1. The K-correction

Quantitatively, the K correction is written as m-M = 5*log(d

If you are observing through a filter that transmits from lambda

Show that the K-correction is given by

2. Galaxy counts

- Using Figure 1 from Metcalfe et al 1996,
calculate the logarithmic slope (dlogN/dm) of the galaxy number count function over the
magnitude range m
_{B}=21-26 for the following:

- the observed number counts;
- no evolution models.
- evolution models for q
_{0}=0.5 and q_{0}=0.05.

- Using this dataset of counts from
the Hubble Ultra Deep Field, create your own version of the
Metcalfe plot, and calculate the slope of the observed galaxy number
count function over that same magnitude range. Make sure to cut out
stars (objects with "stellarity" > 0.8 or so). Note that the UDF
covers
9 square arcmins of the sky.

- Discuss the comparison of your plot with that of Metcalfe etal, in the following terms:
- overall match in the curve
- match of the slopes
- evidence for galaxy evolution

- First, ignore K-corrections and luminosity evolution. Plot the apparent magnitude of the galaxy as a function of redshift for the following three universes (use H0=70 km/s/Mpc throughout):
- Standard CDM (OM=1, OL=0.0)
- Open CDM (OM=0.3, OL=0.0
- Lambda CDM (OM=0.3, OL=0.7)
- Now for the Standard CDM universe only, plot the apparent magnitude of the galaxy as a function of redshift adding a K-correction model that assumes the galaxy has a spectral slope of alpha=-2.0, a reasonable value for a red galaxy.
- Again, for Standard CDM only, assume that M87 was formed instantaneously at a redshift z=5 (how old was the universe at that redshift?), and has evolved passively (ie, just fading over time) according to the expression dM=2.18*log(galaxy_age_in_years)-21.65, where dM=M-M(today), and the M's refer to the absolute magnitude of the galaxy. This expression is a reasonable approximation to the luminosity as a function of time for fading burst of star formation. For the SCDM cosmology only, plot the apparent magnitude of the galaxy as a function of redshift ignoring K-corrections, but using the passive evolution model.
- Explain in words how the evolution correction and the K-correction each changed the curve for SCDM, and why that makes sense given our model for the galaxy.
- Now plot the log of the angular size of the galaxy (in arcsec) as a function of redshift in the three different cosmologies.
- Finally plot the surface brightness as a function of redshift for the three different cosmologies, as well as for the K-corrected and evolution-corrected SCDM cosmology. Explain why the different curves look the way they do! If the V-band surface brightness of the night sky is 21.8 mag/arcsec^2, at what redshift does M87's average surface brightness drop below the sky brightness? below 1/10 of the sky brightness? below 1/100 of the sky brightness?

- A plot of the apparent magnitude of the galaxy as a function of log(z) under two cases: 1) using Hubble's Law and 2) using the correct cosmological luminosity distance. At what redshift does using Hubble's law introduce a magnitude error of 0.1 magnitudes? At what redshift is the magnitude wrong by 0.5 magnitudes?
- A plot of log(apparent size) versus log(z), again plotting both the Hubble's Law case and the proper cosmological case on the same plot. At what redshift does Hubble's law introduce a size error of 10%? At what redshift does Hubble's law introduce a size error of 50%?

Possibilities would include:

- Using the Lyman Alpha forest to constrain cosmology
- The Evolution of the Cosmic Star Formation Rate with Time
- Polarization of the CMB
- etc