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sci.space FAQ - Constants and equations for calcul
Archive-name: space/constants
Last-modified: $Date: 92/09/02 14:48:04 $
CONSTANTS AND EQUATIONS FOR CALCULATIONS
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, etc.)
NUMBERS
7726 m/s (8000) -- Earth orbital velocity at 300 km altitude
3075 m/s (3000) -- Earth orbital velocity at 35786 km (geosync)
6378 km (6400) -- Mean 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 gm) 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.
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 u is gravitational constant, a is
semimajor axis of orbit, P is period.
v^2 = u/a
P = 2pi/(Sqrt(u/a^3))
u = G * M (can be measured much more accurately than G or M)
Vc = sqrt(M * G / r)
Vesc = sqrt(2 * M * G / r) = sqrt(2) * Vc
The period of an eccentric orbit is the same as the period of a
circular orbit with the same semi-major axis
1/2 V**2 - G * M / r = K (conservation of energy)
where
Vc = velocity of a circular orbit (you have something like this)
Vesc = escape velocity
K = -G * M / 2 / a
M = Mass of orbited object
G = Gravitational constant
r = radius of orbit (measured from center of mass of system)
V = orbital velocity
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 (ignoring rotational
velocity of the Earth, 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 (dv = change in velocity, ve = exhaust
velocity, x = reaction mass, m1 = rocket mass excluding reaction
mass):
dv = Ve * ln((m1 + x) / m1)
= Ve * ln((final mass) / (initial mass))
Ve = Isp * g = exhaust velocity, m / s
Isp = specific impulse of engine
g = 9.80665 m / s^2
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
Titius-Bode Law for approximating planetary distances:
R(n) = 0.4 + 0.3 * 2^N Astronomical Units (N = -infinity for
Mercury, 0 for Venus, 1 for Earth, etc.)
This fits fairly well except for Neptune.
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) -- \
3.2616 light years (3) -- --> parsec
3.0856e16 m (3e16) -- /
Black Hole radius (also called Schwarzschild Radius):
2GM/c^2, where G is Newton's Grav Constant, M is mass of BH,
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?
Atmosphere scale height for various planets.
NEXT: FAQ #7/15 - Astronomical Mnemonics
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