evolutionary properties of white dwarf stars

Mem. S.A.It. Vol. 81, 908c© SAIt 2010 Memorie della

Evolutionary properties of white dwarf stars

Leandro G. Althaus

Facultad de Ciencias Astronomicas y Geofısicas – Universidad Nacional de La Plata, Paseodel Bosque S/N, (1900) La Plata, Argentina, e-mail: [email protected]

Abstract. White dwarfs are the most common end-point of stellar evolution. As such, theyhave potential applications to different fields of astrophysics. In particular, the use of whitedwarfs as independent reliable cosmic clocks provides valuable constraints to fundamentalparameters of a wide variety of stellar populations. This is possible in part thanks to therelative simplicity of their evolution. Here, I describe a simple cooling model that capturesthe essentials of the physics of white dwarf evolution. Also, I comment on the results givenby full evolutionary calculations of these stars, with a particular emphasis on the differentenergy sources and processes responsible for chemical abundance changes that occur in thecourse of their evolution.

Key words. Stars: white dwarfs – Stars: evolution – Stars: interiors

1. Introduction

White dwarf (WD) stars constitute the finalstage of stellar evolution for the vast majorityof stars. Indeed, more than 95% of all stars, in-cluding our Sun, are expected to end their livesas WDs. They are very old objects, thereforethe present population of WDs contain valu-able information about the evolution of indi-vidual stars from birth to death and the starformation rate throughout the history of ourGalaxy.

The study of WDs has thus potential appli-cations to different fields of astrophysics (seee.g. Winget & Kepler 2008, for a recent re-view). These applications include the use ofWDs as independent cosmic clocks to infer theage of a wide variety of stellar populations, likethe galactic disk (Hernanz et al. 1994) and sev-eral globular and open clusters (Richer et al.1997; Von Hippel & Gilmore 2000; Von Hippel

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et al. 2006; see also contributions by Bono andMoehler in this volume). In addition, the ex-istence of old and faint WDs belonging to theHalo has been reported (Liebert et al. 1989). Apossible detection of a faint population in theHalo allows to provide a robust lower limit tothe age of our Galaxy.

WDs usually play a role in some inter-acting binaries, such as cataclysmic variables(Ritter, this volume), where mass transfer froma companion star onto the WD gives rise tovery strong energetic events. In addition, WDsare candidates for progenitors of Supernovaeof type Ia, which are thought to be the re-sult of the explosion of a WD exceeding theChandrasekhar limit due to accretion from amass–losing companion star. An understand-ing of the evolutionary and structural proper-ties of WDs may help to improve our knowl-edge of the supernova events, in particular, thevariation in brightness with redshift, with theimportant underlying implications for cosmol-

Althaus: Evolution of white dwarfs 909

ogy. Also, the presence of WDs in binary sys-tems with millisecond pulsar companions pro-vides the opportunity to constrain the age ofmillisecond pulsars and thereby the timescalefor magnetic field decay (van Kerkwijk et al.2005 for a review).

Finally, the high densities and temperaturesthat characterize WDs turn these objects intocosmic laboratories to study numerous phys-ical processes under extreme conditions thatcannot be achieved in terrestrial laboratories.Thanks to the fact that many WDs exhibitpulsational instabilities, these stars constitutepowerful tools for applications beyond stellarastrophysics. In this sense, WDs can be usedto constrain fundamental properties of elemen-tary particles, such as axion and neutrinos, andto study problems related to the variation offundamental constants.

These potential applications of WDs haveled to a renewed interest in the computation offull evolutionary models for these stars. Here,I will discuss the essentials of the physics ofWD evolution, in order to understand the rea-sons that make these stars excellent chronome-ters and potential tools with a wide variety ofapplications. Observational aspects of WDs arebriefly touched upon; the reader is referred toreview articles by Koester (2002) and Hansen& Liebert (2003) for further details.

2. Basic properties of white dwarfstars

A remarkable characteristic of WDs is thatthey are located well below the main sequenceon the HR diagram. This means that these ob-jects are characterized by a very small radius,comparable to the size of the earth. By 1910,it was possible to determine the mass of a WDfor the first time: the WD companion to Sirius,with a stellar mass of about 1 M�. Combinedwith the small radius, this implies very largeaverage densities. With the discovery of otherWDs, it was soon realized that these stars wereindeed a new class of stars, quite different fromordinary stars. In fact, the existence of WDsprovided one of the first test of the quantumtheory of matter and a demonstration of thePauli exclusion principle for electrons. Also,

the detection of the gravitational redshift by1925, resulting from the strong gravity of theWD companion to Sirius, provides an indepen-dent test of the high densities in WDs and aconfirmation of general relativity.

More than 10,000 spectroscopically iden-tified WDs are known. Spectroscopic infer-ences, coupled with theoretical stellar atmo-spheres models, allow to determine their sur-face gravity, on average log g=8, which con-sidering the radii, yields average masses ofM ≈ 0.6M�. Indeed, typical WD mass distri-butions show that the mass values of most WDsare around this value (see Kepler et al. 2007),with a tail extending towards high stellar mass— with several WDs with spectroscopicallydetermined masses within the interval 1.0-1.3M� and most probably being the result fromeither carbon burning in semidegenerate condi-tions during the evolution of progenitor star ormergers of individual WDs of standard masses.There is also an important population of low–mass WDs, which is probably the result of bi-nary evolution.

Spectroscopic observations also reveal thatthe atmospheric constituents of most WDsconsist almost entirely of hydrogen (H) withat most traces of other elements. These arethe DA WDs and they comprise about 80 %of all WDs. Observations also reveal the ex-istence of WDs with helium (He)–rich atmo-spheres (DB), which make up almost 20% ofthe total. There are also WDs with hybrid at-mospheres, or those with peculiar abundances.It is accepted that post–AGB stars evolve intotwo main channels: those with H–rich atmo-spheres, and those with H–deficient compo-sition. The ratio of DA to non-DA changeswith effective temperature. In fact, there is ev-idence that the surface composition of a givenWD may change as it evolves, as a result ofthe interplay of competing processes, such asconvection, mass–loss episodes, accretion andgravitational settling. Individual WDs are be-lieved to undergo spectral evolution.

Values of Teff for WDs range from about150,000 K for the hottest members to 4000 Kfor the coolest degenerate dwarfs. The stellarluminosity ranges from 102−103 to about 10−5

L� for the faintest observed WDs. The major-

910 Althaus: Evolution of white dwarfs

ity of known WDs have temperatures higherthan the Sun, and hence the “white” in theirname. Because the intrinsic faintness of mostWDs, quantitative studies of these stars, tra-ditionally based on photometric and spectro-scopic observations, are restricted to nearbyobjects. Other observational techniques haverevealed the presence of WD populations lo-cated well beyond our own neighborhood, suchas in distant open and globular clusters and inthe galactic halo.

3. Simple treatment of white dwarfevolution

As mentioned above, many applications in-volving WDs require a knowledge of the evo-lutionary properties of WDs. A key feature tounderstand the evolution of WDs is that, indegenerate configurations, the mechanical andthermal structures are more or less decoupledfrom each other and can be treated separately.Thermal properties (T,0) are responsible forradiation and evolution of WDs. For the me-chanical structure, the limit T=0 is usually as-sumed. This property will allow us to derive asimple picture of how WD evolve.

3.1. Zero-temperature approximation:Chandrasekhar’s theory

In this approximation, the structure of a WDcan be described in terms of a zero tempera-ture degenerate electron gas. At zero temper-ature, the structure of a spherically symmet-ric star is determined by the equations of hy-drostatic equilibrium and mass conservation.Chandrasekhar (1939) showed that the prob-lem can be reformulated into a second orderdifferential equation for a dimensionless func-tion.

1η2

ddη

(η2 dΦ

dη

)= −

(Φ2 − 1

y2o

)3/2

(1)

This equation is solved numerically with ap-propriate boundary conditions. For a fixedvalue of the molecular weight per electron, µe,a one-parameter family of models is thus ob-tained, defining implicitly a relation between

mass and radius, such that the more massivethe star, the smaller is its size. In the limitρ → ∞, the relativistic “softening” of theequation of state causes the radius R to be-come zero, while the mass approaches a lim-iting value 5.826/µ2

e[M�], the Chandrasekharlimiting mass. According to our present un-derstanding of stellar evolution, a carbon andoxygen core composition is expected for mostWDs. In this case, the limiting mass becomes1.4 M�. For larger masses, gravitational forcesoverwhelm the electron pressure support, andno stable WD can exist. The existence of a lim-iting mass has strong implications, not only forthe WD themselves, but also for stellar evolu-tion in general. This is particularly true regard-ing the occurrence of mass-loss episodes: starsmust lose a large fraction of their mass at somepoint in their evolution in order to end up asWDs with masses smaller than 1.4 M�.

3.2. Mestel’s model of white dwarfevolution

WD evolution can be understood in terms oftwo main problems:

1) The total energy content E of the star.2) The rate at which this energy is radiated

away.The first problem involves a detailed

knowledge of the thermodynamics of the de-generate core of the WD, which contains morethan 99% of the mass. Because the efficiencyof degenerate electron in transporting energy,the core is essentially isothermic, a fact thatstrongly simplifies the evaluation of E.

The second problem involves the solu-tion of the energy transfer through the non-degenerate envelope above the core. In theenvelope, energy is transferred by radiationand/or convection, which are less efficient thanelectron conduction. Because of this, the enve-lope controls the rate at which energy is trans-ferred from the core into the empty space.

A simple approach to determine the struc-ture of the envelope can be derived by makingthe following assumptions: 1) An ideal equa-tion of state, i. e., a non-degenerate gas, 2)an abrupt transition between the envelope andthe degenerate core and 3) energy is trans-


ported only by radiation, for which a Kramers’law is adopted for the radiative opacity (κ =κ0 % T−3.5, where κ0 depends on the envelope’schemical composition). Under these assump-tions, the thermal structure of the envelope isspecified by

dPdT

=16πac

3κ0

GMT 6.5

%L, (2)

where no energy sources or sinks are assumedin the envelope (L is the surface luminos-ity). By using the equation of state for a non-degenerate gas (% = Pµ/<T ), we arrived ata differential equation which can be integratedfrom the surface (zero temperature and pres-sure) down to the base of the envelope —the transition layer that separates the enve-lope from the degenerate core. The integrationyields

%tr =

32πac8.5 3κ0

µ

<GM

L

1/2

T 3.25tr . (3)

Here, Ttr and %tr are the temperature anddensity at the transition layer. Another relationbetween density and temperature at the tran-sition layer is given by the fact that at thislayer the degenerate electron pressure is equalto the ideal gas pressure (pressure is contin-uous across the transition layer). Finally, be-cause the core is isothermal, we can replace Ttrby Tc (central temperature). We reached a sim-ple power law relation between the surface lu-minosity, stellar mass and central temperatureof the WD (see Shapiro & Teukolsky 1983 fordetails):

L = 5.7 × 105 µ

µ2e

1Z(1 + X)

MM�

T 3.5c , (4)

where Z and X are the envelope metallicityand H abundance by mass respectively. Someimportant inferences can be drawn from thissimple treatment: for typical compositions andM= M� we find that for L= 10L� and L=10−4L�, the central temperature is of the order

of Tc= 100×106K and Tc= 4×106K, respec-tively. The high central temperatures expectedin high-luminosity WDs exclude the presenceof H or He in the degenerate core. Indeed, atthe high densities of the core, stable core nu-clear burning is not possible — the WD wouldbe destroyed by a thermal runaway (see Mestel1952).

If there is no stable nuclear burning, thenwhich energy sources are involved when a nor-mal WD loses energy by radiation ? An impor-tant source of energy is gravitational contrac-tion. Although WDs cannot contract apprecia-bly because of electron degeneracy, the resid-ual contraction — resulting from the gradualdecreasing ion pressure — is very important.This is because WDs are compact objects, sothat a small change of the radius releases animportant amount of gravitational energy. If weneglect nuclear reactions and neutrino losses,the total energy results of contributions fromions, electrons and gravitational energy, i. e.,E = Eion +Eelec +Egrav. The luminosity is givenby the temporal decrease of the total energy E

L = −dEdt

= − ddt

(Eion + Eelec + Egrav). (5)

It can be shown from the virial theo-rem for degenerate configurations that the re-lease of gravitational energy is used to in-crease the Fermi energy of the electrons (aboutthe density-dependent contribution of the elec-tronic energy, see Koester 1978 and alsoPrada Moroni, in this volume). Thus, only thechanges in the thermal contribution to E mustbe considered. This gives rise to an elementarytreatment of WD evolution:

L = − ∂

∂Tc(ET

ion + ETelec)

dTc

dt, (6)

where we have assumed that the core is isother-mal. In terms of the specific heat of ions andelectrons, the luminosity equation becomes

L = − < CV > MdTc

dt, (7)


where < CV >= CelecV + Cion

V and M is theWD mass. Thus, we have a simple relation be-tween the luminosity and the rate of changeof the central temperature. Clearly, the sourceof luminosity of a WD is essentially the de-crease in the thermal energy of ions and elec-trons. Because of this, WD evolution is de-scribed as a cooling process. But at low tem-peratures, Celec

V � CionV , since strongly degen-

erate electrons barely contribute to the specificheat. Thus, the source of luminosity of a coolWD is basically the change of the internal en-ergy stored in the ions.

By using the relation between the luminos-ity and central temperature given by the enve-lope integration (L = CMT 3.5

c , see Eq. 4), andassuming an ideal ion gas (Cion

V = 3</2A, Ais the atomic weight), Eq. 7 can be rewrittenas dTc/dt ∝ T 3.5

c . By integrating this differ-ential equation from an initial time t0 to thepresent time t and introducing the cooling timeas τ = t− t0, i.e., the age from the start of cool-ing at high effective temperatures, we obtain

τ ≈ 108

A

M/M�L/L�

5/7

years, (8)

where it has been assumed that the central tem-perature at the beginning of WD formation ismuch larger than the present central temper-ature of the WD. This is the Mestel’s modelof WD evolution (Mestel 1952). For A = 12(carbon interior), typical cooling ages for thefaintest observed WDs (L ∼ 10−4.5L�) of 1010

yr are derived. The time spent in the WD phasethus turns out to be a major phase in stellar life.WDs are very old objects, and thus are poten-tial candidates to date stellar populations.

Although the Mestel’s model is the sim-plest picture of WD evolve, it captures theessentials of the physics of WD evolution: Itshows that the cooling time of a WD is re-lated essentially to its core chemical compo-sition, its mass and its luminosity. The follow-ing features emerge: a) cooling times dependon the core chemical composition (A) of theWD: oxygen WDs are expected to cool fasterthan carbon WDs, because the specific heat pergram of oxygen is smaller than that of carbon

— there are more ions in a gram of carbon thanin a gram of oxygen. b) Because of their largerthermal content, massive WDs are character-ized by long cooling times and c) cooling timeincreases as luminosity decreases.

3.3. Improvements to the Mestel’s model

The increase in cooling times with decreasingluminosity predicted by the Mestel model im-plies that more and more WDs should be ob-served at fainter luminosities. This is indeedthe case. However, there is an abrupt and steepdrop-off in the numbers of WDs at very lowluminosities (L ∼ 10−4.5L�) that cannot be ex-plained by this model. This discrepancy be-tween theory and observation motivated the ex-ploration of physical effects which might be re-sponsible for shortening the cooling times atvery low luminosities.

The Mestel model is a reasonable descrip-tion of the behavior of real WDs only at in-termediate luminosities, where the assump-tions of ideal gas, isothermal core, and non-degenerate radiative envelope are more or lessvalid. However, it requires major improve-ments if we want to use cool WDs as reliablecosmic clocks . The main improvements to beconsidered are:

Coulomb interactions and Debye cooling:Coulomb interactions modify the specific heatof ions and the cooling times of WDs. Thestrength of Coulomb interactions — relativeto the thermal kinetic energy — is deter-mined by the Coulomb coupling parameter Γ =(Ze)2/aKT = 2.26 × 105Z5/3ρ1/3/T , where ais the inter-ionic separation. For very low Γvalues, the coulomb forces are of minor im-portance (relative to thermal motions) and theions behave like an ideal non-interacting gas.But, once Γ becomes of order unity, ions be-gin to undergo short-range correlations, even-tually behaving like a liquid. For large enoughΓ (∼ 200) the ions form a lattice structure, ex-periencing a first-order phase transition in theprocess and the release of latent heat, of about∼ KT per ion. This results in a new source ofenergy, which introduces an extra delay in theWD cooling. In addition, there is an increasein the specific heat capacity in the crystallized


region (Dulong-Petit law) due to the extra de-gree of freedom associated with lattice vibra-tion. But, most importantly, as the WD coolsfurther, fewer modes of the lattice are excited,and the heat capacity drops according to theDebye relation. This reduction starts to mani-fest itself once the temperature drops below theDebye temperature (θD = 4 × 103ρ1/2), reach-ing fast cooling (CV ∝ T 3) once θD/T > 15.As a result, the thermal content of the WD getssmaller and smaller. This fast cooling is ex-pected to take place in cool WDs, i. e. thosecharacterized by very low surface luminosities.

We can estimate the importance of Debyeeffect on the cooling of WDs as fol-lows: from Eqs. 4 and 7 we can writeCMT 3.5

c = − < CionV > MdTc/dt. We approxi-

mate the specific heat of ions during the Debyeregime by Cion

V ∼ 3.2π4(Tc/θD)3</A. In a firstapproximation we get

C∫ t

todt′ = −16π4

5<A

1θ3

D

∫ T

To

T c−1/2 dTc, (9)

where To is the initial temperature from whichDebye cooling begins (To ∼ θD) and T < To.The integration yields the cooling time duringthe Debye phase

τD ∼ 32π4

5

TθD

3

To

T

1/2

− 1

T M

L<A. (10)

Note that the cooling time during theDebye phase is shorter than the classical resultof Mestel (last factor), whenever T is less thanabout 0.1θD (if T = 0.05θD then τD is a factorof 4 smaller than τMestel). It was hoped that thisreduction in the cooling times would explainthe deficit of observed WDs below ∼ 10−4.5L�.However, it is difficult to attribute the defi-ciency of WDs observed at low luminosity tothe onset of Debye cooling. We can see thisby computing, for a given WD mass, the lu-minosity at which Debye cooling becomes rel-evant from L = CMT 3.5

c , where Tc is the cen-tral temperature at the onset of Debye cooling(of about 0.1θD, as we have recently shown).We use the Chandrasekhar’s model to estime

Table 1. Onset of Debye cooling

M(M�) θD(K) L(L�)0.6 7.0 × 106 10−7

1.0 2.2 × 107 10−5.1

1.2 4.0 × 107 10−4.1

the central density to compute θD. Results areshown in Table 1, which lists, for a given stellarmass, the Debye temperature θD(K) and the lu-minosity for the onset of Debye cooling. Notethat, for a typical WD, fast Debye cooling oc-curs at too much low luminosity, and thus noobservable consequences are expected in thiscase. Debye cooling is relevant at observableluminosities only in very massive WDs, whichare much less abundant.

From these simple estimates, it can be con-cluded that the decrease in the number of dimWDs is difficult to explain in terms of a rapidcooling. The lack of WD at very low luminos-ity suggests instead that the Galaxy is not suf-ficiently old to contain cooler (and thus older)WDs, i.e. it is a result of the finite age of theGalaxy. Hence, the cooling age of oldest WDsprovides an estimate for the age of the galacticdisk (Winget et al. 1987).

Convection: At low temperatures, the as-sumption of purely radiative transfer is notvalid: transport of energy in the outer layersof cool WDs can be due to convection, whichresults from the recombination of the main at-mospheric constituents. With decreasing tem-perature, the base of the convection zone grad-ually moves deeper into the star, following theregion of partial ionization. As long as con-vection does not reach the degenerate core, theconvergence to the radiative temperature gradi-ent causes the central temperature to be inde-pendent of the outer boundary conditions, i.e.chemical stratification or treatment of convec-tion. The heat flow depends in this case on theopacity at the edge of the degenerate core, asassumed in the Mestel treatment.

But at low luminosities, convection reachesthe degenerate core and the radiative gradi-ent convergence is lost. The thermal profile


��

��

−4

−1

8 9 10

−2

−3

Log L/Lsun

0.9 Msun

0.4 Msun

1.2 Msun

Log age

crystallization

Debye cooling

Fig. 1. Surface luminosity as a function of cooling time for carbon-core WDs with different stellar masses.The onset of crystallization and Debye cooling are indicated.

of an old WD becomes strongly constrained1

with the consequence that changes in the at-mospheric parameters are reflected in changesin core temperature. This convective couplingmodifies the L − Tc relationship, and thus therate of cooling of cool WDs. In particular, thecore temperature becomes lower — convectionis much more efficient than radiation in trans-porting energy — as compared with the predic-tions of purely radiative envelopes. Because ofthe lower central temperature, the star has ini-tially an excess of thermal energy which mustbe eliminated, thus causing a delay in the cool-ing times at low luminosities. The convectivecoupling occurs at different stages in the WDevolution depending on the outer layer chem-ical stratification. This will be critical to un-derstand the different cooling behavior of coolWDs with different envelope compositions.

1 In fact, the degenerate core is isothermal, andconvection — which extends from the atmosphereto the core — is essentially adiabatic.

Finally, convection, via mixing episodes,plays a role in the interpretation of the spectralevolution (the observed change in surface com-position as WD evolves) of these stars. This isparticularly true if the WD is formed with athin H envelope, and convection penetrates be-yond the surface H layers.

Specific heat of electrons: The Mestelmodel assumes that electrons are completelydegenerate and so Celec

V � CionV . This is true

for low-luminosity (low Tc) WDs, where thespecific heat of strongly degenerate electron isCelec

V ∝ KT/εF → 0. But at high luminosi-ties, and particularly in low-mass WDs, the en-ergy of electrons is not completely indepen-dent of temperature; so the specific heat ofelectrons must be considered in the computa-tion of the thermal energy content of the WD.For instance, for a 0.5 M� carbon-rich WD atTc ∼ 107 K (L ∼ 10−3L�), Celec

V ∼ 0.25CionV .

These improvements alter the coolingtimes of WDs as given by the simple Mestel’scooling law. The resulting cooling times for


different stellar masses are depicted in Fig. 1.The main results are:

– The dependence of cooling times on thestellar mass is different from that predictedby the Mestel model (τ ∝ M5/7). This istrue at high luminosities, where the contri-bution of electrons to the specific heat in-creases the cooling times of low-mass WDs— because of their lower degree of elec-tron degeneracy — and at low luminositieswhere massive WDs cool faster as a resultof Debye cooling.

– The impact of crystallization (increase inthe ion heat capacity) on the cooling timesis strongly dependent on the stellar massand core composition. In particular, mas-sive WDs crystallize at very high luminosi-ties because of their higher densities.

– Cooling times strongly depend on corechemical composition, as predicted by theMestel’s model (τ ∝ 1/A).

– At observable luminosities, Debye coolingis relevant only for M > 1M�.

4. Detailed models of white dwarfevolution

The Mestel’s model, with the improvementswe have described, provides a reasonable de-scription of WD evolution. However, if wewant to use WDs as reliable cosmic clocks toinfer the age of stellar populations, we needto consider a much more elaborated treatmentthat includes all potential energy sources andthe way energy is transported within the star.The main points to be considered are:

– The thermal and hydrostatic evolution haveto be treated simultaneously. In fact, thestructure of a WD changes with time in re-sponse to the energy radiated. This meansthat pressure is a function of density andtemperature.

– Thermal energy is not the only source ofWD luminosity. There are additional en-ergy sources and sinks: residual contrac-tion of the outer layers, nuclear burning,neutrino losses, energy released by possi-ble chemical redistribution in the core, andthe release of latent heat on crystallization.

-6-5-4-3-2-10Log (1-mr/MWD)

0

0.2

0.4

0.6

0.8

Xi

12C

16O

4He

1H

0.6 Msun

surface

Fig. 2. 1H, 4He, 12C, and 16O distribution withina typical WD in terms of the outer mass fraction1 − mr/mWD.

– The core of a WD is never strictly isother-mal, particularly in hot WDs.

– Changes in chemical composition duringWD evolution, due to convective mixing,diffusion processes, nuclear reactions andaccretion are expected to influence thecooling of WDs.

A proper treatment of these issues requiresa detailed knowledge of the previous history ofthe WD. This is particularly true regarding thecomputation of the early phases of WD evolu-tion at high luminosities, where contraction isnot negligible. In addition, the mass and chem-ical composition of the core and outer layers— which, as we saw, have a marked influ-ence on the WD evolution — are specified bythe evolutionary history of the progenitor star.Thus, an accurate treatment of WD evolutionrequires the complete methods of the stellarevolution theory and calculating the evolution-ary history of the progenitor stars all the wayfrom the Main Sequence, to the mass loss andplanetary nebulae phases. In what follows, Icomment on the main aspects to be consideredin a detailed treatment of WD evolution.

4.1. Chemical abundance distribution

The chemical composition of a WD is deter-mined by the nuclear history of the WD pro-genitor. The chemical composition expected ina typical WD is displayed in Fig. 2 in termsof the outer mass fraction. The use of this co-ordinate strongly emphasizes the outer layersof the model. Three different regions are dis-


tinguished: the carbon-oxygen core (95% ofthe mass), which is the result of convective Hecore burning and the following steady He shellburning in prior evolutionary stages. Possibleextra-mixing episodes during core He burningand existing uncertainties in the 12C(α, γ)16Oreaction rate turn the carbon/oxygen composi-tion of the core into one of the main sourcesof uncertainty weighing upon the determina-tion of WD cooling times.

Overlying the carbon/oxygen core is theHe– and C– rich intershell, built up during thethermally pulsing phase on the AGB. This in-tershell reflects the mixing episode during thelast thermal pulse. Here, the carbon abundancedepends on the occurrence of overshooting andthe number of previous thermal pulses. Aboveon the intershell, there is the He–rich bufferwhich results from prior H burning. During theAGB, the mass of this buffer goes from almostzero — at the pulse peak when the He-flashconvection zone is driven close to the base ofthe H layer — to 0.01M�. The chemical com-position expected in a WD depends also on theinitial stellar mass of progenitor star: massiveWDs are expected to have O/Ne cores and verylow mass WDs, He cores.

The theory of stellar evolution provides up-per limits for the mass of the various inter-shells. For a typical WD of 0.6 M�, the max-imum H envelope mass which survives hotpre-WD stages is about 10−4M� and the to-tal mass of the He buffer and the He-intershellamounts to ∼ 0.02M�. Asteroseismology in-ferences on individual pulsating WDs provide,in some cases, constraints to the thickness ofthe H and He envelopes and the core composi-tion of WDs. In particular, there is evidence forthin H envelopes in some WDs. We concludethat the initial chemical stratification of WDsis not known in sufficient detail from the stel-lar evolution theory or from observations. Thisin turn leads to uncertainties in the evaluationof cooling times.

4.2. Changes in the chemicalabundance distribution

During the WD stage, there are numerousphysical processes which alter the chemical

abundance distribution with which a WD isformed. The effects of these changes on theevolutionary properties of WDs may be moreor less important, depending on the luminos-ity they occur. The most important of theseprocesses is element diffusion. In particular,gravitational settling and chemical diffusionstrongly influence the abundances producedby prior evolution. Gravitational settling is re-sponsible for the purity of the outer layers ofWDs. In fact, because of the extremely largesurface gravity of WDs, and in the absence ofcompeting processes (such as weak mass–lossepisodes), gravitational settling rapidly leadsto the formation of a pure H envelope. Atthe chemical interfaces, characterized by largechemical gradients, chemical diffusion stronglysmoothes out the chemical profile. Here, dif-fusion time scale is comparable to WD evolu-tionary time scale, so equations describing dif-fusion have to be solved simultaneously withthe equations describing WD evolution.

Under certain circunstances, diffusion pro-cesses may trigger the occurrence of thermonu-clear flashes in WDs. Indeed, note from Fig.2 that inside the He buffer, chemical diffusionhas led to a tail of H from the top and a tailof C from the bottom. If the WD is formedwith a thin enough He-rich buffer, a diffusion-induced H shell flash may be initiated, thusleading to the formation of a self-induced nova,during which the WD increases its luminosityby many orders of magnitude in a very shorttime (Iben & MacDonald 1986). During thisprocess, the mass of the H envelope is believedto be strongly reduced.

Another physical process which may al-ter the chemical composition of a WD is con-vection. In particular, for cool WDs with thinenough (< 10−6M�) H envelopes, convectivemixing will lead to dilution of H with the un-derlying He layer, thus leading to the formationof He–rich outer layers.

4.3. Energy sources of white dwarfs

4.3.1. Gravitational energy

Although WDs evolve at almost constant ra-dius, the role of contraction is by no means


negligible. The radius at the beginning ofthe WD phase can be up to twice the zero-temperature fully degenerate radius, and thusthe contribution of compression of the outer,non-degenerate layers to the energy output ofthe star can be important in very hot WDs. Inaddition, changes in the internal density dis-tribution, owing to the increase in the coremass from CNO H burning lead to an impor-tant release of gravitational potential energyfrom the core of pre-WDs (very young WDs).Gravitational contraction is also relevant in thefinal phases when Debye cooling has alreadydepleted the thermal energy content. Here, theresidual contraction of the thin subatmosphericlayers may provide up to 30% of the star lumi-nosity. But, as we mentioned in 3.2, for most ofWD evolution compression barely contributesto the surface luminosity, because the compres-sion energy released in the core is almost com-pletely absorbed by degenerate electrons to in-crease their Fermi energy.

4.3.2. Nuclear energy contribution

Stable CNO H shell burning and He shell burn-ing are the main source of luminosity duringthe evolutionary stages immediately prior toWD formation. As a result of CNO burning,the H–rich envelope is consumed, so that be-low ∼ 100L� nuclear burning is a minor contri-bution to surface luminosity — the density andtemperature at the base of the H–rich envelopebecome too low once MH ∼ 10−4M�. Thus,in a typical WD, the role of nuclear burningas a main energy source ceases as soon as thehot WD cooling branch is reached. But it neverstops completely, and depending on the stellarmass and the exact amount of H left by priorevolution, it may become a non-negligible en-ergy source for DA WDs. Detailed evolu-tionary calculations show that for WDs hav-ing low-metallicity progenitors, stable proton-proton H burning may contribute by more than50% to the surface luminosity by the time cool-ing has proceeded down to L ∼ 10−2 − 10−3L�.It must be noted that a small reduction in the Henvelope by a factor of 2 with respect to theupper theoretical limit is sufficient to stronglyinhibit H burning, a fact that explains the dif-

ferent role of H burning obtained by differentauthors. Hence, a correct evaluation of nuclearburning during the WD stage requires the com-putation of the pre-WD evolution.

Finally, stable H burning may be dominantin low-mass WDs, thus delaying their coolingfor significant periods of time (Driebe et al.1998; Althaus et al. 2001). These stars are theresult of binary evolution and are characterizedby thick H envelopes. In these low-mass WDs,H burning can also become unstable, thus giv-ing rise to thermonuclear flashes at early stagesof WD evolution. This bears important conse-quences for the interpretation of low-mass WDcompanions to millisecond pulsars.

4.3.3. Additional energy sources

The first-order phase transition that occursduring crystallization results in a new energysource, due to the release of latent heat — ofabout KT per ion, Lamb & Van Horn (1975).The impact of this energy release on the WDcooling depends on the stellar mass of theWD. Indeed, because crystallization occurs athigher luminosities in more massive WDs, theimpact of latent heat in these WDs is less rel-evant. An additional source of energy that isexpected to delay the cooling is the releaseof potential gravitational energy resulting fromchanges in the chemical profile upon cyrstal-lization (Salaris et el. 1997). In fact, a par-tial separation between carbon and oxygen,as compared with the initial distribution ofthese elements, is expected to occur after crys-tallization. The resulting release of potentialenergy depends on the initial unknown frac-tions of carbon and oxygen and on their dis-tribution in the interior. The diffusion of mi-nor species in the core may also make a con-tribution to the WD luminosity. In particular,the slow 22Ne diffusion in the WD core re-leases enough potential gravitational energy asto impact the cooling times of massive WDs(Deloye & Bildsten 2002 and Garcıa–Berro etal. 2008). In fact, 22Ne, which is the byproductof nuclear burning during prior evolution, has aneutron excess that results in an imbalance be-tween the gravitational and electric fields. Thisleads to a slow gravitational settling towards


the center. The settling rate is quite uncertainbecause of the poorly known microphysics in-volved in the calculation of the interdiffusioncoefficient of elements at very high densities.

4.3.4. Energy loss by neutrinos

Energy is lost from WDs not only in theform of photons, but also through the emis-sion of neutrinos. Neutrinos are created inthe very deep interior of WDs and provide amain energy sink for hot WDs. Because oftheir extremely small cross–section for inter-action with matter — at a central density of106gr/cm3, typical of WDs, the mean free pathof neutrinos is lµ ≈ 3000R� — once created,neutrinos leave the WD core without interac-tions, carrying away their energy. In youngWDs, neutrinos increase the cooling rate andyield a temperature inversion. Neutrinos resultfrom pure leptonic processes, as a consequenceof the electro–weak interaction. Under the con-ditions prevailing in hot WDs, the plasma-neutrino process is usually dominant. Here,a so-called plasmon, resulting from a collec-tive excitation of the electron gas, decays to aneutrino-antineutrino pair.

4.4. Results from detailed calculations

The time dependence of the luminosity contri-bution due to H burning (Lnuc) via proton pro-ton and CNO nuclear reactions, neutrino losses(Lν), the gravothermal energy (Lgravo) — the re-lease of thermal plus gravitational potential en-ergy — and photon emission (surface luminos-ity, Lsur) for a typical WD is shown in Fig. 3.Evolution is depicted from the planetary nebu-lae stage to the domain of the very cool WDs.Luminosities are in solar units and the age iscounted in years from the moment the progeni-tor departs from the AGB. At early times, i.e. inyoung WDs, nuclear burning via CNO mostlycontributes to the surface luminosity of the star.Here, there is a near balance between Lnuc andLsur and between Lgravo and Lν (like in AGBstar). During this stage, Lgravo results from therelease of potential gravitational energy, ow-ing to the change in the internal density caused

by the increase in the core mass from CNO Hburning. After ∼ 104 yr of evolution, nuclearreactions abruptly cease and the star begins todescend to the WD domain and the surface lu-minosity of the star begins to decline steeply.Thereafter, evolution is dictated by neutrinolosses and the release of gravothermal energy— now essentially the release of internal en-ergy from the core. Nuclear burning (via CNO)is a minor contribution. Note that during thisstage, neutrino losses are the dominant energysink (Lgravo ≈ Lν). In fact, Lν ≈ 10Lsur andas a result the WD cooling is strongly accel-erated. To this stage belong the pulsating DOVand DBV WDs, so an eventual measurement ofthe rate of period change in some of these starswould allow to constrain the plasmon neutrinoproductions rates (Winget et al. 2004). Thetime-scale for CN burning is larger than theevolutionary time-scale. Burning occurs in thetail of H distribution and, as a result, the H en-velope is not reduced by burning during thisstage.

As soon as the structure approaches a cool-ing age of 107 yr, the WD is cold enough thatneutrino and CN burning are no longer rel-evant. Lgravo ≈ Lsur and the WD resemblesthe Mestel idea. This is the best understoodstage in the WD life: no convection, neutrinos,nuclear burning, crystallization, and ion ther-mal energy are the main source of energy. Atmore advanced stages, proton-proton burningshell makes a significant contribution to sur-face luminosity. As a result, the cooling rate ofthe WD is reduced. Eventually, H burning be-comes virtually extinct at the lowest luminosi-ties, and surface luminosity is given by the re-lease of internal energy from the core. Duringthese stages, the contraction of the very outerlayers may contribute to surface luminosity.Finally, we note that the mass of the H enve-lope decreases with increasing metallicity Z ofprogenitor star. Thus, the role of proton-protonburning at advanced stages will be much lessrelevant in WDs with solar metallicity progen-itors. For the case of low metallicity progeni-tors we have analysed, residual proton-protonburning provides an appreciable contributionto luminosity, unless the H envelope is reducedbelow the maximum value predicted by stellar


4

2

0

−2

−6

−4

3 5 7 9 10

Log

L / L

sun

Log Age (yr)

Z=0.001

0.6 Msun

pp

CNO

Gravothermal Neutrino

Photon

Fig. 3. Temporal evolution of different luminosity contribution for a typical WD from the hot pre–WD tothe very advanced stages. The metallicity of the progenitor star is Z = 0.001

evolution. In this context, we note that the massof the H envelope may be orders of magni-tude smaller if the WD progenitor experiencesa late thermal pulse on the WD cooling track(Althaus et al. 2005).

5. Conclusions

There are several reasons that turn WDs intocosmic clocks and laboratories to study dif-ferent kind of problems. Indeed, they consti-tute the final evolutionary stage for vast major-ity of stars. They are a homogeneous class ofstars with a narrow mass distribution and well-known structure, with slow rotation rates andlow magnetic fields. And most importantly,their evolution, as we have seen, is relativelysimple and can be described in terms of acooling problem. However, there are still someopen questions regarding both the evolutionaryhistory of WD progenitors and the physics of

WD cooling, which should be addressed in or-der to use WDs as reliable tools.

Acknowledgements. I appreciate the invitation toparticipate in this School of astrophysics. I thankthe School organizers for financial support. Thiswork has been partially supported by PIP 6521 fromCONICET.

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