Semester 2 - Formulae Flashcards
Pressure
P = 2nkT
density
p = nmp
optical thickness of the corona
τ = n0 σT h
emission measure
EM = ( ∫vol) n^2 dV
nvr^2 =
constant
hence
n ∝ 1/vr^2
critical radius
rc = GM/2c^2
where c^2 = 2kT/mp
dynamic pressure
mp nv^2
ram pressure =
dynamic pressure
photon radiation pressure
P = ℏω/c
Luminosity of photons
L = nℏω
photon flux
Fn = n/4πR^2
force
F = P/A
Eddington luminosity
Ledd = 4πGMmc/σ
terminal wind speed
or CAK velocity profile
v∞ = [2GM/R (Γ-1)]^1/2
where Γ = L/Ledd
Rydberg equation
1/λ = R(1/n1^2 - 1/n2^2) = vn/c
mass loss rate
M(dot) = L/cv∞
conservation of momentum can be used to estimate the mass loss rate
Lω = 1/2 M(dot) v^2∞
multiple scattering affects the mass loss rate
as Σ|pi|
M(dot) = η L/cv∞
Spitzer thermal conductivity
κ = κ0 T^5/2
Heat flux
H = -κ0 T^5/2 dT/dr
Maxwell-Faraday Law
∇ x E = -∂B/∂t
∇ x (v x B) = ∂B/∂t
Lenz Law
Φ =( ∫ S ) B . da
plasma beta
ß = 2µ0nkT/B^2
Alfven speed
vA = B/sqrt(µ0p)
where p is the density
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
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
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
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
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)
mass continuity equation
4πr^2mpn(r)v(r) = M(dot) = constant
nvr^2 = constant
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
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
chapman model
T(r) ∝ r^(-2/7)
so T/T0 = (r0/r)^(2/7)
Outward radiation force
= 𝐿𝜎/4𝜋𝑐𝑟^2
Static (corona/equilibrium)
u = 0 / v = 0
Element abundances
abundance = n/nH
fraction of ions
q(r) = ni/n
If the energy flux of sound waves with amplitude 𝛿𝑣 is conserved
p𝛿𝑣^2/2 c = const
Adiabatic cooling
P ∝ p^gamma
P ∝ n^gamma
P ∝ nT
nvr^2 = const
n ∝ r^-2
T(r) ∝ r^-4/3
Outward radiation force
F = 𝐿𝜎/4𝜋𝑐𝑟^2
Gamma factor
Γ = 𝐿⋆/𝐿𝑐
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
Parker wind solution when r»_space; 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»_space; rc and v»_space; c
take ln
v»_space; c
v = 2c [ln(r/rc)]^(1/2)
Scale height
h = 2kT/mpg
Derive the outward radiation force
Nλ dλ = Lλ λ/hc dλ
Nλ σ/4πr^2 dλ
L = ∫ Lλ dλ
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
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.
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
Multiple scattering at maximum
v(infinity) = (2L/M(dot))^1/2
