Archivename: space/math
Lastmodified: $Date: 96/09/17 15:40:28 $ See reader questions & answers on this topic!  Help others by sharing your knowledge Compilation copyright (c) 1994, 1995, 1996 by Jonathan P. Leech. This document may be redistributed in its complete and unmodified form. Other use requires written permission of the author. 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 blueskying 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 EarthMoon distance 1.496e11 m (15e10)  Mean EarthSun 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 semimajor 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*1.42e4) 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 TitiusBode 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 6.62618e34 Js (7e34)  Planck's Constant "h" 1.054589e34 Js (1e34)  Planck's Constant / (2 * PI), "h bar" 1.3807e23 J/K (1.4e23)  Boltzmann's Constant "k" 5.6697e8 W/m^2/K (6e8)  StephanBoltzmann Constant "sigma" 6.673e11 N m^2/kg^2 (7e11)  Newton's Gravitational Constant "G" 0.0029 m K (3e3)  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. Noncircular 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 0486600610 NASA Spaceflight handbooks (dating from the 1960s) SP33 Orbital Flight Handbook (3 parts) SP34 Lunar Flight Handbook (3 parts) SP35 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 32464, 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 Perturbations for the Design of DeepSpace SolarProbe and Out of Ecliptic Trajectories_, Technical Report 32849, JPL, Pasadena, Calif., 1965. You need to read the first one first to really 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, (WileyInterscience, New York, 1985). Starts with the basics (2body problem, coordinates) and works up to orbit determinations, perturbations, and differential corrections. Taff also briefly discusses stellar dynamics including a short discussion of nbody problems. COMPUTING PLANETARY POSITIONS More net references: "Explanatory Supplement to the Astronomical Almanac" (revised edition), Kenneth Seidelmann, University Science Books, 1992. ISBN 0935702687. $65 in hardcover. Deep math for all the algorthms and tables in the AA. Van Flandern & Pullinen, _LowPrecision Formulae for Planetary Positions_, Astrophysical J. Supp Series, 41:391411, 1979. Look in an astronomy or physics library for this; also said to be available from WillmannBell. 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 19901999. $55 ($80 outside US). Available for IBM (order #PB93500163HDV) or Macintosh (order #PB93500155HDV). From the NTIS sales desk, (703)4874650. I believe this is intended to replace the USNO's Interactive Computer Ephemeris. _Interactive Computer Ephemeris_ (from the US Naval Observatory) distributed on IBMPC floppy disks, $35 (WillmannBell). Covers dates 18002049. "Planetary Programs and Tables from 4000 to +2800", Bretagnon & Simon 1986, WillmannBell. Floppy disks available separately. "Fundamentals of Celestial Mechanics" (2nd ed), J.M.A. Danby 1988, WillmannBell. 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, WillmannBell. "Astronomical Algorithms", J. Meeus 1991, WillmannBell. 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. DuffettSmith 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 142260605 COMPUTING CRATER DIAMETERS FROM EARTHIMPACTING ASTEROIDS Astrogeologist Gene Shoemaker proposes the following formula, based on studies of cratering caused by nuclear tests. Units are MKS unless otherwise noted; impact energy is sometimes expressed in nuclear bomb terms (kilotons TNT equivalent) due to the origin of the model. D = Sg Sp Kn W^(1/3.4) Crater diameter, meters. On Earth, if D > 3 km, the crater is assumed to collapse by a factor of 1.3 due to gravity. Sg = (ge/gt)^(1/6) Gravity correction factor cited for craters on the Moon. May hold true for other bodies. ge = 9.8 m/s^2 is Earth gravity, gt is gravity of the target body. Sp = (pa/pt)^(1/3.4) Density correction factor for target material relative to the Jangle U nuclear crater site. pa = 1.8e3 kg/m^3 (1.8 gm/cm^3) for alluvium, pt = density at the impact site. For reference, average rock on the continental shields has a density of 2.6e3 kg/m^3 (2.6 gm/cm^3). Kn = 74 m / (kiloton TNT equivalent)^(1/3.4) Empirically determined scaling factor from bomb yield to crater diameter at Jangle U. W = Ke / (4.185e12 joules/KT) Kinetic energy of asteroid, kilotons TNT equivalent. Ke = 1/2 m v^2 Kinetic energy of asteroid, joules. v = impact velocity of asteroid, m/s. 2e4 m/s (20 km/s) is common for an asteroid in an Earthcrossing orbit. m = 4/3 pi r^3 rho Mass of asteroid, kg. r = radius of asteroid, m rho = density of asteroid, kg/m^3 3.3e3 kg/m^3 (3 gm/cm^3) is reasonable for a common Stype asteroid. For an example, let's work the body which created the 1.1 km diameter Barringer Meteor Crater in Arizona (in reality the model was run backwards from the known crater size to estimate the meteor size, but this is just to show how the math works): r = 40 m Meteor radius rho = 7.8e3 kg/m^3 Density of nickeliron meteor v = 2e4 m/s Impact velocity characteristic of asteroids in Earthcrossing orbits pt = 2.3e3 kg/m^3 Density of Arizona at impact site Sg = 1 No correction for impact on Earth Sp = (1.8/2.3)^(1/3.4) = .93 m = 4/3 pi 40^3 7.8e3 = 2.61e8 kg Ke = 1/2 * 2.61e8 kg * (2e4 m/s)^2 = 5.22e16 joules W = 5.22e16 / 4.185e12 = 12,470 KT D = 1 * .93 * 74 * 12470^(1/3.4) = 1100 meters More generally, one can use (after Gehrels, 1985): Asteroid Number of Impact probability Impact energy as multiple diameter (km) Objects (impacts/year) of Hiroshima bomb     10 10 10e8 1e9 (1 billion) 1 1e3 10e6 1e6 (1 million) 0.1 1e5 10e4 1e3 (1 thousand) The Hiroshima explosion is assumed to be 13 kilotons. Finally, a back of the envelope rule is that an object moving at a speed of 3 km/s has kinetic energy equal to the explosive energy of an equal mass of TNT; thus a 10 ton asteroid moving at 30 km/sec would have an impact energy of (10 ton) (30 km/sec / 3 km/sec)^2 = 1 KT. References: Clark Chapman and David Morrison, "Cosmic Catastrophes", Plenum Press 1989, ISBN 0306431637. Gehrels, T. 1985 Asteroids and comets. _Physics Today_ 38, 3241. [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, 461494. [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 Earthcrossing asteroids: Orbital classes, collision rates with Earth, and origin. In _Asteroids_, T. Gehrels, ed., pp. 253282, University of Arizona Press, Tucson. Cunningham, C.J. 1988 _Introduction to Asteroids: The Next Frontier_ (Richmond: WillmanBell, 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 Source code for cartographic projections may be found in ftp://charon.er.usgs.gov/pub/PROJ.4/ Two easytofind sources of map projections are the "Encyclopaedia Britannica", (particularly the older editions) and a tutorial appearing in _Graphics Gems_ (Academic Press, 1990). The latter was written with simplicity of exposition and suitability for digital computation in mind (spherical trig formulae also appear, as do digitallyplotted examples). More than you ever cared to know about map projections is in John Snyder's USGS publication "Map ProjectionsA 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)USAMAPS. They can quote you a price and tell you where to send your money. Expect a 68 week turnaround time. A much faster way (about 1 week) is through Timely Discount Topos, (303)4695022, 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 $34 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 NBODY 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, 73:325348, 1987. "An O(N) Algorithm for Threedimensional Nbody 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 Manybody Simulation", SIAM Journal of Scientific and Statistical Computing, Vol. 6, p. 85, 1985. Barnes & Hut, "A Hierarchical O(N log N) ForceCalculation Algorithm", Nature, V324 # 6096, 410 Dec 1986. L. Hernquist, "Hierarchical Nbody 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. NEARBY 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 ftp://ftp.cs.unc.edu/pub/users/leech/FAQ/ (files stars.data and stars.doc) A potentially useful book along these lines is: "Proximity Zero, A Writer's Guide to the Nearest 200 Stars (A 40Lightyear Radius)" Terry Kepner ISBN # 0926895028 Available from the author for $14.95 + $2.90 shipping ($5 outside US): Terry Kepner PO Box 481 Petersborough, NH 03458 NEXT: FAQ #5/13  References on specific areas User Contributions:Comment about this article, ask questions, or add new information about this topic:
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