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Archive-name: space/math
Last-modified: \$Date: 96/09/17 15:40:28 \$

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```    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
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

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

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,

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

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

"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
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
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