Semester 1 - Formulae Flashcards

1
Q

Spherical trig

A

sinA/a = sinB/b = sinC/c

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2
Q

Specific flux

A

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

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3
Q

μ and dμ

A

μ = cosθ
dμ = -sinθdθ

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4
Q

integrating factor

A

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)

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5
Q

intensity emitted

A

dIv = p jv ds

where jv is the emission coefficient

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6
Q

intensity absorbed/gained

A

dIv = -p kv Iv ds

where kv is the absorption coefficient

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7
Q

optical depth

A

dτv = pkvds

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8
Q

angle-dependent definition of optical depth

A

dτv,μ = -dτv/μ

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9
Q

mean intensity

A

Jv = 1/4π ∮ Iv dΩ

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10
Q

equation of transfer

A

μ dIv/dτ = Iv - Sv

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11
Q

relation between eddington flux and k integral

A

Hv = dKv/dτ

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12
Q

total momentum

A

dp = dEv/c cosθ

where dp is the total momentum

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13
Q

dEv =

A

IvcosθdAdΩdt

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14
Q

dΩ =

A

sinθdθdφ

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15
Q

Eddington approximation

A

K = I/3 = J/3

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16
Q

Eddington Barbier relation

A

F = π Sv

where Sv ~ Bv

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17
Q

equation of transfer in terms of absorption and emission coefficient

A

dIv/ds = p jv - pkv Iv

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18
Q

Source function

A

Sv = jv/kv

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19
Q

line depth

A

Aλ = 1 - Rλ

where Rλ = Fλ/Fc

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20
Q

in a naturally broadened line to find the normalisation constant.

A

∆v = v - v0 = 0

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21
Q

HWHM

A

HWHM = ∆v

when

I = Ipeak/2

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22
Q

Heisenberg’s Uncertainty Principle

A

∆E∆t > ℏ/2

23
Q

Planck’s law

A

∆E = h∆v

24
Q

apparent equatorial velocity

A

vm = v0sini

25
rotational broadening HWHM
HWHM = ∆λ
26
rotational broadening ∆λ =
vm λ0 µ/c
27
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
28
Energy density
du = dE/dV where dV = dAcosθl where dt = l/c
29
Components of flux
Fv =Fv+ + Fv- Split the limits
30
Total absorption coefficient
kv = kappa v + σv
31
Mean free path
L = 1/σvp = 1/Nσ
32
Planckian Atmosphere
J = B = S
33
Integrated Planck function B
B = σ/pi T^4
34
Integrated flux F
F = σTe^4
35
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
36
Einstein coefficient Cij and Cji
Cij/Cji = gj/gi exp(-hv0/kT) Collision processes
37
Line formation of a weak atomic line
τc = 2/3 + Δτ and kλ = kl + kc
38
Doppler shifts
Δv/v = -Δλ/λ
39
microturbulence
V0^2 = Vthermal^2 + Vturb^2
40
line of sight velocity
vr = veq sin θ sin φ sin i
41
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
42
At the surface
tau = 0
43
At the edge
mu = 0
44
at the centre
mu = 1
45
Radiation Pressure
dPrad/dr = -pgrad where grad is on the formula sheet
46
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
47
Show that the ratio of intensities at the limb and centre is 40%
I(0,0)/I(0,1) = 0.4
48
Doppler shift
Δ λ/λ = Δ v/v
49
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
50
Solving the equation of transfer ; dIv/dτ = Sv - Iv.
State equation of transfer Use an integrating factor Integration limits are τv and 0.
51
Show that dFv/dτ = 4pi(Jv -Sv)
dHv/dτ = 1/4pi dFv/dτ use equation of transfer
52
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.
53
Eddington-barbier source function.
S(τ) = a0 + a1τ
54
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