AY21 Assignment #2Due by 5pm on Thurs. Feb 4, in Swarnima Manohar’s mailbox in Cahill 249..(1) The controversy over the distances to the so–called “spiral nebulae” was settled by EdwinHubble in 1924 when he successfully resolved individual stars in M31 (The Andromeda Galaxy).If one were to observe a Cepheid variable star in M31 that turns out to have a light curvevariation of 60 days and the average visual magnitude of the star is hmVi = 18, use theperiod–luminosity relationhMVi = −1.35 − 2.78log10P,to determine the distance to M31. Here P is the period of the Cepheid in days and hMVi isits average visual magnitude. Using the Hubble Space Telescope, it would be reasonable toobtain measurements of Cepheids as faint as mV≃ 26. What is the approximate distance outto which Cepheids of similar luminosity to that above could be used for distance measures?(2) Suppose you have a “standard candle” which (somehow) you have established has an intrinsic“scatter” of 0.35 magnitudes (1 sigma), and that it can be used successfully to measure relativedistances (radial velocities) out to cz = 10, 000 km s−1. How many measurements of suchobjects, over the whole sky, would be required to verify that the volume probed is “at rest”with respect to the cosmic mi cr owave background (the net error in velocity should be smallerthan 150 km s−1). How many measurements of the same objects would be necessary, i nprinciple, to measure H0to ∼ 10% accuracy if the volume is not subject to large scale flows orsignificant peculiar velo ci ti es? State all of your assumptions–there is no exact “right” answer,but your reasoning is important. Comment on why, in the real world, things aren’t so easy.[Hint: remember that the direction of peculiar velocities will not always be along our line ofsight...](3) In the attached figure are the velocity profiles of a few spiral galaxies measured in the 21cmline of neutral H. The abscissae have been converted to the radial velocity as seen by us.The Tully–Fisher relation can be expressed asMB= −4.42 − 6.15log10∆v,where ∆v i s the velocity width of the galaxy profile i n km s−1, and MBis the absolute bluemagnitude of the galaxy.(a) Assume that the galaxies NGC 1241 and NGC 5248 have apparent blue magnitudes of mB= 14and mB= 12, respectively. Calculate the distances to these two galaxies. From these and theirradial velocities, estimate the value of the Hubble constant. For simplicity, also assume thatthe galaxies are seen edge–on (inclination angle i = 90◦, so that the velocity width correspondsto the maximum rotational velocity.(b) In most cases, the galaxies observed are not seen edge–on. How do the results obtained in part(a) depend on the inclination angle i?(c) Qualitatively, try to explain the shape of the observed velocity profiles (e.g., why do theyhave these “horns” at the extrema?) [Hint: think about the quantity of gas that will have aparticular radial velocity](4) The co–moving volume out to a redshift z is the volume calculated at the present time enclosedby a 3–sphere centered on Earth and passing through sources from which sp ectr al lines would1AY21 Assignment #2have a redshift z. Calculate the co–moving volume in Mpc3(assuming that H0= 100h) foran Einstein–de Sitter Universe (i.e., Ω = 1) out to z = 1,3,∞. Use a Schechter function as i nproblem set #1 to estimate the number of galaxies brighter than 0.1L∗with redshifts z < 3.The actual number is observed to be ∼ 3 × 1010. What do you think this result
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