Copyright (c) 1996 by Robert P. Stefko 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 5775†K (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 trigenometry, 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).