More derivations

Question 1

how would you find a relation between the fractional density change and fraction radius change

Answer 1

use:
ρ =M/ [4/3 πR³]
dρ/dr = m/[4/3 π] * (-3/r⁴) = -3ρ/r

hence:
dρ/ρ = -3 dr/r

Question 2

how would you find the total velocity change over a Hohmann transfer

Answer 2

Vᵢ = Vₐ - V₂ (V₂: velocity of circular orbit at the aphelion distance)
Vf = V₁ - Vₚ (V₁: velocity of circular orbit at the perihelion distance)

ΔV = Vᵢ + Vf