Vacuum Brainteaser

Ge 151, Spring 2011

Due: Tuesday, April 19, in class.

1. Qualitatively estimate (and briefly explain your reasoning) the hypothetical effects of polar ice traps on the Moon from the following changes. These effects are either negligible or major. See if you can rank them in order of decreasing importance.
   • Moving the Moon to 0.35 AU
   • Increasing the internal heat flow by a factor of 2
   • Increasing the obliquity to 20 degrees
   • Speeding up the rotation velocity by a factor of 2
   • Increasing the albedo of the regolith from 0.1 to 0.2
2. (a) Before 1965 Mercury was believed erroneously to be in synchronous rotation about the sun, i.e. the length of it's day was equal to that of it's year, 88 Earth days. Estimate the approximate surface temperature one would have expected to find on the permanently dark side under those conditions. (You will need to look up or estimate Mercury's heat flow). (b) At what wavelength does the emission from a surface at that temperature peak? (c) In 1962, the first radio emission observations of Mercury were obtained by a group at the University of Michigan, nearly at Inferior Conjunction. They found a brightness temperature around 250K (with large uncertainties), comparable to that of the Moon. List at least two possible explanations for the anomalously high radio brightness temperature.
3. (a) Find the unidirectional energy flux in space due to the 2.7 K background radiation. (Hint: A blackbody in thermal equilibrium with the radiation field will reach the same temperature as the radiation.)  Express your answer in cgs units.  Note that as the background radiation is (almost) isotropic, the net flux summed over all directions is zero. (b) Galactic stars (other than the Sun) occupy a total solid angle of ~ 10^-14 str as seen from the Solar System.  Assuming a typical stellar effective temperature of 10,000 K (this is weighted towards the blue stars, which radiate most of the energy), what is the unidirectional energy flux from galactic stars? (c) Find the solar energy flux at Earth's orbit, at Neptune's orbit and at a typical Oort Cloud distance of 25,000 AU.
4. A sphere of radius R, at a distance r from the observer (along the z-axis), has a uniform brightness B. The specific intensity is equal to B if the ray intersects the sphere, and zero otherwise. Express the flux F in terms of brightness B and the angle θ_c at which a ray from the observer is tangent to the sphere (i.e. 2 θ_c is the angle subtended by the sphere as seen from the observer).  Show that on the surface of the sphere F=πB.