Homework 2

1)
As the planet mass is ~13% that of the star, it matters considerably.
velocity:

r is a fraction of the semi-major axis inversely proportional to relative mass:

Plugging in the numbers, V is 98.75 m/s. A speed of almost 100 m/s seems quite feasible.


2)
The planet orbiting VB10 was discovered by looking for periodicities in astrometric data. It was refuted by the suggested parameters (and a large portion of the physically plausible parameters) being ruled out by improvements in the radial velocity technique. The RV refutation was helped by the astrometric orbital parameters indicating a near edge-on inclination.

3)
a.
For the astrometric method, planet mass (as a fraction of stellar mass) shows an inverse relationship with semi-major axis. For the radial velocity method, it increases with the square root of stellar mass and semi-major axis.

b.
The cross-over depends strongly on stellar mass and distance, ranging from ~1.5e11 m (1 AU) for the plausible near/low mass case to ~3e12 m (20 AU) for a more distant and likely overly high mass star. (The upper left portion of the second graph should probably be considered unphysical)

4)
 Starting with the Planck equation, a spherical star, and a spherical grain in thermal equilibrium (like the in-class derivation, but with variable opacity):

Evaluating the integral in Wolfram Alpha yields:

Substituting:

(The equation given has 2 instead of sqrt(2), but I think it was described as a typesetting error?)

5)
https://github.com/pdn4kd/freezing-tyrion/blob/master/HW2-5.py