9.4 Observational astrophysics

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9.4 Observational astrophysics	179

9.4Observational astrophysics

Astronomical magnitudes

Apparent

m1 − m2 = −2.5log10

(9.27)

apparent magnitude of object i

magnitude

energy ﬂux from object i

m− M = 5log10 D − 5

absolute magnitude

Distance

(9.28)

−

distance modulus

modulusa

= −5log10 p− 5

(9.29)

distance to object (parsec)

annual parallax (arcsec)

Luminosity–

Mbol = 4.75 − 2.5log10

(9.30)

Mbol

bolometric absolute magnitude

magnitude

L 3.04 × 10(28−0.4Mbol)

(9.31)

luminosity (W)

relation

solar luminosity (3.826 × 1026 W)

Flux–

10−(8+0.4mbol )

Fbol

bolometric ﬂux (Wm−2)

magnitude

Fbol

2.559

(9.32)

relation

mbol

bolometric apparent magnitude

bolometric correction

Bolometric

BC = mbol − mV

(9.33)

V -band apparent magnitude

correction

= Mbol − MV

(9.34)

V -band absolute magnitude

Colour

B − V = mB − mV

(9.35)

−

observed B

−

V colour index

U − B = mU − mB

(9.36)

index

U − B

observed U − B colour index

Colour

E = (B

−

V )

−

V )0

(9.37)

V )0

B − V colour excess

excessc

−

intrinsic B

−

V colour index

aNeglecting extinction.

bUsing the UBV magnitude system. The bands are centred around 365 nm (U), 440 nm (B), and 550 nm (V ). cThe U − B colour excess is deﬁned similarly.

Photometric wavelengths

Mean				λR λ dλ

		%			R((λ))dλ
wavelength	λ0 = %				R((λ))dλ
Isophotal						F	λ R(λ) dλ
wavelength	F(λi) =					%	(R)(λ) dλ
E ective						λF%(λ)R(λ) dλ
			%
wavelength	λe = %					F(λ)R(λ) dλ

(9.38)

(9.39)

(9.40)

λ0 mean wavelength

λwavelength

Rsystem spectral response

F(λ) ﬂux density of source (in		9
F(λ) ﬂux density of source (in
	terms of wavelength)
λi	isophotal wavelength
λe	e ective wavelength

180 Astrophysics

Planetary bodies

4 + 3 × 2n

DAU

planetary orbital radius (AU)

Bode’s lawa

DAU =

(9.41)

index: Mercury = −∞, Venus

= 0, Earth = 1, Mars = 2, Ceres

= 3, Jupiter= 4, ...

100M

1/3

(9.42)

satellite orbital radius

central mass

Roche limit

9πρ

> 2.46R

(if densities equal)

(9.43)

satellite density

central body radius

Synodic

synodic period

periodb

−

(9.44)

Earth’s orbital period

planetary orbital period

aAlso known as the “Titius–Bode rule.” Note that the asteroid Ceres is counted as a planet in this scheme. The relationship breaks down for Neptune and Pluto.

bOf a planet.

Distance indicators

Hubble law

v = H0d

(9.45)

Annual

Dpc

= p−1

(9.46)

parallax

Cepheid

log10

1.15log10 Pd + 2.47

(9.47)

variablesa

(9.48)

MV −2.76log10 Pd − 1.40

Tully–Fisher

2vrot

MI −7.68log10

sini − 2.58

relationb

(9.49)

Einstein rings

θ2 =

4GM

ds − dl

(9.50)

dsdl

Sunyaev–

∆T

nekTeσT

Zel’dovich

= −2

(9.51)

mec2

e ectc

... for a

∆T

4RnekTeσT

homogeneous

= −

mec2

(9.52)

sphere

vcosmological recession velocity

H0 Hubble parameter (present epoch)

d(proper) distance

Dpc distance (parsec)

pannual parallax (±p arcsec from mean)

L mean cepheid luminosity

L Solar luminosity

Pd pulsation period (days)

MV absolute visual magnitude

MI I-band absolute magnitude

vrot observed maximum rotation velocity (kms−1)

igalactic inclination (90◦ when edge-on)

θring angular radius

Mlens mass

ds	distance from observer to source
dl	distance from observer to lens
T	apparent CMBR temperature
dl	path element through cloud

Rcloud radius

ne electron number density

kBoltzmann constant

Te electron temperature

σT Thomson cross section me electron mass

cspeed of light

aPeriod–luminosity relation for classical Cepheids. Uncertainty in MV is ±0.27 (Madore & Freedman, 1991, Publications of the Astronomical Society of the Paciﬁc, 103, 933).

bGalaxy rotation velocity–magnitude relation in the infrared I waveband, centred at 0.90µm. The coe cients depend on waveband and galaxy type (see Giovanelli et al., 1997, The Astronomical Journal, 113, 1).

cScattering of the cosmic microwave background radiation (CMBR) by a cloud of electrons, seen as a temperature decrement, ∆T , in the Rayleigh–Jeans limit (λ 1mm).

9.5 Stellar evolution	181

9.5 Stellar evolution Evolutionary timescales

Free-fall	τ =	3π	1/2		τ	free-fall timescale
Free-fall		3π		(9.53)	G	constant of gravitation
timescalea
		32Gρ0
		32Gρ0			ρ0	initial mass density
		−Ug			τKH	Kelvin–Helmholtz timescale
Kelvin–Helmholtz	τKH =	−Ug		(9.54)	Ug	gravitational potential energy
Kelvin–Helmholtz		L			M	body’s mass
timescale		GM2			M	body’s mass
timescale		GM2		(9.55)	R	body’s initial radius
		R0L		(9.55)	L0	body’s luminosity

aFor the gravitational collapse of a uniform sphere.

Star formation

λJ =

λJ

Jeans length

π dp

1/2

constant of gravitation

Jeans lengtha

(9.56)

cloud mass density

Gρ

dρ

pressure

Jeans mass

MJ =

ρλJ3

(9.57)

(spherical) Jeans mass

Eddington luminosity

Eddington

LE =

4πGMmpc

(9.58)

stellar mass

σT

solar mass

limiting

luminosityb

proton mass

1.26 × 10

(9.59)

speed of light

σT

Thomson cross section

aNote that (dp/dρ)1/2 is the sound speed in the cloud.

bAssuming the opacity is mostly from Thomson scattering.

Stellar theorya

Conservation of

dMr

radial distance

= 4πρr2

(9.60)

mass interior to r

mass

mass density

Hydrostatic

−GρMr

(9.61)

pressure

equilibrium

constant of gravitation

Energy release

dLr

= 4πρr2

(9.62)

luminosity interior to r

power generated per unit mass

temperature

Radiative

−3

3ρ

(9.63)

Stefan–Boltzmann constant

transport

16σ

4πr

mean opacity

Convective

γ − 1

(9.64)

ratio of heat capacities, cp/cV

transport

aFor stars in static equilibrium with adiabatic convection. Note that ρ is a function of r. κ and are functions of temperature and composition.

182 Astrophysics

Stellar fusion processesa

		PP i	chain			PP ii chain				PP iii chain
p+ + p+ → 12H + e+ + νe					p+ + p+ → 12H + e+ + νe				p+ + p+ → 12H + e+ + νe
12H + p+ → 23He + γ					12H + p+ → 23He + γ				12H + p+ → 23He + γ
23He + 23He → 24He + 2p+				23He + 24He → 47Be + γ				23He + 24He → 47Be + γ
					47Be + e− → 37Li + νe				47Be + p+ → 58B + γ
					37Li + p+ → 224He					58B → 48Be + e+ + νe
										48Be → 224He
	CNO cycle				triple-α process
12	+	13		4	4		8		γ	photon
6C + p		→ 7N + γ		2He + 2He			4Be + γ		γ	photon
13N		13C + e+ + νe		8Be + 4He			12C	p+		proton
7		→ 6		4	2		6	e	+	positron
136C + p+		→ 147N + γ			126C →		126C + γ	e		positron
147N + p+		→ 158O + γ						e−		electron
15		15	+					νe		electron neutrino
8O		→ 7N + e + νe
157N + p+ → 126C + 24He

aAll species are taken as fully ionised.

Pulsars

Braking

ω˙ −ωn

(9.65)

rotational angular velocity

index

rotational period (= 2π/ω)

n = 2 −

P P

(9.66)

braking index

P˙ 2

characteristic age

Characteristic

T =

(9.67)

luminosity

agea

permeability of free space

n−

1 P

speed of light

m¨

2 sin2 θ

pulsar magnetic dipole moment

(9.68)

pulsar radius

Magnetic

L =

dipole

6πc

magnetic ﬂux density at

magnetic pole

radiation

2πR Bpω sin θ

(9.69)

angle between magnetic and

3c3µ0

rotational axes

DM dispersion measure

Dispersion

DM = 0

ne dl

(9.70)

path length to pulsar

measure

path element

electron number density

dτ

−e2

(9.71)

pulse arrival time

dν

4π

Dispersion

0mecν

∆τ

di erence in pulse arrival time

νi

observing frequencies

∆τ =

−

(9.72)

electron mass

8π2 0mec

ν12

ν22

Assuming

= 1 and that

the pulsar has already slowed signiﬁcantly. Usually n is assumed to be 3 (magnetic dipole

radiation), giving T = P /(2P ).

bThe pulse arrives ﬁrst at the higher observing frequency.

9.5 Stellar evolution	183

Compact objects and black holes

Schwarzschild

2GM

radius

rs =

(9.73)

Gravitational

ν∞

2GM 1/2

(9.74)

redshift

νr

−

rc2

Gravitational

Lg =

32 G4 m12m22(m1 + m2)

(9.75)

wave radiationa

5 c5

Rate of change of

4/3 G5/3

m1m2P −5/3

orbital period

P = −

(4π

)

(m1 + m2)1/3

(9.76)

Neutron star

degeneracy

π2)2/3 ¯h2

5/3

pressure

p =

(9.77)

(nonrelativistic)

¯hc(3π2)1/3

4/3

Relativisticb

p =

(9.78)

Chandrasekhar

MCh 1.46M

(9.79)

massc

Maximum black

GM2

hole angular

Jm =

(9.80)

momentum

Black hole

evaporation time

τe

× 1066

(9.81)

Black hole

T =

¯hc3

10−7

(9.82)

temperature

8πGMk

rs	Schwarzschild radius

Gconstant of gravitation

Mmass of body

cspeed of light

M solar mass

rdistance from mass centre

ν∞	frequency at inﬁnity
νr	frequency at r
mi	orbiting masses

amass separation

Lg gravitational luminosity

Porbital period

ppressure

¯h (Planck constant)/(2π)

mn neutron mass

ρdensity

uenergy density

MCh Chandrasekhar mass

Jm maximum angular momentum

τe	evaporation time

Ttemperature

kBoltzmann constant

aFrom two bodies, m1 and m2, in circular orbits about their centre of mass. Note that the frequency of the radiation is twice the orbital frequency.

bParticle velocities c.

cUpper limit to mass of a white dwarf.

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