Astronomy & Space

Orbits, gravity, stars and cosmology

Explore the key formulas behind the motion of planets, the behavior of stars, and the scale of the universe. This Astronomy & Space collection helps students, exam candidates, and anyone working with physics or astrophysics solve problems involving orbits, gravity, brightness, distance, and cosmic expansion. From escape velocity and Kepler’s laws to redshift and the Hubble relationship, these formulas turn vast celestial ideas into usable calculations.

Escape Velocity

v_e = \sqrt{\frac{2GM}{r}}

Minimum speed to escape a body's gravity.

Orbital Velocity

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

Speed of a circular orbit.

Kepler's Third Law

T^2 = \frac{4\pi^2}{GM}a^3

Relates orbital period to semi-major axis.

Schwarzschild Radius

r_s = \frac{2GM}{c^2}

Event horizon radius of a black hole.

Hubble's Law

v = H_0 d

Recession velocity of galaxies with distance.

Stefan–Boltzmann Law

L = 4\pi R^2 \sigma T^4

Luminosity of a star from radius and temperature.

Redshift

z = \frac{\lambda_{obs} - \lambda_{emit}}{\lambda_{emit}}

Fractional wavelength shift of light.

Parallax Distance

d = \frac{1}{p}

Distance in parsecs from parallax angle in arcseconds.

Surface Gravity

g = \frac{GM}{r^2}

Gravitational acceleration at a body's surface.

Apparent Brightness

b = \frac{L}{4\pi d^2}

Observed brightness from luminosity and distance.

Distance Modulus

m - M = 5\log_{10}(d) - 5

Relates apparent and absolute magnitude to distance.

Light-Year

d = c \times t

Distance light travels in a given time.

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Peak emission wavelength of a blackbody.

Schwarzschild Radius

r_s = \frac{2GM}{c^2}

Radius of the event horizon of a black hole.

Surface Gravity

g = \frac{GM}{r^2}

Gravitational acceleration at a body surface.

Apparent Brightness

b = \frac{L}{4\pi d^2}

Observed flux from a source at distance d.

Wien Displacement Law

\lambda_{max} = \frac{b}{T}

Peak emission wavelength of a blackbody.

Kepler Third Law (General)

T^2 = \frac{4\pi^2}{GM}a^3

Orbital period from semi-major axis.

Light-Year

1\,ly = c\times 1\,\text{year}

Distance light travels in one year.

Parsec

1\,pc \approx 3.26\,ly

Distance at which 1 AU subtends one arcsecond.

Angular Size

\theta = \frac{d}{D}

Apparent angular size of an object (small-angle).

Distance Modulus

m - M = 5\log_{10}(d) - 5

Relates apparent and absolute magnitudes to distance.

Density from Orbit

\rho = \frac{3\pi}{GT^2}

Mean density inferred from a close orbital period.

Tidal Force

F_t \propto \frac{2GMm r}{d^3}

Differential gravitational force across a body.

Hubble Time

t_H = \frac{1}{H_0}

Rough age of the universe from the Hubble constant.

Newton's Law of Gravitation

F = G\frac{m_1 m_2}{r^2}

The attractive force between any two masses.

Gravitational Potential Energy

U = -\frac{GMm}{r}

Energy of a mass in another body's gravitational field.

Mass–Luminosity Relation

\frac{L}{L_\odot} \approx \left(\frac{M}{M_\odot}\right)^{3.5}

How a main-sequence star's brightness scales with its mass.

Eddington Luminosity

L_{Edd} = \frac{4\pi G M m_p c}{\sigma_T}

The maximum steady luminosity before radiation blows a star apart.

Chandrasekhar Limit

M_{Ch} \approx 1.44\,M_\odot

The maximum mass of a stable white dwarf.

Hawking Temperature

T = \frac{\hbar c^3}{8\pi G M k_B}

Temperature of the radiation a black hole emits.

Photon Sphere Radius

r_{ph} = \frac{3GM}{c^2}

Radius where gravity bends light into a circular orbit.

Synodic Period

\frac{1}{S} = \left|\frac{1}{P_1} - \frac{1}{P_2}\right|

Time between repeated alignments of two orbiting bodies.

Gravitational Redshift

z \approx \frac{GM}{rc^2}

Fractional stretching of light climbing out of gravity.

Hill Sphere Radius

r_H \approx a\left(\frac{m}{3M}\right)^{1/3}

The zone where a body's gravity dominates over its host's.