Derivations

Question 1

pressure gradient of the atmosphere derivation (and thus the scale height)

Answer 1

dP/dr = -g*ρ(r)
ρ(r) = Nm̃/V (m̃: molar mass, e.g: 28u)
sub in V= NkT/P

hence dP/dr = -gm̃/kT * P

for the scale height, rearrange and solve the differential equation:
∫ 1/P dP = -gm̃/kT ∫ dr
pressure limits are P₀ –> P and radius limits are R –> R+h
R: planets radius.

then equate that to P₀*e^(-h/h₀) where h₀ is the scale height
h₀ = kT/gm̃

Question 2

tidal force derivation

Answer 2

dF = GMm/(r+dr)² - GMm/r²
= GMm/r² * [(1+dr/r)⁻² -1] (expand as dr/r is small)
= -2GMm/r³ dr

draw the satellite (as a circle). the radius is R. r is the distance from the centre to the planet (which is a dot). s is the distance from the planet to a point on the satellite’s circumference, P. θ is the angle between r and R. φ is the angle between s and r.

ΔF = Fₚ - F꜀
= (GMm/s² cos(φ) - GMm/r²)î - GMm/s² sin(φ)ĵ

s² = (r-Rcos(θ) )² + (Rsin(θ) )²
simplifies to s² = r²[1-2R/r cos(θ)] (using R/r is small so R²/r² is negligible)

φ is small so cos(φ) =1 and sin(φ) = R/r sin(θ)

sub those expressions in to ΔF and simplify. remember R/r is small so 1 + R/r cos(θ) =1

this gives ΔF = GMm/r³ R [2cos(θ)î - sin(θ) ĵ ]

Question 3

Number of daughter nuclei derivation

Answer 3

Number of nuclei is conserved:
Nai + Nbi = Naf + Nbf
hence: Nbf - Nbi = Nai - Naf

sub in Naf = Nai * e^-λt

Nbf - Nbi = (e^λt -1) * Naf

Question 4

Derive Newtons gravitaional law from Kepler’s 3rd

Answer 4

T² = K*r³

sub in T = 2πr/v
rearrange for v²/r then multiply both sides by m
v²m/r = F (centripetal force)

hence F = GMm/r²

Question 5

derive the Roche limit

Answer 5

Gravitational force acting on a rock on the moon’s surface
F = GMₘm/Rₘ²

Tidal force on the rock (at θ = 0):
Fₜ = 2GMₚmRₘ/d³ (d is the planet-moon separation)

the rock is ripped from the moon if Fₜ > F
just rearrange that for d.
Sub in M = 4/3 πR³ρ (for both Mₘ and Mₚ) to get:
d < 2^(1/3) * Rₚ * (ρₚ/ρₘ)^(1/3)

Question 6

derive the optical depth equation

Answer 6

assume the atmosphere removes a fraction of light, K, per meter

dI = -KIdL
dL: length element
I: Intensity

1/I dI = -K dL
I/I₀ = e^(-KL)
I = I₀ * e^(-KL)

compare to I = I₀ * e^(T)
hence T = KL
T: (tau) is the optical depth

Question 7

derive the hydrostatic equilibrium equation

Answer 7

Imagine a shell of thickness dr
outward force: P(r) * dA
Inward force: [ P(r) + dP/dr dr] dA

Net force (Outward - Inward) = F = -dP/dr dr dA
F = Gm(r)dm/r² and dm =ρ(r)drdA

sub those in an rearrange to get:
dP/dr = - Gm(r)ρ(r)/r²

Question 8

derive the decay equation

Answer 8

rate of decay is proportional to the number of nuclei

dN/dt = -λN

rearrange and solve to get:
N = N₀*e^(-λt)

Question 9

derive the free-fall timescale

Answer 9

Use conservation of energy:

-GMm/R = ½mv² - GMm/r

rearrange for v:
v² = 2GM(1/r - 1/R)
v = (2GM)^½ (1/r -1/R)^½
dr/dt = -(2GM)^½ (1/r -1/R)^½(negative cause r is decreasing)

solve the integral:
∫ dt = -(2GM)^-½ * ∫(1/r - 1/R)^-½ dr
limits of dt integral are 0 –> t(sub ff).
limits of dr integral are R –> 0

to solve the integral:
(1/r - 1/R)^-½= (Rr/R-r)^½
and use substitution r=Rsin²(θ)

finally, sub in M = ρ * 4/3 πR³

this gives t(sub ff) = (3π/32Gρ)^½