Blog Post 27, Worksheet 8.1, Problem 2

2. In 1929, astronomer Edwin Hubble discovered that almost all distant galaxies exhibit a positive redshift. Furthermore, it appeared that the farther the galaxy, the larger its redshift. Here we will rediscover Hubble's Law using modern spectroscopic data and supernovae Ia as a our distance indicator to these galaxies. The data we will use come from the Sloan Digital Sky Survey (SDSS), a project that aims to comprehensively map the universe. You can access the relevant data products for this exercise at http://goo.gl/fmIvqc

(a) Below is a list of supernovae observed between 2004 and 2007 and their positions in RA and Dec. You can find the images and spectra of their host galaxies by entering their coordinates in the respective fields. Explore the functions available, including magnifying the image, reading off the photometric measurements (magnitudes in wavebands u, g, r, i, z) of your selected object, and using the 'Explore' button to access more quantitative measurements for these objects. In particular, familiarize yourself with the 'interactive spectrum' feature. 

(b) One of the features for determining distances to Type Ia supernovae is its peak absolute magnitude. You explored the peak bolometric luminosities of SN Ia's in Worksheet 7.1. The peak V-band magnitude for SN Ia's is about -19.3. Use the apparent peak magnitude given in the table above to calculate the distance of these supernovae in unit of Mpc. 

The distance modulus gives the relationship between apparent magnitude (m), absolute magnitude (M), and distance (d): \[m = M = 5\text{log}(d) - 5\] \[d = 10^{\frac{m-M+5}{5}}\] Using an absolute magnitude of M = -19.3 and the apparent magnitudes given, we get the following distances:

Supernova	Right Ascension (RA)	Declination (dec)	extinction-corrected V-magnitude	Distance (Mpc)
SN 2004hu	18.76	0.263	17.28	207
SN 2005gi	13.97	0.505	17.38	216.7
SN 2006rz	56.528	0.39	16.29	131.2
SN 2007hx	31.614	-.899	18.32	334.1

(c) We can use the absorption or emission lines of the host galaxy to find their redshifts which, as you found in Question 1), roughly equals the recessional velocity as a fraction of the speed of light. To measure the redshift to each host galaxy, click on 'Explore' and then on the link 'Interactive Spectrum.' Uncheck the boxes Best Fit and Mark Emission Lines. Zoom in on the absorption line labeled \(H \alpha\), and move your mouse over to the center of the line to read the observed wavelength in Angstroms. The \(H\alpha\) has a rest (i.e. emitted) wavelength of 6563.0 Angstroms. Calculate the redshift, and then derive the radial velocity in kilometers per second, using the relation z = v/c. How close does your redshift measurements compare to the one SDSS reports in the table under the Interactive Spectrum link? Repeat for all galaxies. 

We can find the redshift for each galaxy by using the redshift formula, \[z = \frac{\lambda_{\text{observed}}-\lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}\] We know that \(\lambda_{\text{emitted}}\) for each supernova should be 6563.0 Angstroms, so to find the redshift, we plug in \(\lambda_{\text{emitted}}\) = 6563.0 Angstroms, and \(\lambda_{\text{observed}}\) = the observed wavelength in Angstroms as shown in the table.

To find the velocity, we multiply the redshift by the speed of light: \(v = cz\).

Supernova	Observed Wavelength	Redshift	Reported redshift	Velocity (km/s)
SN 2004hu	6878.599	0.04809	0.047806	14427
SN 2005gi	6897.632	0.05099	0.05078596994	15297
SN 2006rz	6767.06	0.03109	0.0309035	9327
SN 2007hx	7085.981	0.07967	0.0794432	23901

(d) Make a plot of your findings, with distance on the x-axis and velocity on the y-axis. Report the slope of your line in appropriate units. This is the Hubble Constant, \(H_0\). 

\[H_0 = 72.1 \frac{\text{km/s}}{\text{Mpc}}\]

(e) Write an equation for this line in the form of v = __ D, where v is an object's recessional velocity and D is the distance to that object. Express your Hubble Constant in terms of units km/s/Mpc. Congratulations, you have arrived at Hubble's Law!

This question just asks us to plug in the Hubble Constant we found above as the slope for this line: \[v = H_0 D = (72.1 \frac{\text{km/s}}{\text{Mpc}})D\]