Space & Science Fiction Forums

Helpful Constants and Formulae

These will all hopefully be in a file for community access that will be downloadable at a later date.
Resources include but are not limited to

Dunn, Tony. "Astronomical Formulas." Orbitsimulator.com. Tony Dunn. Web. 26 Nov. 2011. .
Cole, G. H. A., and M. M. Woolfson. Planetary Science: the Science of Planets around Stars. Bristol, [U.K.: IoP, 2002. Print.
I'll list more later


I'll finish this post with constants

G is the gravitational constant with a value of 6.67384×10^(-11) m^3/(kg s^2)
σ is the Stefan-Boltzmann constant with a value of 5.670400×10^(-8) J/(m^2s K^4)
L_☉ is 1 Solar Luminosity with the value of 3.846×10^26 W
M_☉ is 1 Solar Mass with the value of 1.98892×10^30 kg
AU us 1 astronomical unit with a current value of 1.495978707×10^11 m
δ_s is the silver ratio 1+2^(1/2)
φ is the golden ratio (1+5^(1/2))/2

Minimum mass of dwarf planet based on the smallest known body Saturn I (Mimas) to be in hydrostatic equilibrium 3.75×10^19 kg, this would require to a similar density.


Feel free to add to this list.

Message has been edited - november 25, 2011

Angular Frequency

ω = 2π/T
ω = 2πf
ω = |v|/|r|

ω is the angular frequency or angular speed (radians per second)
T is the period (seconds)
f is the ordinary frequency (hertz,sometimes symbolised with ν)
v is the tangential velocity of a point about the axis of rotation (meters per second)
r is the radius of rotation (meters)

The first formula is the easiest to use.

Oblateness Constant (simple)

q ≡ (aω^2) / g
q ≡ (aω^2) * (a^2/GM)
q ≡ (a^3 * ω^2)/GM

a is equatorial radius (meters)
ω is the angular frequency or angular speed (radians per second)
g is gravitational acceleration(meters per second)
G is the gravitational constant
M is mass of body (kilograms)
This does not take into account differentiation. To fudge this on can take M and multiply
against either a random number between 0.8 and 1.2 or mulitply against a system wide
value to represent general differentiation.

The second and third formulae, while appearing a little more complex then the first, tend to be easier to use.


Message has been edited - november 23, 2011

Roche limit advanced

d ≈ 2.423R(ρ_M/ρ_m )^(1/3) * (((1+m/3M)+1/3 q(1+m/M))/(1-q))^(1/3)

d is distance from center of primary
ρ_M is density of primary
ρ_m is density of satillite
M is mass in of primary
m is mass in of satillite
q is the oblateness constant (also listed as c/R)

There is a simple version, but it greatly increases the error sigma.

Orbital Period

P=2π*{ a^3 / [G * (M+m) ] }^(1/2)


P is period of orbit (seconds)
a is separation distance (meters)
M is mass in of primary
m is mass in of satillite
G is the gravitational constant

Message has been edited - november 23, 2011

Minimum Rotational Period

P = 2π*( r^3 / GM )^(1/2)

P is period oforbit (seconds)
r is equatorial radius (meters)
M is mass in of body
G is the graviational constant

Synchronous Orbit

r = (GM/ω^2 )^(1/3)


M is mass in of body
G is the gravitational constant
r is the elevation from the center of the body (meters)
ω is the angular frequency or angular speed (radians per second)

Synchronous Orbit Velocity

v=ωr

v is the velocity of the synchronous orbit (meters per second)
r is the elevation from the center of the body (meters)
ω is the angular frequency or angular speed (radians per second)

Effective Planet Temperature

T_eff = {[L_☉ (1-A)]/[16πσa^2 ]}^(1/4)

T_eff is the effective surface temperature not accounting for greenhouse effect (kelvin)
L_☉ is the luminosity in solar units and must be converted to watts L_☉ ∙ 3.846×10^26 W
A is the geometric albedo
σ is the Stefan-Boltzmann constant
a is the semi-major axis(meters)to get minimum and maximum adjust to peri and apo

This does not take into account any greenhouse effect factor,H,to add in H use the following
modification to the T_eff formula

T_wgh=T_eff + 0.15 ∙ T_eff ∙ H/|H| ∙ |H|^(1/2)

It is not unusual for H to be a negative value for bodies who either have a highly reflective surface or a thin conductive atmosphere,Mars and Enceladus are prime examples. For gas planets that are in excess of 12 M_♁ black body value for
planet should be included at equatorial radius as planet will most likely still be radiating heat from formation.

Luminosity Mass Relationship

L=kM^n

L is the luminosity of the stellar object
k is multiplier factor
M is the mass of the stellar object in Solar masses
n is the power factor

k and n vary according to M
M≤0.43,k=0.23,n=2.3
M≤2.00 and M>0.43,k=1,n=4
M≤20.0 and M>2.00,k=1.5,n=3.5
M>20.0,k=1,n=4-M/[44+(M/10.25)^2]