NST II Formulae and Physical Constants (Cambridge)

NATURAL SCIENCES TRIPOS ASTROPHYSICS

Part II

Examinations: 6, 8, 9 & 10 June, 2011

Astrophysics Constants and Formulae Available for examinations in the Natural Sciences Tripos Part II – Astrophysics Physical Constants velocity of light Newtonian gravitational constant Planck’s constant

c = 2.998 × 108 ms−1

G = 6.673 × 10−11 m3 kg−1 s−2 h = 6.626 × 10−34 Js

¯h = 1.055 × 10−34 Js = h/2π

Stefan – Boltzmann constant gas constant Avogadro’s number Boltzmann’s constant atomic mass unit

σ = 5.671 × 10−8 Wm−2 K−4 = ac/4

R∗ = 103 R

R = 8.314 Jmol−1 K−1

NA = 6.022 × 1023 mol−1

k = 1.381 × 10−23 JK−1 = R/NA

mu = 1.661 × 10−27 kg

proton rest mass

mp = 1.673 × 10−27 kg = 938 MeV

neutron rest mass

mn = 1.675×10−27 kg (mn −mp = 1.29 MeV) = 939 MeV

electron rest mass

me = 9.109 × 10−31 kg = 0.511 MeV

classical electron radius

re = 2.818 × 10−15 m

Compton wavelength Thomson cross section

(mp /me = 1836)

λC = 2.426 × 10−12 m = h/me c

σT = 6.652 × 10−29 m2 = (8π/3)re2

inverse fine-structure constant

α−1 = 137.0

radiation density constant

a = 7.566 × 10−16 Jm−3 K−4 = 8π 5 k 4 /15c3 h3

electronic charge Bohr radius

e = 1.602 × 10−19 C

a0 = 5.292 × 10−11 m = h2 /πµ0 c2 me e2

temperature of triple point of water Ttr = 273.16K vacuum magnetic permeability

µ0 = 4π × 10−7 Hm−1 (exact)

vacuum electrical permittivity

ǫ0 = 8.854 × 10−12 Fm−1 = c−2 µ−1 0

ionization potential of hydrogen

χH = 13.598 eV 1

Conversion Factors 1eV = 1.602 × 10−19 J

electron volt erg

1 erg = 10−7 J

year

1 yr = 3.156 × 107 s

Astronomical Measures astronomical unit

1 au = 1.496 × 1011 m

1 pc = 3.086 × 1016 m

parsec

1 ly = 9.461 × 1015 m

light year

1 M⊙ = 1.989 × 1030 kg

solar mass

1 R⊙ = 6.960 × 108 m

solar radius solar luminosity

1 L⊙ = 3.90 × 1026 W

solar effective temperature

5780K

absolute magnitude of sun

MV = 4.83

distance modulus

m − M = 5 log10 D − 5 1M⊕ = 5.976 × 1024 kg

Earth mass

1R⊕ = 6.371 × 106 m

Earth radius Vector Relations

A∧ (B∧ C) = B (A• C) − C(A• B) ∇• (φA) = φ∇• A + A• ∇φ ∇∧ (φA) = φ∇∧ A − A∧ ∇φ ∇(A• B) = (A• ∇)B + (B• ∇)A + A∧ (∇∧ B) + B∧ (∇∧ A) ∇• (A∧ B) = B• ∇∧ A − A• ∇∧ B ∇∧ (A∧ B) = (B• ∇)A − (A• ∇)B + A(∇• B) − B(∇• A) ∇∧ (∇φ) = 0 ∇• (∇∧ A) = 0 ∇∧ (∇∧ A) = ∇(∇• A) − ∇2 A 1 A∧ (∇∧ A) = ∇(A• A) − (A• ∇)A 2 Stirling’s formula: ln n! ∼ n(ln n − 1) as n → ∞ 2

Maxwell’s Equations (in vacuo) ∇• B=0 ∇ • E = ρ/ǫ0 ∇∧ E = − µ−1 0 ∇∧ B = ǫ0

∂B ∂t ∂E +J ∂t

Lorentz force density: f = ρE + J∧ B Electromagnetic energy density: W = Poynting’s Vector:

1 2 ǫ0 E 2 + µ−1 B 0 2 S = E∧ B/µ0

Poynting’s Theorem: ∂W + ∇ • S = −J • E ∂t Candidates may be asked to derive any of the subsequent formulae in this booklet.

Theory of Relativity Contravariant tensor: T ′mn = T rs Covariant tensor: ′ Tmn = Trs

Mixed tensor: Tn′m = Tsr Invariant:

∂x′m ∂x′n ∂xr ∂xs

∂xr ∂xs ∂x′m ∂x′n

∂x′m ∂xs ∂xr ∂x′n

T′ = T

Kronecker delta: δsr

=

1 0

if r = s if r = 6 s 3

Interval:

ds2 = gmn dxm dxn

Metric tensor:

gmr g ms = δrs

gmn = gnm ,

Magnitude of a vector: Orthogonality condition:

X 2 = gmn X m X n gmn X m Y n = 0

Raising and lowering indices: X m = g mn Xn , Xm = gmn X n Christoffel symbols:

1 ∂gmr ∂gnr ∂gmn [mn, r] = + − 2 ∂xn ∂xm ∂xr Γrmn = g rs [mn, s] Christoffel symbols for diagonal metric gmn : (m 6= n 6= r, no sums over repeated indices) 1 ∂gmm 1 ∂gmm m n , Γm , Γm nm = Γmn = nr = 0, Γmm = − n 2gnn ∂x 2gmm ∂xn 1 ∂gmm Γm . mm = 2gmm ∂xm Geodesic:

n m d 2 xr r dx dx + Γ =0 mn ds2 ds ds

or

where ui =

dui 1 ∂gjk j k = u u ds 2 ∂xi dxi ds .

Absolute derivative:

DT i dT i dxk = + Γijk T j Ds ds ds

Parallel propagation:

DT i =0 Ds

Covariant derivative:

∂T r + Γrms T m ∂xs ∂Tr − Γm = rs Tm s ∂x grs;t = 0

T r;s = Tr;s Mixed curvature tensor: Rsrmn =

∂ s ∂ s Γrn − Γ + Γprn Γspm − Γprm Γspn m ∂x ∂xn rm 4

Non-commutative covariant differentiation: Tr;mn − Tr;nm = Rsrmn Ts Covariant curvature tensor: Rrsmn

1 = 2

∂ 2 grn ∂ 2 gsm ∂ 2 grm ∂ 2 gsn + − − ∂xs ∂xm ∂xr ∂xn ∂xs ∂xn ∂xr ∂xm

+ g pq ([rn, p][sm, q] − [rm, p][sn, q]) Identities: Rrsmn = −Rsrmn = −Rrsnm = Rmnrs Rrsmn + Rrmns + Rrnsm = 0 Conditions for flatness: Rrsmn = 0 Bianchi identity: Rrsmn;t + Rrsnt;m + Rrstm;n = 0 Ricci tensor: Rrm = Rmr = Rnrmn Curvature invariant: R = Rnn Einstein tensor:

1 Gnt = Rnt − δtn R 2 Gnt;n = 0

Geodesic Deviation: j l D2 ξ i i k dx dx + R ξ =0 jkl Ds2 ds ds

Schwarzschild metric: 2

ds =

2GM 1− rc2

2GM c dt − 1 − rc2 2

2

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−1

dr 2 − r 2 dθ 2 + sin2 θdφ2

Astrophysical Fluid Dynamics Mass: ∂ρ + ∇. (ρu) = 0 ∂t Momentum: ∂u 1 + u. ∇u = − ∇p − ∇Φ ∂t ρ 1 + J∧ B ρ Energy: ∂E + ∇. [(E + p)u] = −ρQ˙ ∂t where

1 1 p E = ρ( u2 + Φ + e), e = .

2 γ−1ρ

Poisson’s Equation:

∇2 Φ = 4πGρ

Induction: ∂B = ∇∧ (u∧ B) ∂t Rankine-Hugoniot Conditions: ρ 1 u1 = ρ 2 u2 p1 + ρ1 u21 = p2 + ρ2 u22 1 2 p1 1 p2 u1 + e1 + = u22 + e2 + 2 ρ1 2 ρ2 Bernoulli’s constant: 1 H = u2 + 2

Z

dp +Φ ρ

Schwarzschild criterion: d Stability requires dr

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p ργ

>0

Statistical Physics v = ρ−1

T ds = du + pdv, h = u + pv f = u − Ts

g = u − T s + pv

∂ ln p ∂ ln T ∂ ln T −1 γ1 = , 1 − γ2 = , γ3 − 1 = ∂ ln ρ s ∂ ln p s ∂ ln ρ s S = k ln W,

S = kN ln Z + U/T

Boltzmann statistics: W = N!

Y g ni i

ni !

i

ni = N Z −1 gi e−βǫi ,

,

β = (kT )−1

Bose-Einstein statistics: W =

Y (gi + ni − 1)! i

,

ni =

gi ! , ni ! (gi − ni )!

ni =

(gi − 1)!ni !

gi eβ(ǫi −µ) − 1

Fermi-Dirac statistics: W =

Y i

gi β(ǫ −µ) e i

+1

Density of states is h−3 .

Maxwell-Boltzmann distribution: 2

F (p)dp = 4πn(2πmkT )−3/2 e−p Planck radiation law:

Perfect gas: p=

uν dν =

8πh ν 3 dν c3 ehν/kT − 1

R∗ ρT µ

where

h= Radiation:

/2mkT 2

γ p , γ−1ρ

u = aT 4 ,

Mean free path l satisfies nσl = 1.

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R∗ = 103 R γ = cp /cv

p = 31 u

p dp

Cosmology Robertson-Walker metric:

dr 2 2 2 2 2 ds = c dt − R (t) + r dθ + sin θdφ 1 − kr 2 2

2

Friedmann equations:

2

2

kc2 8πG Λc2 R˙ 2 + = ρ + R2 R2 3 3 2 2 ¨ ˙ 2R R kc −8πG + 2+ 2 = p + Λc2 R R R c2

Conservation laws: Dust: ρR3 = constant Hubble constant:

Deceleration parameter:

Radiation: T 4 R4 = constant

R˙

H0 = R 0

¨

RR

q0 = − R˙ 2 0

Proper distance:

dprop (t) = R(t)

Z

0

Luminosity distance for Λ = 0: dL = R2 (t0 )

r1

√

dr 1 − kr 2

h io r1 c n 1/2 = q z + (q − 1) (1 + 2q z) − 1 .

0 0 0 R(t1 ) H0 q02

Angular diameter distance: dθ = R(t1 )r1 = dL (1 + z)−2 Equations of Stellar Structure Gmρ dp =− 2 dr r dm = 4πr 2 ρ dr dLr = 4πr 2 ρǫ dr 3κρL dT =− (radiative) 2 3 dr 16πacr T 1 T dp dT = 1− (convective) dr γ P dr Energy released in conversion of hydrogen to helium: 0.007 mc2

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Stellar Dynamics The Collisionless Boltzmann Equation: Df ∂f ∂f dxi ∂f dvi =0= + + Dt ∂t ∂xi dt ∂vi dt Cartesian coordinates: (x, y, z) r˙ = xi ˙ + yj ˙ + zk ˙ = vx i + vy j + vz k ¨r = v˙ x i + v˙ y j + v˙ z k Df ∂Φ ∂f ∂Φ ∂f ∂f ∂f ∂f ∂f ∂Φ ∂f − − =0= + vx + vy + vz − Dt ∂t ∂x ∂y ∂z ∂x ∂vx ∂y ∂vy ∂z ∂vz ∂f ∂f ∂f +v· − ∇Φ · = ∂t ∂x ∂v Jeans equations: ∂(νvi ) ∂ν + = 0 where ∂t ∂xi

Z

f d3 v Z 1 f vi d3 v. vi = ν Z ∂Φ ∂ ∂ (νvi vj ) 1 vi vj f d3 v (νvj ) + +ν = 0 where vi vj = ∂t ∂xi ∂xj ν

Z 2 ∂ νσij ∂vj ∂Φ 1 ∂vj 2 ν (vi − vi ) (vj − vj ) f d3 v + νvi = −ν − where σij = ∂t ∂xi ∂xj ∂xi ν ν=

Bound orbit in point mass potential:

a 1 − e2 r= 1 + e cos φ Oort’s constants:

1 vc dvc

A≡ −

2 R dR

R0

1 vc dvc

B≡− +

2 R dR

R0

2

κ = −4B(A − B) r −B κ =2 Ω A−B

Last revision: 30 March, 2006

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