This is a collection of equations I've used/gathered over the years for the purpose of creating my planets. They come from a number of sources which I'll try to cite on the bottom; I don't remember them all.

Right now, this page assumes you have some knowledge of astronomy and mathematics, perhaps high-school level, and/or experience with worldbuilding. I might one day edit this page to give instructions instead.

- Interstellar properties
- Star properties
- Star system properties
- Planetary properties
- Moon properties
- Sources

You may find it useful to know certain properties of the region of space around the star system you're working on—particularly if you're making everything up!

These constants assume the galaxy is 87,400 light-years in radius, with a central bulge that is 10,000 light-years and radius. It assumes the galactic thin disk is 1,000 light-years thick and the thick disk is 16,300 light-years thick.

GalDiscVol = 2.368 * 10^{13}ly^{3}(volume of thin disk)

ThickDiscVol = 3.624 * 10^{14}ly^{3}

Stars in thin disk = 95 billion

Stars in thick disk = 16.7 billion

Density of stars in thin disk = 250 ly^{3}/ star

Density of stars in thick disk = 21,675 ly^{3}/ star

The key property is *M*, the mass of the star(s) under consideration relative to the Sun. Other properties can be derived from this one.

Also, if you are doing a binary star system, you will need *a*, the distance between the stars in AU.

Most sources I've seen give the simple luminosity relationship \(L = M^{3.5}\), but in my research I discovered that this relationship only holds for stars with \(M > 2\)—stars that are, generally speaking, too short-lived to have habitable planets.

The full equations are:

$$\begin{equation} L = \begin{cases} 1.4M^{3.5} & \text{if 2 $\le$ M $\lt$ 20} \\ M^{4} & \text{if 0.43 $\le$ M $\lt$ 2} \\ 0.23M^{2.3} & \text{if M $\lt$ 0.43} \end{cases} \label{eq:massluminosity} \end{equation}$$

Alternately, if you're basing your system on a real star, you might have access to *R*, the star's radius, and *T*, its temperature. Using these, you can compute the luminosity:

$$\begin{equation} L = R^2 \left(\frac{T}{5772}\right)^4 \label{eq:luminosityradiustemperature} \end{equation}$$

As with the mass-luminosity relation, most sources I've seen give a relationship that only holds for stars that are too-short lived, where \(M \gt 2\). Under the equations below, such stars have lifespans shorter than 1.25 billion years.

(I.e., stars classified A2V or higher; the earliest known life on Earth formed when the planet was 840 million years old, and Earth didn't have oxygen in its atmosphere until it was 2.1 billion years old.)

The full equations are:

$$\begin{equation} \Omega = \begin{cases} 7.1M^{-2.5} & \text{if 2 $\le$ M $\lt$ 20} \\ 10M^{-3} & \text{if 0.43 $\le$ M $\lt$ 2} \\ 43M^{-1.3} & \text{if M $\lt$ 0.43} \end{cases} \label{eq:masslifespan} \end{equation}$$

These are the only equations I've found for these two properties, and I'm not entirely sure of their validity. Use at your own risk!

The radius relative to our sun, and temperature in Kelvins:

$$\begin{equation} R = M^{0.74} \label{eq:massradius} \end{equation}$$ $$\begin{equation} T = 5772M^{0.505} \label{eq:masstemperature} \end{equation}$$

Properties of the planets and the relationships between them.

This is an generalized version of the Titius-Bode relation, which gives an *estimate* of the distance of each planet from its sun. I usually just use it as-is, and all planets in my system just *happen* to follow it!

The equation is of the following form, where *a _{1}* is the distance to the first planet,

$$\begin{equation} d_n = a_1 C ^{n - 1} \label{eq:titiusbode} \end{equation}$$

For our solar system (using Ceres for the asteroid belt and excluding Pluto), a regression yields the constants *a _{1}* = 0.3533828008 and

The radius, *d _{Tide}*, within which a planet will be tidally locked to its star, i.e. forever have one face towards it and one away. This depends on

I derived this equation myself from Kastings (1993). I am quite proud of it.

For distance in AU and age in billions of years:

$$\begin{equation} d_{Tide} = \sqrt[6]{\frac{A M^2}{479}} \label{eq:tidelockdistance} \end{equation}$$

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