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Oxford Physics

M.Phys. Major Option C1

Astrophysics

Problem Sets 2011-2012

DrC.Gordon, Dr J.Devriendt, DrG.Cotter,Prof. Ph. Podsiadlowski and Prof. P.Roche

Dr E.Bayet, DrN.Gibson, DrR.Houghton, DrK.Maguire,DrM. Sullivan, DrA.Verma and Dr S.Wilkins

Denys Wilkinson Building, Keble Road, Oxford OX1 3RH

[email protected]

9th October 2011—GC

Who’s Who in C1

Lecturers

ChrisGordonEarly Universeand Large-ScaleStructure

JulienDevriendtGalaxies

GarretCotterHigh-EnergyAstrophysics

Philipp Pod-siadlowskiAdvancedStellarAstrophysics

Pat RocheAtomicProcesses andthe InterstellarMedium

Class tutors

Estelle BayetNealeGibson

RyanHoughton

KateMaguire

MarkSullivan

SteveWilkins

Introductory problems

For Tutorial 1

You will need to refer to courses from previous years including the 1st and 3rd yearastrophysics short options.

1. Blackbody spectra and magnitude systems

Explain what is meant by a black-body and describe applications of this conceptin astrophysics.The intensity of radiation emitted per unit wavelength interval by a black-body oftemperature 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 such thathc/kT << 1. In what region of the electromagnetic spectrum would you expectthis approximation to apply to the radiant emission from stars?The apparent magnitude m of a hot star of surface temperature 40 000 K is meas-ured at the two wavelengths 400 nm and 550 nm, corresponding to the colour filtersknown as B and V. The colour index mB −mV is found to be 0.35. Compare thiswith the theoretical colour index expected for a black-body at this temperature.[The apparent magnitude is related to the observed flux f my m = −2.5log10f + kwhere k is a constant for a given wavelength. The colour index for a star of surfacetemperature 11 700 K is zero.]

2. Optics and spectroscopy

Explain what is meant by the angular resolution and magnification of a telescope.Show that the limit of resolution is given approximately by λ/D where λ is thewavelength of the incident radiation and D is the diameter of the telescope aperture.What is the typical angular resolution (in seconds of arc) achieved with a largeground-based telescope operating at visible wavelengths from a good site?Give expressions for the resolving power and angular dispersion of a diffractiongrating, defining all quantities.In order to study the internal motion of galaxies astronomers often need to meas-ure velocity differences differences of order 10 km s−1. Consider a grating spectro-graph equipped with a camera of focal length 2 m and a CCD detector of pixel size

3

C1 Astrophysics problems 2011-2012

20µm. The spectrograph is used to study galaxies in the [Oiii] emission line whosewavelength is 500.7 nm. Given that the light is incident normally on the gratingand the first-order spectrum falls on the detector, estimate the minimum gratingruling, in lines per mm, required to give the desired velocity resolution.

3. Stellar Structure

Explain what is meant by the term hydrostatic equilibrium in stellar structure anddiscuss its importance. Derive an expression for the timescale on which changesoccur if equilibrium conditions are disturbed.A star is completely supported by radiation pressure, and transport of energy is byradiation only. Use the equation of state p = aT 4/3 and the radiative transportequation

L =16πr2acT 3

3κρ

dT

dr

(where L is the luminosity, κ the opacity, ρ the density, r the radius, c the speedof light and a the radiation constant) to show that in hydrostatic equilibrium theluminosity is given by

L =4πGM

κwhere 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. Accretion

Explain what is meant by escape velocity and device an expression for the escapevelocity 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, obtain anexpression for the radius, RS of a black hole of mass M . Evaluate this radius forthe 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 luminosity with100% efficiency. Does your answer support accretion of matter by a black hole asa likely model for quasar energy generation?

5. Cosmological models

State the cosmological principle and define the terms homogeneity and isotropy.Give an example showing that a homogeneous universe need not be isotropic.The Friedmann-Robertson-Walker metric for a homogeneous and isotropic universeis 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, kmeasures the spatial curvature, and r thea and φ are the radial, polar and azimuthalco-ordinates respectively. Discuss the physical significance of a(t), the scale factor,sketching its form for the three cases of a matter-only universe with positive, zero,and negative spatial curvature.

4

Tutorial 1: Introduction

Define the term luminosity distance. By considering the amount of radiation whichis received in a unit area located a co-moving distance away from a source, showthat 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 to con-strain cosmological parameters, and discuss the observations which are requiredand any key assumptions of the method.

6. High-redshift galaxies

Discuss the significance of the observation that special lines of distant galaxies areredshifted 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? Whatparticular problems are encountered in attempting to detect Lyman-α from galaxieswith redshifts greater than about 7?Assuming the Hubble constant H0 to be 70 km s−1 Mpc−1, determine the range ofdistances (in Mpc) corresponding to the range of redshifts you have calculated. Youmay take the relationship between distance d and redshift z to be

d =cz

H0

(1 + z2 )

(1 + z)2.

What sort of celestial objects are observed with redshifts z > 2? Outline furtherevidence which favours these objects being located at cosmological distances.

7. Early Universe

Give an account of the observational evidence for the hot Big Bang model of theUniverse.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 are thederivatives of these quantities with respect to time.) Use these equations to derivethe 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.

5

Early Universe and Large Scale StructureProblems

For Tutorial 2

1. Units (optional)

It is often useful to change between different units when looking at a problem.A common practice is to convert a quantity into completely different units bymultiplying by the fundamental constants such as c, ~, kB and unit conversions.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 ~ to get

~H0 = 1.05h× 10−32K m J Mpc−1.

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

~H01.01h× 10−34.

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

H0 = 1.02h× 10−10yr−1.

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

• 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).

7


• With the above relic number density, now consider the case where one outof three neutrino species has a mass of 1 eV and the rest are massless.What is the energy density of relic neutrinos in units of the critical density,Ων,massive. For what mass is the energy density at the critical value?

2. Expansion of the Universe

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

a as a function of the scale factor. It can depend on a number ofconstants that encode the fractional energy density in the various energy species.These are the fractional energy density in non-relativistic matter, ΩM0, in relativ-istic matter, ΩR0, in the cosmological constant, ΩΛ0 and in curvature, ΩK0. Youwill use the evolution equation for H(a) to solve the following problems.

• Assume the universe today with no curvature or relativistic matter, butwith both non-relativistic matter and a cosmological constant. Write anexpression for the age of the universe. Note that you cannot solve thisexactly.

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

• Assume that there is only relativistic (radiation) and non-relativistic mat-ter in the universe (no cosmological constant) and that the universe is flat.Integrate the age equation to determine the time at which the cosmictemperature was 109 K (remember that the temperature of the Universetoday is 2.7 K) and when it was 1/3 eV (1 eV is approximately 11600K). In this case you can solve the integral exactly. Assume that the ra-tio of radiation energy density to non-relativistic matter energy densitytoday, ΩR0/ΩM0 ' 2.5h−2 × 10−5 where the Hubble constant today isH0 = 100hkm s−1Mpc−1.

3. Growth of inhomogeneities

The equation for the evolutions of small perturbations in an expanding backgroundis:

δ + 2a

aδ − c2s

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 energydensity in non-relativistic matter.

a) In a Universe dominated by non-relativistic matter (or dust), we have thata ∝ t2/3 and c2S ' 0. Find the solutions to δ. Which one dominates?

b) The Newton-Poisson equation in an inhomogeneous universe can be re-written 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 on theexpanding space time). From what you know about the evolution of theenergy-density of non-relativistic matter, find the time evolution of Φ, theNewtonian potential.

8

Tutorial 2: Early Universe / LSS

c) Assume that the universe is a mixture of dust and cosmological constantand that the cosmological constant is constant over all of space. Fur-thermore, assume that the cosmological constant dominates, so that thefractional energy density in dust is effectively zero. Explain why the theequation for the evolution of perturbations in the dust is given by

δ + 2a

aδ = 0

Find the solution of δ as a function of time.d) Compare a Universe in which the cosmological constant dominates with

one 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.

4. A baryon dominated Universe

In the very early Universe, before recombination, baryons and photons interact verystrongly to form a tightly coupled fluid. Let us assume that the transition betweenradiation and matter domination occurs at the same time as recombination. Beforerecombination, the evolution equation for the perturbations in the baryon/photonfluid is

δ′′ +a′

aδ′ − 1

3∇2δ − 4

(a′

a

)2

δ = 0 (1)

with a ∝ η. After recombination the evolution equation is

δ′′ +a′

aδ′ − 3

2

(a′

a

)2

δ = 0 (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 where λJ

is the Jeans length) that separates the large k behaviour from the small kbehaviour. What is it? How does it evolve with time? (Please show thetime evolution for the physical and conformal Jeans wave number).

A very rough approximation to the solution of the evolution equations before re-combination on small wavelengths is

δ = C1 cos

(1√3kη

)+ C2 sin

(1√3kη

)where C1 and C2 are constants.

• In the pre-recombination period, how does a perturbation that starts offwith a k < kJ evolve (answer this question qualitatively)? What type ofbehaviour will it have at sufficiently late times?

• (This is hard). Start off with a perturbation with an amplitude A on scalesmuch larger than Jeans scale at some very early time. Follow its evolu-tion until today. You should find two types of behaviour: modes whosewavenumber k are smaller than kJ at recombination and modes whose

9


wavenumbers are greater than kJ . By matching the large wavelengthsolutions to short wavelength solutions at the time when kJ = k you canget a solution that covers all times. What form will it have at late times?(No need to do the full calculation, just figure out what kind of solutionwill be picked out at late times.)

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

P (k) ≡ |δ(t0,~k)|2

Sketch what you expect the power spectrum of perturbations to be todayif we assume that initially all modes had the same amplitude at some earlytime.

5. Neutrinos and free streaming

Consider a universe with critical density full of nonrelativistic matter. The evolutionequation 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 a ho-mogeneous (ρ0) and imhomogeneous (δρ) part. Find the solution for the evolutionof δ as a function of time.

Now consider a critical density universe which has a mixture of massive neutrinos(with fractional density Ων) and non-relativistic matter (with fractional densityΩM ) so that Ων+ΩM = 1. It will expand at the same rate as in the case consideredabove. On large scales δ will evolve as above but on scales smaller than about40h−1Mpc, the evolution of perturbations in the nonrelativistic matter is set by

δ + 2a

aδ − 3

2

(a

a

)2

ΩM0δ = 0.

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 small scalesfor these two perturbations? Explain the reason for the difference in terms of thebehaviour of the neutrinos.

We are now able to measure the angular positions and redshifts, z of millions ofgalaxies in the universe. In some cases we are even able to measure their distancesfrom us. At low redshifts, a measurement of the distance, d and the redshift of agalaxy can be used to determine its peculiar velocity away from us, v, through

cz = H0d+ v

where H0 is the Hubble constant today and c is the speed of light. From conser-vation of energy we have that the peculiar velocity and the density contrast at agiven point are related through:

~∇ · ~v = −δ

10

Tutorial 2: Early Universe / LSS

Show that, in regions of equal δ, galaxies will have on average lower peculiar ve-locities in a universe with massive neutrinos as compared to a universe with onlynon-relativistic matter.

11

Galaxies Part I

For Tutorial 3

1. Basic Definitions

In a galaxy located at a distance d Mpc from our own Milky Way, what would bethe apparent B−magnitude of a star like our Sun? Show that in such a galaxy,an angle of 1 arcsec on the sky corresponds to a length of 5d pc. If its surfacebrightness is IB = 27 mag arcsec−2, how much B−band light does a patch of onesquare arcsecond of this galaxy emit? Show that although this is equivalent to ∼ 1Lpc−2 in the B−band, it drops to ∼ 0.3 Lpc−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/Aunits.

2. Principles of Galactic Dynamics

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

ΦP (r) = − GMP√r2 + a2

P

, (3)

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

ρP (r) =3a2PMP

4π(r2 + a2P )5/2

. (4)

and calculate UP , its potential energy. What is the mass M(< R) enclosed within aPlummer sphere of radius R? When viewed from a great distance along the z-axis,what is its surface density ΣP (R) at a distance R from the center? Check that thecore radius rc, where ΣP (R) drops to half its central value, is rc ≈ 0.644 aP .

3. Principles of Galactic Dynamics

A simple disk model potential is that of the Kuzmin disk of total mass MK andscale length aK , which reads, in cylindrical polar coordinates:

ΦK(R, z) = − GMK√R2 + (aK + |z|)2

. (5)

Irrespective of whether z is positive or negative, this is the potential of a pointmass MK at R = 0, displaced by a distance aK along the z-axis, on the oppositeside of the z = 0 plane (i.e. negative if z is positive and positive if z is negative).

13


Show that ∇2ΦK = 0 everywhere except in the plane z = 0 and use the divergencetheorem to obtain the surface density ΣK(R) there.

4. Introduction to Hierarchical Galaxy Formation

The Navarro-Frenk-White (NFW) model is used to describe the density profile ofcold dark matter halos with characteristic scale lengths aN and densities ρN thatform in cosmological N-body simulations:

ρNFW (r) =ρN

(r/aN )(1 + (r/aN ))2. (6)

Note the cusp in the center where the density diverges like 1/r and the rapid fall offin r−3 at large radii. Calculate the associated gravitational potential, as a functionof 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 orbit at radius r in thispotential.

5. Our Galactic Backyard: the Milky Way and the Local Group

Using an 8-metre telescope to observe the Galactic center regularly over two dec-ades, you notice that one star moves back and forth across the sky in a straightline: its orbit is edge on. You take one spectra to measure its radial velocity Vr,and find that this repeats exactly each time the star is at the same point in the sky.You are in luck: the furthest points of the star’s motion on the sky are also whenit is closest to the black hole (pericenter) and furthest from it (apocenter). Youmeasure s = 0.248 arcsec, the separation of these two points on the sky, and theorbital period P = 15.24 yr. Assuming that the black hole provides all the grav-itational force, determine the properties of the elliptical orbit of the star aroundit (eccentricity e and semi-major axis a), to find both the mass MBH of the blackhole, and its distance dBH from us.

Data: V apocenterr = 473 km/s; V pericenterr = 7326 km/s.

6. Our Galactic Backyard: the Milky Way and the Local Group

Collision in the local group: the case of the Milky Way and the Andromeda galaxy.The distance r of separation between two point masses m1 and m2 moving undertheir mutual gravitational attraction obeys the same equation as a body of muchsmaller mass attracted by a mass M = m1 + m2. Now the trajectory of a bodyorbiting in the plane z = 0 around this much larger mass M changes according to:

d2r

dt2− L2

z

r3= −GM

r2. (7)

where Lz is the conserved z angular momentum. Show that the solution to thisequation is an ellipse of eccentricity e and semi-major axis a, which can be writtenin the parametric form

r = a(1− e cos η) ; t =

√a3

GM(η − e sin η) (8)

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 current measuredvalues of 770 kpc and -120 km/s respectively, show that η = 4.2 corresponds to

14

Tutorial 3: Galaxies

t0 ≈ 12.5 Gyr and a ≈ 517 kpc for the Milky Way/Andromeda galaxy system.What is the combined mass M then and why is it the smallest possible? Repeatthe calculation for η = 4.25: what conclusion can you draw from this exercise?Show that the Milky Way and M31 will again come close to each other in about 3Gyr.

15

Galaxies Part II; High-Energy Astrophysics Part I

For Tutorial 4

1. The Main Types of Galaxies: S, E and S0

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

M ∝ V 2maxhR (9)

where Vmax is the maximum of the rotation curve of the disk and hR its scale length(see formula for surface brightness profile I(R) below). Show that if the surfacebrightness (averaged over features like spiral arms) in a spiral galaxy follows theexponential profile:

I(R) = I(0) exp(−R/hR) (10)

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 are to follow the sameTully-Fisher relation they must have higher mass-to-light ratios, with approxim-ately M/L ∝ 1/

√I(0)

2. The Main Types of Galaxies: S, E and S0

Looking from a random direction, the fraction of galaxies that we see at an anglebetween i and i + ∆i to the polar axis is just sin i∆i, i.e. the fraction of a spherearound each galaxy corresponding to this viewing directions. If they are all oblatewith axis ratio B/A, then the fraction of galaxies fobl∆q with apparent axis ratiosbetween 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

(11)

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, showthat the number seen in the range 0.95 < q < 1 should be about one third that ofthose with 0.8 < q < 0.85. Finally, show that for smaller values of B/A, an evenhigher proportion of the images will be nearly circular, with 0.95 < q < 1.

3. Galaxy Clustering: Groups, Clusters and Large Scale Structure

For a dark halo potential defined by

4πGρH(r) =V 2H

r2 + a2H

(12)

17


where aH and VH are constants, obtain the surface density Σ(R) projected in acircle of radius R on the sky and derive the corresponding projected mass M(< b)within radius b. The angle bending of a ray of light passing at radius b from agravitational lens writes:

α(b) =4G

c2M(< b)

b. (13)

Show that when both the distance to the source and the lens-source distance aremuch larger than the distance to the lens, a light ray passing far enough from thecenter of the halo, so that b is much larger than aH is bent by an angle:

α ≈ 2πV 2H

c2radians. (14)

A distant source exactly aligned with the center of the “dark halo” or cluster, willappear as a ring of radius θE ≈ α on the sky. Abell 383 shows a large tangential arc16 arcsec from its center, corresponding to VH ≈ 1800 km/s. What is the kineticenergy of a particle in circular orbit in this potential? To what virial temperaturedoes this correspond? Show that this is similar to the observed X-ray temperatureTX ≈ 6× 107 K.

4. High-z Galaxies, Active Galactic Nuclei and Intergalactic Medium

You are observing a high redshift (z ≥ 5) quasar. Suppose that there are n(z) =n0(1 + z)3 damped Lyman-alpha (DLA) clouds per Mpc3 at redshift z, each withcross-sectional area σ. Explain why we expect to see through n(z)σl of them alonga length l of the path towards the quasar. What is the variation ∆l of the path lbetween z and z + ∆z? Deduce from it the number dN/dz∆z of DLAs per unitredshift. Show that in the case of a flat Universe (Ωk = 0, Ωm + ΩΛ = 1) and athigh redshift i.e. when (1 + z)3 ΩΛ/Ωm you can express dN/dz as a function ofΩm 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 thiscompare to the expected value and what is the interpretation of this result?Suppose that these clouds are uniform spheres of radius r with density nH hydro-gen atoms cm−3. Their mass is given by M = (4/3)πr3nHµmH where the meanmass per hydrogen atom is µmH (µ ≈ 1.3 for a 75 % hydrogen 25 % helium mixby weight), and the average column density N(HI) ≈ rnH . Show that, for neutralclouds, M ≈ σµmHN(HI). What is the density ρg of neutral gas at redshift z?If this gas survived unchanged to the present day, what fraction Ωg of the criticaldensity ρcrit = 3H2

0/(8πG) would it represent now if N(HI) = 1021 cm−2 on aver-age?

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

5. 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 flux densityof 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?

18

Tutorial 4: Galaxies / High-Energy Astrophysics

Figure 1. Brightness temperature.

(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 stars atradio 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.

6. Shocks

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

Td =3

16

mv2u

kB

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 this rela-tion?In the lectures we considered the blast wave of SN1993J, which had an inititalexpansion speed of 20 000 km s−1. Assuming the interstellar medium around thesupernova to be hydrogen plasma, estimate the initial temperature behind the blastwave.Hydrogen plasma falls radially onto a white dwarf, passing through a shock veryclose to the surface. Show that the temperature behind the shock is given by

19


kBT

mec2=

3

32

mp

me

RS

R∗where R∗ is the radius of the star and RS is its Schwarzschild radius. What is kBTif R∗ = 6000 km and M = M?

7. Particle acceleration

Outline the physical mechanism by which it is thought that electrons are acceleratedto ultra-relativistic energies in strong non-relativistic shocks such as are found insupernova remnants.Suppose that an electron is involved in a collision which increases the electron’stotal energy by a factor β. Suppose further that there is a probability p that thiselectron remains within the region where further collisions may occur. Show thatthe expected distribution of electron energies is

N(E)dE ∝ EkdEWhere k = −1 + (ln p/ lnβ).Assuming that k = −2 and that the electrons are accelerated to energies at whichthey emit synchrotron radiation, show that the power-law region of the synchrotronspectrum will have a form

Sν ∝ ν−0.5

You may assume that a synchrotron electron emits almost all its radiation at a

characteristic frequency νcrit ∼ γ2eB2πme

and that the power radiated is P = 4γ2β2cσTB2

3µ0

8. Synchrotron spectrum

Describe, with the aid of an annotated sketch, the shape of the spectrum of con-tinuum radio emission from the lobes of a powerful extragalactic radiosource. In-clude in your discussion the optically thick and optically thin regions of the spec-trum, along with an explanation of these terms, and describe how the shape of thespectrum is modified by radiation losses as the radiosource ages.Show that a relativistic electron with Lorentz factor γ, passing through a regioncontaining a uniform magnetic flux density B, has a gyrofrequency

ν =eB

2πγme

Hence show that such an electron emits radiation whose spectrum is strongly peakedat 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 is theThomson cross section. Using these formulae for P and νcrit, calculate a character-istic timescale for the synchrotron lifetime of the electron.

20

Tutorial 4: Galaxies / High-Energy Astrophysics

The powerful giant radiosource 3C236 is 6 Mpc across, with the host galaxy at thecentre. The radio spectrum of the lobes close to the host galaxy shows a cut-off inemission, assumed to be due to radiative losses, at frequencies above about 1 GHz.Taking the magnetic flux density in this region to be 0.3 nT, estimate the age ofthe synchrotron plasma near the host galaxy.On the assumption that this plasma was left behind near the host galaxy by theradiosource jets just as they began to expand into intergalactic space, estimate theexpansion speed of the radiosource. What factors may cause these estimates of ageand expansion speed to be unreliable?

9. Equipartition/Minimum Energy

Describe the evidence that the radio emission from powerful extragalactic radiosources is produced by the synchrotron mechanism.A radio source contains a population of relativistic electrons with Lorentz factorsγ 1 and a uniform magnetic flux density of magnitude B. The number densityof electrons with Lorentz factors in the range γ to γ + dγ is given by

n(γ)dγ = n1

(γ

γ1

)−kdγ

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

Jν ∝ n1γk1γ

1−kB.

You may assume that each electron emits all its synchrotron radiation at a frequencyν = γ2eB/2πme and that the power radiated by each electron is P = 4

3cσTumγ2

where um is the energy density of the magnetic field and σT is the Thomson scat-tering 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 the sourcein the form of relativistic electrons and magnetic field has a minimum value whichoccurs when

ue =4

3um.

The radio source Cygnus A is about 100 kpc across. The flux density of the magneticfield in the radio source is thought to be about 6 nT. Estimate a lower limit to thetotal energy content of the radio source and explain what implications this has forhow Cygnus A is powered.

Appendix: Useful Constants and Unit Conversions

• 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

21

High-Energy Astrophysics Part II; AdvancedStellar Astrophysics Part I

For Tutorial 5

1. Accretion discs

Describe four methods used to determine the masses of black holes in galactic nuclei.What is the physical significance of the Eddington luminosity, LEdd? Derive theexpress ion

LEdd =4πGmpMc

σT

for pure hydrogen plasma accreting onto an object of mass M , where σT is theThomson cross-section, clearly stating any assumptions made.Assume that the emission from a geometrically-thin, optically-thick accretion discis dominated by a component from its innermost part. Take this to be a uniformcircular region of radius r ∼ 6GM/c2. By equating the luminosity emitted from thisregion to the Eddington luminosity, derive an approximate expression for the black-body temperature of radiation from the inner disc. Estimate the temperatures forblack holes of 10 M and 108 M and comment on your results in the context ofthe observed spectra of X-ra y binaries and active galaxies.The galaxy NGC 4258 contains a black hole of mass 3.5 × 107 M. Calculate theEddington luminosity and compare it with the observed X-ray luminosity of 4 ×1033 W.Contrast the properties of accretion discs with M ∼ MEdd and M MEdd.[σT = 6.6× 10−29 m2.

]2. Doppler beaming and boosting

In the lectures we derived the formula for the apparent projected velocity βappc ofmaterial 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 apparent speedis

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.

23


Figure 2. Doppler boosting of a relativistic jet.

Suppose a source is moving along the θ = 0 axis (spherical polar coordinates). Ifthe source emits photons isotropically in its rest frame (the S′ frame) then theangular 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!). Use therelativistic abberation formulae to transform θ′ to the observer’s (S) frame θ andhence show that the distribution of photons in the S frame is given by

P (θ) dθ = D2 1

2sin θ dθ,

where D is the relativistic Doppler factor, [γ(1− βcosθ)]−1.Figure ?? shows how the Doppler factor varies as a function of γ and θ. There arecouple 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 with 0 < θ < 90. Bealert!Second, D becomes less than one while the jet is still pointing towards you! The peakof the boosting is occuring off to the side; you’re looking at the fainter “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 of 0.5. Thedeepest radio maps of 3C273 have a dynamic range of about 5000 and show no signwhatsoever of a counterjet. Is this surprising?

24

Tutorial 5: High-Energy / Advanced Stellar

3. Inverse-Compton scattering

Explain what is meant by Thomson scattering and inverse-Compton scattering.Give two examples where inverse-Compton scattering is important in astrophysics.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 density ofincident radiation. Use this relation to derive an expression for the average powerof radiation inverse Compton scattered by a population of relativistic electrons withLorentz factors γ ∼ 1000. Assume that the distribution of electrons is isotropic andthat the radiation being scattered is at radio frequencies. Justify each step in yourderivation.Ultra-relativistic electrons of energy 1011 eV are observed at the Earth. By con-sidering the effect on the electrons of inverse Compton scattering of the microwavebackground radiation, calculate the maximum length of time for which the elec-trons could have had energies larger than the observed value. Given that the age ofthe Galaxy is ∼ 1010 years, what does this imply? Suggest some possible sourcesof such high-energy electrons.[The energy density of the microwave background is 2.6× 105 eV m−3; σT = 6.6×10−29 m2.]

4. Self-absorbed Synchrotron

By considering the intensity of radiation emitted by a black body, show how anestimate of the magnetic flux density in a compact radio source can be obtainedfrom a measurement of its surface brightness at a frequency at which the source isoptically thick.[You may assume that in the Rayleigh-Jeans part of the spectrum of a black bodyat temperature T , the intensity per unit frequency interval Sν is given by Sν =2kBTν

2/c2]

5. Thermal Bremsstrahlung

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

jνdν ∝ n2e

vln

(mv2

2hν

)dν,

where ne is the electron density.The integrated bremsstrahlung emissivity ε (the power per unit volume) for a hy-drogen 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 of m−3.Considering a virialized cluster of galaxies with a gas mass of 1014M and a sizeof 1 Mpc, estimate the time it takes for the gas to cool, explaining quantitativelyany assumptions you need to make.

25


Why is this estimate an upper limit to the cooling time? What are the implicationsfor the existence of such clusters in the local Universe, given that the time sincethe Big Bang is approximately 13.7 Gyr?The central galaxy in such a cluster is found to harbour a black hole of mass 109M.Describe, with a quantitative calculation, how this may reconcile any difficultiesimplied by your previous calculations.

6. Sunyaev-Zel’dovich Effect and H0

What is meant by the Sunyaev–Zel’dovich effect? How is it measured?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 explains whyδI/I is given by this formula. Estimate the frequency at which this formula isinvalid.The galaxies of the Coma cluster have a mean recession velocity with respect toEarth of 6900 km s−1. Observations of the Coma cluster with X-ray telescopes showthat the intracluster plasma has an angular extent of 30 arcminutes, a temperat-ure of 108 K and an electron density of ∼ 4000m−3. Observations of the CosmicMicrowave Background toward the cluster at 30 GHz show a diminution in tem-perature of 0.6 mK. Use these data to make an estimate of the Hubble parameter,describing any assumptions made.Identify two difficulties involved in the measurement of the Hubble parameter viathe Sunyaev–Zel’dovich effect, and describe how their influence may be minimized.[σT = 6.6×10−29 m2; the mean temperature of the Cosmic Microwave Backgroundis 2.73 K.]

7. Core-Collapse Supernovae [20 points]

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

a) Stating your assumptions, estimate the final spin and the strength of themagnetic field of the neutron star that forms from the collapse of such acore. Compare the spin period to the minimum spin period for a neutronstar.

During the collapse phase, the initial collapse stops when the central core with amass Mcore ' 0.7M reaches a mass density ρ ' 3× 1016 kg m−3. At this densitythe core bounces driving a shock with an energy Ebounce ∼ 1044J into the infallingouter core.

b) Estimate the energy that is required to photodissociate 0.1M of Fe intoneutrons and protons. Compare this energy to the bounce shock energyand comment on the fate of the shock. [Remember that ∼ 0.8 % of therest 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 estimate theneutrino luminosity during the initial neutron-star cooling phase. [Hint:

26

Tutorial 5: High-Energy / Advanced Stellar

assume that all the gravitational potential energy escapes in the formof neutrinos and use a standard random-walk argument to estimate theneutrino 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 neutrinoenergy to reverse the infall of the outer core and drive a successful super-nova explosion (with a typical explosion energy of 1044 J). Compare thistime to the dynamical timescale of the proto-neutron star.

8. The Binary Pulsar PSR J0737-3039: Supernova Kicks [40 points]

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 systemconsists of two pulsars (A and B) in a mildly eccentric orbit with an orbital periodPorb ' 2.4 hr and eccentricity e ' 0.088. The spin periods and spin-down ratesof 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 and the masses have been determinedto be MA ' 1.34M and MB ' 1.25M, respectively.

a) Making reasonable assumptions about the pulsar properties, estimate

the spin-down luminosities and the spin-down ages (i.e. P/2P ) for bothpulsars. Considering the evolutionary history of the system, explain whythe spin-down ages should roughly agree.

b) Assuming that the spin-down is caused entirely by magnetic dipole radi-ation, show that the magnetic field of the pulsars can be estimated from

B ' 1

R3 sin θ

√(3c3µ0

32π3

)PP I,

where R is the radius of the pulsar, θ the (generally unknown) inclinationof the magnetic axis with respect to the rotation axis and I is the momentof inertia of the pulsar (µ0 is the magnetic permeability and c the speedof light in vacuo). Estimate the magnetic fields of the two pulsars.

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 accrete froman accretion disc to be spun-up to the observed spin period. How doesthe actual magnetic field of Pulsar A affect this estimate?

It is reasonable to assume that before the second supernova, in which Pulsar Bwas formed, the immediate pre-supernova binary system was circular and had anorbital separation a0 ' 1.4R.

d) Assuming that in the second supernova an amount of mass ∆M was in-stantaneously ejected and that Pulsar B did not receive a recoil in its ownframe, show that the post-supernova eccentricity e, post-supernova semi-major axis aPSN and post-supernova system velocity vsys (i.e. the velocityof the new centre-of-mass (CM) frame defined by the two pulsars relativeto the pre-supernova CM frame) are given by

e =∆M

MA +MB,

27


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 ' 0.1and estimate vsys.[Hint: You need to compare the energies and momenta of the systembefore and after the supernova. The eccentricity e and semi-major axis aof an eccentric orbit are related to the distance of closest approach, theperiastron separation, rp by rp = (1 − e) a, and the total energy of aneccentric binary is

Ebinary = −GM1M2

2a= −GM1M2

r+

1

2

M1M2

M1 +M2v2,

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

e?) Show that in the limit, where there is no mass loss associated with thesecond supernova but where Pulsar B received an asymmetric supernovakick of magnitude vkick, the post-supernova system velocity is given by

vsys =MB

MA +MBvkick.

What is vsys in this case for a typical vkick ' 250 km s−1?f?) Discuss how the observed eccentricities and system velocities of systems

like the double pulsar may be used to constrain supernova kicks.

28

Advanced Stellar Astrophysics Part II

For Tutorial 6

1. The Last Stable Circular Orbit [25 points]

In General Relativity, the equation for the radial coordinate r of a test particleorbiting 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, (15)

where r = dr/dt and L and E are the angular momentum per unit rest mass andthe energy per unit rest mass of the particle, respectively (the particle is assumed tohave non-zero rest mass). This equation resembles the energy conservation equationin 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 potentialfor a particle near a black hole as a function of radius, both for a small anda large value of L. Characterize the possible types of trajectories/orbitsin both cases.

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

r± =L2 ±

[L4 − 12G2M2L2/c2

]1/22GM

, (16)

provided that L2 > 12G2M2/c2.c) Show that the r+ solution has a minimum value of rmin

+ = 6GM/c2 andargue that this is a stable orbit (i.e. corresponds to a minimum of theeffective potential). What does this imply for the r− solution?

d) Calculate the energy E of a particle at this innermost stable circular orbitand 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 an ac-cretion disc approaches the innermost stable orbit. Compare this case toaccretion onto a non-magnetic neutron star.

2. Gamma-Ray Bursts [25 points]

A popular model for long-duration gamma-ray bursts (GRBs) is the collapsarmodel, in which a massive, relatively rapidly rotating helium (or carbon/oxygen)star collapses to form a compact object (e.g. a black hole) surrounded by a disc of

29


matter at nuclear density. Subsequent accretion from the disc causes the formationof a relativistic jet that penetrates the remaining infalling envelope and generates agamma-ray burst at a large distance away from the star by internal and/or externalshocks.For this problem consider a collapsing helium star of mass MHe = 10M and radiusRHe = 5× 108 m where the inital mass of the central black hole is M0

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 at thelast stable orbit for a 2M black hole, so that an accretion disc can form.Estimate the characteristic dynamical timescales both for the inner discand the collapsing helium star. How do these timescales determine theobservable characteristics of GRBs?

b) Assume that an amount of relativistic energy E = 1044 J is injected bythe central engine of the GRB, driving an expanding relativistic fireball.Estimate the radius at which the fireball becomes optically thin to MeVgamma rays, i.e. the radius at which the optical depth to pair creationγ γ e++e− becomes less than 1. [You may assume that the cross sectionfor pair creation is given by the Thomson cross section σT ' 6.6×10−29 m2

and that the typical photon energy is 1 MeV; argue that the optical depthis then given by τγ ∼ nγ σTR.]

While massive stars are known to be rapidly rotating on the main sequence, they arebelieved to be spun down efficiently during their evolution by hydrodynamical andmagnetohydrodynamical effects and develop cores that are not rotating sufficientlyfast to be consistent with the collapsar model. One way of spinning up a heliumstar is if it is a member of a close binary where it can be spun up by the tidalinteraction with a companion star.

c) Consider a 10M helium star in a close orbit with a compact star (mostlikely a neutron star or a black hole) with an orbital period Porb. Assumethat the spin angular velocity of the helium star is synchronized with theorbital angular velocity of the binary and that the helium star is in solidbody rotation. Estimate the maximum orbital period for which the coreis sufficiently rapidly rotating that only the innermost 2M can collapsedirectly, while the rest collapses first into a disc [take the typical radiusof the inner 2M core as 8× 107 m). [Answer: ∼ 5 hr]

d) Sketch briefly the evolutionary path that can lead to the formation of sucha system.

30

Tutorial 6: Advanced Stellar

3. Mass-Transfer Driving Mechanisms [25 points]

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 written as

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 total an-gular momentum of the system remains constant), the orbital separationis a minimum when M1 = M2. Sketch the evolution of A as a function oftime assuming that M1 > M2 initially and that mass is transferred fromstar 1 to star 2. How does this behaviour of A affect the mass-transferrate, assuming that star 1 attempts to expand at a steady rate?

b) Even in the absence of mass transfer, the orbit of a binary will shrink dueto the emission of gravitational waves, which causes the loss of orbitalangular momentum at a rate

dJ

dt= −32

5

G7/2

c5µ2M5/2

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

c5M2µ

A4.

By setting the orbital period decay time (P/P ) equal to the age of theGalaxy (∼ 1010 yr), determine the maximum separation and hence max-imum orbital period for which a binary consisting of (i) two low-masshelium white dwarfs with M1 = M2 = 0.3M, (ii) two massive car-bon/oxygen white dwarfs with M1 = M2 = 1M and (iii) two neutronstars with M1 = M2 = 1.4M are driven into contact by gravitationalwave emission within the age of the Galaxy. Discuss the likely/possiblefate of the systems in the three cases.

c?) Assume that star 1 loses mass in a stellar wind at a wind mass-loss rate

M = 10−10M yr−1 and that the wind is magnetically coupled to the spinof star 1 up to a radius 10R away from the star (where R is the radiusof the star). Assume further that due to the tidal interaction with thecompanion star, the spin of star 1 is synchronized with the orbital period(i.e. Pspin = Porb). Estimate the orbital period decay time (P/P ) due tothis magnetic braking for a system with M1 = M2 = 1M and A = 3R.[Hint: what is the specific angular momentum lost in the stellar wind?]

4. Neutron-Star Spin-Up by Wind Accretion [25 points]

Consider a massive X-ray binary consisting of a 20M star and a 1.4M neutronstar in a relatively wide orbit where the radius of the massive star is much smallerthan its Roche lobe radius. Assume that the massive star loses mass in a steadyspherical wind with a wind mass-loss rate Mwind = 10−7M yr−1 and a windvelocity vwind = 103 km s−1 and that the neutron star has a magnetic field B =

31


108 T (assumed to be dipolar) and is spun up (or spun down!) by accretion of someof 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 fraction of the windthat is accreted by the neutron star is given by (rBH/2A)2 where A is theorbital separation of the binary.

The matter that is accreted by Bondi-Hoyle accretion typically has a specific angularmomentum that is 1/4 of the specific Keplerian angular momentum at 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 the pulsar(i.e. for which the accreted specific angular momentum is larger than theKeplerian specific angular momentum at the Alfven radius). Determinethe equilibrium spin period of the pulsar for this case.

5. Star Formation [25 points]

a) Explain the concept of the Jeans mass and its importance for star form-ation. Consider two dense cores in a cloud of molecular hydrogen (withµ = 2): (i) a cool core with temperature T = 10 K and (ii) a warm corewith T = 100 K. Taking the number density of molecular hydrogen to ben = 1010 m−3 in both cases, estimate the Jeans mass for both cores.

b) Assume that the pre-collapse core is in hydrostatic equilibrium and can betreated as an isothermal sphere, i.e. a sphere of gas at constant temper-ature 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 of the sphere isapproximately given by

ρ =c2s

4πG

1

r2,

and the mass enclosed within a radius r by

M(r) =c2sGr.

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

M =c3sG

and is independent of r. What is M for the cases (i) and (ii) in part a)?d?) Now consider an unstable molecular cloud core of 1M and initial radius

R = 106R which is in solid body rotation with angular velocity ω =10−13 Hz. Assuming that the size of the protostar forming at the centreof the collapsing core has a radius of 5R and using angular momentumconsiderations, estimate what fraction of the mass of the core can collapse

32

Tutorial 6: Advanced Stellar

directly onto the protostar. Estimate the characteristic size of the proto-stellar disc that forms from the collapse of the bulk of the molecular core.[To obtain these estimates, you may take the density to be constant inthe core and ignore factors of order unity.] Comment on the implicationsof these estimates for the outcome of the star-formation process.

33

Atomic Processes and the ISM

to follow

35

oxford physics m.phys. major option c1 astrophysics ... · m.phys. major option c1 astrophysics...

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