AST 307 · Introductory Astronomy Fall 2003

AST 307
Problem Set #11
Due Friday Dec 5

1. In the central regions of galaxies, the mass is distributed fairly uniformly (except near the central black hole). That is, the density is approximately constant.
a) How does the mass within a radius r depend on the density and r?
b) How does the orbital speed depend on r? Make a sketch of the rotation curve of this region of a galaxy.
c) How does the orbital period depend on r?
d) The rotation pattern in this region is referred to as solid-body rotation. Why?
e) If we were in this region of the Milky Way, what pattern of Doppler shifts would we observe for stars around us?

2. Delta Cephei was the first Cepheid variable star to be recognized. It varies in brightness with a period of 5 days, has a mean apparent magnitude of 4, and a parallax of .003 arcseconds.
a) What is the distance to delta Cephei?
Figure 24.10 in your book shows a Cepheid variable star in the galaxy M100 with a period of 50 days and a mean apparent magnitude of 24.
b) Assuming the period-luminosity relation for Cepheids is linear (or that period is proportional to luminosity) what magnitude would you expect a Cepheid variable in M100 with a period of 5 days to have?
c) What is the ratio of the flux (or apparent brightness) of delta Cephei to the flux of a Cepheid in M100 with a 5 day period (like delta Cephei).
d) From your answer to part c, what is the ratio of the distance to M100 to the distance to delta Cephei?
e) From your answer to part d, what is the distance to M100?
f) The redshift (delta lambda / lambda) of M100 is .004.
What is the speed of M100 relative to the Milky Way?
g) From these numbers, what is the Hubble constant?
(Your answer won't agree with the best number for Hubble's constant. Part of the reason is that the two Cepheid variable stars I chose may not be typical.)

3. The temperature of the gas in the Universe at the time that the microwave background radiation was emitted was about 3000 K.
a) If the wavelengths of the background radiation were stretched by the expansion of the Universe, by what factor have distances in the Universe been stretched since the radiation was emitted?
By what factor have volumes been stretched?
b) The average density of ordinary matter in the Universe (not including dark matter) is currently about 10^-28 kg/m^3.
What was the density of the Universe at the time the background radiation was emitted?
c) Convert your answer in part b to protons per cubic centimeter, and compare it to the average density of the gas in the Milky Way (which you can find in your book).

26 November 2003
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