The_Cambridge_Handbook_of_Physics_Formulas

182 Astrophysics
Stellar fusion processes a
PP i chain PP ii chain PP iii chain
p + +p + → 2 1H+e + +ν e
2
1H+p + → 3 2He+γ
3
2He+ 3 2He → 4 2He+2p +
CNO cycle
12
6 C+p + → 13 7N+γ
13
7 N → 13 6C+e + +ν e
13
6 C+p + → 14 7N+γ
14
7 N+p + → 15 8O+γ
15
8 O → 15 7N+e + +ν e
15
7 N+p + → 12 6C+ 4 2He
p + +p + → 1H+e 2 + +ν e
2
1H+p + → 2He+γ
3
3
2He+ 4 2He → 4Be+γ
7
7
4Be+e − → 3Li+ν 7 e
7
3Li+p + → 2 4 2He
triple-α process
4
2He+ 4 2He ⇀↽ 8 4Be+γ
8
4Be+ 4 2He ⇀↽ 12 6C ∗
12
6 C ∗ → 12 6C+γ
p + +p + → 1H+e 2 + +ν e
2
1H+p + → 2He+γ
3
3
2He+ 4 2He → 4Be+γ
7
7
4Be+p + → 5B+γ
8
8
5B → 4Be+e 8 + +ν e
8
4Be → 2 4 2He
γ
p +
e +
e −
ν e
photon
proton
positron
electron
electron neutrino
a All species are taken as fully ionised.
Pulsars
Braking
index
Characteristic
age a T = 1
n−1
Magnetic
dipole
radiation
˙ω ∝−ω n (9.65)
n =2− P ¨P
Ṗ 2 (9.66)
P
Ṗ
(9.67)
L = µ 0|¨m| 2 sin 2 θ
6πc 3 (9.68)
= 2πR6 Bpω 2 4 sin 2 θ
3c 3 µ 0
(9.69)
ω
P
n
rotational angular velocity
rotational period (= 2π/ω)
braking index
T characteristic age
L luminosity
µ 0 permeability of free space
c speed of light
m
R
B p
θ
pulsar magnetic dipole moment
pulsar radius
magnetic flux density at
magnetic pole
angle between magnetic and
rotational axes
∫
Dispersion
D
DM dispersion measure
D path length to pulsar
measure
DM = n e dl (9.70)
dl path element
0
n e electron number density
dτ
Dispersion b dν = −e 2
τ pulse arrival time
4π 2 DM (9.71)
ɛ 0 m e cν3 ∆τ difference in pulse arrival time
e 2 ( 1
∆τ =
8π 2 ɛ 0 m e c ν1 2 − 1 )
ν i observing frequencies
ν2
2 DM (9.72) m e electron mass
a Assuming n ≠ 1 and that the pulsar has already slowed significantly. Usually n is assumed to be 3 (magnetic dipole
radiation), giving T = P/(2Ṗ ).
b The pulse arrives first at the higher observing frequency.