ASTR/PHYS 328/428 HW #2
1. The K-correction
Quantitatively, the K correction is written as m-M = 5*log(dL) - 5 + K(z)
If you are observing through a filter that transmits from lambda1
to lambda2, the flux in the bandpass from a distant object
is given by
where Llambda is the luminosity per unit wavelength, and te
refers to the spectrum at time of emission.
Show that the K-correction is given by
If a galaxy emits a spectrum (where Lnu is the
luminosity per unit frequency) show that and that the k correction can be written simply as
Since quasars typically have a spectrum with alpha ~ 1, this gives
them a negligible k-correction!
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 mB=21-26 for the following:
- the observed number counts;
- no evolution models.
- evolution models for q0=0.5 and q0=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
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
3. Redshifting M87M87 is a giant elliptical galaxy in the Virgo Cluster. Use the NASA Extragalactic database to figure out the absolute V magnitude, physical size (radius in kiloparsecs of the mu=25 mag/arcsec2 isophote), and average surface brightness (in mag/arcsec2) of the galaxy. Now imagine redshifting it to higher and higher redshift and put the following curves all on one plot, plotting redshift from z=0 to z=2.
- 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?
4. ASTR 428: Hubble's Law vs Cosmological DistancesIn the very local universe, we often will use Hubble's Law (v=H0d) to get distances to galaxies using their observed recession velocity. But we've seen in class that distances take on different definition on cosmological scales. So let's see how far out in redshift we can go using Hubble's Law before the errors introduced by ignoring cosmological effects become big. Let's assume we are observing a galaxy with an absolute magnitude of MV=-21 and a size of r=20kpc. Adopt a cosmology of H0=72 km/s/Mpc, OmegaM=0.3, OmegaL=0.7 and make a plot of the following as a function of log(z) for log(z)=-2 to 0 (so z=0.01 to 1.0):
- 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%?
5. ASTR 428: Project
Pick two possible presentation topics, and give me a one page
written description for each. They should be focused on observational
cosmology and/or structure/galaxy
formation, and something that we aren't going to talk about in class.
They also can't be your own research/thesis topic.
I will look over your possibilities and recommend one for you to follow
Possibilities would include:
You will be asked to write a 5 page critical summary of the topic (due Dec 1), and give a ~ 30 minute in-class
presentation to the class (on Dec 8).
- Using the Lyman Alpha forest to constrain cosmology
- The Evolution of the Cosmic Star Formation Rate with Time
- Polarization of the CMB