Astronomy 722: Radiative Transfer and Gas
Dynamics
c
2014
Andisheh Mahdavi
Thornton 326, San Francisco State University
Spring 2016, TuTh 5:10PM
Homework 3 Due 5:10PM 3/3
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. Consider a star of intrinsic brightness Iν∗ embedded in a cloud of gas cloud of radius R that is
much bigger than it. The cloud does not emit or absorb light, but scatters it with a constant
scattering coefficient σν .This problem aims to guide you through the process of calculating the
resulting scattering output.
(a) Describe qualitatively what happens to the light within the gas cloud. Argue that once the star
has been shining for a long time, and the cloud has reached steady-state, then the radiation
field throughout the cloud can be written as Iν (τν , θ) where the dimensionless optical depth for
scattering is given by
dτν = −σν dr
where r is the distance along the line of sight, and θ is the angle that the radiation field makes
with the radial direction at any point. Show that τν = 0 is the surface of the cloud closest to
us, and τν = τν∗ ≈ Rσν is the optical depth to the surface of the star.
(b) Consider the radiation field between us and the star throughout the cloud of gas. Show that
the appropriate radiative transfer equations can be written as
µ
∂Iν (τν , µ)
= Iν (τν , µ) − Jν (τν )
∂τν
where Jν (τν ) is the mean intensity and µ = cos θ. Argue that this equation only needs to be
integrated between τν = τν∗ and 0 when considering the volume between us and the star.
(c) To analytically solve the above equation, we first need a tool that lets us relate Jν and Iν∗ .
Use the Eddington approximation described in Chapter 1 to arrive at an independent guess for
Jν (τν ). To do this, solve equation 1.119a using the boundary conditions 1.123a and 1.123b,
keeping in mind that there is no light entering the cloud at the surface τν = 0 (for eq 1.123a),
but there *is* light entering the cloud from the surface of the star (for eq 1.123b) at τν = τν∗ .
In other words, do not include the star in the source function; rather use eq 1.123b to take its
brightness into account.
(d) Now argue that at the surface of the star, Iν (τν∗ , µ) = 2Jν (τν∗ )H(µ), where H(µ) is the step
function H(µ < 0) = 0; H(µ > 0) = 1.
(e) Use the boundary condition in (d) together with the Eddington-approximated Jν (τν ) in (c), to
solve the equation you derived in (b) for the radiation field. At this point, you are supposed to
do a general solution, i.e., do not use equation 1.112, but solve for the generic I(τν , µ).
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(f) Use the previous answer to estimate the fraction of the star’s intensity that escapes the cloud
directly along the line of sight to the star; i.e., evaluate Iνobs = Iν (0, 1). The answer should
depend on τ ∗ , and should equal Iν∗ when τ ∗ = 0, i.e., when there is no cloud. What is it when
τν = 1? 10?
(g) If I know σν and Iν∗ , and I can measure, Iνobs , what is the radius of the cloud? Once I know the
radius, how can I measure the distance to the cloud?
2. At t = 0, an electron with zero total mechanical energy is falling along a straight line from a distance
of 1 AU towards the Sun. While doing this, it emits light via Larmor radiation. This is a simple
model of a radiatively inefficient accretion flow.
(a) Show that to very high accuracy, the electron’s radiation doesn’t impact its motion, either with
respect to direction, or with respect to the time taken to reach the surface of the Sun.
(b) Solve for r(t), the motion of the electron as a function of time. Calculate the time in seconds
at which the electron hits the surface of the Sun.
(c) The Fourier transform differentiation rule says that if x(t) has a Fourier transform x̂(ω), then ẋ(t)
has a Fourier transform equal to iωx̂(ω), and ẍ(t) has a Fourier transform equal to −ω 2 x̂(ω).
However, this is not always true—explain the general conditions under which these rules fail
and why.
(d) Calculate the total emitted spectrum of the electron, dW/dω, from the moment it passes 1
AU to the moment it hits the Sun. Argue based on part (c) that you cannot use equations
3.25-3.26 to do this calculation, but must directly take the Fourier transform of equation 3.24.
(e) Plot the spectrum in a log-log plot of power per unit frequency vs. frequency. The spectrum
should look like a power law—find its index by examining the plot (or plotting a power law next
to it).
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