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
5
−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
6
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. 7
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
8
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
9