Equations Flashcards
Keplers 3rd Law
a^3 / P^2 = G(m1 +m2) / (4*pi^2)
Virial Thm
1/2 d^2 I / dt^2 = 2K + U
= 3 integral PdV + U
Magnetic Pressure
P_b = B^2 / (8 Pi)
Radiative Transfer Equation - General
dI_nu/dTau_nu = -I_nu + S_nu
Planck Blackbody
B_nu = (2 h v^3 / c^2) * 1/(exp(h nu / k T)-1)
Stellar Mass Conservation
d M / d r = 4 Pi r^2 rho
Stellar Energy Conservation
d L / d r = 4 Pi r^2 rho epsilon
Stellar HSE
d P / d r = - G M rho / r^2
Stellar Radiative Energy Transp.
dT / dr = - 3 Kappa rho L / (4 a c T^3 4 Pi r^2)
Lamor
dE / dt = 2 q^2 a^2 / (3 c^3)
Fluid - continuity
drho / dt + del dot (rho u) = 0
Fluid - momentum
d(pu) / dt + del dot (rho u u + P - pi) = rho g +f
fluids - energy
dE / dt + del dot [u(E+P)] = -rho u dot del Phi
d(p v^2 / 2 + E) +del dot [v(p v^2 / 2 + (P + E) -
pi + F_c] = rho v Fi / m
eqn of state
P = rho k T / m_bar
Convective energy transport
dT/dR_ad = (1-1/Gamma)T/Pg dP/dr
PE of sphere
- 3 / 5 G M^2 / R
M of elliptical gal or glob cluster
M = 5 (sigma_los)^2 * R / G
FE - a
a_dot^2 - 8 Pi G rho / 3 * a^2 = -kappa c^2
FE - omega
H_0 ^2 (omega_0-1) = kappa * c^2
Luminosity (T)
4 Pi R^2 Sigma T^4
Diffraction Limit (Beta)
1.22 Lamda / D
ISM - J to L
4 Pi J = L / (4 Pi d^2)
Isothermal Slab
I = I0 exp(-Tau) + B(1-exp(-Tau))
Relationship Tau and Sigma
Tau = sigma n d
RJ Limit B
b = 2 nu^2 k T / c^2
Weins Law
lamda = 2900 um K / T
Tb
Tb = c^2 I / (2 k nu^2)
Fermi momentum
p_f = (3 n h^3 /(8 pi))^1/3
Fermi Pressure
P_f = 8 pi / 15 * (pf^5/m_e h^3) goes as n^5/3
Photon momentum
p_gamma = E / c = h nu / c
Bernoullis eqn
v^2/2 - G M / R + integral(P/rho) = const
t_relax
t_relax = N / (6 ln(N/2)) * R / v
kramer opacity
kappa = rho T^(-7/2)
ideal gas adiabatic dT / dr
dT / dr_ab = 2 T / (3 rho) * d rho / d r
ideal gas energy
u = 3 k T / (2 m_bar)
ideal gas sound speed
C_s = ( 5 k T / 3 m_bar)^1/2
fundamental plane
R I sigma = const. (approx.)
schechter function
phi = phi_0 (L/L_star)^alpha exp (-L/L_star) d(L/L_star)
sersic profile
I = I_e exp(-b[(R/R_e)^1/n -1])
Cosmology Fluid Equation
u_dot a / a_dot = -3 (u + P)
Cosmology acceleration eqn
a_dbldot / a = - (4 pi G / 3) * (rho + 3P/c^2)
H J K J
H = F / 4 pi J = c / 4 pi * u = 1/2 (I_plus + I_minus) K = c / 4 pi * p = 1/4 (I_plus + I_minus) J = 3H (Tau + 2/3) = sigma T^4 / pi
T = f(Tau)
T = T_eff ( 3/4(Tau + 2/3))