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
L = luminosity (Suns), M = absolute magnitude
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)
Surface temperature from mass
T(s) = surface temperature (K), m = mass (Suns)
5778 is the Sun’s surface temperature in Kelvin.
Inner rim of habitable zone
r(h(inner)) = inner rim (AU), L = luminosity (Suns). This is a conservative formula resulting in a narrow HZ. Use the “habitable zone limits” formula for more flexibility.
Outer rim of habitable zone
r(h(outer)) = outer rim (AU), L = luminosity (Suns). This is a conservative formula resulting in a narrow HZ. Use the “habitable zone limits” formula for more flexibility.
Main sequence life span from mass
l = life span (years), m = mass (Suns)
Luminosity relative to ZAMS (zero age main sequence)
L = relative luminosity (ZAMS = 1), 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 (m/s²), 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
Energy of matter/antimatter annihilation
E = released energy (J), m = total reaction mass (kg), c = speed of light (m/s)
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)
Earthquake energy
E = released energy (J), M = earthquake magnitude
Time dilation (outside observer) / relativistic mass
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)