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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
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Mathematical Equations & Relationships: A. Orbital Mechanics and Motion Keplers 3rd Law: (MA + MB) = a3/ p2 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). 2 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 2 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 ; Fc = mac ; ac = v2/r = r2 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 (Fc) equals mass (m) times centripetal acceleration (ac), and centripetal acceleration (ac) equals velocity squared (v2) 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 (2). B. Stellar Radiation Wein's Law: max = 2.9 x 107/T This law relates the maximum peak (angstroms) output of radiation from an emitting object (max) to its temperature (T) in Kelvin (K). Stephan-Boltzmann Law: L = 4R2T4 This involves the total luminosity (L) from a stellar surface, which is the produce of its surface area (4R2) and temperature (T) to the fourth power. Another form of this relationship is E = Teff4 where Teff is the effective surface temperature in Kelvin, and E is the energy per unit surface area in erg/cm2. is the Stefan-Boltzmann constant, 5.70 x 10-5 erg/cm2K4s. Other forms of the Stephan-Boltzmann law are as follows: L/Lsun = (R/Rsun)2 x (T/Tsun)4 or R/Rsun = (Tsun/T)2 x L/Lsun These simpler rearrangements express the stellar properties in terms of solar properties. C. Luminosity The Distance Modulus: M = m - 5log10 (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

Inverse Square Law: L = 1/r2 or L = (r/rsun)2 x b/bsun or b/bsun = (r/rsun)2 Light, or luminosity, is one of several phenomena that decrease in brightness as the square of the distance. This can be expressed in many ways - two examples are given above. The distance (r), luminosity (L), or brightness (b) can be written relative to the Sun. Tully-Fisher Relation: L = Vrot4 The luminosity of any spiral galaxy is equal to the 4th power of its rotation, or, the faster a galaxy spins, the more luminous it is. There is a correlation between spin rate and luminosity because the gas and stars are in orbit in the galaxy, so the centripetal and gravitational forces are in balance. Mathematically, v2/r - GMgalaxy/R2 = 0. This shows that the greater the rotation, the more mass the galaxy has to have to maintain a balance between the two forces. So the faster a galaxy rotates, the more massive it must be - and the more luminous. D. Expansion of the Universe Hubble's Law: vr= H0d Hubble's law states that the recessional velocity (vr)of a distant galaxy is equal to its distance (d) times Hubble's constant. (Assume 70km/s/Mpc for H0.) The recessional velocity is determined from the Doppler redshift (z) of the H and K lines in the spectrum of the receding galaxy. Doppler Effect Equation: 0 = r/c ; z = - 0/ 0 ; vr zc The first equation is the combination of the next two. First, z = - 0/ 0 is used to measure the redshift (z) by comparing the spectral lines from the galaxy () and the known spectral lines (0). The recessional velocity (vr)is then equal to the redshift(z) times the speed of light (c). E. Other Important Calculations: Parallax for nearby stars: d = 1/p Frequency, wavelength, and speed of light: f = c Time spent on the main sequence: t(years) = 1010m/L Mass-Luminosity relationship on main sequence: L M4 or in the more expanded form: L/L(Sun) ~ [M/M(Sun)]4 D. Basic Math & Conversion Factors Circumference, Area, Surface Area, and Volume of a Sphere Since most stars, star clusters, clusters of galaxies, etc are spherical, there are many instances where the formulas for the above dimensions are useful. 1 parsec (pc) = 206,265 astronomical units (au) = 3.26 light years (ly) = 3.08 x 1016m ; 1 = 60 arcmin = 60 ; 1 = 60 arcsec = 60

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Derivations and Sample Problems: Parallax

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Derivations and Sample Problems: Orbital Mechanics

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Derivations and Sample Problems: Radiation Laws

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Derivations and Sample Problems: The Distance Modulus and Luminosity

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Derivations and Sample Problems: Hubble's Law and the Doppler Effect

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