The World Builder's Cookbook

Equations

Acceleration and Distance
Basic Planetary Calculations
Stellar Information
Orbits and Planets
Planetary Insolation
Hohmann Transfer Orbits
Temperatures
Titus-Bode Law
Rocket Equations
Black Holes

Constants and Values

Constants
Distances
Masses and Densities

Game Mechanics

Solar Sail info from VDS

References

Game Master/Writer Information

This document contains equations, constants, and other goodies for world-building calculations. They're useful in doing any kind of hard science story, novel, or campaign. Many of these are lifted from the non-copyrighted version of the sci.space FAQ. Others are taken from Stephen Gillett's excellent book, World Building. See the biblography for more details.

Some of these equations and technologies involved are discussed in a sister page, Interplanetary Travel: An Intro (with Equations), though that's more about rocketry. Another good educational link is NASA

Equations

Acceleration and Distance

Where d is distance, v is velocity, and t is time.

For constant acceleration:

Basic Planetary Calculations

Stellar Information

If you're using stars that are somewhat more extreme, you might want to calculate the bolometric magnitude instead. (Bolometric magnitude is the total amount of radiation put out by the star, not just visible radiation.) Add the correction values from this table (stolen from Gilett) to the magnitude of the star. Gillet says, "For a more complete table showing additional classes, refer to Kaler 1997, p. 263." I'll try and dig up the Kaler reference.

You might also go see the On Creating an Earthlike Planet, which is excellent (and I'm not saying that because he links back here; I only just discovered his link).

Class	Main Sequence	Giants	Supergiants
O3	-4.3	-4.2	-4.0
B0	-3.0	-2.9	-2.7
A0	-0.15	-0.24	-0.3
F0	-0.01	0.01	0.14
G0	-0.10	00.13	-0.1
K0	-0.24	-0.42	-0.38
M0	-1.21	-1.28	-1.3
M8	-4.0

If you really need to calculate it, there's an empirical formula and a calculator at http://www.go.ednet.ns.ca/~larry/astro/HR_diag.html.

Orbits and Planets

If you’re interested in learning about orbital mechanics, try the Orbital Mechanics website, http://www.braeunig.us/space/orbmech.htm.

Period of a Circular Keplerian Orbit

T = 2p/(GM/a³)^0.5

where

Delta-vee Between Two Circular Orbits

Normally used for LEO to GEO calculations, this is T.N. Edelbaum's equation. Unless there are simplifications I’m not aware of, it should be valid for differences between any two circular orbits around the same primary:

Delta-Vee = (V₁₂ - 2 V₁ V₂ cos (pi/2 alpha ) + V₂²)^1/2

Where:

V1	circular velocity initial orbit
V2	circular velocity final orbit
alpha	plane change in degrees.

Roche limit and instability

where

Orbital velocities for orbits at a distance r:

Eccentricities of orbits depending on orbit type, with semimajor axis a and semiminor axis b:

Energy of an object of mass m in an orbit around the sun (mass M) with semimajor axis a:

Planetary Insolation

Insolation of a planet determines approximately how much light it gets, and (in solar units) depends on the luminosity of the star and its distance. Dole suggests the acceptable values are between 0.65 and 1.5; see the "fudged temperature" for a more recent measurement.

Hohmann Transfer Orbits

A Hohman transfer orbit is the minimum energy orbit to get from planet A to planet B, assuming they have circular Keplerian orbits. The orbit is circular, with a tangent at the perihelion of one planet and another tangent at the aphelion of the other.

Once you have the semimajor axis, you know transfer time: it's half the orbital period for a circular Keplerian orbit of that radius (use equation above).

To calculate required DV, you need to know the orbital velocity for your transfer orbit at the points where it's tangential to the orbits of the departure and destination planets:

Ignoring for now the problems of calculating the angle that the destination planet needs to subtend and calculating the launch date; sample calculations for Earth to Mars can be found at:

http://www.marsacademy.com/text/angplan.htm

http://www.marsacademy.com/text/ladate.htm

Constant Acceleration Transit

There's a second kind of easily-calculated, efficient orbit, one that assumes a constant low acceleration (the sort you'd expect from an ion drive or a solar sail).

The acceleration must be very much lower than R/P², where R is the distance from the sun and P is the period of the outermost planet. (Note that this is extremelylow; the value of R/P² for Earth is 0.015 m/s²; for Mars, it is 0.0065 m/s², or less than 7 ten-thousandths of a G.)

An acceptable approximation of the travel time is:

2pR₁/(aP₁) * (1 - R₁/R₂)^0.5

where:

R1	Distance from sun of inner planet
R2	Distance from sun of outer planet
P1	Period of the inner planet
a	Aceleration of spacecraft

Be consistent with your units! If a is in m/s², then P must be in seconds.

You can get a value good enough for story or RPG purposes by doubling t=(2d/a)^0.5, where d is half the distance to the other planet. For example, say that Mars to Earth is 7.893E10 meters, its closest approach (2.279E11 - 1.496E11 meters). You have a solar sail that gives you 0.001 G acceleration, or 0.01 m/s². The time to accelerate half-way there is:

(7.83E10/0.01)^0.5 = (7.83E12)^0.5 = 2.8E6 seconds

Or a little over 32 days. Assume the same time tod ecelerate for a total Earth-to-Mars time of about 65 days.

A note from the website http://dutlsisa.lr.tudelft.nl/Propulsion/Data/V_increment_requirements.htm says:

"Transfer or trip time for constant thrust spiral is calculated by dividing total propellant mass by mass flow. Total propellant mass is calculated using the rocket equation also known as Tsiolkowky's equation. In case of negligible propellant mass (constant acceleration), transfer time can be calculated by dividing the velocity change by the acceleration."

Of course, it doesn't look to me that you get to stop at the other end, and you're not necessarily going at the same velocity as your target...but good enough for back-of-the-envelope calculations.

Temperatures

Temperature of a blackbody:

To allow for greenhouse gases, Gillett suggests a fudge factor of about 1.1 for habitable planets:

Intensity of blackbody per unit area:

Titius-Bode Law

GURPS Space uses a variant on this "law" (discovered by Titius, popularized by Bode) for placing planets.

Gillett says that current thinking is this is an example of tidal separations in the protocloud; it holds to lesser extents for moon systems as well, but with different parameters.

The classical formula for our solar system, where r_n is the orbital distance for planet n:

r_n = (0.3 * 2ⁿ) + 0.4 AU

A more general form, suitable for moons around planets, for planet n:

P_n = P₀A_n

Where:

Pn	Period of orbit of nth planet (traditionally in days)
P0	Period of primary's rotation
A	Semimajor axis of the orbit

Warning: I no longer remember where I got the general form, and frankly, looking at it I no longer understand it. So use with caution (or point me to the derivation, so I can do the same for others).

Rocket Equations

Classical rocket equation

Where d_v is the change in velocity, I_sp is the specific impulse of the engine, v_e is the exhaust velocity, x is the reaction mass, m₁ is the rocket mass excluding reaction mass, g is acceleration due to gravity on earth:

Note that (m₁+x)/m₁) is the ratio of the initial mass to the final mass.

The exponent d/v is change in velocity over exhaust velocity.

For a staged rocket where each stage has the same ratio R of initial to final mass and with n stages:

You may notice that's the same as the single stage orbit multiplied by n. Essentially, two stages give you twice the final velocity of a single stage rocket with the same mass ratio, and so on.

Relativistic rocket equations

For constant acceleration:

Black Holes

For a black hole of mass M:

Constants and Values

Some useful constants. Since it's sometimes easier to work things out in solar or terran equivalents, some physical data for our solar system is also included.

For game purposes, one or two significant digits is all you need, but I've gone to four here.

Constants

Distances

Masses and Densities

Game Mechanics

Solar Sail info from VDS

VDS (from BTRC) has the power from a solar sail constant per square kilometer of sail (10 w at Earth orbit). Fiddling with the acceleration equation gives us these two versions:

Where a is the acceleration in meters/s², M is the mass of the vehicle in kilogarms, and P is the power in watts. If you calculate the acceleration for a given vehicle at earth orbit (10 watts), the acceleration at other orbits is proportional to the distance in AU. (Calculating for another star is a different matter.) Sails for different TLs have the following "characteristic acceleration" (acceleration with no payload):

Equations and data were taken from the following references:

World-Building

Stephen L. Gillett, Writer's Digest Books, 1996.

Vehicle Design System

Greg Porter, Blackburg Tactical Research Center, 1997.

"Making Believeable Planets"

Peter Jekel, Strange Horizons (http://www.strangehorizons.com/2002/20020225/planets.shtml)

rec.arts.sf.science

Some posts by Brian Davis (bdavis@pdnt.com) in a thread in December of 2000

GURPS mailing list

The constant acceleration formula came from MA Lloyd, from the archive at http://www.rollanet.org/~bennett/gmsf/respc4.txt

Bolometric Magnitude

from Johnson, H. L.; Morgan, W. W. (1953): Astrophysical Journal, 117:313

Bolometric Magnitude reference

Kaler, James B (1997): Stars and Their Spectra. Cambridge. (Corr3ected paperback ed.) 300 pp.

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