Homework 1

Due September 14

Problem 1.5

Use Equation 1.3 and data from Table 1.1 to show that, when the Sun arrived on the main sequence, its radius was about 87% of its current radius.

Problem 1.7

A star cluster contains 200 F5 stars at the main sequence turnoff, and 20 K0III giant stars. Use Tables 1.2 and 1.4 to show that its absolute <m>V-</m>magnitude <m>M_v \approx -3.25</m> and its color <m>B-V \approx 0.68</m>. (These values are similar to those of the 4 Gyr-old cluster M67: see Table 2.2).

Homework 2

Due September 21

Problem 1.11

Near the Sun, the diffuse interstellar gas has a density of about one atom cm^-3. Show that you would need to compress a cube of gas 30km on a side into 1 cm³ to bring it to Earth's normal atmospheric density and pressure (6 x 10²³ atoms in 22.4 litres: a cube 10cm on a side has a volume of one litre). Assume that each dust grain is a sphere of radius 0.1 μm, and the gas contains one grain for every 10¹² hydrogen atoms. Show that, as light travels through 1 cm layer of the compressed gas in the previous problem, about 1% of it will be intercepted. Show that <m>\kappa=0.015</m>cm^-1, so that a layer about 70cm thick will block a fraction <m>1/e</m> of the rays.

Problem 1.14

In a galaxy at a distance of <m>d</m>Mpc, what would be the apparent <m>B-</m>magnitude of a star like our Sun? In this galaxy, show that 1" on the sky corresponds to 5<m>d</m>pc. If the surface brightness <m>I_B=27</m>mag arcsec^-2, how much <m>B-</m>band light does one square arcsecond of the galaxy emit, compared with a star like the Sun? Show that this is equivalent to L_Sun pc^-2 in the <m>B</m> band, but that <m>I_I=27</m>mag arcsec^-2 corresponds to only 0.3 L_Sun pc^-2 in the <m>I-</m>band.

Problem 1.16

Use Equation 1.25 to show that, for the Universe to be at the critical density, the average ratio of mass to luminosity <m>M/L</m> would have to be approximately 1400hM_Sun/L_Sun in blue light.

the text says 1700hM_Sun/L_Sun, I can get 1400 but not 1700 so I'm assuming this one is correct

Problem 2.7

The ring around supernova 1987a measures about 1.62" x 1.10" across the sky; if its true shape is circular, show that the ring is inclined at <m>i\approx 43^\circ</m> to face-on. If the ring radius is <m>R</m>, use Figure 2.7 to explain why light travelling first to the point A and then to us is delayed by a time <m>t_- = R(1-\sin i)/c</m> relative to light coming straight to us from the supernova. Thus we we a light echo. If <m>t_+</m> is the time delay for light reaching us by way of point B, show that <m>R=c(t_-+t_+)/2</m>. The measured values are <m>t_-=83</m>days, <m>t_+=395</m>days: find the radius <m>R</m> is light-days, and hence the distance <m>d</m> to the supernova. At its brightest, the supernova had apparent magnitude <m>m_V=3</m>; show that its luminosity was <m>L_V\approx 1.4 \times 10^8 L_{sun}</m>.

Homework 3

Due September 30

Problem 2.10

Thin-disk stars make up 90% of the total in the midplane while 10% belong to the thick disk, but <m>h_z</m> for the thin disk is roughly three times smaller than for the thick disk. Starting from Equation 2.8, show that the surface density of stars per square parsec follows <m>\Sigma(R, {\rm thin})\approx 3 \Sigma(R,{\rm thick}).</m>

Problem 2.16

Suppose that gas in the Galaxy does not follow exactly circular orbits, b ut in addition has a velocity <m>U(R,l)</m> radially outward from the Galactic center; stars near the Sun have an outward motion <m>U_0</m>. Show that gas at point P in Figure 2.19 recedes from us at speed

$V_{r}=R_{0}\sin l\left({\frac {V}{R}}-{\frac {V_{0}}{R_{0}}}\right)-R_{0}\cos l\left({\frac {U}{R}}-{\frac {U_{0}}{R_{0}}}\right)+d{\frac {U}{R}}.$

Suppose now that the sun is moving outward with speed <m>U_0>0</m>, but that gas in the rest of the Galaxy follows circular orbits; how should velocities measured in the direction <m>l=180^\circ</m> differ from zero? For gas at a given radius <m>R</m>, in which direction are the extrema (maxima or minima) of <m>V_R</m> shifted away from <m>l=90^\circ</m> and <m>l=270^\circ</m>? Using Figure 2.20 to show that the Sun and the local standard of rest are probably moving outward from the Galactic center.

Problem 3.4

A simple disk mode potential is that of the Kuzmin disk: in cylindrical polar coordinates <m>R,z</m>,

$\Phi _{K}(R,z)=-{\frac {GM}{\sqrt {R^{2}+\left(a_{K}+|z|\right)^{2}}}}.$

Find the mass density corresponding to this potential.

Homework 4

Due October 7

Problem 3.11

Show that, at radius <m>r</m> inside a uniform sphere of density <m>\rho</m>, the radial force <m>F_r=-4\pi G\rho r/3</m>. If the density is zero for <m>r > a</m>, show that

$\phi (r)=-2\pi G\rho \left(a^{2}-{\frac {r^{2}}{3}}\right)$

for <m>r<a</m> so that the potential energy is related to the mass </m>M</m> by

$PE=-{\frac {16}{15}}\pi ^{2}G\rho ^{2}a^{5}=-{\frac {3}{5}}{\frac {GM^{2}}{a}}$

Taking <m>a = R_\odot</m>, the solar radius, and the mass <m>M = M_\odot</m>, show that <m>PE \sim L_\odot \times 10^7</m>yr; approximately this much energy was set free as the Sun contracted from a diffuse cloud of gas to its present size. Since the Earth is about 4.5 Gyr old, and the Sun has been shining for at least this long, it clearly has another energy source – nuclear fusion.

Problem 3.24

Use the divergence theorem to show that the potential at height <m>z</m> above a uniform sheet of matter with surface density <m>\Sigma</m>

$\phi (z)=2\pi G\Sigma |z|$

Show that the vertical force does not depend on z, and check that <m>\nabla^2 \phi= 0</m> when <m>z\neq 0</m>. Suppose that the mass of the Galaxy was all in a flat uniform disk; use Equation 3.91 to find the density <m>n(z)</m> of K dwarfs, assuming that they have a constant velocity dispersion <m>\sigma_z</m>. As in the Earth’s atmosphere, where the acceleration of gravity is also nearly independent of height, show that <m>n(z)</m> drops by a factor of <m>e</m> as <m>|z|</m> increases by <m>h_z = \sigma_z^2/(2\pi G \Sigma)</m>. Estimate <m>h_z</m> near the Sun, taking <m>\sigma_z = 20</m>km s^-1.

Problem 3.26

When the distribution function for stars in a spherical system depends only on their energy, so that <m>f(x,v,t)=f(E)</m>, explain why the velocity dispersion is the same in all directions.

Homework 5

Due October 14

Problem 4.2

The Carina dwarf spheroidal galaxy has a velocity dispersion σ three times less than that at the center of the globular cluster ω Centauri, while Carina's core radius is 50 times greater. Use the virial theorem to show that Carina is about six times as massive as ω Centauri , so M/L must be l5 times larger.

Problem 4.6

The Sagittarius dwarf spheroidal galaxy is now about 20 kpc from the Galactic center: find the mass of the Milky way within that radius, assuming that the rotation curve remains flat with V(R) ≈ 200kms^-1. Show that this dwarf galaxy would need a mass of about 6 x 10⁹ M_sun if stars 5 kpc from its center are to remain bound to it. Show that this requires M/L_V ~ 70, which is much larger than the values listed inTable 4.2.

Problem 4.13

Taking <m>e=1</m> in Equation 4.24, and giving <m>r</m> and <m>dr/dt</m> their current measured values, use that and Equation 4.25 to show that <m>\eta=4.2</m> corresponds responds to <m>t_0=12.8</m>Gyr, and <m>a =520kpc</m>. Use Equation 4.24 to show that the combined mass <m>m+M \approx 4.8 \times 10^{12} M_{sun}</m> Show that the Milky Way and M31 will again come close to each other in about 3 Gyr. Use the data of Table 4.1 to estimate <m>L_V</m> for the Local Group as a whole, and show that the overall mass-to-light ratio is <m>M/L > 80</m> in solar units. By repeating your calculation for <m>\eta = 4.25</m>, show that to <m>t_0 = 14.1</m>Gyr and <m>m + M \approx 4.4 \times 10^{12} M_{sun}</m>: a greater cosmic age corresponds to a smaller mass for the Local Group.

Homework 6

Due October 21

Problem 5.6

In the VLA s c-array configuration, the most widely separated dishes are 3.4km apart, and the closest are 73 m from each other. Show that the resolution in the 2l cm line of HI is roughly 13"; explain why structures larger than 6' are missing from the maps.

Problem 5.10

Ignoring the bulge, use Equation 3.20 to explain why we might expect the mass <m>M</m> of a spiral galaxy to follow approximately

$M\propto V_{\rm {max}}^{2}h_{R}$

Show from Equation 5.1 that <m>L = 2 \pi I(0) h_R^2</m>, and hence that, if the ratio <m>M/L</m> and the central surface brightness <m>I(0)</m> are constant, then <m>L\propto V_{\rm max}^4</m>. In fact <m>I(0)</m> is lower in low-surface-brightness galaxies: show that, if these objects follow the same Tully–Fisher relation, they must have higher mass-to-light ratios, with approximately <m>M/L \propto 1/\sqrt{I(0)}</m>.

Problem 5.12

If the pitch angle i remains constant, show that we have a logarithmic spiral, with <m>f (r, t) tan i = ln R + k</m> for some constant k. Starting from a point on an arm and moving outward at fixed angle <m>\theta</m>, explain why we cross the next arm at a radius <m>{\rm exp}(2\pi {\rm tan} i / m)</m> times larger.

Problem 5.13

Show that, if the rotation curve of the Milky way is flat near the Sun, then <m>\kappa\approx\sqrt{2} \Omega(R)</m> so that locally <m>\kappa=36</m>km s^-1 kpc^-1. Sketch the curves of <m>\Omega, \Omega\pm\kappa/2</m> and <m>\Omega\pm\kappa/4</m> in a disk where <m>v(R)</m> is constant everywhere, and show that the zone where two-armed spiral waves can persist is almost four times larger than that for four-armed spirals.

I haven't done this question yet, but the textbook's errata page (worth looking at, I've discovered) corrects the last part of this question to "three times larger", rather than four times. -EM

Homework 7

Due November 4

Problem 6.4

When a spherical galaxy with stellar density <m>n(r)</m>is viewer from a great distance along the axis <m>z</m>, show that the surface density at distance <m>R</m> from the center is

$\Sigma (R)=2\int _{0}^{\infty }n(r)dz=2\int _{R}^{\infty }{\frac {n(r)rdr}{\sqrt {r^{2}-R^{2}}}}$

If <m>n(r)=n_0(r_0/r)^\alpha</m>, show that as long as <m>\alpha>1</m> we have

$\Sigma (R)=2n_{0}r_{0}\left(r_{0}/R\right)^{\alpha -1}\int _{1}^{\infty }{\frac {x^{1-\alpha }dx}{\sqrt {x^{2}-1}}}=\Sigma (R=r_{0})\left(r_{0}/R\right)^{\alpha -1}.$

Problem 6.5

Use Equation 6.11 to show that if we view them from random directions, the fraction of oblate elliptical galaxies with true axis ratio <m>B/A</m> that appear more flatten than axis ratio <m>q</m> is

$F_{\rm {obl}}(<q)=\int _{B/A}^{q}f_{\rm {obl}}(q')dq'={\sqrt {\frac {q^{2}-(B/A)^{2}}{1-(B/A)^{2}}}}.$

If these galaxies have <m>B/A=0.8</m>, show that the number seen in the range <m>0.95<q<1</m> should be about one-third that of those with <m>0.8<q<0.85</m>. Show that for smaller values of <m>B/A</m> an even higher proportion of the images will be nearly circular, with <m>0.95<q<1</m>. Then in Figure 6.9, count the fraction of objects with <m>-21<M_B<-20</m> that appear rounder that <m>q=0.95</m>, and explain why it is unlikely that galaxies in the luminosity range all have oblate shapes.

Problem 6.7

Assuming that the velocity dispersion <m>\sigma</m> and the ratio <m>M/L</m> constant throughout the galaxy, and that no dark matter is present, show that the kinetic energy <m>T=3M\sigma_r^2/2</m>. Approximating it crudely as a uniform sphere of radius <m>R_e</m>, we have <m>V=-3GM^2/(5R_e)</m> from Problem 3.12. Use Equation 3.44, the virial theorem to show that the mass <m>M\approx 5 \sigma^2 R_e/G</m>. If all elliptical galaxies could be described by Equation 6.1 with the same value of <m>n</m>, explain why we would then have <m>M\propto \sigma^2 R_e</m> and the luminosity <m>L\propto I_e R_e^2</m>, so that <m>M/L=\sigma^2/(I_e R_e)</m>.

Show that, if all ellipticals had the same ratio <m>M/L</m> and surface brightness <m>I(R_e)</m>, they would follow the Faber-Jackson relation.
Show that Equation 6.19, implies that <m>I_e\propto\sigma^{1.5} R_e^{-1.25}</m> and hence that <m>M/L \propto \sigma^{0.5} R_e^{0.25}</m> or <m>M^{0.25}</m>: the mass-to-light ratio is larger in big galaxies.

Homework 8

Due November 13

Problem 6.8

Show that the apparent flattening <m>\epsilon_{\rm app} = 1 - b/a</m> is given by

$\sin ^{2}i(1-B^{2}/A^{2})=1-(b/a)^{2}\approx 2\epsilon _{\rm {app}}$

for small values of <m>\epsilon_{\rm app}</m>.

Problem J2

Calculate the density corresponding to the following potential

$\Phi ={\frac {1}{2}}\left(\omega _{x}^{2}x^{2}+\omega _{y}^{2}y^{2}+\omega _{z}^{2}z^{2}\right).$

Show that there are three integrals of the motion (three conserved energies) for stars moving through this potential. Hint: write out the equations of motion for each of the three directions.

Problem 7.2

Suppose that gas atoms and galaxies in a group move at the same average random speed <m>\sigma</m> along each direction. At temperature, <m>T</m>, the average energy of a gas particle is <m>3k_BT/2</m> where <m>k_B</m> is Boltzmann's constant. If the gas is mainly ionized hydrogen, these particles are protons and electrons; find the temperature of the gas in terms of the velocity dispersion of the galaxies if the atom's kinetic energy is shared equally between the protons and the electrons.

Homework 9

Due November 20

Problem 7.5

Suppose that all galaxy groups share a common form for the density <m>\rho(r)</m>: for example, the Plummer sphere of Equations 3.12 ans 33.7. If all groups have the same radius <m>a_p</m>, and their mass is proportional to the number of members <m>N</m>, show that the virial theorem predicts that <m>\sigma \propto \sqrt{N}</m>. This is roughly what we see in Figure 7.3. Points for the sparsest groups lie above this relation; show that those groups should have smaller radii.

Problem 7.9

Suppose that all galaxy groups and clusters have the same average density. If the gas in a cluster is heated to the virial temperature, show that the cluster's mass <m>M\propto T_X^{3/2}</m>. If hot gas makes up a fixed fraction of the cluster's mass, the average density <m>n</m> is the same for all of them. Show that we expect <m>L_X \propto M \sqrt{T_X}</m> so that <m>L_X \propto T_X^2</m>.

Problem 7.23

Derive equation 7.33 from Euqation 7.26. Why must we specify that <m>\theta \gg \theta_E</m>. Show that if the surface density <m>\Sigma(R)</m> is constant, then we have <m>\theta \propto \beta</m> and that such a uniform sheet of matter does not contribute to the shear.

Homework 10

Due November 27

Problem 8.2

The free-fall time <m>t_{ff}=1/\sqrt{G\rho}</m> provides a rough estimate of the time taken for a galaxy of cluster to grow to density <m>\rho</m>. Show that a cluster of galaxies with density <m>200\rho_{\rm crit}</m> can barely collapse with the age of the Universe. This density divides structures like the Local Group that are still collapsing from those that might have settled into an equilibrium.

Problem 8.3

The Local Group moves at 600 km s^-1 relative to the cosmic background radiation. At this speed, show that an average galaxy would take <m>\sim 40 h^{-1}{\rm Gyr}</m> to travel from the center to the edge of a typical void. Whatever process removed material from the voids must have taken place very early when the Universe was far more compact.

Problem 8.6

Show that the power spectrum <m>P(k)\propto k^n</m> corresponds to a correlation function <m>\xi(r) \propto r^{-3 + n}</m>. Hence <m>\gamma \approx 1.5</m> implies <m>n\approx -1.5</m>. Figure 8.17 shouws that when <m>k</m> is large <m>P(k)</m> declines roughly as <m>k^{-1.8}</m> about as expected.

Homework 11

Due December 4

Problem 9.2

As a mass <m>m</m> of gas falls into a black hole, at most <m>0.1 m c^2</m> is likely to emerge as radiation; the rest is swallowed by the black hole. Show that the Eddington luminosity for a black hole of mass <m>M</m> is equivalent to <m>2 \times 10^{-9} M c^2</m>yr^-1. Explain why we expect the black hole's mass to grow by at least a factor of <m>e</m> every <m>5 x 10^7</m> years if it accretes at the Eddington rate.

Problem 9.3

Show that 10¹²L_sun corresponds to an energy output of 0.1 M_sunc² per year. As they age, stars like those in the solar neighbolhood eject about M_sun per year of gas for each 10¹⁰L_sun of stars. If all the gas lost by stars in our Galaxy could be funnelled into the center, and 10% of its mass released as energy, how bright would the Milky Way's nucleus be?

Problem 9.6

Defining <m>\gamma=1/\sqrt{1-v^2/c^2}</m>, show that <m>v_{\rm obs} \leq \gamma v</m>, with equality when <m>cos\theta= v/c</m> and that <m>v_{\rm obs}</m> can exceed <m>c</m> only if <m>v>c/\sqrt{2}</m>.