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Astr 498V Fall 2001

Galaxies Sylvain Veilleux

October 30, 2001 DUE: November 8 + 5 = 13, 2001


Problem 1.

(a) What is your personal mass-to-light ratio $\gamma = M/L$ in solar units, $M_\odot/L_\odot$? Assume that you are emitting like a black body with Tbb = 310 K.

(b) Using your answer in (a), find out how many ASTR 498V students per cubic parsec are needed to close the universe.

Problem 2.

The requirement that the highest velocity stars in the solar neighborhood are bound to our Galaxy provides a limit on the mass of our Galaxy. Show that the escape velocity for a star in a galaxy with a flat rotation curve like our own is

V^2_{\rm esc} = 2~V^2_c \left[ {\rm ln}\left({GM_{\rm tot}\over V^2_c R_0}\right) + 1 \right]\end{displaymath}

where Vc is the circular velocity at the galactocentric radius of the Sun, R0, and $M_{\rm tot}$ is the total mass of the Galaxy (= Rmax Vc2/G where Rmax is the radius at which the halo is truncated, i.e. $\rho_{halo}(R \gt R_{max}$) = 0). [Hint: Recall that $V_{esc} = \sqrt{-2 \Phi(R_0)}$ where $\Phi(R_0)$ is the gravitational potential at R0.] Note the very weak dependence of Vesc on Mtot. What is $M_{\rm tot}$ if $V_{\rm esc}$ $\sim$ 600 km s-1? Refer to the class notes for the values of R0 and Vc. Note the factor of 2 missing in the class notes, p. 54 (also missing on p. 76 of the Science 2000 article by Alcock).

Problem 3.

Linder's ``? Think About 4.1'' on p. 64.

Problem 4.

(a) Show that the particle horizon at z = 103 (epoch of last scattering when the cosmic background radiation can finally flow freely) is rH = 200 h-1 Mpc as measured today (= 0.2 h-1 Mpc at z= 103) if there is no inflation period.

(b) Show that the angle subtended today by the particle horizon at z = 103 is $\theta_H = 2\arcdeg~(10^3/z)^{1/2}$ in an Einstein-de-Sitter universe with $\Omega$ = 1. (hint: see p. 70 in Linder's). The cosmic background radiation is observed to be uniform in about one part in 105 on a much larger scale than this. This is an indirect argument in favor of an inflationary period in the early universe.

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Sylvain Veilleux