Archive-name: space/math
Last-modified: $Date: 93/04/01 14:39:12 $

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:

    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).
