Semester 2 - Formulae

Question 1

Pressure

Answer 1

P = 2nkT

Question 2

density

Answer 2

p = nmp

Question 3

optical thickness of the corona

Answer 3

τ = n0 σT h

Question 4

emission measure

Answer 4

EM = ( ∫vol) n^2 dV

Question 5

nvr^2 =

Answer 5

constant

hence

n ∝ 1/vr^2

Question 6

critical radius

Answer 6

rc = GM/2c^2

where c^2 = 2kT/mp

Question 7

dynamic pressure

Answer 7

mp nv^2

Question 8

ram pressure =

Answer 8

dynamic pressure

Question 9

photon radiation pressure

Answer 9

P = ℏω/c

Question 10

Luminosity of photons

Answer 10

L = nℏω

Question 11

photon flux

Answer 11

Fn = n/4πR^2

Question 12

force

Answer 12

F = P/A

Question 13

Eddington luminosity

Answer 13

Ledd = 4πGMmc/σ

Question 14

terminal wind speed

or CAK velocity profile

Answer 14

v∞ = [2GM/R (Γ-1)]^1/2

where Γ = L/Ledd

Question 15

Rydberg equation

Answer 15

1/λ = R(1/n1^2 - 1/n2^2) = vn/c

Question 16

mass loss rate

Answer 16

M(dot) = L/cv∞

Question 17

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

Answer 17

Lω = 1/2 M(dot) v^2∞

Question 18

multiple scattering affects the mass loss rate

Answer 18

as Σ|pi|

M(dot) = η L/cv∞

Question 19

Spitzer thermal conductivity

Answer 19

κ = κ0 T^5/2

Question 20

Heat flux

Answer 20

H = -κ0 T^5/2 dT/dr

Question 21

Maxwell-Faraday Law

Answer 21

∇ x E = -∂B/∂t

∇ x (v x B) = ∂B/∂t

Question 22

Lenz Law

Answer 22

Φ =( ∫ S ) B . da

Question 23

plasma beta

Answer 23

ß = 2µ0nkT/B^2

Question 24

Alfven speed

Answer 24

vA = B/sqrt(µ0p)

where p is the density

Question 25

radiative instability (derivation)

Answer 25

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

Question 26

downward heat conduction in the corona

Answer 26

-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

Question 27

show that n(r) = n0 exp(-a(1-r0/r))

Answer 27

start from hydrostatic equilibrium

ideal gas pressure p = 2nkT

integrate to obtain expression

Pull factor of r0 out and create a new constant

Question 28

non-isothermal static atmosphere

Answer 28

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

Question 29

chapman model derivation

Answer 29

starting from heat flux has a factor 4pir^2

rearrange for T

and integrating with T -> 0 as r -> ∞

T(r) ∝ r^(-2/7)

Question 30

mass continuity equation

Answer 30

4πr^2mpn(r)v(r) = M(dot) = constant

nvr^2 = constant

Question 31

derive the parker wind solution

Answer 31

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

Question 32

Multiple scattering derivation

Answer 32

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

Question 33

chapman model

Answer 33

T(r) ∝ r^(-2/7)

so T/T0 = (r0/r)^(2/7)

Question 34

Outward radiation force

Answer 34

= 𝐿𝜎/4𝜋𝑐𝑟^2

Question 35

Static (corona/equilibrium)

Answer 35

u = 0 / v = 0

Question 36

Element abundances

Answer 36

abundance = n/nH

Question 37

fraction of ions

Answer 37

q(r) = ni/n

Question 38

If the energy flux of sound waves with amplitude 𝛿𝑣 is conserved

Answer 38

p𝛿𝑣^2/2 c = const

Question 39

Adiabatic cooling

Answer 39

P ∝ p^gamma

P ∝ n^gamma

P ∝ nT

nvr^2 = const

n ∝ r^-2

T(r) ∝ r^-4/3

Question 40

Outward radiation force

Answer 40

F = 𝐿𝜎/4𝜋𝑐𝑟^2

Question 41

Gamma factor

Answer 41

Γ = 𝐿⋆/𝐿𝑐

Question 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

Answer 42

Stellar breeze

-2ln(v/c) = 4ln(r/rc)

v/c ~ (rc/r)^2

Question 43

Parker wind solution when r&raquo_space; rc

starting from v^2/c^2 -2ln v/c = 4ln r/rc + 4 rc/r + const

Answer 43

Const = -3

divide through by 2 and -1 and take exp

r&raquo_space; rc and v&raquo_space; c

take ln

v&raquo_space; c

v = 2c [ln(r/rc)]^(1/2)

Question 44

Scale height

Answer 44

h = 2kT/mpg

Question 45

Derive the outward radiation force

Answer 45

Nλ dλ = Lλ λ/hc dλ

Nλ σ/4πr^2 dλ

L = ∫ Lλ dλ

Question 46

Show that mv dv/dr = GMm/r^2 (Γ-1)

Answer 46

mv dv/dr = -GMm/r^2 + Lσ/4πcr^2

Γ = L/Lc

where Lc = 4πGMmc/σ

Question 47

show that v(infinity) = v(esc) sqrt(Γ-1)

Answer 47

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.

Question 48

Show that T ∝ r^-1

Answer 48

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

Question 49

Multiple scattering at maximum

Answer 49

v(infinity) = (2L/M(dot))^1/2