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From: [e-mail] (Jon Leech)
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Subject: Space FAQ 04/13 - Calculations
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Archive-name: space/math
Last-modified: $Date: 96/09/17 15:40:28 $

    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
    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?


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 Perturbations 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 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, (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. 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 Earth-crossing
    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 S-type 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 nickel-iron meteor
    v = 2e4 m/s        Impact velocity characteristic of asteroids
                in Earth-crossing 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     10e-8            1e9 (1 billion)
      1               1e3     10e-6            1e6 (1 million)
      0.1           1e5     10e-4            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 0-306-43163-7.

    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

    Source code for cartographic projections may be found in

    ftp://charon.er.usgs.gov/pub/PROJ.4/

    Two easy-to-find 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 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.


    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
        40-Lightyear Radius)"
    Terry Kepner
    ISBN # 0-926895-02-8

    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