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A natural explanation for periodic X-ray outbursts in Be/X-ray binaries

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A natural explanation for periodic X-ray outbursts in Be/X-ray binaries
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    a  r   X   i  v  :  a  s   t  r  o  -  p   h   /   0   1   0   8   0   3   7  v   1   2   A  u  g   2   0   0   1 Astronomy & Astrophysics   manuscript no.(will be inserted by hand later) A natural explanation for periodic X-ray outbursts in Be/X-raybinaries A.T. Okazaki 1 , 2 and I. Negueruela 3 1 Faculty of Engineering, Hokkai-Gakuen University, Toyohira-ku, Sapporo 062-8605, Japan 2 Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK 3 Observatoire de Strasbourg, 11 rue de l’Universit´e, Strasbourg F67000 FranceReceived ; accepted Abstract.  When applied to Be/X-ray binaries, the viscous decretion disc model, which can successfully accountfor most properties of Be stars, naturally predicts the truncation of the circumstellar disc. The distance at whichthe circumstellar disc is truncated depends mainly on the orbital parameters and the viscosity. In systems withlow eccentricity, the disc is expected to be truncated at the 3:1 resonance radius, for which the gap between thedisc outer radius and the critical lobe radius of the Be star is so wide that, under normal conditions, the neutronstar cannot accrete enough gas at periastron passage to show periodic X-ray outbursts (Type I outbursts). Thesesystems will display only occasional giant X-ray outbursts (Type II outbursts). On the other hand, in systemswith high orbital eccentricity, the disc truncation occurs at a much higher resonance radius, which is very closeto or slightly beyond the critical lobe radius at periastron unless the viscosity is very low. In these systems, disctruncation cannot be efficient, allowing the neutron star to capture gas from the disc at every periastron passageand display Type I outbursts regularly. In contrast to the rather robust results for systems with low eccentricityand high eccentricity, the result for systems with moderate eccentricity depends on rather subtle details. Systemsin which the disc is truncated in the vicinity of the critical lobe will regularly display Type I outbursts, whereasthose with the disc significantly smaller than the critical lobe will show only Type II outbursts under normalconditions and temporary Type I outbursts when the disc is strongly disturbed. In Be/X-ray binaries, materialwill be accreted via the first Lagrangian point with low velocities relative to the neutron star and carrying highangular momentum. This may result in the temporary formation of accretion discs during Type I outbursts,something that seems to be confirmed by observations. Key words.  stars: circumstellar matter – emission-line, Be – binaries: close – neutron – X-ray: stars, bursts 1. Introduction Be/X-ray binaries are X-ray sources composed of a Be starand a neutron star. The high-energy radiation is believedto arise owing to accretion of material associated with theBe star by the compact object (see Negueruela 1998; seealso Bildsten et al. 1997).A “Be star” is an early-type non-supergiantstar, whichat some time has shown emission in the Balmer series lines(Slettebak 1988, for a review). Both the emission linesand the characteristic strong infrared excess when com-pared to normal stars of the same spectral types are at-tributed to the presence of circumstellar material in a disc-like geometry. The causes that give rise to the disc are notwell understood. Different mechanisms (fast rotation, non-radial pulsation, magnetic loops) have been proposed, butit is still unclear whether any of them can explain the ob-served phenomenology on its own. The discs are rotation- Send offprint requests to : okazaki@elsa.hokkai-s-u.ac.jp ally dominated and motion seems to be quasi-Keplerian(Hanuschik 1996). However, some kind of global outflowis needed to explain the X-ray emission from Be/X-raybinaries (Waters et al. 1988).Some Be/X-ray binaries are persistent X-ray sources(see Reig & Roche 1999), displaying low luminosity ( L x  ∼ 10 34 ergs − 1 ) at a relatively constant level (varying by up toa factor of   ∼ 10). On the other hand, most known Be/X-ray binaries (though this is probably a selection effect)undergo periods in which the X-ray luminosity suddenlyincreases by a factor  > ∼ 10 and are termed Be/X-ray tran-sients.Be/X-raytransients fall within a relatively narrowareain the  P  orb / P  spin  diagram (see Corbet 1986; Waters & vanKerkwijk 1989), indicating that some mechanism must beresponsible for the correlation. Those systems with fast-spinning neutron stars do not show pulsed X-ray emissionduring quiescence (though non-pulsed radiation could becaused by accretion on to the magnetosphere) because of   2 Okazaki & Negueruela: X-ray outbursts from Be/X-ray binaries the centrifugal inhibition of accretion (Stella et al. 1986).Systems with more slowly rotating pulsars show X-rayemission at a level  L x  < ∼ 10 35 erg s − 1 when in quiescence.Transients show two different kinds of outbursts: –  X-ray outbursts of moderate intensity ( L x  ≈  10 36 − 10 37 erg s − 1 ) occurring in series separated by the or-bital period (Type I or normal), generally (but notalways) close to the time of periastron passage of theneutron star. In most cases, the duration of these out-bursts seems to be related to the orbital period. –  Giant (or Type II) X-ray outbursts ( L x  > ∼ 10 37 erg s − 1 )lasting for several weeks or even months. GenerallyType II outbursts start shortly after periastron pas-sage, but do not show any other correlation with or-bital parameters (Finger & Prince 1997). In systemslike 4U0115+63 the duration of the Type II outburstsseems to be to some degree correlated with its peakintensity, but in A0535+262 Type I outbursts may beas long as much brighter Type II outbursts (Finger etal. 1996a). 2. Radial outflows vs. quasi-Keplerian discs Attempts at modelling the X-ray luminosities of Be/X-ray transients during outbursts have made use of a simplewind accretion model, in which the neutron star accretesfrom a relatively fast radial outflow. The disc of the centralBe star is supposed to have a power-law density distribu-tion ρ ( r ) =  ρ 0   rR ∗  − n (1)where  ρ 0  is the density at the stellar surface and  R ∗  is theradius of the star. This results in a power velocity law of the form v ( r ) =  v 0   rR ∗  n − 2 (2)where the values of   v 0  and  n  have to be determined obser-vationally (see Waters et al. 1989 and references therein).The rotational velocity of the outflow takes the form v rot ( r ) =  v rot , 0   rR ∗  − α (3)with 0 . 5  ≤  α  ≤  1 (respectively the Keplerian case andconservation of angular momentum).Accretion is considered to follow the classical Bondi-Hoyle-Littleton (BHL) approximation. The most impor-tant parameter, the relative velocity between the outflowand the neutron star, can be written as v 2rel  = ( v − V  rad ) 2 + ( v rot − V  rot ) 2 (4)where  V  rad  and  V  rot  are the radial and tangential compo-nents of the orbital velocity of the neutron star. Materialis supposed to be accreted when it is within a captureradius defined as r c  = 2 GM  x v − 2rel  (5)where  M  x  is the mass of the neutron star and  G  is thegravitational constant. The X-ray luminosity in the BHLapproximation can then be expressed as L x  = 4 πG 3 M  3x R − 1x  v − 4rel F  m  ∝ ρv − 3rel  (6)where  R x  is the radius of the neutron star and  F  m  =  ρv rel is the mass flow. In order to explain the wide range of observed X-ray luminosities, large changes in the valueof the radial velocity have to be invoked. For example,Waters et al. (1989) deduced that the relative velocity was v rel  ≈ 300kms − 1 during a Type I outburst of V0332+53in 1983, while it was  ≪ 100kms − 1 during a Type II out-burst in 1973.The use of the BHL approximation implies a numberof simplifying assumptions which are not always easy to justify. For example, it neglects any effect of the mass-losing star, which for periastron distances of   ∼ 10 R ∗  andmass ratios q   ∼ 0 . 1 seems to be an excessive simplification.In addition, while the use of the accretion radius formalismis adequate for an accreting object immersed in a medium,its application to an outflow characterised by a relativelysmall scale-height ( H  ), such as a Be disc, is dubious.Moreover, when the low outflow velocities required toexplain Type II outbursts are considered, the formalismbreaks down completely, since the capture radius becomesfar too large to have any physical meaning. For example,for  v rel  ≃ 20kms − 1 ,  r c  ≃ 9 . 6 × 10 11 m ≈ 1400 R ⊙ , whichis one order of magnitude larger than the binary separa-tion. A crude way round this problem is to consider thatthe radius of the effective Roche lobe of the neutron star r R  should be used instead of   r c  whenever the calculatedvalue is larger than  r R  (e.g., Ikhnasov 2001). In spite of these shortcomings, the model has been repeatedly usedin an attempt to model lightcurves of Be/X-ray binaries(Raguzova & Lipunov 1998; Reig et al. 1998) with onlymoderate success.Beyond the purely formal aspects, one obvious diffi-culty for the model is the fact that many Be/X-ray bina-ries (e.g., A0535+262) show low-luminosity X-ray emis-sion when they are not in outburst. It is believed thatall Be/X-ray binaries for which centrifugal inhibition of accretion is not effective display this emission. Motch etal. (1991) detected A0535+262 on several occasions at lu-minosities of   ≈  2 × 10 35 ergs − 1 . In order to explain thisluminosity within the framework described above and tak-ing into account that optical and infrared observations donot show any sign of the large variations that would beassociated with a change of several orders of magnitudein the density of material, enormous relative velocities (of the order of   ∼ 10 4 kms − 1 ) are needed.One further complication comes from the fact thatBe/X-ray binaries spend most of their time in the qui-escent state described in the previous paragraph and onlyoccasionally show series of outbursts. The model does notoffer any explanation as to why there could be a changefrom quiescence to outburst, unless again very large andsudden changes in the density and velocity of the flow areassumed. Given the changes in relative velocities needed  Okazaki & Negueruela: X-ray outbursts from Be/X-ray binaries 3 to account for the observed range of X-ray luminositiesand the lack of any physical mechanism that could ex-plain them, it is clear that direct accretion from a wind-like outflow is not the best approximation to the way inwhich matter is fed on to the neutron star.But the major objection to the model is simply thefact that there is no observational evidence whatsoeversupporting the existence of such fast outflows. All obser-vations of Be stars imply bulk outflow velocities smallerthan a few kms − 1 (Hanuschik 2000). The evidence forrotationally dominated quasi-Keplerian discs around Bestars is overwhelming (see Hanuschik et al. 1996; Hummel& Hanuschik 1997; Okazaki 1997), especially owing to thesuccess of the one-armed global oscillation model to ex-plain V/R variability in the emission lines of Be stars(Kato 1983; Okazaki 1991, 1996; Papaloizou et al. 1992;Hummel & Hanuschik 1997).Therefore it seems necessary to attempt an explana-tion of the outburst behaviour of Be/X-ray binaries thatdoes not imply large outflow velocities. 3. The viscous disc model Whatever the mechanism srcinating the Be phenomenon,the model which at present appears more applicable toexplaining the discs surrounding Be stars is the viscousdecretion disc model (Lee et al. 1991; see Porter 1999 andOkazaki 2001 for detailed discussion). In this scenario, an-gular momentum is transferred from the central star bysome mechanism still to be determined (perhaps associ-ated with non-radial pulsations) to the inner edge of thedisc, increasing its angular velocity to Keplerian. Viscositythen, operating in a way opposite to an accretion disc,conducts material outwards. In this scenario, material inthe disc moves in quasi-Keplerian orbits and the radialvelocity component is highly subsonic until the materialreaches a distance much larger than the line-emitting re-gions (Okazaki 2001). The outflow is very subsonic for thedistances at which neutron stars orbit in close Be/X-raytransients  v ( r )  <  1kms − 1   and still subsonic for the or-bital sizes of all Be/X-ray binaries for which there is anorbital solution. The viscous decretion disc model success-fully accounts for most of the observational characteristicsof Be discs.Negueruela & Okazaki (2001, henceforth Paper I) havemodelled the disc surrounding the Be primary in theBe/X-ray transient 4U0115+63 as a viscous decretiondisc and found that the tidal interaction of the neutronstar naturally produces the truncation of the circumstel-lar disc, as it does for accretion discs in close binaries(Paczy´nski 1977). The result of Negueruela & Okazaki(2001) is in agreement with the results of Reig et al.(1997), who showed that there is a correlation betweenthe orbital size and the maximum equivalent width of H α ever observed in a system. Even though it is clear that theequivalent width of H α  is not an effective measurement of the size of the disc owing to several effects (see, for ex-ample, Negueruela et al. 1998), the maximum equivalentwidth ever observed becomes a significant indicator if thesystem has been monitored during a period which is longin comparison with the typical time-scale for changes inthe disc (which, if viscosity is dominant, should be onlya few months). Therefore the result of Reig et al. (1997)clearly indicates that the neutron star has some sort of effect on the size of the disc.In this paper we apply the model presented in Paper Ito several Be/X-ray transients for which orbital solutionsexist and investigate how the truncation radius dependson different orbital parameters. 4. Model description and limitations The model developed in Paper I describes a binary systemin which a primary Be star of mass  M  ∗  and radius  R ∗  isorbited by a neutron star of mass  M  x  = 1 . 4  M  ⊙  whichmoves in an orbit of eccentricity  e  and period  P  orb . TheBe star is assumed to be surrounded by a near-Kepleriandisc which is primarily governed by pressure and viscosity.For simplicity, the disc is assumed to be isothermal andShakura-Sunyaev’s viscosity prescription is adopted.In such a disc, angular momentum is added to thedisc by the viscous torque, whereas it is removed from thedisc by the resonant torque exerted by the neutron starcompanion, which becomes non-zero only at radii wherethe ratio between the angular frequency of disc rotationand the angular frequency of the mean binary motion isa rational number. As a result, the disc decretes outwardowing to the transfer of angular momentum by viscosityuntil the resonant torque becomes larger than the viscoustorque at a resonant radius.Therefore, the criterion for the disc truncation at agiven resonance radius is written as T  vis  + T  res  ≤ 0 ,  (7)where  T  vis  and T  res  are the viscous torque and the resonanttorque, respectively.The viscous torque  T  vis  is written as T  vis  = 3 παGM  ∗ σr  H r  2 (8)(Lin & Papaloizou 1986), where  σ  is the surface densityof the disc,  α  is the Shakura-Sunyaev viscosity parameterand  H   is the vertical scale-height of the disc given by H r  =  c s V  K ( R ∗ )   rR ∗  1 / 2 (9)for the isothermal disc. Here,  c s  is the isothermal soundspeed and  V  K ( R ∗ ) is the Keplerian velocity at the stellarsurface. In the systems we will discuss later,  c s /V  K ( R ∗ )ranges 3 . 1 − 4 . 1 · 10 − 2 ( T  d /T  eff  ) 1 / 2 , where  T  d  and  T  eff   arethe disc temperature and the effective temperature of theBe star, respectively. Note that the viscous torque is pro-portional to the disc temperature.The resonant torque  T  res  is calculated by usingGoldreich & Tremaine’s (1979, 1980) torque formula, after  4 Okazaki & Negueruela: X-ray outbursts from Be/X-ray binaries decomposing the binary potential Φ into a double Fourierseries asΦ( r,θ,z ) =  − GM  ∗ r  −  GM  x [ r 2 + r 22 − 2 rr 2  cos( θ − f  )] 1 / 2 + GM  x rr 22 cos( θ − f  )=  m,l φ ml  exp[ i ( mθ − l Ω B t )] ,  (10)where  r 2  is the distance of the neutron star from the pri-mary,  f   is the true anomaly of the neutron star,  m  and l  are the azimuthal and time-harmonic numbers, respec-tively, and Ω B  = [ G ( M  ∗  + M  x ) /a 3 ] 1 / 2 is the mean motionof the binary with semimajor axis  a . The pattern speed of each potential component is given by Ω  p  = ( l/m )Ω B . Thethird term in the right hand side of the first equation is theindirect potential arising because the coordinate srcin isat the primary.For each potential component, there can be three kindsof resonances, i.e., the outer and inner Lindblad reso-nances at radii [( m ± 1) /l ] 2 / 3 (1 +  q  ) − 1 / 3 a , where Ω  p  =Ω  ±  κ/m , and a corotation resonance (CR) at the ra-dius ( m/l ) 2 / 3 (1 +  q  ) − 1 / 3 a , where Ω  p  = Ω. Here,  κ  isthe epicyclic frequency and the upper and lower signscorrespond to the outer Lindblad resonance (OLR) andinner Lindblad resonance (ILR), respectively. In circum-stellar discs, however, the resonant torque from the innerLindblad resonance, which is given by( T  ml ) ILR  = − m ( m − 1) π 2 σ ( λ  + 2 m ) 2 φ 2 ml 3 l 2 Ω 2 B ,  (11)where  λ  =  d ln φ ml /d ln r , always dominates the reso-nant torques from the corotation resonance and the outerLindblad resonance. The resonant torque at a given reso-nance radius is then given by T  res  =  ml ( T  ml ) ILR  +  m ′ l ′ ( T  m ′ l ′ ) OLR  +  m ′′ l ′′ ( T  m ′′ l ′′ ) CR ≃  ml ( T  ml ) ILR .  (12)Since high-order potential components contribute little tothe total torque, the summation in Eq. (12) is safely takenover several lowest-order potential components which givethe same radius.For a given set of stellar and orbital parameters andthe disc temperature, criterion (7) at a given resonance ismet for  α  smaller than a critical value  α crit  and we assumethat the disc is truncated at the resonance if   α < α crit .In Paper I it was shown that for any subsonic outflow,the drift time-scale  τ  drift  ∼  ∆ r/v r  ∼ M − 1 r  (∆ r/H  )Ω − 1 was considerably longer than a typical truncation time-scale  τ  trunc  ∼ ( α/α crit ) τ  vis  ∼ α − 1crit (∆ r/H  ) 2 Ω − 1 , where ∆ r is the gap size between the truncation radius and the ra-dius where the gravity by the neutron star begins to dom-inate, and  v r  and  M r  are the radial velocity and Machnumber, respectively. As a consequence, the tidal and res-onant interaction with the neutron star led to disc trun-cation.One important consequence of the above is that thediscs surrounding the primaries in Be/X-ray binaries can-not reach a steady state. Most of the outflowing mate-rial loses angular momentum and falls back towards thecentral star. As a consequence of the interaction with thematerial which is coming outwards from the inner regions,it is likely that the disc becomes denser and the densitydistribution in the radial direction becomes flatter withincreasing time. We note that we do not expect the trun-cation effect to be one hundred per cent efficient. Thisis not only owing to theoretical considerations (see be-low), but probably required by the existence of pulsedlow-luminosity X-ray emission during quiescence.In the formulation above only torques integrated overthe whole orbit are considered. Given the large eccentric-ities observed in Be/X-ray binaries (mostly larger than0.3 and sometimes approaching 0.9), the gravitational ef-fect of the neutron star is very strongly dependent on theorbital phase. This means that in such systems the disc ra-dius and the truncation radius are also phase-dependent.The disc would shrink at periastron, at which the trunca-tion radius becomes smallest, while it would spread whenthe neutron star is far away and its gravitational effect isnot felt so strongly. The spread of the disc would continueuntil the truncation radius becomes smaller than the discradius at some phase before the next periastron passage.This variation in disc radius would be larger for a largereccentricity. For a longer orbital period, the effect wouldbe yet stronger. Given that, for some systems considered(see Section 5), the model truncation radius is close tothe critical lobe radius at periastron, this will provide amechanism by which disc material can reach the neutronstar (see also Fig. 3 for scenarios for two families of Type Ioutbursts). 5. Modelling the Be/X-ray binaries Table 1 shows a list of known X-ray binaries which haveexhibited outbursting behaviour or some sort of orbitalmodulation in their X-ray lightcurve. The top panel con-tains Be/X-ray binaries with an identified Be counterpartwhich have displayed Type I X-ray outbursts, i.e., a seriesof outbursts separated by their orbital period (note thatin many cases there is no orbital solution for the systemand the recurrence period of the outbursts is taken to bethe orbital period). The middle panel shows other sys-tems without identified optical counterparts whose X-raybehaviour marks them out as Be/X-ray binaries. Finallythe bottom panel contains a few other Be/X-ray binarieswith identified counterparts whose X-ray behaviour devi-ates slightly from what is considered typical (and whichwill be discussed individually in Section 6).In this section, we apply our model to systems forwhich exact orbital solutions have been deduced from theanalysis of Doppler shifts in the arrival times of X-ray pho-  Okazaki & Negueruela: X-ray outbursts from Be/X-ray binaries 5 tons. They include five Be/X-ray binaries in the top panel(V0332+53, A0535+262, GROJ1008 − 57, 2S1417 − 624,and EXO2030+375) and a likely Be/X-ray binary in themiddle panel (2S1845 − 024). These systems are discussedindividually in the following subsections. The case of theBe/X-ray transient 4U0115+63 has been carefully dis-cussed in Paper I and therefore it will not be includedhere.As mentioned in the previous section, we adoptShakura-Sunyaev’sviscosity prescription, in which the vis-cosity parameter  α  is a free parameter, and assume theBe disc to be isothermal. In what follows, we adopt  T  d  = 12 T  eff  . Note that our assumption of the disc temperatureis consistent with the results by Millar and Marlborough(1998, 1999), who computed the distribution of the disctemperature within 100 R ∗  around the B0 star  γ   Cas andthe B8–9 star 1 Del by balancing at each position therates of energy gain and energy loss and found that thedisc is roughly isothermal at a temperature about half theeffective temperature of the star.When adopting a particular model for a system, themain source of uncertainty comes from the choice of massfor the primary, a parameter which can only be guessedfrom the spectral type. We find two main difficulties. Insome cases, the spectral type of the primary is not welldetermined. Moreover, there is some evidence that fast ro-tators may be moderately over-luminous for their masses(see Gies et al. 1998). For this reason, in general we havetaken masses slightly lower than those given in the calibra-tion of Vacca et al. (1996). The spectral distribution of pri-maries of Be/X-ray binaries (Negueruela 1998) is stronglypeaked at B0. Therefore for systems without exact deter-mination of the spectral type, we have calculated modelscorresponding to B0V and B0III primaries. In any case,our results show that the exact mass of the primary isnot one of the main factors in the X-ray behaviour of thesources. 5.1. V0332+53  This transient pulsar has an orbital period  P  orb  = 34 . 25dwith a relatively low eccentricity of   e  = 0 . 31 (Stella etal. 1985). The optical component of this system is an un-evolved star in the O8-9 range (Negueruela et al. 1999).For O8.5V stars Vacca et al. (1996) give an spectroscopicmass  M  ∗  = 23 . 6 M  ⊙ and a theoretical mass  M  ∗  = 28 M  ⊙ .Here we will assume as a conservative model the lowerlimit of   M  ∗  = 20 M  ⊙  and  R ∗  = 8 . 8 R ⊙ . The mass func-tion  f  ( M  ) = 0 . 1 then implies sin i  = 0 . 17, resulting in anorbital separation  a ≃ 130 R ⊙  ( r per  ≈ 10 R ∗ ).This system has been rarely detected in X-rays. AType II outburst was observed in 1973 (see Negueruelaet al. 1999 for references). Ten years later, it was ob-served during a series of three Type I outbursts. Finallyit was observed during another Type II outburst in 1989.Between 1991 and 2000 it has not been detected by eitherthe BATSE instrument on board the  ComptonGRO   satel- Fig.1.  Critical values of   α  at some resonance radii forsystems discussed in the text. Annotated in the figure arethe locations of the  n  : 1 commensurabilities of disc andmean binary orbital frequencies.  T  d  =  12 T  eff   is adopted forall models. For other disc temperatures,  α crit  should bemultiplied by a factor of   T  eff  / 2 T  d .lite or the All Sky Monitor on board  RossiXTE   and it isbelieved to be in a dormant state (Negueruela et al. 1999).In Fig. 1, we plot  α crit  at the  n  : 1 resonance radii forthose Be/X-raybinaries which will be discussed in this sec-tion. The resonant torques at the  n  : 1 radii are strongerthan those at radii with other period commensurabilitieslocated nearby. Note that we have adopted a particulardisc temperature,  T  d  =  12 T  eff  , for each stellar model. Forother disc temperatures,  α crit  should be multiplied by afactor of   T  eff  / 2 T  d , taking account of the fact that the vis-cous torque is proportional to  T  d .Fig. 1 shows that the Be disc in V0332+53 is expectedto be truncated at the 3:1 resonance radius ( r t /a ≃ 0 . 47,where  r t  is the truncation radius) for 0 . 099  < ∼  α < ∼  0 . 60and at the 4:1 resonance radius ( r t /a ≃ 0 . 39) for 0 . 019  < ∼ α < ∼ 0 . 099.It is interesting to see how closethe truncation radius isto the size of the critical lobe at periastron. Fig. 2 showsorbital models for the systems discussed in this section.The potential  ψ  describing the effects of the gravitationaland centrifugal forces on the motion of test particles or-biting the Be star is given by ψ ( r,θ,z ) = Φ( r,θ,z ) −  12Ω 2 ( r ) r 2 ,  (13)where Φ is the potential defined by Eq. (10). Also shown isthe distance scale corresponding to 0 . 1 c s P  orb . This scale
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