Structure formation/Gravitational collapse

[utopiaNow]

Homework Statement

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 $$\Omega _{m0} = 1, \Omega _{\Lambda} = \Omega _{K} = 0$$ )

Homework 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:
$$H^2(a) = H_0^2 [\Omega _{\Lambda 0} + \frac{\Omega_{mo}}{a^3} + \frac{\Omega_{ro}}{a^4} + \frac{\Omega_{Ko}}{a^2}]$$

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?

 

[member38644]

Yes, your reasoning is correct. The maximum redshift at which we could see a galaxy with comparable v and R_halo to our own would be when the minimum time for gravitational collapse, t_min, is equal to the age of the universe at that redshift. This is because any galaxy that formed after that redshift would have taken longer than the minimum time to collapse, and therefore would not have comparable v and R_halo to our own.

Using the Friedmann equation, we can find the age of the universe at a given redshift using t(z) = (2/3H_0)(1/(1+z))^3/2. We can then set this equal to t_min = R_halo/v and solve for the maximum redshift, which gives us the answer of t(z) = R_halo/v. This means that the maximum redshift at which we could see a galaxy comparable to our own would be when t(z) = R_halo/v.

Overall, your understanding and approach to the problem are correct. Well done!