SPACE/MATH - Constants and Equations for Calculations
SCI.SPACE FAQ No. 04 - space/math
Last-modified: $Date: 94/03/01 17:24:39 $
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, 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:
- explorer.arc.nasa.gov:pub/SPACE/FAQ/MoreEquations
- 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 = 9.80665 m / s^2
- Ve = Isp * g
- dv = Ve * ln((m1 + x) / m1)
- = Ve * ln((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) -- \
- 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?
PERFORMING CALCULATIONS AND INTERPRETING DATA FORMATS
- COMPUTING SPACECRAFT ORBITS AND TRAJECTORIES
- References that have been frequently recommended on the net are:
- "Fundamentals of Astrodynamics" Roger Bate, Donald Mueller, Jerry White
- 1971, Dover Press, 455pp $8.95 (US) (paperback). ISBN 0-486-60061-0
- NASA Spaceflight handbooks (dating from the 1960s)
- SP-33 Orbital Flight Handbook (3 parts)
- SP-34 Lunar Flight Handbook (3 parts)
- SP-35 Planetary Flight Handbook (9 parts)
- These might be found in university aeronautics libraries or ordered
- through the US Govt. Printing Office (GPO), although more
- information would probably be needed to order them.
- M. A. Minovitch, _The Determination and Characteristics of Ballistic
- Interplanetary Trajectories Under the Influence of Multiple Planetary
- Attractions_, Technical Report 32-464, Jet Propulsion Laboratory,
- Pasadena, Calif., Oct, 1963.
- The title says all. Starts of with the basics and works its way up.
- Very good. It has a companion article:
- M. Minovitch, _Utilizing Large Planetary Perubations for the Design of
- Deep-Space Solar-Probe and Out of Ecliptic Trajectories_, Technical
- Report 32-849, JPL, Pasadena, Calif., 1965.
- You need to read the first one first to realy understand this one.
- It does include a _short_ summary if you can only find the second.
- Contact JPL for availability of these reports.
- "Spacecraft Attitude Dynamics", Peter C. Hughes 1986, John Wiley and
- Sons.
- "Celestial Mechanics: a computational guide for the practitioner",
- Lawrence G. Taff, (Wiley-Interscience, New York, 1985).
- Starts with the basics (2-body problem, coordinates) and works up to
- orbit determinations, perturbations, and differential corrections.
- Taff also briefly discusses stellar dynamics including a short
- discussion of n-body problems.
- COMPUTING PLANETARY POSITIONS
- More net references:
- "Explanatory Supplement to the Astronomical Almanac" (revised edition),
- Kenneth Seidelmann, University Science Books, 1992. ISBN 0-935702-68-7.
- $65 in hardcover.
- Deep math for all the algorthms and tables in the AA.
- Van Flandern & Pullinen, _Low-Precision Formulae for Planetary
- Positions_, Astrophysical J. Supp Series, 41:391-411, 1979. Look in an
- astronomy or physics library for this; also said to be available from
- Willmann-Bell.
- Gives series to compute positions accurate to 1 arc minute for a
- period + or - 300 years from now. Pluto is included but stated to
- have an accuracy of only about 15 arc minutes.
- _Multiyear Interactive Computer Almanac_ (MICA), produced by the US
- Naval Observatory. Valid for years 1990-1999. $55 ($80 outside US).
- Available for IBM (order #PB93-500163HDV) or Macintosh (order
- #PB93-500155HDV). From the NTIS sales desk, (703)-487-4650. I believe
- this is intended to replace the USNO's Interactive Computer Ephemeris.
- _Interactive Computer Ephemeris_ (from the US Naval Observatory)
- distributed on IBM-PC floppy disks, $35 (Willmann-Bell). Covers dates
- 1800-2049.
- "Planetary Programs and Tables from -4000 to +2800", Bretagnon & Simon
- 1986, Willmann-Bell.
- Floppy disks available separately.
- "Fundamentals of Celestial Mechanics" (2nd ed), J.M.A. Danby 1988,
- Willmann-Bell.
- A good fundamental text. Includes BASIC programs; a companion set of
- floppy disks is available separately.
- "Astronomical Formulae for Calculators" (4th ed.), J. Meeus 1988,
- Willmann-Bell.
- "Astronomical Algorithms", J. Meeus 1991, Willmann-Bell.
- If you actively use one of the editions of "Astronomical Formulae
- for Calculators", you will want to replace it with "Astronomical
- Algorithms". This new book is more oriented towards computers than
- calculators and contains formulae for planetary motion based on
- modern work by the Jet Propulsion Laboratory, the U.S. Naval
- Observatory, and the Bureau des Longitudes. The previous books were
- all based on formulae mostly developed in the last century.
- Algorithms available separately on diskette.
- "Practical Astronomy with your Calculator" (3rd ed.), P. Duffett-Smith
- 1988, Cambridge University Press.
- "Orbits for Amateurs with a Microcomputer", D. Tattersfield 1984,
- Stanley Thornes, Ltd.
- Includes example programs in BASIC.
- "Orbits for Amateurs II", D. Tattersfield 1987, John Wiley & Sons.
- "Astronomy / Scientific Software" - catalog of shareware, public domain,
- and commercial software for IBM and other PCs. Astronomy software
- includes planetarium simulations, ephemeris generators, astronomical
- databases, solar system simulations, satellite tracking programs,
- celestial mechanics simulators, and more.
- Andromeda Software, Inc.
- P.O. Box 605
- Amherst, NY 14226-0605
- COMPUTING CRATER DIAMETERS FROM EARTH-IMPACTING ASTEROIDS
- Astrogeologist Gene Shoemaker proposes the following formula, based on
- studies of cratering caused by nuclear tests.
- (1/3.4)
- D = S S c K W : crater diameter in km
- g p f n
- (1/6)
- S = (g /g ) : gravity correction factor for bodies other than
- g e t Earth, where g = 9.8 m/s^2 and g is the surface
- e t
- gravity of the target body. This scaling is
- cited for lunar craters and may hold true for
- other bodies.
- (1/3.4)
- S = (p / p ) : correction factor for target density p ,
- p a t t
- p = 1.8 g/cm^3 for alluvium at the Jangle U
- a
- crater site, p = 2.6 g/cm^3 for average
- rock on the continental shields.
- C : crater collapse factor, 1 for craters <= 3 km
- in diameter, 1.3 for larger craters (on Earth).
- (1/3.4)
- K : .074 km / (kT TNT equivalent)
- n empirically determined from the Jangle U
- nuclear test crater.
- 3 2 22
- W = pi * d * delta * V / (12 * 4.185 * 10 )
- : projectile kinetic energy in MT TNT equivalent
- given diameter d, velocity v, and projectile