Astron. Astrophys. 358, 154–168 (2000) ASTRONOMYAND

ASTROPHYSICS

Accretion disc models for AM Canum Venaticorum systems

W. El-Khoury and D. Wickramasinghe

Astrophysical Theory Centre, Department of Mathematics, Australian National University; ACT 0200, Australia(walid,dayal@maths.anu.edu.au)

Received 16 November 1999 / Accepted 16 March 2000

Abstract. Using a model of the vertical structure of accretiondiscs, we calculate disc spectra across a range of system param-eters for the helium-rich AM CVn cataclysmic variables, andinvestigate thermal instabilities in their discs.

Applying aχ2-minimisation technique, predicted HeI andHI continuum and line profiles are compared with those ob-served for two such systems, AM CVn and CR Boo, to estimateseveral system parameters. We find:

– mean accretor masses close to, but somewhat higher than,the0.7–0.8M range predicted by present population syn-thesis studies;

– typical helium-to-hydrogen number density ratios of102–104, consistent with a helium-degenerate donor sur-rounded by a helium-rich envelope; and

– mass-transfer rates of about1017gs−1, which may place thesystems in the region of potential disc instability.

S-curves calculated for the best-fit models are consistent witha thermally-stable disc in AM CVn, and a thermally unstabledisc in CR Boo.

Key words: stars: individual: AM CVn, CR Boo – stars: novae,cataclysmic variables – stars: evolution – stars: binaries: general

1. Introduction

AM CVn systems are puzzling variable stars with hydrogen-deficient spectra and ultrashort period photometric variations inthe range1000–3000s. Only six have been observed to date. It iscurrently believed that they are interacting white dwarf binary(IWDB) systems (Faulkner et al. 1972) in which a low-masswhite dwarf secondary of massM2 ∼ 0.04M fills it Rochelobe and transfers helium-rich gas to a white dwarf primary,around which a hydrogen-deficient accretion disc is formed.They are the white dwarf counterparts of the better studied neu-tron star binary systems, and are an interesting but little under-stood end-product of close binary evolution.

The original motivation behind the IWDB model was to ex-plain a peculiar object, HZ 29 (later designated AM Canum Ve-naticorum, or AM CVn), following observations by Malmquist

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(1936) and Humason & Zwicky (1947). Greenstein & Mathews(1957) had classified HZ 29 as a DB white dwarf since it washydrogen deficient, exhibiting only strong HeI absorption lines.Later observations suggested a quasi-stellar object (Burbidge etal. 1967) or a hot star (Wampler 1967). However, the discoveryof periodic variability (∼ 17min) by Smak (1967) suggesteda binary system and, after discovering flickering in the high-speed photometry, Warner & Robinson (1972) suggested thatAM CVn was a close binary in mass-transfer mode.

More objects with similar characteristics were identified:G61-29 (GP COM) was thought by Burbidge & Strittmat-ter (1971) to be an isolated white dwarf, but later photome-try showed flickering activity, suggestive of a binary system(Warner 1972). PG1346+082, later designated CR Bootis, or CRBoo (Nather 1985), and V803 Cen were other objects showingsimilar characteristics (O’Donoghue et al. 1987). CP Eri wasfound by Howell et al. (1991) to be a short-period photomet-ric variable of the AM CVn type. The latest object exhibitingAM CVn class characteristics is EC15330-1403, discovered byO’Donoghue et al. (1994).

Some class members exhibit high and low states in their opti-cal flux, which is also strongly suggestive of a mass-transferringclose binary, while others are more stable. The characteristics ofthis long-term variability are reminiscent of what is seen in cat-aclysmic variables of different sub-types – dwarf novae, nova-likes, SU UMa etc. – where a combination of thermal and tidaldisc instabilities appear to play the major role in explaining thelong-term variability. A similar interpretation may be possiblefor AM CVn systems (Warner 1995). AM CVn and EC153330seem to be in a permanent high state, showing broad and shal-low absorption lines presumably originating from a stable highmass-transfer rateM and optically-thick accretion disc – sim-ilar to nova-like variables and Z Cam systems. GP Com is theonly member with a stable low state, and exhibits an emissionline spectrum indicative of optically-thinner discs with lowerM . The other three members – CR Boo, V803 Cen and CP Eri– alternate between high and low states, with their spectra show-ing absorption lines in the former case, and continuous or weakemission lines in the latter case. Such behaviour is expected ifM is in an intermediate range where the disc becomes thermallyunstable – as in dwarf novae. Over and above the thermal in-stabilities, the discs in AM CVn stars are also expected to be

W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO 155

tidally-unstable by virtue of their low donor-to-accretor mass ra-tio q = M2/M1. Thus it has been suggested that AM CVn maybe in a permanent superoutburst state exhibiting a superhumpperiod that is slightly larger than the orbital period, making itsimilar to the high mass-transfer rate SU UMa variables (Osaki1996).

In this paper we investigate the viability of a disc-instabilityexplanation for some observed behaviour in these systems, andto constrain their parameters by modelling the spectra expectedfrom helium-rich accretion discs and then fitting the results toobserved spectra. The paper is arranged as follows. In Sect. 2we outline the current theoretical and observational status ofAM CVn and CR Boo. Sect. 3 presents the equations used tocalculate the vertical structures and emergent flux of accretiondiscs, describing both modelling techniques and assumptions.Numerical results for disc structures and line profiles are pre-sented in Sect. 4, together with spectral-fitting results for AMCVn and CR Boo. Finally, in Sect. 5 results are discussed, andconclusions drawn, in relation to disc instability models andindependent estimates of binary masses.

2. Observations

2.1. AM CVn

Observations of AM CVn have shown the presence of multipleperiodicities (Patterson et al. 1992). The1051s period is farfrom stable, changing erratically from year-to-year, making ita poor candidate for the orbital period of the system. Pattersonet al. (1992) suggested that this was the period of a permanentsuperhump due to a precessing disc, which is consistent with thelow mass ratio indicated for this system. A second more coherentperiod at1028.7325s± 0.0004 is also seen in this system, butit sometimes changes to a period of1011s. Harvey et al. (1998)proposed the1028.7325s period as the actual orbital period ofthe system. Although the periods show complex behaviour, thedisc in the system appears to be stable in the long-term, with noreports of large changes in the mean apparent magnitude. Indeedit seems that AM CVn is a system with a disc in a permanenthigh state, and most likely in permanent superoutburst.

In what follows we take1028.7325s as the orbital periodPorb. We use as our basic source of spectral data the phase-averaged optical spectrum published by Patterson et al. (1993)– hereafter PHS – which shows characteristic features such asthe shape of the continuum, line widths and the double-peakedfeature of the lines. From the lack of radial velocity variations inthe spectrum of AM CVn, PHS deduced that the radial velocityamplitude of the primaryK1 < 50km s−1.

2.2. CR Bootis

CR Boo is one of the three known unstable AM CVn systems,and exhibits variations in mean apparent magnitude stronglyreminiscent of those caused by thermal disc instabilities in dwarfnovae.

Extensive photometric analysis – Patterson et al. (1997) andreferences therein – led to the current belief that its orbital period

Porb = 1471s, and that a1490s variation is a good candidate fora superhump period which seems to stabilise at1487.29s±0.02after300–600 orbits.

For our purposes, we assumePorb = 1471s and adopt aspectrum for CR Boo in its high state kindly provided to us byJ. Patterson.

3. The model

3.1. Accretion disc structure

We assume an axisymmetric disc – extending from inner radiusRin to outer radiusRout – around a central accretor, and use thegeometrically-thin approximation withα-prescription for vis-cosity (Shakura & Sunyaev 1973) to describe the disc structure.Under these assumptions radial pressure gradients are negligi-ble, and angular velocity is effectively the Keplerian value. Insystems with extreme mass ratios, tidal effects due to the com-panion star will distort outer regions of the disc beyond the 3-1resonance radius. While these distortions may be important incalculating and interpreting phase-dependent line spectra, weneglect them here as we model only phase-averaged spectra.

For mass-transfer at rateM through a thin accretion discaround a primary with negligible rotation, the effective temper-atureTeff at disc radiusR is independent of viscosity parameterα and given by (Frank et al. 1992)

T 4eff =

38πσ

(GM1M

R3

)1 −

(R1

R

)1/2

, (1)

whereσ andG are the Stefan-Boltzmann and gravitational con-stants respectively. Note that the radiusR1 of the white dwarfmay be related to its massM1 and the Chandrasekhar limitingmassMch by (Nauenberg 1972)

R1 = 7.79 × 108

(M1

Mch

)−2/3

−(

M1

Mch

)2/31/2

cm, (2)

and thatTeff may be expressed as

Teff = 7.837 × 104(

M1

M

)1/2(

M

1017

)1/4

×(

R

R1

)−7/8(

R

R1

)1/2

− 1

1/4

K, (3)

whereM is measured in gs−1 and we have used a simplifiedversion of (2)

R1

R= 0.013

(M1

M

)−1/3

. (4)

Typically for AM CVn stars,Teff falls from Tin = 5 × 104Kto Tout = 104 K asR increases fromRin to Rout.

We calculate the radial and vertical structure of an accretiondisc – ignoring convection – as follows. We first divide the discinto a number of elemental rings of widthδR: the choice of this

156 W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO

number, as well as ofRin andRout, will be discussed below.For each ring, radiative transfer in the vertical directionz istreated in the two-stream grey approximation, using a formu-lation similar to that in Wehrse et al. (1994) – see also Shaviv& Wehrse (1986). They assume the two streams of radiation totravel normal to the disc, but we assume they travel at angleθ = cos−1(1/

√3) to the normal, which means we may use

conventional Gaussian quadrature to approximate the flux inte-gral. We stress that variations in disc structures between the twoformulations are minimal.

The frequency-integrated flux,F , and mean intensity,J , aredefined in terms of the frequency-integrated intensityI by

F =∫ 1

−1Iµdµ, (5)

J =12

∫ 1

−1Idµ, (6)

where µ = cos θ. Using n-point Gaussian quadrature withweighting-factorsai and corresponding outgoing and incomingintensitiesI+

i andI−i respectively, these expressions become

F = 2π

n∑i=1

aiµiI+i − µiI

−i , (7)

J =12

n∑i=1

aiI+i − I−

i . (8)

For the two-stream approximationn = 1, a1 = 1 andµ1 =1/

√3, in which case

F = 2πµ1(I+ − I−), (9)

J =12(I+ + I−). (10)

With no irradiation and assuming the LTE approximation,the model equations for an elemental ring at disc radiusR withlocal gas densityρ become

dF

dz= 4πκR(

σT 4

π− J), (11)

dJ

dz= − χF

4πµ21, (12)

dPg

dz= −Ω2

Kzρ +χ

cF, (13)

4πκR(σT 4

π− J) =

32αΩKPg, (14)

which give the grey fluxF , mean intensityJ , gas pressurePg

and temperatureT as functions ofz: κR andχ are Rosselandmean of absorption and extinction coefficients respectively,ρ =ρ(T, Pg) assuming an ideal gas, andΩK is the angular velocityat disc radiusR given by

ΩK =

√GM1

R3 = 0.423(

M1

M

)1/2(R

R1

)−3/2

s−1, (15)

−10 −8 −6 −4 −2 0 2 44.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

5.1

log τR

log

T

(K)

Fig. 1.Vertical temperature distribution at disc radiusR = 7.4×109cmwith log M(gs−1) = 17.0, M1 = 1M, Γ = 103 and Teff =1.5 × 104K. Dotted line:α = 0.05. Dashed line:α = 0.3. Solid line:α = 1.0

the latter form assuming the white dwarf mass-radius relation-ship (4).

The above constitute a closed algebraic-differential equationsystem for aboundary-valueproblem with eigenvaluez0, thesemi-thickness of the disc. Boundary conditions at the surfaceof the disc,z = z0, are

F (z0) = σT 4eff ,

J(z0) =σT 4

eff

4πµ1,

Pg(z0) = P0,

with P0 chosen such that the line formation region is fully withinthe calculated structure (e.g.,log P0 ∼ −3) andT (z0) is deter-mined via the energy-balance equation (14). From symmetry,the boundary condition at the disc mid-plane,z = 0, is

F (0) = 0. (16)

The eigenvaluez0 is calculated iteratively until the model con-verges to the solution satisfying (16), which is the solution forthe vertical structure of the accretion disc ring at radiusR. Theadaptive finite-difference code FORTRAN DASSL (Brenan etal. 1989) is used to solve the equations. Although DASSL is pre-sented as aninitial-valuealgebraic-differential equation solver,we treat the boundary conditions atz0 as initial values, and theniteratively solve for the central boundary condition atz = 0using Newton’s method.

We specifyα = 0.3 as a typical value for a high-state viscos-ity parameter. But it is important to subject this assumption tosensitivity analysis because, whileTeff is independent ofα, theenergy equation (14) does incorporateα. Fig. 1 shows the tem-perature structure for an accretion disc ring atR = 7.4×109cmfor the three valuesα = 0.05, 0.3 and1.0. Clearly, while disccoronal structures are sensitive toα, the line formation regions

W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO 157

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

8

10

12x 10

−3

Tr

ener

gy g

ener

atio

n pe

r un

it vo

lum

e (e

rg/s

/cm

3 )

AB

−8 −6 −4 −2 0 2 43.9

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

log τR

log

T

(K)

Fig. 2. Model solutions for a disc ring withTeff = 1.1 × 104K atR = 1.2×1010cm with log M(gs−1) = 17.0, M1 = 1M, Γ = 103

andi = 15o. Top: the right-hand (dashed line) and left-hand (solid line)sides of (14) versusTr = T/Teff Bottom: the vertical temperaturestructure.

in the photosphere – and hence the model spectra – are relativelyinsensitive. Therefore we are justified in usingα = 0.3 in ourinvestigations.

It is well known – e.g., Adam et al. (1988) – that the energyequation sometimes yields more than one solution for tempera-ture at a given pressure. Typically there are two thermally-stablesolutions, one of which starts with a hot corona at low opticaldepths but matches, with increasing optical depth, onto a coolerdisc structure: the other is a cold solution at all optical depths.The top panel of Fig. 2 shows a typical solution for (14): thedashed straight line represents the right-hand side of the equa-tion, and the solid line the left-hand side – the two peaks aredue to HeI and HeII ground-state jumps. The intersection ofthese two lines is the solution to (14) so, in general, there arefour solutions, two of which are unstable – those on negative-

sloping portions of the solid line. The other two stable solutions– termed here “hot” and “cold” – are on positive-sloping por-tions of the solid line. There is a transition from a hot to a coldsolution at the tangent point A, where (14) becomes too diffi-cult to solve numerically, causing a jump to the cold solution atB. This transition is also evident as a discontinuity in the tem-perature distribution shown in the bottom panel of Fig. 2, but itoccurs at much shallower optical depths than the line formationregion and so does not influence the emitted spectrum.

3.2. Line and continuum flux calculations

The disc structure calculations described above yieldP, ρ, TandJ as functions ofR andz. In order to estimate line andcontinuum fluxes, the disc is divided into elements boundedby R, R + δR andφ, φ + δφ whereφ is the rotational azimuthmeasured from the projected line-of-sight on the disc. Radiationat a given frequencyν and inclination anglei in the observer’sframe comes from Doppler-shifted frequencyνd(R, φ) in therest frame of each disc element where, ifc is the velocity oflight

νd(R, φ) =ν

(1 − vproj/c), (17)

vproj = RΩK(R) sin i sinφ. (18)

Monochromatic line and continuum opacitiesκνd(R, z) are

then calculated, and the full plane-parallel atmosphere radia-tive transfer equation is solved in the frame of the disc element.This yields the emergent intensityIνd

(µ, R, z0) at the surfacez = z0 of each element(R, φ) of the disc, whereµ = cos i.The flux radiated from the surface of the disc is then calculatedas

Fν =∫

R

∫φ

Iνd(µ, R, z0)µRdφdR, (19)

and the apparent monochromatic flux which is observed

fν =Fν

d2 , (20)

whered is the distance of the system from the Earth. Theseresults are used to calculate absolute visual magnitudes for themodel disc in order to estimated, as well as mean colours B-Vand U-B for direct comparison with observation.

In calculating the monochromatic and Rosseland opacities,we have included bound-free and free-free continuum opacitiesdue to hydrogen, HeI and HeII but have neglected metals. Giventhe uncertainties in the treatment of viscosity along with the LTEapproximation, this assumption is expected to be of secondaryimportance in determining details of the vertical structure. Op-tical spectra are calculated including the HeI lines shown inTable 1 and theHβ line λ4861, allowing for Stark and thermalDoppler-broadening. Although most of the strong HeI lines havebeen included, some lines near series limits have been omittedbecause of inadequate line data. For this reason, some discrep-ancies in line modelling could arise with blended lines such asλ3867 andλ3889.

158 W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO

Table 1.HeI lines included in the model

3820 2p-6d triplet3889 2s-3p triplet3965 2s-4p singlet4026 2p-5d triplet4121 2p-5s triplet4144 2p-6d singlet4169 2p-6s singlet4388 2p-5d singlet4472 2p-4d triplet4713 2p-4s triplet4922 2p-4d singlet5016 2s-3p singlet5048 2p-4s singlet5876 2p-3d triplet

We use aχ2-minimisation technique to fit observed andmodelled spectra, seeking the minimum helium-to-hydrogennumber density ratioΓ which gives negligibleHβ line com-pared with the HeI lines, and which also reproduces the generalshape of the observed spectrum. The fitting is conducted in thatpart of the spectrum containing most of the above lines, with aconsiderable part of the continuum also included so as to prop-erly fit the continuum slope.

The best part of the spectrum forχ2-minimisation fittingturns-out to be4350 ≤ λ(A) ≤ 5800 for AM CVn, and4350 ≤λ(A) ≤ 5100 for CR Boo. In the latter case, this excludes theregion aroundλ5170, where Patterson et al. (1997) note thelikelihood of a Fe/Mg-blend feature: use of this region wouldtherefore require an extension of the model to include metalopacities. Interestingly, inclusion of the intervalλ < 4350 A inthe fitting routine yields, for both AM CVn and CR Boo, eitherno solution or the physically unrealistic oneM1 > Mch: thismay be because of insufficient lines in our model spectra orcalibration difficulties with the observed data. To examine thepossibility that a blue spectral contribution from the primarycould influence the fitting, we included in some model runs acontribution to the spectrum from a black-body at the sameTeff

as the accretor: we found the contribution to be typically lessthan10% and therefore negligible.

3.3. Parameters for the secondary

The secondaries in helium-rich cataclysmic variables could bedegenerate helium white dwarfs (Faulkner et al. 1972), non-degenerate helium stars (Savonije et al. 1986) or helium-mantlestars with CO cores (Tutukov & Yungelson 1996 – hereafterTY96). Although predicted birth rates of these three types dif-fer only by factors of2–10, current estimates based on binaryevolution models suggest that the degenerate category will pre-dominate in magnitude-limited samples: predicted numbers aretypically higher by factors of103 compared to systems with non-degenerate helium stars, and typically105 compared to systemswith helium mantle stars (TY96). The most promising modelfor AM CVn systems appears therefore to be an IWDB model

where the donor is a low-mass helium-degenerate white dwarf.However for the sake of completeness, we also consider thepossibility that the donor may be a semi-degenerate helium star– see Warner (1995) for a review and discussion.

Assuming a semi-degenerate donor, its mass-radius rela-tionship is (Savonije et al. 1986)

R2

R= 0.029

(M2

M

)−0.19

, (21)

Eliminating binary separationa between the expression for or-bital period

Porb = 2π

a3q

GM2(1 + q)

1/2

, (22)

and the approximate expression whenq < 0.8 for the donor’sRoche lobe radius (Paczynski 1971)

RL2 = 0.462a

(q

1 + q

)1/3

, (23)

we find

R32L =

(0.462)3

4π2 GM2P2orb. (24)

Therefore, in the mass-transfer situation whenR2 = RL2, wefind from (24) and (21) that

M2

M=(

0.029R0.462M3

)1.91( 4π2

GP 2orb

)0.64

(25)

= 629.0P−1.27orb (s),

which givesM2 = 0.0921M andM2 = 0.058M for AMCVn and CR Boo respectively.

On the other hand, if we assume a fully-degenerate sec-ondary, then as previously noted

R2

R= 0.013

(M2

M

)−1/3

, (26)

and, following the same procedure as above, we find thatM2 =0.046M for AM CVn andM2 = 0.032M for CR Boo. In ourmodels we assume fully-degenerate mass values for the donorbut discuss, towards the end of the paper, the ramifications ofvarying this assumption.

3.4. Range limits for the accretion disc

We have assumed in all our calculations that the disc extends tothe surface of the white dwarf accretor, i.e.,Rin = R1. As weadoptM1 as the main free-parameter in the model we findR1,and henceRout, from (2), i.e.,

Rout = 7.79 × 108

(M1

Mch

)−2/3

−(

M1

Mch

)2/31/2

. (27)

An upper limitfor Rout is clearly the accretor’s Roche loberadiusRL1 which (Eggleton 1983) is well-approximated for allq by

W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO 159

RL1 =0.49q−2/3a

0.6q−2/3 + ln(1 + q−1/3)(28)

or, using (22), by

RL1 =0.49q−1

0.6q−2/3 + ln(1 + q−1/3)

×

GM2(1 + q)4π2 P 2

orb

1/3

. (29)

We have estimates ofM2 from above and knowPorb from ob-servations so, withM1 as the main free-parameter, we may findRL1 and hence an upper limit forRout. More refined estimatesof Rout are discussed next in the context of setting a range gridfor solving the disc-structure equations.

3.5. Setting the model grid

Given viscosity parameterα, disc structure – and thus the emer-gent spectrum – may be calculated onceRout, M , M1, Γ andiare specified. To do this, for a given value ofRout, we form thefour parameter grid:0.5 ≤ (M1/M) ≤ 1.4, 2 ≤ log Γ ≤ 4,16.5 ≤ log M(gs−1) ≤ 18.0 and15o ≤ i ≤ 75o. Then, aidedby theχ2-minimisation technique, we use this grid to seek thoseparameters generating model spectra which best fit the observedspectra of AM CVn and CR Boo.

Note thatRout is not used as a grid parameter: this is becausethe concomitant increase in computing time is not warranted byincreased precision in comparison with simple models of discsize. The above analysis showing thatRout ≤ RL1 is one suchsimple model, while thetidal radiusconcept (Frank et al. 1992)suggesting thatRout = 0.9RL1 is a related one. Yet anotherarises by considering streamlines at the outer edge of a closebinary accretion disc – where disc pressure and viscosity arevery small – as simple periodic orbits in the restricted three-body problem. Paczynski (1977) found in these circumstancesthat the maximum radius of the last stable orbit

Rst = rmaxa (30)

for tabulatedrmax = rmax(q). For allq corresponding to stellarmasses of interest here – i.e.,0.5 ≤ (M1/M) ≤ 1.3 and0.03 ≤ (M2/M) ≤ 0.1 – the tables show thatrmax = 0.48to within 1%. So, using (28), alower limit for Rout is

Rst = 0.980.6 + q2/3 ln(1 + q−1/3)RL1, (31)

in which case, for theq values of interest here,0.71 ≤(Rst/RL1) ≤ 0.92. HenceRst ≤ Rout ≤ RL1 and accord-ingly, when using the model, we construct two range grids; onewith Rout = RL1 and the other withRout = Rst. While bestfits from these two grids are not greatly different, indicationsare thatRout is in fact closer toRL1, and that is the value weuse to present spectral fits.

Note also that, becauseRL1 from (28) is not very sensi-tive to changes inq for typical AM CVn system configurationsand hence – from (31) – neither isRst, our model results forAM CVn systems are relatively insensitive toq. In particular,

4000 4500 5000 5500 6000 65001

1.1

1.2

1.3

1.4

1.5

1.6x 10

17

λ (angstrom)

tota

l flu

x (e

rgs/

s/H

z/st

rd)

Fig. 3.Disc spectra withRout = 1.3×1010cm,log M(gs−1) = 16.5,M1 = 1M, Γ = 103 andi = 15o. Upper to lower: 30, 20 and 10elemental rings

changing from fully to semi-degenerate values forM2 does notgreatly change predicted disc spectra.

As a practical compromise between model accuracy andcomputing time we divide the disc into10 equally-spaced el-emental rings. To demonstrate the reasonableness of this com-promise, Fig. 3 shows spectra for an accretion disc which havebeen calculated with10, 20 and30 elemental rings: the differ-ence in half-maximum line widths forλ4471 between a 10-ringand 30-ring disc is less than0.1%.

As a further measure to economise on computing resources,all discs are initially constructed withRout = 1.3 × 1010cm –larger than any disc anticipated for either AM CVn or CR Boo.Then, for eachM1 point in the grid, this large disc is cut-downin size to the appropriateRout = Rst or RL1, both of whichdepend onq. In other words, for givenM , M1, Γ andi, we usethe sameRout = 1.3 × 1010cm disc to construct discs withvariousRout ≤ 1.3 × 1010cm.

4. Results

To help understand theχ2-minimisation fitting results presentedlater in this section, we first examine whether theν1/3 spectrumcommon in accretion discs – e.g., Wade 1984 – is expected inAM CVn systems. This is followed by an investigation of theinfluence of parametersRout, M , M1, Γ andi on the continuumenergy distribution and line profiles for radiation emitted byaccretion discs.

4.1. The general shape of the spectrum

In an optically-thick accretion disc, one can approximate theflux from an elemental ring at radiusR by that of a black bodywith effective temperature given by (1), and so – see Frank etal. 1992 – thetotal discflux is

160 W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO

0 0.5 1 1.5 2 2.5 362.5

63

63.5

64

log hν/kTout

log

Fν (

arbi

trar

y un

its)

−0.5 0 0.5 1 1.5 2 2.560

61

62

63

64

65

66

log hν/kTout

log

Fν (

arbi

trar

y un

its)

Fig. 4. The continuum fluxFν of a disc radiating as a black body(solid line) with log M(gs−1) = 17.0, M1 = 1M, Γ = 103 andi = 15o. The dashed line is∝ ν1/3 and the dotted line is∝ ν2. Top:Rout = 1012cm. Bottom:Rout = 1011cm.

Fν =4πhν3 cos i

c2

∫ Rout

R1

RdR

exp(hν/kTeff ) − 1, (32)

whereh is Planck’s constant. Note that the flux is indepen-dent ofα, which is characteristic of optically-thick discs in asteady-state. For small frequencies –ν kTout/h – we havethe Rayleigh-Jeans behaviourFν ∝ ν2, while for large fre-quencies –ν kTout/h – we have the Wien formFν ∝ν2 exp(−hν/kTeff ). Moreover if Rout R1, the limits ofthe integral in (32) tend to0 and∞, and (1) tends to

T 4eff =

38πσ

(GM1M

R3

), (33)

in which caseFν ∝ ν1/3.Fig. 4 shows the emergent flux from accretion discs of differ-

ent size which are emitting as black bodies; spectra proportional

3000 3500 4000 4500 5000 5500 6000 6500 70000.5

1

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λ (angstrom)

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nsity

(er

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rd)

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3000 3500 4000 4500 5000 5500 6000 6500 70002.5

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nsity

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rd)

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Fig. 5a and b.Optical spectra (solid line) from the outer ring in ac-cretion discs withlog M(gs−1) = 17.0, M1 = 1M, Γ = 103 andi = 15o, superimposed on black-body spectra of the same effectivetemperature (dashed line). Top: ring atR = 2.9 × 109cm, Teff =2.85×104K. Bottom: ring atR = 8.6×109cm,Teff = 1.35×104K.

toν2 andν1/3 are superimposed. That part of the spectrum pro-portional toν1/3 is clearly not very extensive, and it becomessmaller asRout decreases from1012 to 1011 cm – i.e., asTout

increases. For this reason we do not expect the model spectra tohave theν1/3 form for Rout ∼ 109cm, which is a typical AMCVn accretion disc size.

We would also expect departures from the black-body spec-trum to be more evident indisc regionswhere the grey condi-tions assumed in the model clearly do not hold. For example,Fig. 5a shows the optical spectrum of a ring at2.9 × 109cmsuperimposed on a black-body spectrum of the same effectivetemperature as the ring,Teff = 2.85 × 104K; the fit betweenthe two is good. However, Fig. 5b shows the spectrum for a ringatR = 8.6× 109cm whereTeff = 1.35× 104K, and it is clearthat the corresponding black-body spectrum does not match the

W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO 161

3500 4000 4500 5000 5500 6000 6500 70000.5

1

1.5

2

2.5

3

3.5x 10

17

λ (angstrom)

tota

l flu

x (

ergs

/s/H

z/st

rd)

Rout

≈ 6.3× 109 cm

Rout

≈ 1.3× 1010 cm

Fig. 6. Optical spectra for two accretion discs withlog M(gs−1) =17.0, M1 = 1M, Γ = 103 and i = 15o. Upper:Rout = 1.3 ×1010cm. Lower:Rout = 6.3 × 109cm.

model spectrum nearly as well. This is not surprising consider-ing the non-grey nature of the opacities – e.g., HeI continuumjumps – in the cooler region.

4.2. Effects of disc parameters on the spectra

4.2.1. Disc sizeRout

Fig. 6 shows spectra for discs withRout = 1.3 × 1010cm and6.3 × 109cm. In general, if all other parameters are held con-stant, line profiles become deeper and narrower with increasingdisc radius. This is expected, because contributions to spectrallines come predominantly from the cooler outer disc rings –with their larger surface area – where lines are less affected byStark-broadening as well as Doppler-broadening, and thus havesharper features.

4.2.2. Mass-transfer rateM

Fig. 7 shows spectra for an accretion disc at three different mass-transfer rateslog M(gs−1) = 17, 17.5and18. The increase inTeff with increasingM steepens the slope of the blue con-tinuum and deepens the lines whileTeff is less than the peakionisation temperature for HeI. At higher values ofM , HeI isfully-ionised and this effect is reversed.

4.2.3. Accretor massM1

Fig. 8 shows that, as expected from (1), the effect of increasinggravity asM1 increases is similar to that ofM noted above; i.e.,the continuum slope and line depths increase with increasingM1. But, asM1 ∝ R−3

1 , theM1-effect is even more pronouncedbecause the(R1/R)1/2 term in (1) – although small – decreasesasM1 increases. In addition, both pressure and rotational linebroadening (fori /= 0o) increase withM1.

3500 4000 4500 5000 5500 6000 6500 70001

2

3

4

5

6

7

8

9

10

11x 10

17

λ (angstrom)

tota

l flu

x (

ergs

/s/H

z/st

rd)

Fig. 7.Optical spectra for an accretion disc withRout = 1.3×1010cm,M1 = 1M, Γ = 103 andi = 15o. Upper to lower:log M(gs−1) =18.0, 17.5and17.0.

3500 4000 4500 5000 5500 6000 6500 70001

1.5

2

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17

λ (angstrom)

tota

l flu

x (

ergs

/s/H

z/st

rd)

M1=0.5

M1=0.75

M1=1.0

M1=1.25

Fig. 8.Optical spectra for an accretion disc withRout = 1.3×1010cm,log M(gs−1) = 17.0, i = 15o. Upper to lower:(M1/M) =1.25, 1.0, 0.75and0.5.

4.2.4. Hydrogen-to-helium number ratioΓ

Although Γ is clearly important in determining the strengthsand shapes of spectral lines (indeed we use it as a key fitting-parameter), its effects may be masked by other factors. For in-stance, a weak but not negligibleHβ line may not be observedin an inclined disc because of thesin i-dependence of Doppler-broadening. Again, a particularΓ might give a significantHβ

contribution from the middle of a disc but not, because of thedifferent fractional ionisations, from the inner (hotter) and outer(colder) regions. An absence of observedHβ may then be dueto either lack of hydrogen, or masking of lines emitted from themiddle by the continuum from elsewhere in the disc.

162 W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO

4000 4500 5000 5500 6000 6500 70000.8

0.9

1

1.1

1.2

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17

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/s/H

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a

Hβ

4000 4500 5000 5500 6000 6500 70001.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8x 10

17

λ (angstrom)

tota

l flu

x (

ergs

/s/H

z/st

rd)

b

Fig. 9a and b.Optical spectra for an accretion disc withRout = 8.6×109cm,M1 = 1M andi = 15o. Panela: log M(gs−1) = 16.5, with(upper)Γ = 102 and (lower)Γ = 103. Panelb: log M(gs−1) = 17.0,with (upper)Γ = 102 and (lower)Γ = 103.

The spectral influence ofΓ is strongly-dependent onTeff

– and henceM – as is shown in Fig. 9. Fig. 9a shows spectrafrom two discs differing only in thatΓ = 102 for the upperone and103 for the lower: theHβ line, clearly visible withΓ = 102, almost disappears forΓ = 103. The disc configurationfor Fig. 9b is similar except for a higher mass-transfer rate: withalmost total hydrogen ionisation, theHβ line disappears evenwhenΓ = 102.

4.2.5. Inclination anglei

Fig. 10 shows the effect of varying inclination angle on theemergent spectrum from an accretion disc. Because of thesin i-dependence of Doppler effects:

4000 4500 5000 5500 6000 6500 70000.5

0.6

0.7

0.8

0.9

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1.1

1.2

1.3

1.4

1.5

1.6x 10

17

λ (angstrom)

tota

l flu

x (

ergs

/s/H

z/st

rd)

Hβ

Hβ

i=15o

i=45o

Fig. 10. Optical spectra for an accretion disc withRout = 1.3 ×1010cm, log M(gs−1) = 16.5, M1 = 1M andΓ = 102. Upper:i = 15o. Lower: i = 45o.

– the double-peaked feature of Doppler-broadened lines isclear fori = 45o but not fori = 15o, and

– the Hβ line for i = 45o is relatively weak compared toi = 15o.

The enhanced total flux from the disc wheni = 15o arisesprimarily because the projected surface area of the disc seen bythe observer depends oncos i.

4.3. Models of best fit for AM CVn

In Figs. 11 and 12 we present confidence contours from theχ2-fitting for AM CVn with Rout = Rst and RL1. Best-fitparameters, with one standard deviation error estimates, are:

– Rout = Rst givesM1 = 0.77M ± 0.11, log M(gs−1) =17.0 ± 0.2, i = 46.0o ± 2.6 andlog Γ > 2.5; while

– Rout = RL1 givesM1 = 0.90M ± 0.2, log M(gs−1) =17.1 ± 0.2, i = 43.7o ± 3.5 andlog Γ > 2.4.

Averaged values areM1 = 0.84M, log(M(gs−1) = 17.0 andi = 44.9o. The value forM1 is near the0.65M peak valueof the distribution of primary masses predicted for AM CVnsystems in TY96.

The difference in fit between the two cases is not great,although an increase inM1 with increasingRout is evident.This may be explained by two competing effects noted earlier:

– the deepening and narrowing of line profiles with increasingRout, and

– the increase in both pressure and rotational line broadening(for i /= 0o) with increasingM1.

Therefore, when fitting a particular observed line profile, a largerRout needs an increase inM1 to compensate for the narrow-linecontribution from the outer disc regions. So more precise esti-mates ofM1 – and to a lesser extent the other disc parameters too

W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO 163

Fig. 11.Confidence contours fromχ2-fitting for AM CVn with Rout =Rst. Inner to outer:99%, 90% and66% confidence level.

Fig. 12.Confidence contours fromχ2-fitting for AM CVn with Rout =RL1. Inner to outer:99%, 90% and66% confidence level.

– are dependent on more accurate estimation ofRout, includingpossible orbital-eccentricity effects. Nevertheless, we believethat parameter estimates based onRst andRL1 give reasonablelower and upper limits respectively. Another interaction shownin the results is a correlation betweenM andM1, evident fromthe slopingM1 − M confidence contours. Eq. (1) shows thatthis correlation arises because, for fixedTeff at disc radiusR,an increase inM1 requires a corresponding decrease inM .

We chooseRout = RL1 to present final fitting results.Fig. 13a shows that part of the observed spectrum whereχ2-

4400 4600 4800 5000 5200 5400 5600 58007

8

9

10

11

12

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tota

l flu

x (

× 10

26)(

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/cm

2 /s/H

z)

a

3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 60007

8

9

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11

12

13

λ (angstrom)

tota

l flu

x (

× 10

26)(

ergs

/cm

2 /s/H

z)

b

Fig. 13a and b.Observed spectrum of AM CVn (thin line) superim-posed on the best-fit spectrum (thick line) withRout = RL1. Panela shows that part of the spectrum whereχ2-minimisation is applied.Panelb shows the full spectrum.

minimisation is applied, on which is superimposed the best-fitmodel spectrum whenRout = RL1. Fig. 13b shows the samebest-fit model spectrum superimposed on the full observed spec-trum. Using (20), these results indicate that the distanced of AMCVn from the Earth lies in the interval338–495pc. Mean coloursB-V and U-B from observed and predicted data for AM CVnare compared in the upper part of Table 2: despite a possiblebias caused by the3550 ≤ λ ≤ 6150 bandpass for observeddata, there is less than1.5% difference between correspondingvalues.

4.4. Models of best fit for CR Boo

In Figs. 14 and 15 we present confidence contours from theχ2-fitting for CR Boo withRout = Rst andRL1. Best-fit pa-rameters, with one standard deviation error estimates, are:

164 W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO

Fig. 14.Confidence contours fromχ2-fitting for CR Boo withRout =Rst. Inner to outer:99%, 90% and66% confidence level.

Table 2.Mean colours for AM CVn and CR Boo

mean colour observed predicted

AM CVn

B-V -0.074 -0.073U-B -0.955 -0.966

CR Boo

B-V -0.050 -0.072U-B -0.972 -0.966

– Rout = Rst givesM1 = 0.98M ± 0.1, log M(gs−1) =16.8 ± 0.3 andi = 44.6o ± 3.2; while

– Rout = RL1 givesM1 = 1.0M ± 0.1, log M(gs−1) =17.0 ± 0.2 andi = 43.0o ± 3.2.

Averaged values areM1 = 0.99M, log M(gs−1) = 16.9 andi = 43.8o. Again the value forM1 is near the0.65M peakvalue of the distribution of primary masses predicted for AMCVn systems in TY96.

Unfortunately it is not possible to find a lower limit forΓ, as indicated by the openi − Γ confidence contours. Thisis because adverse signal-to-noise levels in the CR Boo datamake it difficult to accurately measureHβ line widths. And thisinherent data limitation means that the computing-time cost ofextending theΓ grid below102 is not warranted. In any case,additional fitting based on metal lines may be needed for moreaccurate constraints onΓ, which requires extension of the modelto include metal opacities.

Once again, difference in fit between the two cases are notgreat. Indeed, in comparison with the AM CVn fitting, whilesimilar interaction effects between CR Boo disc parameters areseen, they are not nearly as significant. UsingRout = RL1

Fig. 15.Confidence contours fromχ2-fitting for CR Boo withRout =RL1. Inner to outer:99%, 90% and66% confidence level.

to present final fitting results, Fig. 16a shows that part of theobserved spectrum whereχ2-minimisation is applied, on whichis superimposed the best-fit model spectrum whenRout = RL1.Fig. 16b shows the same best-fit model spectrum superimposedon the full observed spectrum. Note the crude fitting of thecontinuum around the Fe/Mg-blend feature nearλ5170, which– as mentioned earlier – is not within the spectral interval usedfor χ2-minimisation.

Using (20), these results indicate that the distanced of CRBoo from the Earth lies in the interval191–221pc. Mean coloursB-V and U-B from observed and predicted data for CR Booare compared in the lower part of Table 2: note that there is apossible bias caused by the3450 ≤ λ ≤ 6450 bandpass forobserved data. The difference between predicted and observedmean colours is less than1% for U-B but, probably because theFe/Mg-blend feature is lacking in our model, is about44% forB-V. Nevertheless, our estimates of mean colours for CR Booare well within those of Wood et al. (1987). Furthermore, ourresults are consistent with observational data (Warner 1995)suggesting that mean colours for CR Boo in a high state aresimilar to those of AM CVn.

5. Discussion and conclusions

It is widely believed (Warner 1995) that AM CVn systems arecataclysmic variables at a late evolutionary stage. A plausiblescenario (TY96) has them originating as high-mass wide bina-ries which evolve through two common envelope stages, leavingcomponents which are remnant cores from evolved stars. Thepresent accretor’s progenitor would have been initially the moremassive, typically leaving a C-O white dwarf after ejection ofthe first common envelope. The less massive – and hence lessevolved – progenitor of the present donor enters the second com-

W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO 165

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65

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a

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b

Fig. 16a and b.Observed spectrum of CR Boo (thin line) superimposedon the best-fit spectrum (thick line) withRout = RL1. Panela showsthat part of the spectrum whereχ2-minimisation is applied. Panelbshows the full spectrum.

mon envelope stage with a helium-rich core. They emerge fromthe second common envelope stage as a short-period binaryand, when orbital angular momentum losses from gravitationalradiation bring them close enough for the present donor to ex-perience Roche lobe overflow, mass transfer at rateM througha helium-rich accretion disc commences.

During this latter quasi-stable mass-transfer phase, assum-ing mass conservation but orbital angular momentum lossesfrom gravitational radiation, the donor’s mean mass-loss rateM2 is related to the orbital periodPorb by (Warner 1995)

M2 = 8 × 1017 q2

(1 + q)1/3

(56

+ξad

2− q

)−1

×(

M1

M

)8/3(Porb

1hr

)−8/3

gs−1 (34)

where, for entropyS

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.815

15.5

16

16.5

17

17.5

18

log Σ ( g cm −2 )

log

M (

g s

.−

1 )

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.83.6

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

4.5

log Σ ( g cm −2 )

log

Tef

f (K

)

Fig. 17.S-curves for the best-fit AM CVn model withRout = 8.7 ×109 cm,M1 = 0.84M andlog Γ = 3.

ξad =∂ lnR2

∂ lnM2

∣∣∣∣∣S

(35)

is the donor’s adiabatic radius-mass coefficient. If the accretiondisc is in astablethermal regime then we expectM2 ≈ M butif, on the other hand, it is in anunstableregime then we expect

– M2 > M if the system is in a low state, and– M2 < M if it is in a high state.

We now consider whether our results are consistent with thisgeneral picture and, in particular whether the results

– support the thermal disc-instability explanation of thebrightness variations observed in CR Boo but not in AMCVn, and

– throw any light on the level of degeneracy in the helium-richbinary component.

166 W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO

5.1. Thermal stability of the best-fit AM CVn disc

For the purposes of studying disc stability we take the aver-age of relevant best-fit parameters from Sect. 4.3 for the AMCVn model. The mean valueM1 = 0.84M and we knowthatPorb = 1028.7s, so with the fully-degenerate donor massM2 = 0.046M and ξad = −1/3 from Sect. 3.3 we findlog M2(gs−1) = 16.8. As this is approximately the same asthe average disc mass-transfer ratelog M(gs−1) = 17.0, ourresults are consistent with the observed stable behaviour of AMCVn.

If we assume a semi-degenerate donor massM2 =0.092M with ξad = −0.19 then log M2(gs−1) = 17.4. Asthis is a factor of2.5 greater than the average disc mass-transferrate from modelling, a semi-degenerate donor seems inconsis-tent with the observed stable behaviour of AM CVn. Furthersupportive evidence for a fully-degenerate donor in AM CVncomes from the detectable upper limit50 km s−1 on K1 notedby PHS. Using averaged best-fit values for AM CVn param-eters, we find thatK1 = 32 km s−1 with the fully-degeneratedonor and62 km s−1 with the semi-degenerate one: so a fully-degenerate donor seems more plausible on these grounds too.

If the above arguments about thermal stability of the best-fitAM CVn model hold, then we should be able to find anRout

within the allowableRst ≤ Rout ≤ RL1 interval for which thewhole best-fit disc is thermally stable. In fact using the averagebest-fit parameters,Rout = 8.7 × 109cm – which is within the7.1×109 ≤ Rout(cm) ≤ 9.2×109 allowable range – generatesan S-curve which places AM CVn just on the stable branch. Thisis shown in Fig. 17, where it is clear that the upper turning-pointoccurs atlog M(gs−1) = 17.0, and any part of the disc insidethis radius will also be stable at that mass-transfer rate. Henceour best-fit parameters are able to give disc models for AM CVnthat are thermally stable, which is in accord with observations.

5.2. Thermal stability of the best-fit CR Boo disc

For CR Boo, the mean valueM1 = 0.99M and we know thatPorb = 1471s, so with the fully-degenerate donor massM2 =0.032M andξad = −1/3 we find log M2(gs−1) = 16.1. Asthis is considerably less than the average disc mass-transfer ratelog M(gs−1) = 16.9, our results are consistent with CR Boobeing in the instability regime but in a high state.

If we assume a semi-degenerate donor massM2 =0.058M with ξad = −0.19, thenlog M2(gs−1) = 16.7. Asthis is also considerably less than the average disc mass-transferrate from modelling, a semi-degenerate donor is also consistentwith the observed unstable behaviour of CR Boo. The predictedvalue of K1 for CR Boo using averaged best-fit parametersare18 km s−1 for a fully-degenerate donor and32 km s−1 forthe fully-degenerate donor. Both values are too small to be de-tectable at present, so no conclusion may be drawn from themabout the nature of the secondary in CR Boo.

If the best-fit CR Boo model is in the instability regimeas suggested above, then we should be able to find a best-fitdisc satisfyingRst ≤ Rout ≤ RL1, within which there is a

1.4 1.6 1.8 2 2.2 2.4 2.6 2.815

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16.5

17

17.5

18

log Σ (g cm −2 )

log

M (

g s

.−

1 )

1.4 1.6 1.8 2 2.2 2.4 2.6 2.83.7

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3.9

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

log Σ ( g cm −2 )

log

Tef

f (K

)

Fig. 18.S-curves for the best-fit CR Boo model withM1 = 0.99Mand log Γ = 3. Dashed line:R = 7 × 109 cm. Solid line:R =5 × 109 cm.

region whereM2 lies on the unstable branch of the S-curvebut M lies on the hot branch. We find thatR = 7.0 × 109cm– which is within the9.4 × 109 ≤ Rout(cm) ≤ 1.3 × 1010

allowable range – generates an S-curve which does just this.This is shown in Fig. 18, where the upper turning-point occursat aboutlog M(gs−1) = 16.9 andlog M2(gs−1) is on the un-stable branch. Any part of the disc inside this radius will alsobe unstable at the average CR Boo mass-transfer rate as is alsoshown in Fig. 18 forR = 5×109 cm. Hence our best-fit param-eters are able to give disc models for CR Boo that are thermallyunstable, which is in accord with observations.

However, S-curves also depend onΓ which, in our fitting,has a large error range: Fig. 19 shows CR Boo S-curves forR = 1.0 × 1010 cm with log Γ = 2, 3 and4. It is clear that ahigher hydrogen content can reduce the effective temperatureof the instability region. The kink in thelog Γ = 2 curve showsthat the helium instability strip – evident in thelog Γ = 4 case

W. El-Khoury & D. Wickramasinghe: Models for AM CVn and CR BOO 167

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.815

15.5

16

16.5

17

17.5

18

log Σ ( g cm −2 )

log

M (

g s

.−

1 )

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.83.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

4.5

log Σ ( g cm −2 )

log

Tef

f (K

)

Fig. 19.S-curves for the best-fit CR Boo model withM1 = 0.99MatR = 1.0×1010 cm. Dotted line:log Γ = 2. Dashed line:log Γ = 3,Solid line:log Γ = 4.

– has almost disappeared. Hence the nature of the instabilityregion does not only depend onR, but also onΓ.

5.3. The nature of the secondary

We have seen from above there is some evidence that the donorin AM CVn is fully-degenerate, although there is no such evi-dence for CR Boo. A possible source of independent informa-tion on the mass – and therefore degeneracy status – of the donorin AM CVn systems comes from tidal resonance models of thesuperhump phenomenon. These models predict a beat periodPb

arising between superhump periodPs and orbital periodPorb,where1Pb

=1

Porb− 1

Ps. (36)

There is strong evidence for the tidal resonance model, bothfrom photometric and spectroscopic analysis, in hydrogen-rich

cataclysmic variables with superhumps such as OY Car, HT Casand Z Cha (PHS, Warner 1995).

Furthermore, if we identify the1051s periodicity seen in AMCVn with a superhump period, then applying (36) yieldsPb =13.36hrs for the AM CVn orbital periodPorb = 1028.7s. Andsimilarly the hypothesised1490s superhump period in CR BoogivesPb = 32.0 hrs. Now (Warner 1995) the tidal resonancemodel for superhumps leads to the relationship

Pb

Porb=

3.73(1 + q)q

(37)

for low mass ratiosq. Hence, for AM CVn we findq = 0.087,which, using our average best-fit estimateM1 = 0.84M,results inM2 = 0.07M, between our estimates of fullyand semi-degenerate secondary masses. Similarly for CR Booq = 0.05 which, for M1 = 0.99M, givesM2 = 0.05M –also between fully and semi-degenerate estimates.

Finally, we stress while the quoted errors are formal errorsof the fitting procedure, they may be an underestimate of the truesystematic errors when our model approximations are relaxed.

5.4. Conclusions

We conclude that a simple helium-rich disc model may be fit-ted to observed spectra of AM CVn and CR Boo. Resultingmean accretor masses are close to, but somewhat higher than,the0.7–0.8M range predicted by present population synthesisstudies, while modelled helium-to-hydrogen number density ra-tios are consistent with a helium-degenerate donor surroundedby a helium-rich envelope.

Predicted disc mass-transfer rates of about1017gs−1 mayplace the systems in the region of potential disc instability. Infact, S-curves calculated for the best-fit models are consistentwith a thermally-stable disc in AM CVn, and a thermally un-stable disc in CR Boo.

Acknowledgements.We thank Dr Stephen Roberts for help with com-putational aspect of this paper, and Dr Stephane Vennes for valuablediscussions and for providing line and continuum opacity routines.Professor Joseph Patterson kindly supplied spectral data on CR Boo.We are also grateful to both Professor Ronald Webbink and MichaelBurgess for their helpful comments and advice.

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