spaceformulae.txt

Copyright (c) 1996 by Robert P. Stefko <stefko@westol.com>
Stellar Mass
	The mass values given in GURPS Space are approximations.  For the
(more or less) exact mass of a star in solar masses, use the following
formulas.  (Note:  You need to know the luminosity of the star; see
below.)
	For a star dimmer than the sun, mass equals roughly the 4.5 root of
the star's luminosity.
	For a star brighter than the sun, mass equals roughly the 3.5 root of
the star's luminosity.

	Example:  Sirius is 23.5 times brighter than the sun (luminosity =
23.5).  The 3.5 root of 23.5 is about 2.5, so Sirius is 2.5 times more
massive than the sun.

Stellar Luminosity
	If you don't know a star's luminosity, but you do know its absolute
magnitude (which is given in most astronomy books -- check one out
from your local library), use the following formula to find
luminosity:
	L = 1/(2.5^(M-4.8)).  L = luminosity, M = absolute magnitude.

	Example:  For Epsilon Eridani, M = 6.1.  Using the formula, we find
that L is about .3.  Epsilon Eridani is .3 times as bright as the sun.

Stellar Diameter
	A star's diameter in solar diameters can be determined if its surface
temperature is known.  The problem is, I can't find a way to
accurately determine a star's temperature without some pretty
expensive observation equipment.  (If you've found a book with that
kind of information, tell me!  My E-mail address is listed with this
posting.)  Anyway, the formula is:
	D = (5750/T)^2 x the square root of L.  D = diameter in solar
diameters, T = surface temperature in degrees Kelvin, and L =
luminosity.

	Example:  Alpha Centauri A has a surface temperature of about 5775ºK
(this is the only one I'm halfway sure about) and is 1.3 times as
luminous as the sun.  Therefore, using the formula, its diameter is
1.13 solar diameters.

Absolute Magnitude
	On the rare occasion where a star's absolute magnitude isn't given,
but its luminosity is, use this formula:
	M = 4.8 + 2.5 log 1/L.  M = absolute magnitude, L = luminosity.

	Example:  61 Cygni A's luminosity is .082.  Using the formula, we find
that M = 7.5.

	If the apparent magnitude and distance (in parsecs) are given, use
this formula:
	M = (((log D-1)/.2) - m) x -1.  M = absolute magnitude, D = distance
in parsecs, m = apparent magnitude.

	Example:  Fomalhaut is 6.8 parsecs away and has an apparent magnitude
of 1.2.  Using the formula, we find that M = 2.0.

Stellar Distances
	If you don't know how far away a star is, but you do know the star's
absolute and apparent magnitudes (or figured them out with these
formulas), use this formula:
	D = 10^(.2(m-M)+1).  D = distance in parsecs, m = apparent magnitude,
M = absolute magnitude.

	Example:  Rasalhague has an absolute magnitude of .8 and an apparent
magnitude of 2.1.  Using the formula, we find that D = 18.2.
Rasalhague is 18.2 parsecs away.

Sunlight Received by Planet
	If you want to know exactly how much sunlight a planet is receiving
from its primary, all you need to know is the star's luminosity and
the distance of the planet from the star in AU . . . and, of course,
this formula:
	S = L/D^2.  S = sunlight received by planet (equals 1 for Earth), L =
star's luminosity, D = distance of planet from star in AU.
	Though there is some disagreement on this issue, it is commonly held
that life as we know it can develop on worlds where S is between 1.35
and .65.

	Example:  A planet circling Alpha Centauri A (luminosity 1.3) at 1.14
AU would receive almost exactly the same amount of sunlight as Earth
(S = 1).

Binary Separation
	The distance between two stars in a binary system can be determined if
their individual masses and period of revolution are known.
	P^2/(M+m) = the cube root of A.  P = period of revolution in years, M
= mass of heavier star, m = mass of lighter star, A = average
separation in AU.

	Example:  Alpha Centauri A and B have a period of revolution of 80
years and mass 1.92 solar masses total.  Using the formula, we find
that the two stars orbit each other at an average of about 15 AU (the
actual distance ranges between 11 and 19 AU).

Stellar Coordinates
	To find the coordinates of a star on a three-dimensional grid map, you
must know its declination and right ascension in degrees and its
distance in light years.  These are usually given in an astronomy book
to make locating them in the sky easier.  Before you find the actual
coordinates, you need to find a value I like to call the "h-bar" (it
stands for horizontal bar, the line along the horizontal plane that
leads to the point directly above or below the stars position in 3D
space).  The h-bar will be used in the formulas used to determine
actual coordinates.  Note that this is basically just high school
trigonometry, so drawing and labeling a right triangle may help you
visualize this process better.
	h-bar = cos D x d.  D = declination in degrees, d = distance in light
years.
	x = cos R x h.  x = X-coordinate, R = right ascension in degrees (1
hour = 15 degrees), h = h-bar.
	y = sin R x h.  y = Y-coordinate, R = right ascension in degrees, h =
h-bar.
	z = sin D x d.  z = Z-coordinate, D = declination in degrees, d =
distance in light years.

	Example:  For Alpha Centauri, declination is -60 degrees, right
ascension is 14 hours 40 minutes (~211 degrees), and distance is 4.37
light years.  Therefore, h-bar = 2.2, x = -1.9, y = -1.1, z = -3.8.
The coordinates for Alpha Centauri are -1.9, -1.1, -3.8 (in light
years).