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Astr 498V Fall 2001

Galaxies Sylvain Veilleux

September 11, 2001 DUE: September 18, 2001

PROBLEM SET # 1

Problem 1.

Estimate the average density of the Local Group of galaxies and compare it with the critical density $\rho_c = 3 H^2_0 / 8 \pi G = 1.9
\times 10^{-29} h^2$ g cm-3. What are the implications of your result? Mention the references you use for your calculations.

Problem 2.

The time taken for a galaxy or cluster to grow to density $\rho$ must be at least as great as the free-fall time $t_{ff} = 1/\sqrt{G\rho}$(G = gravitational constant), since cosmic expansion must first be halted locally. What is tff for (a) the Milky Way ($\rho \sim
10^5 \rho_{c}$) and (b) a cluster of galaxies ($\rho \sim 10^2
\rho_{c}$). Compare with the Hubble time H-10.

Problem 3.

We saw in class that the power spectrum

\begin{displaymath}
P({\bf k}) \equiv \int \xi(r)~{\rm exp}(i{\bf k} \cdot {\bf ...
 ...^3r = 4 \pi \int^{\infty}_0 \xi(r) {{\rm sin}kr\over kr} r^2 dr\end{displaymath}

(a)
Prove the last equality in the previous equation. [Hint: one method is to write the volume integral for P(k) in spherical polar coordinates $r, \theta, \phi$ and then set ${\bf k} \cdot {\bf r} = kr
{\rm cos} \theta$.
(b)
Show that because $\xi(r)$ describes departures from the mean density, the above equation implies $\int^{\infty}_0 \xi(r) r^2 dr = 0$, and hence $P(k) \rightarrow 0$ as $k \rightarrow 0$.

(c)
Show that the power spectrum $P(k) \propto k^n$ corresponds to a correlation function $\xi(r) \propto r^{-(3+n)}$. Hence $\gamma
\approx 1.5$ implies $n \approx -1.5$, approximately as observed.

Problem 4.

Linder's ``? Think About 1.3'' on p. 16

Problem 5.

Linder's ``? Think About 1.4'' on p. 17



 
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Sylvain Veilleux
9/12/2001