Axions and the cooling of white dwarf stars

0.15 MB·English

by  J. Isern

Draftversion January12,2009
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
AXIONS AND THE COOLING OF WHITE DWARF STARS
J. Isern1,2, E. Garc´ıa–Berro3,2, S. Torres3,2, S. Catala´n1,2
Draft version January 12, 2009
ABSTRACT
White dwarfs are the end product of the lifes of intermediate- and low-mass stars and their evo-
lution is described as a simple cooling process. Recently, it has been possible to determine with
an unprecedented precision their luminosity function, that is, the number of stars per unit volume
and luminosity interval. We show here that the shape of the bright branch of this function is only
9
sensitive to the averaged cooling rate of white dwarfs and we propose to use this property to check
0
0 the possible existence of axions, a proposed but not yet detected weakly interacting particle. Our
2 resultsindicatethattheinclusionoftheemissionofaxionsintheevolutionarymodelsofwhitedwarfs
noticeably improves the agreement between the theoretical calculations and the observational white
n
dwarf luminosity function. The best fit is obtained for macos2β ≈ 5 meV, where ma is the mass of
a
the axion and cos2β is a free parameter. We also show that values larger than 10 meV are clearly
J
excluded. The existing theoretical and observational uncertainties do not yet allow the confirmation
2
of the existence of axions, but our results clearly show that if their mass is of the order of few meV,
1
the white dwarf luminosity function is sensitive enough to detect their existence.

] Subject headings: elementary particles — stars: luminosity function, mass function — white dwarfs
h
p
- 1. INTRODUCTION aresurroundedbyathinheliumlayer,withamassrang-
ro One solution to the strong CP problem of quantum ing from10−2 to 10−4 M⊙ which, inturn, is surrounded
ast ccheiro&moQduyinnanm1ic9s77isa,th19e77Pbe)c.ceiT-Qhuisinsnymsymmemtreytriys s(pPoenc-- btwyeaenn1e0v−en4 atnhdin1n0e−r1l5ayMer⊙,oaflthhyodurgohgeanbowuitth25a%moafswshbitee-
[ taneously broken at an energy scale that gives rise to dwarfsdonothavehydrogenatmospheres. Whitedwarfs
the formation of a light pseudo-scalar particle named displaying hydrogen in their spectra are called DA and
2 the remaining ones are known as non-DAs. Because of
the “axion” (Weinberg 1978; Wilczek 1978). This scale
v
thedifferentopacities,DAwhitedwarfscoolmoreslowly
7 of energies is not defined by the theory but it has to
than the non-DA ones.
0 be well above the electroweak scale to ensure that the
The standard theory of white dwarf cooling can be
8 coupling between axions and matter is weak enough to
summarized as follows (Isern et al. 1998). When the lu-
2 account for the lack of a positive detection up to now.
minosity is large, M < 8, the evolution is dominated
. The mass of axions and the energy scale are related by bol
06 ma ≈0.6(107GeV/fa)eV.Astrophysicalandcosmologi- btiyesnceoumtreinforoemmiossuironp.ooInr tkhnioswplhedagseetohfethmeaiinnituinaclecrotnadini--
calarguments(Raffelt2007)havebeenusedtoconstrain
8
0 this mass to the range 10−2 eV ≥ ma ≥ 10−4 eV. For tions. Fortunately, it has been shown that all the ini-
tial thermal structures converge toward a unique one
: this mass range,axionscan escape fromstars andact as
v (D’Antonna & Mazzitelli 1989). For smaller luminosi-
a sink of energy.
i ties, 8 ≤ M ≤ 12, the main source of energy is of
X White dwarfs are the final evolutionary phase of low- bol
gravothermalorigin. In this phase, the Coulomb plasma
r and intermediate-mass stars (M ≤ 10±2 M⊙). Since couplingparameterisnotlargeandthecoolingcanbeac-
a they are degenerate objects, they cannot obtain energy
curatelydescribed. Furthermore,theenergyfluxthrough
from thermonuclear reactions and their evolution can
the envelope is controlled by a thick nondegenerate or
be described just as a gravothermal process of cool-
partially degenerate layer with an opacity dominated by
ing. Therefore, if axions exist, the properties of these
hydrogen,whenpresent,andhelium,anditisweaklyde-
starswouldbenoticeablyperturbed. Furthermore,white
pendent on the metal content since metals sink towards
dwarfs have a relatively simple structure: a degenerate
the base of the envelope by gravitationally induced dif-
core that contains the bulk of the mass and acts as an
fusion. Below these luminosities, white dwarfs evolve
energyreservoirandapartiallydegenerateenvelopethat
into a region of densities and temperatures where the
controls the energy outflow. The vast majority of white
plasma crystallizes. When this happens, two additional
dwarfs have masses in the range 0.4≤M/M⊙ ≤1.05 —
sources of energy appear. The first one is the release
although these figures are still uncertain — and have a
of latent heat during crystallization. The second one is
coremadeofamixtureofcarbonandoxygen. Allofthem
the releaseofgravitationalenergyinducedbyphasesep-
Electronicaddress: [email protected] aration of the different chemical species (Garc´ıa-Berro
1Institut de Ci`encies de l’Espai, CSIC, Facultat de Ci`encies, et al. 1988a, 1988b; Isern et al. 1997, 2000). When the
CampusUAB,08193Bellaterra,Spain bulk of the star is solid the white dwarf enters into the
2Institutd’EstudisEspacialsdeCatalunya,c/GranCapita`2–4,
Debye cooling phase and the only important source of
08034Barcelona,Spain
3Departament de F´ısica Aplicada, Escola Polit`ecnica Superior energy comes from the compression of the outer layers.
de Castelldefels, Universitat Polit`ecnica de Catalunya, Avda. del These late phases of cooling are not yet well understood
CanalOl´ımpics/n,08860Castelldefels,Spain (Isern et al. 1998).
2
2. THEWHITE DWARFLUMINOSITYFUNCTION
One way to test the evolutionary properties of white
dwarfs is using their luminosity function, which is de-
finedasthenumberofwhitedwarfsperunitvolumeand
magnitude. The first luminosity function was derived
four decades ago (Weidemann 1968) and since then it
has been noticeably improved. The most recent deter-
minations use data from the Sloan Digital Sky Survey
(SDSS). The first of these (Harris et al. 2006) was built
from a sample of 6000 DA and non-DA white dwarfs
with accurate photometry and proper motions culled
from the SDSS Data Release 3 and the USNO-B cat-
alogue, whereas the second one (DeGennaro et al. 2008)
was constructedfroma sample of 3528spectroscopically
identified DA white dwarfs from the SDSS Data Release
4 (see Fig. 1). The monotonic behavior of this function
clearly proves that the evolution of white dwarfs is just
acoolingprocess. Thesharpcutoffatlowluminositiesis
a consequence of the finite age of the Galactic disk. The
luminosityfunctionofwhite dwarfscanbe computedas:
Fig. 1.— Luminosity functions of white dwarfs. Filled squares
Mu correspond to the luminosity function of white dwarfs (DA and
n(Mbol)=Z φ(M)ψ(TG−tcool−tps)τcool dM, (1) non-DAtypes)obtainedusingthereducedpropermotionmethod
Ml (Harrisetal.2006)andopensquarestothatobtainedusingspec-
troscopicallyidentifiedDA whitedwarfs(DeGennaroetal.2008).
where Mbol is the bolometric magnitude, M is the mass Thesolidlinesarethetheoreticalluminosityfunctionsobtainedfor
oftheparentstar—forconvenienceallwhitedwarfsare different ages of the Galaxy — from left to right: 10, 11, 12 and
labeledwiththemassofitsprogenitor—T istheageof 13 Gyr — and a constant star formation rate. The dotted lines
G are the same for an exponentially decreasing star formation rate
thepopulationunderstudy,tcool isthetimethatawhite — ψ(t)=exp(−t/τ) withτ =0.5, 3 and 5 Gyr — and the same
dwarfwithaprogenitorofmassM takestocooldownto ageoftheGalaxy,11Gyr. Alltheluminosityfunctionshavebeen
a bolometric magnitude M , τ is the characteristic normalizedtothesameobservational datapoint.
bol cool
cooling time of the white dwarf, τ = dt /dM ,
cool cool bol
and t is the lifetime of the parent star. In equation
ps
(1)M is the maximummassofwhite dwarfprogenitors
and Mu is the mass of the progenitor that satisfies the Mu
l n∝hτ i φ(M)ψ(T −t −t )dM. (2)
condition TG −tcool −tps = 0, i.e., the minimum mass cool ZM cool ps
l
of a star able to produce a white dwarf of the required
luminosity. Theremainingquantitiesaretheinitialmass Restrictingourselvestobrightwhitedwarfs—namely,
function, φ (here we have used Salpeter’s law), and the those forwhichtcool is small— the lowerlimit ofthe in-
star formation rate, ψ, which is not known a priori and tegralissatisfiedbylow-massstarsand,asaconsequence
depends on the population under study. of the strong dependence of the main-sequence lifetimes
Sinceneitherthestarformationratenorthetotalden- with mass, it takes a value that is almost independent
sity of white dwarfs are known, theoretical luminosity of the luminosity under consideration. Therefore, if ψ
functions are normalized to a given observational bin, is a well-behaved function and TG is large enough, the
usually the one with the smallest error bar. Figure 1 integralis notsensitiveto the luminosity, its value is ab-
displays the observed luminosity functions and several sorbed by the normalization procedure, and the shape
luminosity functions obtainedwith the sameDA cooling of the luminosity function only depends on the averaged
sequences (Salaris et al. 2000) and different star forma- physical properties of white dwarfs. It is important to
tion rates and ages of the Galaxy. The cooling models mention here that the initial-final mass relationship en-
assume a nonhomogeneous distribution of carbon and tersasaweightintothecalculationofthisaverage. Nev-
oxygen in the core (Salaris et al. 1997), a pure helium erheless,sinceonlythosefunctionsabletoprovideagood
layer of 10−2M∗ and on top of it a pure hydrogen layer fit to the mass distribution of white dwarfs are accept-
of 10−4M∗, where M∗ is the mass of the white dwarf. able,itsinfluenceontheshapeofthebrightbranchofthe
Since a relationship connecting the mass of the white luminosity function is minor. Here we use this property
dwarf and the mass of its progenitor is also necessary ofthewhitedwarfluminosityfunction,togetherwiththe
we have adopted the one that best reproduces the mass recentlyobtainedhigh-precisionobservationalluminosity
distribution of white dwarfs (Catal´an et al. 2008). function, to study the influence of the emission of ax-
An interesting feature of Figure 1 is that the bright ions,andto checkwhichmassofthe axionis compatible
partofthe white dwarfluminosity function —that with withobservations. Theideaofusingthecoolingtimesof
bolometric magnitude M < 13 — is almost indepen- white dwarfs to constrain the properties of axions is not
bol
dent of the assumed star formation rate. This can be new (Raffelt 1986), but the crudeness of the theoretical
explained with simple arguments. Since the characteris- modelsandoftheobservationaldatapreventedadefinite
ticcoolingtimeisnotstronglydependentonthemassof conclusion and just a loose upper bound was obtained,
the white dwarf, equation (1) can be written as ma <30 meV.
3
Fig. 2.—Energylossesfora0.61M⊙ whitedwarfasafunction Fig. 3.— White dwarf luminosity functions for different values
ofthebolometricmagnitude. Thedashedlinesrepresenttheaxion of the axionmass. Theluminosityfunctions have been computed
luminosityfordifferentvaluesofmacos2β —fromtoptobottom: assumingmacos2β=0(solidline),5(dashedline)and10(dotted
macos2β =10, 5, 1, 0.1 and 0.01 meV. The thick solid line is line)meV.
thephotonluminosity,whilethethinsolidlineshowstheneutrino
Figure 2 shows that in the regionM ∼10 the axion
luminosity. bol
luminosity can be comparable with the photon and neu-
3. AXIONSANDTHEWHITEDWARFLUMINOSITY trinoones,dependingontheadoptedaxionmass. Italso
FUNCTION showsthat the regionaroundM ∼12 providesa solid
bol
Axions can couple to photons, electrons and nucle- anchor point to normalize the luminosity function be-
ons with a strength that depends on the specific im- causetheretheobservationaldatahavereasonablysmall
plementation of the Peccei-Quinn mechanism. The two errorbars,modelsarereliable,neutrinosarenotrelevant
mostcommonimplementationsaretheKVSZ(Kim1979; and axions, if they exist, are not dominant.
Shifman et al. 1980) and the DFSZ models (Dine et al.
1981;Zhitnitskii1980). Inthe firstcaseaxionscoupleto 4. RESULTSANDCONCLUSIONS
hadrons and photons, while in the second they also cou- Figure3displaysseveralluminosityfunctionsobtained
ple to charged leptons. For the temperatures and densi- using different axion masses, adopting a constant star
ties of the white dwarfs under consideration,only DFSZ formation rate and an age of the Galactic disk of 11
axions are relevant and in this case they can be emit- Gyr. As already mentioned, it is important to real-
ted by Compton, pair annihilation and bremsstrahlung ize that the bright branch of the luminosity function is
processes, but only the last mechanism turns out to not sensitive to these last assumptions. All the lumi-
be important. Figure 2 shows the energy losses for nosity functions have been normalized to the luminosity
an otherwise typical 0.61M⊙ white dwarf as a func- bin at log(L/L⊙) ≃ −3 or, equivalently, Mbol ≃ 12.2.
tion of the bolometric magnitude. The dashed lines The best-fit model — namely that which minimizes the
represent the axion luminosity for different values of χ2 test in the region −1 > log(L/L⊙) > −3 (that is,
macos2β. The axion emission rate (in erg g−1 s−1) 7.2 < Mbol < 12.2), which is the region where both the
has been computed (Nakagawa et al. 1987, 1988) as observationaldataandthetheoreticalmodelsarereliable
εa = 1.08 × 1023α(Z2/A)T74F, where F is a function — is obtainedfor macos2β ≈5 meV and solutions with
of the temperature and the density which takes into ac- macos2β > 10 meV are clearly excluded. Figure 4 dis-
count the properties of the plasma, and α = ga2e/4π is plays the behavior of χ2 as a function of the mass of the
related to the axion-electron coupling constant gae = axion in our fiducial case (solid line) and in the case in
2.8 × 10−11macos2β/1 eV. Since the core is almost which we use the initial-final mass relationship of Wood
isothermal it turns out that La ∝ T4 in the region in (1992), which is marginally compatible with the white
whichaxionsarethe dominantsink ofenergy. The thick dwarf mass distribution (Catal´an et al. 2008). In both
solidline representsthe photonluminosity(Salaris et al. cases the behavior of the luminosity function is similar
a
2000). For the region of interest Lν ∝T , with a∼2.6, and gives similar values for the mass of the axion once
although this value changes as the white dwarf cools one takes into account the present uncertainties. It is
down. The thin solid line represents the neutrino lu- also important to notice that the largest contribution to
minosity, whichscales asLν ∝T8 andis alsodominated thelackofaccuracycomesfromthe brightestbinsofthe
by the plasma and bremsstrahlung processes (Itoh et al. luminosity function, which have large error bars.
1996). Therefore, since the temperature dependence of This result is completely compatible with the previ-
the different energy-loss processes is not the same, the ously existing constraints (Raffelt 2007). Furthermore,
luminosity function allows to disentangle the different these values are also compatible with the bounds im-
contributions. posed by the drift of the pulsational period of the ZZ
4
to test the existence of weakly interacting massive par-
ticles because it is only sensitive to the averagedcooling
rate of white dwarfs and not to the details of the star
formation rate or the initial mass function, as shown in
§2. Moreover,ourresultsareproblablythefirstevidence
that the shape of the white dwarf luminosity function
couldbe affected by the emissionof axionsandthat this
change can be measured. If this is indeed the case the
mass of the axion would be of about 5 meV. In addi-
tion, we have derived an upper bound to the mass of
the axions of 10 meV, which is compatible with other
recent determinations (Bischoff-Kim et al. 2008). It is
worth mentioning that this result is relevant for other
researchfields and, in particular, for cosmology. Specifi-
cally, assuming cos2β =1, the contribution of axions to
darkmatterwouldbeoftheorderof0.2%(Raffelt2007).
In view of this, we consider it of the largest importance
to improve the observationaldetermination of the white
dwarfluminosityfunction, especiallyinthe regionofthe
hottest white dwarfs (see Fig. 3). Thus, the extension
of the SDSS is of the maximum interest not only for as-
Fig. 4.—Valueofχ2asafunctionofthemassoftheaxionforthe tronomers and cosmologists, but also for particle physi-
case in which the initial–final mass relationship of Catala´netal. cists. However,notonlyobservationaleffortsareneeded,
(2008)(solidline)andthatofWood(1992)(dottedline)areused. since in order to obtain a reliable determination of the
mass of the axion it is also important to decrease the
Ceti star G117−B15A (Isern et al. 1992; C´orsico et al.
uncertainties in the plasmon neutrino emission rates at
2001;Bischoff-Kim et al.2008). Itisworthwhiletomen-
the relevant temperature range. Furthermore, it would
tion here that axions with ma ≈ 5 meV would change
be also convenient to intensify the study of the drift of
the expected period drift of variable DB white dwarfs
the pulsational periods of variable white dwarfs in order
— which have values between P˙ ∼ 10−13 and 10−14
to obtain additional independent evidence.
s s−1 (C´orsico & Althaus 2004) — by a factor 2, the ex-
act value depending on the adopted temperature of the
stellar core and that the detection of such a drift would
provide a strong additional argument in favor of the ex-
istence of axions. Part of this work was supported by the MEC grants
The results presented here are not a definite proof of AYA05-08013-C03-01and02,andESP2007-61593bythe
the existence of axions, since there are still some ob- EuropeanUnionFEDERfunds andbythe AGAUR.We
servational and theoretical uncertainties. However, the thankM.Salarisforprovidinguswiththeneutrinoemis-
calculations reported here show that the hot branch of sion files corresponding to the models of Salaris et al.
the white dwarf luminosity function is a powerful tool (2000).
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