Chapter 11 White Dwarfs and Neutron...

Chapter 11

White Dwarfs and Neutron Stars

Red giants will eventually consume all their accessible nu-clear fuel.

• After ejection of the envelope, the cores of these starsshrink to the very hot, very dense objects that we callwhite dwarfs.

• An even more dense object termed a neutron star canbe left behind after the evolution of more massivestars terminates in a core-collapse supernova explo-sion.

Technically, white dwarfs and neutron stars are stellarcorpses, not stars, but it is common to refer to them looselyas stars.

505

506 CHAPTER 11. WHITE DWARFS AND NEUTRON STARS

11.1 Sirius B

The bright star Sirius, in Canis Major, is actually a doublestar.

• The brighter component is labeled Sirius A and thefainter companion tar is known as Sirius B.

• Sirius B is an example of a white dwarf.

• Because of its proximity to Earth, Sirius B is not par-ticularly dim (visual magnitude 8.5), but it is difficultto observe because it is so close to Sirius A.

• Sirius B is clearly not a normal star; its spectrum andluminosity indicate that it is hot (about 25,000 K sur-face temperature) but very small.

• This spectrum contains pressure-broadened hydro-gen lines, implying a surface environment with muchhigher density than that of a normal star.

11.1. SIRIUS B 507

• Assuming the spectrum of Sirius B to be blackbodyand using the well-established distance to Sirius, weconclude from its luminosity that Sirius B has a ra-dius of only about 5800 km.

• But Sirius is a visual binary with a very well studiedorbit.

• Therefore, we may use Kepler’s laws to infer that themass of Sirius B is about 1.03M⊙.

• We conclude that a white dwarf like Sirius B packsthe mass of a star in an object the size of the Earth.

Sirius B is the nearest and brightest whitedwarf and we shall often use it as illustration.But it is in some respects not so representativebecause its mass of about 1.03M⊙ is muchlarger than the average mass of about 0.58M⊙

observed for white dwarfs (Sirius B is in the98th percentile of white dwarf masses).


11.2 Properties of White Dwarfs

The preceding discussion allows us to make some imme-diate estimates that will shed light on the nature of whitedwarfs even before we carry out any detailed analysis.

11.2. PROPERTIES OF WHITE DWARFS 509

11.2.1 Density and Gravity

• Since white dwarfs contain roughly the mass of theSun in a sphere the size of the Earth, we expect thatwhite dwarfs have densities in the vicinity of 106

g cm−3.

• For Sirius B the average density calculated from theobserved mass and radius is about2.5×106 g cm−3.

• The gravitational acceleration and the escape veloc-ity at the surface for Sirius B are

g=GmR2 ≃ 3.7×108 cm s−2 vesc

c=

√

2GmRc2

≃ 0.02,

respectively, indicating that

– the gravitational acceleration is almost 400,000times larger than at the Earth’s surface, but

– general relativity effects, while not completelynegligible, are sufficiently small to be ignored ininitial approximation.


11.2.2 Equation of State

• We conclude from the preceding that hydrostaticequilibrium under Newtonian gravitation is adequateas a first approximation for the structure of whitedwarfs.

• What about the microphysics of the gas?

– Can we apply a Maxwell–Boltzmann descrip-tion, or will the quantum statistical properties ofthe gas play a crucial role?

– Will electron velocities be describable classi-cally or will velocities become relativistic?


Let us assume initially nonrelativistic velocities and that electrons areprimarily responsible for the internal pressure of the white dwarf.

• For simplicity we shall also assume that the white dwarf is com-posed of a single kind of nucleus having atomic numberZ, neu-tron numberN, and atomic mass numberA = Z+N.

• Then the average electron velocity is̄ve = p̄/me where p̄ is theaverage momentum andme is the electron mass.

• By the uncertainty principle, the average momentum may be es-timated as

p̄≃ ∆p≃ h̄/∆x≃ h̄n1/3e ,

wherene is the electron number density.

• We may expect the gas to be completely ionized and the corre-sponding electron number density is

ne =

(

numbere−

nucleon

)(

number nucleonsunit volume

)

=

(

ZA

)(

ρmH

)

.

• Therefore, the average electron velocity may be approximatedby

v̄e

c=

p̄mec

=h̄n1/3

e

mec=

h̄mec

(

ZρAmH

)1/3

≃ 0.25,

where we assume thatA = 2Z, as would be true for12C, 16O, or4He, which are the primary constituents of most white dwarfs.

• We conclude on general grounds thatelectron velocities willbecome relativistic (significant fraction of c) for higher-densitywhite dwarfs.


• The average spacing between electrons in the gas is

d ≃ n−1/3e ≃ 1.5×10−10 cm,

usingZ/A = 0.5 and the average density of Sirius B.

• The deBroglie wavelength of electrons in the gas is on average

λ̄e =hp

=h

v̄eme

≃ 9.6×10−10 cm.

• Because particle separation is comparable to deBroglie wave-length, the electron gas will be degenerate, provided that thetemperature is not too high.

• For a degenerate fermion gas the fermi energy is (h̄= c= 1units)

Ef =√

k2f +m2.

The gas will remain degenerate as long asEf is much larger thanthe characteristic energykT of particles in the gas.

• Since from the preceding equationEf ≥ mec2 = 0.511 MeV, thisimplies that a temperatureT = E/k= 0.511 MeV/k≃ 6×109 Kis required to break the degeneracy.

• Detailed calculations indicate that interior white dwarftempera-tures are typically in the range 106–107 K, so we conclude thatwhite dwarfs contain cold, degenerate gases of electrons.

• Therefore they may be approximated by polytropic equations ofstate having the form

P = Kργ ,

1. γ = 53 for nonrelativistic degenerate electrons

2. γ = 43 for ultrarelativistic degenerate electrons.


• While we expect the electrons to be degenerate and to becomerelativistic at higher densities, the ions are much more massivethan the electrons.

• The ions are neither relativistic nor degenerate, and are well de-scribed by an ideal gas equation of state.

• Because the ions move slowly, they contribute little to thepres-sure.

• However, calculations indicate that most of the heat energy storedin the white dwarf is associated with motion of the ions.

• Finally, photons in the white dwarf constitute a relativistic gasapproximated by a Stefan–Boltzmann equation of state,

P = 13 aT4,

wherea is the radiation density constant andT is the tempera-ture.

Thus, the picture that emerges for a white dwarf is of ahot, dense object for which the mechanical properties (ex-emplified by the pressure, which is generated primarily bythe degenerate electrons) are decoupled from the thermalproperties (which are associated primarily with the ions atnormal temperatures).


11.2.3 Ingredients of a White Dwarf Description

An initial description of a white dwarf may be afforded by a theoryfor which

1. The stable configurations correspond to hydrostatic equilibriumunder Newtonian gravitation.

2. The ions carry most of the mass and store most of the thermalenergy of the white dwarf, but the electrons are responsibleforthe bulk of the pressure.

3. The electron equation of state will be that of a cold degenerategas, conveniently approximated in the polytropic formP = Kργ

with γ = 53 for nonrelativistic andγ = 4

3 for relativistic electrons,respectively.

4. Ions are nonrelativistic and may be described by an ideal gasequation of state.

5. Photons are described by a Stefan–Boltzmann equation of state.

6. Because the degenerate electron gas is primarily responsible forthe pressure but its equation of state does not depend on temper-ature, the thermal and mechanical properties of the white dwarfare decoupled.

7. As density increases the velocity of the electrons increases andspecial relativity becomes important, corresponding to a transi-tion

P≃ Kρ5/3 −→ P≃ K′ρ4/3.

in the electron equation of state.

Let us now turn to a theoretical description embodying thesebasicideas in a relatively simple formulation.

11.3. LANE–EMDEN EQUATIONS 515

11.3 Lane–Emden Equations

The equations of hydrostatic equilibrium may be com-bined to give the differential equation

dm= 4πr2ρ(r)dr

dPdr

= −Gm(r)

r2 ρ

→1r2

ddr

(

r2

ρdPdr

)

= −4πGρ .

Assuming a completely degenerate electron gas, we adopta polytropic equation of state with

P = Kργ = Kρ1+1/n γ ≡ 1+1n.

Introducing dimensionless variablesξ andθ through

ρ = ρcθn r = aξ a =

√

(n+1)Kρ(1−n)/nc

4πG,

whereρc ≡ ρ(r = 0) is the central density, the differentialequation embodying hydrostatic equilibrium for a poly-tropic equation of state may be expressed in terms of thenew dependent variableθ(ξ ) and independent variableξas,

1ξ 2

ddξ

(

ξ 2dθdξ

)

= −θn.


In terms of these new variables the boundary conditionsare

θ(0) = 1 θ ′(0) ≡dθdξ

∣

∣

∣

∣

ξ=0= 0,

• The first follows from the requirement that the cor-rect central densityρc = ρ(0) be reproduced.

• The second follows from requiring that the pressuregradientdP/dr vanish at the origin (necessary condi-tion for hydrostatic equilibrium).

Then we may integrate

1ξ 2

ddξ

(

ξ 2dθdξ

)

= −θn.

outward from the origin (ξ = 0) until the pointξ = ξ1

whereθ first vanishesto define the surface of the star,since at this pointρ = P = 0 because

ρ = ρcθn P = Kργ .

Solutions having this property generally exist forn < 5.

11.3. LANE–EMDEN EQUATIONS 517

Table 11.1: Lane–Emden constants

n γ ξ1 ξ 21 |θ

′(ξ1)|

0 ∞ 2.4494 4.8988

0.5 3 2.7528 3.7871

1.0 2 3.14159 3.14159

1.5 5/3 3.65375 2.71406

2.0 3/2 4.35287 2.41105

2.5 1.4 5.35528 2.18720

3.0 4/3 6.89685 2.01824

4.0 5/4 14.97155 1.79723

4.5 1.22 31.83646 1.73780

5.0 1.2 ∞ 1.73205

• The equation

1ξ 2

ddξ

(

ξ 2dθdξ

)

= −θn.

has analytical solutions for the special casesn =

0,1, and 5, but

• in the physically most interesting cases the equationsmust beintegrated numericallyto define the Lane–Emden constantsξ1 andξ 2

1 |θ′(ξ1)| for givenn.

These are tabulated for various values ofn and γ in Ta-ble 11.1.


The transformation equations

ρ = ρcθn r = aξ a =

√

(n+1)Kρ(1−n)/nc

4πG,

may then be used to express quantities of physical interest in termsof these constants for definite values of the polytropic index n. Forexample, the radiusR is

R= aξ1 =

√

(n+1)K4πG

ρ(1−n)/2nc ξ1,

and the massM is given by (Exercise 11.4)

M ≡ 4πa3ρc

[

−ξ 2dθdξ

]

ξ=ξ1

= 4π[

(n+1)K4πG

]3/2

ρ(3−n)/2nc ξ 2

1 |θ′(ξ1)|,

Eliminatingρc between these two equations gives a general relation-ship between the mass and the radius,

M = 4πR(3−n)/(1−n)

(

(n+1)K4πG

)n/(n−1)

ξ (3−n)/(n−1)1 ξ 2

1 |θ′(ξ1)|.

for a solution with polytropic indexn.

11.4. POLYTROPIC MODELS OF WHITE DWARFS 519

11.4 Polytropic Models of White Dwarfs

We expect that white dwarfs are approximately describedby systems in hydrostatic equilibrium having degenerateelectron equations of state.

• Thus, we may expect that solutions of the Lane–Emden equation with polytropic indexn = 3

2, cor-responding toγ = 5

3, are relevant for the structure oflow-mass white dwarfs where electron velocities arenonrelativistic.

• Likewise, we may expect that in more massive whitedwarfs the electrons become relativistic and the cor-responding structure is related to a Lane–Emden so-lution with polytropic indexn = 3, corresponding toγ = 4

3.

• Between these extremes the electron equation of statemust generally be described in numerical terms per-mitting an arbitrary level of degeneracy and degreeof relativity.


11.4.1 Low-Mass White Dwarfs

Let us first consider a low-mass white dwarf.

• Assuming aγ = 53 polytropic equation of state (n = 3

2), the pre-ceding equation

M = 4πR(3−n)/(1−n)

(

(n+1)K4πG

)n/(n−1)

ξ (3−n)/(n−1)1 ξ 2

1 |θ′(ξ1)|.

then givesMR3 = constant,

since

R(3−n)/(1−n) = R(3−3/2)/(1−3/2) = R(3/2)/(−1/2) = R−3.

Thus the product of the mass and the volume of a white dwarf isconstant.

We obtain the surprising result that, contrary to the be-havior of normal stars,increasing the mass of a low-masswhite dwarf causes its radius to shrink.

• This behavior is a direct consequence of a degenerate electronequation of state.


11.4.2 The Chandrasekhar Limit

If we continue to add mass to a white dwarf, the electrons willmovefaster by uncertainty principle arguments and eventually will becomerelativistic. If γ = 4

3 (corresponding ton = 3),

M = 4πR(3−n)/(1−n)

(

(n+1)K4πG

)n/(n−1)

ξ (3−n)/(n−1)1 ξ 2

1 |θ′(ξ1)|.

then implies that

M = constant×R0 = constant

This even more surprising resultdefines theChan-drasekhar limiting mass, which implies that there isanupper limit for the mass of a white dwarf.

Inserting the constants, we find that for the radius of a high-masswhite dwarf described by a relativistic, degenerate electron equationof state,

R= 3.347×104(

ρc

106 g cm−3

)−1/3(µe

2

)−2/3km,

and for the Chandrasekhar mass,

M0 = 1.457

(

2µe

)2

M⊙ ≃ 1.4M⊙,

where the last estimate follows because2/µe is of order unity.

Thus the Chandrasekhar limiting mass is slightly compo-sition dependent but implies an upper limit on the mass ofa white dwarf of approximately1.4M⊙.


0.0 0.2 0.4 0.6 0.8 1.0 1. 2 1.4 1.6 1.8 2.0

Mass (Solar Units)

0

2000

4000

6000

8000

10000

Ra

diu

s (

km

)

Figure 11.1:The dependence of radius on mass for a white dwarf. The limitingmass of 1.44 solar masses is indicated by the onset of numerical instability in thecalculation. Calculated assumingYe = 0.5 and a central temperatureTc = 5.0×106 K. (The electron fractionYe is the ratio of the number of electrons to the totalnumber of nucleons. For symmetric matterZ = N, so for fully-ionized symmetricmatter,Ye = 1

2.)

• In Fig. 11.1 the radius versus mass for white dwarfs in hydro-static equilibrium is shown for a numerical simulation.

• This calculation uses a numerical equation of state that accountsfully for arbitrary degrees of electron degeneracy and arbitraryrelativity for electrons.

• Thus, for electrons this equation of state approximates aγ = 53

polytrope at low mass and aγ = 43 polytrope at high mass, with

a smooth transition in between.


0.0 0.2 0.4 0.6 0.8 1.0 1. 2 1.4 1.6 1.8 2.0

Mass (Solar Units)

0

2000

4000

6000

8000

10000

Ra

diu

s (

km

)

• The ions of the white dwarf are assumed to obey anideal gasequation of state.

• The photonsare described by aStefan–Boltzmann equation ofstate.

• We see clearly in the above figure the behavior implied by thepreceding equations.

1. Lower masses:the radius of the white dwarf decreases steadilywith increase in mass, in accord withMR3 = constant.

2. Higher masses:the curve turns over and approaches a verti-cal asymptote given byM = M0 = constant, with the calcu-lation becoming numerically unstable as the limiting massis approached.


103

104

105

106

107

108

Density (Central Solar Units)

0.001

0.010

0.100

1.000

10.000

Mass o

r R

adiu

s (

Sola

r U

nits) Chandrasekhar Limit = 1.44

Mass

Radius

Y(e) = 0.5

Figure 11.2:The variation of mass and radius for white dwarfs as a function ofthe central density in units of central solar densities.

• In Fig. 11.2 the variation of the mass and radius of white dwarfsas a function of the central density in central solar units isplottedfor calculations similar to those described in Fig. 11.1.

• Note the steady trend to zero radius as the white dwarf approachesthe limiting mass asymptotically.


11.4.3 Heuristic Derivation of the Chandrasekhar Limit

The Chandrasekhar limiting mass was obtained above asa consequence of the Lane–Emden equations, which em-body polytropic equations of state and hydrostatic equi-librium. It will prove useful in understanding the limitingmass for white dwarfs to obtain the Chandrasekhar resultin a somewhat more intuitive way.


• Assume a fully-ionized sphere of symmetric(equal numbers ofprotons and neutrons)matter containingN electrons.

• The mass of the sphere is thenM = 2mpN,

• The average spacing between electrons isd ∼ R/N1/3, and

• The average momentum of the electrons is (uncertainty princi-ple)

pf ∼h̄d∼

h̄M1/3

Rm1/3p

.

• By estimating the total energy of the degenerate electronsandbalancing that against the gravitational energy of the protons (seeExercises 11.1–11.2), one obtains for energy balance in thenon-relativistic and relativistic limits,

E = aM5/3

R2 −bM2

R(nonrelativistic)

E = cM4/3

R−d

M2

R(relativistic)

wherea, b, c, andd are positive constants.

• Notice that the two terms in the nonrelativistic case havediffer-ent dependence onR.

• Thus, by setting∂E/∂R= 0, one finds an equilibrium configu-ration in the nonrelativistic case that generally satisfiesMR3 = constant.


• On the other hand, in the relativistic case

E = cM4/3

R−d

M2

R(relativistic)

the two terms have thesame dependence onR (a result that fol-lows directly from the requirementγ = 4

3).

• Thus, attempting to solve∂E/∂R= 0 for R corresponding toa gravitationally stable configuration leads to anindeterminateresult(the resulting equation does not depend onR).

• The meaning of this result is clarified if we note that both termsin this equation vary asR−1, but the first term depends onM4/3

while the second varies asM2.

• The second term has a net negative sign and a stronger depen-dence onM than the first term, sothe total energy of the systembecomes negative if the mass is made large enough.

• But since the total energy scales asR−1, once the total energy be-comes negativethe system can minimize its energy by shrinkingto zero radius:

For the relativistic degenerate gas there is alimiting mass beyond which the system is un-stable against gravitational collapse.

• We may estimate this critical mass by equating the two termsinE = cM4/3/R−dM2/R, yielding (see Exercise 11.1)

M0 =

(

h̄c

Gm4/3p

)3/2

≃ 1 M⊙,

which is correct to order of magnitude.


11.4.4 The Adiabatic Indexγγγ and Gravitational Stability

The preceding results are another variation of the themeintroduced previously in conjunction with the collapse ofprotostars.

• There we found that an adiabatic index ofγ ≤ 43 im-

plies aninstability against expansion or contraction.

• From the polytropic equation of stateP = Kργ ,

dPdρ

= Kγργ−1 →ρP

dPdρ

= γρP

Kργ−1 = γKργ

P= γ

so we maydefinean effectiveγ for any equation ofstateP(ρ) by

γ ≡ρP

dPdρ

=dlnPdlnρ

.

• Taking this logarithmic derivative as the definition ofan effective adiabatic indexγeff, we may expect thatin any simulation of hydrostatic equilibrium,

γeff ≡ρP

dPdρ

≃43

heralds the onset of a radial scaling instability.


0 5000 10000 15000

Radius (km)

1.0

1.2

1.4

1.6

1.8

2.0

Gam

ma

0.32

0.54

0.83

1.10

1.291.39

γ = 5/3

γ = 4/3

Figure 11.3:Values of the parameterγ ≡ d lnP/d lnρ at constant temperature forwhite dwarfs of various masses (solar units). The values ofγ corresponding tononrelativistic (γ = 5/3) and relativistic (γ = 4/3) polytropes are indicated. Thecalculation becomes unstable as the mass approaches the Chandrasekhar limitingmass which is 1.44 solar masses for this calculation (for which Ye = 0.5). Thecentral temperature is assumed to be 5×106 K in all calculations.

• In Fig. 11.3 the value ofγeff as a function of radius is calculatednumerically using

γ =ρP

dPdρ

=dlnPdlnρ

.

for white dwarf solutions that have been obtained with an equa-tion of state allowing arbitrary electron degeneracy and relativ-ity.


0 5000 10000 15000

Radius (km)

1.0

1.2

1.4

1.6

1.8

2.0

Gam

ma

0.32

0.54

0.83

1.10

1.291.39

γ = 5/3

γ = 4/3

• For low-mass white dwarfs the effective value ofγ is near thenonrelativistic expectation of53 for the entire interior.

• However, as the mass of the white dwarf is increased, the effec-tive value ofγ in the deep interior begins to drop.

• As the mass approaches the Chandrasekhar limit,γeff tends to43

and the numerical solution becomes very unstable.

• These numerical fluctuations reflect the incipient gravitationalinstability that in this case occurs at 1.44 solar masses.


The roles ofrelativity andquantum mechanicsare centralto the preceding results.

• Nonrelativistic degenerate matterhasγ ∼ 53, which

is gravitationally stable.

• But quantum mechanics (the uncertainty principle)requires that theelectrons move faster as the densityincreases,implying that the velocities eventually be-come relativistic as the white dwarf mass increases.

• Relativistic degenerate matterhasγ ∼ 43, which in-

herently isgravitationally unstable.

• Because theelectron speed is limited by the speed oflight, there is a mass beyond which even the degen-eracy pressure cannot prevent gravitational collapseof the system.

This critical point is theChandrasekhar limiting mass.


0.0 2000 4000 6000 8000 10,000

Radius (km)

0.0

0.2

0.4

0.6

0.8

1.0

Mass, D

ensity, or

Tem

pera

ture

T/T(0)

M (Solar Units)

ρ/ρ(0)

Figure 11.4: Behavior of density, enclosed mass, and temperature for a whitedwarf. In this calculation the white dwarf has a central density of 2.9×106 g cm−3,a central temperature of 5.0× 106 K, a total mass of 0.595 solar masses, and aradius of 9234 km.

11.5 Internal Structure of White Dwarfs

• A numerical calculation of the internal structure of a white dwarfis illustrated in Fig. 11.4, which plots the density, enclosed mass,and temperature as a function of radius.

• The calculation corresponds to hydrostatic equilibrium with arealistic electron equation of state in which the electronshavearbitrary degeneracy and degree of relativity.

• The ions are assumed to obey an ideal gas equation of state andradiation to obey a Stefan–Boltzmann equation of state.

11.5. INTERNAL STRUCTURE OF WHITE DWARFS 533

0 2000 4000 6000 8000 10000

Radius (km)

10-3

10-4

10-2

10-1

100

101

102

103

104

105

106

107

Pre

ssure

(centr

al sola

r units)

Electronic Contribution

Ionic Contribution

Figure 11.5:Relative contributions of the electronic pressure and ionic pressurefor the calculation described in Fig. 11.4. The contribution to the pressure fromradiation under these conditions is completely negligiblerelative to the electronicand ionic contributions. The electronic contribution is very nearly that expected fora fully degenerate gas.

Figure 11.5 illustrates the relative contribution of elec-trons and ions to the pressure in the preceding calculation,and provides strong justification for our earlier assump-tion that the pressure in white dwarfs is dominated by thecontribution from degenerate electrons.


0.0 2000 4000 6000 8000 10,000

Radius (km)

0.0

0.2

0.4

0.6

0.8

1.0

Mass, D

ensity, or

Tem

pera

ture

T/T(0)

M (Solar Units)

ρ/ρ(0)

The internal temperature variation in the calculation shown above isdetermined as follows.

• Because degenerate matter is such a good conductor of thermalenergy, the interior of a white dwarf cannot support a substantialtemperature gradient and we assume all but a thin surface layerto be isothermal and strongly heat conducting.

• On the other hand, near the surface the density drops to zeroandthe nearly ideal gas expected there is a very good insulator.

• This suggests that a good model of how white dwarfs cool is oneof a conducting sphere with no temperature gradient surroundedby a thin layer of normal gas with a gradient set by its transportproperties (that is, by its opacity).

• This model is analogous mathematically to the cooling of a hotmetal ball surrounded by a thin insulating jacket, since degener-ate gases have many of the properties of metals.

11.5. INTERNAL STRUCTURE OF WHITE DWARFS 535

0.0 2000 4000 6000 8000 10,000

Radius (km)

0.0

0.2

0.4

0.6

0.8

1.0

Mass, D

ensity, or

Tem

pera

ture

T/T(0)

M (Solar Units)

ρ/ρ(0)

In the “metal ball plus insulating blanket” model for the above figure,

• The interior is assumed fully conductive,

• The surface is assumed insulating with a radiative opacitygivenby the Kramers bound–free opacity, and

• the transition between the two is governed by the degeneracyparameter

α =µ −mec2

kT,

whereµ is the chemical potential for the electrons.


0 5,000 10,000 15,000

Radius (km)

0

50

100

150

200

250

300

De

ge

ne

racy P

ara

me

ter

Figure 11.6:The degeneracy parameterα ≡ (µ−mec2)/kT as a function of radiusin a white dwarf, whereµ is the electron chemical potential andme the electronmass. The white dwarf has a calculated mass of 0.54 solar masses, a radius of10,000 km, a central temperature of 5× 106 K, and a central density of 2.135×106 g cm−3. The figure was calculated with the equation of state used in Fig. 11.2.

• The variation of the degeneracy parameter

α =µ −mec2

kT,

with radius is illustrated in Fig. 11.6.

• In the interiorα is large, indicating ahighly degenerate gasofelectrons.

• But very near the surfaceα → 0, implying that in a thin surfacelayer the electrons obey approximately anideal gas law.

11.6. COOLING OF WHITE DWARFS 537

11.6 Cooling of White Dwarfs

Although white dwarfs have no internal heat source, theycan remain luminous for long periods of time as the heatleft over from their glory days slowly leaks away.

• The cooling curve for a white dwarf should then re-flect both the internal structure and the age of the star.

• As we have seen, white dwarfs are well describedby a spherical ball of electron degenerate matter sur-rounded by a very thin surface layer that obeys anideal gas equation of state.

This can serve as a simple but quantitative model for cool-ing rates in white dwarfs.


11.7 Beyond White Dwarf Masses

The preceding discussion of limiting masses for whitedwarfs assumes all pressure to derive from electrons.

• However, if the Chandrasekhar mass is exceeded andthe system collapses, eventually a density will bereached where the nucleons (also fermions) will be-gin to produce a strong degeneracy pressure.

• Whether this nucleon degeneracy pressure can haltthe collapse depends on the mass.

• Calculations indicate that for a mass less than about2–3 solar masses (depending weakly on details suchas the equation of state), the collapse converts essen-tially all protons into neutrons through the weak in-teractions, producing a neutron star.

• The degeneracy pressure of the neutrons halts thecollapse at neutron-star densities and radii approxi-mately 500 times smaller than for white dwarfs.

• Calculations, and general considerations for stronggravity, indicate that for masses greater than this eventhe neutron degeneracy pressure cannot overcomegravity and the system collapses to a black hole.

• These considerations also indicate that white dwarfsand neutron stars are theonly possible stable config-urations lying between normal stars and black holes.

Therefore, let us now consider neutron stars.

11.8. BASIC PROPERTIES OF NEUTRON STARS 539

11.8 Basic Properties of Neutron Stars

• Neutron stars were predicted in 1933 by Baade andZwicky as a possible end result of what we wouldnow call a core-collapse supernova.

• Oppenheimer and Volkov worked out equations de-scribing their general structure and properties in1939. (This will be covered in Astro 616, since itrequires general relativity.)

• However, they were not taken very seriously untilthe discovery of radio pulsars in the 1960s pointedto rapidly rotating neutron stars as their most likelyexplanation.

• It is estimated that there are some 108 neutron starsin our galaxy.

• Of order 1000 of these have actually been observedby astronomers so far.


• Most neutron stars have been discovered as radio pulsars butthe vast majority of the energy emitted by neutron stars is invery high-energy photons (X-rays andγ-rays), rather than radiowaves.

• Typically only about10−5 of their radiated energy is in the radio-frequency spectrum.

• Most neutron stars have masses of1–2M⊙ and diameters of10–20 km. Very loosely, a neutron star packs the mass of a normalstar like the Sun into a volume of order 10 km in radius.

• From the density of a little over1 g cm−3 and radius of about7×105 kmfor the Sun, we may estimate immediately an averagedensity of order1014 g cm−3 for neutron stars (it can actually beabout an order of magnitude larger than that).

• Thus, they have enormousdensities similar to those of an atomicnucleus.

• In fact, in certain ways (but not all),a neutron star is similar toa giant atomic nucleus the size of a city.

• Their enormous densities imply strong gravitational fields andthe possibility of significant general relativistic deviations fromNewtonian gravity(to be discussed in Astro 616).


Box 11.2 Electron Capture and Neutronization

The formation of a neutron star results from a process calledelectron capture(a form of beta decay), which can follow thecore collapse of a massive star late in its life to produce a super-nova (see further Ch. 14).

• The process is also calledneutronization,because its ef-fect is to destroy protons and electrons and create neutrons.The basic reaction is

e−+ p+ → n0+νe.

• It is slow under normal conditions (because it is mediatedby the weak interaction), but very fast in the high densityand temperature environment produced by core collapse ina massive star.

• In the supernova explosion the enormous amount of en-ergy released gravitationally in the collapse of the coreblows off the outer layers of the star and leaves behindan extremely dense, hot remnant.

• As the neutronization reaction proceeds, the neutrinos es-cape carrying off energy and leave behind the neutrons.

• Because neutrons carry no charge, there is no electricalrepulsion as in normal matter and the core can collapse tovery high density once it has become mostly neutrons.

• The structure of actual neutron stars is more complex thanthis, and they are not composed entirely of neutrons, butthis simple picture captures the basic idea.


10 km

Atmosphere

Outer Crust

Inner Crust

Outer Core

Inner Core?

200 m thick fluid or solid lattice of heavy nuclei; pressure: degenerate electrons.

600 m deep. Lattice of heavy nuclei; superfluid free neutrons; pressure: degenerate electrons.

Superfluid neutrons; some superconducting protons; pressure: degenerate neutrons.

Uncertain, but there may be a core of elementary particles. Density of order 1015 g cm-3.

Hot plasma.

Figure 11.7:Internal structure of a typical neutron star.

Internally, we believe that a neutron star can be divided into the fol-lowing general regions (see Fig. 11.7).

• The atmosphere is thin (∼ 1 cm thick) and consists of very hot,ionized gas.

• The outer crust is only about 200 meters thick and consists ofa solid lattice or a dense liquid of heavy nuclei. The dominantpressure in this region is from electron degeneracy. The densityis not high enough here to favor neutronization.

• The inner crust is from12 to 1 kilometer thick. The pressure is

higher and the lattice of heavy nuclei is now permeated by freesuperfluid neutrons that begin to “drip” out of the nuclei. Thepressure is still mostly from degenerate electrons.


10 km

Atmosphere

Outer Crust

Inner Crust

Outer Core

Inner Core?

200 m thick fluid or solid lattice of heavy nuclei; pressure: degenerate electrons.

600 m deep. Lattice of heavy nuclei; superfluid free neutrons; pressure: degenerate electrons.

Superfluid neutrons; some superconducting protons; pressure: degenerate neutrons.

Uncertain, but there may be a core of elementary particles. Density of order 1015 g cm-3.

Hot plasma.

• The outer core is composed primarily of superfluid neutronsandthe neutrons supply most of the pressure through neutron degen-eracy, though there are some free superconducting protons.Thisregion is what gives the neutron star its name.

• The structure of the inner core is less certain than that of theouter portion of the star because we are less certain about howmatter behaves under the intense pressure at the center (that is,theequation of statefor matter under these conditions is not wellunderstood).

• It might even consist of a solid core of particles more elementarythan nucleons (pions, hyperons, quarks, . . . ).

Although much of a neutron star consists of closely packed neutronsand thus has some resemblance to a giant atomic nucleus, it isimpor-tant to remember that it is gravity, not the nuclear force, that holds aneutron star together (see the following box).


Box 11.3 Neutron Stars Are Bound by Gravity

In some ways a neutron star is like 20-km diameter atomic nu-cleus, but there is one important difference:

A neutron star is bound bygravity, and thestrength of that binding is such that the den-sity of neutron stars is even greater than thatof nuclear matter.

• How can the weakest force (gravity) produce an objectmore dense than atomic nuclei, which are held togetherby a diluted form of the strongest force?

• The answer:range and sign of the forces involved.

– Gravity is weak, but long-ranged and attractive.

– The strong nuclear force is short-ranged, acting onlybetween nucleons that are near neighbors.

– The normally attractive nuclear force becomes repul-sive at very short distances. (A neutron star wouldexplodeif gravity were removed.)

• This is a kind of Tortoise and Hare fable:

– Gravity is weak, but relentless and always attractive.

– Thus, over large enough distances and long enoughtime, gravity—the plodding Tortoise of forces—always wins.

That is why the material in a neutron star can be compressed tosuch high density by the most feeble of the known forces.

11.9. PULSARS 545

11.9 Pulsars

In 1967 something remarkable was discovered in the sky:a star that appeared to be pulsing on and off with a pe-riod of about a second. Shortly, other such “pulsars” ofeven faster variation were discovered and the fastest nowknown (themillisecond pulsars) pulse on and off at nearlya thousand times a second.

Pulsars exhibit several common characteristics:

1. They have well-defined periods that challenge the ac-curacy of the best atomic clocks.

2. The measured periods range from about 4.3 secondsdown to 1.6 milliseconds.

1.6 ms corresponds to approximately 640 rev-olutions per second, implying a 20-km wideobject spinning as fast as a kitchen blender.

3. The period of a pulsar decreases slowly with time.The typical rate of decrease is a few billionths of asecond each day, which implies that the frequencyof pulsation will drop to zero after about 10 millionyears for typical pulsars.

What could cause this rather remarkable behavior? Weshall now argue that only a neutron star can do the job.


11.9.1 The Pulsar Mechanism

The observational details for pulsars are inconsistent withan actual pulsation on that timescale for realistic objectsbut a rotating star couldappear to pulseif it had someway to emit light in a beam that rotated with the source(just as a lighthouse appears to pulse as the beam sweepsover an observer).

• What kind of object would be consistent with ob-served pulsar periods?

• Simple calculations show that only a very dense ob-ject could rotate fast enough and not fly apart becauseof the forces associated with the rapid rotation.

• A white dwarf is not dense enough. The minimumrotational period for a typical white dwarf would beseveral seconds; for shorter periods it would fly apart.

• But a neutron star is so dense that it could rotatemore than a thousand times a second and still holdtogether.

This qualitative inference, augmented bymuch more detailed considerations, leads tothe conclusion that the only plausible expla-nation for pulsars is that they are rapidly spin-ning neutron stars, with a mechanism to beamradiation in a kind of lighthouse effect.

11.9. PULSARS 547

Box 11.4 The Lighthouse Mechanism

A magnetic field varying in time produces an electrical field.

• Thus, the rapidly spinning magnetic field of the pulsar gen-erates a very strong electrical field around the neutron star.

• This field accelerates electrons away from the surface at“hot spots” near the magnetic poles and these acceleratedelectrons produce radiation by the synchrotron effect.

• The synchrotron radiation is beamed strongly in the direc-tion of electron motion.

• These beams rotate with the star, but the magnetic axisdoes not generally coincide with the rotation axis (recallEarth), so the beams rotate in a kind of corkscrew fashion:

• If these gyrating beams sweep over the Earth, they act sim-ilar to a lighthouse and we observe flashes of light.

Thus, the neutron star appears to be pulsing, even though it isneither pulsing nor is it really a star.


Table 11.2: Some typical magnetic field strengths

Object Strength (gauss)

Earth’s magnetic field 0.6

Simple bar magnet 100

Strongest sustained laboratory fields 4×105

Strongest pulsed laboratory fields 107

Maximum field for ordinary stars 106

Typical field for radio pulsar 1012

Magnetars 1014–1015

11.9.2 Magnetic Fields

Some pulsars contain the strongest magnetic fields knownin our galaxy, and many of their basic properties arethought to derive from these fields.

• Some typical magnetic field strengths for various ob-jects are listed in Table 11.2 (1 tesla = 104 gauss).

• From the table, the two classes of objects with thelargest known magnetic fields are seen to be

1. radio pulsars and

2. magnetars (magnetars will be discussed below),

both of which involve rotating neutron stars.

We infer that very strong magnetic fields are likely to bevery common for neutron stars in general, although deduc-ing that is more difficult if the neutron star is not observedas a pulsar or magnetar.

11.9. PULSARS 549

11.9.3 The Crab Pulsar

The first pulsar was found by Jocelyn Bell and AnthonyHewish at the Cambridge radio astronomy observatory in1967. The most famous pulsar was discovered shortly af-ter that.

• It lies in the Crab Nebula (M1), which is about 7000light years away in the constellation Taurus.

• The Crab Pulsar rotates about 30 times a second,emitting a double pulse in each rotation in the radiothrough gamma-ray spectrum.

• In visible light, the Crab Pulsar appears to be a mag-nitude 16 star near the center of the nebula, but stro-boscopic techniques reveal it to be pulsing.


1250 1200 1150 11000

0.1

0.2

0.3

0.4

Time (milliseconds)

Rela

tive Inte

nsity

Light Curve

Time-Lapse Pulsar Image

Primary Pulse Secondary Pulse

Figure 11.8:Light pulses from the Crab Pulsar. In this composite of EuropeanSouthern Observatory data, the pulsar is shown in a time lapse image at the top andthe light curve is displayed at the bottom on the same timescale.

Figure 11.8 shows the Crab Pulsar in action.

• The sequence is a composite of images taken through3 different filters, all in the visible spectrum.

• Both the image sequence and the light curve showclearly the “double pulsing” of the Crab: in eachcycle there is a strong primary pulse followed by amuch weaker secondary pulse.

• The period (time between successive primary or sec-ondary pulses) implies one primary and one sec-ondary pulse about 30 times every second.

11.9. PULSARS 551

1250 1200 1150 11000

0.1

0.2

0.3

0.4

Time (milliseconds)

Rela

tive Inte

nsity

Light Curve

Time-Lapse Pulsar Image

Primary Pulse Secondary Pulse

• This double pulsing effect can be explained by thelighthouse model if the geometry is such that thebeam from one magnetic pole sweeps more directlyover the Earth but the beam from the other pole doesso only partially.

Although the Crab Pulsar emits visible light (and X-raysand gamma rays), most pulsars are detectable only by theirradio frequency radiation. However, a few pulse stronglyin other wavelength bands.


1969 1971 1973 1975 1975

Date

0.089230

0.089220

0.089210P

eri

od

(µ

-se

c)

0.089240

10-5 sec

Glitch

Glitch

Glitch Vela Pulsar

Figure 11.9:Glitches in the Vela Pulsar.

11.10 Pulsar Spindown and Glitches

In some pulsars “glitches” are observed where the spinrate suddenly jumps to a higher value (Fig. 11.9).

• The fractional change in period caused by a glitch istypically from 10−6 to 10−9 of the original period.

• Glitches indicate some internal rearrangement has al-tered the rotation rate by a small amount.

– Proposal: “starquakes” in the dense crust causethe neutron star to contract slightly and thus tospin faster (angular momentum conservation).

– Another: angular momentum stored in circula-tion of an internal superfluid liquid is suddenlytransferred to the crust, altering the rotation rate.

11.11. MILLISECOND PULSARS 553

11.11 Millisecond Pulsars

As a pulsar radiates away its energy, its spin rate decreasesslowly. This change is small but can be measured veryprecisely.

• The rate of change in the rotational period for a radiopulsar is important because it can be used to estimatethe strength of the magnetic field associated with theneutron star.

• Since pulsars are slowing down with time as theyemit energy both in electromagnetic and gravitationalwaves, we may expect that the fastest pulsars are theyoungest.

• For example, the Crab Pulsar is young (less than1000 years), and pulses 30 times a second.

• However, this reasoning breaks down for the pulsarswith millisecond periods.

• For many of these fast pulsars there is evidence thatthey are old, not young as we would expect for thefastest spin rates.

• This evidence consists primarily of the rate at whichthe pulsar spin is slowing, and where the millisecondpulsars are found.


• For example, the first millisecond pulsar discovered,PSR 1937 + 21, is very fast but it is spinning downvery slowly.

This is an example of the standard pulsarnaming system where the designation PSRindicates a pulsar, the first part of the num-ber gives the approximate right ascension inhours and minutes, and the second part of thenumber gives the declination (with a plus ornegative sign) in degrees.

• This slow spindown rate implies that it has a weakmagnetic field and is old. (Older pulsars should haveweaker fields and these should be less effective thanyounger, stronger fields in braking their motion.)

• Also, many of the millisecond pulsars that have beendiscovered are found in globular clusters, which con-tain an old population of stars.

• Therefore, they are not likely to be sites of recent su-pernova explosions that could have produced youngpulsars since core collapse supernovae occur in veryshort-lived, massive stars.

11.11. MILLISECOND PULSARS 555

AccretionDisk

PrimaryStar

AccretionStream

Angular momentum imparted toneutron star by impact of theaccretion particles as they strike the surface

Figure 11.10:The spin-up mechanism for producing millisecond pulsars.

• The most plausible way of explaining the contradic-tion that the fastest pulsars seem very old is thatmil-lisecond pulsars have been “spun up” since birth.

• The proposed mechanism involves mass transfer inbinary systems that adds angular momentum to theneutron star (Fig. 11.10).

• This accretion mechanism (binary spinup) transfersangular momentum from the orbital motion of the bi-nary to rotation of the neutron star.

• Later, after the neutron star has been spun up to highrotational velocity, the primary star may become asupernova and disrupt the binary system.

• This leaves the rapidly spinning but old neutron staras a millisecond pulsar that defies the systematics ex-pected from the evolution of isolated neutron stars.


11.12 Magnetars

Neutron stars have extremely strong magnetic fields. However, a newclass of spinning neutron stars with abnormally large magnetic fields,even for a neutron star, have been discovered.

• These have been calledmagnetars.

• The magnetar SGR 1900+14 is estimated to have a magneticfield so strong (at least of order 1015 gauss) that if a magnet ofcomparable strength were placed halfway to the Moon, it couldpull a metal pen out of your pocket on Earth!

SGR (soft gamma-ray repeater) indicates amagnetar. Like pulsars, the 1st part of thenumber gives the right ascension in hours andminutes, and the 2nd part of the number givesthe declination (±) in degrees.

• In these rotating neutron stars it thought that the huge magneticfields act as a kind of brake, slowing the rotation of the star.

• This slowing of the rotation disturbs the interior structure of theneutron star and “starquakes” in the star or magnetic field re-connection events release energy into the surrounding gases thatcause emission of bursts of gamma rays.

11.12. MAGNETARS 557

Observationally, these are calledsoft gamma ray repeaters(SGR):

• “Soft” means that the gamma rays are of low energy (in fact,they lie more in the X-ray portion of the spectrum);

• “Repeater” means that the bursts of gamma rays can repeat, un-like ordinary gamma ray bursts, which have not been observedto repeat.

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