Simulations of spectral lines from an eccentric precessing accretion disc

0.44 MB·English

by  S. B. Foulkes

Mon.Not.R.Astron.Soc.000,1–16(2004) Printed2February2008 (MNLATEXstylefilev2.2)
Simulations of spectral lines from an eccentric precessing
accretion disc
Stephen B. Foulkes1, Carole A. Haswell1, James R. Murray2,3, Daniel J. Rolfe 1,2
1Department of Physics & Astronomy, The Open University,Walton Hall, Milton Keynes,MK7 6AA, UK.
4 2Department of Physics & Astronomy, Universityof Leicester, UniversityRoad, Leicester,LE1 7RH, UK.
0 3Department of Astrophysics & Supercomputing, Swinbourne Universityof Technology, Hawthorn, VIC 3122, Australia.
0 Email SBF: [email protected], CAH: [email protected], JRM: [email protected], DJR: [email protected]
2

n
a Accepted. Received
J

9
1 ABSTRACT
Two dimensional SPH simulations of a precessing accretion disc in a q = 0.1 binary

1 system (such as XTE J1118+480) reveal complex and continuously varying shape,
v
kinematics, and dissipation. The stream-disc impact region and disc spiral density
0
wavesareprominentsourcesofenergydissipation.Thedissipatedenergyismodulated
7 ontheperiodP =(P −1−P −1)−1withwhichtheorientationofthediscrelative
3 sh orb prec
to the mass donor repeats. This superhump modulation in dissipation energy has a
1
variation in amplitude of ∼10% relative to the total dissipation energy and evolves,
0
4 repeatingexactlyonlyafterafulldiscprecessioncycle.Asharpcomponentinthelight
0 curveisassociatedwithcentrifugallyexpelledmaterialfallingbackandimpactingthe
/ disc. Synthetic trailed spectrograms reveal two distinct ‘S-wave’ features, produced
h
respectivelybythestreamgasandthediscgasatthestream-discimpactshock.These
p
S-waves are non-sinusoidal, and evolve with disc precession phase. We identify the
-
o spiral density wave emission in the trailed spectrogram. Instantaneous Doppler maps
r showhowthestreamimpactmovesinvelocityspaceduringanorbit.Inourmaximum
t
s entropy Doppler tomogram the stream impact region emission is distorted, and the
a
spiral density wave emission is suppressed. A significant radial velocity modulation
:
v of the whole line profile occurs on the disc precession period. We compare our SPH
i simulation with a simple 3D model: the former is appropriate for comparison with
X
emissionlineswhilethelatterispreferableforskewedabsorptionlinesfromprecessing
r discs.
a
Key words: accretion: accretion discs - X-rays: binaries - binaries: low mass X-ray
binaries, cataclysmic variables - stars: individual: A0620-00, XTE J1118+480,GRO
J1655-40- methods: numerical
1 INTRODUCTION modulated tidal stresses (Osaki1985; Murray 1998). This
modulation is accompanied by a change in the solid angle
Superhumps are photometric modulations in the observed subtended by the disc at the central X-ray source, and it
lightcurvesfoundinsomeshortperiodcataclysmicvariables is this shape change which leads to the irradiation-powered
(CVs) (Patterson 1998), and in some of the transient low superhumpsin X-ray bright SXTs (Haswell et al. 2001).
mass X-ray binaries known as soft X-ray transients (SXTs)
MaterialfromthedonorstarleavesthefirstLagrangian
(Zurita et al. 2002). In general the superhump period, P ,
sh point(L1)andflowsalongaballistictrajectorytowardsthe
isafewpercentlongerthanthebinaryorbitalperiod,P ,
orb centralprimaryobject.Whenthisgasstreamencountersthe
andisattributedtotheprogradeapsidalprecessionofanon-
accretion disc material, large amounts of energy are dissi-
axisymmetric accretion disc. Such precession is expected in
patedinabrightspotregionontheouteredgeofthedisc.As
extreme mass ratio systems (q . 0.3, where q = M2/M1
the eccentric accretion disc precesses in the inertial frame,
is the mass ratio) in which the disc encompasses the 3:1
and the orbital motion of the binary proceeds, the energy
resonance (Lubow 1991). If the disc extends to the 3:1 res-
dissipated at thebrightspot will bemodulated as described
onant radius, it becomes eccentric and non-axisymmetric
byRolfe, Haswell & Patterson (2001 hereafter RHP2001).
(Murray 1998)anditwillthenprecessunderthetidalinflu-
enceofthesecondarystar.InCVs,superhumpsinthelight Spectroscopic observations of superhumping CVs
curve are understood as being powered directly from the (Vogt 1982; Hessman et al. 1992; Patterson et al. 1993a)
2 S. B. Foulkes et al.
and SXTs (in particular XTEJ1118+480) have shown that The model includes a simple 3D structure for the
trailedspectrogramsofsuchobjectsarecomplexandchange brightspotregion.Anellipticalcross-sectioninther-zplane
from night to night (Torres et al. 2002; Zurita et al. 2002; is assumed, with size r and h 1 centred on the outer
spot spot
Haswell et al. 2004).Overintervalsofdaysormonths(many discedge.r ,h andthespotbrightnessdecreasedown-
spot spot
times the orbital period) the spectral line profiles evolve streamfrom theinitialimpactase−θ2/∆θs2pot whereθ isthe
from almost symmetric to enhanced red-shifted or blue- angular distance downstream from the initial impact point.
shifted. These changes have been attributed to a precess- Upstream of the impact the brightspot surface is rounded
ingaccretion disc,andhereinwepresentcalculations which offwithahemisphereofuniformbrightnessequaltothatat
strongly support this interpretation. theinitialimpact.Weusedr =0.03aandh =0.012a
spot spot
In this paper we describe the results from two simula- at θ=0 (aisthebinaryseparation). Theangularextentof
tions of a precessing eccentric accretion disc. The first sim- thebrightspotregion isset by∆θspot =∆θ0adisc/r0,where
ulation consisted of a simple analytical three-dimensional r0 is the disc radius at the impact point and adisc is the
representation of the disc and brightspot region based semi-major axis of the disc. This keeps the arc length of
on RHP2001. The second model was a high-resolution the brightspot region roughly constant at ∆θ0adisc as the
smoothed particle hydrodynamics (SPH) simulation. SPH eccentric disc precesses. Weused ∆θ0=30◦.
is a Lagrangian numerical method for modelling fluid flow The disc was modelled as an ellipse with a =0.31a
disc
using a set of moving particles. The fluid properties at any and eccentricity 0.093 at the outer edge2. Streamlines and
point within the flow field are determined by interpolating brightnesscontourswithinthediscareellipseswithsizeand
from the local particle properties. This interpolation takes eccentricity decreasing smoothly tozero at theprimary ob-
the form of a summation over the local particles, weighted ject,withthevelocitygivenby|V~ |.Theinnerboundaryis
disc
accordingtotheirdistancefrom theevaluation point.Fora near-circular with radius 0.025a. The disc brightness varies
comprehensive review of the method see Monaghan (1992). as a−1.2. The disc is flared with total thickness increasing
orbit
In section 2 we detail the binary system modelled and linearly from 0 at the primary object to 0.02a at the outer
the two numerical methods employed to model the system. edge.Anorbitalinclinationof70◦wasusedtoavoideclipses.
Section 3 presents our results and Section 4 is devoted to In the brightspot region, where the stream and disc
discussion and conclusions. flows merge, we adopted a velocity distribution which was
non-zerobetweenthelocaldiscvelocityatthatpoint,V~ ,
disc
and the velocity V~disc + (V~stream,0 − V~disc)e−θ2/∆θs2pot,v.
2 SIMULATIONS Thus, at the impact point, velocities between V~ and the
disc
2.1 Binary Parameters ballistic stream velocity at theimpact, V~stream,0 contribute
to the Doppler broadening of the local line profile. Down-
TosimulateatypicalSXT,weadoptedamassratioq=0.1. stream the local velocity smoothly approaches the disc ve-
Theorbitalperiod,P ,is 4π2a3/GM 1/2,whereaisthe locity. Upstream of the impact the exponential factor was
orb t
orbital separation, and Mt(cid:0)is the total (cid:1)mass of the binary set to 1. ∆θspot,v behaves as ∆θspot as the eccentric disc
system. The mass transfer rate was 1.46×10−9Mtyr−1. precesses. Weused ∆θ0,v =5◦.
The local linewidths were assumed to have Gaussian
profiles, with the disc width set to the disc thermal veloc-
2.2 Simple Analytical Approximation to a ity. The brightspot profile width was the quadratic sum of
Precessing Eccentric Disc thediscthermalvelocityandthethermalvelocityofa104K
Herewebrieflydescribeasimplemodelfortheeccentricdisc sourcedecayingase−θ2/4∆θs2pot downstream.Thedecaywas
and stream-disc impact developed from that used to model imposed for consistency with the brightspot brightness, as-
latesuperhumpsintheSUUMaCVIYUma,seeRHP2001. suming brightspot brightness ∝Tb4rightspot.
The model assumes the brightspot emission is directly pro-
portional tothekineticenergy oftherelativemotion of the
2.3 SPH Simulation
stream and disc flows at the impact point, i.e. the rate of
energy dissipation at thebrightspot is: The accretion disc was also modelled using a two-
dimensional SPH computer code that has been described
E˙ = 21M˙Vr2el⊥+ 12M˙Vr2elk VVrelk in detail in Murray (1996, 1998). Modifications have been
relk made to convert the code from FORTRAN to C++. C++
(cid:12) (cid:12)
where M˙ is themass flowrate of thestr(cid:12)eam(cid:12)and V and is an Object Orientated computer language allowing each
rel⊥
particle to have its own private attributes (e.g. position,
V aretherelativevelocityofthestreamanddiscresolved
relk
velocity) and particle interactions can be coded in a more
perpendicularandparalleltothediscrespectively.Thedisc
flexiblemanner.Thecodewasalsoextendedtoallowitsex-
velocity at radius r from the primary object is assumed to
ecutiononanarrayofprocessorsusingtheMessage-Passing
bethatofanellipticalorbitaroundtheprimaryobject,that
Interface Standard3. The SPH results presented here were
is:
generated on a small private Linuxnetwork.
|V~disc|=rGM1 2r − a 1 , In the SPH simulation the total system mass, Mt, and
(cid:16) orbit(cid:17)
where aorbit is the semi-major axis of the orbit and M1 is 1 SeeFigure8ofRHP2001.
themassoftheprimarilyobject.Thevelocityofthestream 2 TheeccentricitywaschosetomatchthatofourSPHsimulation
V~stream is simply that of theballistic trajectory. 3 http://www-unix.mcs.anl.gov/mpi/
Spectroscopic simulations of superhumps 3
Figure 1. The top row (a, b, c) are accretion disc surface density maps using a logarithmic colourscale for binary phases 0.00, 0.36,
and0.64respectively. Thesecondary orbitsanti-clockwisewithrespecttothe inertialframe,withmassbeingadded fromthe L1 point
attherightofeachmap.ThesolidcurveistheRochelobeoftheprimary,plottedinaframethatco-rotateswiththebinary.Thearrow
in the lower right-hand corner of each map indicates a fixed direction to an observer. The middle row (d, e, f) are the corresponding
dissipationmapsplottedwithalogarithmiccolourscale.Thebottomrow(g,h,i)arethematchingDopplermapsplottedusingalinear
colourscale. The letter “G” on map i, indicates the point at which the gas stream impacts on the outer edge of the accretion disc and
theletter “H”indicatesthearcofemissionproducedbythediscparticlesintheimpactregion.
4 S. B. Foulkes et al.
the binary separation, a, were both scaled to unity. The occur and hence the hot spot region will have a reduced
binaryorbital period, P ,was scaled to2π.Theaccretion contributiontotheenergydissipated.However,thehotspot
orb
disc had an open inner boundary condition in the form of region will still contribute a significant amount to thetotal
a hole of radius r1 = 0.025a centred on the position of the dissipation observed from the accretion disc.
primary object. Particles entering the hole were removed
fromthesimulation.Particlesthatre-enteredthesecondary
Roche lobe were also removed from the simulation as were
3 RESULTS
particlesthatwereejectedfrom thediscandhadadistance
>0.9a from thecentre of mass of theprimary. 3.1 Disc morphology: orbital modulation
Theartificial viscosityterm ofMurray(1996) was used
In this subsection we follow the evolution of the SPH sim-
toimprovetheeffectiveshearviscosityintheSPHequations
ulated accretion disc through a complete binary orbit. The
of motion. In the inner regions (<∼0.12a) of the accretion
images(a,b,c)inFigure1areasetofsurfacedensitymaps,
disctheparticlesgenerallyexecutedcircularorbits,andthe
plottedintheco-rotatingframe,fororbitalphases0.00,0.36
radialsheartermdominatedtheinnerdiscdissipation.How-
and 0.64 respectively. Each map uses the same logarithmic
ever,intheouterdiscthetidalinteraction ofthesecondary
colourscale,withbluerepresentinglowdensityandredhigh,
was stronger and the bulk term was more significant. The
we have also plotted the primary Roche lobe. Since we can
discenergydissipationwascalculatedusingtheSPHenergy
view the disc from any angle, we have arbitrarily fixed the
equationasinMurray(1996).TheShakura-Sunyaev(1973)
conjunctionoftheprimary(andhencezeroorbitalphase)to
viscosityparametersweresettoα of0.1andα of1.0,
low high
correspondtoFigure1(a).Thearrowinthelowerright-hand
and theviscosity switched from low to high as described in
cornerofeachmapindicatesafixeddirectionintheinertial
Truss et al. (2000). The SPH smoothing length, h, was al-
frame and we calculate radial velocities as they would be
lowed to vary in both space and time and had a maximum
observed by an observer in thisdirection.
valueof 0.01a.
These density maps show the disc to be asymmetric,
The SPH code was started with no particles, i.e. zero
eccentric and continuously changing shape under the tidal
mass in the accretion disc. A single particle was injected
into the simulation every 0.01Ω−1 at the first Lagrangian influence of the donor star; its overall position is moving
orb only very slowly in the inertial frame. There are wrapped
point. The sound speed in the donor atmosphere was used
spiraldensitywavesthatextendfromtheoutermostregions
astheinitialspeedofeach particleandtheyfollowed abal-
ofthediscdowntoapproximatelythecircularisation radius
listic trajectory. Within twenty binary orbits the resulting
of the system. These produce shear and dissipation in the
discbecameeccentricandencounteredtheLindblad3:1res-
outer regions of the disc. The spiral density waves propa-
onance.Afterthedischadreachedamassequilibriumstate,
gate angular momentum outwards, thus allowing the disc
wefollowed itsevolutionthroughfourdiscprecessioncycles
gastomoveinwardtowardsthecentralprimaryobject.The
(∼180 binary orbits) to produce the results described here.
precession of the disc means that the density maps do not
There were approximately fifty-three thousand particles in
repeat from one orbit to another; instead the maps repeat
thedisc,andtheaveragenumberof‘neighbours’,thatisthe
on a period a few percent longer than the orbital period.
average number of particles used in the SPH update equa-
We have placed a Microsoft AVI movie of the change
tions, was 8.9 particles.
in surface density as function of orbital phase on an In-
Thecomplicated hydrodynamicsofthestream-discim-
ternet web-site (see http://physics.open.ac.uk/FHMR/ for
pacthavebeeninvestigatedanalyticallybyHessman(1999)
more information).
and numerically by Armitage & Livio, first with the SPH
technique (1996) and later with the grid based ZEUS code
(1998). In the latter paper, they found the character of the
3.2 Light curve
stream-overflow in particular depended strongly upon the
equationofstate.Kunze,Speith&Hessman(2001)compre- InFigure1(d,e,f),wehaveplottedviscousdissipationmaps
hensively studied the impact with much higher resolution correspondingtothedensitymapsinFigure1(a,b,c).Each
SPH simulations, for a range of binary mass ratios. They dissipation map is represented using the same logarithmic
showedthatasubstantialfraction(perhapsmorethanhalf) colourscalewithbluelowdissipationandredhigh.Itisclear
of the stream overflowed the outer disc edge, to impact the from this set of images that the inner disc is in a state of
disc close to the circularisation radius. The latter paper is constanthighdissipation.Thespiralcompressionwavesand
particularly relevent as it demonstrates that the detailed the impact of the gas stream on the outer edge of the disc
shock structure of the impact region can be captured us- also generate high levels of dissipation.
ingtheSPHtechnique,ifsufficientnumbersofparticlesare We are unable to directly resolve the morphology of
deployed. The computational cost is however so great as to the accretion disc in observed interacting binary star sys-
make a calculation following the evolution of a disc pro- tems. The simplest directly observable quantity which re-
hibitively expensive. veals some of the underlying properties is the light curve.
As our SPH calculations are two dimensional, stream- Hence we use the SPH model to simulate the disc lumi-
overflowcannotoccurandthebrightnessofthehotspotwill nosity and generate an artificial light curve. We assume
beoveremphasisedbyafactordeterminedbycooling inthe thattheluminosityisgeneratedthroughviscousdissipation
stream. The SPH results presented in the following section which is promptly radiated away from the point at which
will show that the complex stream-disc interaction region it was generated. The simulated light curve can be directly
dominates the energy dissipation signature of the accretion compared with the observations. We must, however, bear
disc. In a three dimensional SPH code stream-overflow will in mind that in observed systems there will generally be
Spectroscopic simulations of superhumps 5
an orbital modulation in the visibility of each location in
theco-rotating frame. Hence, oursimulated light curves,in
which we have essentially assumed 100% visibility for each
location throughout theorbit, are likely to be simpler than
observed light curves.
The upper left plot of Figure 2 shows five simulated
light curves. The upper curve corresponds to the total vis-
cous dissipation from the whole disc; the lower four curves
correspond to the four smaller annuli defined in Figure 2
caption.Thedissipationintheinnerdisc(.0.12a,interme-
diatebetween‘c’and‘d’inFigure2)isconstant,exceptfor
stochastic noise. The lower left-hand plot is the dissipation
from the innermost disc region and was generated by sub-
tractingdissipationlightcurvebfromdissipationlightcurve
a.Subtractingcurvecfromaandsubtractingcurvecfromb
would also produce similar constant dissipation curves. Su-
perimposed on this, for disc regions (r > 0.2a) there is a
repeating series of “humps.” These humps recur on the su-
perhumpperiod,P .Themodulation consistsoftwosepa-
sh
ratecomponents.Thefirstistherelativelysmoothcontinu-
ous modulation, which has a minimum at orbital phase 0.0 Figure 3. Power spectrum of the SPH dissipation light curve
in Figure 2 (corresponding to Figure 1(a,d)) and reaches a Figure2(a).Thetimeseriescoveredapproximately60superhump
maximum at orbital phase ∼0.64, (corresponding to Figure periodswitharesolutionof0.01Ω−or1b.Thepeaklabelled0isthe
1(c,f)). The second is the much sharper spiky signal with superhumpfrequency,thesuperhumpharmonicsarealsolabelled
a short duration which appears at orbital phase ∼0.36 and (1-5).ThelowerpeakscorrespondtolinearcombinationsofΩorb,
ω andΩ.
corresponds to Figure 1(b,e). Photometric superhumps are
invariably detected in optical light. Since the hot inner re-
gionsofaviscously-heateddisccontributerelativelylittleto
placedMicrosoftAVImoviesshowingthesurfacedissipation
the total optical light, the comparison between observation
in the SPH simulation and in the analytical model on the
and our simulation might best be made by comparing one
Internetsite http://physics.open.ac.uk/FHMR/ .
of the lower curves in Figure 2.
To determine the superhump period, P , we took a
sh
Figure 1(d) corresponds to the light curve minimum. Fourier transform of approximately sixty superhump cycles
Fromthismapitisclearthatthedissipationatthestream- ofthesimulatedlightcurve.Thepowerspectrumisshownin
discimpactisataminimum.Thisisbecausethespiralcom- Figure 3. The spectrum is plotted using a logarithmic scale
pression arm in the outer disc has moved such that gas is torevealthelowerlevelpeaks.Alargepeakthatcorresponds
escaping from theprimary Rochelobe and returningto the tothesuperhumpperiod,labelled0inthefigure,dominates
secondary.ThegasleavingtheL1 pointfromthesecondary the spectrum; the superhump harmonics have also been la-
encounters the disc gas instantly and the two flows have belled(1-5).Usingthepowerspectrumweestimatedthesu-
a low relative velocity. Hence relatively little dissipation is perhumpperiodtobe(1+0.0222±0.0005)P ;thisimplies
orb
generated by the converging flows. At this phase the dissi- P =(46±1)P .Thesuperhumpperiodmanifestinour
prec orb
pation is dominated by the shear flow in the inner regions simulation is in agreement with the analysis of Mineshige,
of the disc. Hirose&Osaki(1992).Theothermuchloweramplitudepe-
Asthebinaryorbitproceeds,alargevoidopensupbe- riodicitiespresentoccuratfrequenciesΩrelatedtoΩ and
orb
tweentheL1pointandtheaccretiondisc,asshowninFigure disc apsidal precession frequency ω by:
1 (b) and (e) and the spiral arm which previously extended
towardstheL1regionbecomesdetachedfromthediscunder k(Ω−ω)=j(Ω−Ωorb)
centrifugalforcesandthenfallsbacktowardsthedisc.When
where k,j are positive integers.
thisgashitsthedisc,theimpactcausesasuddenriseindis-
sipation, corresponding to thespikypeak in thelight curve
atorbitalphase∼0.36,andtotheupperinhomogeneousarc
3.3 Trailed Spectra
on Figure 1(e). Once this gas has been re-assimilated into
thediscflowthedissipation falls again. Thesecondary con- Kinematics in observed systems are revealed primarily by
tinuesitsorbitaroundthedisc,andthestreamflowandthe thespectral line profiles. To compare the results of the two
disc flow at the stream-disc impact point become increas- simulations with CV and SXT observations we therefore
inglymisaligned.Astherelativevelocityofthetwoflowsin- generated syntheticemission line profiles and used these to
creases,thedissipation inthebrightspotregioncorrespond- build up synthetictrailed spectrograms.
inglyincreases,untilitreachesamaximumatorbitalphase An accretion disc emission line is Doppler broadened.
∼0.64,showninFigure1(c)and(f).Astheorbitproceeds, Horne&Marsh(1986) calculated spectrallineprofilesfrom
the angle between the stream and disc flows at the impact discswithsmoothazimuthallysymmetriclineemissivitydis-
point decreases again, until the next minimum is reached tributions, and Horne (1995) further included the effects of
at orbitalphase1.022. Similarbehaviourisseen inthesim- anisotropic turbulence on the local line broadening profile.
ple3Dmodel,andwasfullyexploredinRHP2001.Wehave We generated artificial emission line spectra for emissivity
6 S. B. Foulkes et al.
Figure2. ThetopleftplotisSPHviscousdissipationlightcurvesfordifferentregionsoftheaccretiondisc.Thetopcurve,labelleda,is
forthewholeaccretiondisc,theneachcurveindescendingordercorrespondstodissipationfromthediscatradii>0.05a(b),>0.1a(c),
>0.2a(d) and>0.3a(e)respectively. Thelightcurveminimumandtwomaximumpointsareindicated. Thelowerleft-handplotisthe
signalfromthediscinnerregionandwasgeneratedbysubtractinglightcurvebfromlightcurvea.Theplotsontheright-handsideare
discdissipationmapsthatcorrespondtothelightcurveminimumandtwomaxima.Thecircleslabelledb,c,dandeshowthedistances
fromtheprimarilythatcorrespondtothelightcurvesb,c,d,erespectively.
distributions calculated from both the simple 3D code and is shown in Figure 5 and the trailed spectra for a full disc
the2D SPH code. precession cycle is shown in Figure 6.
For the simple 3D model we adopted Gaussian local UsingtheSPHmodelthedissipationfromtheaccretion
line profiles with widths proportional to the disc thermal discisdividedintoradialvelocitybins.Theemissionineach
velocity; the brightspot emission was assumed to be ther- velocity bin is the sum of the viscous dissipation from each
mally broadened using a brightspot temperature of 104K. particle within the velocity bin. For each SPH time step,
We did not check for occultation of the accretion disc by 0.01Ω−1, an emission line profile was generated and shown
orb
the brightspot, but the 70◦ inclination was chosen to avoid in the trailed spectrogram in Figure 7. We have used a lin-
eclipses by the secondary. Figure 4 shows three dissipation eargreyscale to representdissipation intensity.Thebottom
mapsfromtheanalyticalmodelfororbitalphases0.00,0.36 of the spectrogram corresponds to the minimum flux point
and 0.64 which correspond to the same orbital phases used on the artificial light curve of Figure 2 (labelled ‘min’). At
in Figure 1. These maps show the dissipation plotted using approximately orbital phase 0.36 (‘max1’) there is a large
alogarithmicgreyscalewithwhitelowdissipationandblack emission associated with the first large spiky hump on the
high.TheprimaryRochelobehasbeenplottedusingasolid light curve. At about orbital phase 0.64 (‘max2’) there are
curve.Thearrows indicatea fixeddirection, as in Figure 1. two distinctive emission arcs. These arcs are generated by
The extent of the hotspot region is clearly visible in these theparticlesinthebrightspot region. Thelower arcisfrom
maps. The simulated trailed spectra for two orbital periods the particles in the gas stream that collide with the parti-
Spectroscopic simulations of superhumps 7
Figure4. Dissipationmapsfortheaccretiondiscusedintheanalyticalmodel.Thedissipationisdiplayedusingalogarithmicgreyscale
forbinary phases 0.00, 0.36, and 0.64respectively. The solidcurve isthe Roche lobeof the primary.The secondary’s orbital motion is
vertiallyupwards.Thearrowdisplayedineachmapindicatesafixeddirectiontoanobserver,withphases definedasinFigure1.
clesintheaccretiondisc.Theotherarcistheemissionfrom tion in the line profile which occurs in the SPH simulation.
theparticlesintheaccretion discthatareinthebrightspot At first glance the pattern appears to repeat from orbit to
region. The disc spiral density compression waves produce orbit, but in fact there is a gradual evolution which can be
enhanced emission that can be seen in Figure 7 as features clearlyseeninFigures8and9.Thespectralpatternrepeats
in the trailed spectrogram along the velocity lines ∼300 after a complete disc precession period.
kms−1and ∼−700 kms−1. Figure 8 shows the trailed spec-
trogram for a complete disc precession cycle, equivalent to
3.4 Doppler tomography
that of thesimple 3D model in Figure 6.
Allofoursimulatedtrailedspectrogramsaredominated Dopplertomographywasdevelopedtointerprettrailedspec-
bystrongdouble-peakedemission whichexhibitssignificant trograms of CVs (Horne& Marsh 1988; Marsh 2000). Gen-
variability.Emissionfromthebrightspotregionisprominent erally observers use a maximum-entropy inversion process
in all spectrograms. InFigures 5to 8thesummation of the to generate the most likely Doppler map from the spectra.
emission horizontally and vertically is shown, so the upper Dopplertomography condenses theinformation in atrailed
panel is the mean line profile, and the right-hand panel is spectrogramintoavelocitymapoftheemissionregionsina
the light curve. The light curve in Figure 6 clearly shows system. The observed spectral line profiles as a function of
the modulation in the superhump profile which was fully orbital phaseareinverted torevealthebest-fittingDoppler
explored in RHP2001 and Rolfe (2001). map of the line flux as a function of velocity in the binary
Figure 9is amontage of SPH trailed spectra for differ- frame. It implicitly assumes:
ent orbits sampling a complete disc precession. Each spec-
• the emission regions are fixed within the co-rotating
trogram uses the same linear greyscale and the number in
frame
the upper left-hand corner of each image indicates the disc
• theemissiondoesnotvarywithtimeoverthecourseof
precession phase. This figure clearly demonstrates the vari-
a binary orbit
ability of accretion disc spectra with respect todisc preces-
• all points are visible at all times
sionphase.WehaveplacedontheInternetaMicrosoft AVI
• all motion is in theorbital plane
movieshowing thespectrogram’s evolution overa complete
• the width of theprofiles from any point is small
disc precession cycle (http://physics.open.ac.uk/FHMR/).
Inordertocompareourhighresolutiontrailedspectrawith Oursimulations, thosedescribedin (Murray 2000)and
currentobservations,wereducedthespectralandtimereso- eclipsemappingofV348Pup(Rolfe et al. 2000)revealdiscs
lutionofeachspectrogram ofFigure9byafactorof25and which violate the first two of these assumptions. Hence,
theresult is displayed in Figure 10. for precessing non-axisymmetric discs, it is safer to work
In all of our trailed spectrograms we see the effects of with trailed spectrograms directly rather than construct-
theprecessingdisc.InFigure6weseealongperiodS-wave ingDopplertomograms. Nevertheless,manyof therelevant
intheenvelopeofthediscemission,causedbythedisc’spre- observational publications in the last decade have included
cession on P . The orbital modulation in the brightspot tomography, so we have constructed tomograms from our
prec
emission causes the shorter timescale S-wave which is most simulated line profiles.
clearly apparent where it crosses each of the two peaks in Doppler maps represent the binary configuration in
theline profile. two-dimensional velocity-space. Figure 11 shows an exam-
InFigure7weshowthemorecomplex orbitalmodula- ple of an instantaneous Doppler map generated from the
8 S. B. Foulkes et al.
Figure 5. Trailed spectrogram using a linear greyscale for the analytical accretion disc model. The spectrogram is for two orbital
periods. The curves on the top and right of the spectrogram are the average line flux vertically and horizontally respectively, i.e. they
constitute ameanlineprofileandalightcurverespectively.
SPHdissipationdata.Theprimaryandsecondaryvelocities cellmultipliedbytheparticle dissipation,isplottedusinga
are overlaid on the Doppler map y-axis with the secondary linear grey intensity scale.
having a positive y-velocity. The lower of the two overplot- Figure 1(d, e, f) demonstrates that the emission from
ted arcs indicates the ballistic trajectory of thegas stream; the disc changes substantially in the course of a binary or-
the upper arc indicates the velocity of a circular Keplerian bit, and is far from fixed in the co-rotating frame. Hence
disc at positions along the ballistic stream trajectory: the theinstantaneous Dopplermap varies significantly with or-
stream’s ‘Kepler shadow’. The velocity of each particle is bital phase. Figure 1(g, h, i) shows detail of instantaneous
used to transformed each particle into velocity space. The Dopplermapscorrespondingtothedissipationsnapshotsin
velocity density at each velocity cell in the Doppler map Figure 1(d, e, f).
is the sum of the dissipation from each particle mapped to
Theprominentdissipationneartheletter“G”onFigure
thatcell,i.e.thenumberofparticlesoccupyingeachvelocity
1(i) arises from thegas stream as it impacts theouteredge
of the accretion disc. This gas is accelerated until it has
Spectroscopic simulations of superhumps 9
Figure 6. Trailedspectrogram usingalineargreyscaleforthesimple3Daccretiondiscmodel.Thespectrogramisforacompletedisc
precession.Thecurvesonthetopandrightofthespectrogram aretheaveragefluxverticallyandhorizontallyrespectively.
the same speed as the gas in the disc, producing the small InFigure1(g)therearethreearcsofenhancedemission.
very bright arc close to the impact point. The letter “H” Two of them are produced by the stream particles and the
onthisDopplermapindicatesthearcofemission produced disc particles at the impact; the third, uppermost, velocity
by the disc particles in the impact region. As the two flows arc is produced by emission from the centrifugally expelled
converge (progressing to velocity locations which move in material, which can be seen extending outside the primary
ananti-clockwisesense)therelativevelocityofthetwoarcs Roche lobe in Figure 1(d). The material outside the lobe is
decreasestozero.Emission from thegasdownstream ofthe moving up and slowly to the right in Figure 1(d), and so it
impactpointcanbeseentrailingaway(anti-clockwise)from appears in the upper right quadrant of the velocity map in
thebrightspotimpactpoint.Theouterdiscinthesimulation Figure1(g),whereitextendsacrossthedonorvelocitylobe.
does not consist of particles executing circular Keplerian
The disc in our simulation is eccentric, precessing
orbits, so the disc velocity at the impact point does not
rapidlywithrespecttotheco-rotatingframe(sinceitispre-
generally lie on the Kepler shadow trajectory overplotted
cessing only slowly in the inertial frame), and continuously
on themap.
flexing and relaxing on the superhump period. This causes
10 S. B. Foulkes et al.
Figure 7. TrailedspectrogramusingalineargreyscalefortheSPHaccretiondiscmodel.SeeFigure5caption.
theappreciable changes intheinstantaneousDopplermaps particles.Thesetwobrightarcsshowthechangeofvelocity
produced at different orbital phases, Figure 1(g,h and i). (and position) of the impact region as a function of orbital
phase.Thereisalsoanotherlowerintensityarcinthelower
Since observers cannot produce an instantaneous
left-handquadrant.Thesourceofthisarcisoneofthespiral
Dopplermap,weshowinFigure12,middleimage,aDoppler
density waves seen in Figure 1(d).
dissipationmapforacompleteorbitalperiod.Thismapwas
TheSPH trailed spectrogram, shown in Figure12,was
generatedbyaveragingtheinstantaneousDopplermapsfor
eachmodeltimestep,0.01Ω−1,i.e.629(≈100×2π)instan- inverted using doppler, a maximum entropy Doppler to-
orb mographypackagedevelopedbyTomMarsh4,andisshown
taneous maps corresponding to a complete orbit were aver-
in thebottom image ofFigure 12.Aconstant Dopplermap
aged.Twolargeemissionarcsdominatethismap.Thelower
wasgeneratedusingmakimgandthenscaledusingoptscl.
arcissituated justabovethegasstream ballistictrajectory
The Doppler map had a starting chi-squared value of 26.6
and was generated by the gas stream impacting the outer
edge of the accretion disc. The merging streams generate a
large dissipation. The upper arc is generated by the parti-
cles in the accretion disc that interact with the gas stream 4 http://www.astro.soton.ac.uk/∼trm/tar/doppler.tar.gz