An eLIMA model for the 67 s X-ray periodicity in CAL 83

0.77 MB·

by  A. Odendaal

MNRAS000,1–17(2016) Preprint25January2017 CompiledusingMNRASLATEXstylefilev3.0
An eLIMA model for the 67 s X-ray periodicity in CAL 83
A. Odendaal1⋆ and P. J. Meintjes1
1Department of Physics, Universityof the Free State,P.O. Box 339, Bloemfontein, 9300, South Africa
AcceptedXXX.ReceivedYYY;inoriginalformZZZ
7
1
0 ABSTRACT
2 Supersoft X-ray sources (SSSs) are characterized by their low effective temperatures

n and high X-ray luminosities. The soft X-ray emission can be explained by hydrogen
a nuclear burning on the surface of a white dwarf(WD) accretingat an extremely high
J rate.Apeculiar∼67speriodicity(P67)waspreviouslydiscoveredintheXMM-Newton
4 light curves of the SSS CAL 83. P67 was detected in X-ray light curves spanning
2 ∼9 years, but exhibits variability of several seconds on time-scales as short as a few
hours, and its properties are remarkably similar to those of dwarf nova oscillations

] (DNOs). DNOs are short time-scale modulations (.1 min) often observed in dwarf
E
novae during outburst. DNOs are explained by the well established low-inertia mag-
H
neticaccretor(LIMA)model.Inthispaper,weshowthatP67 anditsassociatedperiod
h. variabilitycanbe satisfactorilyexplainedbyanapplicationofthe LIMAmodeltothe
p more ‘extreme’ environment in a SSS (eLIMA), contrary to another recent study at-
- tempting to explain P67 and its associated variability in terms of non-radial g-mode
o
oscillations in the extended envelope of the rapidly accreting white dwarf in CAL 83.
tr In the eLIMA model, P67 originates in an equatorial belt in the WD envelope at the
s
boundary with the inner accretiondisc, with the belt weakly coupledto the WD core
a
[ by a ∼105 G magnetic field.
NewopticallightcurvesobtainedwiththeSutherlandHigh-speedOpticalCamera

1 (SHOC) are also presented, exhibiting quasi-periodic modulations on time-scales of
v ∼1000 s, compatible with the eLIMA framework.
1
9 Key words: white dwarfs – stars:oscillations – accretion, accretion discs – stars:
7 individual: CAL 83 – binaries: close – X-rays:binaries
6
0
.
1
0
1 INTRODUCTION Chandra and XMM-Newton X-ray spectra, and the best-
7
:1 Supersoft X-ray sources (SSSs) were established as a fiLtt∼ing3.4W×D10m3a7sesrgwas−s1M. O1d∼en1d.a3aMl &⊙,Mweiitnhtjeaslu(2m0i1n5obs)itpyroo-f
v quite unique class of objects after observations by the
vided a general review of the properties of CAL 83, while
i Einstein Observatory (Longet al. 1981; Seward & Mitchell
X Odendaal& Meintjes(2015a)consideredthesourcevariabil-
1981) and ROSAT (Tru¨mperet al. 1991). Their defin-
ity.
r ing characteristics are their low X-ray temperatures of
a CAL 83 exhibits long-term quasi-periodic modulations
kTeff ∼ 20–100 eV, and their extreme soft X-ray
luminosities, from ∼1036 erg s−1 up to ∼1038 erg s−1 in the optical (P ∼ 450 d), cycling between an optical low
state and an optical high state (Rajoelimanana et al. 2013
(e.g. Kahabka& van denHeuvel 2006). The vast majority
and references therein). Several X-ray off-states have been
of observed SSSsare binary systems. Vanden Heuvelet al.
observed during optical high states, while the X-ray high
(1992)showedthattheirobservationalpropertiescanbeex-
states were observed during optical low states. This long-
plained bythenuclearburningof accreted hydrogen on the
termanti-correlationbetweenX-rayandopticalfluxarealso
surface of a whitedwarf (WD).
observedinanotherLMCSSS,namelyRXJ0513.9–6951,al-
CAL 83 was one of the first SSSs to be discovered,
thoughthelatterhasashortercycleperiodof∼168d.Sev-
and with its relatively short orbital period of 1.047529 ±
eral authors have discussed different aspects of a so-called
0.000001 d (Rajoelimanana et al. 2013) it forms part of
‘limit cycle’modelthatcanbeemployedtoexplainthisbe-
the so-called ‘close binary supersoft source’ (CBSS) sub-
haviourinbothsources.Insuchamodel,theX-rayhigh,op-
class. Lanz et al. (2005) fitted WD atmosphere models to
ticallowstateisassociatedwiththeWDphotospherebeing
in a contracted state (higher effective temperature), while
theX-raylow,opticalhighstatecorrespondstoanexpanded
⋆ E-mail:[email protected] WDphotosphere(lowereffectivetemperature)duetoanen-
(cid:13)c 2016TheAuthors
2 A. Odendaal and P. J. Meintjes
hanced accretion rate (Southwell et al. 1996; Reinsch et al. accreted material with hydrogen burningat the base of the
2000; Hachisu & Kato 2003). envelope.Inthispaper,wethereforerefertothisapplication
CBSSs are closely related to another class in the WD of the LIMA model in the ‘extreme’ accretion limit by the
binarypopulation,namelythecataclysmicvariables(CVs). acronym ‘eLIMA’.
In CVs, the primary WD is more massive than the donor, In §2, a brief summary of thedifferent short time-scale
i.e. the mass ratio q = M2/M1 is smaller than 1. How- quasi-periodic modulations observed in CVs is given, while
ever, it has been shown that an accretion rate of the order §3providesanoverviewofthoseaspectsoftheLIMAmodel
m˙acc ∼10−7 M⊙ yr−1isrequiredtodrivepersistentsurface for DNOsthat are themost relevantfor thecurrent discus-
nuclearburninginSSSs,whichcanonlybesustained(inthe sion. The properties of P67 in CAL 83 are reviewed in §4,
caseofRochelobeoverflow)ifthemassofthedonoriscom- where the analysis of Odendaalet al. (2014) has been ex-
parable to or greater than theWD mass (q&1). The main tendedtotakeintoaccounttheeffectoftheopticalblocking
characteristics of SSSs that distinguish them from CVs are filtersofXMM-Newton, andtobetterillustrate thedynam-
therefore the inverted mass ratios of the former, their high ical nature of P67. The proposed properties of the eLIMA
m˙acc andtheextremeluminosities derived from surface nu- model in CAL 83 are discussed in §5. New optical light
clear burning. curves of CAL 83 have been obtained with the Sutherland
To provide some context for the high m˙acc in SSSs, it High-speedOpticalCamera(SHOC)onthe1.9-mTelescope
is useful to keep in mind the range of m˙acc that is typi- at the South African Astronomical Observatory (SAAO),
cal of different subclasses in the related class of CVs: from and themodulations observed in these light curvessupport
∼(2–50)×10−11 M⊙ yr−1 indwarfnovaeduringquiescence, the proposed eLIMA model. These are presented in §6, fol-
to∼(3–10)×10−9 M⊙ yr−1 innovaremnants,novalikesand lowed bythe conclusion in §7.
dwarfnovaeduringoutburst(e.g.Warner1995a,pp.64–66).
It is well known that CVs exhibit (often quite drastic)
variability on various time-scales, including quasi-periodic
2 QUASI-PERIODIC MODULATIONS IN
modulations with periods ranging from a few seconds to a
CATACLYSMIC VARIABLES
few thousand seconds. These modulations have been well
studiedinCVs,butduetothegreatsimilaritybetweenCVs ManyCVshavebeenobservedtoexhibitquasi-periodicvari-
and supersoft X-ray binaries, one can expect similar quasi- ability.Thetime-scalesoftheseoscillationsaretypicallybe-
periodic modulations from thelatter class. tweenafewsecondsandafewthousandseconds(seee.g.the
Odendaalet al. (2014) reported the discovery of a review papers of Warner 2004 and Warner & Woudt 2008).
∼67 s X-ray periodicity (hereafter P67) in archival XMM- Thefirst typeof oscillation is knownas ‘dwarf novaoscilla-
Newton data of CAL 83. The peculiar variability observed tions’ (DNOs),as they were first discovered in dwarf novae
in this period precluded a straightforward interpretation (DNe)duringoutburst.However, DNOshavealso been ob-
as the WD rotation period. The favoured interpretation of served in other CVs with high accretion rates, but neverin
Odendaalet al. (2014) was that the period may originate intermediatepolars(IPs).IPsarecharacterizedbymagnetic
in an extended WD envelope of which the rotation is not fieldsof the orderof 107 G.
quitesynchronizedwiththerotationoftheWDitself,caus- The typical periods associated with DNOs are in the
ing the observed drift in the period. Since the discovery, ∼5–40 s range. They have primarily been observed in the
a detailed comparison of P67 with the numerous different optical, with typical oscillation amplitudes of <1 per cent,
types of periodicities observed in CVs has shown that it is andthepulseprofileishighlysinusoidalinshape.However,
remarkablysimilartodwarfnovaoscillations(DNOs),which DNOs have also been discovered in the ultraviolet and soft
are variable, short time-scale oscillations observed in dwarf X-rayswithmuchlargeramplitudes,evenupto100percent
novae during outburst, as well as in other high-m˙acc CVs. insoftX-rays.Therateofchangeofaquasi-periodicperiod
This provides a suitable theoretical framework to investi- P is often described by thequality factor
gatethepossiblecorrelation betweenP67 anditsassociated dP −1
variability in CAL83, and DNOs,which differs from there- Q≡ , (1)
(cid:12) dt (cid:12)
cent study (Ness et al. 2015) which attempted to explain (cid:12) (cid:12)
(cid:12) (cid:12)
this phenomenon in terms of oscillating non-radial g-mode which(cid:12) wil(cid:12)lhaveasmallvalueiftherearerapidchangesinP,
oscillations intheextendedenvelopeoftherapidlyrotating whileahighlycoherentperiodwillhavealargeQ-value.The
white dwarf. value of Q for DNOs is usually in the 103–107 range, while
The most widely accepted model for DNOs involves e.g.thecoherentWDrotation period intheIPDQHerhas
anisotropic emission from a region close to the surface of Q∼1012.
a WD of which the magnetic field is too weak to enforce DNO periods exhibit variability of both a continuous
rigidco-rotationofthecoreandexteriorregions,butstrong andadiscontinuousnature.Thecontinuousvariabilityobeys
enough to significantly influence accretion close to the WD a period-luminosity relation, with the shortest period cor-
surface. This is known as the low-inertia magnetic accre- responding to the highest luminosity, i.e. the highest m˙acc
tor (LIMA) model (Warner & Woudt 2002). In this paper, during the DN outburst, and the period increases as the
the application of a similar LIMA model to the observed post-outburst luminosity decreases. However, this relation
P67 in CAL 83 is discussed. Although the same underlying is actually not one-to-one, but rather has a ‘banana loop’
principles of this model as applied to dwarf novae are still nature(seePatterson 1981 for moredetails). Discontinuous
applicable, one must keep in mind that the conditions on jumpsof ∼0.01percentarealso sometimes observedin the
the surface of the WD in SSSs are probably more extreme, period.
withthehighm˙acc andtheassociated extendedenvelopeof Quasi-periodic modulations belonging to the second
MNRAS000,1–17(2016)
An eLIMA model for CAL 83 3
class are known as ‘longer period DNOs’ (lpDNOs) becoupledrigidlytotheinterior,allowingtheformationofa
(Warneret al. 2003). They haveperiods typically ∼4 times beltstructureintheequatorialregion onthesurfaceonthe
longer than DNOs, and slightly larger amplitudes, without WD that is rotating rapidly at the local Keplerian velocity,
the strong period-luminosity relationship of DNOs. These in other words faster than the central WD core. Actually,
lpDNOs are also only observed in CVs with high accretion due to its inertia, this ‘equatorial belt’ would most likely
rates. haveaperiodslightlylongerthanthelocalKeplerianperiod
The third type is simply called ‘quasi-periodic oscilla- (Warner 1995b). In fact, evidence supporting the existence
tions’(QPOs),withperiodsandoscillationamplitudesmuch of a hot, rapidly rotating belt around a WD in relatively
largerthanthoseofDNOs.QPOsarealsolesscoherentthan slow rotation has been found in HST spectra of the dwarf
DNOs, with Q typically between 5 and 20. QPOs have not novaeVWHyi,UGem,ALComandRUPegduringandaf-
only beenobserved in high m˙acc CVs, butalso in DNedur- teroutburst(Sion et al.1996;G¨ansicke & Beuermann1996;
ing quiescence. In cases where QPOs do occur simultane- Chenget al. 1997; Szkodyet al. 1998; Sion & Urban 2002).
ously with DNOs, the relation PQPO ∼ 15PDNO has been This concept was initially suggested by Paczyn´ski (1978),
observed, and then the term ‘DNO-related QPO’ is often where-after Warner (1995b) and Warner& Woudt (2002)
used (Warneret al. 2003). developed it into a low-inertia magnetic accretor (LIMA)
Another type of QPO also exists, having much longer model toexplain theDNOphenomenon.
periods of ∼1000–3000 s. Many CVs with such a QPO are WithintheLIMAmodel,thetotalaccretionluminosity
suspected to be IPs, where the coherent WD spin period is isusedtoestimate thetotalmass ofthematerial contained
not observed directly,possibly dueto ahigh accretion rate, in the equatorial belt as <10−10 M⊙. Therefore, the iner-
butreprocessedbyaQPOsourcewithvaryingperiod.This tia of the belt is much lower than the inertia of the whole
is supported by these QPOs having periods comparable to WD,andthebeltcanreadilybetuggedaroundduetomag-
that of the WD rotation periods in canonical intermediate netic coupling with the inner disc. Then, the DNO period-
polars,i.e.∼15min.However,incaseswhereQPOsareob- luminosity relation can beexplained as follows: Asm˙acc in-
servedsimultaneouslywithDNOs,thesystemcannotbean creases,theWDmagnetosphereiscompressedandtheinner
intermediatepolar (seebelow). Inthesesystems,thelonger disc boundarymoves further inward, yielding a shorter Ke-
period QPOs may represent the Keplerian periods of blobs plerian (or quasi-Keplerian) period for the belt (spin-up).
at the outer edge of an accretion disc. Another source of As m˙acc decreases during the final stages of outburst, the
QPOsmaybeamodulationinmasstransfercausedbynon- inner disc boundary moves outward rapidly. It is expected
radial oscillations of the secondary. that the belt would not be able to also slow down rapidly
enoughtoremainnearequilibrium,andsomeofthematerial
in the belt is centrifuged to larger radii, removing angular
3 THE EXISTING LIMA MODEL FOR DWARF momentum from thebelt (spin-down).
NOVA OSCILLATIONS Althoughthespin-upandspin-downmechanismsabove
couldexplainthecontinuouschangesinDNOperiods,itcan
IthasbeenproposedthatDNOsoriginateinsystemswhere
notexplainthediscontinuous‘jumps’thatarealsoobserved.
the accreting WD has a weak magnetic field, thus repre-
The latter are most likely caused by magnetic reconnection
sentinganextensionoftheintermediatepolarclasstolower
events, through which the system adjusts to seek the ac-
magneticfield values.Inthisregard,it isimportant tocon-
cretion configuration with the lowest energy. Even though
siderthefindingsofKatz(1975).Heconsideredtherotation
themagneticfieldisrelatively weak,it will stillhavesignif-
period P∗ of the WD in DQ Her, and argued that the high icantinteractionwiththeaccretingmaterial.Magneticfield
degreeofcoherencyoftheobservedperiod mustbearesult
lines threading the slipping equatorial belt will be wound
of rigid body rotation of the WD, with a magnetic field as
up, until reconnection starts occurring after differential ro-
themechanismcouplingthestableclockoftherotatingcore
tation of ∼2π. The whole equatorial belt is not necessarily
to theouterenvironment.
associatedwithasingleperiod.Latitudinalvariationsinro-
The magnetic field required to transmit this angular
tationalperiodmaybepresentinthebelt,withlongerperi-
accelerationfromtheWDsurfaceregionstothecorecanbe
odsathigherlatitudes(analogoustothelatitudinaldifferen-
estimatedbyequatingtheMaxwellandmechanicalstresses,
tialrotationobservedintheSun,e.g.Carroll & Ostlie2007,
i.e.
p. 364), approaching the rotation period of the underlying
B4rπBφ ≈ρwd Ω˙∗R12 , (2) pbyrimreacroyn.nMecatgionnetmohayydbroedaysnsoacmiaicte(dMwHitDh)stou-crabluleledn‘caecccraeutsioend
where B and B represent the radial and azimuthal com- curtains’wheregascanpotentiallybefedthroughaccretion
r φ
ponents of the internal WD magnetic field respectively. A arcs at differentlatitudes.
typicalWDradiusofR1 ∼109 cmcanbeusedintheequa- One can expect that many such accretion arcs would
tion above, and assuming that a significant portion of the beinvolvedsimultaneously,andif thesearcs are spread out
deep interior of the WD is still in fluid form and not yet over different latitudes in a quite regular manner, then a
crystalline, a density of ρwd ∼ 106 g cm−3 can be used. relatively coherent period may not be observable at all. On
AdoptingtheseparametersyieldsB B ∼1010 G2,andap- the other hand, the coincidental domination of one or two
r φ
proximatingthetwocomponentsasbeingequal,oneobtains arcs may yield an observable DNO. A reconnection event
B ∼ B ∼ 105 G (Katz 1975). This provides an approxi- wouldcauseareconfigurationofoneormoreaccretionarcs,
r φ
matelowerlimitforthemagneticfieldtopreventdifferential andtherapidtransferenceoftheaccretion streamfrom one
rotation of theouter regions. arctotheotherwouldbeabletoexplaintheobservedjumps
Forweakermagneticfields,theaccretiontorquewillnot in theobserved DNO period.
MNRAS000,1–17(2016)
4 A. Odendaal and P. J. Meintjes
In several systems, two DNOs with slightly different theeffect of theoptical blocking filter was not discussed by
periods have been observed, together with a DNO-related Odendaalet al. (2014), therefore we providedetails here.
QPO, and it has been found that the slightly longer DNO To estimate theeffect of the optical blocking filter, the
corresponds to the beat period between the shorter DNO Chandra proposal planning tool PIMMS v4.7b was used,
andtheQPO.Insuchacase,theshorterDNOcouldbeas- whichprovidesgeneralfunctionalityforconversionsbetween
sociated with the equatorial belt as explained above, and source flux and CR for a variety of X-ray instruments2. By
the QPO with a rotating structure (like a thickening in usingablackbodyapproximation,theshapeandfluxofthe
the accretion disc, e.g. at the point of impact of the accre- CAL83X-rayspectrumwereprovidedasinputtoPIMMS.
tion stream) revolving around the primary in a prograde The tool was then used to estimate the expected CRs in
direction. The beat period is then caused by the reflec- EPIC pn in the 0.15–1.0 keV band, using different optical
tion/reprocessing of the shorter DNO beam by the QPO blockingfilters.Itwasfoundthat,forEPICpndata,multi-
structure. plicativefactorsof1.2and2.6couldbeusedtoconvertCRs
The lpDNOs are also thought to be associated with obtained with the medium and thick filters respectively to
such systems, but are found to have periods equal to ap- an equivalent CR with the thin filter. The same estimates
proximately one half of the spin period of the WD itself. were performed for EPIC MOS, yielding factors of 1.2 and
Warner(2004)suggested thatthisconnectsthelpDNOsdi- 2.2.Thesefactorsaremodel-dependent,andkeepinginmind
rectlytotheWDspinperiod,withthefactorof2beingdue that the long-term X-ray flux changes in CAL 83 are prob-
tomagneticallychannelledtwo-poleaccretionontotheWD. ablyrelated totemperaturemodulations,thesefactors may
Because PlpDNO ∼ 4PDNO, this implies that the equatorial not bevalid for a wide range of CRs.
belt at the inner disc is rotating at an angular velocity 8 Alternatively, one could assume for the moment that
times larger than that of theWD itself. theaverageX-rayluminosityofCAL83wasidenticalduring
eachofthethreeEPICsegmentsinobservation0123510101.
By calculating the ratios between the mean observed CRs
in the 0.15–1.0 keV band for the 3 segments, multiplica-
4 THE NATURE OF THE 67 s PERIODICITY tive factors of 1.2 and 2.3 for pn, and 1.3 and 2.4 for MOS
IN CAL 83 were obtained, for conversion from medium and thick fil-
ter counts respectively to ‘thin filter counts’. These values
There are 23 observations of CAL 83 in the XMM-Newton
compare quitewell with thefactors predicted byPIMMS.
Science Archive. One of these observations was obtained
Subsequently, the factors derived from observation
in 2000, five in 2007, sixteen in 2008 and one in 2009
0123510101 were applied to all the EPIC observations ob-
(see Table 1). During all the observations, data were ob-
tained with medium and thick optical blocking filters, to
tained with all the X-ray detectors on board, i.e. the three
yield the approximate CR that would have been obtained
EPICdetectors(MOS1,MOS2andpn)(Stru¨deret al.2001;
if a thin filter was used. The benefit of this transforma-
Turneret al. 2001) and the Reflection Grating Spectrome-
tion is that the EPIC CRs for different observations can
ters (RGS). All the observations, except the first, also in-
now be compared directly.Because this was a simple linear
cluded Optical Monitor (OM) data, but these OM data do
transformation of the CRs and errors, it did not affect the
not form part of thecurrent discussion.
subsequentperiod analysis.Considering themean EPICpn
Four of the observations were performed during an
CR per observation as given in Table 1, one can broadly
X-rayoff-state.The19X-rayon-stateobservationswerere-
divide the XMM-Newton observations into two categories
ducedbyfollowingstandarddatareductionprocedureswith
(apartfromthefourX-rayoff-states):anX-raybrightstate
the XMM-Newton Science Analysis System (sas) Version with CR > 5 counts s−1, and an X-ray faint state with
13.0.11 (XMM-Newton Science OperationsCentre Team CR<2 countss−1.
2014).InTable1,themeanEPICpncountrate(CR)inthe
Themain period analysis methodapplied in thisstudy
0.15–1.0keVband(the‘broadband’CR)isprovidedforeach
wasthewell-knownLomb-Scargle(LS)periodogram,asim-
observation (converted to ‘thin filter counts’ as explained plementedintheStarlinkperiod3code,version5.0-2.Ade-
below) to enable a comparison of the relative strength of
tailed LS analysis of the EPIC light curves was performed,
theX-rayemission between different observations.
experimenting with different detrending methods and time
During observation 0123510101, three exposures were
bin sizes. As the EPIC pn detector has the highest effec-
obtained with each EPIC detector, with thin, medium and
tiveareaandisthereforemost suitedtotiming,thecurrent
thick optical blocking filters respectively. The gaps be-
discussion will mostly focus on theEPIC pn results.
tween the exposures were almost 5 ks, therefore they were
A very strong peak at ∼15 mHz was immediately evi-
analysed separately, and were assigned observation IDs of
dent in the periodogram of EPIC pn dataset 0506531501,
0123510101A, 0123510101B and0123510101C. Asthereare
corresponding to a period of ∼67 s. This peak was also
19 on-state observations, this subdivision of observation
present at a >99.73 per cent significance level in observa-
0123510101 yieldsa total of 21 on-statedatasets to becon-
tions 0500860601 and 0506531701. By simply scrutinizing
sidered. For the other EPIC datasets, either a thin or a
therest of theperiodograms by eye and comparing the sig-
medium filter was used. It must be kept in mind that the
nal strength at 15 mHz with the local noise level, it was
choiceofopticalblockingfilteralsoinfluencestheX-rayCR
significantly, especially at soft energies. The correction for
2 http://cxc.harvard.edu/toolkit/pimms.jsp
3 http://www.starlink.rl.ac.uk/docs/sun167.htx/node16.
1 http://xmm.esac.esa.int/sas/ html
MNRAS000,1–17(2016)
An eLIMA model for CAL 83 5
Table 1.Archival XMM-Newton observations ofCAL83. The last3columns provideinformationregardingthe ∼67s periodicity, for
thoseobservationswhereaperiodogrampeakatthispositionwasdetected.
Observation Startdate& Duration MeanEPIC X-ray Period Significance Modulation
ID time(UT) (s) pnCRa state (s) (percent) semi-amplitude(percent)
0123510101A 2000Apr23 07:34:01 45021 6.471±0.021 Bright 68.69±0.24 29.94 2.5
0123510101B 66.86±0.22 59.47 2.8
0123510101C 66.63±0.22 98.16 5.0
0500860201 2007May13 22:03:32 11949 7.324±0.036 Bright 65.45±0.19 81.69 3.1
0500860301 2007Jul06 23:31:52 10920 6.764±0.036 Bright 67.55±0.22 99.02 4.0
0500860401 2007Aug21 15:11:21 7915 5.208±0.037 Bright 66.55±0.30 23.25 3.9
0500860501 2007Oct05 23:49:21 17615 5.034±0.028 Bright 70.55±0.20 51.74 4.0
0500860601 2007Nov24 21:10:14 23173 6.445±0.026 Bright 67.67±0.11 99.99 3.7
0500860701 2008Jan16 13:24:33 10914 Off/Low
0500860801 2008Mar10 10:14:38 6916 Off/Low
0506530201 2008Mar20 00:33:22 7715 0.619±0.013 Faint
0506530301 2008Apr03 18:40:55 14918 Off/Low
0506530401 2008Apr11 06:02:24 5914 Off/Low
0506530501 2008Apr16 12:32:01 10873 1.680±0.025 Faint 66.76±0.49 56.03 8.6
0506530601 2008Apr17 13:40:13 11217 1.426±0.015 Faint
0506530801 2008Apr19 06:43:35 5916 0.350±0.012 Faint
0506530901 2008Apr20 22:38:43 11617 0.403±0.008 Faint
0500860901 2008Apr21 02:10:24 12716 0.234±0.008 Faint
0506531001 2008Apr21 18:47:57 9076 0.507±0.012 Faint
0506531201 2008Apr23 11:20:22 7418 0.166±0.008 Faint
0506531301 2008Apr25 08:13:32 9614 0.438±0.010 Faint
0506531401 2008Apr29 00:40:23 14117 0.249±0.006 Faint
0506531501 2008Aug12 14:50:27 6918 7.846±0.045 Bright 66.87±0.35 ≫99.99 7.6
0506531601 2008Sep17 11:10:21 6817 0.101±0.007 Faint
0506531701 2009May30 08:00:48 46115 7.687±0.017 Bright 65.173±0.047 ≫99.99 2.5
aConvertedto‘thinfiltercounts’(seetext).
concluded that the ∼67 s periodicity can also be consid- periodograms of the much lower signal-to-noise MOS light
ered to be ‘present’ in observations 0123510101 (all 3 seg- curves did not exhibit this peak at a >99.73 per cent level,
ments), 0500860201, 0500860301, 0500860401, 0500860501 although it did appear slightly above the local noise level
and0506530501, whileitis‘absent’intherestoftheobser- in observations 0123510101A, 0123510101B, 0500860301,
vations. (It is noted that this ‘absence’ is probably a selec- 0500860401 and especially in 0506531701. This discovery of
tion bias rather than an intrinsic source property, as one is P67wasreportedbyOdendaalet al.(2014),whoalsoshowed
more likely to detect a short time-scale variation in a light thatthisperiodicityisinherenttoCAL83andisnotfound
curve with higher counts and therefore better statistics. A in theperiodograms of thebackgroundcounts.
detectorwithalargereffectiveareamighthaverevealedthe
same periodicity in thefaint observations.) The 11 EPIC light curves exhibiting P67 were subse-
quently folded on their corresponding periods as given in
The periodograms of all the EPIC pn on-state light
Table 1, using the starting point of each light curve as the
curvesofCAL83inthe2–50mHzrangeareshowninFig.1.
reference point. For each of these, a sine curve was also fit-
(The 0–100 mHz pn periodograms of some of these obser-
tedtothefoldedlightcurve,keepingtheperiodfixedtothe
vations were presented in figure 1 of Odendaalet al. 2014,
valueinTable1.Thesemi-amplitudeofthemodulationwas
togetherwiththeMOSperiodogramswhereP67wasevident, calculated by expressing the semi-amplitude of the sine fit
but here the full set of pn periodograms is provided.) The
totheEPICpndataasapercentageof themean ofthefit,
exact periods associated with the∼15 mHzpeak in each of
andisalsogiveninTable1.ThefoldedEPICpnlightcurves
these observations are also listed in Table 1, together with
arepresentedinFig.2.Onecanseethatmostofthesedoin-
thestatisticalsignificanceofthepeak.Theerrorestimatein
deedfollow aquasi-sinusoidalmodulation pattern,with the
theperiod valuewas obtained from thesimple relation
semi-amplituderangingfrom2.5to8.6percent.(Thefolded
P2 pnlightcurvesofsomeoftheseobservationswerepresented
∆P = , (3)
2T in figure 3 of Odendaalet al. 2014, together with the MOS
which representstheFourierresolution oftheperiodogram, light curves where P67 was evident, but here the full set of
pnlightcurvesisprovided,alsoincorporatingthecorrection
where P is the value of the period itself, and T is the total
tothin filter counts.)
length of thedataset.
When considering the mean EPIC pn CRs in Table 1 The other 10 datasets where the periodicity was not
togetherwiththeresultsoftheLSanalysis,onecanseethat detected in the periodograms, were also folded, but on the
P67 was detected in all the X-ray bright observations, with meanperiodof67s,asanadditionalcheckofwhetherthey
an additional detection in the brightest of the faint state exhibitanytracesofsuchamodulation.Nosinecurveswere
observations:0506530501, withCR=1.680±0.025. TheLS fitted for these. In several of these datasets, a ∼67 s mod-
MNRAS000,1–17(2016)
6 A. Odendaal and P. J. Meintjes
0123510101Apn(CAL83) 0123510101Bpn(CAL83) 0123510101Cpn(CAL83)
40
30
20
99.73% 99.73% 99.73%
10
0
0500860201pn(CAL83) 0500860301pn(CAL83) 0500860401pn(CAL83)
40
30
20
99.73% 99.73% 99.73%
10
0
0500860501pn(CAL83) 0500860601pn(CAL83) 0500860901pn(CAL83)
40
30
20
99.73% 99.73% 99.73%
10
0
c
tisti 40 0506530201pn(CAL83) 0506530501pn(CAL83) 0506530601pn(CAL83)
a
t
s 30
S
L
ed 20
z
mali 10 99.73% 99.73% 99.73%
r
o
N 0
0506530801pn(CAL83) 0506530901pn(CAL83) 0506531001pn(CAL83)
40
30
20
99.73% 99.73% 99.73%
10
0
0506531201pn(CAL83) 0506531301pn(CAL83) 0506531401pn(CAL83)
40
30
20
99.73% 99.73% 99.73%
10
0
0506531501pn(CAL83) 0506531601pn(CAL83) 0506531701pn(CAL83)
40
30
20
99.73% 99.73% 99.73%
10
0
15 30 45 15 30 45 15 30 45
Frequency(mHz)
Figure1.Lomb-Scargle(LS)periodogramsofthebroadbandEPICpnlightcurvesofCAL83inthe2–50mHzrange.The99.73percent
significancelevel isindicated. Alightcurvebinningof5swas used,andthe lightcurves weredetrended bysubtractingasecond-order
polynomialfit.
MNRAS000,1–17(2016)
An eLIMA model for CAL 83 7
0123510101Apn 0123510101Bpn 0123510101Cpn
6.955 7.018 7.426
6.507 6.558 6.619
6.058 6.098 5.812
0500860201pn 0500860301pn 0500860401pn
8.030 7.333 5.730
7.505 6.833 5.290
6.980 6.333 4.851
0500860501pn 0500860601pn 0500860901pn
5.634 7.008 0.315
5.183 6.611 0.251
)
1
− 4.731 6.214 0.188
s
s
t
n
u 0506530201pn 0506530501pn 0506530601pn
co 0.756 2.233 1.669
(
e
t
a
r
nt 0.632 1.809 1.478
u
o
c
d
ne 0.509 1.385 1.287
n
bi
e- 0506530801pn 0506530901pn 0506531001pn
as 0.488 0.536 0.711
h
P
0.383 0.435 0.542
0.279 0.333 0.372
0506531201pn 0506531301pn 0506531401pn
0.285 0.604 0.314
0.188 0.480 0.267
0.091 0.357 0.220
0506531501pn 0506531601pn 0506531701pn
9.368 0.240 8.071
8.265 0.126 7.770
7.162 0.012 7.469
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
Phase
Figure 2.TheEPICpnbroadbandlightcurvesofCAL83foldedonthe∼67speriod.Forthedatasets inwhichtheperiodicitycould
be detected in the periodograms (see Fig. 1), the best fitting sine curve with the period fixed to the corresponding value provided in
Table1isoverplotted.Theother10datasetswerefoldedonthemeanperiodof67s.Thestartingpointforthephasefoldingwastaken
atthestartofeachdataset.Twocyclesareplottedforclarity.
MNRAS000,1–17(2016)
8 A. Odendaal and P. J. Meintjes
ulation may indeed be present in the data, even if not at a >95.45 per cent detection significance. The value of Q was
very significant level. Onemust also keep in mind that, if a evaluated for every set of two measurements (t ,P ) and
i i
similarmodulationisinherentlycontainedintheseobserva- (ti+1,Pi+1) in the light curve (provided that Pi 6= Pi+1),
tions, it may well be at a period slightly different from the using
∼67smean,whichwillchangetheappearanceofthefolded
light curve. Q= Pi+1−Pi −1 , (4)
Asanadditionalinvestigationofthepossibleoccurrence (cid:12)(cid:12) ti+1−ti (cid:12)(cid:12)
of the same periodicity in the MOS data, each MOS light yieldi(cid:12)(cid:12)nga value(cid:12)(cid:12)for Q at time (ti+ti+1)/2. Standardprop-
curve was folded on the same period as the corresponding
agation of theerrors in theperiods were used to obtain the
pn observation, using the same reference point as the pn
errors in Q. Considering only Q-values for which the error
light curvesso thattheirphasingcan becompared. Forthe
bars did not include zero, the resultant Q-values were be-
11 datasets containing P67, the folded pn and MOS light tween210±100and460±120,withameanof343±87.The
curves were found to be in phase, with the exception of
second-longest continuous observation, 0500860601, yielded
0123510101B, where theMOS light curvewas verynoisy.
similar results.
It is quiteobviousthat thereis considerable variability
Odendaalet al. (2014) also investigated the possibility
in both the period and the amplitude of the periodicity in
that P67 might represent the beat period between a very
differentobservations.Withinasingleobservation,thepeak
shortWDrotationperiodandtheKeplerianperiodofdense
often seems to be multiperiodic (see Fig. 1, especially ob-
blobs in the inner accretion disc close to the WD. In such
servation 0506531701). By using Eq. (3), it was determined
a scenario, the latter period might only be quasi-periodic,
that, to obtain a ±1 s error in a measurement of a period
duetovariationsintheKeplerianradiusandassociatedKe-
with a value near 67 s, the length of the light curve needs
plerian period of such orbiting inhomogeneities, yielding a
to be 2245 s. To investigate this variability within a single
variable beat period. Odendaalet al. (2014) showed that a
observation,theEPICpnlightcurvescontainingtheperiod-
67 s beat period could in principle result from a Keplerian
icitywerethendividedintoaseriesofconsecutivesegments,
period in the ∼4–15 s range, and a WD rotation period
each with a length of exactly 2245 s. The starting point of
in the ∼4–12 s range (just above the break-up period of
aparticularsegmentwasdisplacedbyavaluecloseto600s
theWD),butnoconsistent X-raymodulation wasfoundat
relative to the starting point of the previous segment, with
suchhighfrequenciesduringthetiminganalysis.TheeLIMA
the exact displacement determined by the total length of
modeldescribed inthenextsection providesamoreplausi-
the observation, so that all the data points in the observa-
bleframework for theinterpretation of P67.
tioncouldbeused.Theseoverlappingsegmentsenabledthe
calculation of a ‘moving average’ of the period through the
course of the observation. A LS analysis and accompany-
ingsignificancecomputationforfrequenciesbetween10and
5 THE eLIMA MODEL FOR CAL 83
20 mHzwere performed for each segment.
The dynamical periodogram created in this way is ThepropertiesoftheobservedP67 areverysimilartothose
shown in Fig. 3 for the EPIC pn light curve of the longest of DNOs. The observed modulation amplitudes fall in the
observation,0506531701, containing73segments.Theaver- range that has previously been observed for DNO light
age CR persegment is also shown for comparison. Theseg- curves,althoughP67 seemstobeslightlylesscoherent(with
ments in which a period with frequency between 13.5 mHz Q of the order of 102) than typically expected for DNOs
and16.5mHzwasfoundata>95.45percentlevelareanno- (with Q usually in the 103–107 range). This leads one to
tatedwiththeperiodanditssignificance.ThenatureofP67 consider the possibility that, like DNOs, P67 may originate
in the dynamical periodograms of the shorter observations inanequatorialbeltaroundtheWD.Thelowercoherenceof
was similar to that in 0506531701. P67 canbeexpected,duetothemoredynamicnatureofthe
It is evident that there is significant variability of at environmentontheWDsurfaceinSSSs.Withthenatureof
least ∼3stoeachsideofthemeanP67,evenontime-scales DNOsindwarfnovaebeingexplainedbythewell-established
of a few hours, and that P67 also tends to disappear and LIMAmodel(summarizedin§3),wenowproposetheappli-
reappear again after some time. The length of observation cationofasimilarmodeltothemore‘extreme’environment
0506531701 is approximately one half of the orbital period, on the surface of the WD in SSS CAL 83, i.e. the eLIMA
andiftherewasadirectcorrelationbetweentheperiodvalue model.
and the orbital motion, one would expect the period varia- In this section, it is illustrated in a broad quantitative
tiontotaketheapproximateshapeofonehalfofasinusoid. manner that the eLIMA model for P67 is indeed plausible.
However, the pattern of the variability of the modulation First,somecontextisprovidedforthismodelinCAL83by
inthisobservationdoesnotsuggestanycorrelationwhatso- someremarksonthebeltlocationin§5.1,followedin§5.2by
everwiththeorbitalmotion.Atfirstglance,Fig.3seemsto adiscussionofthemagnitudeoftheWDmagneticfield,and
suggest that theremay be a correlation between theperiod its implications regarding the flow of material in the vicin-
value and the CR, but a correlation study did not reveal a ity of thebelt.The mass of theequatorial belt is estimated
statistically significant correlation. in §5.3. The generation of a toroidal magnetic field within
Asexplained in §2,thequalityfactor Q can beusedto the rotating belt is discussed in §5.4, as well as a poten-
quantifythecoherenceofaperiodicity.Toquantifythevari- tial explanation of the variations in P67 within the eLIMA
ability in P67,theperiod valuesand correspondingtimes of framework. The section concludes with the investigation of
observation t were extracted from observation 0506531701, possible longer time-scale X-ray periodicities related to P67
utilizing data only from the segments in Fig. 3 with a in §5.5.
MNRAS000,1–17(2016)
An eLIMA model for CAL 83 9
LSpower Period(significance)
65.9 (99.99)
65.9 (>99.99)
12 64.9 (>99.99)
64.9 (99.99)
24 64.9 (99.31)
65.9 (>99.99)
65.4 (>99.99)
64.9 (>99.99)
65.4 (99.94)
21
10
66.4 (99.72)
66.4 (99.62)
18
8
66.9 (99.48)
67.9 (>99.99)
67.9 (>99.99)
15 67.9 (>99.99)
66.9 (>99.99)
65.9 (99.69)
T0 65.4 (99.99)
nce 6644..99 ((9999..3873))
si
12 Hours 6 666444...559 (((999997...954660)))
64.9 (99.87)
65.9 (99.99)
66.4 (99.08)
67.9 (99.88)
68.4 (99.79)
9 64.0 (99.56)
64.5 (99.87)
64.5 (99.91)
4 64.9 (99.95)
66.4 (99.56)
66.4 (>99.99)
66.4 (>99.99)
66.4 (>99.99)
65.9 (99.89)
6
71.1 (97.93)
69.5 (>99.99)
68.9 (99.60)
2 66.4 (95.57)
64.0 (99.87)
3 66.9 (98.06)
66.9 (99.99)
66.9 (>99.99)
66.4 (>99.99)
66.4 (99.99)
66.9 (>99.99)
67.9 (99.34)
68.9 (99.98)
68.4 (99.36)
0
11 13 15 17 19 9.0 7.4 5.8
T0=BJDTDB2454981.839 Frequency(mHz) CR
Figure3.Lomb-Scargle(LS)periodogramsofEPICpndataset0506531701 ofCAL83(thelongestcontinuous XMM-Newton observa-
tion), illustratingthe variabilityin the value of the ∼67 s period. The colour represents the LS power, and in cases where a peak was
detected between 13.5 and 16.5 mHz at a >95.45 per cent significance level, the corresponding period in seconds and its significance
percentage are given. The error in the period values is ±1 s. The BJDTDB reference represents the start of the observation. For a
comparisonwiththeX-raycountrate(CR),thebroadbandlightcurveisalsoprovided.
5.1 The location of the equatorial belt while the Keplerian period associated with Rphot is signifi-
cantly smaller, i.e.
According to Lanzet al. (2005), the photospheric radius of
theWDinCAL83isRphot ∼(7.0±0.7)×108 cm,andthis
willberegardedasanapproximateouterlimitfortheequa-
atonrdiatlhbeelrte.sLulatninzgeKt aepl.leersitaimnartaeddiutsheasWsoDciamteadsswaisth∼a1.p3eMrio⊙d, PK≈9(cid:18)1.3MM1⊙(cid:19)−1/2(cid:18)7.0×Rp1h0o8t cm(cid:19)3/2 s . (6)
of 67 s is
Therefore, an equatorial belt with rotation period ∼67 s at
∼7×108 cm rotates more slowly than the local Keplerian
RK =2.7×109(cid:18)1.3MM1⊙(cid:19)1/3(cid:18)6P7Ks(cid:19)2/3 cm , (5) vweiltohciatyW, pDoscsoibrelythdautehtaostahneemveangnloetnigcecroruoptalitnigonofpetrhieodb.elt
MNRAS000,1–17(2016)
10 A. Odendaal and P. J. Meintjes
5.2 The magnetic field of the white dwarf One can see that the value of Ω˙∗ is proportional to B1 to
the power 2/7 (through its dependence on RM), and there-
If the belt now has a larger angular velocity than the WD
fore does not change too drastically with large changes in
core, it indicates that the WD magnetic field is too weak
magneticfield.E.g.,whilekeepingtheotherparametersthe
torigidly couple thebelt tothecore. Anupperlimit tothe same aspreviously,thevalueof Ω˙∗ changes byonly oneor-
magnetic field strength can be calculated with Eq. (2), but derofmagnitudefrom10−15 s−2 to10−14 s−2whenvarying
an estimate of the potential spin-up rate Ω˙∗ as a result of thedipole field strength from B1 ∼3×102 Gto ∼106 G.
accretiontorquesmustfirstbeobtained.Toperformthises- Once again assuming ρwd ∼ 106 g cm−3, the upper
timate,oneneedstoconsiderthemass,radiusandmagnetic
limit for the product of the interior magnetic field compo-
field of the WD.
nents to avoid rigid rotation can now be calculated with
One must keep in mind that Rphot includes not only Eq.(2).WhenvaryingΩ˙∗between10−15 s−2and10−14 s−2,
theWDcore,butalsoanextendedaccretedenvelopeonthe and the radius between R1 ∼ 4 × 108 cm and Rphot ∼
WD surface, with hydrogen burningat its base. In orderto
7×108 cm, the value of B ∼ B varies from a few times
determinetheactualradiusoftheWDcore,onecanconsider r φ
104 Gtoafewtimes105 G.Inturn,theinnerfieldstrengths
theradiusR1ofazero-temperatureWDthatisrelatedtoits can be considered as an upper limit to the strength of
mass M1 by (Hamada& Salpeter 1961; Eracleous & Horne the surface field B1. As an order of magnitude estimate,
1996)
the primary field strength should therefore obey the limit
R1 =4.0×108(cid:18)1.3MM1⊙(cid:19)−0.8 cm . (7) Bst1ru.ctu10re5aGt tthoeailnlonwertehdegeexoisfttehnecedoisfc.a non-corotating belt
The WD radius derived with Eq. (7) can be considered
as a lower limit when considering hot WDs, as shown by 5.3 The mass of the equatorial belt
Panei, Althaus& Benvenuto(2000).Thereforetheradiusof
theWDin CAL 83canbeestimated asR1 &4.0×108 cm. If the observed decreases in P67 result from spin-up of the
equatorial belt, this places certain constraints on the mo-
At a certain distance from the WD, the magnetic pressure
ment of inertia and thus the mass of the belt. The maxi-
will be equal to the ram and gas pressure of the accreting
material.ThislimitingradiusisknownastheAlfv´enradius. mum spin-up rate Ω˙b of the belt can be estimated by con-
sidering the period variations as illustrated in Fig. 3. The
Foraspherical accretion flow, theAlfv´en radiusis givenby
(Elsner & Lamb 1977) rapid decrease in P67 from 5 to 5.8 hours elapsed since T0
inobservation0506531701willbeassumedtoberepresenta-
B4R12 1/7 tiveofthemostrapidchangeintheperiodoftheequatorial
rM=(cid:18)8GM11m1˙2acc(cid:19) , (8) obfelΩt˙b(P. bT)h,eanpderwioildl bdeiffuesreedncteobeesttiwmeaentetthheetmwaoxismegummenvtasluies
where B1 represents the dipole field strength of the WD. ∆P =(68.4−64.5) s=3.9 s,andthetimeelapsedbetween
For a cylindrical configuration, the Alfv´en radius (also themis∆t=3000 s.Thus,P˙b =∆P/∆t=1.3×10−3,and
known as the magnetospheric radius) is typically given by
since Ωb =2π/Pb,
(Ghosh & Lamb 1979)
RM ∼0.5rM . (9) Ω˙b ∼1.8×10−6(cid:18)1.3×P˙b10−3(cid:19)(cid:18)6P7bs(cid:19)−2 s−2 . (13)
IafnRdMfor>raRd1ii,Rth<e aRccMr,ettihoenadcicsrcetsitnrgucmtuarteerwiailllwbileldflioswruapltoendg, In Eq. (11), the factor (GM1RM)1/2 represents specific an-
gularmomentuminaKeplerianrotatingmasselement.The
the magnetic field lines, and be channelled onto the polar
specific angular momentum of the belt magnetically linked
caps of the WD. For example, in the case of CAL 83, if
wbeeloawss)u,mEqe.m(˙8a)ccyi∼eld1s0−7 M⊙ yr−1, and if B1 ∼ 105 G (see ittostahzeimWutDhaisl veestloimciatyt.edThbeysRpvinb,-φu,pwrhaetereΩ˙vbb,φof=th2eπbRe/ltPbcains
thusbe expressed as
rM∼2×108(cid:18)10B51G(cid:19)4/7(cid:18)4×1R018 cm(cid:19)12/7× Ω˙b =m˙acc(cid:18)2πPRb2(cid:19) Ib−1 , (14)
(cid:18)1.3MM1⊙(cid:19)−1/7(cid:18)10−7mM˙a⊙ccyr−1(cid:19)−2/7 cm , (10) oorb,tasionlsving for the moment of inertia Ib of the belt, one
icmaspelywinhgerReMthe∼W10D8rcomtaatecscoarsdainrgigtiodEbqo.d(y9,)a.nCdoenxspideerrieinngceas Ib= m˙Ω˙abcc (cid:18)2πPRb2(cid:19)
‘spin-up’ due to the transfer of angular momentum from
the accretion disc, the rate of change of the spin angular ∼1.6×1041 m˙acc Ω˙b −1×
velocity of the WD can be expressed as (e.g. Frank et al. (cid:18)10−7 M⊙ yr−1(cid:19)(cid:18)1.8×10−6 s−2(cid:19)
2002, pp.162–163): R 2 Pb −1 g cm2 , (15)
Ω˙∗ =m˙acc(GM1RM)1/2I−1 , (11) (cid:18)7.0×108 cm(cid:19) (cid:18)67 s(cid:19)
which is ∼9 orders of magnitude smaller than that of the
where I is the moment of inertia of the WD. For a solid
spherical body,I = 25M1R12, i.e. in CAL 83, WtheD.obTsherisverdepΩ˙rebseisnttsoabemeaxxpimlauinmedvablyuethfoerspthine-uinperotfiathief
I ∼1.6×1050 M1 R1 2 g cm2 . (12) belt. One can now proceed to estimate the mass of the
(cid:18)1.3 M⊙(cid:19)(cid:18)4.0×108 cm(cid:19) belt. For simplicity, it will be assumed that the geometry
MNRAS000,1–17(2016)