Articles to help you discover the wonders of the Universe

# Equations

By Tim Trott, 14th December 2008 in Astronomy

In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true.

## Geometry & Trigonometry

Circle, circumference of

Equation 1 - Circle, circumference of
Calculating the circumference of a circle.

How Large is the Earth?

Sphere, Surface Area of

Equation 21 - Sphere, Surface Area of
The surface area of a sphere of radius r.

Luminosity and Flux of Stars

## Size & Distance

Distance Calculation using Parallax

Equation 10 - Distance Calculation using Parallax
The distance to an object in parsecs is equal to 1 AU over the parallax in arc seconds.

Parallax, Distance and Parsecs

Angular Size Calculation

Equation 12 - Angular Size Calculation
Calculate the angular size of an object based on its apparent size and distance between measure and observer, where theta is the angular size in radians, Sap is the apparent size in mm and l is the distance between the measure and the observer.

What is Angular Size?

## Magnitude

Pogsons Relation

Equation 20 - Pogsons Relation
Pogson's Relation is used to find the magnitude difference between two objects expressed in terms of the logarithm of the flux ratio

The Astronomical Magnitude Scale

Absolute Magnitude Relation

Equation 23 - Absolute Magnitude Relation
Very similar to Pogsons Relation for apparent magnitudes, this equation shows the relation for absolute magnitudes.

The Astronomical Magnitude Scale

Distance Modulus

Equation 25 - Distance Modulus
Distance modulus can be used to find a relationship between a stars absolute magnitude and its apparent magnitude given a distance, d, in parsecs.

The Astronomical Magnitude Scale

Absolute Magnitude

Equation 31 - Absolute Magnitude
Calculating the Absolute Magnitude of a star within our galaxy.

The Astronomical Magnitude Scale

Bolometric Magnitude

Equation 39 - Bolometric Magnitude
Bolometric magnitude is related to the luminosity ratio

The Astronomical Magnitude Scale

Luminosity ratio of magnitudes

Equation 40 - Luminosity ratio of magnitudes

The Astronomical Magnitude Scale

## Gravity & Orbits

Kelpers Law of Planetary Motion

Equation 2 - Kelpers Law of Planetary Motion
Keplers law of Planetary Motion is simple and strait forward: The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of their orbits.

Kepler's Laws of Planetary Motion

Gravitational interaction

Equation 43 - Gravitational interaction
Gravitational interaction is the most important force, and so the most useful definition of the potential energy. For two-body problems, potential energy, U, is given by this formula.

The Physics Governing the Universe

Centripetal force

Equation 44 - Centripetal force
An orbiting body, at radius r, experiences the centripetal force Fc, which keeps it moving in a circle.

The Physics Governing the Universe

Gravitational force

Equation 45 - Gravitational force

The Physics Governing the Universe

Equation 50 - Schwarzschild radius
The Schwarzschild radius is the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from the surface of the sphere would equal the speed of light.

Black Holes

## Energy

Energy of a Photon

Equation 29 - Energy of a Photon
The energy of a photon in Joules is equal to h (Planck constant) times by its frequency.

Spectroscopy and Spectrometry

Frequency of Light

Equation 30 - Frequency of Light
The frequency of light is dependant on the speed of light (c) and it's wavelength.

Spectroscopy and Spectrometry

Kinetic energy of a particle

Equation 42 - Kinetic energy of a particle
The fundamental definition of the kinetic energy of a particle is given by the formula, where v is the velocity of the particle

The Physics Governing the Universe

The energy of a photon

Equation 46 - The energy of a photon
The energy, E, of a photon where h is Planck's constant, v is the frequency of the wave, c is the speed of light and λ is the wavelength.

The Nature of Light

Normal gas pressure

Equation 49 - Normal gas pressure

Electron Degeneracy Pressure

## Time

Local Sidereal Time

Equation 14 - Local Sidereal Time
Use this formula to calculate local sidereal time given the Greenwich sidereal time plus your longitude (east of Greenwich)

Sidereal Time, Civil Time and Solar Time

Relationship between LST, HA and RA

Equation 17 - Relationship between LST, HA and RA
These equations show the relationsip between Local Sidereal Time, Hour Angle and Right Ascension.

Sidereal Time, Civil Time and Solar Time

## Solar Physics

Ratio of Fluxes

Equation 18 - Ratio of Fluxes
Ratio of flux between two stars with apparent magnitudes m and n and measured fluxes of Fm and Fn.

Luminosity and Flux of Stars

Flux and Luminosity

Equation 22 - Flux and Luminosity
This equation shows the relationship between the observed flux of a star and its luminosity.

Luminosity and Flux of Stars

Inverse Square Law for Flux

Equation 24 - Inverse Square Law for Flux
This equation gives a stars absolute flux (FM) using the inverse square law for a star at distance d (parsecs), with observed flux Fm.

The Astronomical Magnitude Scale

Stefan-Boltzmann Law

Equation 26 - Stefan-Boltzmann Law
Also known as Stefan's law, states that the total energy radiated per unit surface area of a black body in unit time, is directly proportional to the fourth power of the black body's thermodynamic temperature T.

Hertzsprung-Russell Diagram and the Main Sequence

Electron degeneracy pressure

Equation 38 - Electron degeneracy pressure
Electron degeneracy pressure in a material can be computed, where h is Planck's constant, me is the mass of the electron, mp is the mass of the proton, ρ is the density, and μe = Ne / Np is the ratio of electron number to proton number.

Electron Degeneracy Pressure

## Cosmology

Redshift

Equation 27 - Redshift
Redshift calculation based on observed wavelengths.

Redshift and Blueshift

Temperature scale factor

Equation 32 - Temperature scale factor
Temperature of the universe through time as a function of scale factor.

What is the Cosmic Microwave Background Radiation?

Temperature redshift.

Equation 33 - Temperature redshift.
Temperature of the universe expressed as a function of redshift. T0 represents the temperature of the universe today (~2.7K).

What is the Cosmic Microwave Background Radiation?

Proper Distance

Equation 35 - Proper Distance
Proper distance x(t) corresponding to different epochs in terms of the comoving radial distance coordinate r.

Cosmic Scale Factor

Redshift and Scale Factor

Equation 36 - Redshift and Scale Factor
Relationship between redshift and scale factor for another epoch.

Cosmic Scale Factor

Hubble's Law

Equation 47 - Hubble's Law
The linear relationship between recession velocity, v, and the distance, x, of a local galaxy

The Expansion of the Universe

## Relativity

Special theory of relativity

Equation 41 - Special theory of relativity
The special theory of relativity states that mass and energy are equivalent.

The Physics Governing the Universe

Lorentz factor

Equation 48 - Lorentz factor
The factor by which time, length, and relativistic mass change for an object while that object is moving.

The Speed of Light