Astr/Phys 328/428 Homework #1

1. Volume of the Universe

Step 1: Find the volume of the Universe out to a comoving distance r. Using the Robertson-Walker metric, we can define the differential volume element as

which means the volume out to a comoving distance r is given by

Solve to get an analytic expression for V(r) for k=-1,0,1.

Step 2: Find the relationship between comoving distance r (which you've calculated the volume for, but is unobservable) to redshift z (which is observable).

Start by using the R-W metric and integrating along the path of a light ray.  Do this exercise using a flat, matter dominated OmegaM=1 universe, remembering that R(t)~t2/3 in such a universe. While this cosmology is certainly not the correct one, it is a useful "benchmark" to compare to (and it is analytically tractable!). This should give you a relationship like this: r=f(R,t0). Then use the Lemaitre equation and the expression for t0 to turn it into r=f(z,H0).

Step 3: Combine the relationship in steps 1 and 2 to show that

2. Lookback Times and Age Constraints

Use either the astropy cosmology package or Ned Wright's Cosmology Calculator for this exercise.
3. The Flatness of the Universe

Starting with the Friedmann equantion for a Lambda=0 universe, show that the Hubble parameter can be written as:

Helpful hints:

Then use that to show that

Helpful Hints:

So we see that Omega varies with time. Now do the following:

4. Distant objects

Do this calculation twice. Once analytically for a OmegaM=1, OmegaL=0 universe (show your work!), and then use the astropy cosmology calculator to get the values for a OmegaM=0.3, OmegaL=0.7 universe. Compare how the values differ in the different universes.

One of the most distant radio galaxies is 8C 1435+63, at a redshift of z=4.25. Answer the following:

ASTR 428 additional items:

  1. Read the following (books on reserve in the Astronomy Library):
  2. A star cluster is made out of 106 solar-type stars, and is located 12 kpc from the Sun. It has a radius of 5 pc. Calculate the following:
  3. The giant elliptical galaxy NGC 4874 sits at the center of the Coma cluster, has an apparent V magnitude of 11.73, and a B-V color of 0.87. Its effective radius is about 32 kpc. The recession velocity of the Coma cluster is 7190 km/s. Calculate the following for NGC 4874:
  4. Using the relationship between magnitude and flux, show that for small uncertainties, the uncertainty in magnitudes is essentially the same as the relative flux error. In other words, show that sigma(mag) ~ sigma(f)/f.