Mass Transfer in Binary Stars using SPH. II. Eccentric Binaries
aa r X i v : . [ a s t r o - ph . S R ] N ov Draft version October 23, 2018
Preprint typeset using L A TEX style emulateapj v. 03/07/07
MASS TRANSFER IN BINARY STARS USING SPH. II. ECCENTRIC BINARIES
Charles-Philippe Lajoie & Alison Sills
Department of Physics and Astronomy, McMaster University,Hamilton, ON L8S 4M1, Canada
Draft version October 23, 2018
ABSTRACTDespite numerous efforts to better understand binary star evolution, some aspects of it remainpoorly constrained. In particular, the evolution of eccentric binaries has remained elusive mainlybecause the Roche lobe formalism derived for circular binaries does not apply. Here, we report theresults of our Smoothed Particle Hydrodynamics simulations of mass transfer in eccentric binariesusing an alternate method in which we model only the outermost layers of the stars with appropriateboundary conditions. Using this technique, along with properly relaxed model stars, we characterizethe mass transfer episodes of binaries with various orbital parameters. In particular, we show thatthese episodes can be described by Gaussians with a FWMH of ∼ .
12 Porb and that the peak ratesoccur after periastron, at an orbital phase of ∼ .
58, independently of the eccentricity and mass of thestars. The accreted material is observed to form a rather sparse envelope around either or both stars.Although the fate of this envelope is not modeled in our simulations, we show that a constant fraction( ∼ Subject headings: binaries: close — stars: evolution — hydrodynamics — methods: numerical INTRODUCTIONThe study of exotic stellar populations (e.g. blue strag-glers, low-mass X-ray binaries, helium white dwarfs) re-quires the understanding of close binary evolution. Inturn, the study of binary evolution involves the useof many physical mechanisms occurring over dynami-cal, thermal, and nuclear timescales, which quickly ren-der the problem at hand complicated. The analyticaltools generally used to study binary stars have diffi-culty resolving all of these timescales; instead, they usu-ally incorporate only some of the mechanisms or relyon analytical approximations. On the one hand, stel-lar evolution codes can evolve single stars over manybillion years while taking into account convective mix-ing and different nuclear reactions networks. Only re-cently have they been used to evolve binary stars, al-though most of them are still in one dimension and onlya handful use two dimensions (Han, Tout, & Eggleton2000; Deupree & Karakas 2005). These binary evo-lution codes, along with population synthesis codes,which evolve many millions of stars at once overnuclear timescales (Portegies Zwart & Verbunt 1996;Hurley et al. 2002; Ivanova et al. 2005), rely on analyt-ical prescriptions for the mass transfer and accretionrates. None of these techniques actually models the masstransfer itself as it often occurs on timescales that aretoo short. On the other hand, hydrodynamics is wellsuited for purposes such as mass transfer. However, itcan be difficult to incorporate physical ingredients suchas convective mixing, magnetic fields, radiative transferor nuclear reactions. Hydrodynamics is also usually notdesigned to evolve stars over long periods of time. Thesedifficulties therefore render the modeling of the long-term
Electronic address: [email protected],[email protected] hydrodynamical evolution of interacting binaries ratherchallenging.The ideal case of circular orbits and conservative masstransfer has been studied intensively over the years. Sem-inal work by Morton (1960), Paczy´nski (1965, 1971) andPaczy´nski & Sienkiewicz (1972), among others, on masstransfer and its consequences on the stars and the orbitalparameters have opened the way to a more quantitativestudy of binary evolution. Iben (1991) and Iben & Livio(1993) have more recently laid out the overall evolution-ary paths of many different binary populations and ex-plained the formation scenarios of many exotic objects.But, from a theoretical point of view, a detailed descrip-tion of some aspects of close interactions are still lack-ing. Of particular interest are the rate at which massis transferred from one star to the other, the amountof mass accreted by the secondary stars, and the de-gree of mass loss from these systems, if any. To date,these quantities have usually been either approximatedfrom theoretical estimates or arbitrarily fixed. But be-cause these quantities are critical for understanding thelong-term evolution of binary stars, as they are someof the mechanisms that drive the change of orbital sep-aration and ultimately dictate the fate of binaries, itis important to get better estimates for the more real-istic, non-idealized cases. In particular, recent studies(Sepinsky et al. 2007a, 2007b, 2009) have suggested thateccentric binaries may evolve differently when comparedto circular binaries. Given that a non-negligible frac-tion ( ∼ BRIEF THEORY OF ECCENTRIC BINARYSYSTEMSAlthough most short-period binaries are on circular or-bits, many relatively close binaries also have eccentric or-bits in which mass transfer can occurs only close to pe-riastron. Such episodes of mass transfer are usually nottaken into account in binary population synthesis studiessince rapid circularization at the onset of mass transferis often assumed. To some extent, such episodes of masstransfer could modify the general picture of exotic starpopulations, especially in dense clusters where the forma-tion of eccentric binaries through captures is more likely.However, most of the theoretical background generallyused applies only to circular orbits, and one must relyon other approximations to estimate the rate of masstransfer in eccentric binaries, as we now discuss.2.1.
Roche lobe and equipotentials
Using analytics to investigate the equations of motionin binary systems, Sepinsky et al. (2007a) showed thateccentric binaries undergoing mass transfer can behavequite differently when compared with circular binaries.Indeed, the authors found that the Roche lobe radius (de-noted R L ) can be smaller than the circular case by morethan 20% for binaries with mass ratios close to unity androtating faster than the orbital velocity at periastron.The reverse is also true, as binaries rotating slower thanthe orbital velocity at periastron can have a Roche loberadius ∼ larger than the circular case. Moreover,depending on the degree of asynchronicity and eccentric-ity, the geometry of the equipotential surfaces is found tochange significantly and allow for some mass to be ejectedfrom the system through the L point. Sepinsky et al.(2007a) found that the usual Roche lobe can sometimesopen up around the secondary star, allowing for somemass loss through the L point, and that the potentialat the L point can sometimes be only slightly larger thanthat at the L point, also allowing for some more massloss (see Figure 7 of Sepinsky et al. 2007a). Althoughwe expect some mass to be ejected from the system, thetotal amount lost is hard to estimate and authors havegenerally used some parameterizations to study the ef-fect of non-conservative mass transfer on the evolutionof binaries (Sepinsky et al. 2009; hereafter, SWKR09).These results suggest that eccentric and asynchronousbinaries are likely to undergo mass transfer at earlierphases of their life (compared to circular binaries) andthat the latter is most likely non-conservative. Note thatDermine et al. (2009) finds similar changes to the shapeof equipotentials when considering the effect of radiationpressure in circular and synchronized binaries. These recent works emphasize that the classical Rochemodel is not adequate in most instances. The addition ofrealistic physical ingredients (e.g. asynchronism, eccen-tricity, radiation pressure) in the models of binary starsmodifies the structure of equipotentials. The Roche lobemodel derived for circular orbits therefore does not apply.2.2. Secular evolution
Based on their previous results, Sepinsky et al. (2007b)(hereafter, SWKR07) and SWKR09 went on to studythe secular evolution of eccentric binaries undergoingmass transfer, with the assumptions of instantaneousmass transfer ( ˙M = 10 − M ⊙ /yr) centered at perias-tron and both conservative and non-conservative masstransfer. The authors found that depending on the massratio and eccentricity, the secular changes of orbital sep-aration and eccentricity can be positive or negative, andcan occur on timescales ranging from a few million yearsto a few billion years. Moreover, these timescales can,in some cases, be comparable to the orbital evolutiontimescales due to tidal dissipation, which can be addi-tive or competitive. Based on these findings, the authorssuggest that the usual rapid circularization assumptionis not always applicable and, in some cases, very unlikely.Finally, SWKR09 conclude that relaxing the assumptionof conservative mass transfer does not change the overallconclusions of their previous work (SWKR07). The ratesof secular evolution for a and e found by SWKR07 andSWKR09 are directly proportional to the assumed masstransfer rate. However, this can be hard to constrainwith analytical prescriptions only. Indeed, when masstransfer occurs periodically, binaries can remain on ec-centric orbits for long periods of time, making the Rochelobe radius and the mass transfer rate difficult to de-termine, as the latter depends on the degree of overflow(∆ R = R ∗ − R L , where R ∗ is the radius of the star).2.3. Previous simulations of mass transfer
One way to better estimate the rates of mass trans-fer is by using hydrodynamical simulations. Only ahandful of such simulations have been done to this day.Despite usually not being suited for long, thermal- ornuclear-timescale studies, hydrodynamical simulationscan be useful for understanding transient phenomena andepisodes of dynamical mass transfer.Only a few authors have investigated the hydrodynam-ics of eccentric binaries. Reg¨os et al. (2005) (see alsoLayton et al. 1998) studied the shape of the equipoten-tial surfaces in eccentric binaries using both analyticaland numerical (SPH) approaches. Their findings agreewith those of Sepinsky et al. (2007a) in that mass trans-ferred through the L point close to periastron passagesmay leave the system (as well as through the L point).However, their estimates for the Roche radius are largerand similar to the Roche lobe radii for the circular andsynchronized case. Interestingly, the authors also studythe onset of mass transfer along the orbit for one binaryand different eccentricities. The low resolution of thesesimulations (10,000 particles), however, does not allowfor accurate mass transfer rate determinations.Church et al. (2009) have partially circumvented thisproblem using an innovative SPH technique for modelingmass transfer in cataclysmic variables, where the leastASS TRANSFER IN BINARIES. II. 3massive star is losing mass to a compact white dwarf.With the aim of getting better estimates of mass trans-fer rates, their innovative approach allows for high mass resolution in the outer parts of the star and therefore forthe resolution of low mass transfer rates. Despite usinga relatively low number of particles ( ∼ , q = 0 . L point and did not encompass either the donor or the ac-cretor and did not assess whether mass was lost from thesystem.Finally, in some simulations to date, the accreting staris not realistically modeled but rather often modeled asa point mass or with surface boundary conditions. Thesesimplifications prevent from drawing any quantitativeconclusions regarding the accretion process. Moreover,as pointed out by Sills & Lombardi (1997), the use ofpolytropes instead of realistic models may lead to sig-nificantly different internal structures for collision prod-ucts, which may arguably be applicable to interactingbinaries. Therefore, more work remains to be done inorder to better understand how mass transfer operatesand affects the evolution of eccentric binary systems. COMPUTATIONAL METHODFor a more realistic modeling of hydrodynamical masstransfer, it is better to use hydrodynamics techniquessince they can easily be adapted to model binary systemsin three dimensions and physically follow the transfer ofmass from one star to the other. Here, we use an SPHcode based on the version of Benz (1990) and Bate et al.(1995) with a recent updated treatment of boundary con-ditions specifically designed to model boundary stars pre-sented in Paper I. We model our stars from theoreticalprofiles obtained from our stellar evolution code (YREC;Guenther et al. 1992) and distribute SPH particles onan hexagonal lattice while iteratively assigning particlemasses so that the density profile matches that from ourstellar evolution code. Binaries are relaxed in their owngravitational force (and centrifugal force) prior to thestart of the mass transfer simulations (see Paper I). Us-ing our treatment of boundary conditions, we replacethe inner particles with a central point mass and modelonly the outermost layers of the stars. The location ofthe boundary is, at this point, arbitrary but should beplaced at least a few smoothing lengths from the sur-face. Use of our boundary conditions allows for betterspatial and mass resolutions in the mass transfer streamas well as the use of less CPU time. Note that each par-ticle’s smoothing length is also consistently evolved intime, following the prescription of Benz (1990), allowingfor a better spatial resolution in regions of high density.Finally, we use Monaghan’s viscosity (Monaghan 1989)with α = 1 . β = 2 . P = ( γ − ρu , where γ = 5 / ρ is the density and u the internal energy (per unit mass). MASS TRANSFER IN ECCENTRIC BINARIESWe now present the results of our simulations of masstransfer for two different binary systems and discuss theoverall behaviours observed in our simulations. In par-ticular, we are interested in the mass transfer rates andproperties involved in such close interactions. We havemodeled binary systems with stars of different masses,semi-major axes, and eccentricity. The different orbitalparameters modeled for both system are summarized inTable 1 along with some preliminary results.4.1. ⊙ + 0.48 M ⊙ Our first model consists of a low-mass binary systemrepresentative of the turn-off mass of globular clusters.The initial separation of this binary system, at apastron,is set to 4 R ⊙ such that the stars do not initially over-flow their Roche lobe. The boundary is set at 0.8 R ⊙ for the 0.8-M ⊙ star and 0.35 R ⊙ for the 0.48-M ⊙ star,both corresponding to ∼
75% of the stars’ radii. Atthis radius, most of the mass of the stars is encompassedwithin the central point mass. The total number of par-ticles is ∼ ,
000 and the total mass of SPH particlesis ∼ × − M ⊙ and ∼ × − M ⊙ for the 0.8- and0.48-M ⊙ stars respectively. Using SPLASH (Price 2007), apublicly available visualization tool for SPH simulations,we show in Figure 1 the logarithm of the density, in the XY plane, for the case with e = 0 .
25. The time is shownin units of the dynamical time ( τ dyn = q R ⊙ /GM ⊙ ≃ . ∼ τ dyn .Each image is 12 R ⊙ by 12 R ⊙ , and the density scaleranges from 10 − g cm − (dark) to 1 g cm − (white).The 0.8-M ⊙ star is the larger of the two stars and thelarge density contrast between the two stars is obviousfrom these plots. Mass transfer occurs only periodically,close to periastron, and shuts off when the stars are fur-ther apart. Moreover, the secondary is retaining some ofthe transferred mass, forming an envelope, whereas theprimary does not seem to be affected strongly from losingmass. The density of the accreted material around thesecondary is much lower than that of the secondary’s sur-face layer and this may have some implications for thelong-term accretion of this material (see § L point whereas no mass is lost through the pri-mary’s far side through the L point. The whole sys-tem eventually becomes engulfed in a relatively warm butlow-density envelope that extends for many solar radii.Finally, the mass transfer proceeds relatively smoothlyand no shocks are observed at the surface of the sec-ondary. Moreover, it is observed that the mass transferstream in between the two stars is relatively cooler thanthe surrounding envelope, since its expansion comes atthe expense of its own internal energy.Figure 2 shows the different energies, normalized totheir initial value, as a function of time for the samesystem. The total energy is fairly well conserved dur-ing the whole duration of the simulation. It varies byat most ∼
3% and seems to do so periodically. The ec-centricity of this system is obvious from the shape ofthe curve of the kinetic energy as it peaks at periastron, LAJOIE & SILLS
Fig. 1.—
Logarithm of the density in the XY -plane for the0 .
80 + 0 .
48 M ⊙ binary and e = 0 .
25. Each image is 12 R ⊙ by 12R ⊙ and the central point masses are not shown. The time is shownin units of the dynamical timescale ( τ dyn ) and the orbital periodis ∼ τ dyn . halfway through the orbital period, and decreases almostto its initial value. The different values of the extrema ofthe kinetic energy suggest that the orbital separation ischanging. Similar behaviours are also observed for mod-els with different eccentricities. As for the gravitationalenergy, it varies in the same way as the kinetic energy,whereas the thermal energy stays constant to within lessthan 0 .
5% over the whole duration of the simulation.The total angular momentum, on the other hand, variesaround its initial value, by no more than 2.5% over thewhole duration of the simulation. We use the binary’scentre of mass as the rotation axis to calculate the totalangular momentum of the system and all of the angularmomentum is, as expected, in the z -direction, i.e. per-pendicular to the orbital plane. The angular momentumin the other directions is at least 4 orders of magnitudesmaller and remains negligible for the whole duration ofthe simulations. These variations of the total angularmomentum observed in our simulations are acceptablegiven that the angular velocity of the ghosts is artificially Fig. 2.—
Different energies as a function of time for the 0.80+0.48M ⊙ binary with e = 0 . maintained at a fixed value (see Paper I).4.2. ⊙ + 1.40 M ⊙ The second system we model is a higher-mass binaryrepresentative of the population of relatively old openclusters. Also, since the secondary is much larger andits density is of the same magnitude as the primary, weexpect the infalling material to interact much more dy-namically with the envelope of the secondary. The twostars are initially set at a separation of 6 R ⊙ , at apastron,which places them well within their Roche lobe. The lo-cation of the boundaries is chosen at 75% of the totalradius of the stars, corresponding to a radius of 1 .
05 and0 .
90 R ⊙ for the primary and the secondary respectively.The total number of particles is ∼ ,
000 and, conse-quently, the total mass in SPH particles in the primaryamounts to ∼ . × − M ⊙ whereas the secondary con-tains ∼ . × − M ⊙ of SPH particles. The remainderof the mass is contained in the central point masses. Fig-ure 3 shows the logarithm of the density in the XY planefor the e = 0 .
25 case at different times. The orbital pe-riod for this system is ∼ τ dyn . The interaction betweenthe two stars is much stronger here, as material fromboth stars is lost, and a clear spiral pattern is observedand most prominent towards the end of each mass trans-fer episode (i.e. after each periastron passage). At loweccentricity, the mass transfer is rather smooth and haslittle effect on the secondary, whereas for our largest ec-centricity runs, the systems almost come into contact atperiastron and material from the primary plows throughthe secondary’s envelope, which is pushed around thewhole system. Most of the envelope surrounding bothstars is relatively hot as it gets heated up after the firstperiastron passage. Also, unlike the low-mass binary, weobserve mass loss through both the L and L points,which may be enhanced by the fact that asynchronism issubstantial at periastron, thus lowering the potential atthe L point (see § Fig. 3.—
Logarithm of the density in the XY -plane for the1 .
50 + 1 .
40 M ⊙ binary and e = 0 .
25. Each image is 18 R ⊙ by 18R ⊙ and the central point masses are not shown. The time is shownin units of the dynamical timescale ( τ dyn ) and the orbital periodis ∼ τ dyn . trema at periastron. The kinetic energy always peaks atthe same value and comes back to its initial value when atapastron, suggesting that the binary is well relaxed andthat it follows the orbit it was initially put on. Also, thetotal energy changes by no more than ∼ .
5% over thewhole duration of the simulation. We also notice that thetotal internal energy slowly increases, by 8% at the endof the simulation. This change in thermal energy comesat the expense of gravitational energy, but although 8%seems substantial, we emphasize that the total thermalenergy represents roughly only 1 part in 1000 of both thekinetic and gravitational energies. Therefore, it wouldbe hard to observe such a small change in gravitationalenergy on the scale of Figure 4. The total angular mo-mentum of the system and of the two stellar componentsalso remains constant during the entire simulation to a1% level for the whole duration of the simulation.4.3.
Mass transfer rates
We now determine the mass transfer rates from oursimulations. We use the method based on the total en-
Fig. 4.—
Different energies as a function of time for the 1.50+1.40M ⊙ binary with e = 0 . Fig. 5.—
Mass transfer rates as a function of orbital period andeccentricity for a selection of runs from the 0 .
80 + 0 .
48 M ⊙ (uppertwo panels) and the 1 .
50 + 1 .
40 M ⊙ (lower two panels) binaries.The solid and dotted lines represent the mass transfer and accretionrates of the primary and secondary respectively. ergy of each SPH particle, as discussed in Paper I (seealso Lombardi et al. (2006)), to determine to which com-ponent SPH particles are bound. Particles are assignedto one of the following components: the primary andsecondary stars, the binary envelope, and the ejecta.4.3.1. Rate and duration of mass transfer
Figure 5 shows some of the mass transfer and accre-tion episodes a function of time and eccentricity for thestellar components in the 0 .
80 + 0 .
48 M ⊙ and 1 .
50 + 1 . ⊙ systems. For the primary (solid lines), we plot thenegative of the mass transfer rates so that we can com-pare it to the (positive) accretion rate of the secondary. LAJOIE & SILLSIn most cases, the mass transfer rates are well definedand peak right after the periastron passages. We notethat mostly particles from the outer envelope of the pri-mary only are transferred during each episode and thatthe boundary never becomes involved in the interaction.For the low-mass binary, mass transfer occurs only for ec-centricity greater than ∼ .
20. In these cases, the masstransferred from the primary is almost totally accretedby the secondary, as shown by the reciprocity of the solidand dotted lines. We note also that both rates sometimedip in the negative part of the plots, meaning that somematerial is falling back onto the primary or that the sec-ondary is losing some of its newly-accreted mass. Forlower eccentricities (e.g. e = 0 . − . e = 0 .
20 case is rather noisy and there seems to be a sig-nificant fraction of the mass transferred that falls backonto the primary and secondary after the main episodesof mass transfer. This seems to be important only in thesmaller eccentricity cases. The primary’s mass trans-fer rates rarely becomes negative, unlike the secondary’saccretion rates, which are mostly negative in betweenperiastron passages, suggesting that the secondary losesmass. We note however that mass becomes bound to thesecondary (and the primary) only temporarily as subse-quent episodes of mass transfer are sometimes energeticenough to plow through the surrounding envelope of thesecondary and eject some of this material. Likewise, themaximum mass transfer rates are observed to increaseboth with time and eccentricity. Given that both starsremain very close to their initial eccentric orbit (see Fig-ure 4), this increase in the peak mass transfer rate islikely due to an increase of the primary star radius. Fig-ure 6 shows the radii enclosing different fractions of thetotal bound mass for the primary. The periastron pas-sages are clearly visible and, most importantly, the radii in between the mass transfer episode gradually increase,which is indicative of the expansion of the primary’s en-velope as matter is being lost. This increase in radiusinevitably leads to an increase in the degree of overflowand, consequently, the mass transfer rate. We also note
Fig. 6.—
Radii enclosing different fractions of the total boundmass (in SPH particles) to the primary star as a function of timefor the 1 .
40 + 1 .
50 M ⊙ binary with e = 0 .
25. The dotted linerepresents the location of the boundary. that the boundary is well within the star and the ra-dius containing 60% of the mass in SPH particles barelychanges with time, indicative that tidal effects are neg-ligible at this location. Finally, we note the similaritiesbetween the two systems modeled in the position, dura-tion, and shape of the mass transfer rate episodes. Inparticular, their shape is suggestive of a Gaussian func-tion.The range of mass transfer rates observed in our sim-ulations ranges from a few 10 − M ⊙ yr − , for the high-mass binary with e = 0 .
15, to 0 . ⊙ yr − for the low-mass binaries. We emphasize that these relatively highmass transfer rates last only for a short period of time(i.e. ∼ ∼ − − − M ⊙ yr − per periastron pas-sage. In all cases, the number of particles transferredranges from a few hundreds to many thousands per masstransfer episode. Binaries where the number of parti-cles transferred is less than ∼
100 are considered as nottransferring mass on the basis of the poor SPH treatmentfor such low numbers (see Table 1). Given the least mas-sive particles in our simulations, the lowest possible masstransfer we can model (notwithstanding the numericalnoise), is of the order of 10 − − − M ⊙ yr − , whichis comparable to that of D’Souza et al. (2006). How-ever, given the low number of particles that would betransferred in such instances, the SPH approach fails atproperly evaluating the hydrodynamical forces on theseisolated particles. We compare our results with theoret-ical expectations in § Gaussian fits to mass transfer episodes
Using our mass transfer rate profiles of Figure 5,we now fit a Gaussian function to every mass transferepisode. The general Gaussian we use has the followingform: ˙ M ( t ) = A exp (cid:16) − ( t − µ ) σ (cid:17) + D (1)where A is the maximum amplitude, µ is the centreof the Gaussian, σ is proportional to the width of theGaussian and D is the background (or continuum) masstransfer rate. The latter parameter is used to measurethe background noise as in some case the mass transferrates do not fall back to zero in between mass transferepisodes. All the free parameters are fitted using theASS TRANSFER IN BINARIES. II. 7 Fig. 7.—
Gaussian fits to the primary’s mass transfer episodesfor the 0.80+0.48 M ⊙ binary with e = 0 .
25. The values of thefitted parameters are reported in Table 2.
Fig. 8.—
Gaussian fits to the primary’s mass transfer episodesfor the 1.50+1.40 M ⊙ binary with e = 0 .
25. The values of thefitted parameters are reported in Table 2. nonlinear least-squares method of Levenberg-Marquardt(Press et al. 1992). We fit the height of the Gaussian ex-tended wings so that the width of the Gaussian matchesmore closely the data points. However, in cases wherematter falls back onto the stars between periastron pas-sages, the fitting procedure is to be taken with care. Ex-amples of the Gaussian fits to the mass transfer episodesare shown in Figures 7 and 8 for two of our simulations.The parameters obtained from the fitting procedure aregiven in Table 2. Some data points are assigned a rela-tively large error since they are part of the pre-periastronmass transfer episodes and do not contribute to the mainepisode of mass transfer nor to the fitting procedure.Note that doing so does not significantly change the val-ues of both µ and σ . Moreover, for every first episode ofmass transfer, we do not fit the continuum (parameter Fig. 9.—
Amplitude, position, and width of the Gaussian fits tothe mass transfer episode for the primary as a function of eccen-tricity for both binary systems. Solid dots are for the 0 .
80 + 0 . ⊙ binary while open dots are for the 1 .
50 + 1 .
40 M ⊙ binary. Seealso Table 2. D ) as we expect the value of the mass transfer rate priorto the first periastron passage to be identically zero. Wedo fit this parameter for any subsequent peak however.For most of our simulations, Gaussians fit the datapoints remarkably well. In most cases, the amplitude,centre, and width all closely match the data points.Again, for cases where matter is observed to fall backonto the stars, the fits to the height of the extended wingsis obviously not as reliable. For example, the Gaussianfit for the low-mass binary with an eccentricity e = 0 . A and D , is the noise on either side of thepeaks seen in the data. On the other hand, the widthand amplitude of most (if not all) of the mass transferepisodes are well matched by Gaussians.We plot, in Figure 9, the amplitude, centre, and widthof all the Gaussians we fitted as a function of eccen-tricity. Many trends can be seen in this plot. First, themaximum mass transfer rate increases with the eccentric-ity. This is expected since as the eccentricity increases,the periastron distance gets smaller and the two starsget closer to each other, thus facilitating mass transfer.Our results also suggest that the maximum mass trans-fer rate increases linearly with the eccentricity, althoughwe also expect a cut-off at low eccentricity where theprimary will not fill its Roche lobe even when at peri-astron. Also, although we only have two data pointsfrom our low-mass binary simulations, these two simula-tions suggest a similar trend. As for the position wherethe maximum mass transfer rate occurs, the results fromour high-mass binary simulations clearly show that masstransfer rates peak at an orbital phase slightly larger thanperiastron, around 0 . − .
57. Although mass transferstarts around periastron, it only peaks later when thetwo stars have already started getting further away from LAJOIE & SILLSeach other. This is in contrast with one of the basicassumptions of SWKR07 and SWKR09, who assumedthat mass transfer occurred instantaneously at perias-tron. The observed delays in the peak mass transfer ratesare consistent with a free-fall time ( τ ff ≃ . G ¯ ρ ) − / )into the secondaries’ potential well at periastron. Follow-ing Eggleton (1983), the free fall times for our systemsshould be about 0 . − .
065 P orb , similar to the delaysobserved in our simulation (see e.g. Table 2), with an ex-pected slight increase for the low-mass binaries. It seemslikely therefore that the small differences in the positionof the mass transfer rate peaks observed in our simu-lations are real. Our method for determining the masstransfer rates does not tell whether a particle will betransferred but rather if it has been transferred, whichis what we define as mass transfer, and our results seemto indicate that this occurs over a free-fall time. Onlyat periastron is the tidal force large enough to strip thedeeper layers of the primary. Since this material hasto travel to the Roche surface before being assigned tothe secondary, a delay in the peak mass transfer rate isto be expected. We note that such delays in the peakmass transfer rates should be intrinsic to eccentric bina-ries as these systems never exactly fill their Roche lobebut rather periodically shrink within and expand beyondit. Although the results from our low-mass binary sim-ulations are less suggestive, the delays observed in theposition of the maximum mass transfer rate also suggestthat the maximum degree of overflow should occur laterthan periastron.Finally, we also observe in both sets of simulations thatthe width (or duration) of the mass transfer episode is fi-nite in time and arguably independent of the eccentricity.Our results suggest that the full width at half maximum(FWHM ≈ √ σ ) is approximately 0 . − .
13 Porb.One could argue that there is a small negative slope sug-gesting that the higher the eccentricity, the faster themass transfer occurs, which is plausible since stars onhigh eccentric orbits spend less time around periastroncompared to star on low eccentric orbits. No matter thetrend, this value of the width of the mass transfer ratesis also in contrast with another basic assumption used bySWKR07 and SWKR09, who assumed an instantaneousmass transfer rate. Our results clearly show that this isnot the case and that the mass transfer occurs over anextended but finite period of time.4.4.
Accretion onto the secondary
Most of the mass lost by the primary eventually be-comes bound to the secondary. One way to look at theaccretion is to look at the origin of the particles makingup each component. This is shown in Figure 10, wherewe plot the origin of the particles in the orbital planeand where red and blue dots are particles that were ini-tially bound to the primary and secondary respectively.This colour-coded representation allows us to track theparticles as they are shuffled around and become boundto any of the components of the system (i.e. the sec-ondary, the binary envelope, or the ejecta). In the caseof the low-mass binary, we see that the secondary is notstrongly affected by the infalling material as none of itsown particles are being mixed up with the material fromthe primary. As a matter of fact, the secondary is so
Fig. 10.—
Origin of particles in the orbital plane for the0.80+0.48 M ⊙ binary with e = 0 .
25. Red and blue dots are par-ticles that initially come from the primary and secondary respec-tively. The time is shown in units of the dynamical timescale ( τ dyn )and the orbital period is ∼ τ dyn .In this case, the secondary isnot affected by the infalling material. dense that it is not perturbed at all by the mass trans-fer episodes and its accreted material simply forms anenvelope around it. Despite this large density gradientat the surface, we note that the interaction between theinfalling material from the primary and the secondary’senvelope is relatively smooth and no shocks are observedat the surface of the boundary. The smoothing lengthsof the transferred particles are consistently adjusted intime (see §
3) and are of the same order of magnitudeas those of the particles at the surface of the secondary.Also, the difference in mass between the particles beingtransferred and the particles forming the outer layers ofthe secondary differs by less than one order of magnitudeand we do not observe spurious motions in the envelopesuggesting interactions between particles with extrememass ratios (see e.g. Lombardi et al. 1999). The den-sity contrast can also be observed in Figure 11, whichshows a
SPLASH three-dimensional surface rendition ofthe 0 .
80 + 0 .
48 M ⊙ system with e = 0 .
25. The sur-faces are rendered by defining a critical surface throughwhich we can not see, similar to an optical depth. Thisthree-dimensional rendition allows for the visualizationof the whole system. We see that the material trans-ferred from the primary initially forms a thick torus-likecloud around the secondary, and subsequent episodes ofASS TRANSFER IN BINARIES. II. 9
Fig. 11.—
Surface rendition (with τ = 0 .
08) of the densityfor the 0.80+0.48 M ⊙ binary with e = 0 .
25 showing the accretionaround the secondary. The time is shown in units of the dynamicaltimescale ( τ dyn ) and the orbital period is ∼ τ dyn . Initially, athick disk forms around the secondary, but later engulfs it almostcompletely. Mass loss occurs also primarily from the secondary’sfar side, at the L point. mass transfer eventually form an envelope that rather en-gulfs the secondary and corotates with it. Moreover, theprimary also becomes engulfed by a thin envelope. Sincethe secondary does not lose any mass, this envelope, aswell as the binary ejecta, is made up of the material fromthe primary.In the case of the high-mass binary, the similar den-sities of the two stars allows for the material beingtransferred to interact much more strongly with the sec-ondary’s envelope, as shown in Figure 12 (for e = 0 . ∼
5% for the e = 0 .
30 and ∼
10% for the e = 0 .
25 case,at the end of both simulations). Again, we emphasizethat the fact that the secondary loses some mass is theresult of both the secondary slightly overfilling its Rochelobe (see upper right panel of Figure 12 for example) andthe interaction of the infalling material plowing throughthe envelope of the secondary. This is more analogousto a so-called “direct impact” where the secondary fillsmost of its Roche lobe and therefore is hit almost rightafter the transferred material passes the L point. In-terestingly, we do not observe such a mass loss from thesecondary for smaller eccentricities (e.g. e = 0 .
15 and e = 0 . Fig. 12.—
Origin of particles in the orbital plane for the1.50+1.40 M ⊙ binary with e = 0 .
30. Red and blue dots areparticles that initially come from the primary and secondary re-spectively. The time is shown in units of the dynamical timescale( τ dyn ) and the orbital period is ∼ τ dyn . In this case, the sec-ondary loses material because of partial Roche lobe overflow andthe interaction of the infalling material with its envelope. terial being lost from both stars eventually engulfs thewhole system rather than forming an envelope around thesecondary only. This envelope is substantially denser andthicker than in the low-mass binary case (see Figure 11)as the surface rendition of Figure 13 uses a much largeroptical depth in order to peer through the envelope andsee the surface of the stars. We observe the formation ofsuch an envelope in all of our simulations for this binary,although the envelope for the e = 015 − .
20 cases isthinner as less material is lost from either star. We notethat in the high-mass binaries, the particle mass profilesin both stars are almost identical and we expect and, infact, observe no spurious motion in the envelope. Simi-larly, the smoothing lengths are consistently evolved suchthat particles in regions of similar density have similarsmoothing length, allowing for a better spatial resolution.As in the case of the mass transfer episodes, the accre-tion episodes also display similar positions and durationand characteristic shapes of Gaussian functions. Our re-sults are again displayed in Table 2. The position of thepeaks and the width of the Gaussian are similar to themass transfer episodes of the primary. The Gaussian pa-rameters for the accretion onto the secondary follow thesame trends as shown in Figure 9, with only minor differ-0 LAJOIE & SILLS
Fig. 13.—
Surface rendition (with τ = 25 .
0) of the densityfor the 1.50+1.40 M ⊙ binary with e = 0 .
30 showing the accretionaround the secondary. The time is shown in units of the dynamicaltimescale ( τ dyn ) and the orbital period is ∼ τ dyn . In this case,an envelope engulfs both star and mass loss occurs also from thefar side of both stars, at the L and L points ence in the width of the Gaussians ( σ ). Indeed, we noticeonly a slightly larger spread in σ for the accretion ratewhen compared with the mass transfer rates. Althoughwe could expect the maximum mass transfer rate ( A )to be larger than the maximum accretion rate ( A ), wesee that it is not always the case. The reason is simplythat the values of the continuum (parameter D ) are dif-ferent for the primary and the secondary, thus yielding adifferent zero level from which the peak values are mea-sured. In any case however, the total mass accreted bythe secondary is always less (or equal) to the mass lostby the primary. The remaining mass is obviously lost tothe binary as a whole or to the ejecta, which is what wediscuss next. 4.5. Mass loss
We now quantify the amount of mass lost to either thebinary envelope or the ejecta in our simulations. Parti-cles are assigned to either of these two components if theyare far enough from either stellar components and/ortheir total (relative) energy is positive (see Paper I).4.5.1.
Escaping Particles
Escaping particles are particles that are found far (i.e.many smoothing lengths) from the bulk of the particles.By design, the code, and more specifically the tree build-ing and the neighbours search, run into some problemswhen particles escape and/or are found in between thetwo stars, with only a few other close particles. Indeed,when particles are ejected from the system, the searchfor the required number of neighbours become lengthyand sometimes unsuccessful. To circumvent this issue,
Fig. 14.—
Changes in mass in the binary envelope and ejectanormalized by the change in mass of the primary for selected runsfor both the 0.80+0.48 M ⊙ (upper two panels) and the 1.50+1.40M ⊙ (lower four panels) binaries. In all cases, ∼
5% of the masslost by the primary ends up in the ejecta. we set a maximum smoothing length ( ∼ ⊙ ) such thatthe code does not unnecessarily spend CPU time iterat-ing and adjusting the smoothing length of a small set ofescaping particles (see also § Binary envelope and ejecta
We now assess the degree of mass loss during the masstransfer episodes in our simulations. We find that inall cases, the mass contained in the binary envelope isgreater than that in the ejecta, by a factor of at least two.Both the binary envelope and the ejecta grow in mass asa function of time, and although no clear episodes of massgrowth is observed for the ejecta, we observe a stepwiseincrease in the mass bound to the binary envelope. Thetotal mass in each component ranges from a few 10 − M ⊙ for the high-mass binary to ∼ × − M ⊙ for thelow-mass binary. This amounts to ∼ . −
1% of the totalinitial mass (in SPH particles) of the primary.Figure 14 shows the change in bound mass of the bi-nary envelope and the ejecta normalized by the change inmass of the primary. Essentially, this shows the fractionof the mass lost by the primary that ends up in the bi-nary envelope or the ejecta. Interestingly, the fraction ofmass bound to the binary envelope shows some periodicbehaviour, peaking shortly before periastron, where thestars are at their closest separation along the orbit. Thislag comes from the fact that (some of) the infalling ma-terial only temporarily becomes bound to the secondarybefore becoming bound to the envelope or the ejecta. Bythe end of our simulations, around 20% of the mass lostby the primary end up in the binary envelope, and ourresults suggest that this fraction slowly increases as aASS TRANSFER IN BINARIES. II. 11function of time.The mass in the ejecta, on the other hand, is roughlyconstant around 5% for all of our simulations. This isan unexpected result, given that we observe this trendin all of our simulations, no matter the mass of the starsor the eccentricity. Moreover, given that the degree ofmass loss in binary evolution is rather unconstrained (e.g.SWKR09), this result is suggestive of an almost uniformmass loss over different binary masses. Although conser-vative mass transfer is usually employed as an idealizedcase, the constant and small fraction of mass lost in oursimulations suggests that mass transfer is indeed non-conservative but only to a small degree. This is alsointeresting in the context of two (slightly) different ac-cretion scenarios, where accretion occurs via an accretiondisk or through a direct impact. In the latter case, whenthe secondary fills a significant fraction of its Roche lobe,the matter falling in from the L point hits the accretoralmost directly, whereas in the former case, material fallsdeep in the potential well of the secondary and forms adisk. Although the direct impact scenario is more rep-resentative of our high-mass binary rather than the low-mass binary, and whether or not the infalling materialinteracts with the secondary’s envelope, we still get thatroughly 5% of the mass lost by the primary ends up inthe ejecta. Note, however, that our simulations do notallow us to assess the fate of the binary envelope. i.e.whether it is going to be expelled from the system andbecome part of the ejecta or be accreted by either star. COMPARISONS WITH PREVIOUS WORKSimilarly to Church et al. (2009), our results show thatthe mass transfer rates get increasingly larger as the starsget closer to each other at periastron. Moreover, ourresults indicate that the mass transfer episodes do notoccur precisely at periastron and last for a constant frac-tion of the orbital period, independent of the eccentricity.However, unlike Church et al. (2009), our mass resolu-tion does not allow us to resolve mass transfer rates aslow as ∼ − − − M ⊙ yr − . Having a better massresolution would help increase the number of particles inthe stream of material for our low-eccentricity binaries,but we do not expect that this would drastically changeour conclusions.Our binaries are set up such that they are in corota-tion at apastron, therefore making them subsynchronousat periastron. In both of our sets of simulations, the ra-tio of the angular velocity, which is fixed for the wholeduration of the simulations, to the orbital velocity atperiastron ranges from around 0 .
30 to 0 .
60. Thus, ac-cording to Sepinsky et al. (2007a), this has the effect ofslightly increasing the Roche lobe radius, by ∼ de-crease the degree of overflow and, consequently, the masstransfer rate.Comparing the magnitude of the mass transfer rate ob-served in our simulations with theoretical expectationsis difficult because estimates of the actual radius of theprimary from our simulations are uncertain. Neverthe-less, we build a simple model for mass transfer usingthe (dynamical) mass transfer rate derived for polytropesof index n by Paczy´nski & Sienkiewicz (1972) (see alsoEdwards & Pringle 1987; Eggleton 2006; Gokhale et al. Fig. 15.—
Comparison between the instantaneous mass transferrate of Equation 2 (dotted) and our results (solid) for the the 1 . .
40 M ⊙ binary with e = 0 . M = − ˙ M (cid:16) R − R instL R (cid:17) n +3 / , (2)where ˙ M is a canonical mass transfer rate. R instL is theinstantaneous Roche lobe radius, i.e. a simple generaliza-tion of the Roche lobe radius for circular and synchronousbinaries (Eggleton 1983): R instL = D ( t ) 0 . q / . q / + ln(1 + q / ) , (3)where D ( t ) is the instantaneous separation of the twostars. This mass transfer rate, which applies when thedonor can be approximated by a polytrope, dependsstrongly on the degree of overflow, as expected, and isequally zero when ∆ R ≤ R ≃ . R ⊙ , we calculate the instantaneous degree ofoverflow, ∆ R , based on the instantaneous separationof the stars, and a mass transfer rate. This is shownin Figure 15 for the high-mass binary with e = 0 . M ≃ − M ⊙ yr − ) such that the first peak of masstransfer matches that from our simulation. Moreover,to mimic the slightly increasing peak mass transfer rate,as can be observed in our simulations, we assume thatthe radius of the star increases at apastron passage byincrements of 0 .
05 R ⊙ . We do not expect the radius ofthe primary to change by much over the course of oursimulations since the total mass transferred is small, andthis artificial (and rather large) change in radius sim-ply allows for a better match with the increasing peakmass transfer rates. We also note that we see no reasonswhy both the canonical mass transfer rate ˙ M and thepolytropic index n should remain constant as mass trans-fer proceeds, although we have assumed so here, whichcould compensate for changes in radius. The difficultyin doing such a comparison with eccentric orbits lies inthe facts that we are using the instantaneous Roche loberadius derived for circular and synchronous binaries andthat the theoretical mass transfer rate used here was de-rived for polytropes of constant n . Although this simplemodel agrees qualitatively well with our simulations (e.g.Gaussian-like episodes of mass transfer), we emphasize2 LAJOIE & SILLS Fig. 16.—
Semi-major axis normalized by the initial radius ofthe primary as a function of eccentricity for all of our simulations.Solid dots are our successful runs, open dots are runs for whichmass transfer was too large for our boundary conditions to handle,and open-crossed symbols are for cases when mass transfer was toolow or not resolved. The dotted line represents an approximatedelimitation above which our boundary conditions can be applied. that we arbitrarily fixed the canonical mass transfer rateso that the peaks match. However, we observe that theposition and width of the mass transfer episode strik-ingly differ from the theoretical expectation. The widthof the peaks depends on the star’s radius, as a shrink-ing star would delay the start of mass transfer (as wellas decrease the degree of overflow), therefore decreas-ing the mass transfer rate. On the other hand, there isno parameter that could account for the position of themaximum rates as, by construction, the largest degree ofoverflow occurs when the stars are closest to each other,i.e. at periastron. CONCLUSIONS AND FUTURE WORKThe evolution of binary stars has grown into an intensefield of study since it has become clear that many popula-tions of stars have to form through interactions with closestellar companions. Although the main phases of binaryevolution are nowadays well understood, these evolution-ary paths usually rely on the (idealized) formalism de-rived for circular and synchronized orbits. This so-calledRoche lobe formalism does not apply for close and inter-acting eccentric binaries, in which the rotation is asyn-chronous and the gravitational potential time-dependent.Given the relatively large number of binary stars, and inparticular, of binary stars with eccentric orbits, it is im-perative to better understand the interactions of thesesystems in order to further constrain the different galac-tic populations of exotic stars. Recent breakthroughsby Sepinsky et al. (2007a), SWKR07 and SWKR09, inparticular, have allowed to extend the knowledge of thelong-term evolution of eccentric binaries. Although theseworks clearly show that eccentric binaries behave differ-ently from circular ones, their conclusions are based ona number of assumptions. In this paper, we have pre-sented the results of SPH simulations with the aim ofconstraining these assumptions.The results from our large-scale simulations are inter-esting both for the performances of our alternate ap-proach (see Paper I) and for the characterization of the mass transfer episodes. Our boundary conditionscan effectively handle intermediate mass transfer rates( ∼ − − − M ⊙ yr − ), although particles penetratethe boundary when the periastron distance is such thatthe mass transfer rates become too large. On the otherhand, our code, by design, does not handle cases whereonly a handful of particles are transferred. The parame-ter space where our technique can be applied is thereforerestricted by these two conditions on the number of par-ticles. Figure 16 summarizes the orbital parameters forwhich our technique is well suited. We see that whenthe normalized semi-major axis is greater than ∼ . ⊙ , our boundary conditions behave well as the masstransfer rates are not excessively large. Also, the use ofmore lower-mass particles would help resolve lower masstransfer rates and allow for the modeling of systems withhigher a/R ⊙ values.The results from our simulations of mass transfer alsoshow clear trends. In particular, we show that theepisodes of mass transfer can be described by Gaussianswith a FWMH of ∼ . − .
15 Porb, and the peak masstransfer rates occur after periastron, around an orbitalphase of ∼ . − .
56. It is interesting to note thatthese results apply for both of the binary systems mod-eled and for any eccentricity. The technique used in thiswork represents an interesting alternative to previouswork (e.g. Edwards & Pringle 1987, Reg¨os et al. 2005,Church et al. 2009) and we suggest that our results onthe properties of interacting eccentric main-sequence bi-naries could be used in analytical work such as that ofSWKR07 and SWKR09 to further constrain the evolu-tion of such stars. We also discussed the accretion ontothe secondary and showed that it is also well character-ized by similar Gaussians. The accreted material is ob-served to form a rather sparse envelope around the sec-ondary, in the low-mass binary, and around both stars,in the high-mass binary. Although the fate of this en-velope is not determined using our method (whether itis going to be accreted onto either stars or ejected fromthe system), we showed that a constant fraction of thematerial lost by the primary is ejected from the sys-tems. Although poorly constrained, the concept of non-conservative mass transfer is generally accepted nowa-days and our results may help constrain the degree ofmass conservation in binary evolution. In the future, wehope to cover more of the parameter space ( q , a , and e REFERENCESBate, M.R., Bonnell, I.A., & Price, N.M. 1995, MNRAS, 277, 362Benz, W. 1990, in Buchler J.R., ed. The Numerical Modeling ofNonlinear Stellar Pulsations: Problems and Prospects. Kluwer,Dordrecht, p.26 9 Church, R.P., Dischler, J., Davies, M.B., Tout, C.A., Adams, T.,& Beer, M.E. 2009, MNRAS, 395, 1127D’Souza, M.C.R., Motl, P.M., Tohline, J.E., & Frank, J. 2006, ApJ,643, 381
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TABLE 1Parameter space explored of both binaries modeled in this work.
System r ap e a r peri ⊙ ) (R ⊙ ) (R ⊙ ) orbits transfer0 .
80 + 0 .
48 M ⊙ .
50 + 1 .
40 M ⊙ Note . — 1: only a few particles transferred. 2: mass transfer rate too largefor the boundary to handle at first periastron passage; particles penetration.3: mass transfer rate becomes too large after three orbits. See text for moredetails.
TABLE 2Gaussian parameters for the mass transfer and accretion episodes of bothbinaries modeled in this work.
System e A A µ µ σ σ (M ⊙ /yr) (M ⊙ /yr) (Porb) (Porb) (Porb) (Porb)0 .
80 + 0 .
48 M ⊙ × − × − × − × − × − × − × − × − × − × − × − × − × − × − .
50 + 1 .
40 M ⊙ × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × − × −2