Semester 1 - Formulae

Question 1

Spherical trig

Answer 1

sinA/a = sinB/b = sinC/c

Question 2

Specific flux

Answer 2

Fv = (2π ∫ 0) (π/2 ∫ -π/2) Iv (θ,φ) cosθsinθdθdφ

Question 3

μ and dμ

Answer 3

μ = cosθ
dμ = -sinθdθ

Question 4

integrating factor

Answer 4

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)

Question 5

intensity emitted

Answer 5

dIv = p jv ds

where jv is the emission coefficient

Question 6

intensity absorbed/gained

Answer 6

dIv = -p kv Iv ds

where kv is the absorption coefficient

Question 7

optical depth

Answer 7

dτv = pkvds

Question 8

angle-dependent definition of optical depth

Answer 8

dτv,μ = -dτv/μ

Question 9

mean intensity

Answer 9

Jv = 1/4π ∮ Iv dΩ

Question 10

equation of transfer

Answer 10

μ dIv/dτ = Iv - Sv

Question 11

relation between eddington flux and k integral

Answer 11

Hv = dKv/dτ

Question 12

total momentum

Answer 12

dp = dEv/c cosθ

where dp is the total momentum

Question 13

dEv =

Answer 13

IvcosθdAdΩdt

Question 14

dΩ =

Answer 14

sinθdθdφ

Question 15

Eddington approximation

Answer 15

K = I/3 = J/3

Question 16

Eddington Barbier relation

Answer 16

F = π Sv

where Sv ~ Bv

Question 17

equation of transfer in terms of absorption and emission coefficient

Answer 17

dIv/ds = p jv - pkv Iv

Question 18

Source function

Answer 18

Sv = jv/kv

Question 19

line depth

Answer 19

Aλ = 1 - Rλ

where Rλ = Fλ/Fc

Question 20

in a naturally broadened line to find the normalisation constant.

Answer 20

∆v = v - v0 = 0

Question 21

HWHM

Answer 21

HWHM = ∆v

when

I = Ipeak/2

Question 22

Heisenberg’s Uncertainty Principle

Answer 22

∆E∆t > ℏ/2

Question 23

Planck’s law

Answer 23

∆E = h∆v

Question 24

apparent equatorial velocity

Answer 24

vm = v0sini

Question 25

rotational broadening HWHM

Answer 25

HWHM = ∆λ

Question 26

rotational broadening ∆λ =

Answer 26

vm λ0 µ/c

Question 27

Intensity is constant along a ray path

Answer 27

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

Question 28

Energy density

Answer 28

du = dE/dV

where dV = dAcosθl

where dt = l/c

Question 29

Components of flux

Answer 29

Fv =Fv+ + Fv-

Split the limits

Question 30

Total absorption coefficient

Answer 30

kv = kappa v + σv

Question 31

Mean free path

Answer 31

L = 1/σvp = 1/Nσ

Question 32

Planckian Atmosphere

Answer 32

J = B = S

Question 33

Integrated Planck function B

Answer 33

B = σ/pi T^4

Question 34

Integrated flux F

Answer 34

F = σTe^4

Question 35

Rosseland Mean Opacity

Answer 35

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

Question 36

Einstein coefficient Cij and Cji

Answer 36

Cij/Cji = gj/gi exp(-hv0/kT)

Collision processes

Question 37

Line formation of a weak atomic line

Answer 37

τc = 2/3 + Δτ

and kλ = kl + kc

Question 38

Doppler shifts

Answer 38

Δv/v = -Δλ/λ

Question 39

microturbulence

Answer 39

V0^2 = Vthermal^2 + Vturb^2

Question 40

line of sight velocity

Answer 40

vr = veq sin θ sin φ sin i

Question 41

Show S = 3F/4pi (tau + C)

starting from dKv/dtau= Hv = Fv/4pi

Answer 41

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

Question 42

At the surface

Answer 42

tau = 0

Question 43

At the edge

Answer 43

mu = 0

Question 44

at the centre

Answer 44

mu = 1

Question 45

Radiation Pressure

Answer 45

dPrad/dr = -pgrad

where grad is on the formula sheet

Question 46

Integration limits for solutions to the equation of transfer

Answer 46

(τ2 ∫ τ1) for the outward direction τ2 = τ0 and τ1 -> infinity

for the inward direction replace mu with -mu

Question 47

Show that the ratio of intensities at the limb and centre is 40%

Answer 47

I(0,0)/I(0,1) = 0.4

Question 48

Doppler shift

Answer 48

Δ λ/λ = Δ v/v

Question 49

Show Fv = πIvR^2/r^2

Answer 49

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

Question 50

Solving the equation of transfer ; dIv/dτ = Sv - Iv.

Answer 50

State equation of transfer

Use an integrating factor

Integration limits are τv and 0.

Question 51

Show that dFv/dτ = 4pi(Jv -Sv)

Answer 51

dHv/dτ = 1/4pi dFv/dτ

use equation of transfer

Question 52

Determine the constant, C, if I = 3F/4pi (τ+mu+C) by considering the conditions at the surface of a star.

Answer 52

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.

Question 53

Eddington-barbier source function.

Answer 53

S(τ) = a0 + a1τ

Question 54

Show that A = 2/3 kL/kc d(lnBv)/dτ

Answer 54

Line depth use Eddington barrier relation Fv = piBv

Taylor expand around small t

replace τ through definition of opacity