SPACE/MATH - Constants and Equations for Calculations

Last-modified: $Date: 94/03/01 17:24:39 $

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:

explorer.arc.nasa.gov:pub/SPACE/FAQ/MoreEquations

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 = 9.80665 m / s^2

Ve = Isp * g

dv = Ve * ln((m1 + x) / m1)

= Ve * ln((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) -- \

3.2616 light years (3) -- --> parsec

3.0856e16 m (3e16) -- /

Black Hole radius (also called Schwarzschild Radius):

2GM/c^2, where G is Newton's Grav Constant, M is mass of BH,

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 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:

"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.

(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 22

W = pi * d * delta * V / (12 * 4.185 * 10 )

: projectile kinetic energy in MT 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 as

diameter (km) (impacts/year) multiple of

Hiroshima bomb

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. The Hiroshima explosion is assumed to be

.013 MT TNT equivalent, or about 5*10^13 joules.

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

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.

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

ftp://explorer.arc.nasa.gov/pub/SPACE/FAQ/ (files stars.data,stars.doc)

NEXT: FAQ #5/13 - References on specific areas

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This is a static and possibly incomplete copy of the USEnet SCI.SPACE Frequently Asked Questions (FAQ) document dated March 1994 (before the material was copyrighted). The official copyrighted and and up-to-date FAQ is maintained by Jon Leech (leech@cs.unc.edu) and is posted to Network News and available via the World Wide Web . KSC's Hypertext converter last run Thursday June 15 10:22:40 EDT 1995