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Astrophysics Formulas

Complete collection of astrophysics formulas with detailed explanations.

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Orbital Mechanics

Kepler's Third Law

T2=4π2GMa3T^2 = \frac{4\pi^2}{GM}a^3

Square of orbital period equals 4π² divided by gravitational constant times mass times cube of semi-major axis.

Notation:

T:Orbital period
G:Gravitational constant
M:Central mass
a:Semi-major axis

Units:

T:s
G:6.674 × 10⁻¹¹ N·m²/kg²
M:kg
a:m

Applications:

  • •Planetary orbits
  • •Satellite motion
  • •Binary star systems

Limitations:

Two-body problem

Escape Velocity

vescape=2GMrv_{escape} = \sqrt{\frac{2GM}{r}}

Escape velocity equals square root of twice gravitational constant times mass divided by radius.

Notation:

v_escape:Escape velocity
G:Gravitational constant
M:Mass
r:Radius

Units:

v_escape:m/s
G:N·m²/kg²
M:kg
r:m

Applications:

  • •Rocket launches
  • •Black hole physics
  • •Space exploration

Limitations:

Point mass approximation

Orbital Velocity

v=GMrv = \sqrt{\frac{GM}{r}}

Circular orbital velocity equals square root of gravitational constant times mass divided by radius.

Notation:

v:Orbital velocity
G:Gravitational constant
M:Central mass
r:Orbital radius

Units:

v:m/s
G:N·m²/kg²
M:kg
r:m

Applications:

  • •Satellite orbits
  • •Planetary motion
  • •Space station calculations

Limitations:

Circular orbit approximation

Gravitational Force

F=GMmr2F = \frac{GMm}{r^2}

Gravitational force between two masses equals gravitational constant times product of masses divided by square of distance.

Notation:

F:Gravitational force
G:Gravitational constant
M, m:Masses
r:Distance between centers

Units:

F:N
G:N·m²/kg²
M, m:kg
r:m

Applications:

  • •Planetary attraction
  • •Tidal forces
  • •Binary star systems

Limitations:

Point masses or spherical bodies

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Stellar Physics

Stefan-Boltzmann Law

L=4πR2σT4L = 4\pi R^2\sigma T^4

Luminosity equals 4π times radius squared times Stefan-Boltzmann constant times temperature to the fourth power.

Notation:

L:Luminosity
R:Radius
σ:Stefan-Boltzmann constant
T:Temperature

Units:

L:W
R:m
σ:5.67 × 10⁻⁸ W/m²K⁴
T:K

Applications:

  • •Star brightness
  • •Blackbody radiation
  • •Stellar evolution

Limitations:

Perfect blackbody approximation

Mass-Luminosity Relation

L∝M3.5L \propto M^{3.5}

Luminosity is proportional to mass raised to the power of 3.5 for main sequence stars.

Notation:

L:Luminosity
M:Mass

Units:

L:L☉ (solar luminosities)
M:M☉ (solar masses)

Applications:

  • •Stellar classification
  • •Distance determination
  • •Stellar evolution models

Limitations:

Main sequence stars only

Hydrostatic Equilibrium

dPdr=−ρGM(r)r2\frac{dP}{dr} = -\rho\frac{GM(r)}{r^2}

Pressure gradient equals negative density times gravitational constant times enclosed mass divided by radius squared.

Notation:

dP/dr:Pressure gradient
ρ:Density
G:Gravitational constant
M(r):Enclosed mass
r:Radius

Units:

dP/dr:Pa/m
ρ:kg/m³
G:N·m²/kg²
M(r):kg
r:m

Applications:

  • •Stellar structure
  • •Planetary interiors
  • •White dwarf physics

Limitations:

Spherical symmetry assumed

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Cosmology

Hubble's Law

v=H0dv = H_0 d

Recessional velocity equals Hubble constant times distance.

Notation:

v:Recessional velocity
H₀:Hubble constant
d:Distance

Units:

v:km/s
H₀:km/s/Mpc
d:Mpc

Applications:

  • •Universe expansion
  • •Distance measurements
  • •Cosmological models

Limitations:

Large distances only

Redshift Formula

z=λobserved−λemittedλemittedz = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}}

Redshift equals observed wavelength minus emitted wavelength divided by emitted wavelength.

Notation:

z:Redshift
λ_observed:Observed wavelength
λ_emitted:Emitted wavelength

Units:

z:Dimensionless
λ_observed:m
λ_emitted:m

Applications:

  • •Distance measurement
  • •Expansion rate
  • •Galaxy surveys

Limitations:

Relativistic effects at high z

Critical Density

ρc=3H028πG\rho_c = \frac{3H_0^2}{8\pi G}

Critical density equals 3 times Hubble constant squared divided by 8π times gravitational constant.

Notation:

ρ_c:Critical density
H₀:Hubble constant
G:Gravitational constant

Units:

ρ_c:kg/m³
H₀:s⁻¹
G:N·m²/kg²

Applications:

  • •Universe geometry
  • •Dark energy studies
  • •Cosmological parameters

Limitations:

Homogeneous universe model

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Black Hole Physics

Schwarzschild Radius

Rs=2GMc2R_s = \frac{2GM}{c^2}

Schwarzschild radius equals twice gravitational constant times mass divided by speed of light squared.

Notation:

R_s:Schwarzschild radius
G:Gravitational constant
M:Mass
c:Speed of light

Units:

R_s:m
G:N·m²/kg²
M:kg
c:3 × 10⁸ m/s

Applications:

  • •Event horizon size
  • •Black hole classification
  • •Gravitational collapse

Limitations:

Non-rotating black holes

Hawking Temperature

TH=ℏc38πGMkBT_H = \frac{\hbar c^3}{8\pi GM k_B}

Hawking temperature equals Planck constant times speed of light cubed divided by 8π times gravitational constant times mass times Boltzmann constant.

Notation:

T_H:Hawking temperature
ℏ:Reduced Planck constant
c:Speed of light
G:Gravitational constant
M:Mass
k_B:Boltzmann constant

Units:

T_H:K
ℏ:J·s
c:m/s
G:N·m²/kg²
M:kg
k_B:J/K

Applications:

  • •Black hole evaporation
  • •Quantum gravity
  • •Information paradox

Limitations:

Semi-classical approximation

Time Dilation Near Black Hole

t=t01−Rsrt = \frac{t_0}{\sqrt{1 - \frac{R_s}{r}}}

Time dilation factor equals proper time divided by square root of one minus Schwarzschild radius divided by radial distance.

Notation:

t:Dilated time
t₀:Proper time
R_s:Schwarzschild radius
r:Radial distance

Units:

t:s
t₀:s
R_s:m
r:m

Applications:

  • •Gravitational time dilation
  • •GPS corrections
  • •Relativistic effects

Limitations:

Static spacetime