1. [50 pts] For this problem set, you will simulate the effects of Malmquist bias using a computer program to generate the sample and to select galaxies from the sample. You may find it useful to link to Numerical Recipes to generate random numbers.
(a) Assuming a Gaussian form for the luminosity function (say M0 = -19.0, sigma = 2.0) and a constant space density, simulate the effects of apparent magnitude limits at m= 12.0, 14.0, and 16.0. n = 10,000 to 100,000 is a good sample size.
(i) Use supermongo or IDL to plot M vs distance (Mpc) for each limiting apparent magnitude. Plot the data out to 100 Mpc. (NOTE: you can either link your program directly to supermongo, or use supermongo or IDL interactively after writing the data to a file).
(ii) Plot N(M) using a reasonable bin size for the galaxies which satisfy the sample selection. What is the mean absolute magnitude for the galaxies in each of your samples? What is the average distance to the galaxies in each of your samples? For the local volume (d < 30 Mpc), to what absolute magnitude are your samples complete?
(b) Now assume that you have some parameter P such that M = C1 * P + C2. [Note: This is similar to the functional form of the Tully-Fisher Relation, if P = log(width) - 2.5; C1= -7.27; C2 = -20.] Compute "true" values for P by assuming C1 and C2 and then add some random Gaussian distributed noise such that the corresponding scatter in M is 0.3 mag (i.e., sigma(P) = 0.3/C1).
(i) Plot M vs P.
(ii) Compute the predicted absolute magnitudes < M > = C1 * P + C2 and
plot the difference
(M - < M >) vs M, P, and distance.
(iii) Now perform least-squares fits on the magnitude limited samples and show that the slope recovered is (within the errors) the correct one if the minimization is done in P but incorrect if done minimizing in M. Make a table of the slopes and show the fitted solutions on M vs P plots.
(iv) Use the best-fit slopes and intercepts to compute the predicted values
of M and plot the difference
(M - < M >) vs M, P, and distance for
any one of the magnitude limited samples but for both the correct
(minimizing in P) and incorrect (minimizing in M) slopes and intercepts.
(v) Briefly summarize your findings.