The physics of white dwarfs - Semantic Scholar · The physics of white dwarfs Increasingly sensitive and detailed astronomical observations coupled with calculations of the properties

The physics of white dwarfsIncreasingly sensitive and detailed astronomical observations

coupled with calculations of the properties of matter under extreme conditionshave given us new insights into their structure and evolution.

Hugh M. Van Horn

White dwarf stars, so called because of thecolor of the first few to be discovered, oc-cupy a key position in astrophysicaltheory. Together with neutron stars andblack holes, they are the terminal pointsof stellar evolution. Their propertiesthus provide clues to the physical pro-cesses that take place during the rapidand often spectacular evolutionary stagesnear the ends of stellar lifetimes. In ad-dition, white dwarfs provide astrophysical"laboratories" for "measuring" thephysical properties of matter under ex-treme conditions. These extend fromconditions like those in laser-producedplasmas to those typical of the solid crustsof neutron stars. White-dwarf stars alsooccur as components of cataclysmic bi-nary systems—novae, dwarf novae andrelated objects—and knowledge of theproperties of white dwarfs is essential tothe development of satisfactory theoret-ical models for these systems.

In this article, I shall describe our cur-rent understanding of the nature of thewhite dwarfs. The past ten years haveseen a number of remarkable new dis-coveries about these stars. We now know,for example, that some white dwarfspossess magnetic fields of several hundredmegagauss, much larger than are attain-able in terrestrial laboratories, someothers display periodic light variations,and still others may have solid cores thathave crystallized at temperatures of sev-eral million kelvins. These discoveriesand theoretical interpretations have re-sulted from a close interplay between as-tronomical observations and develop-ments in physical theory.1 This closerelationship has characterized white-

Hugh M. Van Horn is a professor of physics andastronomy and a member of the C. E. KennethMees Observatory at the University of Roches-ter.

dwarf research from its beginnings, andimportant contributions to our under-standing of these stars have come fromalmost every subdiscipline of physics,including statistical mechanics, nuclearand particle physics, solid state, and fluiddynamics.

Stars as small planets

White dwarfs were first recognized asa distinct class of stars, having stellarmasses but planetary dimensions, in theearly years of the present century. Byabout 1920, the gravitational redshift oflight predicted by Einstein's generaltheory of relativity had been measured inthe white dwarf Sirius B (figure 1), con-firming the small size and high density ofthis star (see the box on page 25). Theexistence of such compact stars consti-tuted one of the major puzzles of astro-physics until the quantum-statisticaltheory of the electron gas was worked outby Enrico Fermi and P. A. M. Dirac in themid-1920's. In accordance with the Pauliexclusion principle, the Fermi-Diractheory showed that even the coldestelectrons cannot all accumulate in thequantum state of lowest energy, and thusthe total energy and pressure of an elec-tron gas remain non-zero even at zerotemperature. R. H. Fowler and othersrealized immediately that this was theexplanation of the puzzle of the whitedwarfs, and Fowler showed that thepressure of a degenerate electron gas, inwhich the low-lying quantum states aresuccessively filled, is sufficient to supportan object of stellar mass against its ownself-gravitation at precisely the radii ofthe white dwarfs (see the box on page 28).For this reason white dwarfs are some-times also called "degenerate dwarfs," andI shall use these two terms interchange-ably.

By the early 1930's, Subrahmanyan

Chandrasekhar, then a young graduatestudent at Cambridge, had extended thetheory of the degenerate electron gas toinclude the effects of special relativity.As a result of this work, Chandrasekharfound that there is a critical stellar massabove which stable degenerate dwarfscannot exist (see the box on page 28).This was a major discovery, for the exis-tence of this critical mass—now called the"Chandrasekhar limit," Mch. about1.44MO—has profound consequences forthe final stages of stellar evolution. Starsthat end their evolution with masses inexcess of this limit have gravitationalself-attraction too great to be counter-balanced by the pressure of the degener-ate electrons and are doomed to ultimatecollapse. Supernovae are externalmanifestations of such catastrophes. Insome cases, even the violence of a super-nova explosion does not wholly disruptthe star, however, and a stellar remnantof even greater density than a white dwarfmay be left behind: a neutron star orblack hole. Investigations of evolutionarypaths by which stars become white dwarfsare therefore important not only ofthemselves, but also because they imposelimitations on the possible pathwaysleading to supernovae and their exoticstellar remains.

Knowledge of the thermal structure ofwhite dwarfs began to develop early in the1940's. The first step was Robert E.Marshak's application of the quantumtheory of electron conduction in metals,worked out earlier by Arnold Sommerfeldand Hans Bethe, to conditions appropri-ate to the electron gas in a degeneratedwarf. Marshak, then a student withBethe at Cornell, found the thermal con-ductivity to be so high that the electron-degenerate core of a white dwarf is virtu-ally isothermal, while, in contrast, heattransfer through the thin, non-degenerate

0031-92?A/7q/nmo23-O9IOO 50 © 1979 American Institute ol Physics PHYSICS TODAY / JANUARY 1979 23

The white dwarf Sirius B is the companion of the ten-thousand-times brighter star Sirius. SiriusB is below and to the right of the bright image of Sirius in this photo by Irving Lindenblad of the USNaval Observatory (reference 2). The hexagonal shape of the image of Sirius as well as the su-pernumerary images to the left and right are caused by a hexagonal diffraction grating. Figure 1

surface layers is very slow and inefficient.Numerical calculations showed that fortypical white-dwarf luminosities ofroughly 10~2L©, the temperature of theisothermal core is about 107 K. Even atthese high temperatures—as hot as thecenter of the Sun—the electron gas isdegenerate because of the high density ofmatter in the white-dwarf core. Thus, awhite dwarf came to be pictured as amassive, hot, conducting sphere sheathedin a thin, but very effective, blanket ofinsulation.

By this time, astronomical observationshad long established that hydrogen is themost abundant element in the universeand that it makes up about 70 percent ofthe mass of ordinary stars. In 1938,Bethe calculated the rates of thermonu-clear reactions involving hydrogen underconditions appropriate to stellar interiors,and he showed that these reactions pro-vide the energy source from which theluminosities of most ordinary stars arederived. For this work, Bethe wasawarded the Nobel Prize for Physics in1967. Application of Bethe's results towhite dwarfs, however, led to a startlingcontradiction: the predicted energygeneration rates led to stellar luminositiesmany orders of magnitude greater thanthose observed! The inescapable con-clusions were:• Thermonuclear reactions are not thesource of white-dwarf luminosities, and• Hydrogen must be entirely absent from

the white-dwarf interiors.These results showed clearly that the

white dwarfs are stars that have ex-hausted their thermonuclear energysources and are indeed among the termi-nal stages of stellar evolution—theburned-out embers of dying stars. What,then, is the source of the energy they ra-diate? This question went unanswereduntil 1952, when Leon Mestel3 pointedout that the thermal energy content of thehot white-dwarf interior, leaking slowlyaway through the insulating surfacelayers, is entirely sufficient to explain theobserved luminosities. Mestel's theoryalso predicts a statistical distribution ofthe luminosities of the white dwarfs thatagrees satisfactorily with existing obser-vations. Except for details, this theory ofwhite-dwarf evolution has remained un-changed over the past quarter century,and it is the accepted theory today.

Spectra and surface layers

Observations of white dwarfs requirethe light-gathering power of a large tele-scope, because these stars are intrinsicallyvery faint. Completion of the 5-meter(200-inch) Hale reflector on Mt Palomarin 1948 provided the first opportunity fordetailed studies of these stars. In thefollowing decade, Jesse Greenstein andhis collaborators at the Hale Observa-tories began extensive spectroscopic andphotometric investigations using this in-strument, and their work has provided

much of our present observationalknowledge of the white dwarfs. In thepast two decades, other large telescopeshave been constructed in various parts ofthe world, and several other groups havealso begun white-dwarf observations. Asa result, detailed spectroscopic informa-tion has now been acquired for more than400 of these stars. More recently, satel-lite and rocket-borne observations havebegun to open up the far ultraviolet andx-ray regions of the spectrum that areinaccessible from ground-based obser-vations, and important new results havealready been obtained from this re-search.

It became apparent quite early thatwhite-dwarf spectra are very differentfrom those of ordinary stars. One dif-ference is that the spectral lines are verybroad, regularly being 20 to 50 A wide athalf the central line depth. This is a di-rect consequence of the high surfacegravities of degenerate dwarfs: g ~ 108

cm/sec2, some 104 times larger than thatof the Sun. The high gravities result inhigh pressures in the atmospheres of thesestars that in turn cause large Starkbroadening of the spectral lines.

A second difference from ordinary starsis the striking abnormality of the elementabundances in white-dwarf atmospheres,for reasons that are not yet fully under-stood. In ordinary stars, differencesamong the spectra are primarily due todifferences in effective surface tempera-tures Teff (and, to a lesser extent, in sur-face gravities), rather than to large in-trinsic differences in abundances. NearTef(« 10 000 K, normal stellar spectra aredominated by the strong Balmer absorp-tion lines of hydrogen. At higher tem-peratures most of the hydrogen becomesionized, causing the Balmer lines toweaken; stars with Te(c > 15 000 K arecharacterized by strong absorption linesof helium, the second most abundant el-ement, which is less easily ionized thanhydrogen. Conversely, in stars muchcooler than 10 000 K (like the Sun, withTeff = 5800 K), the spectra are dominatedby the lines of various metals and, in ex-treme cases, by molecular bands. Thesespecies dissociate or ionize much morereadily than hydrogen with increasingTeff, and their lines accordingly disappearin the hotter stars.

In marked contrast to this well-under-stood behavior of ordinary stellar spectra,the white dwarfs display few explicablespectral regularities. By far the largestnumbers of these stars, comprising abouttwo-thirds of all white dwarfs, exhibitonly the pressure-broadened Balmer linesof hydrogen. These spectra are all themore remarkable because we know thathydrogen must be completely absent fromthe deep interiors of these stars! Thisgroup of hydrogen-line white dwarfs istermed spectral class DA. The spectrumof Sirius B, which is a DA white dwarf, isshown in figure 2. The remaining third

24 PHYSICS TODAY / JANUARY 1979

of the degenerate dwarfs, which displayother than pure hydrogen spectra, aredistributed among a number of differentspectral classes. About 8% of all whitedwarfs have essentially pure helium linespectra and are classified as spectral typeDB. Another 14% of the total displayonly featureless continua; these aretermed spectral type DC. Still othergroups of white dwarfs, containing only adozen or so stars each, exhibit weak me-tallic lines or molecular bands, generallyof carbon-bearing molecules.

In the DA white dwarfs, the absence oflines due to elements other than hydrogencannot be simply the result of peculiari-ties of ionization equilibria in dense stellaratmospheres. The enormous range ofTeff covered by the DA spectral class ex-tends from less than 5000 K to more than70 000 K (for HZ43 and Feige 24, thehottest white dwarfs known). The asso-ciated variations in ionization balancewould inevitably lead to spectral changesanalogous to those in ordinary stars ifother elements were present in significantamounts. Instead, the elements otherthan hydrogen appear to be truly defi-cient in the DA white dwarfs, relative tonormal cosmic abundances. The proba-ble explanation of the remarkable purityof these hydrogen atmospheres, whichwas proposed some time ago by theFrench astrophysicist Evry Schatzman,is that gravitational settling has takenplace in the strong gravitational fields ofthe degenerate dwarfs, with heavier ele-ments sinking below the atmosphericlayers and the hydrogen floating up to thestellar surface. The estimated timescalefor this process appears short enough togive rapid element separation in the highgravity fields of the white dwarfs, thussatisfactorily accounting for the spectralappearance of the DA stars.

The existence of white dwarfs withnon-DA (hydrogen-deficient) spectra,however, has not yet been satisfactorilyexplained. The complete absence of hy-drogen from these stars appears to requireejection of the residual, hydrogen-bearingsurface layers sometime after the com-pletion of thermonuclear hydrogen-burning and before these stars becomewhite dwarfs. Stars known to be amongthe ancestors of white dwarfs and that areobserved to have ejected as much as sev-eral tenths of a solar mass of material, doexist. They are the stellar nuclei of theso-called "planetary nebulae." It is notclear whether the nuclei of planetarynebulae are completely devoid of hydro-gen, however, and it also appears possiblethat their rate of formation may signifi-cantly exceed the estimated birthrate ofthe non-DA white dwarfs. The evolu-tionary connections between the plane-tary nuclei and specific white dwarfspectral classes thus remain still to beestablished. In addition, the situation iscomplicated further by the existence ofseveral other types of hydrogen-deficient

FREQUENCY(Hz)

The spectrum of Sirius B. The curve in the main figure gives the emergent flux distribution computedby Harry Shipman for a pure hydrogen stellar atmosphere with Teff = 32 000 K and log g = 8.65.The bars with arrows give upper limits of extreme-ultraviolet emission from rocket-borne mea-surements (references given in reference 4); they indicate a temperature somewhat lower than30 000 K for the star. Insets a and b show line profiles of the Balmer lines Hex and Hy with intensityplotted as a function of | A \ | , the magnitude of the distance to the line center (data from reference5; theoretical curves for 7" = 33 000 K, log g = 8.8). Inset c shows the Lyman lines La and Lfiwith data from Copernicus satellite measurements (reference 6) and a theoretical curve calculatedby Francois Wesemael for a T= 27 000 K log g = 8.65 pure hydrogen atmosphere. Figure 2

Sirius BSirius B, the faint companion of the "dog star" Sirius, was one of the first white dwarfs to bediscovered. Some of its properties are given in the accompanying table. For comparison,the corresponding values for the Sun, the standard astronomical "yardstick" for measure-ments of stellar quantities, are also given. It is interesting to note that the radius of the Earthis 6.371 X 108 cm = 0.00915R0, somewhat larger than Sirius B!

Quantity

MassRadiusLuminosityEffective (blackbody) temperatureGravitational redshiftMean densityCentral density"Central temperature"

• Model-dependent quantities

Sirius B

1.05M©0.008Ro

0.03/.Q27 000 K89 ± 16 km/sec2.8 X 106gm/cm3

3.3 X 107 gm/cm3

2.2 X 107K

Sun

Me = 1.989 X 1033gmfio = 6.96 X 1010cmL o = 3.90X 1033 erg/sec5800 K0.6 km/sec1.41 gm/cm3

1.6 X 102 gm/cm3

1.6 X 107K

PHYSICS TODAY / JANUARY 1979 25

4 5 6 7 8

TEMPERATURE Log (T/kelvin)

The temperatures and densities relevant to white-dwarf atmospheres. The irregularly shapedregion in color is the region of helium-dominated white-dwarf envelopes. The lower left-hand corner(light color) is the region of incomplete helium ionization. On the high-temperature boundary,ionization is dominated by thermal excitation of the electrons, and on the upper, high-density,boundary, ionization results from the increased Fermi momentum of the electrons and the over-lapping of ion cores. The line marked rj = 0 is the degeneracy boundary where the Fermi energyis approximately kT. The line marked F = 1 shows where the electron-ion Coulomb interactionenergy is on the order of kT. The gray region shows the parameters of interest for laser-fusionplasmas. (Figure adapted from reference 7) Figure 3

stars that may also be precursors of someof the non-DA white dwarfs.

To make quantitative analyses of thevarious white-dwarf spectra observed, weneed detailed numerical models of theatmospheres of these stars. Such calcu-lations, first begun about a decade ago,have become increasingly sophisticatedin more recent years. Calculations byPeter Strittmatter and D. T. Wickram-asinghe at Steward Observatory, by HarryShipman first at the Hale Observatoriesand now at the University of Delaware, byKarl-Heinz Bohm and his collaboratorsat the University of Washington (Seattle),and by Volker Weidemann and his groupin Germany have now provided quanti-tative evaluations of the effective tem-peratures, surface gravities and atmo-spheric compositions for many of thesestars.

An early result of these model atmo-sphere calculations that has had far-reaching consequences was Bohm's dis-covery in 1968 that convection occurs inthe surface layers of these stars. Becauseconvection produces efficient mixing instars, Strittmatter and Wickramasinghesubsequently suggested that mixing of athin superficial hydrogen layer into a deepsubsurface helium convection zone mightbe capable of transforming DA (H-line)white dwarfs into DB (He-line) whitedwarfs at a particular stage in the coolingof these stars. Although this hypothesisis now believed to be incorrect (it appears

to be in conflict with the observed tem-perature distribution of the DA and DBstars, as pointed out by Shipman) thework of Strittmatter and Wickramasinghewas the first to consider the possibility ofevolutionary transitions between dif-ferent spectral classes of the whitedwarfs.

More recent investigations have latelybegun to establish a reliable frameworkfor understanding the chemical evolutionof white-dwarf surface layers. In partic-ular, Shipman has pointed out that atsome stage in the cooling of DB stars thehelium lines must disappear. If the at-mospheres of these stars contain nothingother than helium, their spectra at stillcooler temperatures will thus display onlyfeatureless continua. Accordingly,Shipman suggests that the DC (purecontinuum) white dwarfs are simplycooled-off DB's. Much additional workremains to be done before the many re-maining spectral classes can be fitted intothis scheme, however, and efforts are justbeginning to explore the effects of con-vection, gravitational settling, accretion,and the thermal evolution of the star itselfupon the spectral evolution of the whitedwarfs.

A second important consequence ofconvection in white dwarfs is its effectupon heat transfer through the non-de-generate outer layers of these stars. Be-cause the rate of heat transfer from theisothermal degenerate core of a white

dwarf through the non-degenerate "en-velope" to the stellar surface controls therate of cooling of the star, and becauseconvection is generally a much more ef-ficient heat transfer process than radia-tive diffusion, the presence of a subsurfaceconvection zone can greatly affect the rateof cooling of a white dwarf. Indeed,Bohm found that in the cooler whitedwarfs the convection zones may actuallypenetrate into the degenerate cores,causing a great reduction in the insulatingabilities of the stellar envelopes.

Models of the convective envelopes ofthe white dwarfs, however, involve theo-retical calculations of the opacities,thermal conductivities, and thermody-namic properties of plasmas under rathercomplex physical conditions. Not only isionization incomplete and pressure-ion-ization important, but also the electronsare partially degenerate and Coulombinteractions are strong enough that ionmotions in the plasma begin to exhibitcorrelations. Rather interestingly, theseconditions are very similar to those en-countered in the laser-produced plasmasnow being studied for thermonuclear fu-sion applications (figure 3). For thewhite-dwarf problem, the most extensivecalculations of thermodynamic propertiesfor this complex physical regime havebeen carried out by Gilles Fontaine' (nowat the Universite de Montreal), H. C.Graboske, Jr (of Lawrence LivermoreLaboratories), and myself; by FrancesesD'Antona, Italo Mazzitelli, and G. Magni;by M. A. Sweeney (now at Sandia Labo-ratories); and by Bohm and his collabo-rators. In some ranges of stellar tem-peratures, the white-dwarf envelopemodels constructed by the differentgroups agree rather well. However, at lowtemperatures serious differences arise dueto the uncertainties in the calculations ofthe thermodynamic and transport prop-erties for the deeper parts of white-dwarfenvelopes. The resolution of theseproblems is important for both white-dwarf and laser-fusion calculations andprovides a point of contact betweenwhite-dwarf research and a developingarea of physical theory.

Diamonds in the sky

An important milestone in the theoryof degenerate dwarfs was Edwin Salpet-er's publication8 in 1961 of his calcula-tions of the thermodynamic properties offully degenerate matter, including thecorrections to Chandrasekhar's theoryarising from electrostatic interactionsamong the ions and degenerate electrons.These corrections are primarily attractiveand reduce the energy and pressure of theplasma by as much as several percent atlow temperatures. The reduced pressuresupport in turn causes similar reductionsboth of the radius of a star of a given massand of the limiting mass of a stable whitedwarf, compared to the values given byChandrasekhar's theory (see figure 4).


Because the electrostatic interactionenergy depends upon the charge +Ze ofthe positive ions, Salpeter's results de-pend upon the composition of the de-generate matter. Unfortunately, exceptfor the certainty that there can be no ap-preciable hydrogen in a white dwarf, thereis meager information about the internalcompositions of these stars. Evolution-ary calculations indicate that the pro-genitors of the white dwarfs have proba-bly undergone both hydrogen- and he-lium-burning, while it appears unlikelythat carbon ignition occurs in evolution-ary sequences leading to the white dwarfs.Thus the most plausible composition forthe core of a degenerate dwarf is a mixtureof carbon and oxygen, the main productsof helium-burning reactions.

One of the most interesting conse-quences of Salpeter's work was his rec-ognition (also pointed out independentlyby A. Abrikosov and D. A. Kirzhnits in theSoviet Union) that at very low tempera-tures the Coulomb interactions in ahigh-density plasma cause the ions tofreeze into a regular crystalline latticewithin the nearly uniform "sea" of de-generate electrons. This crystallizationoccurs because an ion lattice has lowerenergy than a random distribution ofpoint charges under these conditions. Athigh temperatures, of course, the thermalmotions of the nuclei are sufficiently en-ergetic to prevent lattice formation.

[ Thus the interesting question arises: Atwhat temperature does white-dwarf

{ matter crystallize?A preliminary answer to this question

was obtained in 1966 in a now-classicwork9 by Stephen Brush, H. L. Sahlin andEdward Teller of the Lawrence RadiationLaboratory. In this paper, the statisticalmechanics of a classical, one-component

' plasma, consisting of positive point1 charges in a uniform neutralizing back-

ground-charge distribution, was investi-gated numerically by Monte Carlo cal-culations. The properties of this systemdepend only upon the ratio of the Cou-lomb interaction energy to kT (effectivelythe ratio of potential to kinetic energy),and Brush and his colleagues accordinglyfound it convenient to express their re-sults in terms of the dimensionless pa-rameter

= (Ze)2/akT (1)

where a is the radius of a sphere contain-ing one single ion at the given ion density.High temperatures (or low densities)correspond to small T, and the ions be-have as an ideal gas with weak electro-static corrections. For T S; 2.5, however,the ion distribution begins to developshort-range order, as in a conventionalliquid, and at T ~ 125, Brush, Sahlin andTeller found direct evidence in theirMonte Carlo results of the expectedfirst-order phase transition from this

' Coulomb liquid to a regular crystalline1 solid. More recent Monte Carlo calcu-

o<

MASSM/M®

Mass-radius relation for white dwarfs. Chandrasekhar's pressure-density relation for a zero-temperature, non-interacting electron gas with /j.e = 2 is shown by the colored line (see the boxon page 28). The black curves show Salpeter's pressure-density relations for zero-temperaturedwarfs for the composition indicated. CCM is cold-catalyzed matter, in which the iron nuclei havecaptured electrons from the Fermi sea to produce the lowest energy state of nuclei plus electrons.(Adapted from reference 8) Figure 4

lations, carried out principally by Jean-Pierre Hansen and his coworkers inFrance, have greatly refined and extendedthis work, and have made the one-com-ponent plasma now one of the best un-derstood nonelementary systems in sta-tistical mechanics. As one result of thesecalculations, the F-value of the liquid-solid phase transition has been evaluatedwith considerably greater accuracy, andit is now believed that crystallization oc-curs near T = 150.

To appreciate the relevance of this re-sult for white-dwarf evolution, we can useequation 1 to express T = 150 as a tem-perature-density relation for the plasmacrystallization curve. For a C12 plasmawe obtain

T107K

= 0.238106gm cm '•'-

1/3(2)

For densities and compositions that aretypical of white-dwarf interiors, crystal-lization thus takes place at temperaturescharacteristic of the cores of the coolerdegenerate dwarfs. The coolest whitedwarfs are thus, almost literally, "dia-monds in the sky," except with a body-centered cubic lattice (that is, the latticecell is a cube with ions located at the eight

corners and at the center), rather than anactual diamond lattice structure.

Crystallization of the plasma in a de-generate dwarf has a number of conse-quences for the evolution of these stars, aspointed out some time ago by Mestel,Malvin Ruderman, and myself.10 First,the crystallization temperature given byequation 2 is quite sensitive to the corecomposition of the white dwarf, varyingapproximately as Z5/3. Perhaps thissensitivity may eventually enable us toplace more stringent limits upon the al-lowable range.of core compositions inthese stars. In addition, crystallizationleads to the release of latent heat associ-ated with the liquid-solid phase transi-tion, which slightly reduces the rate ofcooling of the star during this stage.Early calculations raised the hope thatthis would slow the evolution enough to beobservable, thus marking the composi-tion-dependent locations of the crystal-lizing white dwarfs. However, the effecthas turned out to be relatively small, andobservations have so far been unable todistinguish it. Still another consequenceof crystallization is the rapid decline ofthe heat capacity when the core temper-ature of the white dwarf falls below theDebye temperature, which is determined


0 -2 -4

LUMINOSITY Log (L/LJ

Theoretical and observed luminosity functionsfor white dwarfs. The luminosity function isproportional to the number of stars in each rangeof luminosities. The various curves are dis-placed vertically by arbitrary amounts for clarity.The straight diagonal lines are the theoreticalluminosity function predicted from Mestel'scooling theory. Curve a is the result of a theo-retical calculation (reference 11). The experi-mental curves are based on measurements by:(b) Luyten (1958), (c) Luyten (1963-66) and (d)Eggen and Greenstein (1965). Figure 5

by the characteristic lattice vibrationfrequencies of the ions. This decline ofthe heat capacity greatly accelerates therate of white-dwarf cooling at low tem-peratures and may even lead to completecooling of the more massive white dwarfswithin the 10-billion-year age span of theGalaxy.

In recent years, Don Lamb and I andAttay Kovetz and Giora Shaviv in Israelhave carried out detailed evolutionarycalculations11 of the later phases ofwhite-dwarf cooling. These calculationswere the first to incorporate full and ac-curate representations of the effects of theelectrostatic interactions upon the ther-modynamic properties of white dwarfmatter, including crystallization, latentheat release, and Debye cooling at lowtemperatures. Some results of thesecalculations for a 1 Mo , pure C12 whitedwarf are shown in figure 5.

From the figure it is clear that the mostsensitive tests of the new theoreticalmodels will come at high luminosities(where the cooling times, which aredominated by energy losses due to copiousneutrino emissions in these phases arevery short and consequently the stellarstatistics small) and at very low lumi-nosities (where the stars are intrinsicallyfaint, sometimes difficult to identifyclearly as white dwarfs, and thus veryhard to study). Progress is being made atboth extremes, however. Recent space-borne observations in the extreme ultra-

violet have revealed a (small) number ofwhite dwarfs with much higher surfacetemperatures than previously realized.At the other end of the range, a systematicsearch for very faint white dwarfs hasbeen undertaken by James Liebert (atSteward Observatory), Conrad C. Dahn(at the United States Naval Observatory),and their collaborators. This work hasalready begun to yield results that suggestdiscrepancies between the observationsand the theories. Although it is prema-ture to conclude that there is a seriousconflict, further investigations are clearlynecessary. Undoubtedly the FaintObject Spectrograph currently beingplanned for the Space Telescope willprovide important data bearing on thisproblem when it is orbited early in thenext decade.

Magnetic white dwarfs

In 1970, James Kemp and John Swe-dlund of the University of Oregon to-gether with John Landstreet and RogerAngel, then of Columbia University, madethe remarkable discovery12 that the op-tical continuum radiation from the whitedwarf Grw +70°8247is circularly polar-ized. This was immediately interpretedas evidence for a strong magnetic field inthis star. Since then, Angel (now atSteward Observatory), Landstreet (nowat the University of Western Ontario),and others have discovered13 about adozen magnetic white dwarfs, half of them

The radii of white dwarfs and the Chandrasekhar limitThe structure of any star is determined bythe condition of hydrostatic balance betweeninternal pressure and self-gravitation. In anordinary star like the Sun, the pressure is theideal-gas pressure of the ions and electronsat the temperatures prevailing in the stellarinterior. In a white dwarf, the pressure isdominated by the degenerate electron gasand is insensitive to the temperature. Themechanical equilibrium of a white dwarf canthus be calculated to a very good approxi-mation by taking the pressure to be that of afully degenerate electron gas at zero tem-perature. In this case, the radii of degen-erate dwarfs and the value of the Chandra-sekhar limit can be roughly estimated by thefollowing argument, which emphasizes thephysical principles underlying the structuresof these stars.

Suppose the electron gas is compressedto a density ne, and let r0 be the radius of asphere containing one single electron, thatis, r0 = [3/(47rne)]

1/3. From the Heisenberguncertainty principle, the momentum of anelectron confined to a region of dimension/o is of order h/r0. The quantum statesavailable to the degenerate electrons thusrange from states of nearly zero momentum(corresponding to electrons with macro-scopic de Broglie wavelengths) up to statesof momentum pF (the Fermi momentum), pF

~ h/ro- The Fermi energy eF, which is, for

an electron gas, roughly the same as themean energy per electron, is just the kineticenergy for electrons with momentum pF.For extremely relativistic electrons (pF »me) the Fermi energy is proportional to ne

V3,while for non-relativistic electrons (pp « me)it is proportional to ne

213. We can now usethermodynamics to compute the pressure ofthe gas, using the usual definition of pressureas the derivative of energy with respect tovolume:

For non-relativistic electrons

P~h2 ne5/3/m for ne « (mclhf

and for relativistic electrons

P~ hcne4'3 for ne » (mc/h)3

For the fully ionized matter in a white dwarf,the mass density and electron density arerelated by

P = ne/ixeH

where fig = A/Z\s the so-called mean mo-lecular weight per electron, A and Zare theatomic mass and charge, respectively, andH = 1.66 X 1(T24g is the unit of atomicmass. Together, these expressions forP(ng) and f){ng) comprise an approximateparametric equation of state for white-dwarfmatter.

The condition of hydrostatic equilibriumis, for self-gravitating objects,

where Mr is the mass interior to a sphere ofradius r. A rough dimensional analysis ofthis equation replaces dP/dr by P/R, and rbyR. In equilibrium the pressure and gravita-tional force must balance.

In the non-relativistic limitP~ GM2//?4 ~ h2(M/neH)5/3/mR5

and we can always find a value of the stellarradius such that balance is achieved. For astar of one solar mass this argument givesthe somewhat low value R ~ 2 X 108 cm.

In the relativistic limit

i n e )

and equilibrium can exist only for a specialvalue of the mass

M~(hclGf'2l(neHfFor Me = 2 this mass is roughly 1/2 /W© or 1033

gm. Exact calculation yields the value Mch

= 1.44 MQ quoted in text. For masses M>MCh, the gravitational potential, Gkfi/R4,overwhelms the pressure and leads to col-lapse. For M < MCh, the pressure domi-nates and expands the star to lower (lessrelativistic) densities, where more exactcalculations show that hydrostatic balancecan again be achieved.


from polarization studies and half by ob-servations of Zeeman splitting in spectrallines of H, He, or CH. The inferredmagnetic field strengths range from about5 X 106 to more than 108 gauss. In thehigher fields, the Zeeman splitting is soextreme that field gradients over thestellar surfaces often broaden the linesinto unrecognizability. In addition, someof the magnetic white dwarfs exhibit at-mospheric composition anomalies, whichare presumably caused in some way by thestrong fields.

Quantitative analyses of the spectra ofmagnetic white dwarf's require calcula-tions of the Zeeman shifts of atomic en-ergy levels. Although the shifts are linearwith magnetic field strength for weakfields, the term in the Hamiltonian thatis quadratic in the field intensity domi-nates when the fields become as large asthose in the magnetic white dwarfs. Forsuch cases numerical computations of thelevel shifts are required. Calculationshave been carried out for the levels givingrise to optical transitions in hydrogen andhelium, the most detailed work havingbeen done by S. B. Kemic and RoyGarstang at the Joint Institute for Labo-ratory Astrophysics. An example of thecomputed splittings of the Balmer linesHfi, H v and Ha (which have unperturbedwavelengths of 486lA,4340A, and 410lA,respectively) is shown in figure 6 for arelatively weak field of 5 X 106 gauss.The widths of the line subcomponentsresult from averaging over a uniformdistribution of fields ranging from 4.5 to5.5 X 106 gauss. Also shown is an ob-served spectrum of the magnetic whitedwarf GD90, with the same wavelengthsmarked as in the theoretical spectrum.Evidently the magnetic shifts accountvery well for the locations of the observedfeatures.

The origin of the magnetic white dwarfsposes an interesting problem. Calcula-tions have shown that the time scale forohmic dissipation of the magnetic field isof the order of several billion years. Thisis long enough for the white dwarfs to cooldown to surface temperatures near Teff <=5000 K, and it may be significant that nomagnetic white dwarf has yet been dis-covered with temperature much lowerthan this. The long magnetic-field decaytimes are consistent with the view that thefields are primordial (frozen in when thestar became a white dwarf), rather thanbeing generated later by some processintrinsic to cool degenerate stars. Howthe fields were produced in the ancestorsof the white dwarfs is still not known,however. Are they frozen-in relics datingback to the main sequence origins of themagnetic white dwarfs? Were they pro-duced in a stellar dynamo perhaps drivenby the final stage of thermonuclearburning prior to the white dwarf phase?The answers to such questions, which arestill being sought, can teach us muchabout the relationship of the white dwarf's

4000 5000

WAVELENGTH (A)

Observed and predicted spectra of the magnetic white dwarf G 90. The theoretical spectrum(lower curve) is based on calculations of the Zeeman splitting of the Balmer lines in a magneticfield of 5 X 106 gauss. (Adapted from reference 13) Figure 6

to the earlier stages of stellar evolution.

Oscillating white dwarfs

In 1968, rapid, quasi-periodic oscilla-tions in the light from the white dwarf HLTaurus 76 were discovered14 by ArloLandolt of Louisiana State University.Since then, rapid variations have alsobeen detected in more than 30 other de-generate stars. These discoveries haveresulted from the application of high-speed data-recording techniques to as-tronomical observations and from theutilization of the Fast Fourier Transformalgorithm to permit efficient powerspectrum analyses of these data. Suchmethods have been employed with nota-ble success by R. Edward Nather, EdwardL. Robinson and their collaborators, atMcDonald Observatory; by Brian Warnerand his colleagues at Capetown, and byJames E. Hesser, Barry M. Lasker andtheir coworkers at the Cerro Tololo In-teramerican Observatory. These studieshave shown that the variable white dwarfsare either single, normal degeneratedwarfs of spectral type DA or else mem-bers of cataclysmic binary systems. Theexcitation mechanisms are almost cer-tainly different in the two cases, and onecan therefore expect to learn about dif-ferent properties of degenerate dwarfsfrom the two types of systems.

The single white-dwarf oscillators,named ZZ Ceti stars after the type-member of the class, undergo luminosityvariations that have been shown bypower-spectrum analyses to involve sev-eral different frequencies excited simul-taneously (see figure 7). The periods ofthese oscillations are of the order of sev-eral hundred to a few thousand seconds,much too long for radial pulsations of thewhite dwarfs. On the other hand, the ZZCeti stars are concentrated in a very nar-

row range of effective temperatures, fromabout 10 000 to 14 000 K; this is preciselywhere the instability strip defined by thewell-known Cepheid variable stars ex-trapolates into the domain of the whitedwarfs. Perhaps the same excitationmechanism that drives the radial pulsa-tions in the Cepheids gives rise to non-radial oscillations in the surface ionizationzones of ZZ Ceti stars. Why the samephysical process should excite two verydifferent types of oscillations is currentlybeing studied by several groups.

To elucidate the nature of the oscilla-tions observed in the ZZ Ceti stars andcataclysmic variables, theoretical calcu-lations of the free, non-radical oscillationspectra have recently been carried out fora number of different, realistic white-dwarf models. The most detailed anduseful theoretical studies to date are thosedone by Yoji Osaki and Carl J. Hansen ofthe Joint Institute of Laboratory Astro-physics, by Anthony Brickhill of Cape-town Observatory, and by WojtekDziembowski of the Copernicus Astron-omy Center in Poland.16 Stellar oscilla-tions in general, and those of degeneratedwarfs in particular, can be grouped intoa number of different classes, of which themost thoroughly studied have been thespheroidal oscillations. For these modes,the pulsation eigenfunction is composedof a spherical harmonic Yim{(),ip) multi-plied by a function of the radius, in closeanalogy with quantum-mechanicalwavefunctions. Two classes of thesemodes appear to play a role in whitedwarfs. In one of these the pressure is thedominant restoring force (the "p-modes"), as in ordinary sound waves.Gravity is the dominant restoring force inthe "g-modes," which have somewhat thecharacter of long ocean waves. Radialpulsations are a special case; they corre-


spond to p-modes with I = 0. The ve-locity distributions for all other modes—the remaining p-modes and all g-modes-have transverse components.

Because the g -modes have the longestperiods of any of the possible oscillationsof white dwarfs, it is generally believedthat the oscillations observed in the ZZCeti stars are of this type. However,theoretical calculations have yet to dem-onstrate the existence of the suspectedCepheid-variable-like driving mechanismfor modes that are compatible with theobserved pulsations.

In contrast to the ZZ Ceti stars, rapidoscillations are not present at all times inthe cataclysmic binaries. These systemsconsist of an ordinary star and a whitedwarf that are so close together thatmatter can spill over from the normal staronto its degenerate companion. Some ofthese types of systems, the dwarf novae,undergo frequently recurring outbursts.In these eruptions, the luminosity of thesystem increases manyfold, risingabruptly and then decaying again in abouta week or ten days. It is during the out-bursts that oscillations have been de-

2 3

TIME (10 2days)

FREQUENCY (mHz)

Oscillations of the rapid variable white dwarf G29-38. Part a shows a portion of the light curveof this star taken during one observing run. (The data are continuous from the right part of the uppergraph to the left of the lower.) Part b shows the power spectrum derived from the full data fromthis run. Some of the prominent frequencies giving rise to the regularities seen in part a are markedwith colored dots and several of the beat frequencies that result from their combinations are markedwith black dots. (Adapted from reference 15). Figure 7

tected in the dwarf novae. The oscilla-tion periods are generally about 10 to 35seconds, and they also appear to besomewhat too long for radial modes. Inaddition, the periods change graduallywith the system luminosity (which seemsincompatible with identifications as radialmodes), and the oscillations appear anddisappear much more rapidly than seemspossible for radial pulsations. Rotatingstars like the white dwarfs in the cata-clysmic variable systems can also sustaintoroidal oscillations. John Papaloizu andJames E. Pringle of the Institute forTheoretical Astronomy in Cambridgehave suggested that these modes, whichmay have periods as short as the rotationperiods of the white dwarfs, may be re-sponsible for the observed oscillations.The mechanism of excitation of the os-cillations in the cataclysmic binary sys-tems appears to be related to processes,such as accretion or thermonuclearburning, that are associated with theoutburst, rather than processes operatingin the surface ionization zones, as in theZZ-Ceti stars. The details, however, areat present completely unknown.

New challengesThe discoveries about degenerate

dwarfs that have occurred within the pastdecade have created new problems andraised new challenges. Observationsfrom space have already brought aboutchanges in our understanding of thesestars, and additional observational dataat the short-wavelength end of the spec-trum will soon become available from theInternational Ultraviolet Explorer satel-lites. Such observations provide essentialdata for questions about the origins of thewhite dwarfs. In addition, because thedegenerate dwarfs are low-luminositystars, detailed observations of the faintestones can be carried out only from abovethe background of scattered light due tothe terrestrial atmosphere. This willbecome possible with the advent of theSpace Telescope to be orbited aboard theSpace Shuttle early in the 1980's. Suchobservations will permit investigations ofthe very faintest—and thus oldest andcoolest—of the degenerate dwarfs. Someof these stars are expected to date back tothe earliest periods in the evolution of ourGalaxy and may provide new insights intoevents that happened then. In addition,observations of the faintest degeneratedwarfs will sharpen the comparison withtheoretical cooling models and may helpto constrain the internal composition ofthese stars. This would in turn provideinformation about the processes takingplace in the final phases of stellar evolu-tion preceding the white dwarf stage.

Research in the theory of degeneratedwarfs is currently confronted with a va-riety of problems of considerable interest.For example, we do not at present un-derstand the reasons even for the exis-tence of the non-DA (hydrogen-deficient)


What suppresses fuel cell output more than any-thing else? The oxygen electrode. Mysteriously, it re-fuses to develop the electrical potential that ther-modynamics says is possible.

In the H,-O2 fuel cell, for example, theory prom-ises 1.23 volts. Yet, reality produces only 1.06.

Many explanations have been proposed to accountfor this anomaly. But none have withstood scientificscrutiny. Here at the General Motors Research Lab-

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ oratories, however,m m ^ ^ ^ ^ M H S I ^ ^ ^ ^ ^ ^ O we've devised a hy-

pothesis that we'reconvinced will holdup. It stems fromour fundamentalstudies of theelectrochemistryof oxygen.

Our reason-Swelling (area within circular impression} caused U l g ' 1 h e O9by oxygen dissolving into platinum .

trode is catalyzedwith platinum, whose surface adsorbs oxygen. But theadsorbed oxygen continuously dissolves into the Ptlattice, limiting adsorption to about 30% of the Pt sur-face. Consequently, minute corrosion cells form in theremaining areas, and their combined potential bucks the1.23 volts.

To test the logic, we charged a pure platinumdiaphragm with oxygen. The oxygen did indeed ad-sorb on the surface and dissolve into the metal, asx-ray analysis and the expanded diaphragm (photo-graph) later showed.

When it became saturated, the diaphragm couldno longer dissolve any oxygen. This allowed adsorp-tion to spread over the entire Pt surface, thus prevent-ing corrosion. Now unopposed, the potential betweenthe platinum and a reference electrode climbed to

Effect of Oxygen Saturationon Platinum-Catalyzed Oxygen Electrode

1.227 volts, the highest ever reported for oxide-freeplatinum.

Granted, understanding the problem doesn'tnecessarily solve it. But. . . first things first.

We currently have openings for Ph.D.s in engineeringor the physical, mathematical, or biomedical sciences. Ifinterested, please send your resume to: GMR Personnel,Dept. 114. An Equal Opportunity Employer.

WhyO2electrodeslimit fuelcell voltage.

1.0

bo

0.5 -

1

0 50

mm^m Platinum Saturated with 0• « Unsaturated Platinum

1 1100 150

HOURS

I

200I

250

1.23

1.2271.06

VVV

300

General MotorsResearch LaboratoriesWarren, Michigan 48090


highspeedspectroscopy

ADCleCroy3511

13 BITAOC

^ HI

DISCRIM

.? 5 1

VOLTS

INPUT

MODE

GATE

Model 3511 is a high-speed, high-resolu-tion, ultra-linearnuclear spectroscopyADC, offeringCAMAC compati-bility, making it idealfor multi-parameter/multi-user applica-tions. Outstandingspecifications in-clude:

• 8K resolution withprogrammable con-version gain.

1 <10 fisec conversiontime for 8K channels.

• Better than ± 0.025%integral non-linearityover upper 99.5% of8K range.

• Better than ± 0.75%differential non-linearity over upper99.5% of 8K range.

• Exclusive LEDreadout of upper andlower discriminatorvoltage levels

» High-density single-width CAMAC modulepermits up to 23 ADCchannels in a singleCAMAC crate.

> CAMAC design offerssimplified and flexibleinterfacing to mini-computers or micro-processors forautomatic control ofdata acquisition/processing and ADCparameters.

1 Any number of 3511'scan be cascaded tovirtually eliminatedeadtime.

BUSYOUT Write or call for full

details on 3511 andfree reprint on theCAMAC interfaceconcept.

ifiCroyRESEARCH SYSTEMS »

LeCroy Research Systems, Inc.; 700 South Main Street;Spring Valley, N.Y. 10977 (914) 425-2000; TWX710-577-2832. International Offices: Geneva, Switzerland;Heidelberg, W. Germany; Paris, France; Wheatley, Oxford,England.

Booths #38, #39, #40Annual Physics Show

Circle No. 17 on Reader Service Card


white dwarfs. Related to this is theproblem of the spectral evolution of thesestars: How do gravitational settling,convection, accretion, and perhaps otherprocesses interact to produce the ob-served types of white dwarf spectra?Another class of problems concerns thephysical properties of matter under ex-treme conditions: How can these prop-erties be calculated for partially ionizedplasmas in the complex physical regimeintermediate between rarefied gases andnear-solid densities? The answer to thisquestion, in particular, is essential notonly for understanding the nature ofwhite-dwarf surface layers and the exci-tation mechanisms for white-dwarf os-cillations, but also for analyses of variouslayer-produced plasmas and certain typesof thermonuclear fusion experiments.Yet additional problems concern the ef-fects of rotation, the origins of white-dwarf magnetic fields, and the role ofphase separation during core crystalliza-tion in degenerate dwarfs.

Finally, a subject about which I havesaid little in this article, but which in-creasingly involves knowledge of thestructure and properties of degeneratedwarfs, is that of the cataclysmic vari-ables. Despite recent successes in theo-retical modelling of the common novaoutburst,17 severe problems remain. Inparticular, why do current thermonuclearexplosion models fail to "turn off," asnovae are observed to do, shortly after theinitial outburst? Furthermore, in thecase of the dwarf novae, the mechanism ofthe outburst itself is rather controversial:Is it an accretion event, as suggested byGeoffrey Bath and his colleagues in En-gland, or is it a thermonuclear out-burst—a scaled-down version of a nova?Related to this are a whole host of ques-tions pertaining to accretion disks in ca-taclysmic binaries, including those relat-ing to the structure and properties of thedisks themselves and to the nature of thedisk-white-dwarf interface. In addition,within the past few years a completelynew class of cataclysmic variables hasbeen discovered—the AM Herculis sys-tems. These appear to resemble thedwarf novae, except that the degeneratedwarfs are believed to possess magneticfields of order 108 gauss, which preventthe formation of accretion disks in thesesystems. The study of these systems hasevidently just begun.

A brief overview of white-dwarf re-search such as I have given here cannothope to do justice to the many fascinatingproblems that arise in this area of astro-physical investigation. The variety ofphysical conditions present in these starsand their intrinsic faintness combine tomake observational and theoretical in-vestigations both challenging and re-warding. We thus anticipate the con-tinuing development of interest in thestudy of degenerate dwarfs as investiga-tions extend the frontiers of research.

It is a pleasure to thank Malcolm Savedoff,David Douglass, Sheridan Simon, FranqoisWesemael and Mark Bocko for their helpfulcomments on an earlier version of this article.The author is also grateful to the NationalScience Foundation and The National Aero-nautics and Space Administration for theircontinuing grant support of white-dwarf re-search at the University of Rochester.

References1. General background material on white-

dwarf stars may be found in the followingsources: S. Chandrasekhar, An Intro-duction on the Study of Stellar Structure,Univ. of Chicago, Chicago (1939); reprintedby Dover, New York (1957); E. Schatzman,White Dwarfs, North-Holland, Amster-dam (1958); M. Schwarzschild, Structureand Evolution of the Stars, PrincetonU.P., Princeton (1958), chapter 7; J. L.Greenstein, in Stars and Stellar Systems,vol. 6: Stellar Atmospheres (J. L.Greenstein, ed.), Univ. of Chicago, Chicago(1960), page 676; V. Weidemann, Ann. Rev.Astron. Astrophys. 6,351 (1968); J. P. Os-triker, Ann. Rev. Astron. Astrophys. 9,353(1971), and H. M. Van Horn, in The Un-certainty Principle and the Foundationsof Quantum Mechanics (W. C. Price and

• S. S. Chissick, eds.), Wiley, London (1977),page 441.

2. I. W. Lindenblad, Astron. J. 75, 841(1970).

3. L. Mestel, Mon. Not. R. Astron. Soc. 112,583 (1952).

4. W. Cash, S. Bowyer, M. Lampton, Astro-phys. J. 221, L87 (1978).

5. J. L. Greenstein, J. B. Oke, H. L. Shipman, •Astrophys. J. 169, 563 (1971).

6. M. P. Savedoff, H. M. Van Horn, F.Wesemael, L. H. Auer, T. P. Snow, D. G.York, Astrophys. J. 207, L45 (1976).

7. G. Fontaine, PhD thesis, University ofRochester (1973); G. Fontaine, H. C. Gra-boske, H. M. Van Horn, Astrophys. J.Suppl. 35, 293 (1977).

8. E. E. Salpeter, Astrophys. J. 134, 669(1961); T. Hamada, E. E. Salpeter, Astro-phys. J. 134,683(1961).

9. S. G. Brush, H. L. Sahlin, E. Teller, J.Chem. Phys. 45, 2102 (1966).

10. L. Mestel, M. Ruderman, Mon. Not. R.Astron. Soc. 136, 27 (1967); H. M. VanHorn, Astrophys. J. 151, 227 (1968).

11. D. Q. Lamb, PhD thesis, University ofRochester (1974); D. Q. Lamb, H. M. VanHorn, Astrophys. J. 200, 306 (1975); G.Shaviv, A. Kovetz, Astron. Astrophys. 51,383 (1976).

12. J. C. Kemp, J. Swedlund, J. D. Landstreet,J. R. P. Angel, Astrophys. J. 161, L77(1970).

13. J. R. P. Angel, Astrophys. J. 216,1 (1977);also Ann. Rev. Astron. Astrophys. 16,(1978), in press.

14. A. U. Landolt, Astrophys. J. 153, 151(1968).

15. J. T. McGraw, PhD thesis, University ofTexas (Austin), (1977).

16. Y. Osaki, C. J. Hansen, Astrophys. J. 185,277 (1973); A. J. Brickhill, Mon. Not. R.Astron. Soc. 170, 405 (1975); W. Dziem-bowski, Acta Astron. 27,1 (1977).

17. J. S. Gallagher, S. G. Starrfield, Ann. Rev.Astron. Astroph'ys. 16, (1978), in press. D

Documents

The physics of white dwarfs - Semantic Scholar · The physics of white dwarfs Increasingly sensitive and detailed astronomical observations coupled with calculations of the properties