Semester 1 - Formulae Flashcards
Spherical trig
sinA/a = sinB/b = sinC/c
Specific flux
Fv = (2π ∫ 0) (π/2 ∫ -π/2) Iv (θ,φ) cosθsinθdθdφ
μ and dμ
μ = cosθ
dμ = -sinθdθ
integrating factor
dy/dx + P(x)y = Q(x)
where the integrating factor is μ(x) = exp^( ∫P(x)dx)
hence the ODE becomes
d/dx (μ(x)y) = μ(x)Q(x)
intensity emitted
dIv = p jv ds
where jv is the emission coefficient
intensity absorbed/gained
dIv = -p kv Iv ds
where kv is the absorption coefficient
optical depth
dτv = pkvds
angle-dependent definition of optical depth
dτv,μ = -dτv/μ
mean intensity
Jv = 1/4π ∮ Iv dΩ
equation of transfer
μ dIv/dτ = Iv - Sv
relation between eddington flux and k integral
Hv = dKv/dτ
total momentum
dp = dEv/c cosθ
where dp is the total momentum
dEv =
IvcosθdAdΩdt
dΩ =
sinθdθdφ
Eddington approximation
K = I/3 = J/3
Eddington Barbier relation
F = π Sv
where Sv ~ Bv
equation of transfer in terms of absorption and emission coefficient
dIv/ds = p jv - pkv Iv
Source function
Sv = jv/kv
line depth
Aλ = 1 - Rλ
where Rλ = Fλ/Fc
in a naturally broadened line to find the normalisation constant.
∆v = v - v0 = 0
HWHM
HWHM = ∆v
when
I = Ipeak/2
Heisenberg’s Uncertainty Principle
∆E∆t > ℏ/2
Planck’s law
∆E = h∆v
apparent equatorial velocity
vm = v0sini
rotational broadening HWHM
HWHM = ∆λ
rotational broadening ∆λ =
vm λ0 µ/c
Intensity is constant along a ray path
show that dIv/ds = 0
This occurs when the emission coefficient and total absorption coefficient = 0
which occurs when there are no sources or sinks
Energy density
du = dE/dV
where dV = dAcosθl
where dt = l/c
Components of flux
Fv =Fv+ + Fv-
Split the limits
Total absorption coefficient
kv = kappa v + σv
Mean free path
L = 1/σvp = 1/Nσ
Planckian Atmosphere
J = B = S
Integrated Planck function B
B = σ/pi T^4
Integrated flux F
F = σTe^4
Rosseland Mean Opacity
Start from K integral and definition of opacity
Kv = Jv/3 = Bv/3
rewrite for Hv
dB/dT dT/ds
Integrate Hv over all frequencies
divide two versions of H
use relation for B
Einstein coefficient Cij and Cji
Cij/Cji = gj/gi exp(-hv0/kT)
Collision processes
Line formation of a weak atomic line
τc = 2/3 + Δτ
and kλ = kl + kc
Doppler shifts
Δv/v = -Δλ/λ
microturbulence
V0^2 = Vthermal^2 + Vturb^2
line of sight velocity
vr = veq sin θ sin φ sin i
Show S = 3F/4pi (tau + C)
starting from dKv/dtau= Hv = Fv/4pi
grey atmosphere means no frequency dependence
Eddington’s approximation K = I/3 = J/3
J = B = S in a Planckian atmosphere
dB/dtau = 3F/4pi
integrating gives as required
At the surface
tau = 0
At the edge
mu = 0
at the centre
mu = 1
Radiation Pressure
dPrad/dr = -pgrad
where grad is on the formula sheet
Integration limits for solutions to the equation of transfer
(τ2 ∫ τ1) for the outward direction τ2 = τ0 and τ1 -> infinity
for the inward direction replace mu with -mu
Show that the ratio of intensities at the limb and centre is 40%
I(0,0)/I(0,1) = 0.4
Doppler shift
Δ λ/λ = Δ v/v
Show Fv = πIvR^2/r^2
Draw a star with a resolvable disc
Start with definition of flux with limits pi/2 to -pi/2 and theta to -theta.
Use spherical trig. relationship and change variables
Solving the equation of transfer ; dIv/dτ = Sv - Iv.
State equation of transfer
Use an integrating factor
Integration limits are τv and 0.
Show that dFv/dτ = 4pi(Jv -Sv)
dHv/dτ = 1/4pi dFv/dτ
use equation of transfer
Determine the constant, C, if I = 3F/4pi (τ+mu+C) by considering the conditions at the surface of a star.
At the surface of the star τ = 0 and looking radially outwards.
Definition of flux insert definition of I. Solve for C. Where C = 2/3.
Eddington-barbier source function.
S(τ) = a0 + a1τ
Show that A = 2/3 kL/kc d(lnBv)/dτ
Line depth use Eddington barrier relation Fv = piBv
Taylor expand around small t
replace τ through definition of opacity
