Formulas
This is an index of most formulas used internally by SFCalcSheet. If you find a mistake here, don’t hesitate and file an issue.
Luminosity from absolute magnitude
M = Absolute magnitude, L = Luminosity (Suns)
Absolute magnitude from luminosity
M = Absolute magnitude, L = Luminosity (Suns)
Visible magnitude from absolute magnitude and distance
m = Visible magnitude, M = Absolute magnitude, d = Distance (pc)
Luminosity difference from absolute magnitudes
L = Luminosity, M = Absolute magnitude
Luminosity from mass
L = Luminosity (Suns), m = Mass (Suns)
Radius from mass (stars less massive than the Sun)
r = Radius (Suns), m = Mass (Suns)
Radius from mass (stars as massive or more massive than the Sun)
r = Radius (Suns), m = Mass (Suns)
Inner rim of habitable zone
r(h(inner)) = Inner rim (AU), L = Luminosity (Suns). This is a conservative formula resulting in a narrow HZ.
Outer rim of habitable zone
r(h(outer)) = Outer rim (AU), L = Luminosity (Suns). This is a conservative formula resulting in a narrow HZ.
Main sequence life span from mass
l = Life span (years), m = Mass (Suns)
Luminosity relative to ZAMS (zero age main sequence)
L = Luminosity (Suns), m = Mass (Suns), t = Age of star (years)
Orbital period of binary orbit from distance and mass
T = Orbital period (s), d = Separation of bodies’ centers/sum of semi-major axes (m), G = Gravitational constant, m(1) and m(2) = Masses of orbiting bodies (kg)
Barycenter of binary orbit
d(b) = Distance of barycenter to center of first body (m), d = Distance between bodies’ centers (m), m(1) and m(2) = Masses of orbiting bodies (kg)
Radius from mass and density
r = Radius (m), m = Mass (kg), ρ = Density (kg/m³)
Gravitational acceleration from mass and radius
g = Gravitational acceleration (m/s²), G = Gravitational constant, m = Body mass (kg), r = Body radius (m)
Escape velocity from mass and radius
v(e) = Escape velocity (m/s), G = Gravitational constant, m = Mass (kg), r = Radius (m)
Roche limit (rigid bodies) from densities
d = Roche limit (m), r(p) = Planet radius (m), ρ(p) = Density of planet (kg/m³), ρ(s) = Density of satellite (kg/m³)
Roche limit (fluid/loose bodies) from densities
d = Roche limit (m), r(p) = Planet radius (m), ρ(p) = Density of planet (kg/m³), ρ(s) = Density of satellite (kg/m³)
Orbital period from distance and mass
T = Orbital period (s), d = Semi-major axis (m), G = Gravitational constant, m = Mass of orbited body (kg)
Hill sphere
r(h) = Radius of Hill sphere (m), d = Distance to orbited body (m), m = Mass of body (kg), m(o) = Mass of orbited body (kg)
Tidal force
F(t) = Tidal force (per unit mass), G = Gravitational constant, m = Mass of causing body (kg), r = Radius of affected body (m), d = Distance between bodies (m)
Habitable zone limits
d = Distance of HZ limit to the star (AU), L = Luminosity (Suns), S(e) = Normalized stellar flux (runaway greenhouse = 1.41; moist greenhouse = 1.107; 1st condensation = 0.53; maximum greenhouse = 0.356)
Solar irradiance
R = Irradiance (W/m²), L = Luminosity of star (Suns), G(SC) = Solar constant (1,361 W/m²), d = Distance of planet to star (AU)
Effective temperature
T(e) = Effective temperature (K), A = Bond albedo of planet, L = Luminosity of star (W), d = Distance to star (m), σ = Stefan-Boltzmann constant
Surface temperature
T(s) = Surface temperature (K), T(e) = Effective temperature (K), ε = Atmospheric absorption (0-1)
Land area
A(l) = Land area (km²), r = Planet radius (km), A(l(p)) = Land percentage (0-1)
Visual diameter
d(v) = Visual diameter (rad), r = Radius of satellite (km), d = Distance to satellite (km)
Visual area from visual diameter
A(v) = Visual area (rad²), d(v) = Visual diameter (rad)
Illuminance
E(v) = Illuminance (full moons; ≈ 0.25 lux), A(v) = Visual area (rad²), A(b) = Bond albedo (0-1)
48900 is a shorthand for 1 divided by half the Moon’s angular diameter (in radians) squared. 0.12 is the Moon’s bond albedo.
Schwarzschild radius
r(s) = Schwarzschild radius (m), G = Gravitational constant, m = Mass (kg), c = Speed of light (m/s)
Black hole lifetime
t(l) = Lifetime (s), m = Mass (kg), G = Gravitational constant, ℏ = Dirac constant, c = Speed of light (m/s)
Impactor mass
m = Impactor mass (kg), ρ = Impactor density (kg/m³), r = Impactor radius (m)
Impactor kinetic energy
K = Impactor kinetic energy (J), m = Impactor mass (kg), v = Impactor velocity (m/s)
Impact crater approximate size
d(c) = Approximate crater diameter (m), K = Impactor kinetic energy (J)
4.184 × 10^12 is the explosion energy of 1 kt TNT in joules. 46 is the diameter of the crater left by the explosion of 1 kt TNT (in meters).
Impact crater size lower bound
d(crlower) = Crater diameter lower bound, d(c) = approximate crater diameter
Impact crater size upper bound
d(crupper) = Crater diameter upper bound, d(c) = approximate crater diameter
Bomb blast maximum fireball radius
r(max) = Maximum fireball radius (m), E = Bomb yield (kt TNT)
This is an intermediate formula between the approximate fireball radii of aerial and ground-based detonations. Use a multiplicator of 33.5 for aerial and 44 for ground-based detonations.
Bomb blast fireball duration
t = Fireball duration (s), E = Bomb yield (kt TNT)
Bomb blast crater diameter
d = Crater diameter (m), E = Bomb yield (kt TNT)
Bomb blast shockwave radius
r = Shockwave radius (m), E = Explosion energy (J), Δt = Time since detonation (s), ρ = Mass density of medium (kg/m³)
Time dilation (observer time)
t = Time, v = Velocity (m/s), c = Speed of light (m/s)
Uniform acceleration time
t(a) = Acceleration time (s), v(f) = Final velocity (m/s), v(i) = Initial velocity (m/s), a = Acceleration (m/s²)
Uniform acceleration distance
d = Distance (m), v(i) = Initial velocity (m/s), v(f) = Final velocity (m/s), t(a) = Acceleration time (s)