Semester 1 - Formulas

Question 1

Gravity

Answer 1

g(r) = Gm(r)/r^2

Question 2

Hydrostatic Equilibrium

Answer 2

dP/dr = -p(r)g(r)

Question 3

Gravitational Potential Energy

Answer 3

Ω = -GM^2/R

Question 4

number density

Answer 4

n = p/mH

Question 5

Mass

Answer 5

M = Vp

M = 4/3 πR^3 p

M = ( R ∫ 0) dm(r)

Question 6

Core Pressure

Answer 6

P(c) = P(gas,i) + P(deg,e)

i = ions
e = electrons

Question 7

Adiabatic Index

Answer 7

gamma = 1 + 1/n

Question 8

Mean Free Path Derivation

Answer 8

s(bar) = l1(bar) + l2(bar) + l3(bar) + … + lN(bar)

s^2 = Nl^2

t = l/c

l = 1/nσ

Question 9

Mean Free Path

Answer 9

l = 1/nσ

Question 10

Gas Pressure

Answer 10

Pg = nkT

Question 11

Radiation Pressure

Answer 11

Prad = 1/3 aT^4

a = 4σ/c

Question 12

Degeneracy Pressure (non-relativistic and relativistic)

Answer 12

P = Kn^5/3 (non-relativistic)

P = Kn^4/3 (relativistic)

Question 13

characteristic energy-loss timescale

Answer 13

τ = E/L

Question 14

kelvin-helmholtz timescale

Answer 14

τ = Ω/L

Question 15

potential energy of a star

Answer 15

m(r) = ( r ∫ 0) dm

if m = 4/3πr^3 p

dm = 4πp(r)r^2dr

Ω = - ( M* ∫ 0) Gm(r)dm/r

Question 16

hydrostatic equilibrium derivation

Answer 16

A[P(r) - P(r+∆r)] - g(r)p(r)A∆r = 0

Taylor expand P(r+∆r)

P(r) + dP/dr ∆r + …

Question 17

free-fall timescale derivation

Answer 17

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

Question 18

Virial Theorem Derivation

Answer 18

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

Question 19

mean atomic mass

Answer 19

1/µi = Σj Xj/Aj

where Xj is the fraction by mass
and Aj its atomic weight

Question 20

the average number of free electrons per nucleon

Answer 20

1/µe = Σj Xjqj/Aj

where qj is the no. of free electrons per nucleon

Question 21

mean molecular mass

Answer 21

1/µ = 1/µi + 1/µe

Question 22

minimum stellar mass

Answer 22

d/dpc = 0

Question 23

polytropic model

Answer 23

P = Kp^(gamma)

Question 24

lane-emden equation derivation

Answer 24

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

Question 25

opacity

Answer 25

dτv = -kvpIvds

Question 26

Rosseland mean opacity

Answer 26

F = ( ∞ ∫ 0) Fvdv

Fvkvp/c = -dPv/dr

and Pv = 4π/3c Bv

dBv/dT dT/dr

Question 27

kramers opacity

Answer 27

κ = κ0 p^pT^q

Question 28

Coloumb repulsion

Answer 28

kbTign > z1z2e^2/4π εr

Question 29

Eddington quartic equation

Answer 29

1-β = 0.003(M/Msun)^2 µ^4β^4

Question 30

Number density

Answer 30

n = 2 electron density

Question 31

Volume of shell

Answer 31

4πr^2dr

Question 32

Energy release rate from the P-P chain

Answer 32

ε = p^2 T^4

Question 33

Energy release rate for CNO-cycle

Answer 33

ε = p^2 T^16

Question 34

Main branch of the P-P chain

Answer 34

PP-1

p+ + p+ -> 2H + e+ + v

2H + p+ -> 3He + gamma

3He + 3He -> 4He + 2p+

Question 35

Temperature gradient derivation

Answer 35

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

Question 36

A star that is stable against convection

Answer 36

dT/dr|star < dT/dr|temp gradient (adiabatic)

where gamma = 5/3 for non relativistic

Question 37

A uniform density

Answer 37

p = p0

and p0 = M/4/3piR^3

Question 38

Mass luminosity relationship

Answer 38

L/Lsun =(M/Msun)^3.5

Question 39

To find the slope of the main sequence

Answer 39

Looking for the log relationship of L and Teff

Question 40

If p(r) = constant than alpha = 3/5 for the potential energy.

Answer 40

Start from definition of potential energy.

Ω = ∫ Gmdm/r

Substitute dm and m(r) assuming uniform density

integrate for r

Question 41

Show <P> ~ GM^2/R^4

Answer 41

Ω/3 = -<P>V

substitute for Ω

Question 42

Derive the temperature in a contracting protostar

Answer 42

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.

Question 43

Derive the radiation pressure

Answer 43

L = (inf int 0) Lv dv
Fv = Lv/4𝜋𝑟^2
p = hv/c

Hence = L/4𝜋𝑟^2𝑐

Question 44

Lower limit for the central core pressure

Answer 44

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

Question 45

Show that Pv = 4π/3c Bv

If P = 1/3 (∞ ∫ 0) pvn(p) dp

Answer 45

p = hv/c

and v -> c in P

n(v) = 4πBv/hvc

Prad = (∞ ∫ 0) Pv dv

Pv = 4π/3c Bv

Question 46

Derive the temperature gradient in the case of radiative transport

Answer 46

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

Question 47

Derive the Eddington quartic equation

Answer 47

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