Formulas

Celestial Mechanics & Kepler’s Laws

Kepler’s Third Law (General Form)
P² = (4π² / G(M₁ + M₂)) * a³
P Orbital Period
G Gravitational Constant
M₁ M₂ Masses of the Two Bodies
a Semi Major Axis

Kepler’s Third Law (Solar System Approximation)
P² ≈ a³
P Orbital Period in Years
a Semi Major Axis in AU

Orbital Velocity (Circular Orbit)
v = √(GM / r)
v Orbital Velocity
G Gravitational Constant
M Mass of Central Body
r Orbital Distance

Escape Velocity
v_esc = √(2GM / R)
v_esc Escape Velocity
G Gravitational Constant
M Mass of the Body
R Radius of the Body

Newton’s Law of Universal Gravitation
F = G * (M₁ * M₂ / d²)
F Gravitational Force
G Gravitational Constant
M₁ M₂ Masses of the Two Objects
d Distance Between Centers

Vis Viva Equation (Orbital Energy)
v² = GM(2/r – 1/a)
v Relative Orbital Speed
G Gravitational Constant
M Mass of Central Body
r Current Distance
a Semi Major Axis

Angular Momentum (Orbital)
L = m * v * r * sin(θ)
L Angular Momentum
m Mass of Orbiting Body
v Velocity
r Distance
θ Angle Between Vectors

Centripetal Force
F_c = m * v² / r
F_c Centripetal Force
m Mass
v Velocity
r Radius

Sidereal Period
1/P_syn = 1/P₁ – 1/P₂
P_syn Synodic Period
P₁ Sidereal Period of Inner Body
P₂ Sidereal Period of Outer Body

Hill Sphere Radius
R_H ≈ a * (m / (3M))^(1/3)
R_H Hill Sphere Radius
a Semi Major Axis
m Mass of Smaller Body
M Mass of Larger Body

Stellar Properties & Evolution

Distance Modulus
m – M = 5 * log₁₀(d) – 5
m Apparent Magnitude
M Absolute Magnitude
d Distance in Parsecs

Absolute Magnitude
M = m – 5 * log₁₀(d/10)
M Absolute Magnitude
m Apparent Magnitude
d Distance in Parsecs

Luminosity
L = 4πR²σT⁴
L Luminosity
R Radius of the Star
σ Stefan Boltzmann Constant
T Effective Temperature

Stefan Boltzmann Law
F = σT⁴
F Energy Flux
σ Stefan Boltzmann Constant
T Temperature

Inverse Square Law (Light)
F = L / (4πd²)
F Flux Received
L Luminosity of Source
d Distance to Source

Magnitude Intensity Relation
m₁ – m₂ = -2.5 * log₁₀ (I₁ / I₂)
m₁ m₂ Apparent Magnitudes
I₁ I₂ Intensities or Fluxes

Mass Luminosity Relation (Main Sequence)
L ∝ M^3.5
L Luminosity
M Mass

Wein’s Displacement Law
λ_max = b / T
λ_max Peak Wavelength
b Wien’s Displacement Constant
T Temperature

Planck’s Law
B_λ(T) = (2hc² / λ⁵) * 1 / (e^(hc / λkT) – 1)
B_λ(T) Spectral Radiance
h Planck Constant
c Speed of Light
λ Wavelength
k Boltzmann Constant
T Temperature

Schwarzschild Radius (Event Horizon)
R_s = 2GM / c²
R_s Schwarzschild Radius
G Gravitational Constant
M Mass
c Speed of Light

Hubble’s Law
v = H₀ * d
v Recessional Velocity
H₀ Hubble Constant
d Proper Distance

Hubble Time
T_H = 1 / H₀
T_H Hubble Time
H₀ Hubble Constant

Critical Density of the Universe
ρ_c = 3H² / (8πG)
ρ_c Critical Density
H Hubble Parameter
G Gravitational Constant

Density Parameter
Ω = ρ / ρ_c
Ω Density Parameter
ρ Actual Density
ρ_c Critical Density

Redshift
z = (λ_obs – λ_rest) / λ_rest
z Redshift
λ_obs Observed Wavelength
λ_rest Rest Wavelength

Redshift Velocity Relation (Non Relativistic)
z ≈ v / c
z Redshift
v Recessional Velocity
c Speed of Light

Redshift Velocity Relation (Relativistic)
z = √((1 + v/c) / (1 – v/c)) – 1
z Redshift
v Recessional Velocity
c Speed of Light

Lookback Time
t_L = d / c
t_L Lookback Time
d Distance
c Speed of Light

Parallax Angle
d (pc) = 1 / p (“)
d Distance in Parsecs
p Parallax Angle in Arcseconds

Small Angle Formula
θ = D / d
θ Angular Size in Radians
D Physical Size
d Distance

Roche Limit (Rigid Body)
d = 1.26 * R * (ρ_M / ρ_m)^(1/3)
d Roche Limit Distance
R Radius of Primary Body
ρ_M Density of Primary Body
ρ_m Density of Satellite

Roche Limit (Fluid Body)
d ≈ 2.44 * R * (ρ_M / ρ_m)^(1/3)
d Roche Limit Distance
R Radius of Primary Body
ρ_M Density of Primary Body
ρ_m Density of Satellite

Tidal Force
F_tide ≈ (2GMmR / d³)
F_tide Tidal Force
G Gravitational Constant
M Mass of Primary Body
m Mass of Secondary Body
R Radius of Secondary Body
d Distance Between Bodies

Black Body Temperature (Planet)
T_p = T_* * √(R_* / (2d))
T_p Planetary Equilibrium Temperature
T_* Stellar Temperature
R_* Stellar Radius
d Orbital Distance

Radial Velocity Semi Amplitude
K = (2πG / P)^(1/3) * (M_p sin i) / (M_* + M_p)^(2/3)
K Radial Velocity Semi Amplitude
G Gravitational Constant
P Orbital Period
M_p Planet Mass
i Orbital Inclination
M_* Stellar Mass

Transit Depth
ΔF / F = (R_p / R_*)²
ΔF / F Fractional Flux Decrease
R_p Planetary Radius
R_* Stellar Radius

Transit Duration
T = (P R_*) / (π a)
T Transit Duration
P Orbital Period
R_* Stellar Radius
a Semi Major Axis

Saha Ionization Equation
N_{i+1} / N_i = (2 / n_e) * (2π m_e k T / h²)^(3/2) * e^{-χ_i / kT}
N_{i+1} Number Density of Ions
N_i Number Density of Lower Ionization State
n_e Electron Density
m_e Electron Mass
k Boltzmann Constant
T Temperature
h Planck Constant
χ_i Ionization Potential

Jeans Mass
M_J ≈ (kT / (Gμm_p))^(3/2) * (1 / ρ)^(1/2)
M_J Jeans Mass
k Boltzmann Constant
T Temperature
G Gravitational Constant
μ Mean Molecular Weight
m_p Proton Mass
ρ Density

Free Fall Time
t_ff = √(3π / (32 G ρ))
t_ff Free Fall Time
G Gravitational Constant
ρ Density

Eddington Luminosity
L_Edd = (4π G M m_p c) / σ_T
L_Edd Eddington Luminosity
G Gravitational Constant
M Mass
m_p Proton Mass
c Speed of Light
σ_T Thomson Scattering Cross Section

Salpeter Initial Mass Function
dN / dM ∝ M^{-2.35}
dN / dM Number of Stars per Mass Interval
M Stellar Mass

Chandrasekhar Limit
M_Ch ≈ 1.4 M_☉
M_Ch Chandrasekhar Limit
M_☉ Solar Mass

Tully Fisher Relation (Spirals)
L ∝ v_max^n
L Luminosity
v_max Maximum Rotation Velocity
n Exponent (approx 4)

Faber Jackson Relation (Ellipticals)
L ∝ σ^4
L Luminosity
σ Velocity Dispersion

Virial Theorem (for a Galaxy/Cluster)
2K + U = 0
K Total Kinetic Energy
U Total Potential Energy

Mass from Virial Theorem
M = (2 * * R) / G
M Mass
Mean Square Velocity
R Characteristic Radius
G Gravitational Constant

Hubble Classification Tuning Fork
Descriptive Classification Scheme

Oort Constants
A = 1/2 ( (V_0/R_0) – (dV/dR)_0 )
B = -1/2 ( (V_0/R_0) + (dV/dR)_0 )
A B Oort Constants
V_0 Circular Speed at Sun
R_0 Galactocentric Distance of Sun
dV/dR Radial Shear

Poisson’s Equation (Gravity)
∇²Φ = 4πGρ
Φ Gravitational Potential
G Gravitational Constant
ρ Density

Friedmann Equations
(ȧ / a)² = (8πG / 3)ρ – (kc² / a²) + (Λc² / 3)
ä / a = -(4πG / 3)(ρ + 3p/c²) + (Λc² / 3)
a Scale Factor
ȧ ä Derivatives of Scale Factor
G Gravitational Constant
ρ Density
k Curvature Parameter
c Speed of Light
Λ Cosmological Constant
p Pressure