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Archive-name: space/math Last-modified: $Date: 93/06/02 23:13:58 $ 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) 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 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. Additional more specialized equations are available from: ames.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 (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? 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 19 W = pi * d * delta * V / (12 * 4.185 * 10 ) : projectile kinetic energy in kT TNT equivalent given diameter d, velocity v, and projectile density delta in CGS units. delta of around 3 g/cm^3 is fairly good for an asteroid. An RMS velocity of V = 20 km/sec may be used for Earth-crossing asteroids. Under these assumptions, the body which created the Barringer Meteor Crater in Arizona (1.13 km diameter) would have been about 40 meters in diameter. More generally, one can use (after Gehrels, 1985): Asteroid Number of objects Impact probability Impact energy diameter (km) (impacts/year) (* 5*10^20 ergs) 10 10 10^-8 10^9 1 1 000 10^-6 10^6 0.1 100 000 10^-4 10^3 assuming simple scaling laws. Note that 5*10^20 ergs = 13 000 tons TNT equivalent, or the energy released by the Hiroshima A-bomb. References: Gehrels, T. 1985 Asteroids and comets. _Physics Today_ 38, 32-41. [an excellent general overview of the subject for the layman] Shoemaker, E.M. 1983 Asteroid and comet bombardment of the earth. _Ann. Rev. Earth Planet. Sci._ 11, 461-494. [very long and fairly technical but a comprehensive examination of the subject] Shoemaker, E.M., J.G. Williams, E.F. Helin & R.F. Wolfe 1979 Earth-crossing asteroids: Orbital classes, collision rates with Earth, and origin. In _Asteroids_, T. Gehrels, ed., pp. 253-282, University of Arizona Press, Tucson. Cunningham, C.J. 1988 _Introduction to Asteroids: The Next Frontier_ (Richmond: Willman-Bell, Inc.) [covers all aspects of asteroid studies and is an excellent introduction to the subject for people of all experience levels. It also has a very extensive reference list covering essentially all of the reference material in the field.] MAP PROJECTIONS AND SPHERICAL TRIGNOMETRY Two easy-to-find sources of map projections are the "Encyclopaedia Brittanica", (particularly the older volumes) and a tutorial appearing in _Graphics Gems_ (Academic Press, 1990). The latter was written with simplicity of exposition and suitability of digital computation in mind (spherical trig formulae also appear, as do digitally-plotted examples). More than you ever cared to know about map projections is in John Snyder's USGS publication "Map Projections--A Working Manual", USGS Professional Paper 1395. This contains detailed descriptions of 32 projections, with history, features, projection formulas (for both spherical earth and ellipsoidal earth), and numerical test cases. It's a neat book, all 382 pages worth. This one's $20. You might also want the companion volume, by Snyder and Philip Voxland, "An Album of Map Projections", USGS Professional Paper 1453. This contains less detail on about 130 projections and variants. Formulas are in the back, example plots in the front. $14, 250 pages. You can order these 2 ways. The cheap, slow way is direct from USGS: Earth Science Information Center, US Geological Survey, 507 National Center, Reston, VA 22092. (800)-USA-MAPS. They can quote you a price and tell you where to send your money. Expect a 6-8 week turnaround time. A much faster way (about 1 week) is through Timely Discount Topos, (303)-469-5022, 9769 W. 119th Drive, Suite 9, Broomfield, CO 80021. Call them and tell them what you want. They'll quote a price, you send a check, and then they go to USGS Customer Service Counter and pick it up for you. Add about a $3-4 service charge, plus shipping. A (perhaps more accessible) mapping article is: R. Miller and F. Reddy, "Mapping the World in Pascal", Byte V12 #14, December 1987 Contains Turbo Pascal procedures for five common map projections. A demo program, CARTOG.PAS, and a small (6,000 point) coastline data is available on CompuServe, GEnie, and many BBSs. Some references for spherical trignometry are: _Spherical Astronomy_, W.M. Smart, Cambridge U. Press, 1931. _A Compendium of Spherical Astronomy_, S. Newcomb, Dover, 1960. _Spherical Astronomy_, R.M. Green, Cambridge U. Press., 1985 (update of Smart). _Spherical Astronomy_, E Woolard and G.Clemence, Academic Press, 1966. PERFORMING N-BODY SIMULATIONS EFFICIENTLY "Computer Simulation Using Particles" R. W. Hockney and J. W. Eastwood (Adam Hilger; Bristol and Philadelphia; 1988) "The rapid evaluation of potential fields in particle systems", L. Greengard MIT Press, 1988. A breakthrough O(N) simulation method. Has been parallelized. L. Greengard and V. Rokhlin, "A fast algorithm for particle simulations," Journal of Computational Physics, 73:325-348, 1987. "An O(N) Algorithm for Three-dimensional N-body Simulations", MSEE thesis, Feng Zhao, MIT AILab Technical Report 995, 1987 "Galactic Dynamics" J. Binney & S. Tremaine (Princeton U. Press; Princeton; 1987) Includes an O(N^2) FORTRAN code written by Aarseth, a pioneer in the field. Hierarchical (N log N) tree methods are described in these papers: A. W. Appel, "An Efficient Program for Many-body Simulation", SIAM Journal of Scientific and Statistical Computing, Vol. 6, p. 85, 1985. Barnes & Hut, "A Hierarchical O(N log N) Force-Calculation Algorithm", Nature, V324 # 6096, 4-10 Dec 1986. L. Hernquist, "Hierarchical N-body Methods", Computer Physics Communications, Vol. 48, p. 107, 1988. INTERPRETING THE FITS IMAGE FORMAT If you just need to examine FITS images, use the ppm package (see the comp.graphics FAQ) to convert them to your preferred format. For more information on the format and other software to read and write it, see the sci.astro.fits FAQ. SKY (UNIX EPHEMERIS PROGRAM) The 6th Edition of the Unix operating system came with several software systems not distributed because of older media capacity limitations. Included were an ephmeris, a satellite track, and speech synthesis software. The ephmeris, sky(6), is available within AT&T and to sites possessing a Unix source code license. The program is regarded as Unix source code. Sky is <0.5MB. Send proof of source code license to E. Miya MS 258-5 NASA Ames Research Center Moffett Field, CA 94035-1000 eugene@orville.nas.nasa.gov THREE-DIMENSIONAL STAR/GALAXY COORDINATES To generate 3D coordinates of astronomical objects, first obtain an astronomical database which specifies right ascension, declination, and parallax for the objects. Convert parallax into distance using the formula in part 6 of the FAQ, convert RA and declination to coordinates on a unit sphere (see some of the references on planetary positions and spherical trignometry earlier in this section for details on this), and scale this by the distance. Two databases useful for this purpose are the Yale Bright Star catalog (sources listed in FAQ section 3) or "The Catalogue of Stars within 25 parsecs of the Sun" (in pub/SPACE/FAQ/stars.data and stars.doc on ames.arc.nasa.gov). NEXT: FAQ #5/13 - References on specific areas