Semester 2 - Formulas

Question 1

Break-up rotation rate

Answer 1

Fcent = Fgrav

Question 2

Schwarszschild radius

Answer 2

Rs = 2GM/c^2

Question 3

Potential energy

Answer 3

Ω = GM^2/R

Question 4

Accretion Luminosity

Answer 4

Lacc = M(dot)Ω

Question 5

density

Answer 5

p = n mH

Question 6

virial theorem

Answer 6

2U + Ω = 0

Question 7

total internal kinetic energy

Answer 7

U = 3/2 nkT V

remember p = nµmH

where V = M/p

Question 8

Binding Energy

Answer 8

E = ∆mc^2

where ∆m = ([Zmp + Nmn]-M)

Question 9

mass loss rate

Answer 9

M(dot) = L/Q

Question 10

escape velocity

Answer 10

vesc = (2GM/R)^(1/2)

Question 11

birth function

Answer 11

dN = Φ(M)dM

where Φ(M) is the birth function

Φ(M) ∝ M^(-2.35)

Question 12

initial mass function

Answer 12

ξ(M) ∝ (M/M☉)^(-1.35)

Question 13

effect of rotation

Answer 13

dv/dt = ω^2r -GM/r^2

and dv/dt = 1/2 dv^2/dr

L = mωr^2

Integrate

v -> 0 when r = r1 i.e v = 0

Question 14

main sequence lifetime

Answer 14

τ ~ 1/M^2

Question 15

Hydrostatic Equilibrium

Answer 15

dP/dr = -pg = -pGM/r^2

Question 16

Larmor Gyro-radius

Answer 16

r = mv/|q|B

Question 17

Number of hydrogen

Answer 17

NH = M/Zmp

Question 18

Magnetic pressure

Answer 18

P = B^2/2μ0

which is equal to the magnetic energy density such that

UB = uB V

Question 19

Fully ionised hydrogen

Answer 19

μ = 0.5

Question 20

Mean Separation between particles

Answer 20

Δx = 1/n^1/3

Question 21

Fraction of mass converted to energy

Answer 21

f = Q value / rest mass

rest mass being i.e for 56Fe -> 56mpc^2

Question 22

Nuclear energy

Answer 22

E = f Mc^2

where f is the fraction of mass converted to energy

Question 23

Triple alpha process

Answer 23

4He + 4He -> 8Be

4He + 8Be -> 12C

Question 24

End of the isothermal phase

Answer 24

once all of the hydrogen and helium in the cloud is dissociated and ionised.

NH2ϵd + NHϵi,H + NHeϵi,He = Ω1 - Ω2

Question 25

Show that MJ ∝ p^1/2

Answer 25

Pg ∝ p^gamma

pkbT/μmH ∝ p^gamma

T ∝ p^gamma-1

MJ ∝ T^3/2/p^1/2

where gamma =5/3

Question 26

Mean molecular mass of neutral H2 and ionised H

Answer 26

μH2 = 2 and μH =0.5

Question 27

Derive the temperature at which H.E is established

Answer 27

Virial theorem
Assume dissociation and ionisation is complete
U = Ω(R2) = -GM^2/R2

and Ω(R2) = N/mH ϵi + N/2mH ϵd

U = 3/2NkT where N = M/mH and NH = 2N

T ~ 30,000K

Question 28

Sound speed

Answer 28

c = (P/p)^1/2

c^2 ∝ T/μ

Question 29

nuclear timescale

Answer 29

τ nuc = (fEfM Mc^2)/L

fE = 0.007
fM ~ 0.1

Question 30

Total energy

Answer 30

Etot = U + Ω

U = -Ω/2

Etot = -U

Question 31

two reactions between neutrinos and deuterium

Answer 31

v + 2H -> v + p + n : neutral current

ve + 2H -> e + p + p : charged current

Question 32

Schondberg-Chandrasekhar limit

Answer 32

M core ~ 0.37(μenv/μcore)^2 M

Question 33

Triple alpha process

Answer 33

4He + 4He + 4He -> 12C

Question 34

Derive the mass radius relationship of a white dwarf starting from the Lane Emden equation.

Answer 34

Start from lane emden equation looking specifically at the constant alpha.
Replace alpha with R = alpha ξ and rearrange for pc. Substitute this back into the lane emden equation and substitute for pc.

M ∝ R^(3-n)/(1-n)

where gamma = 1 + 1/n

Question 35

Chandrasekhar limiting mass

Answer 35

MCH = 1.457(2/μe)^2 M ☉

Question 36

Photodisintegration

Answer 36

56Fe + gamma -> 13 4He + 4n

4He + gamma -> 2p + 2n

Question 37

Energy of a core collapse supernova

Answer 37

Ep ~ GM^2(NS)/R(NS)

Question 38

Derive the Jeans Mass

Answer 38

Start from the virial theorem

Look at potential including alpha term, replace R by rearranging mass assuming uniform density

Looking at U replace N and V

Alpha = 3/5 for uniform density

Rearrange for mass