Equations

Question 1

Parallax?

Answer 1

d = 1 / p, where d is distance [parsecs], and p is parallax [arcseconds]

This tells you how far away a star is. If you know how much it appears to “wiggle” (its parallax angle p) as the Earth moves around the Sun, this formula tells you the distance in parsecs.
Smaller angle = farther away.

Question 2

What is the relation between flux, F, and apparent magnitude, m?

Answer 2

m₁ - m₂ = -2.5 log (F₂/ F₁)

Question 3

What is the relation between Flux, F and luminosity, L?

Answer 3

L = 4 pi d² F

Question 4

How are apparent and absolute magnitudes related?

Answer 4

m − M = 5 log₁₀(d / 10 pc)

Question 5

Temperature of a star based on its emission

Answer 5

T_eff = ( L / (4 pi R²) ) ^1/4

Question 6

Photosphere radius of a star

Answer 6

R = sqrt(L / (4 pi sigma T ⁴)

Question 7

What is Wien’s Law?

Answer 7

λ_max × T= 2.9 × 10⁻³ m·K

Question 8

What is the equation of hydrostatic equilibrium?

Answer 8

dP/dr = -G × m(r) × ρ(r) / r²

This is the balance of gravity pulling inward and pressure pushing outward inside a star.
Without this balance, a star would collapse or explode.

Question 9

What is the mass continuity equation?

Answer 9

dm/dr = 4 × π × r² × ρ(r)

Question 10

What is the luminosity gradient equation?

Answer 10

dL/dr = 4 × π × r² × ρ(r) × ε

•	L(r) : the luminosity generated within radius r.
•	ρ(r) : mass density at radius r.
•	ε : energy generation rate per unit mass (in watts per kilogram).

Question 11

What is the radiative temperature gradient equation?

Answer 11

dT/dr = - (3 × κ × ρ × L) / (16 × π × a × c × r² × T³)

• a = radiation constant
• c =speed of light
• κ = opacity
• ρ = density

Question 12

What is the ideal gas law for stars?

Answer 12

P = n k T
kKnetic theory (particle number density)

P = (ρ k T)/( μ m_H}
Stellar astrophysics (mass density + particle mass)

P = (R / μ) ρ T
Thermodynamics (molar mass units)

Question 13

What is the formula for mean molecular weight, μ?

Answer 13

1/μ = Σ(Xᵢ / Aᵢ)

Question 14

What is the pressure relation for non-relativistic degenerate electrons?

Answer 14

P ∝ ρ^(5/3)

Question 15

What is the pressure relation for relativistic degenerate electrons?

Answer 15

P ∝ ρ^(4/3)

Question 16

What is the energy released during fusion?

Answer 16

Q = (mass of reactants − mass of products) × c²

Question 17

What is the nuclear reaction rate formula?

Answer 17

r = n₁ × n₂ × ⟨σv⟩

Question 18

What is the power per unit mass formula?

Answer 18

ε = r × Q / ρ

Question 19

What reactions occur in the triple-alpha process?

Answer 19

2 He-4 → Be-8

Question 20

Which stars undergo the CNO cycle?

Answer 20

Stars with mass > 1.3 solar masses (M☉)

Question 21

What is the Virial Theorem?

Answer 21

2U + Ω = 0

Question 22

What is the dynamical timescale?

Answer 22

τ_dyn = √(R³ / G × M)

Question 23

What is the Kelvin–Helmholtz timescale?

Answer 23

τ_KH = G × M² / (R × L)

Question 24

What is the nuclear timescale?

Answer 24

τ_nuc = ε × M / L

Question 25

What triggers a Type Ia supernova?

Answer 25

White dwarf exceeds the Chandrasekhar limit (~1.4 M☉)

Question 26

What happens in a Type II supernova?

Answer 26

Iron core collapses

Question 27

What is the solar neutrino flux at Earth?

Answer 27

~7 × 10¹⁴ νₑ / m² / s

Question 28

Why were fewer solar neutrinos detected than expected?

Answer 28

Because of neutrino oscillations (νₑ → ν_μ / ν_τ)

Question 29

What is the Chandrasekhar mass limit?

Answer 29

Approximately 1.4 solar masses (M☉)

Question 30

What are common polytropic indices?

Answer 30

n = 3/2 for white dwarfs