Mathematical Equations & Relationships: A. Orbital Mechanics and Motion Kepler’s 3rd Law: (MA + MB) = a 3 / p 2 Kepler's law is useful for any orbital motion, such as two stars in a binary system. It is a relationship among mass (M), period (p), and distance of separation in au's (a). 2π r = vP This equation is used to determine the rotational periods of an object. During one rotation a point in the equatorial region will travel a distance equal to 2π r. This distance is equal to the velocity of the point times the time elapsed during one rotation. It is a relationship among radius, velocity, and period. If you know any two of the variables, you can solve for the third variable. v = d/t ; a = v/t ; F c = ma c ; a c = v 2 /r = rω 2 These fundamental physics of motion equations should not be forgotten. Everything is moving in space, and for even stars and galaxies velocity (v) equals the rate at which distance (d) changes over time (t), and acceleration (a) is equal to the rate at which velocity changes over time. Everything also rotates in space, and therefore centripetal forces also apply. Centripetal force (F c ) equals mass (m) times centripetal acceleration (a c ), and centripetal acceleration (a c ) equals velocity squared (v 2 ) divided by the radius (r). Since velocity on a spinning object is an angular displacement, angular acceleration is also equal to radius times angular velocity squared (ω 2 ). B. Stellar Radiation Wein's Law: λ max = 2.9 x 10 7 /T This law relates the maximum peak (angstroms) output of radiation from an emitting object (λ max ) to its temperature (T) in Kelvin (K). Stephan-Boltzmann Law: L = 4πR 2 σT 4 This involves the total luminosity (L) from a stellar surface, which is the produce of its surface area (4πR 2 ) and temperature (T) to the fourth power. Another form of this relationship is E = σT eff 4 where T eff is the effective surface temperature in Kelvin, and E is the energy per unit surface area in erg/cm 2 . σ is the Stefan-Boltzmann constant, 5.70 x 10 -5 erg/cm 2 K 4 s. Other forms of the Stephan-Boltzmann law are as follows: L/L sun = (R/R sun )2 x (T/T sun ) 4 or R/R sun = (T sun /T) 2 x √L/L sun These simpler rearrangements express the stellar properties in terms of solar properties. C. Luminosity The Distance Modulus: M = m - 5log 10 (d)/10 This is a relationship among absolute magnitude (M) - or luminosity, apparent magnitude (m), and distance (d). If you know any two of these three variables, you can use this relationship to find the third variable. Used with Cepheid and RR Lyrae variable stars, and the other standard candles that measure cosmological distances. 82