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Astro 405/505, fall semester 2005
Additional problems
Problem 1: plasma
You intend to study a gas cloud that collapses to form a star. Suppose the cloud has a radius R = 10^15 cm, a temperature T = 1000 K, a density of neutral atoms nn = 10^12 cm−^3 , and a density of ionized atoms (and electrons) ne = 10^8 cm−^3.
Explain with what technique you would describe the collapse.
Problem 2: emission spectra
Suppose an emission region is homogeneous and the radiation coefficients have the following form: jν = A ν−^1 αν = B ν−^3 σ = 0 A, B = const.
Calculate the angular distribution of the intensity, Iν , that would be seen by an observer at a fixed position at a distance D from the emission region, and the total flux.
a) Assume the emission region has the form of a cube with sidelength R, that is observed exactly along the normal of one of the sides.
b) Assume the emission region has the form of a sphere with radius R.
Problem 3: stellar emission
The conditions in stellar photospheres are often well represented by a Local Thermodynamic Equilibrium (LTE). Neglecting scattering please derive a quantitative estimate for the angular distribution of the intensity, Iν , that would be seen by an outside observer.
a) Assume the temperature T is constant, independent of the vertical optical depth, which itself is assumed independent of frequency.
b) Assume the temperature is constant in each of three zones with values
T (τz) =
T 0 0 ≤ τz ≤ τ 1 2 T 0 τ 1 ≤ τz ≤ τ 2 4 T 0 τ 2 ≤ τz
Problem 4: pinhole camera
A pinhole camera consists of a small circular hole of diameter d on the front side of a box, which is at a distance L from the film plane on the back side of the box. If θ denotes the angle to the optical axis (perpendicular to the front and back planes of the camera), show that the flux at the film plane depends on the intensity field Iν (θ, φ) as
Fν =
π cos^4 θ 4 f 2
Iν (θ, φ)
Problem 5: radiation transport
Consider a cloud of gas in LTE with temperature Tg and diameter and thickness D (box geometry). Assume emission and absorption processes for continuum and lines to operate in the cloud.
Derive the solution of this radiation transport problem and discuss it for the limiting cases of very large and very small optical depth.
Suppose a background star with Teff Tg was located behind the gas cloud. What would the spectrum of the star look like?
Problem 6: Review questions
a) What conservation laws do the hydrodynamical equations describe?
b) What is the difference between intensity and flux?
c) How would you estimate the level occupation density of hydrogen in the solar photosphere and in a diffuse interstellar gas cloud? Can the two be treated the same way?
d) Why do accretion disks form and how do they help the accretion of matter on compact objects?
e) What is the color of blackbody radiation?
f) What can you say about the directionality of the radiation field, if the scattering optical depth τσ =
∫ (^) x 0 ds σ(s)^ is very large, and absorption is inefficient?
