HW1

1)
a.
Estimating the maximum angular separation of Earth and Jupiter from the Sun at a distance of 10     parsecs: Earth's Aphelion is 1.01673 AU, Jupiter's is 5.45492 AU.1 AU at 1 pc is 1", so scaling proportionally, these are (to excessive significant figures) 0.101673" and 0.545492" from the Sun. Earth is questionable with the nominal HST resolution of 0.05" (the star and planet would be adjacent), but Jupiter (some 20 pixels away) could be distinguished.
b.
Using the planet-to-star flux ratio approximation R_p²/a²,

Planet	Radius (m)	Semi-Major Axis (m)	Flux Ratio	Magnitudes Fainter
Earth	6.371e6	1.49598023e11	1.814e-9	21.9
Jupiter	6.9911e7	7.78299e11	8.069e-9	20.2

2)
The problems of defining a planet at the high mass end are how they become increasingly star-like. Newly formed brown dwarfs are notable for having effective temperatures (few thousand kelvin), compositions (hydrogen-rich), and densities very close to still-forming red dwarfs. Possible solutions include looking at the spectra for deuterium, helium-3, and lithium abundances (higher in brown dwarfs than stars, at least once the stars have had time to burn those elements/isotopes).

3)
a.
Using Kepler's 3rd law in its full Newtonian glory:
T² == 4*pi²/(G*(M1+M2))*a³, or T == 2*pi*a^1.5/sqrt(G(M1+M2)).
For the stars (a == 1200 AU, M1 == M2 == 0.9 M_sol), T == 9.78e11 s, or 31000 (Julian) years.
b.
For the planet, T == 1.004e7 s, or 0.318 (Julian years), so Tstar/Tplanet ~= 97400 orbits

4)
Using Kepler's 3rd law in simplified form:
T² == a³,  a == T^2/3 == (120/365.25)^2/3 == 0.47613 AU
Periastron == a(1-e)
e == 1 - Periastron/a == 0.36992 ~= 0.37

5)

 I would not expect the decreasing density vague power law guess (mass ~ radius^2.5) to hold for higher masses (topping out at perhaps 2x Jupiter's radius even with >10x Jupiter's mass), given predictions for brown dwarf sizes and the relatively small radii of low mass red dwarfs.