Chapter 1 Some Properties of Stars - University of...

Chapter 1

Some Properties of Stars

This chapter is assigned reading for review.

1

2 CHAPTER 1. SOME PROPERTIES OF STARS

Chapter 2

The Hertzsprung–Russell Diagram

This chapter is assigned reading for review.

3

4 CHAPTER 2. THE HERTZSPRUNG–RUSSELL DIAGRAM

Chapter 3

Equations of State

Our fundamental initial task in understanding astrophysicsis to understand the structure of stars. At a minimum, anunderstanding of stars will require

• A set of equations describing the behavior of stellarmatter in gravitational fields.

• A set of equations governing energy production andassociated compositional changes driven by ther-monuclear reactions.

• A set of equations describing how energy is trans-ported from the energy-producing regions deep in thestar to the surface.

• Equations of state that carry information about themicroscopic physics of the star and that relate macro-scopic thermodynamic variables to each other.

5

6 CHAPTER 3. EQUATIONS OF STATE

Furthermore, these sets of equations are coupled to each other inhighly non-trivial ways.

Example: The hydrodynamics is influencedby the energy production in the thermonu-clear processes and the thermonuclear pro-cesses are in turn strongly dependent on vari-ables such as temperature and density con-trolled by the hydrodynamical evolution.

• The full problem will correspond to a set of coupled, non-linear,partial differential equations that can only be solved by large-scale numerical computation.

• In many cases simplifying assumptions can be made that allowsimpler solutions illustrating many basic stellar features.

• We begin the discussion by considering typical equations of statein this chapter.

Equation of State:A relationship among ther-modynamic variables for a system that

• Carries information beyond what isknown about the system on purely ther-modynamic grounds.

• Often based on detailed microscopicstructure input from nuclear, atomic, orparticle physics.

7

An equation of state is of the general form

P = P(T,ρ ,Xi , . . .),

whereP is the pressure,T is the temperature,ρ is thedensity, theXi are concentrations variables, and so on.The preceding equation is intended to be highly schematicat this point:

• An equation of state can take many forms.

• It need not even be specified analytically.

Example: Equations of state for large-scaleastrophysics simulations may be specified bydirect interpolation in tables that have beenconstructed numerically.

• Fortunately, for many (not all!) applications in astro-physics, relatively simple equations of state suffice.


3.1 The Pressure Integral

Except possibly at extremely high densities, we are primarily con-cerned withequations of states for gasesin astrophysics.

If quantum effects can be neglectedthe pressure in a gasmay be expressed in terms of thepressure integralderivedin Exercise 3.6:

P = 13

∫ ∞

0vpn(p)dp,

where

• v is the velocity,

• p is the momentum,

• n(p) is the number density of particles with momen-tum in the intervalp to p+dp.

This formula represents a very general result that can beshown to be valid for gas particles withany velocity, up toand including v= c (see Exercise 3.18).

3.2. IDEAL GASES 9

3.2 Ideal Gases

If the particles in a gas may be treated aspoints that inter-act weakly enoughthe gas obeys theideal gas equation ofstate,which may be expressed in a variety of equivalentforms:

P = nkT =NV

kT =NmH

VRT = ρ

kTµ

,

where

• P is the pressure,

• n is the number density of gas particles,

• V is the volume,

• N = nV is the number of particles contained in a vol-umeV,

• the Boltzmann constant isk and the temperature isT,

• the number of moles in the gas volumeV is NmH,

• the universal gas constant isR = kNA (where Avo-gadro’s number isNA = m−1

H , with mH the atomicmass unit),

• µ = ρ/nmH is the average molecular weight for thegas particles and the mass density isρ .


f(v)

in u

nits o

f 1

0-8

Velocity (cm/s)

0 4x107 8x107 1.2x108 1.6x108 2x108

2.0

1.0

0.5

0

1.5

1x107 K

2x107 K

3x107 K

4x107 K

Figure 3.1:Maxwell velocity distribution for hydrogen gas at various tempera-tures.

The ideal gas equation follows from the general pressureintegral

P = 13

∫ ∞

0vpn(p)dp,

evaluated specifically for aMaxwellian velocity distribu-tion,

n(p)dp=4πnp2dp

(2πmkT)3/2e−p2/2mkT,

(see Exercise 3.7). The Maxwell velocity distributionfor hydrogen gas at various temperatures is illustrated inFig. 3.1.

3.2. IDEAL GASES 11

For an ideal gas theinternal energyU is given by

U =∫ T

0CV(T) dT,

with theheat capacity at constant volumeCV(T) is

CV(T) =

(∂U∂T

)

V= T

(∂S∂T

)

V,

whereSis the entropy and thefirst law of thermodynamics,

dU = δQ−PdV = TdS−PdV,

has been used. Theenergy densityu is given by

u =UV

=1V

∫ T

0CV dT,

and thespecific energy(energy density per unit mass) is

ε =uρ

=UρV

=1

ρV

∫ T

0CV dT.

For the special case of amonatomic, nonrelativistic idealgas,

CV = 32Nk U = CVT = 3

2NkT.


Expressing the internal energy in differential form,

U =

∫ T

0CV(T) dT −→ dU = CV(T)dT,

introducing the heat capacity at constant pressure,

CP =

(∂U∂T

)

P= T

(∂S∂T

)

P,

and using the first law

dU = TdS−PdV,

we have for an ideal gas (see Exercise 3.1)

CP = CV +Nk.

3.2. IDEAL GASES 13

Theadiabatic indexγ is defined by

γ ≡CP

CV,

• For an ideal gas the heat capacities are independentof temperature and if the gas is monatomic,

γ =CP

CV=

CV +NkCV

=32Nk+Nk

32Nk

=53.

• γ for an ideal gas is directly related to the number ofdegrees of freedom for each particle (Ch. 6).

• The relationship between the pressureP and energydensityu for an ideal gas may be expressed in termsof γ (Exercise 3.21):

P = (γ −1)u.

• This equation may be used to define an effective adi-abatic indexγ for the general case, but only in theideal gas limit isγ given by the ratioCP/CV .

• The adiabatic speed of sound in an ideal gas is

vs =

√

γPρ

,

(Exercise 3.10.) whereρ is density andP is pressure.


3.3 Average Molecular Weights in the Gas

We are concerned with gases consisting of more than oneatomic species that may be partially or totally ionized.

• For example, the gas in a star may contain

– hydrogen atoms and ions,

– helium atoms and ions,

– various heavier elements as atoms or ions,

– the electrons produced by the ionization.

• In many cases we can treat these mixtures as a singlegas with an effective molecular weight.

Example:If the density is low, a mixture of

– hydrogen ions,

– fully-ionized helium ions,

– electrons produced by the ionization

will behave as three ideal gases, each con-tributing a partial pressure to the total pres-sure(Dalton’s law of partial pressures).

Then we can treat the system asa gas of a single kindof fictitious particle having an effective molecular weightrepresenting the relative contributions of each individualgas to the system properties.

3.3. AVERAGE MOLECULAR WEIGHTS IN THE GAS 15

3.3.1 Concentration Variables

The mass densityρi of a speciesi is given by

ρi = niAimH = niAi

NA

,

where

• Ai is the atomic mass number,

• mH is the atomic mass unit,

• ni is the number density,

• NA is Avogadro’s number.

Let us introduce themass fractionXi by

Xi ≡ρi

ρ=

niAimH

ρ=

niAi

ρNA

,

where

• ρ is the total mass density.

• The labeli may refer to ions, atoms, or molecules.

• The mass fractions sum to unity:∑Xi = 1.

We also will use themole fractionor abundanceYi,

Yi ≡Xi

Ai=

ni

ρNA

.

Generally, the sum of theYi will not be unity.


3.3.2 Total Mean Molecular Weight

If the energy density and momentum carried by radiationis ignored the average mass of a gas particle (atoms, ions,and electrons) is given by

µ =

(

∑i(1+yiZi)Yi

)−1

,

• The sum is over isotopic speciesi.

• yi is thefractional ionizationof the speciesi (yi = 0for no ionization andyi = 1 if the speciesi is com-pletely ionized).

• Zi is the atomic number for isotopic speciesi.

In very hot stars the momentum and energy density carriedby photons is non-trivial and we will see later that this willfurther modify the effective mean molecular weight of thegas.

With this formalism we have replaced theactual gas (a mixture of electrons and dif-ferent atomic, possibly molecular, and ionicspecies) with a gas containing a single kindof fictitious particle having an effective massµ (often termed themean molecular weight)that is given by the preceding equation.

3.3. AVERAGE MOLECULAR WEIGHTS IN THE GAS 17

Example 3.1

The composition of many white dwarfs may be approximated by acompletely ionized gas consisting of equal parts12C and16O by mass.The mass fractions areX12C = X16O = 0.5, so the abundances are

Y12C =X12C

12=

0.512

= 0.04167 Y16O =X16O

16=

0.516

= 0.03125.

Therefore,

µ =1

∑i(1+yiZi)Yi

=1

(1+6)0.04167+(1+8)0.03125

= 1.745 amu

if we assume complete ionization(yi = 1).


3.3.3 Common Notation

A common shorthand notation is to define

X ≡ Xhydrogen Y ≡ Xhelium Z ≡ Xmetals

where“metals” refers to the sum of all elements heavierthan helium andX +Y +Z = 1.

For a typical Pop I star just entering the mainsequence(termed a Zero-Age Main Sequenceor ZAMS star)we find

X ≃ 0.7 Y ≃ 0.3 Z ≃ 0.03.

The metal concentrationZ will be less in PopII stars.

3.4. POLYTROPIC EQUATIONS OF STATE 19

3.4 Polytropic Equations of State

An ideal gas equation of state (with an effective meanmolecular weightµ) is often a very realistic approxima-tion for many astrophysical processes.

• But there are other possible equations of state thatplay an important role in particular contexts.

• One is example is apolytropic equation of state.

• Generally, apolytropic processis defined by the re-quirement

δQδT

= c

where

– Q is the heat,

– T is the temperature,

– andc is a constant.


Some important properties of polytropes are explored inExercise 3.11. For example,

• From the first law of thermodynamics and

δQδT

= c

we may prove that polytropic processes in ideal gasesobey

dTT

= (1− γ)dVV

where thepolytropicγ is defined by

γ ≡CP−cCV −c

.

The polytropicγ reduces to the ideal gas adi-abatic parameterγ only if c = 0.

• You may verify by substitution that the differentialequation

dTT

= (1− γ)dVV

has three classes of solutions:

PVγ = C1 P1−γTγ = C2 TVγ−1 = C3,

(polytropic equations of state) withCn constants.

3.4. POLYTROPIC EQUATIONS OF STATE 21

The most common form of a polytropic equation of stateemployed in astrophysics is

P(r) = Kργ(r) = Kρ1+1/n(r),

where thepolytropic indexn is parameterized by

n =1

γ −1

with the parameterγ not necessarily defined by the idealgas expression.

Note: This is of the formPVγ = C1, since thedensityρ ∝ 1/V.

1. The polytropic approximation implies physically thatthe pressure is independent of the temperature, de-pending only on density and composition.

2. This approximation often makes solution of the equa-tions for stellar structure easier because itdecou-ples the differential equations describing hydrostaticequilibrium from those governing energy transferand the temperature gradients.


Examples where polytropes are appropriate:

1. For acompletely ionized star, fully mixed by convec-tion with negligible radiation pressure,

P = Kρ5/3,

which corresponds to a polytrope withγ = 5/3 andn= 3/2. The parameterK is constant for a given star,but can differ from star to star.

2. For acompletely degenerate gas of nonrelativisticfermions (defined below)

P = Kρ5/3,

again corresponding to a polytrope withγ = 53 and

n = 32, but nowK is fixed by fundamental constants .

3. For a completelydegenerate gas of ultrarelativisticfermions(defined below)

P = Kρ4/3,

corresponding to a polytrope withγ = 4/3 andn= 3,with K again fixed by fundamental constants.

These three polytropic equations of statewill be relevant for homogenous stars thatare completely mixed by convection, whitedwarfs, and neutron stars, respectively.

3.5. ADIABATIC PROCESSES 23

3.5 Adiabatic Processes

In terms of the heatQ and the entropyS, adiabatic pro-cessesare defined by the condition that

δQ = TdS= 0,

• From the first law,

dU = δQ−PdV = TdS−PdV,

we see that in adiabatic processes the change in in-ternal energy

dU = −PdV

comes only fromPdV work.

• Since adiabatic processes do not exchange heat withtheir environment, they are fully reversible(dS= 0).

• Most phenomena in astrophysics are not adiabaticbut many are at least approximately so.


It is standard practice to introduce threeadiabatic expo-nentsΓ1, Γ2, andΓ3 through

Γ1 ≡

(∂ lnP∂ lnρ

)

S

Γ2

Γ2−1≡

(∂ lnP∂ lnT

)

S

Γ3−1≡

(∂ lnT∂ lnρ

)

S

where the subscriptsS remind us that adiabatic processescorrespond to constant entropy, and where the logarithmicderivatives are equivalent to∂ lnA = ∂A/A. This impliesequations of state having one of the three forms

PVΓ1 = c1 P1−Γ2TΓ2 = c2 TVΓ3−1 = c3.

where thecn are constants.

From the above definitions,

Γ1(Γ2−1) = Γ2(Γ3−1),

so theΓi are not independent.

We also introduce,

∇ad ≡Γ2−1

Γ2,

which will play a role in our later discussion of convectivegas motion in stars.

3.5. ADIABATIC PROCESSES 25

For the special case of ideal gases

Γ1 ≡

(∂ lnP∂ lnρ

)

S

Γ2

Γ2−1≡

(∂ lnP∂ lnT

)

S

Γ3−1≡

(∂ lnT∂ lnρ

)

S

are equal and equivalent to the ideal gasγ (Exercise 3.13),

Γ1 = Γ2 = Γ3 = γ (ideal gas).

But for more general equations of stateΓ1, Γ2, and Γ3

are distinct and carry information emphasizing differentaspects of the gas thermodynamics:

1. Because it specifies how changes in pressure are re-lated to changes in density,Γ1 enters intodynamicalproperties of the gassuch as the speed of sound.

2. Γ2, and the quantity∇ad defined through it, relatechanges in pressure to changes in temperature, whichis important inunderstanding convective gas motion.

3. Γ3 depends on the derivative of temperature with re-spect to density, which influences theresponse of thegas to compression.

An example of these differences is given be-low for a mixture of ideal gas and photons inthe adiabatic limit.


3.6 Quantum Mechanics and Equations of State

Stellar equations of state reflect microscopic properties ofthe gas in stars.

• At low densitiesthis gas tends to behave classically,

• But the correct microscopic theory of matter is quan-tum mechanics andat higher densities a quantum de-scription becomes essentialto an accurate treatment.

The requisite physics can be understood conceptually interms of four basic ideas.

1. deBroglie Wavelength:The foundation of a quantum descriptionof matter isparticle–wave duality:

• A particle at the microscopic level takes on wave propertiescharacterized by adeBroglie wavelengthλ = h/p, wherepis the momentum andh is Planck’s constant.

• Thus in quantum mechanics thelocation of a particle be-comes fuzzy,spread out over an interval comparable to thedeBroglie wavelength.

2. Heisenberg Uncertainty Principle:The uncertainty principlequan-tifies the fuzziness of particle–wave duality,requiring that

• ∆p·∆x≥ h̄, where∆p is the uncertainty in momentum,∆xis the uncertainty in position, and̄h≡ h/2π .

• ∆E ·∆t ≥ h̄, where∆E is the uncertainty in energy and∆tis the uncertainty in the time over which the energy is mea-sured.

3.6. QUANTUM MECHANICS AND EQUATIONS OF STATE 27

3. Quantum Statistics:All elementary particles may be classifiedas eitherfermionorbosons. These classifications have to do withhow aggregates of elementary particles behave.

• Fermions (such as electrons, or neutrons and protons if weneglect their internal quark and gluon structure) obeyFermi–Dirac statistics.

– The most notable consequence is thePauli exclusionprinciple: no two fermions can have an identical set ofquantum numbers.

– All elementary particles ofhalf-integer spinare fermions.

• Bosons (photons are the most important example for ourpurposes) obeyBose–Einstein statistics.

– Unlike for fermions, there isno restriction on how manybosons can occupy the same quantum state.

– All elementary particles ofinteger spinare bosons.

• Matter is made from fermions(electrons, protons, neutrons,. . . ).

• Forces between particles are mediated by the exchange ofbosons(for example, the electromagnetic force results froman exchange of photons between charged particles).


4. Degeneracy:The exclusion principle implies that in a many-fermion systemeach fermion must be in a different quantumstate.

• Thus the lowest-energy state results from filling energy lev-els from the bottom up.

• Degenerate mattercorresponds to a many-fermion state inwhich all the lowest energy levels are filled and all the higherones are unoccupied.

• Degenerate matter occurs frequently at high densities andhas a very unusual equation of state with a number of impli-cations for astrophysics.

3.7. EQUATIONS OF STATE FOR DEGENERATE GASES 29

3.7 Equations of State for Degenerate Gases

Degenerate equations of state play an important role in avariety of astrophysical applications. For example,

• In white dwarf stars the electrons are highly degen-erate.

• In neutron stars the neutrons are highly degenerate.

Let us look at this in a little more detail for the case ofdegenerate electrons.

• We first demonstrate that (as a consequence of quan-tum mechanics)most stars are completely ionizedover much of their volumebecause ionization canbe induced by sufficiently high pressure, even at lowtemperature.

• This implies the possibility of producing a (rela-tively) cold gas of electrons,which is the necessarycondition for a degenerate electron equation of state.


r

d

Atom

V = πd343

Figure 3.2:Schematic illustration of average atomic spacing in dense stellar mat-ter. These are slices of 3-dimensional spherical volumes

3.7.1 Pressure Ionization

Consider the schematic diagram shown in Fig. 3.2, where

• Atoms occupy the darker spheres of radiusr .

• The average spacing between atoms is represented interms of the lighter spheres with radiusd.

• To illustrate simply we assume for that the stellar ma-terial consists only of

– Ions of a single species

– Electrons produced by ionizing that species


r

d

Atom

V = πd343

• Electrons in the atoms obey Heisenberg relations of the form

p ·∆x≥ h̄.

• Taking an average volume per electron ofV0 ≃ (∆x)3, the uncer-tainty relation becomes

p≥ h̄/V1/30 .

• The uncertainty principle produces ionization when the effectivevolume of the atoms becomes too small to confine the electrons.

• The volume per electronV0 and the volume per ionVi are relatedby ZV0 = Vi, since there areZ electrons per ion. Thus

p≥ h̄/V1/30 −→ p≥ h̄Z1/3/V1/3

i .


r

d

Atom

V = πd343

• From atomic physics the atomic radius may be approximated byr ≃ a0Z−1/3, wherea0 = 5.3×10−9 cm is the Bohr radius.

• If the star is composed entirely of an element with atomic num-berZ and mass numberA, there areZ electrons in each sphereof radiusd and the average number density of electrons is

ne =Z

43πd3

−→ d ≃

(3Z

4πne

)1/3

.


• Provided that

d =

(3Z

4πne

)1/3

< r,

we may expect pressure ionization, as illustrated in the followingfigure:

One bound state

Increasing density

Multiple bound

states

No bound states

With increasing density fewer locally boundstates are possible until none remain and theelectrons are all ionized. Thus,sufficientlyhigh density can cause complete ionization,even at zero temperature.


r

d

Atom

V = πd343

• Since there areA nucleons in each volume of radiusd in theabove figure, the mass densityρ is

ρ =AmH

43πd3

and requiring thatd ≃ r ≃ a0Z−1/3 defines a critical density

ρcrit ≃ZAmH

43πa3

0

.

• We may expect thatfor densities greater than this there will bealmost complete pressure ionization, irrespective of the temper-ature.


Table 3.1: Critical pressure-ionization densities

Element (Z,A) Density (g cm−3)

Hydrogen (1,1) 3.2

Helium (2,4) 26

Carbon (6,12) 230

Oxygen (8,16) 410

Iron (26,56) 4660

• The density condition

ρcrit ≃ZAmH

43πa3

0

.

is fulfilled rather easily. Consider the case of pure hydrogen.InsertingZ = A = 1 gives a critical density of3.2 g cm−3, onlya little larger than that of water.

• The critical pressure ionization densities for gases composed ofa few representative elements are summarized in Table 3.1.

• These critical densities may be compared with typical actualdensities of order

– ∼ 100 g cm−3 for the center of the Sun,

– ∼ 106 g cm−3 for a carbon–oxygen white dwarf,

– ∼ 109 g cm−3 for the iron core of a massive pre-supernovastar.

• These considerations imply that Saha ionization equations, whichare derived assuming ionization to be caused by thermal effects,are no longer reliable in the deep interior of stars.


The Saha equations would predict approximately 24% ofthe hydrogen in the core of the Sun to be neutral.

• However, comparison of the preceding table withproperties of the solar interior indicates that the den-sity is sufficiently high to pressure ionize hydrogenover the inner 40% of the Sun.

• Note that increased pressure favors pressure ioniza-tion but disfavors thermal ionization because of in-creased recombination.

Between thermal and pressure effects, the so-lar interior is almost entirely ionized, in con-trast to what we would expect from the Sahaequations alone.


3.7.2 Classical and Quantum Gases

Let us now consider the distinction between a classical gasand a quantum gas, and the corresponding implications forstellar structure.


Identical fermions are described statistically in quantummechanics by theFermi–Dirac distribution

f (εp) =1

exp[(εp−µ)/kT]+1(Fermi–Dirac),

whereεp is given by

εp = mc2+p2

2m(nonrelativistic)

ε2p = p2c2+m2c4 (relativistic)

and thechemical potentialµ is introduced by adding tothe First Law a term accounting for a possible change inthe particle numberN:

dU = TdS−PdV+ µdN,

whereT, P, andµ are taken as the macroscopic thermody-namical variables for the gas andSis the entropy. Identicalbosons are described by theBose–Einstein distribution

f (εp) =1

exp[(εp−µ)/kT]−1(Bose–Einstein).


We shall consider a gas to be aquantum gasif it is de-scribed by one of the distributions

f (εp) =1

exp[(εp−µ)/kT]+1(Fermi–Dirac),

f (εp) =1

exp[(εp−µ)/kT]−1(Bose–Einstein).

and aclassical gasif the condition

e(mc2−µ)/kT >> 1

is fulfilled. If the gas is classical,

• The states of lowest energy haveεp ∼ mc2.

• For either fermions or bosons the distributionfunction becomes well approximated byMaxwell–Boltzmann statistics,

f (εp) = e−(εp−µ)/kT (Maxwell–Boltzmann),

where generallyf (εp) << 1 for the classical gas.

• Thus, in the classical gas

– The lowest energy states are scarcely occupied

– The Pauli principle plays little role

– The gas obeys Maxwell–Boltzmann statistics


We now demonstrate that the conditions for forming aclassical gas are equivalent to a constraint on the densitysuch that the deBroglie wavelength of the particles in thegas is considerably less than the average interparticle spac-ing in the gas.


Non-Relativistic Classical and Quantum Gases:

Let us introduce acritical (number) density variablenQ

nQ ≡

(2πmkT

h2

)3/2

=(2π)3/2

λ 3 ,

where thedeBroglie wavelengthλ for non-relativistic par-ticles is given by

λ =hp≃

(h2

mkT

)1/2

.

The number of particles in the gas is

N =

∫ ∞

0f (εp)g(p) dp,

where the integration measure may be expressed as

g(p)dp= gs

Vh3 4π p2dp,

with p the momentum andgs = 2 j + 1 = 2 the spin de-generacy factor for electrons. Substituting the Maxwell–Boltzmann distribution with a nonrelativistic energy,

f (εp) = e−(εp−µ)/kT εp = mc2 +p2

2m

integrating, and rearranging yields

mc2−µ = kT ln(gsnQ

n

)

,

where the number densityn is given byn = N/V.


Therefore, the classical gas condition

e(mc2−µ)/kT >> 1

is, by virtue of

mc2−µ = kT ln(gsnQ

n

)

,

a requirement that at a given temperaturen << nQ.

In a classical gas, the actual number densitynis small on the scale set by the quantum den-sity nQ.


The preceding result has an alternative interpretation:

• The average separation between particles in the gasis d ∼ n−1/3 → 1/n∼ d3,

• The conditionn << nQ defining a classical gas im-plies that1/n >> 1/nQ,

• Because

nQ =(2π)3/2

λ 3 → nQ ∼ λ−3→ 1/nQ ∼ λ 3

the conditionn << nQ is equivalent to requiring that

1n

>>1nQ

→ d3 >> λ 3→ d >> λ .

For a classical gas, the average separation be-tween particles must be be much larger thanthe average deBroglie wavelengthλ for par-ticles in the gas.

• This makes sense conceptually:

– The “quantum fuzziness” of a particle extendsover a distanceλ .

– If particles are separated on average by distanceslarger thanλ , quantum effects are minimized.


Classical gas Quantum gas

λ

d

Illustration of classical and quantum gases.

• In the classical gas the average spacingd between gas particlesis much larger than their deBroglie wavelengthsλ .

• In the quantum gas the interparticle spacing is comparableto orgreater than the deBroglie wavelength. distribution.

The gas particles have a range of deBroglie wavelengths because theyhave a thermal velocity distribution.


Ultrarelativistic Classical and Quantum Gases:

Proceeding in a manner similar to that for the non-relativistic case, for ultrarelativistic particles (v ∼ c) therest mass of the particle may be neglected and from

εp ≃ kT =√

m2c4+ p2c2 ≃√

p2c2

andf (εp) = e−(εp−µ)/kT,

we obtain

µ = −kT ln

(gsnR

Q

n

)

,

where therelativistic quantum critical density variableisdefined by

nRQ = 8π

(kThc

)3

.

Hence in the ultrarelativistic case

• The condition that the gas be a classical one is equiv-alent to a requirement thatn << nR

Q.

• This again is equivalent to requiring that the de-Broglie wavelength

λ =hp≃

hckT

be small compared with the average separation ofparticles in the gas.


3.7.3 Transition from Classical to Quantum Gas Behavior

We conclude from the preceding results that at highenough density the gas behaves as a quantum gas. Noticefrom

nQ ≡

(2πmkT

h2

)3/2

=(2π)3/2

λ 3 ,

that with increasing gas density

• The least massive particles in the gas will be mostprone to a deviation from classical behavior becausethe critical density is proportional tom3/2.

• Thus photons, neutrinos, and electrons are most sus-ceptible to such effects.

• The massless photonsnever behave as a classicalgasand the neutrinos (also massless or nearly so) in-teract so weakly with matter that they leave the starunimpeded when they are produced.a

• It follows that in normal stellar environments it istheelectrons that are most susceptibleto a transitionfrom classical to quantum gas behavior.

aAn exception to this statement occurs for core-collapse supernovae, where thedensities and temperatures become high enough to trap neutrinos for significanttimes relative to the dynamical timescales for the problem.


Presently in the center of the Sun

• The average number density for electrons isn∼ 6×1029 m−3

• For a temperature of16× 106 K the nonrelativisticcritical quantum density isnQ ∼ 3×1031 m−3.

• Thus,n/nQ ∼ 0.02 and electrons in the Sun are rea-sonably well approximated by a dilute classical gas.

However, the core of the Sun, as for all stars, will contractlate in its life as its nuclear fuel is exhausted.

• The approximate relationship between a star’s tem-peratureT and radiusR is kT ≃ 1/R, which impliesthat

nQ ≡

(2πmkT

h2

)3/2

=

(2πmh2R

)3/2

≃ R−3/2

and, because the actual number density behaves asn∼ R−3, that

nnQ

≃R−3

R−3/2≃ R−3/2.

As the core of the Sun contracts later in itsevolution, eventually the electrons in it willbegin to behave as a quantum rather than clas-sical gas.


3.7.4 The Degenerate Electron Gas

From the definition

nQ ≡

(2πmkT

h2

)3/2

the quantum gas conditionn>> nQ is equivalent to atem-perature constraint,

kT <<h2n2/3

2πm.

A quantum gas is a cold gas, butcold on a temperaturescale set by the right side of the preceding equation.

• If the number densityn is high enough, this equationindicates that the gas could be “cold” and still have atemperature of billions of degrees!

• The precise quantum mechanical meaning of a coldelectron gas is thatthe electrons are all concentratedin the lowest available quantum states.

• We say that such a gas isdegenerate.

• Degenerate gases have much in common with themetallic state in condensed matter.

Example: Ideal gases have very low thermalconductivity (that is why down parkas keepyou warm). Degenerate electron gases, as formetals, are very good thermal conductors.


Occu

pa

tio

n p

rob

ab

ility

1

0 ε(p)εf = ε(p

f)

Fermi surface (T=0)

Finite T

Occupied

states

Unoccupied

states

In the limit that the temperature may be neglected, theFermi–Dirac distribution becomes a step function in en-ergy space (see figure above),

ff(εp) =1

e(εp−µ)/kT +1−→T→0

{

f (εp) = 1 εp ≤ εf

f (εp) = 0 εp > εf

where

• The value of the chemical potentialµ at zero temper-ature is denoted byεf and is termed thefermi energy.

• The corresponding value of the momentum is de-noted bypf and is termed thefermi momentum.

• Thus, the fermi energy gives theenergy of the highestoccupied statein the degenerate fermi gas.


The density of states as a function of momentump is givenby g(p) in

g(p)dp= gs

Vh3 4π p2dp,

and the number of electrons in the degenerate gas at zerotemperature is just the number of states with momentumless than the fermi momentumpf,

N =

∫ pf

0g(p) dp

= 4πVgs

h3

∫ pf

0p2 dp

=8πV3h3 p3

f ,

wheregs = 2 has been used for electrons. Solving for thefermi momentumpf and introducing the number densityn = N/V, we find that thefermi momentumis determinedcompletely by theelectron number density

pf =

(3h3

8π·NV

)1/3

=

(3n8π

)1/3

h.

For a number densityn the interparticle spac-ing is∼ n−1/3, so this result implies that thedeBroglie wavelength of an electron at thefermi surface,λ = h/pf ∼ n−1/3, is compara-ble to the average spacing between electrons.


We may construct the equation of state for the degenerateelectron gas by evaluating the internal energy of the gas.Let us first do this in the nonrelativistic and then in theultrarelativistic limits for degenerate electrons.


Nonrelativistic Degenerate Electrons:

In the nonrelativistic limitpf << mc, which implies (sincepf ∼ h̄n1/3) that

n <<(mc

h̄

)3=

(1λc

)3

,

whereλc ≡ h̄/mc is theCompton wavelengthfor an elec-tron. In this limit the internal energy for the degenerateelectron gas (for whichf (εp) is a step function) is

U =

∫ ∞

0εp f (εp)g(p) dp≃ Nmc2

︸︷︷︸

potential

+3N10m

p2f

︸︷︷︸

kinetic

,

where

εp = mc2+p2

2mg(p) = gs

Vh3 4π p2 N =

8πV3h3 p3

f

andgs = 2 have been used. For a nonrelativistic gas thepressure is given by23 of the kinetic energy density;iden-tifying the second term of

U = Nmc2+3N10m

p2f ,

divided by the volumeV as the kinetic energy density,

P=23× (kinetic energy density) =

23

(NV

3p2f

10m

)

= np2

f

5m=

h2

5m

(3

8π

)2/3

n5/3 (γ = 53 polytrope),

wheren = N/V andpf = (3n/8π)1/3h have been used.


Example:For low-mass white dwarfs withρ <∼ 106 g cm−3

the electrons are nonrelativistic and the electron pressureis given by then = 3

2 polytrope implied by the precedingequation,

Pe =h2

5m

(3

8π

)2/3( ρmpµe

)5/3

with the mean molecular weightµe defined throughne =ρ/mpµe. As noted earlier, the constantK in the polytropicform

P = Kργ(r)

is fixed by fundamental constants.


Ultrarelativistic Degenerate Electrons:

For ultrarelativistic electrons,n >> nRQ implies thatn >>

(mc/h)3. Utilizing the ultrarelativistic limitεp = pcand

g(p) = gs

Vh3 4π p2

the internal energy is

U =

∫ ∞

0εp f (εp)g(p) dp

≃8πVc

h3

∫ pf

0p3 dp= 3

4Ncpf.

For an ultrarelativistic gas thepressure is13 of the kinetic

energy density. Identifying the kinetic energy density asU = 3

4Ncpf divided by the volumeV, for ultrarelativisticparticles

P= 13 × (kinetic energy density)

= 13 × (3

4cnpf) =hc4

(3

8π

)1/3

n4/3,

wheren = N/V and we have usedpf = (3n/8π)1/3h.


Example: For higher-mass white dwarfs havingρ >

∼ 106 g cm−3 the electrons are highly relativisticand the corresponding degenerate equation of state takesthe form implied by the preceding equation,

Pe =hc4

(3

8π

)1/3( ρmpµe

)4/3

which is a polytrope withn = 3. As in the nonrelativisticcase, we see that the constantK multiplying ρ4/3 is fixedby fundamental considerations.


3.7.5 Summary: High Gas Density and Stellar Structure

This preceding discussion implies that increasing the density can havea large impact on the structure of stars. Generally, we may identifyseveral important consequences of high densities in stellar environ-ments:

• An increase in the gas density above a critical amount enhancesthe probability for pressure ionization, thereby creatinga gas ofelectrons and ions irrespective of possible thermal ionization.

• An increase in the gas density, by uncertainty principle argu-ments, increases the average momentum of gas particles andmakes them more relativistic.

• An increased density raises the fermi momentum. This, for ex-ample, influences the weak interaction processes that can takeplace in the star.

• An increase in the gas density decreases the interparticlespac-ing relative to the average deBroglie wavelength, making itmorelikely that the least massive particles in the system make a tran-sition from classical to degenerate quantum gas behavior.


• Increased density enhances the strength of the gravitational fieldand makes it more difficult to maintain stability of the star againstgravitational collapse. Higher density also makes it more likelythat general relativistic corrections to Newtonian gravitation be-come important.

• Higher density (often implying higher temperature) tendsto changethe rates of thermonuclear reactions and to alter the opacity ofthe stellar material to radiation.

1. The former changes the rate of energy production;

2. the latter changes the efficiency of how that energy is trans-ported in the star.

Both can have large consequences for stellar structure and evo-lution.

These consequences of increased densityhave large implications for the structure andevolution of stars because all stars are ex-pected to go through late evolutionary stagesthat may dramatically increase their centraldensities.


From the preceding, a gas can have a pressure that is of purelyquantum-mechanical origin, independent of its temperature.

• Assume pressure dominated by non-relativistic electronsand dropfactors like1

2. For an ideal gas the average energy of an electronis E ∼ kT = 1

2mev2, so electrons have a velocity

vthermal≃

(kTme

)1/2

of purely thermal origin.

• But even at zero temperature the electrons have a velocityvQM

implied by the uncertainty principle, since

p∼ ∆p∼ h̄/∆x∼ h̄n1/3e → vQM ≃

pme

≃h̄n1/3

e

me

.

• Thus, the average velocity of particles in the gas has two contri-butions,

1. one from the finite temperature and

2. one from quantum mechanics,

with the thermal contribution vanishing identically at zero tem-perature.

• The pressure contributed by the thermal motion is

Pthermal= nekT = nemev2thermal

and the pressure contributed by quantum mechanics is (up tosome constant factors)

PQM ≃h̄2

me

n5/3e = neme

(

h̄n1/3e

me

)2

= nemev2QM

• A degenerate gasis one for whichPQM >> Pthermal.


• The thermal pressure is proportional toT and density, but thequantum-mechanical pressure is independent ofT and propor-tional to a power of the density:

Pthermal= nekT PQM ≃h̄2

me

n5/3e .

• Therefore, degeneracy is favored in low-temperature, dense gases,and a gas can have a high pressure of purely uncertainty-principleorigin, even atT = 0.

• Furthermore, changingT in a highly-degenerate gas will havelittle effect on the pressure, which is dominated by a term in-dependent ofT (as long as changing the temperature does notsignificantly change the degeneracy of the gas).

All of these properties have profound consequences for stellar struc-ture and stellar evolution when high densities are encountered.


3.8 Equation of State for Radiation

We may view electromagnetic radiation in stars as anul-trarelativistic gas of massless bosons.The equation ofstate associated with radiation follows from substitutingthe blackbody (Planck) frequency distribution

n(ν)dν =8πν2dν

c3(ehν/kT −1),

into the pressure equation

P = 13

∫ ∞

0vpn(p)dp,

and evaluating the integral (see Exercise 3.8). This yieldsfor the radiation pressure,

Prad = 13aT4,

wherea is the radiation density constant. The correspond-ing energy density of the radiation field is

urad = aT4 = 3Prad.

3.8. EQUATION OF STATE FOR RADIATION 61

Gravitational Stability and Adiabatic Exponents for Radiation:

You are asked to show in Exercise 3.14 that the adiabaticexponentsΓ1, Γ2, andΓ3 for a pure radiation field are allequal to4

3.

• As we shall see in more detail later, an adiabatic ex-ponent less than43 generally implies aninstabilityagainst gravitational collapse.

• Therefore, admixtures of radiation contributions(more generally, of any relativistic component) topressure often signaldecreased gravitational stabil-ity for a gas.


3.9 Equation of State for Matter and Radiation

For a simple stellar model, it is often a good starting pointto assume

• An ideal gas equation of state for the matter (pro-vided that the density is not too high).

• A blackbody equation of state for the radiation.

In that case we may write for the pressureP and internalenergyU

P=NV

kT︸︷︷︸Ideal Gas

+aT4

3︸︷︷︸

Radiation

U = uV = CVT︸︷︷︸Ideal Gas

+aT4V︸︷︷︸Radiation

,

where the first term in each case is the contribution of theideal gas and the second term is that of the radiation.

3.9. EQUATION OF STATE FOR MATTER AND RADIATION 63

3.9.1 Mixtures of Ideal Gases and Radiation

For mixtures of gas and radiation (common in high-temperature stellar environments) it is convenient to de-fine a parameterβ that measures the relative contributionsof gas pressurePg and radiation pressurePrad to the totalpressureP:

β =Pg

P1−β =

Prad

PP = Pg +Prad.

Therefore

• β = 1 implies that all pressure is generated by thegas.

• β = 0 implies that all pressure is generated by radia-tion.

• All values in between correspond to situations wherepressure receives contributions from both gas and ra-diation.


Example: Assuming an ideal gas equation of state, thepressure generated by the gas alone in a mixture of idealgas and radiation is

Pg = nkT = βP.

Solving this equation for the total pressure,

P =nkTβ

=ρkTβ µ

=NkTβV

,

which is of ideal gas form. Thus, the formal effect of mix-ing the radiation with the gas

• Produces an ideal gas equation of state

• But the gas has an effective mean molecular weightβ µ , whereµ is the mean molecular weight for thegas alone.

Thus mixtures of ideal gases and radiationmay be treated as modified ideal gases, butnormally the relative contribution of radiationand gas to the pressure varies through the vol-ume of a star soβ in realistic cases is a localfunction of position.

3.9. EQUATION OF STATE FOR MATTER AND RADIATION 65

3.9.2 Adiabatic Systems of Gas and Radiation

The preceding discussion of gas and radiation mixturesdepends only on the ideal gas assumption.

• Suppose we further confine discussion to adiabaticprocesses.

• From the adiabatic conditionδQ = 0, the first lawof thermodynamics, and the definition ofβ we mayshow that at constant entropy (see Exercise 3.16)

dlnTdlnV

=−(γ −1)(4−3β )

β +12(γ −1)(1−β ).

• These logarithmic derivatives may then be used toevaluate the adiabatic exponents with the results

Γ1 =dlnPdlnρ

= β +(4−3β )2(γ −1)

β +12(1−β )(γ −1)

Γ2 =

(

1−dlnTdlnP

)−1

= 1+(4−3β )(γ −1)

β 2+3(γ −1)(1−β )(4+β )

Γ3 = 1+dlnTdlnρ

= 1+(4−3β )(γ −1)

β +12(1−β )(γ −1)


1.3

1.4

1.5

1.6

1.7

0.0 0.2 0.4 0.6 0.8 1β

γ = 5/3

Γ1

Γ3

Γ2

γ = 4/3

Figure 3.3:Adiabatic exponents in a mixture of ideal gas and radiation.β = 0implies all pressure from radiation andβ = 1 implies all pressure from gas.

The adiabatic exponentsΓ1, Γ2, and Γ3 are plotted inFig. 3.3 as a function of the parameterβ governing therelative contribution of gas and radiation to the pressure.They have the expected limiting behavior:

• Assumingγ = 53 for a monatomic ideal gas andβ = 1

(no radiation contribution to pressure) gives

Γ1 = Γ2 = Γ3 = 53,

• For β = 0 (all pressure generated by radiation) wefind that the adiabatic exponentsΓ1, Γ2, andΓ3 allconverge to a value of43.

• For other values ofβ the adiabatic exponents are gen-erally not equal and lie between43 and5

3.

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