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.
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 * 1013 ly3 (volume of thin disk)
ThickDiscVol = 3.624 * 1014 ly3
Stars in thin disk = 95 billion
Stars in thick disk = 16.7 billion
Density of stars in thin disk = 250 ly3 / star
Density of stars in thick disk = 21,675 ly3 / 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 a1 is the distance to the first planet, C is a constant, and n is the number of the planet under consideration (asteroid belts count as planets). You have to decide on the constants yourself!
$$\begin{equation} d_n = a_1 C ^{n - 1} \label{eq:titiusbode} \end{equation}$$
For our solar system (using Ceres for the asteroid belt and excluding Pluto), a regression yields the constants a1 = 0.3533828008 and C = 1.733068308. The r2 is 0.9933. On average, the equations estimates are 10% off of the real values, higher or lower; Venus, Ceres, and Uranus are 15% off; Mars is 20% off; and Saturn is almost spot on (0.08%).
The radius, dTide, 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 M and A, the age of the star.
I derived this equation myself from Kastings (1993). I am quite proud of it.
For distance in AU and age in billions of years:
$$\begin{equation} d_{Tide} = \sqrt[6]{\frac{A M^2}{479}} \label{eq:tidelockdistance} \end{equation}$$
[ Colony list | Planets | Home ]