Semester 1 - Formulas Flashcards
Gravity
g(r) = Gm(r)/r^2
Hydrostatic Equilibrium
dP/dr = -p(r)g(r)
Gravitational Potential Energy
Ω = -GM^2/R
number density
n = p/mH
Mass
M = Vp
M = 4/3 πR^3 p
M = ( R ∫ 0) dm(r)
Core Pressure
P(c) = P(gas,i) + P(deg,e)
i = ions
e = electrons
Adiabatic Index
gamma = 1 + 1/n
Mean Free Path Derivation
s(bar) = l1(bar) + l2(bar) + l3(bar) + … + lN(bar)
s^2 = Nl^2
t = l/c
l = 1/nσ
Mean Free Path
l = 1/nσ
Gas Pressure
Pg = nkT
Radiation Pressure
Prad = 1/3 aT^4
a = 4σ/c
Degeneracy Pressure (non-relativistic and relativistic)
P = Kn^5/3 (non-relativistic)
P = Kn^4/3 (relativistic)
characteristic energy-loss timescale
τ = E/L
kelvin-helmholtz timescale
τ = Ω/L
potential energy of a star
m(r) = ( r ∫ 0) dm
if m = 4/3πr^3 p
dm = 4πp(r)r^2dr
Ω = - ( M* ∫ 0) Gm(r)dm/r
hydrostatic equilibrium derivation
A[P(r) - P(r+∆r)] - g(r)p(r)A∆r = 0
Taylor expand P(r+∆r)
P(r) + dP/dr ∆r + …
free-fall timescale derivation
a = dv/ dt = -Gm/r^2
dv/dt = 1/2 dv^2/dr
integrate and solve for v
free fall timescale = (0 ∫ r0) dt/dr dr
substitute in 1/v , change variables x = r/r0 and integrate
x = sin^2 theta => pi^2/4
Virial Theorem Derivation
start from hydrostatic equilibrium
multiply both sides by volume and integrate over r
RHS
dm = 4πr^2pdr
Ω = (M ∫ 0) Gmdm/r
LHS
integrate by parts at R, P=0
dm = 4πr^2pdr
- (V ∫ 0) PdV
P = nkT
mean atomic mass
1/µi = Σj Xj/Aj
where Xj is the fraction by mass
and Aj its atomic weight
the average number of free electrons per nucleon
1/µe = Σj Xjqj/Aj
where qj is the no. of free electrons per nucleon
mean molecular mass
1/µ = 1/µi + 1/µe
minimum stellar mass
d/dpc = 0
polytropic model
P = Kp^(gamma)
lane-emden equation derivation
star from H.E. rearranged for m
insert dm/dr for m (include d/dr) and substitute polytropic relationship
dP/dr = dP/dp dp/dr chain rule
trial solutions of the form p = pcθ^n
chain rule
adiabatic index = 1 + 1/n
Const = alpha ^2
opacity
dτv = -kvpIvds
Rosseland mean opacity
F = ( ∞ ∫ 0) Fvdv
Fvkvp/c = -dPv/dr
and Pv = 4π/3c Bv
dBv/dT dT/dr
kramers opacity
κ = κ0 p^pT^q
Coloumb repulsion
kbTign > z1z2e^2/4π εr
Eddington quartic equation
1-β = 0.003(M/Msun)^2 µ^4β^4
Number density
n = 2 electron density
Volume of shell
4πr^2dr
Energy release rate from the P-P chain
ε = p^2 T^4
Energy release rate for CNO-cycle
ε = p^2 T^16
Main branch of the P-P chain
PP-1
p+ + p+ -> 2H + e+ + v
2H + p+ -> 3He + gamma
3He + 3He -> 4He + 2p+
Temperature gradient derivation
P = Kp^gamma
and P = pkT/mumH
p = PmumH/kT
insert p into P = p^gamma
Take derivative with respect to r
Eventually giving the temperate gradient
A star that is stable against convection
dT/dr|star < dT/dr|temp gradient (adiabatic)
where gamma = 5/3 for non relativistic
A uniform density
p = p0
and p0 = M/4/3piR^3
Mass luminosity relationship
L/Lsun =(M/Msun)^3.5
To find the slope of the main sequence
Looking for the log relationship of L and Teff
If p(r) = constant than alpha = 3/5 for the potential energy.
Start from definition of potential energy.
Ω = ∫ Gmdm/r
Substitute dm and m(r) assuming uniform density
integrate for r
Show <P> ~ GM^2/R^4
Ω/3 = -<P>V
substitute for Ω
Derive the temperature in a contracting protostar
Pc = Pg + PD
replace n with p = nmH and assume μ = 1
Pc ~ GM^2/R^4 = 𝜒 GM^2/R^4
rearrange for kTc remember to divide both sides.
pc = 𝛽 M/R^3 find R^4 and replace
kbTc = Apc^1/3 - Bpc^2/3
find the minimum M by taking derivative which gives max T.
Derive the radiation pressure
L = (inf int 0) Lv dv
Fv = Lv/4𝜋𝑟^2
p = hv/c
Hence = L/4𝜋𝑟^2𝑐
Lower limit for the central core pressure
dP/dr = -pg
dP/dm = -Gm/4pir^4
integrate between M* and 0 where r < R*
Pc - Psurf > GM^2/8piR^4 and Psurf > 0
Show that Pv = 4π/3c Bv
If P = 1/3 (∞ ∫ 0) pvn(p) dp
p = hv/c
and v -> c in P
n(v) = 4πBv/hvc
Prad = (∞ ∫ 0) Pv dv
Pv = 4π/3c Bv
Derive the temperature gradient in the case of radiative transport
F = ∫ Fv dv
where Fv = -c/kp dP/dr
dP/dT dT/dr = dP/dr
dPrad = aT^4/3 so dP/dT = 4aT^3/3
and F = L/4πr^2
Derive the Eddington quartic equation
𝜅𝜂 ≈ const ≡ 𝜅𝑠
dPrad/dP = 𝐿𝜅/4𝜋𝑐𝐺𝑚
Integrate
𝛽 =𝑃𝑔𝑎𝑠/𝑃
(1-𝛽) =𝑃rad/𝑃
Prad = aT^4/3 and assuming the ideal gas law we can rearrange for T
Substituting constants for Eddington quartic equation
