equationsheet

Equationsheet - Physics 160 Fall 2011 Midterm Exam Equation Sheet Physical constants Quantity Value Units

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Unformatted text preview: Physics 160 Fall 2011 Midterm Exam Equation Sheet Physical constants Quantity Value Units Stefan-Boltzmann constant (σ) Radiation constant (a) Electron mass (me) Proton mass (mp) Neutron mass (mn) Hydrogen mass (mH) Atomic mass unit (u) Avogadro's number (NA) Gas constant (R) Bohr radius (a0) 6.673x10-11 3.000x108 1.602x10-19 6.626x10-34 1.381x10-23 8.617x10-5 5.670x10-8 7.566x1016 9.109x10-31 1.67262x10-27 1.67493x10-27 1.67353x10-27 1.66054x10-27 6.022x1023 8.314 5.292x10-11 N m2 kg- 2 m s- 1 C Js J K- 1 ev K- 1 W m- 2 K- 4 J m- 3 K- 4 kg kg kg kg kg mol-1 J mol-1 K-1 m Solar Solar Solar Solar Solar Solar Earth Earth 1.989x1030 3.839x1026 6.955x108 5800 4.74 G2 V 5.974x1024 6.378x106 kg W m K λ0 for Lyα line (HI n=1 → n=2) λ0 for Hα line (HI n=2 → n=3) 121.6 656.3 nm nm Astronomical unit (AU) Light year (ly) Parsec (pc) Electron volt (eV) Radian π e 1.496x1011 9.46x1015 3.086x1016 1.602x10-19 57.296 3.14159 2.71828 m m m J º Gravitational constant (G) Speed of light (c) Electric charge (e) Planck's constant (h) Boltzmann's constant (k) mass (M) luminosity (L) radius (R) effective temperature abs. bolometric magnitude Spectral Type mass (ME) radius (RE) kg m Physics 160 Fall 2011 Midterm Exam Equation Sheet Equations Definition of magnitude: m is the apparent magnitude, f is the flux Distance modulus: m1 – m2 =  ­2.5 log10 (f1/f2) M is the absolute magnitude and d is distance; note that MVega = 0 in all bands M – m =  ­5 log10 (d/10 pc) Bolometric magnitude: Mbol = 4.74  ­ 2.5 log10 (L/L) Bolometric correction: BCV = mbol – V = Mbol  ­ MV Angular separation: θ (radians) = separation/distance θ (“) = separation (AU) / distance (pc) For radians, separation and distance must be in same units Parallax: π(“) = 1/d(pc) Tangential speed: Vtan (km/s) = 4.74 µ (“/yr) d(pc) µ is the magnitude of proper motion Doppler shift: λ0 is the rest frame wavelength, Vrad is the radial motion, positive for motion away from observer ∆λ = λ 0 Stefan ­Boltzmann Law: F = σ T 4 L = 4π R2 σ T 4 F is the flux, T the temperature, L the luminosity and R the radius Blackbody distribution: 2hc2 1 Bλ (T ) = 5 hc/λkT λe − 1 B(T) is a spectral radiance: energy per unit time per unit area per unit wavelength per unit angular area Wein’s Displacement Law: λpeak is the peak wavelength of the blackbody distribution Vrad c λpeakT = 2898 µm K En = −χi /n2 Potential energy for e ­ orbital state n in a Hydrogen ­like atom: Degeneracy for orbital state n: gn = 2n2 χi = 13.6 eV for Hydrogen Maxwell ­Boltzmann Distribution: n(v) is the number density of particles with velocity v, n is the total number density, m is the particle mass n(v )dv = n Ideal gas pressure law: n and ρ are the number and mass densities, m = µmH is the individual mass M ≈ uNnucleon is the molar mass P = nkT = m 2π kT 3/2 e−mv ρRT ρkT = µmH M 2 /2kT 4π v 2 dv Physics 160 Fall 2011 Radiation Pressure Law Mean molecular weight: Aj = mj/mH, Nj is the number density and zj is the proton number of species j P= Boltzmann equation: Ni is the number of atoms in a given orbital state i Midterm Exam Equation Sheet 14 aT 3 j µn = Nj Aj j µi = Nj for neutrals Nj Aj j 1+zj j Nj for ionized gas Ni gi = e−(Ei −Ej )/kT Nj gj ∞ Partition function: gi e−(Ei −E1 )/kT Z1 (T ) = i=1 Saha equation: Nm is the number of atoms in ionization state m (m = 1 is a neutral atom), χm is the energy required to remove m electrons from a neutral atom 2 Zm+1 Nm+1 = Nm ne Zm 2π me kT h2 3 2 e− Mean free path: n and ρ are the number and mass densities of scatterers, σ is the cross section, κ is the absorption coefficient l = 1/nσ = 1/ρκ Definition of optical depth: dτλ = −nσλ ds = −ρκλ ds Optical depth is inward from surface Equation of Radiative Transfer: Sλ is the source function, jλ is the emission coefficient, κλ is the absorption coefficient, all wavelength ­dependent Eqn. of Hydrostatic Equilibrium: dIλ = Iλ − Sλ dτλ Sλ ≡ jλ /κλ P is the local pressure, ρ the mass density, M(r) the mass within radius r, g(r) the surface gravity at radius r M ( r ) ρ( r ) dPr = −ρ(r)g (r) = −G dr r2 Equation of Mass Conservation: dMr = 4π r2 ρ(r) dr Equation of Energy Transfer: L is the luminosity at radius r, ε is the energy generation rate per unit mass dLr = 4π r2 ρ(r) (r) dr Equation of Radiative Transport: κ is the mean opacity ¯ 3 κ(r)ρ(r) L(r) dT ¯ =− dr 4ac T 3 (r) 4π r2 (χm+1 −χm ) kT ...
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This note was uploaded on 02/26/2012 for the course PHYS 160 taught by Professor Norman,m during the Fall '08 term at UCSD.

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