17 Constants and equations for calculations (Space)
Description
This article is from the Space FAQ, by Jon Leech leech@cs.unc.edu and Mark Bradford tla@surly.org with numerous contributions by others.
17 Constants and equations for calculations (Space)
This list was originally compiled by Dale Greer. Additions would be
appreciated.
Numbers in parentheses are approximations that will serve for most
blue-skying purposes.
Unix systems provide the 'units' program, useful in converting between
different systems (metric/English, CGS/MKS etc.)
NUMBERS
7726 m/s (8000) -- Earth orbital velocity at 300 km altitude
3075 m/s (3000) -- Earth orbital velocity at 35786 km (geosync)
6371 km (6400) -- Mean radius of Earth
6378 km (6400) -- Equatorial radius of Earth
1738 km (1700) -- Mean radius of Moon
5.974e24 kg (6e24) -- Mass of Earth
7.348e22 kg (7e22) -- Mass of Moon
1.989e30 kg (2e30) -- Mass of Sun
3.986e14 m^3/s^2 (4e14) -- Gravitational constant times mass of Earth
4.903e12 m^3/s^2 (5e12) -- Gravitational constant times mass of Moon
1.327e20 m^3/s^2 (13e19) -- Gravitational constant times mass of Sun
384401 km ( 4e5) -- Mean Earth-Moon distance
1.496e11 m (15e10) -- Mean Earth-Sun distance (Astronomical Unit)
1 megaton (MT) TNT = about 4.2e15 J or the energy equivalent of
about .05 kg (50 g) of matter. Ref: J.R Williams, "The Energy Level
of Things", Air Force Special Weapons Center (ARDC), Kirtland Air
Force Base, New Mexico, 1963. Also see "The Effects of Nuclear
Weapons", compiled by S. Glasstone and P.J. Dolan, published by the
US Department of Defense (obtain from the GPO).
EQUATIONS
Where d is distance, v is velocity, a is acceleration, t is time.
Additional more specialized equations are available from:
ftp://ftp.cs.unc.edu/pub/users/leech/FAQ/MoreEquations.gz
For constant acceleration d = d0 + vt + .5at^2 v = v0 + at v^2 = 2ad
Acceleration on a cylinder (space colony, etc.) of radius r and
rotation period t:
a = 4 pi**2 r / t^2
For circular Keplerian orbits where: Vc = velocity of a circular orbit Vesc = escape velocity M = Total mass of orbiting and orbited bodies G = Gravitational constant (defined below) u = G * M (can be measured much more accurately than G or M) K = -G * M / 2 / a r = radius of orbit (measured from center of mass of system) V = orbital velocity P = orbital period a = semimajor axis of orbit Vc = sqrt(M * G / r) Vesc = sqrt(2 * M * G / r) = sqrt(2) * Vc V^2 = u/a P = 2 pi/(Sqrt(u/a^3)) K = 1/2 V**2 - G * M / r (conservation of energy)
The period of an eccentric orbit is the same as the period
of a circular orbit with the same semi-major axis.
Change in velocity required for a plane change of angle phi in a
circular orbit:
delta V = 2 sqrt(GM/r) sin (phi/2)
Energy to put mass m into a circular orbit (ignores rotational
velocity, which reduces the energy a bit).
GMm (1/Re - 1/2Rcirc) Re = radius of the earth Rcirc = radius of the circular orbit.
Classical rocket equation, where
dv = change in velocity Isp = specific impulse of engine Ve = exhaust velocity x = reaction mass m1 = rocket mass excluding reaction mass g = 9.8 m / s^2 Ve = Isp * g dv = Ve * log((m1 + x) / m1) = Ve * log((final mass) / (initial mass))
Relativistic rocket equation (constant acceleration)
t (unaccelerated) = c/a * sinh(a*t/c) d = c**2/a * (cosh(a*t/c) - 1) v = c * tanh(a*t/c)
Relativistic rocket with exhaust velocity Ve and mass ratio MR:
at/c = Ve/c * ln(MR), or t (unaccelerated) = c/a * sinh(Ve/c * ln(MR)) d = c**2/a * (cosh(Ve/C * ln(MR)) - 1) v = c * tanh(Ve/C * ln(MR))
Converting from parallax to distance:
d (in parsecs) = 1 / p (in arc seconds) d (in astronomical units) = 206265 / p
Miscellaneous
f=ma -- Force is mass times acceleration w=fd -- Work (energy) is force times distance
Atmospheric density varies as exp(-mgz/kT) where z is altitude, m is
molecular weight in kg of air, g is local acceleration of gravity, T
is temperature, k is Bolztmann's constant. On Earth up to 100 km,
d = d0*exp(-z*1.42e-4)
where d is density, d0 is density at 0km, is approximately true, so
d@12km (40000 ft) = d0*.18 d@9 km (30000 ft) = d0*.27 d@6 km (20000 ft) = d0*.43 d@3 km (10000 ft) = d0*.65
Atmospheric scale height Dry lapse rate (in km at emission level) (K/km) ------------------------- -------------- Earth 7.5 9.8 Mars 11 4.4 Venus 4.9 10.5 Titan 18 1.3 Jupiter 19 2.0 Saturn 37 0.7 Uranus 24 0.7 Neptune 21 0.8 Triton 8 1
Titius-Bode Law for approximating planetary distances:
R(n) = 0.4 + 0.3 * 2^N Astronomical Units
This fits fairly well for Mercury (N = -infinity), Venus
(N = 0), Earth (N = 1), Mars (N = 2), Jupiter (N = 4),
Saturn (N = 5), Uranus (N = 6), and Pluto (N = 7).
CONSTANTS
6.62618e-34 J-s (7e-34) -- Planck's Constant "h"
1.054589e-34 J-s (1e-34) -- Planck's Constant / (2 * PI), "h bar"
1.3807e-23 J/K (1.4e-23) - Boltzmann's Constant "k"
5.6697e-8 W/m^2/K (6e-8) -- Stephan-Boltzmann Constant "sigma"
6.673e-11 N m^2/kg^2 (7e-11) -- Newton's Gravitational Constant "G"
0.0029 m K (3e-3) -- Wien's Constant "sigma(W)"
3.827e26 W (4e26) -- Luminosity of Sun
1370 W / m^2 (1400) -- Solar Constant (intensity at 1 AU)
6.96e8 m (7e8) -- radius of Sun
1738 km (2e3) -- radius of Moon
299792458 m/s (3e8) -- speed of light in vacuum "c"
9.46053e15 m (1e16) -- light year
206264.806 AU (2e5) -- one parsec
3.2616 light years (3) -- one parsec
3.0856e16 m (3e16) -- one parsec
Black Hole radius (also called Schwarzschild Radius):
2GM/c^2, where G is Newton's Gravitational Constant, M is mass of
black hole, c is speed of light
Things to add (somebody look them up!)
Basic rocketry numbers & equations
Aerodynamical stuff
Energy to put a pound into orbit or accelerate to interstellar
velocities.
Non-circular cases?
Continue to:
My Books
