Semester 2 - Formulae Flashcards

1
Q

Pressure

A

P = 2nkT

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2
Q

density

A

p = nmp

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3
Q

optical thickness of the corona

A

τ = n0 σT h

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4
Q

emission measure

A

EM = ( ∫vol) n^2 dV

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5
Q

nvr^2 =

A

constant

hence

n ∝ 1/vr^2

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6
Q

critical radius

A

rc = GM/2c^2

where c^2 = 2kT/mp

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7
Q

dynamic pressure

A

mp nv^2

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8
Q

ram pressure =

A

dynamic pressure

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9
Q

photon radiation pressure

A

P = ℏω/c

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10
Q

Luminosity of photons

A

L = nℏω

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11
Q

photon flux

A

Fn = n/4πR^2

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12
Q

force

A

F = P/A

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13
Q

Eddington luminosity

A

Ledd = 4πGMmc/σ

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14
Q

terminal wind speed

or CAK velocity profile

A

v∞ = [2GM/R (Γ-1)]^1/2

where Γ = L/Ledd

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15
Q

Rydberg equation

A

1/λ = R(1/n1^2 - 1/n2^2) = vn/c

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16
Q

mass loss rate

A

M(dot) = L/cv∞

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17
Q

conservation of momentum can be used to estimate the mass loss rate

A

Lω = 1/2 M(dot) v^2∞

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18
Q

multiple scattering affects the mass loss rate

A

as Σ|pi|

M(dot) = η L/cv∞

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19
Q

Spitzer thermal conductivity

A

κ = κ0 T^5/2

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20
Q

Heat flux

A

H = -κ0 T^5/2 dT/dr

21
Q

Maxwell-Faraday Law

A

∇ x E = -∂B/∂t

∇ x (v x B) = ∂B/∂t

22
Q

Lenz Law

A

Φ =( ∫ S ) B . da

23
Q

plasma beta

A

ß = 2µ0nkT/B^2

24
Q

Alfven speed

A

vA = B/sqrt(µ0p)

where p is the density

25
radiative instability (derivation)
power in = Cn power out - f(T)n^2 d/dt (3nkT) = Cn - n^2f(T) equilibrium dT(0)/dt = 0 T = T0 + ∆T we have f(T0+∆T) so taylor expand the differential equation has a solution ∆T = ∆T(0)exp[-nf'(T0)t/3k] only stable if f' > 0
26
downward heat conduction in the corona
-r0^2F0 = -r^2 κ0 T^5/2 dT/dr have r on one side and T the other Integrate between r and r0 and T and T0 rearrange for T
27
show that n(r) = n0 exp(-a(1-r0/r))
start from hydrostatic equilibrium ideal gas pressure p = 2nkT integrate to obtain expression Pull factor of r0 out and create a new constant
28
non-isothermal static atmosphere
start from hydrostatic equilibrium with ideal gas pressure differentiate both n and T with respect to r 2k(T dn/dr + n dT/dr) … Divide by nT n(r) -> 0 and T(r) -> 0 as r -> ∞ T(r) ∝ r^-a with alpha > 1
29
chapman model derivation
starting from heat flux has a factor 4pir^2 rearrange for T and integrating with T -> 0 as r -> ∞ T(r) ∝ r^(-2/7)
30
mass continuity equation
4πr^2mpn(r)v(r) = M(dot) = constant nvr^2 = constant
31
derive the parker wind solution
hydrostatic equilibrium dp/dr = -pg where newton's second law g = dv/dr dr/dt => v dv/dr and p = nmp dP/dr + gp = Newton’s second law d(2nkt)/dr - GMmpn/r^2 = - nmp v dv/dr divide by nmp v dv/dr = - 2kt/nmp dn/dr -GM/r^2 take n into dn/dr to get d(ln n) Use mass continuity to obtain expression for ln(n) then insert sound speed c = sqrt(P/p) = sqrt(2kT/mp) equation on formula sheet
32
Multiple scattering derivation
mv dv/dr = Lsigma/4picr^2 Multiply by 4pir^2n note M(dot) and v(infinity) Integrate sigma n dr = tau giving tau L/c
33
chapman model
T(r) ∝ r^(-2/7) so T/T0 = (r0/r)^(2/7)
34
Outward radiation force
= 𝐿𝜎/4𝜋𝑐𝑟^2
35
Static (corona/equilibrium)
u = 0 / v = 0
36
Element abundances
abundance = n/nH
37
fraction of ions
q(r) = ni/n
38
If the energy flux of sound waves with amplitude 𝛿𝑣 is conserved
p𝛿𝑣^2/2 c = const
39
Adiabatic cooling
P ∝ p^gamma P ∝ n^gamma P ∝ nT nvr^2 = const n ∝ r^-2 T(r) ∝ r^-4/3
40
Outward radiation force
F = 𝐿𝜎/4𝜋𝑐𝑟^2
41
Gamma factor
Γ = 𝐿⋆/𝐿𝑐
42
When the Parker wind solution v -> 0 and r -> infinity starting from v^2/c^2 -2ln v/c = 4ln r/rc + 4 rc/r + const
Stellar breeze -2ln(v/c) = 4ln(r/rc) v/c ~ (rc/r)^2
43
Parker wind solution when r >> rc starting from v^2/c^2 -2ln v/c = 4ln r/rc + 4 rc/r + const
Const = -3 divide through by 2 and -1 and take exp r >> rc and v >> c take ln v >> c v = 2c [ln(r/rc)]^(1/2)
44
Scale height
h = 2kT/mpg
45
Derive the outward radiation force
Nλ dλ = Lλ λ/hc dλ Nλ σ/4πr^2 dλ L = ∫ Lλ dλ
46
Show that mv dv/dr = GMm/r^2 (Γ-1)
mv dv/dr = -GMm/r^2 + Lσ/4πcr^2 Γ = L/Lc where Lc = 4πGMmc/σ
47
show that v(infinity) = v(esc) sqrt(Γ-1)
mv dv/dr = GMm/r^2 + L/σ/4πcr^2 Γ + L/Lc integrate and let v0 = 0 giving v(infinity) and v(esc) on formula sheet.
48
Show that T ∝ r^-1
H = -ΚdT/dr Κ = 1/3plv(bar)Cp l ∝ 1/nσ v(bar) ∝ T^1/2 1/2 mv^2 = e^2/4piϵr^2 and 1/2 mv^2 = 3/2 kT T ∝ r^-1
49
Multiple scattering at maximum
v(infinity) = (2L/M(dot))^1/2