Physics 722: Radiative Transfer and Gas
Dynamics
c
2016
Andisheh Mahdavi
TH 326, San Francisco State University
Spring 2016, TuTh 5:10PM
Homework 4 Due 3/31
While I may have consulted with other students in the class regarding this homework, the solutions presented
here are my own work. I understand that to get full credit, I have to show all the steps necessary to arrive
at the answer, and unless it is obvious, explain my reasoning using diagrams and/or complete sentences.
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1. You observe the hot gas in a cluster of galaxies with an X-ray telescope. The gas distribution looks
roughly spherical with a radius of 1 Mpc. You find a total X-ray luminosity of 1045 erg/s and a
constant temperature of kT = 15 keV.
Assume that the cluster is in hydrostatic equilibrium:
∇P = −ρg ∇Φ(r)
where P (r) is the ideal gas pressure, ρg (r) is the gas density, and Φ(r) is the gravitational potential.
None of these functions is constant.
Assume that the gas in the cluster makes up the bulk of the luminous matter, and that it consists of
a uniform, thermal plasma made of 75% hydrogen and 25% helium by mass. There are also enough
free electrons to make the whole cluster electrically neutral. The emission process in the cluster is
thermal Bremsstrahlung. The frequency-averaged Gaunt factor at 15 keV is 1.31.
(a) (10 points) Under the ideal gas law, we know that P = nkT , where n is the number density
of all free particles in the system (n = ne + nH + nHe ). The ratio of the gas density to the
number density is then ρg /n = µmp , where mp is the proton mass. Find the numerical value
of the constant of proportionality µ for this particular cluster. We call this the mean molecular
weight. Also write down the ratio of the number density of electrons ne to the number density
of ions ni .
(b) (20 points) Consider the Cauchy-Schwarz integral inequality
2
Z
f (x)g(x)dx
≤
Z
2
f (x) dx
Z
g(x)2 dx
Use this inequality together with the Bremsstrahlung assumption to determine an upper limit to
the total gas mass of the cluster. This limit is purely mathematical and so valid even without
the assumption of hydrostatic equilibrium.
(c) (20 points) Assume the gas makes up 100% of the mass of the cluster. Integrate the hydrostatic
equilibrium equation to solve analytically for ρg (r) (you will get a nonlinear second order differential equation, but instead of trying to solve it directly, try an educated guess at the solution).
This will give you the hydrostatic mass of the cluster. Show that this mass is in violation of the
upper limit derived in (3), i.e., that there must be dark matter in the cluster.
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(d) (10 points) Now assume that the gas fraction is everywhere constant, i.e.
ρg (r) = fgas ρtotal (r)
Solve for fgas . What fraction of the cluster is dark matter?
2. Consider a spherical star. In its (primed) rest frame, has radius R0 and is rotating relativistically with
angular frequency ω 0 about the x0 axiis; it emits blackbody radiation with temperature T 0 . In your
frame, you observe the star from a distance D >> R along your z axis. Figure 1.6 in the book is
the relevant schematic, such that lines of constant θ are also lines of constant latitude on the star.
(a) (10 points) Let your position be (x, y, z) = (0, 0, 0) and the star’s position be (x, y, z) =
(0, 0, D). You observe the star in angular coordinate φ, θ, such that θ = 0 points towards the
star’s equator and φ goes from 0 to 2π. Let α0 , δ 0 be the longitude and colatitude on the star
in its rest frame, such that δ 0 = 0 lies along the x axis. Argue that α0 = α and δ 0 = δ despite
length contraction, and relate the observer angles (φ, θ) to α and δ. You may use D >> R to
simplify the algebra.
(b) (20 points) Find Iν (φ, θ), the intensity you observe as a function of the angles from the line of
sight. For this problem it will be useful to know that Iν /ν 3 is a Lorentz invariant (i.e., has the
same value in all reference frames; see section 4.9 for details).
(c) (10 points) Use the ParametricPlot3D function in Mathematica to plot a picture of the star
color [approximated as the peak frequency of Iν (θ, φ)] as observed by you; you may choose any
temperature and angular frequency you wish, but you need to make sure all the colors of the
rainbow are clearly visible.
over