Justify each step in your derivation.Ultra-relativistic electrons of energy 1011 eV are observed at the Earth. Byconsidering the effect on the electrons of inverse Compton scattering of themicrowave background radiation, calculate the maximum length of time forwhich the electrons could have had energies larger than the observed value.Given that the age of the Galaxy is ∼ 1010 years, what does this imply?Suggest some possible sources of such high-energy electrons.[The energy density of the microwave background is 2.6× 105 eV m−3; σT =6.6× 10−29 m2.]

25


5.4 Self-absorbed Synchrotron

By considering the intensity of radiation emitted by a black body, show howan estimate of the magnetic flux density in a compact radio source can beobtained from a measurement of its surface brightness at a frequency at whichthe source is optically thick.[You may assume that in the Rayleigh-Jeans part of the spectrum of a blackbody at temperature T , the intensity per unit frequency interval Sν is givenby Sν = 2kBTν

2/c2]

5.5 Thermal Bremsstrahlung

What is the mechanism that produces bremsstrahlung radiation?Beginning with the result that an interaction between a stationary proton andan electron with speed v at impact parameter b emits a total energy ∝ 1/b3v,show that the spectral energy distribution of bremsstrahlung radiation in ahydrogen plasma as a function of frequency ν is given by

jνdν ∝n2

e

vln

(mv2

2hν

)dν,

where ne is the electron density.The integrated bremsstrahlung emissivity ε (the power per unit volume) fora hydrogen plasma is given by

ε = 1.7× 10−40T 1/2n2e W m−3,

where T is the temperature of the plasma in units of K and ne is in units ofm−3. Considering a virialized cluster of galaxies with a gas mass of 1014Mand a size of 1 Mpc, estimate the time it takes for the gas to cool, explainingquantitatively any assumptions you need to make.Why is this estimate an upper limit to the cooling time? What are theimplications for the existence of such clusters in the local Universe, giventhat the time since the Big Bang is approximately 13.7 Gyr?The central galaxy in such a cluster is found to harbour a black hole of mass109M. Describe, with a quantitative calculation, how this may reconcileany difficulties implied by your previous calculations.

5.6 Sunyaev-Zel’dovich Effect and H0

What is meant by the Sunyaev–Zel’dovich effect? How is it measured?

26

Tutorial 5: High-Energy II

Under certain conditions, the magnitude of the Sunyaev–Zel’dovich effect is

δI

I= −2

∫ σTnekBTemec2

dl.

Give an account of the physical origin of this effect that qualitatively explainswhy δI/I is given by this formula. Estimate the frequency at which thisformula is invalid.The galaxies of the Coma cluster have a mean recession velocity with re-spect to Earth of 6900 km s−1. Observations of the Coma cluster with X-raytelescopes show that the intracluster plasma has an angular extent of 30 ar-cminutes, a temperature of 108 K and an electron density of ∼ 4000m−3.Observations of the Cosmic Microwave Background toward the cluster at30 GHz show a diminution in temperature of 0.6 mK. Use these data to makean estimate of the Hubble parameter, describing any assumptions made.Identify two difficulties involved in the measurement of the Hubble parametervia the Sunyaev–Zel’dovich effect, and describe how their influence may beminimized.[σT = 6.6× 10−29 m2; the mean temperature of the Cosmic Microwave Back-ground is 2.73 K.]

27


28

Chapter 6

Avanced Stellar Astrophysics I

For Tutorial 6

6.1 Core-Collapse Supernovae

Consider the final iron core of a massive star with a mass MFe ' 1.5M andradius RFe ' 3× 106 m, spinning with a spin period P ' 500 s and having amagnetic field at its outer edge BFe ' 2× 103 T.

a) Stating your assumptions, estimate the final spin and the strength ofthe magnetic field of the neutron star that forms from the collapse ofsuch a core. Compare the spin period to the minimum spin period fora neutron star.

During the collapse phase, the initial collapse stops when the central corewith a mass Mcore ' 0.7M reaches a mass density ρ ' 3× 1016 kg m−3. Atthis density the core bounces driving a shock with an energy Ebounce ∼ 1044Jinto the infalling outer core.

b) Estimate the energy that is required to photodissociate 0.1M of Feinto neutrons and protons. Compare this energy to the bounce shockenergy and comment on the fate of the shock. [Remember that ∼ 0.8 %of the rest mass energy of protons is released in the conversion 561H→56Fe.]

c) In the proto-neutron star (with an initial radius ∼ 30 km), the meanfree path of neutrinos is lν ∼ 0.3 m. Estimate the diffusion time forneutrinos to escape from the proto-neutron star and hence estimatethe neutrino luminosity during the initial neutron-star cooling phase.

29


[Hint: assume that all the gravitational potential energy escapes inthe form of neutrinos and use a standard random-walk argument toestimate the neutrino diffusion time.]

d) Assuming that 5 to 10 % of the neutrino luminosity is absorbed by theinfalling outer core, estimate how long it takes to absorb enough neut-rino energy to reverse the infall of the outer core and drive a successfulsupernova explosion (with a typical explosion energy of 1044 J). Com-pare this time to the dynamical timescale of the proto-neutron star.

6.2 The Binary Pulsar PSR J0737-3039: Su-

pernova Kicks

Recently, the first binary pulsar was discovered (Lyne, A.G. et al. 2004,Science, 303, 1153), which provides a rare laboratory for relativistic physics.The system consists of two pulsars (A and B) in a mildly eccentric orbit withan orbital period Porb ' 2.4 hr and eccentricity e ' 0.088. The spin periodsand spin-down rates of the two pulsars have been measured to be PA '22.7 ms, PB ' 2.77 s, PA ' 1.7× 10−18 s s−1 and PB ' 0.88× 10−15 s s−1 andthe masses have been determined to be MA ' 1.34M and MB ' 1.25M,respectively.

a) Making reasonable assumptions about the pulsar properties, estimatethe spin-down luminosities and the spin-down ages (i.e. P/2P ) for bothpulsars. Considering the evolutionary history of the system, explainwhy the spin-down ages should roughly agree.

b) Assuming that the spin-down is caused entirely by magnetic dipoleradiation, show that the magnetic field of the pulsars can be estimatedfrom

B ' 1

R3 sin θ

√√√√(3c3µ0

32π3

)PP I,

where R is the radius of the pulsar, θ the (generally unknown) inclina-tion of the magnetic axis with respect to the rotation axis and I is themoment of inertia of the pulsar (µ0 is the magnetic permeability andc the speed of light in vacuo). Estimate the magnetic fields of the twopulsars.

30

Tutorial 6: Advanced Stellar I

c) It is believed that Pulsar A was spun up by accretion of matter from theprogenitor of Pulsar B. Neglecting magnetic fields during the accretionphase, estimate how much mass Pulsar A would have had to accretefrom an accretion disc to be spun-up to the observed spin period. Howdoes the actual magnetic field of Pulsar A affect this estimate?

It is reasonable to assume that before the second supernova, in which PulsarB was formed, the immediate pre-supernova binary system was circular andhad an orbital separation a0 ' 1.4R.

d) Assuming that in the second supernova an amount of mass ∆M wasinstantaneously ejected and that Pulsar B did not receive a recoil in itsown frame, show that the post-supernova eccentricity e, post-supernovasemi-major axis aPSN and post-supernova system velocity vsys (i.e. thevelocity of the new centre-of-mass (CM) frame defined by the twopulsars relative to the pre-supernova CM frame) are given by

e =∆M

MA +MB

,

aPSN =a0

1− e,

vsys = v0orb

∆M

MA +MB

MA

MHe +MA

,

where MHe is the mass of the progenitor of Pulsar B just before thesupernova (i.e. MB+∆M) and v0

orb is the pre-supernova orbital velocity.Determine ∆M assuming that the post-supernova eccentricity was e '0.1 and estimate vsys.[Hint: You need to compare the energies and momenta of the systembefore and after the supernova. The eccentricity e and semi-major axisa of an eccentric orbit are related to the distance of closest approach,the periastron separation, rp by rp = (1− e) a, and the total energy ofan eccentric binary is

Ebinary = −GM1M2

2a= −GM1M2

r+

1

2

M1M2

M1 +M2

v2,

where r is the separation and v the relative orbital velocity at a particu-lar binary phase, and M1 and M2 are the masses of the two components.See, e.g., Carroll & Ostlie, Chapter 2.3.]

31


e?) Show that in the limit, where there is no mass loss associated withthe second supernova but where Pulsar B received an asymmetric su-pernova kick of magnitude vkick, the post-supernova system velocity isgiven by

vsys =MB

MA +MB

vkick.

What is vsys in this case for a typical vkick ' 250 km s−1?

f?) Discuss how the observed eccentricities and system velocities of systemslike the double pulsar may be used to constrain supernova kicks.

6.3 The Last Stable Circular Orbit

In General Relativity, the equation for the radial coordinate r of a testparticle orbiting a non-rotating black hole of mass M can be written as

1

2r2 +

1

2

(1− 2GM

c2r

) (L2

r2+ c2

)=

1

2

E2

c2, (6.1)

where r = dr/dt and L and E are the angular momentum per unit restmass and the energy per unit rest mass of the particle, respectively (theparticle is assumed to have non-zero rest mass). This equation resembles theenergy conservation equation in Newtonian dynamics, EN = 1/2 r2 + Veff(r),except for the additional term −GML2/c2 r3 in the effective potential Veff

that becomes dominant at small radii.

a) Treating the problem like a Newtonian one, sketch the effective po-tential for a particle near a black hole as a function of radius, bothfor a small and a large value of L. Characterize the possible types oftrajectories/orbits in both cases.

b) Show that for each value of L there are two possible circular orbits

r± =L2 ± [L4 − 12G2M2L2/c2]

1/2

2GM, (6.2)

provided that L2 > 12G2M2/c2.

c) Show that the r+ solution has a minimum value of rmin+ = 6GM/c2 and

argue that this is a stable orbit (i.e. corresponds to a minimum of theeffective potential). What does this imply for the r− solution?

32

Tutorial 6: Advanced Stellar I

d) Calculate the energy E of a particle at this innermost stable circularorbit and show that it’s binding energy per unit rest mass EB is

EB = (1− (8/9)1/2) c2 ' 0.06 c2.

e) Discuss briefly what happens as matter orbiting a black hole in anaccretion disc approaches the innermost stable orbit. Compare thiscase to accretion onto a non-magnetic neutron star.

33


34

Chapter 7

Advanced Stellar AstrophysicsII

For Tutorial 7

7.1 Gamma-Ray Bursts

A popular model for long-duration gamma-ray bursts (GRBs) is the col-lapsar model, in which a massive, relatively rapidly rotating helium (or car-bon/oxygen) star collapses to form a compact object (e.g. a black hole) sur-rounded by a disc of matter at nuclear density. Subsequent accretion fromthe disc causes the formation of a relativistic jet that penetrates the remain-ing infalling envelope and generates a gamma-ray burst at a large distanceaway from the star by internal and/or external shocks.For this problem consider a collapsing helium star of mass MHe = 10M andradius RHe = 5 × 108 m where the inital mass of the central black hole isM0

BH = 2M.

a) Show that the specific angular momentum of the infalling material hasto be larger than 2 × 1012 m2 s−1, the specific angular momentum atthe last stable orbit for a 2M black hole, so that an accretion disccan form. Estimate the characteristic dynamical timescales both forthe inner disc and the collapsing helium star. How do these timescalesdetermine the observable characteristics of GRBs?

b) Assume that an amount of relativistic energy E = 1044 J is injectedby the central engine of the GRB, driving an expanding relativisticfireball. Estimate the radius at which the fireball becomes optically

35


thin to MeV gamma rays, i.e. the radius at which the optical depth topair creation γ γ e+ +e− becomes less than 1. [You may assume thatthe cross section for pair creation is given by the Thomson cross sectionσT ' 6.6×10−29 m2 and that the typical photon energy is 1 MeV; arguethat the optical depth is then given by τγ ∼ nγ σTR.]

While massive stars are known to be rapidly rotating on the main sequence,they are believed to be spun down efficiently during their evolution by hy-drodynamical and magnetohydrodynamical effects and develop cores that arenot rotating sufficiently fast to be consistent with the collapsar model. Oneway of spinning up a helium star is if it is a member of a close binary whereit can be spun up by the tidal interaction with a companion star.

c) Consider a 10M helium star in a close orbit with a compact star(most likely a neutron star or a black hole) with an orbital period Porb.Assume that the spin angular velocity of the helium star is synchronizedwith the orbital angular velocity of the binary and that the helium staris in solid body rotation. Estimate the maximum orbital period forwhich the core is sufficiently rapidly rotating that only the innermost2M can collapse directly, while the rest collapses first into a disc [takethe typical radius of the inner 2M core as 8×107 m). [Answer: ∼ 5 hr]

d) Sketch briefly the evolutionary path that can lead to the formation ofsuch a system.

36

Tutorial 7: Advanced Stellar II

7.2 Mass-Transfer Driving Mechanisms

Consider a binary consisting of two stars of mass M1 and M2 with an orbitalseparation A and orbital period P .

a) Show that the total angular momentum of the binary can be writtenas

J = µA2 2π

P= µ

√G (M1 +M2)A,

where µ ≡M1M2/(M1 +M2) is the reduced mass of the system. Showthat for conservative mass transfer (where the total mass and the totalangular momentum of the system remains constant), the orbital sep-aration is a minimum when M1 = M2. Sketch the evolution of A asa function of time assuming that M1 > M2 initially and that mass istransferred from star 1 to star 2. How does this behaviour of A affectthe mass-transfer rate, assuming that star 1 attempts to expand at asteady rate?

b) Even in the absence of mass transfer, the orbit of a binary will shrinkdue to the emission of gravitational waves, which causes the loss oforbital angular momentum at a rate

dJ

dt= −32

5

G7/2

c5

µ2M5/2

A7/2,

where M = M1 + M2 is the total mass of the binary. Show that thisimplies that the orbital period decreases as

1

P

dP

dt= −96

5

G3

c5

M2µ

A4.

By setting the orbital period decay time (P/P ) equal to the age ofthe Galaxy (∼ 1010 yr), determine the maximum separation and hencemaximum orbital period for which a binary consisting of (i) two low-mass helium white dwarfs with M1 = M2 = 0.3M, (ii) two massivecarbon/oxygen white dwarfs with M1 = M2 = 1M and (iii) twoneutron stars with M1 = M2 = 1.4M are driven into contact bygravitational wave emission within the age of the Galaxy. Discuss thelikely/possible fate of the systems in the three cases.

c?) Assume that star 1 loses mass in a stellar wind at a wind mass-loss rateM = 10−10M yr−1 and that the wind is magnetically coupled to thespin of star 1 up to a radius 10R away from the star (where R is the

37


radius of the star). Assume further that due to the tidal interactionwith the companion star, the spin of star 1 is synchronized with theorbital period (i.e. Pspin = Porb). Estimate the orbital period decaytime (P/P ) due to this magnetic braking for a system with M1 = M2 =1M and A = 3R. [Hint: what is the specific angular momentum lostin the stellar wind?]

7.3 Neutron-Star Spin-Up by Wind Accre-

tion

Consider a massive X-ray binary consisting of a 20M star and a 1.4Mneutron star in a relatively wide orbit where the radius of the massive staris much smaller than its Roche lobe radius. Assume that the massive starloses mass in a steady spherical wind with a wind mass-loss rate Mwind =10−7M yr−1 and a wind velocity vwind = 103 km s−1 and that the neutronstar has a magnetic field B = 108 T (assumed to be dipolar) and is spun up(or spun down!) by accretion of some of the wind material.

a) Explain the meaning of Alfven radius (rAlf) and Bondi-Hoyle radius(rBH). Calculate rBH for the above system and show that the fractionof the wind that is accreted by the neutron star is given by (rBH/2A)2

where A is the orbital separation of the binary.

The matter that is accreted by Bondi-Hoyle accretion typically has a specificangular momentum that is 1/4 of the specific Keplerian angular momentumat the Bondi-Hoyle radius.

b) Estimate the characteristic size of a disc that would form around theneutron star in the absence of a magnetic field.

c?) Now considering the actual magnetic field of the pulsar, estimate themaximum orbital separation A for which a disc can form around thepulsar (i.e. for which the accreted specific angular momentum is largerthan the Keplerian specific angular momentum at the Alfven radius).Determine the equilibrium spin period of the pulsar for this case.

7.4 Star Formation

a) Explain the concept of the Jeans mass and its importance for starformation. Consider two dense cores in a cloud of molecular hydrogen

38

Tutorial 7: Advanced Stellar II

(with µ = 2): (i) a cool core with temperature T = 10 K and (ii) awarm core with T = 100 K. Taking the number density of molecularhydrogen to be n = 1010 m−3 in both cases, estimate the Jeans massfor both cores.

b) Assume that the pre-collapse core is in hydrostatic equilibrium andcan be treated as an isothermal sphere, i.e. a sphere of gas at constanttemperature T where the supporting thermal pressure is given by P =

ρ c2s and where cs =

√kT/µmH is the isothermal sound speed of the

gas. Show that the density as a function of radius r from the centre ofthe sphere is approximately given by

ρ =c2

s

4πG

1

r2,

and the mass enclosed within a radius r by

M(r) =c2

s

Gr.

c) The collapse of an unstable isothermal molecular cloud core occurs fromthe inside out. Assume that the innermost mass m(r) within a radiusr has already collapsed and that the infall velocity at r is given by cs.Show that the mass-infall rate M is given by

M =c3

s

G

and is independent of r. What is M for the cases (i) and (ii) in parta)?

d?) Now consider an unstable molecular cloud core of 1M and initial ra-dius R = 106R which is in solid body rotation with angular velocityω = 10−13 Hz. Assuming that the size of the protostar forming at thecentre of the collapsing core has a radius of 5R and using angularmomentum considerations, estimate what fraction of the mass of thecore can collapse directly onto the protostar. Estimate the character-istic size of the proto-stellar disc that forms from the collapse of thebulk of the molecular core. [To obtain these estimates, you may takethe density to be constant in the core and ignore factors of order unity.]Comment on the implications of these estimates for the outcome of thestar-formation process.

39


40

Chapter 8

Early Universe / Large ScaleStructure

For Tutorial 8

8.1 Units (optional)

It is often useful to change between different units when looking at a prob-lem. A common practice is to convert a quantity into completely differentunits by multiplying by the fundamental constants such as c, h, kB and unitconversions.For example, take the Hubble constant

H0 = 100hkm s−1Mpc−1.

We can re-express it as a length by dividing by c to get

H0

c= 3.334× 10−4Mpc−1.

To re-express as an energy (in eV) we must first multiply by h to get

hH0 = 1.05h× 10−32K m J Mpc−1.

We then need to convert Mpc to Km and J to eV to get

hH01.01h× 10−34.

Finally to convert it into an inverse timescale (in years) we can simply convertMpc to Mm and s into yr

H0 = 1.02h× 10−10yr−1.

41


These correspond to an upper limit on the mass of the dark energy particle,the inverse Hubble length, inverse approximate age.Now do the same for the following quantities. Note that you will have tolook up the values of the fundamental constants and the conversions betweenunits.

• ρcrit = 3H20/8πG into (a) g cm−3, (b) GeV4, (c) eV cm−3, (d) protons

cm−3, (e) MMpc−3. If the cosmological constant has ρΛ = 2ρcrit/3,

what is its energy scale in eV (i.e. ρ1/4Λ ). Compare to the Planck mass,

MPl = (8πG)−1/2

• The photon temperature, TCMB=2.728K to (a) eV4. Assuming a blackbody distribution, convert this to a number density, nγ in photons cm−3

and energy density, ργ in (a) eV, (b) g cm−3 and Ωγ = ργ/ρcrit.

• The neutrino temperature, Tν = (4/11)1/3TCMB. Use this to express nν ,ρν and Ων in the above units assuming that the neutrinos are relativistic(and have three species).

• With the above relic number density, now consider the case where oneout of three neutrino species has a mass of 1 eV and the rest are mass-less. What is the energy density of relic neutrinos in units of the criticaldensity, Ων,massive. For what mass is the energy density at the criticalvalue?

8.2 Expansion of the Universe

In the lectures you have been given an expression for the evolution of theHubble constant, H = a

aas a function of the scale factor. It can depend on a

number of constants that encode the fractional energy density in the variousenergy species. These are the fractional energy density in non-relativisticmatter, ΩM0, in relativistic matter, ΩR0, in the cosmological constant, ΩΛ0

and in curvature, ΩK0. You will use the evolution equation for H(a) to solvethe following problems.

• Assume the universe today with no curvature or relativistic matter, butwith both non-relativistic matter and a cosmological constant. Writean expression for the age of the universe. Note that you cannot solvethis exactly.

• Assume instead that the universe is open (hyperbolic) with non-relativisticmatter but no cosmological constant or relativistic matter. Once againyou cannot solve this exactly.

42

Tutorial 8: Early Universe / LSS

• Assume that there is only relativistic (radiation) and non-relativisticmatter in the universe (no cosmological constant) and that the universeis flat. Integrate the age equation to determine the time at which thecosmic temperature was 109 K (remember that the temperature of theUniverse today is 2.7 K) and when it was 1/3 eV (1 eV is approximately11600 K). In this case you can solve the integral exactly. Assume thatthe ratio of radiation energy density to non-relativistic matter energydensity today, ΩR0/ΩM0 ' 2.5h−2 × 10−5 where the Hubble constanttoday is H0 = 100hkm s−1Mpc−1.

8.3 Growth of inhomogeneities

The equation for the evolutions of small perturbations in an expanding back-ground is:

δ + 2a

aδ − c2

s

a2∇2δ − 3

2

(a

a

)2

Ωδ = 0

where a is the scale factor, c2s is the speed of sound, and Ω is the fractional

energy density in non-relativistic matter.

a) In a Universe dominated by non-relativistic matter (or dust), we havethat a ∝ t2/3 and c2

S ' 0. Find the solutions to δ. Which one domin-ates?

b) The Newton-Poisson equation in an inhomogeneous universe can berewritten as

∇2Φ = 4πGa2ρδ

where ρ is the mean energy density, and (as above) the gradient is takenin terms of conformal coordinates (i.e. coordinates which are fixed onthe expanding space time). From what you know about the evolutionof the energy-density of non-relativistic matter, find the time evolutionof Φ, the Newtonian potential.

c) Assume that the universe is a mixture of dust and cosmological con-stant and that the cosmological constant is constant over all of space.Furthermore, assume that the cosmological constant dominates, so thatthe fractional energy density in dust is effectively zero. Explain whythe the equation for the evolution of perturbations in the dust is givenby

δ + 2a

aδ = 0

43


Find the solution of δ as a function of time.

d) Compare a Universe in which the cosmological constant dominates withone in which it is absent. Consider a structure with a density contrastδ(t0) today. It can be a galaxy or a cluster of galaxies. Will it havebeen around for longer in the Universe with or without the cosmologicalconstant? Think of a way for testing for the presence of delayed growthof structure by looking for galaxies at large redshifts.

8.4 A baryon dominated Universe

In the very early Universe, before recombination, baryons and photons in-teract very strongly to form a tightly coupled fluid. Let us assume thatthe transition between radiation and matter domination occurs at the sametime as recombination. Before recombination, the evolution equation for theperturbations in the baryon/photon fluid is

δ′′ +a′

aδ′ − 1

3∇2δ − 4

(a′

a

)2

δ = 0 (8.1)

with a ∝ η. After recombination the evolution equation is

δ′′ +a′

aδ′ − 3

2

(a′

a

)2

δ = 0 (8.2)

with a ∝ η2.

• Rewrite these equations in Fourier space by replacing ∇ by −i~k.

• Solve both of these equations in the long wavelength limit.

• Before recombination, there is a scale (let us call it kJ = 2π/λJ whereλJ is the Jeans length) that separates the large k behaviour from thesmall k behaviour. What is it? How does it evolve with time? (Pleaseshow the time evolution for the physical and conformal Jeans wavenumber).

A very rough approximation to the solution of the evolution equations beforerecombination on small wavelengths is

δ = C1 cos

(1√3kη

)+ C2 sin

(1√3kη

)

where C1 and C2 are constants.

44

Tutorial 8: Early Universe / LSS

• In the pre-recombination period, how does a perturbation that startsoff with a k < kJ evolve (answer this question qualitatively)? Whattype of behaviour will it have at sufficiently late times?

• (This is hard). Start off with a perturbation with an amplitude Aon scales much larger than Jeans scale at some very early time. Fol-low its evolution until today. You should find two types of behaviour:modes whose wavenumber k are smaller than kJ at recombination andmodes whose wave numbers are greater than kJ . By matching the largewavelength solutions to short wavelength solutions at the time whenkJ = k you can get a solution that covers all times. What form will ithave at late times? (No need to do the full calculation, just figure outwhat kind of solution will be picked out at late times.)

• (This is hard). Define the power spectrum of perturbations to be

P (k) ≡ |δ(t0, ~k)|2

Sketch what you expect the power spectrum of perturbations to betoday if we assume that initially all modes had the same amplitude atsome early time.

8.5 Neutrinos and free streaming

Consider a universe with critical density full of nonrelativistic matter. Theevolution equation for density perturbations, is

δ + 2a

aδ − 3

2

(a

a

)2

δ = 0.

with δ = δρ/ρ0, where we have expanded the total density ρ = ρ0 + δρ into ahomogeneous (ρ0) and imhomogeneous (δρ) part. Find the solution for theevolution of δ as a function of time.

Now consider a critical density universe which has a mixture of massive neut-rinos (with fractional density Ων) and non-relativistic matter (with fractionaldensity ΩM) so that Ων + ΩM = 1. It will expand at the same rate as inthe case considered above. On large scales δ will evolve as above but onscales smaller than about 40h−1Mpc, the evolution of perturbations in thenonrelativistic matter is set by

δ + 2a

aδ − 3

2

(a

a

)2

ΩM0δ = 0.

45


Find the solution for the evolution of δ as a function of time.

What is the difference in the growth rate of density perturbations on smallscales for these two perturbations? Explain the reason for the difference interms of the behaviour of the neutrinos.

We are now able to measure the angular positions and redshifts, z of millionsof galaxies in the universe. In some cases we are even able to measure theirdistances from us. At low redshifts, a measurement of the distance, d andthe redshift of a galaxy can be used to determine its peculiar velocity awayfrom us, v, through

cz = H0d+ v

where H0 is the Hubble constant today and c is the speed of light. Fromconservation of energy we have that the peculiar velocity and the densitycontrast at a given point are related through:

~∇ · ~v = −δ

Show that, in regions of equal δ, galaxies will have on average lower peculiarvelocities in a universe with massive neutrinos as compared to a universewith only non-relativistic matter.

46

Chapter 9

Galaxies I

For Tutorial 9

9.1 Basic Definitions

In a galaxy located at a distance d Mpc from our own Milky Way, whatwould be the apparent B−magnitude of a star like our Sun? Show that insuch a galaxy, an angle of 1 arcsec on the sky corresponds to a length of 5dpc. If its surface brightness is IB = 27 mag arcsec−2, how much B−bandlight does a patch of one square arcsecond of this galaxy emit? Show thatalthough this is equivalent to ∼ 1 Lpc−2 in the B−band, it drops to ∼ 0.3Lpc−2 in the I−band.

Data: MV = 4.83; B−V = 0.65; V − I = 0.72; mB = -2.5 log10 (FB) + 8.29

if the B−band flux FB is in 10−12 erg/s/cm2/A units.

9.2 Principles of Galactic Dynamics - Spher-

oids

The Plummer sphere of total mass MP and scale radius aP is a simple ifcrude model for star clusters and round galaxies. Its gravitational potential:

ΦP (r) = − GMP√r2 + a2

P

, (9.1)

47


approaches that of a point mass when r aP . Show that its density is given:

ρP (r) =3a2

PMP

4π(r2 + a2P )5/2

. (9.2)

and calculate UP , its potential energy. What is the mass M(< R) enclosedwithin a Plummer sphere of radius R? When viewed from a great distancealong the z-axis, what is its surface density ΣP (R) at a distance R from thecenter? Check that the core radius rc, where ΣP (R) drops to half its centralvalue, is rc ≈ 0.644 aP .

9.3 Principles of Galactic Dynamics - Disks

A simple disk model potential is that of the Kuzmin disk of total mass MK

and scale length aK , which reads, in cylindrical polar coordinates:

ΦK(R, z) = − GMK√R2 + (aK + |z|)2

. (9.3)

Irrespective of whether z is positive or negative, this is the potential of apoint mass MK at R = 0, displaced by a distance aK along the z-axis, on theopposite side of the z = 0 plane (i.e. negative if z is positive and positive ifz is negative). Show that ∇2ΦK = 0 everywhere except in the plane z = 0and use the divergence theorem to obtain the surface density ΣK(R) there.

9.4 Introduction to Hierarchical Galaxy Form-

ation: Dark Matter Halos

The Navarro-Frenk-White (NFW) model is used to describe the density pro-file of cold dark matter halos with characteristic scale lengths aN and densitiesρN that form in cosmological N-body simulations:

ρNFW (r) =ρN

(r/aN)(1 + (r/aN))2. (9.4)

Note the cusp in the center where the density diverges like 1/r and the rapidfall off in r−3 at large radii. Calculate the associated gravitational potential,as a function of aN and the characteristic velocity dispersion σN = 4πGρNa

2N .

Use this result to obtain the speed V (r) of a test particle on a circular orbitat radius r in this potential.

48

Tutorial 9: Galaxies I

9.5 The Milky Way: Our Very Own Super

Massive Black Hole

Using an 8-metre telescope to observe the Galactic center regularly over twodecades, you notice that one star moves back and forth across the sky in astraight line: its orbit is edge on. You take one spectra to measure its radialvelocity Vr, and find that this repeats exactly each time the star is at the samepoint in the sky. You are in luck: the furthest points of the star’s motion onthe sky are also when it is closest to the black hole (pericenter) and furthestfrom it (apocenter). You measure s = 0.248 arcsec, the separation of thesetwo points on the sky, and the orbital period P = 15.24 yr. Assuming thatthe black hole provides all the gravitational force, determine the propertiesof the elliptical orbit of the star around it (eccentricity e and semi-major axisa), to find both the mass MBH of the black hole, and its distance dBH from us.

Data: V apocenterr = 473 km/s; V pericenter

r = 7326 km/s.

49


50

Chapter 10

Galaxies II

For Tutorial 10

10.1 The Milky Way and the Local Group:

Collision with Andromeda

Collision in the local group: the case of the Milky Way and the Andromedagalaxy. The distance r of separation between two point masses m1 and m2

moving under their mutual gravitational attraction obeys the same equationas a body of much smaller mass attracted by a mass M = m1 + m2. Nowthe trajectory of a body orbiting in the plane z = 0 around this much largermass M changes according to:

d2r

dt2− L2

z

r3= −GM

r2. (10.1)

where Lz is the conserved z angular momentum. Show that the solution tothis equation is an ellipse of eccentricity e and semi-major axis a, which canbe written in the parametric form

r = a(1− e cos η) ; t =

√a3

GM(η − e sin η) (10.2)

with a = L2z/(GM(1−e2)) and time t is measured from one of the pericenter

passages where η = 0. Taking e = 1 and giving r and dr/dt their currentmeasured values of 770 kpc and -120 km/s respectively, show that η = 4.2corresponds to t0 ≈ 12.5 Gyr and a ≈ 517 kpc for the Milky Way/Andromedagalaxy system. What is the combined mass M then and why is it the smallestpossible? Repeat the calculation for η = 4.25: what conclusion can you draw

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from this exercise? Show that the Milky Way and M31 will again come closeto each other in about 3 Gyr.

10.2 The Main Types of Galaxies: Tully-Fisher

Ignoring the presence of a bulge, explain why we might expect the mass Mof a spiral galaxy to follow approximately

M ∝ V 2maxhR (10.3)

where Vmax is the maximum of the rotation curve of the disk and hR its scalelength (see formula for surface brightness profile I(R) below). Show that ifthe surface brightness (averaged over features like spiral arms) in a spiralgalaxy follows the exponential profile:

I(R) = I(0) exp(−R/hR) (10.4)

then its luminosity is given by L = 2πI(0)h2R and hence that if the ratio

M/L and central surface brightness are constants then L ∝ V 4max. In fact,

I(0) is lower in low-surface-brightness galaxies: show that if these objects areto follow the same Tully-Fisher relation they must have higher mass-to-light

ratios, with approximately M/L ∝ 1/√I(0)

10.3 The Main Types of Galaxies: Spheroids

in the Sky

Looking from a random direction, the fraction of galaxies that we see at anangle between i and i + ∆i to the polar axis is just sin i∆i, i.e. the fractionof a sphere around each galaxy corresponding to this viewing directions. Ifthey are all oblate with axis ratio B/A, then the fraction of galaxies fobl∆qwith apparent axis ratios between q and q + ∆q is given by:

fobl(q)∆q =sin i∆q

|dq/di|=

q∆q√1− (B/A)2

√q2 − (B/A)2

(10.5)

What is the fraction of oblate elliptical galaxies with true axis ratio B/A thatappear more flattened than axis ratio q? If these galaxies have B/A = 0.8,show that the number seen in the range 0.95 < q < 1 should be about onethird that of those with 0.8 < q < 0.85. Finally, show that for smaller valuesof B/A, an even higher proportion of the images will be nearly circular, with0.95 < q < 1.

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Tutorial 10: Galaxies II

10.4 Galaxy Clustering: Clusters and Lens-

ing

For a dark halo potential defined by

4πGρH(r) =V 2H

r2 + a2H

(10.6)

where aH and VH are constants, obtain the surface density Σ(R) projected ina circle of radius R on the sky and derive the corresponding projected massM(< b) within radius b. The angle bending of a ray of light passing at radiusb from a gravitational lens writes:

α(b) =4G

c2

M(< b)

b. (10.7)

Show that when both the distance to the source and the lens-source distanceare much larger than the distance to the lens, a light ray passing far enoughfrom the center of the halo, so that b is much larger than aH is bent by anangle:

α ≈ 2πV 2H

c2radians. (10.8)

A distant source exactly aligned with the center of the “dark halo” or cluster,will appear as a ring of radius θE ≈ α on the sky. Abell 383 shows a largetangential arc 16 arcsec from its center, corresponding to VH ≈ 1800 km/s.What is the kinetic energy of a particle in circular orbit in this potential?To what virial temperature does this correspond? Show that this is similarto the observed X-ray temperature TX ≈ 6× 107 K.

10.5 High-z Galaxies: Intergalactic Medium

You are observing a high redshift (z ≥ 5) quasar. Suppose that there aren(z) = n0(1 + z)3 damped Lyman-alpha (DLA) clouds per Mpc3 at redshiftz, each with cross-sectional area σ. Explain why we expect to see throughn(z)σl of them along a length l of the path towards the quasar. What is thevariation ∆l of the path l between z and z+∆z? Deduce from it the numberdN/dz∆z of DLAs per unit redshift. Show that in the case of a flat Universe(Ωk = 0, Ωm+ΩΛ = 1) and at high redshift i.e. when (1+z)3 ΩΛ/Ωm youcan express dN/dz as a function of Ωm only. Locally we find dN/dz ≈ 0.045.If the cross section σ does not change, what is the value we expect at z = 5?We measure dN/dz ≈ 0.4. How does this compare to the expected value andwhat is the interpretation of this result?

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Suppose that these clouds are uniform spheres of radius r with density nHhydrogen atoms cm−3. Their mass is given by M = (4/3)πr3nHµmH wherethe mean mass per hydrogen atom is µmH (µ ≈ 1.3 for a 75 % hydrogen 25% helium mix by weight), and the average column density N(HI) ≈ rnH .Show that, for neutral clouds, M ≈ σµmHN(HI). What is the density ρg ofneutral gas at redshift z? If this gas survived unchanged to the present day,what fraction Ωg of the critical density ρcrit = 3H2

0/(8πG) would it representnow if N(HI) = 1021 cm−2 on average?

Data: H0 = 70 km/s/Mpc, Ωm = 0.25, ΩΛ = 0.75

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