# Structure formation/Gravitational collapse

1. Apr 4, 2009

### utopiaNow

1. The problem statement, all variables and given/known data
Suppose that the stars in a disk galaxy have a constant orbital speed v out to the edge of its spherical dark halo, at a distance $$R_{halo}$$ from the center of the galaxy.

1. What is the average density $$\rho$$ for the galaxy, including its dark halo?

2. If a bound structure, such as a galaxy, forms by gravitational collapse of an initially small density fluctuation, the minimum time for collapse is

$$t_{min} \approx t_{dyn} \approx 1/ \sqrt{G \rho}$$

Show that $$t_{min} \approx R_{halo}/v$$ for a disk galaxy.

3. What is the maximum possible redshift at which you would expect to see galaxies comparable in v and $$R_{halo}$$ to our own galaxy?(use $$\Omega _{m0} = 1, \Omega _{\Lambda} = \Omega _{K} = 0$$ )

2. Relevant equations
Total Mass inside R is given by:
$$M(R) = \frac{v^2 R}{G}$$

And the volume inside R is given by $$V = \frac{4}{3} \pi R^3$$

Friedmann Equation:
$$H^2(a) = H_0^2 [\Omega _{\Lambda 0} + \frac{\Omega_{mo}}{a^3} + \frac{\Omega_{ro}}{a^4} + \frac{\Omega_{Ko}}{a^2}]$$
3. The attempt at a solution
1. Combining the first two equations basically doing M/V I get $$\rho = \frac{3v^2}{4 \pi G R^2}.$$
2. Here its basically just plugging my expression I got for #1 into the approximate equivalence they give to show that its approximately equal to that thing.

3. Here is where I have the trouble. I guess I'm looking for the farthest out redshift which will still have a comparable minimum time for gravitational collapse to our own galaxy. And I'm supposed to use the Friedmann equation somehow. And that's about all I understand at the moment. Any insights will be appreciated. Thanks.

Edit: My only guess I can make is from the Freidmann Equation we can get time as a function of redshift(z). To find the formula:
$$t(z) = \frac{2}{3H_o} (\frac{1}{1+z})^{3/2}$$

And we set $$t_{min} = t(z)$$ and find the corresponding redshift. However I don't know why that would be the answer, I'm simply plugging and playing to find an answer.

Edit 2: I think I understand why that would give the answer, It's because that's minimum time it would take for a galaxy of comparable v and $$R_{halo}$$ to collapse and form. So if a comparable galaxy to ours forms in this minimal time, the maximum redshift we would be able to view such a galaxy would be given by the t(z) formula I gave above. We need to minimize time to maximize redshift because we can see from t(z) function I gave that time and redshift are inversely proportional. So if we could see galaxies of comparable v and $$R_{halo}$$ to ours at farther redshift, they would've had to collapse faster than the minimum time, which is not possible.

Does this seem reasonable?

Last edited: Apr 4, 2009