The formation of high-field magnetic white dwarfs from common envelopes

0.29 MB·English

by  J. Nordhaus

The formation of high-field magnetic white dwarfs
from common envelopes
J. Nordhaus∗, S. Wellons∗, D. S. Spiegel∗, B. D. Metzger∗, & E. G. Blackman†
∗DepartmentofAstrophysicalSciences,PrincetonUniversity,Princeton,NJ08544,U.S.A.,and†DepartmentofPhysicsandAstronomy,UniversityofRochester,Rochester,
NY14627,U.S.A
SubmittedtoProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica
Theoriginofhighly-magnetizedwhitedwarfshasremainedamystery ion is a low-mass main-sequence star) contains a HFMWD
since their initial discovery. Recent observations indicate that the [11, 12, 13]. If the magnetic field strengths of white dwarfs
1
formation of high-field magnetic white dwarfs is intimately related were independent of binary interactions, then the observed
1
to strong binary interactions during post-main-sequence phases of distributionofisolatedWDsshouldbesimilartothoseinde-
0 stellarevolution. Ifalow-masscompanion,suchasaplanet,brown
tached binaries. In particular, within 20 pc, there are 109
2 dwarf, or low-mass star is engulfed by a post-main-sequence giant,
knownWDs(21ofwhichhaveanon-degeneratecompanion),
gravitationaltorquesintheenvelopeofthegiantleadtoareduction
n and SDSS has identified 149 HFMWDs (none of which has
ofthecompanion’sorbit. Sufficientlylow-masscompanionsin-spiral
a until they are shredded by the strong gravitational tides near the a non-degenerate companion). Assuming binomial statistics,
J white dwarf core. Subsequent formation of a super-Eddington ac- the maximum probability of obtaining samples at least this
2 cretion disk from the disrupted companion inside a common enve- differentfromthesameunderlyingpopulationis5.7×10−10,
1 lope can dramatically amplify magnetic fields via a dynamo. Here, suggestingatthe6.2-σ levelthatthetwopopulationsaredif-
we show that these disk-generated fields are sufficiently strong to ferent (for details on this kind of calculation, see Appendix
] explain the observed range of magnetic field strengths for isolated, B of [14]). Furthermore, SDSS identified 1253 WD+M-dwarf
R high-field magnetic white dwarfs. A higher-mass binary analogue binaries (none of which are magnetic). As was previously
mayalsocontributetotheoriginofmagnetarfields.
S pointed out, this suggests that the presence or absence of
. binarity is crucial in influencing whether a HFMWD results
h WhiteDwarfs | Magnetars | Magnetohydrodynamics [15]. These results initially seem to indicate that HFMWDs
p preferentially form when isolated. However, unless there is a
- Abbreviations: HFMWD,high-fieldmagneticwhitedwarf;SDSS,SloanDigitalSky mechanismbywhichverydistantcompanionspreventthefor-
o
Survey;MG,Megagauss;WD,whitedwarf;CV,cataclysmicvariable;CE,common mationofastrongmagneticfield,amorenaturalexplanation
r
t envelope;MS,mainsequence;AGB,asymptoticgiantbranch;BD,browndwarf is that highly-magnetized white dwarfs became that way by
s
engulfing (and removing) their companions.
a
In the binary scenario, the progenitors of HFMWDs are
[ Introduction
those systems that undergo a common envelope (CE) phase
2 A significant fraction of isolated white dwarfs form with during post-MS evolution. In particular, magnetic CVs may
v strong magnetic fields. These high-field magnetic white be the progeny of common envelope systems that almost
9 dwarfs (HFMWD), the majority of which were discovered mergebutejecttheenvelopeandproduceaclosebinary. Sub-
2 via the Sloan Digital Sky Survey (SDSS; [1, 2, 3]), comprise sequentorbitalreductionviagravitationalradiationandtidal
5 ∼10% of all isolated white dwarfs [4]. Surface field strengths forcesturndetachedsystemsintothosewhichundergoRoche-
1 range from a few to slightly less than a thousand megagauss lobeoverflow. ForCEsinwhichthecompanionisnotmassive
. (MG)whilethebulkoftheisolatedWDpopulationhavemea- enoughtoejecttheenvelopeandleaveatightpost-CEbinary,
0
sured weak fields or non-detection upper limits of typically thecompanionisexpectedtomergewiththecore. Itwassug-
1 (cid:46)104−105 G [5, 6, 7]. gested that these systems may be the progenitors of isolated
0
In WD-binary systems, the companion’s surface is an HFMWDs [15].
1
equipotential. In tight binaries, this equipotential extends In this paper, we calculate the magnetic fields generated
:
v towardtheWD,leadingtomasstransferfromthecompanion during the common envelope phase. We focus on low-mass
i ontotheWD,aprocesscalled“Roche-lobeoverflow.” Among companions embedded in the envelope of a post-MS giant.
X systems such as these, a possibly even larger fraction of the During in-spiral, the companion transfers orbital energy and
r WDprimariesarehighlymagnetic(i.e.,∼25%ofcataclysmic angular momentum, resulting in differential rotation inside
a variables;[8]). Magneticcataclysmicvariables(CVs)aregen- the CE. Coupled with convection, a transient α−Ω dynamo
erallydividedintotwoclasses: AMHerculis(polars)andDQ amplifies the magnetic field at the interface between the con-
Herculis (intermediate polars). For reviews on AM Her and vectiveandradiativezoneswherethestrongestshearisavail-
DQ Her systems see [9] and [10]. Generally, AM Her systems
are those in which both components of the binary are syn-
chronously rotating at the orbital period. In this scenario,
the formation of an accretion disk is prevented and material ∗[email protected]
Reserved for Publication Footnotes
is funneled onto the WD via the magnetosphere. Polars have
strongmagneticfields(∼107−108 G),copiousX-rayemission
andstablepulsationsintheirlightcurves. Ontheotherhand,
DQ Her systems (intermediate polars) are not synchronously
lockedandhaveweakermagneticfieldsthantheirpolarcoun-
terparts, often by an order of magnitude or more. For these
systems, an accretion disk forms from which the WD is spun
up.
Remarkably, not a single observed close, detached binary
system (in which the primary is a WD and the compan-
www.pnas.org/cgi/doi/10.1073/pnas.0709640104 PNAS IssueDate Volume IssueNumber 1–7
able. Thefieldsproducedfromthisinterfacedynamoaretran- companion around the proto-WD. The disk-generated fields
sientandunlikelytoreachthewhitedwarfsurfacewithsuffi- naturally accrete onto the WD surface and are sufficiently
cient strength to explain HFMWD observations. If, however, strong to match observations.
thecompaniontidallydisrupts,theresultantsuper-Eddington
accretiondiskcanamplifymagneticfieldsviaadiskdynamo.
The disk-generated fields are strong (∼ 101 −103 MG) and
accretion provides a natural mechanism with which to trans- EnvelopeDynamo.Theshearprofileinagiantstarthatispro-
port the fields to the WD surface. For a range of disrupted- duced by a low-mass companion’s CE-induced in-spiral leads
companion masses, the fields generated in the disk are suffi- toadynamointhepresenceoftheturbulentconvectivezone.
cient to explain the range of observed HFMWDs. The field primarily amplifies at R = Rconv, the interface be-
tweentheconvectiveandradiativeregions. Theback-reaction
of field-amplification on the shear is included with differen-
tial rotation and rotation depleting via loss of Poynting flux
Common Envelope Evolution and turbulent dissipation. For the precise equations solved,
Common envelopes are often invoked to explain short period generic features of envelope dynamos, and a pictorial repre-
systems in which one component of the binary is a compact sentation of the dynamo geometry in isolated stellar and CE
star [16, 17, 18]. The immersion of a companion in a CE settings see [28]. Here we calculate an upper limit to the
with a post-main sequence star can occur via direct engulf- magnetic field that may be generated by this mechanism, by
ment or through orbital decay due to tidal dissipation in the taking the shear profile in the stellar envelope to be Keple-
giant’s envelope [19, 20, 21]. Once engulfed, the companion rian (which produces stronger shear than could possibly be
in-spirals due to hydrodynamic drag until it either survives maintained in a spherical-hydrostatic star).
(ejects the envelope leaving a post-CE close binary) or is de- Before presenting details of these calculations, we note a
stroyed. While CEs can form for various mass-ratio binaries, few generic features of this type of scenario that might make
we focus on low-mass stellar and substellar companions such the envelope dynamo an unlikely explanation of HFMWDs,
that the remnant system is expected to be an isolated white irrespective of details. There are a few reasons why a field
dwarf. Since low-mass companions do not release enough or- generated at the radiative-convective boundary may not be
bitalenergytoejecttheenvelope,astheorbitalseparationis able to produce strongly magnetized material in the vicinity
reduced, the differential gravitational force due to the proto- oftheWD.First,asshownbelow,arobustupperlimittothe
WD tidally shreds the companion. The disrupted companion magnitude of the envelope dynamo generated field is a few
then forms a disk inside the CE that subsequently accretes times 104 G. To explain megagauss fields at the WD surface,
onto the proto-WD [22]. For more detail on the onset and this requires an inwardly increasing gradient of B (e.g. via
dynamics of the CE phase for post-MS giants and low-mass field amplification from flux freezing), which might lead the
companions (i.e. low, mass-ratio binaries), see [18] and [19]. highest-fieldregionstorisebuoyantly,thereforeneverreaching
the WD surface. Second, since the envelope is transient (op-
erates for ∼100 years) and the typical AGB lifetime is ∼105
years, the envelope would need to be ejected at the time of
Amplification of Magnetic Fields
formation of a HFMWD. Without fine tuning, this results in
Asaconsequenceofcommonenvelopeevolution,thetransfer atight, post-CEbinarywithanunder-massivewhitedwarf–
oforbitalenergyandangularmomentumduringin-spiralgen- the exact opposite of the observed HFMWDs. Finally, even
erates strong shear. Coupled with convection, shear leads to if the field could penetrate to the WD surface, some mass
large-scalemagneticfieldamplificationviaanα−Ωdynamo. would need to remain and/or fallback during envelope ejec-
During in-spiral, the free energy in differential rotation avail- tion to anchor the field. A further subtlety is that the trans-
able to the dynamo is proportional to the companion mass. port of magnetic flux into the radiative layer is likely to in-
In general, the more massive the companion, the more free volve not merely isotropic diffusion but anisotropic diffusion,
energyindifferentialrotation[23]. Thedynamoconvertsfree asadownwardpumpingofmagneticfluxinvolvesanisotropic
energyindifferentialrotationintomagneticenergyandthere- convection. We are led to what seems to be a more natural
fore, strong shear leads to strong magnetic fields. formationexplanation,whichisdescribedinthenextsection.
Someinvestigationsofthedynamointhiscontexthaveim- Nevertheless,despitetheaforementionedpotentialdifficulties,
posedavelocityfieldtodeterminewhatsteady-statemagnetic we now investigate the viability of the envelope dynamo sce-
fieldmightariseifthevelocityweresteadilydriven[24,25,26]. nario.
However, as the magnetic field amplifies, differential rotation Our stellar evolution models were computed using the
decreases. In general, for systems in which shear is not re- “Evolve Zero-age Main Sequence” (EZ) code [30].1 We em-
supplied, a transient dynamo and decay of the magnetic field ployazero-agemain-sequenceprogenitorof3M ,withsolar
(cid:12)
result[27]. Thisscenariowasinvestigatedasawaytogenerate metallicity at the tip of the AGB. Under the assumption of
the strong fields necessary to power bipolar outflows in post- Keplerian rotation, the saturated toroidal field strength at
AsymptoticGiantBranch(post-AGB)andplanetarynebulae the base of the 3-M progenitor’s convective zone (R ∼
(cid:12) conv
(PNe) [28, 29]. A similar, but weaker, dynamo (akin to the 6×1011 cm) is roughly B ∼2×104 G. To survive past the
φ
Solardynamo)mayoperateinisolatedRedGiantBranchand PNstage,thefieldsmustreachtheWDsurface(whichisthe
Asymptotic Giant Branch stars [23]. core of the AGB star, a distance L (cid:39) R interior to the
conv
Here, we investigate two scenarios for magnetic field gen- base of the convective zone; see Fig. 3 of [18] for a pictorial
eration during a CE phase between a post-MS giant and a representation), anchor, and sustain or induce a field in the
low-mass companion. First, we calculate the magnetic fields WD. The latter scenario has been investigated by [31], who
at the interface between the convective and radiative zones concludethatanorderedexternalfield(potentiallygenerated
in the CE (hereafter referred to as the envelope dynamo sce- from a dynamo) can induce a surface field on the WD that
nario) and show that this scenario, while potentially viable decaystoafewpercentofitsinitialvalueafteramillionyears.
under special circumstances, is unlikely in general to explain
the origin of HFMWDs. As an alternative, we estimate the
magnetic fields generated in the disk of a tidally disrupted 1http://www.astro.wisc.edu/∼townsend/static.php?ref=ez-web
2 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Nordhausetal.
For the fields to reach the WD surface, radially inward companionstosolar-typestarsaresignificantlymoreplentiful
diffusionofmagneticfluxmustactonafastertimescalethan than brown dwarfs [38, 39, 40], disks at the upper or lower
magneticbuoyancy,whichtransportsfluxoutward. Thebuoy- mass range may be more common than those in the interme-
antrisevelocity,u,isfoundbyequatingtheupwardbuoyancy diate mass range.
force on a flux tube to the downward viscous drag, thereby The disk is ionized and susceptible to the development of
obtainingtheupwardterminalvelocity. Itmayberepresented magnetizedturbulence(e.g. viathemagneto-rotationalinsta-
as u∼(3Q/8)(a/H )2(cid:0)v2/v(cid:1), where v =B/(4πρ)1/2 is the bility[41]). Accretiontowardsthecentralproto-WDoccurson
p a a
Alfv´envelocity,v istheconvectivefluidvelocity,aistheflux a viscous timescale given by tvisc (cid:39)R2/ν (cid:39)Porb/αss(H/R)2,
tube radius, Hp is a pressure scale height, and Q is a di- where Porb is the Keplerian orbital period at radius R, ν =
mensionless quantity of order unity [32]. The time to buoy- αsscsH = αss(H/R)2R2Ω is the effective kinematic viscosity,
antly rise a distance L from Rconv, assuming a ∼ L/2, is H is the disk scaleheight, cs = HΩ is the midplane sound-
tb = L/u ∼ 0.1 years. The diffusion timescale to traverse a speed,Ω=2π/PorbistheKeplerianorbitalfrequency,andαss
distance L, given a turbulent diffusivity β , is t = L2/β is the dimensionsless Shakura-Sunyaev parameter that char-
φ d φ
and must be less than the buoyant-rise time t . This im- acterizes the efficiency of angular momentum transport [42].
b
plies a constraint on the turbulent diffusivity: β > Lu. For The initial accretion rate can be approximated as
φ
L ∼ 6×1011 cm, the buoyant-rise velocity of a 2×104-G
M
flux tube in an AGB star with ρ ∼ 10−4 g cm−3, requires a M˙ ∼ c ≈7M yr−1× [1]
diffusivity of at least a few times 1017 cm2 s−1; a very large tvisc|rs (cid:12)
and physically unlikely value. Note however that if weaker (cid:16) α (cid:17)(cid:18) M (cid:19)3/2(cid:18)r (cid:19)−1/2(cid:18)H/R(cid:19)2
fields are generated, the requirement on the diffusivity would ss c c ,
10−2 30M r 0.5
lower correspondingly. Furthermore, during one cycle half- J J
period τ0.5 (defined as the time for one reversal of the field), where we have scaled rc to the radius of Jupiter4 and have
magneticfluxofagivensign(positiveornegative)candiffuse scaled H/R to a large value ∼ 0.5 because the disk cannot
into the radiative layer.2 After reversal, field of the opposite cool efficiently at such high accretion rates and would be ge-
sign amplifies and diffuses in the radiative layer. Therefore, ometrically thick. Since the companions under consideration
the diffusion timescale must not be greater than the cycle lead todisks much moredense than thestellar envelope, it is
half-period: td ≤ τ0.5. This implies another a theoretical reasonabletoignoreanyinteractionbetweenthediskandthe
constraint on the turbulent diffusivity for consistency of our star.
dynamo model, namely βφ ≥ L2/τ0.5. This latter constraint Equation (1) illustrates that for Mc ∼ 0.1−500MJ, and
yieldsβφ (cid:38)4×1017 cm2 s−1. Asubtletyisthatthetransport for typical values of α ∼ 0.01−0.1, M˙ is ∼3 to 9 orders
ss
would not be a strictly isotropic diffusion but could be the
of magnitude larger than the Eddington accretion rate of the
result of down pumping by anisotropic convection [33]. proto-WD (i.e., M˙ ∼ 10−5M yr−1). At first, it might
Onewaytoinfertheturbulentdiffusivityinevolvedstars Edd (cid:12)
seemapparentthatinflowofmassontotheWDsurfacewould
is to note that isotopic anomalies in low-mass Red Giant
be inhibited by radiation pressure in such a scenario. How-
Branch and AGB stars require that material from the base
ever, this neglects the fact that, at sufficiently high accretion
of the convective zone be transported to near the H-burning
rates, photons are trapped and advected to small radii faster
shell, processed, and returned to the convective envelope
than they can diffuse out [43, 44, 45, 46, 47]. In this ‘hyper-
[34, 35]. This so-called “cool bottom processing” (CBP) is
critical’ regime, accretion is possible even when M˙ (cid:29)M˙ .
thought to be magnetically driven [36, 23]. In the magnetic Edd
We evaluate the possibility of hyper-critical accretion by
mixingscenario,alowerboundontheturbulentdiffusivityis
∼7×1015 cm2 s−1 [23]. Whiletheactualdiffusioncoefficients estimating the radius interior to which the inward accretion
timescale, t , is less than the timescale for photons to dif-
inevolvedstarinteriorsareunknown,thevaluesinferredfrom visc
fuse out of the disk midplane, t (which is approximately
CBP are more than an order of magnitude lower than what diff
Hτ/c,whereτ istheverticalopticaldepth[44,48,45]). This
are required for the envelope dynamo scenario to transport
∼104-G fields all the way to the WD surface. Higher field “trapping radius” is given by Rtr = M˙κH/4πRc where κ is
strengths(inparticular,∼MGfields)leadtoshorterbuoyant- the opacity. An important quantity is the ratio of the trap-
risetimesandsmallercyclehalf-periods,andthereforerequire ping radius to the outer disk radius Rs (coincident with the
even greater diffusivity. In short, a substantial macroscopic tidal shredding radius) :
diffusivity would be needed to transport megagauss fields to R (cid:16) α (cid:17)(cid:18) κ (cid:19)
the WD surface in AGB stars. We presently do not have tr ≈ 1.0×104 ss × [2]
R 10−2 κ
independent constraints on these coefficients. s es
(cid:18) M (cid:19)11/6(cid:18) M (cid:19)−1/3(cid:18)r (cid:19)−3/2(cid:18)H/R(cid:19)3
c WD c ,
Disk Dynamo.If the field generated in the envelope cannot 30M 0.6M r 0.5
J (cid:12) J
diffusetotheWDsurface,analternativepossibilityisampli-
ficationinanaccretiondiskthatformswhenthecompanionis where we have used equation (1) and scaled κ to the elec-
tidally disrupted inside the common envelope [22]. As shown tron scattering opacity κes =0.4 cm2 g−1. If Rs <Rtr, then
below,thisscenarioisattractivebecauseitprovidesanatural tvisc <tdiff and photons are advected with the matter.
mechanismfortransportingthefieldtotheproto-WDsurface. From equation (2) we conclude that Rtr > Rs for Mc (cid:38)
We consider disks formed from companions spanning the 0.1MJ. This implies that photon pressure will be unable to
range from sub-Jupiter-mass planets (see below) to brown halt accretion initially and that the local energy released by
dwarfs to low-mass stars (i.e. M between ∼0.1M and a
c J
few times 102M ).3 Tidal disruption of the companion re-
J
sults in formation of a disk inside the AGB star. This oc- 2Thoughthecycleperioddoesincreaseinthedynamicalregime,itdoesnotdeviatesignificantly
fromthehalf-cyclekinematicvalueweusehere(τ ∼0.03years);[28]). Notethatwhilethe
curs at the tidal shredding radius, which we estimate as radiativelayerisconvectivelystableitmaystillpossessnon-negligibleturbulence.
Rs (cid:39) Rc(2M/Mc)1/3, where rc and Mc are the radius and 3MJisthemassofJupiter. Thisclassificationasplanetorbrowndwarfisbasedsolelyonmass
mass of the companion and M is the stellar mass interior to [37]
R [19]. We note that, since planetary and low-mass stellar 4rJistheradiusofJupiter. Notethatrcdependsonlyweaklyonmassforthecompanionswe
s consider.
Nordhausetal. PNAS IssueDate Volume IssueNumber 3
WD surface (∼0.1 MeV nucleon−1), burning that occurs ex-
plosively could have a dynamical impact. For the accretion
ratesofrelevance,weestimatethatthemidplanetemperature
ofthediskexceeds∼108 K,suchthathydrogenwillburnvia
thehotCNOcycle. ThehotCNOcycleoccursataratethat
is independent of temperature and is itself thermally stable.
However,hydrogen-richmaterialdepositedintheHeburning
layermay,undersomecircumstances,triggerathermonuclear
runaway[55],whichcouldremovetheaccretedmassfromthe
proto-WDsurface,orevenejectthecommonenvelopeentirely.
In what follows we assume stable burning and set aside this
importantcaveat. Additionalworkwill,however,berequired
to address what conditions are required for stable versus ex-
plosive burning.
Due to the presence of shear in the disk,6 the MRI is a
likely source of turbulence, which in turn amplifies magnetic
field. Large-scalefieldsproducedbytheMRIhavebeenmod-
eledviaα−Ωdynamosatvariouslevelsofsophistication(see
[57] for the most recent example). However, for the purposes
Fig. 1. Toroidalmagneticfieldstrength,Bφ(eq.[3]),attheWDsurfaceasafunc- of estimating orders of magnitude of the fields, approximate
tionofthemassofthetidallydisruptedcompanionMc. Toroidalfieldstrengthsare values from less sophisticated treatments can be employed.
presentedfortwovaluesoftheviscosity,αss=0.01(solidline)andαss=0.1 Although this situation is qualitatively similar to that de-
(dotted line)andassuminganaccretionefficiencyηacc=0.1. Thewhitedwarf scribed in the previous subsection, the dynamo we envision
massandradiusareMWD = 0.6M(cid:12) andrWD = 109 cm,respectively. The hereoperatesintheaccretiondiskandnottheenvelope. The
verticallinesshowthecompanionmassabovewhichphotonsaretrappedintheac-
Alfv´en velocity in the disk obeys v2 = α c2 [58], such that
cretionflow(i.e.,rtr > rs;eq. [2]),suchthatsuper-Eddingtonaccretionoccurs. a ss s
the mean toroidal field at radius R is given by [58]
accretion must be removed by advection [52]. Advection acts
likeaconveyorbelt,nominallycarryingthegastosmallradii
asitsangularmomentumisremoved. If,however,thereisno (cid:18)M˙Ω R(cid:19)1/2
B ∼ [3]
sinkforthehotgas,thisconveyormay“jam”. Thisisanim- φ R H
portantdistinctionbetweenwhitedwarfsandneutronstarsor (cid:16)η (cid:17)1/2(cid:16) α (cid:17)1/2(cid:18) M (cid:19)3/4
black holes, as the latter two can remove the thermal energy ≈ 160MG acc ss c ×
by neutrino cooling or advection through the event horizon, 0.1 10−2 30MJ
respectively. IncontrasttoaWD,neutrinocoolingisineffec- (cid:18) M (cid:19)1/4(cid:18)r (cid:19)−1/4(cid:18)H/R(cid:19)1/2(cid:18) R (cid:19)−3/4
tiveandthermalpressurebuilds,suchthatradiationpressure WD c ,
0.6M r 0.5 109cm
may again become dynamically significant [47]. This in turn (cid:12) J
shuts down the accretion to at most, the Eddington rate.
However,thisscenarioneglectsthepossibilityofoutflows5,
which can sustain inflow by carrying away the majority of
whereinthesecondequalitywehavesubstitutedequation(1)
the thermal energy, thereby allowing a smaller fraction of
the material to accrete at a higher than Eddington rate. forM˙,andmultipliedbythefactor0.1(cid:46)ηacc ≤1toaccount
Though more work is needed to assess the efficiency of out- forthepossibilityofoutflowsasdescribedabove. Ifanα−Ω
flows in the present context, radiatively-inefficient accretion dynamooperates,themeanpoloidalfieldBp isrelatedtothe
flows are widely thought to be prone to powerful outflows toroidal field via B = α1/2B [58]. However, regardless of
p ss φ
[49, 50, 51, 52, 53, 54]. Even if accretion occurs at, or near, whether a large scale field is generated, a turbulent field of
the Eddington rate, the fields produced ((cid:38) 10 MG) are still magnitudeB islikelytobepresentandcontributingsignifi-
φ
strongenoughtoexplainthebulkofthemagnetizedWDpop- cantly to the Maxwell stresses responsible for disk accretion.
ulation. The origin of the strongest field systems (∼100-1000 Figure 1 shows the toroidal field evaluated near the WD
MG)maybeproblematicifaccretionislimitedtotheEdding- surface R ≈ R ≈ 109 cm as a function of companion
WD
ton rate. mass M , calculated for two different values of the viscos-
c
Nevertheless, the material deposited outside the WD will ity (α = 0.01 and 0.1). In both cases, we assume that
ss
be hot and virialized, forming an extended envelope with a η = 0.1. Note that for the range of relevant companion
acc
lengthscale comparable to the radius. Though initially hot, massesM ∼0.1−500M ,B ∼10−1000MG,inprecisely
c J φ
thismaterialwilleventuallycool(onlonger,stellartimescales) thecorrectrangetoexplaintheinferredsurfacefieldstrengths
andbecomeincorporatedintothestellarlayersneartheproto- of HFMWDs.
WD surface. If this material cools at some fraction of the Although our results suggest that companion accre-
local Eddington luminosity, it will be incorporated onto the tion can produce the field strengths necessary to explain
WD surface on a timescale ∼102−104 years, which is much HFMWDs, for the field to be present on the surface of the
less than typical AGB lifetimes. WDwhenitforms,itmustataminimumsurvivetheremain-
A final complication may arise because the composition ing lifetime of the star. In particular, the field will decay due
of the companion is likely to be hydrogen-rich and will begin
to burn thermonuclearly as material accretes or after it set-
tles on the proto-WD surface. Because the energy released 5It’sinterestingtonotethatthepresenceofsuchoutflowsinthepresentcontextiscoincidentwith
in burning hydrogen to helium (∼ 7 MeV nucleon−1) vastly theprevalenceofbipolarproto-planetarynebulae.
exceeds the gravitational binding energy per nucleon at the 6Forblackholedisks,differentialrotationnearthediskmidplaneispreservedeveninathick-disk,
super-Eddingtoncontext[56].However,forwhitedwarfdisks,photonsarenotadvectedinto,and
lostto,theblackhole.
4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Nordhausetal.
to ohmic diffusion on the timescale via dynamo action. For the range of disrupted companions
consideredhere,thediskinitiallyaccretesatsuper-Eddington
τ ∼ 3(∆R)2 ∼4×106yr(cid:18) T (cid:19)3/2× values. Inthishyper-criticalregime,thetimescaleforphoton
decay ηs 108K diffusion out of the disk is longer than the viscous timescale.
(cid:18)lnΛ(cid:19)−1(cid:18) R (cid:19)2(cid:18) M (cid:19)−2(cid:16)η (cid:17)2(cid:18) M (cid:19)2 For a range of disrupted-companion disks, we find that the
WD WD acc c , saturated toroidal mean-field attains values between a few
10 109cm 0.6M 0.1 30M
(cid:12) J and a few thousand megagauss. Amplification of the mag-
[4] neticfieldinasuper-Eddingtonaccretiondiskisattractiveas
it reproduces the observed range of HFMWDs and naturally
where ηs ≈5×1012(lnΛ/10)T3/2 cm2s−1 is the Spitzer resis- transports magnetic flux to the WD surface.
tivity, lnΛ is the Coulomb logarithm, T is the stellartemper- High-mass stars may also undergo common envelope in-
atureneartheproto-WDsurface,and∆R=ηaccMc/4πrW2 Dρ¯ teractionsinthepresenceofclosecompanions. Ifthecommon
is the final thickness of the accreted companion mass after envelopefieldmechanismsdescribedhereoperateinhigh-mass
it is incorporated into the surface layers of proto-WD, where stars, then the result could be strong field neutron stars or
ρ¯≡MWD/(4πrW3 D/3) is the mean density of the proto-WD. magnetars (neutron stars with magnetic fields in excess of
Equation(4)showsthatfortypicalcoretemperaturesdur- ∼1014−1015 G).Inparticular,formationofanaccretiondisk
ingtheAGBphaseT ∼108 K,τdecay exceedsthemeanAGB from an engulfed companion during a red supergiant phase
lifetimeτAGB ∼MyrforMc (cid:38)20MJ. Thissuggeststhatfields could produce a magnetized WD core. In the eventual core-
may survive Ohmic decay, at least for very massive compan- collapse and stellar supernova explosion (possibly driven by
ions. Furthermore, the magnetized companion material may the neutrino mechanism; [60]), the magnetized WD core col-
become incorporated into the degenerate WD core long be- lapsestoaneutronstar. Ifsimplefluxfreezingoperates(itself
foretheAGBphaseends,inwhichcasethemuchhighercon- an open question) and the initial magnetized core is on the
ductivity due to degenerate electrons substantially increase order of ∼100-1000 MG, homologous collapse to a neutron
the Ohmic decay timescale over that given by equation (4), star would generate ∼1014−1015 G fields. Typical neutron
thereby ensuring long-term field survival. stars that possess modest field strengths may originate from
core-collapse supernova of single stars or stars without hav-
ing incurred a CE phase. Note that what we are proposing
Conclusions
here is an alternative to the neutrino-driven convection dy-
RecentobservationalevidencefromtheSloanDigitalSkySur- namo described in [61, 62]. In our model, the engulfment of
vey strongly suggests that high-field magnetic white dwarfs acompanionandtheformationofanaccretiondisknaturally
originate from binary interactions. In particular, it was pro- providesfastrotation,magnetizedturbulenceanddifferential
posed that the progenitors of HFMWDs are binary systems rotation. We emphasize that the viability of this mechanism
that evolve through a common envelope phase [15]. In this dependsonthelengthscaleofthemagneticfielddepositedin
scenario,companionsthatsurviveCEevolution(leavingtight the pre-collapse core, which must be sufficiently large to pro-
binaries) may produce magnetic cataclysmic variables, while duce, upon collapse, the dipole-scale, volume-encompassing
those that do not survive (i.e. that merge) may produce iso- fieldsnecessarytoaccountforthespin-downbehaviorofmag-
lated HFMWDs. netars and the energy budget of their giant flares [63].
Toinvestigatethishypothesis,wehaveestimatedthemag- In summary, common envelope evolution as the ori-
neticfieldsresultingfromalow-masscompanionembeddedin gin of strongly-magnetized, compact objects seems plausible.
a CE phasewith a post-MSgiant. During in-spiral, the com- Whether this hypothesis is ultimately found to be viable will
panion transfers energy and angular momentum to the enve- depend on the statistics of low-mass stellar and substellar
lope. The resulting shearing profile, coupled with the pres- companionstostarsofsimilarmassesto(orsomewhathigher
enceofconvection,canamplifylarge-scalemagneticfieldsvia massesthan)theSun. Thenumerousradialvelocitysearches
dynamo action in the envelope. We incorporate the back- of the last 20 years have revealed a number of such compan-
reaction of the magnetic field growth on the shear, which ions [64, 65, 66]. The precise occurrence rate of companions
results in a transient dynamo. The fields generated at the in orbits that could lead to the kind of disk-dynamo mech-
interfacebetweentheconvectiveandradiativezonesareweak anism described above remains unclear (though if such com-
(∼2×104 G)andwouldhavetodiffusetotheWDsurfaceand panionsturnouttoberarerthantheHFMWDfraction,then
anchorthereiftheyaretoexplaintheHFMWDs. Asuccess- theSDSSdataindicatingabinaryoriginwouldbepuzzling).
ful envelope dynamo scenario must also survive to the end of Further theoretical work into the binary origin of HFMWDs
theAGBphase,whichprobablymeansonlyifthecompanion andmagnetarsrequiresthedevelopmentofmulti-dimensional,
ejectstheturbulentenvelope;thedynamoistransientaslong magnetohydrodynamicsimulationsoftheCEphase. Suchan
as there is a source of turbulent diffusion. The bulk of the approach has already had initial success for purely hydrody-
systems formed this way might be post-CE, tight binaries – namic adaptive mesh refinement simulations [67].
potentially magnetic CVs.
ToexplainisolatedHFMWDs,analternativetoadynamo
ACKNOWLEDGMENTS.WethankAlbertoLopez,JayFarihi,JimStone,Jeremy
operating in the envelope is a dynamo operating in an ac-
Goodman,KristenMenou,DeepakRaghavan,AndreiMesinger,AdamBurrowsand
cretion disk. During the CE phase, if the companion is of Fergal Mullally for thoughtful discussions. JN acknowledges support for this work
sufficiently low mass, it avoids prematurely ejecting the en- from NASA grant HST-AR-12146. DSS acknowledges support from NASA grant
velope. Instead, in-spiral proceeds until the companion is NNX07AG80G.SupportforBDMisprovidedbyNASAthroughEinsteinPostdoctoral
FellowshipgrantnumberPF9-00065awardedbytheChandraX-rayCenter, which
tidally shredded by the gravitational field of the proto-WD.
isoperatedbytheSmithsonianAstrophysicalObservatoryforNASAundercontract
The subsequent formation of an accretion disk, which also NAS8-03060. EGBacknowledgessupportfromgrantsNSFPHY-0903797andNSF
posesses turbulence and shear, can amplify magnetic fields AST-0807363.
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