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).