Astro 300B

Fall 2011

Homework #3

DUE: Friday, Feb. 4, 2011

  1. Limb darkening in the Sun. (10 points). Consider a semi-infinite, plane-parallel slab of gas. Determine the emergent intensity as a function of direction from the normal, I(), where is the anglebetween I and the normal. Assume the source function, S() = a + b , where  is the optical depth into the gas normal to the surface.
  1. Opacity. (10 points). Calculate how far you could see through Earth’s atmosphere if it had the opacity of the solar photosphere. Assume the Sun’s opacity is κ = 0.264 cm2g−1, and the density of the Earth’s atmosphere is 1.2×10−3g cm−3.
  1. Equilibrium Temperature of the Planets. The temperature of objects in the solar system can be estimated by assuming that they absorb radiation from the Sun, and then radiate like blackbodies, at temperatures which turn out to be such that the peak of the black body typically is in the infrared.

a. (10 points) Assume that planets reflect a fraction of the incident solar radiation, A, called the albedo, and absorb the rest, and that they are rotating rapidly enough that the temperature is the same over the entire surface – the thermal conductivity is infinite.

Derive an expression for the equilibrium temperature achieved, as a function of distance from the Sun.

b. (10 points) Calculate the equilibrium temperature of Mercury, the Earth, Neptune and Pluto, under these assumptions, assuming A=0.1. Compare what you get with the measured average temperatures of 533 K, 288 K, 72K and 44K, respectively.

c. (10 points) Now suppose that the thermal conductivity is zero, for a non-spinning planet at a distance of 1 AU from the Sun, with a radius equal to the radius of the Earth. Derive an expression for the temperature as a function of location on the planet defined as the polar angle from the subsolar point. Sketch an isothermal region of the surface. (seeBradt, Astrophysical Processes, question 6.24). What is the temperature at the sub-solar point  = 0, at the tangent point  = /2, and at the latitude of Tucson? Ans: 400 cos1/4

d. (10 points) Now assume zero thermal conductivity and a rapidly spinning planet with spin axis normal to the planet-sun line. Show that the temperature depends on as 300 cos1/4Again, sketch an isothermal temperature region.

What is the temperature at the equator (=0) and =45 degrees?

e. (5 points). Which of the models described in (a) to (d) apply to the Earth, and to the Moon?