Circumbinary Disk Evolution Using Only a Small Number of Particles All Having the Same Binary

arXiv:astro-ph/0204175v1 10 Apr 2002 hi rpriswt bevtoa data. observational and with disks compare properties circumstellar we envelope their and evolution the circumbinary the when developed modeling fully evolution By cir- cleared. the created in been of the interested has phase are of We following properties disks. the derive circumstellar and and cumbinary code SPH with system an protobinary a onto collapsing cloud molecular Hartigan accretion 1998; 1995). al. estimate al. et et (Gullbring to stars the binary used of the part onto typically luminosities rates optical are the and layer of Observations infrared boundary spectrum. the the from of in- data usually emission binaries 1998). spectroscopic al. clude et the (Duvert of Aur GG observations UY namely The and 1994) disks, al. opti- et circumbinary (Dutrey have wide with Tau imaged there systems directly Sco, binary of AK cal observations and two Tau only DQ been like disks surrounding Mathieu 2001). in Mathieu systems (Reviews & Zinnecker systems 2000; multiple multiple al. in all et nearly or born that are indicate to binary stars seem stars sequence to that pre-main observations of recent shown However, 1991). belong Mayor have & (Duquennoy stars 60% sequence about main of Observations Introduction 1. e-mail: edopitrqet to requests offprint Send DI ilb netdb adlater) hand by inserted be will (DOI: Astrophysics & Astronomy ae&Bnel(97 oe h al hs fa of phase early the model (1997) Bonnell & Bate with systems binary spectroscopic many from Apart bet osri h aaeeso h ytm ymthn b words. matching Key by cent for systems available. the are where the onto findings data of accreted observational main parameters to is Our the distributions clos which Aur. constrain gap the UY to disk model and able Tau the to GG through chosen of material are systems setup wider the the approac of numerical implemented. parameters novel been Physical has A systems however applied. thin, coordinate is different infinitesimally cooling be o to radiative consisting assumed and system is a disk model The to disk. employed are simulations ical Abstract. 2002 14, March accepted 2002; 21, January Received Germany Computationa Abt. Astrophysik, & für Astronomie Institut [email protected] esuyteeouino icmiaydsssronigcl surrounding disks circumbinary of evolution the study We crto ik iais eea yrdnmc metho – hydrodynamics – general binaries: – disks accretion .Günther, R. : aucitno. manuscript icmiayDs evolution Disk Circumbinary ihr ¨nhr&WlemKley Wilhelm Günther & Richard ec o ceti iaiswsetmtd oee,when However, estimated. was binaries leads eccentric for potential dence the of points saddle a the facilitated to along is which flow gas process This a accreted stars. by the eventually binary into the become of to penetrate one disk to by the able of tempera- is region gap and disk inner the viscosities from high material sufficiently tures demon- for could numerical that (1996) Applying Lubow the strate & hydrodynamics. Artymowicz using scheme particle a been such smoothed have of far method so disks cumbinary Artymowicz 1991; 1994). al. et Lubow (Artymowicz & well (eccentricity) as temperature parameters binary as orbital the the of such on of parameters and form disk viscosity and on and width depend The of binary. gap shrinkage the the secular of a axis to semi-major leads the momentum angular of loss h iay evn eino eylwdniyaon the around called density so low a very binary, from of central receding region is a momentum, material leaving the binary, angular and the up, this speeds receiving disk inner Upon the to disk. binary the outer con- from a the momentum is angular there energy of Hence, their transfer disk. tinuous the deposit in dissipating, momentum waves upon angular momentum and which, angular disk ma- positive disk the the generating in than is period it shorter the a terial As with disk. orbiting an- the is of and binary exchange binary continuous the a between momentum is gular there disk accretion an by hsc,AfdrMresel 0 -27 Tübingen, D-72076 10, Morgenstelle der Auf Physics, l eta ceti iaysa ihna accretion an within star binary eccentric central a f sn aallzdDa-rdtcnqeo two on technique Dual-Grid parallelized a using h ealdeeg aac nldn icu heating viscous including balance energy detailed a is acltosdaigwt h vlto fcir- of evolution the with dealing calculations First nasse ossigo tla iaysurrounded binary stellar a of consisting system a In usdaccretion pulsed a tr napaedpnetpoes eare We process. dependent phase a in stars ral t crto ae n eie pcrlenergy spectral derived and rates accretion oth ytm fD a n KSo swl as well as Sco, AK and Tau DQ of systems e sia a tr.Hg eouinnumer- resolution High stars. Tau T assical s Numerical ds: h ih iaisasbtnilflwof flow substantial a binaries tight the hs antd n hs depen- phase and magnitude whose ne gap inner ttesm ie the time, same the At . oebr2,2018 20, November 2 Richard Günther & Wilhelm Kley: Circumbinary Disk evolution using only a small number of particles all having the same binary. Hence, the timescale for a change in a and e is mass only a limited resolution is achievable. indeed much longer than the timescales considered here Later Rozyczka & Laughlin (1997) used a finite dif- (Artymowicz et al. 1991). For the majority of the compu- ference method in attacking this problem. Using a con- tations we employ an inertial coordinate system, however stant kinematic viscosity coefficient and eccentric binaries test calculations were performed in a system corotating they also found, in agreement with Artymowicz & Lubow with the binary as well. (1996), a pulsed accretion flow across the gap, with most of the accretion occurring onto the lower mass secondary 2.2. Equations star. In this paper we extend these computations and solve The evolution of the disk is given by the two-dimensional explicitly a time dependent energy equation which in- (x, y or r, ϕ) evolutionary equations for the density Σ, the cludes the effects of viscous heating and radiative cooling. velocity field u ≡ (ux,uy) = (ur,uϕ), and the temperature We aim at the structure and dynamics of the disk as well T . In a coordinate-free representation the equations read as the gas flow in the close vicinity of the binary star. To ∂Σ that purpose we utilize a newly developed method which + ∇ · (Σu) = 0 (1) ∂t enables us to cover the whole spatial domain. For the first time we perform long-time integration of the complete sys- ∂Σu + ∇ · (Σu · u)= −∇P − Σ∇Φ+ ∇ · T (2) tem covering several hundred orbital periods of the binary ∂t and compare the properties of the evolved systems with ∂Σεtot observational data such as spectral energy distributions +∇·[(Σεtot + P ) u]= −Σu·Φ−∇·(F − u · T)(3) in the infrared and optical bands and accretion rates esti- ∂t 2 mated from luminosities. Here εtot = 1/2u + εth is the sum of the specific in- In the next section we layout the physical model fol- ternal (εth = cvT ) and kinetic energy. P is the vertically lowed by some remarks concerning important numerical integrated (two-dimensional) pressure issues (Sect. 3). We describe the setup of the various nu- RΣT merical models in Sect. 4. The main results are presented P = in Sect. 5 and our conclusions are given in Sect. 6. µ with the midplane temperature T and the mean molecular 2. Physical Model weight µ, which is obtained by solving the Saha rate equations for Hydrogen dissociation and ionization and Helium 2.1. General Layout ionization. The gravitational potential Φ, generated by the binary The system consists of a stellar binary system and a sur- stars, is given by rounding circumbinary accretion disk. We assume a ge- ometrically thin disk and describe the disk structure by GMi r r r r for| − i| > R∗ means of a two-dimensional, infinitesimal thin model in  | − i| Φ= −  2 2 (4) the z = 0 plane with the origin at the center of mass  GMi 3R − |r − ri| iX=1,2  ∗ for|r − r |≤ R (COM) of the binary. We use vertically averaged quanti- 2 R3 i ∗  ∗ ties, such as the surface mass density  where ri are the radius vectors to the two stars, and R∗ is ∞ the stellar radius assumed to be identical for both stars. Σ= ρdz, Z−∞ The second case in Eq. (4) gives the potential inside a star with radius R⋆ having a homogeneous (constant) density. where ρ is the regular three-dimensional density. We work with two different coordinate systems. Firstly, a cylindrical coordinate system (r, ϕ) centered on the COM of the 2.3. Viscosity and Disk Height binary stars. As such a coordinate system does not al- The effects of viscosity are contained in the viscous stress low for a full coverage of the plane, we overlay the center tensor T. Here we assume that the accretion disk may be additionally with a Cartesian (x, y) grid (see Sec. 3.2). described as a viscous medium driven by some internal The gas in the disk is non-self-gravitating and is or- turbulence which we approximate with a Reynolds ansatz biting a binary system with stellar masses M1 and M2 for the stress tensor. The components of T in different which are modeled by the gravitational potential of two coordinate systems are spelled out explicitly for example solid spheres. The binary is fixed on a Keplerian orbit in Tassoul (1978). A useful form for disk calculations con- with given semimajor axis a and eccentricity e. Hence, we sidering angular momentum conservation is given in Kley neglect self-gravity and the gravitational backreaction of (1999). the disk onto the stars. This is justified because the total For the kinematic shear viscosity ν we assume in this mass of the disk Md within the simulated region, which work an α-model (Shakura & Sunyaev 1973) in the form extends up to several tens of semi-major axis a of the binary, is typically much smaller than that of the central ν = αcsH (5) Richard Günther & Wilhelm Kley: Circumbinary Disk evolution 3 where cs and H are the local sound-speed and vertical 3. Numerical Issues height, respectively. The local vertical height H(r) is com- In order to study the disk evolution, we utilize a finite puted from the vertical hydrostatic equilibrium difference method to solve the hydrodynamic equations ∂Φ 1 ∂p = (6) outlined in the previous section. As we intend to model ∂z ρ ∂z possible accretion flow onto the binary stars and to achieve where ρ and p are the standard three dimensional density a very good resolution in their vicinity, we use a Dual- and pressure. Substituting Eq. (4) for the potential with Grid-technique. |r − ri| > R∗, assuming an ideal gas law p = RρT/µ, and integrating over z yields 3.1. General 1 z2 − 2 H2 ρ = ρ0e (7) The code is based on the hydrodynamics code RH2D 1/2 suited to study general two-dimensional systems includ- where ρ0 =Σ/[(2π) H] is the midplane density, and H RH2D is the vertical height given by ing radiative transport (Kley 1989). uses a spa- tially second-order accurate, mixed explicit and implicit − 1 − 1 2 2 method. Due to an operator-splitting method it is semi- GMi H(r)=   =  H−2(r) (8) second order accurate in time. The advection is computed 2 r r 3 i iX=1,2 cs | − i| iX=1,2 by means of the second order monotonic transport al-     gorithm, introduced by van Leer (1977), which guaran- which can be split into the single star disk heights given 3 tees global conservation of mass and angular momentum. |r−ri| by Hi(r)= cs . The sound speed is given here by Advection and forces are solved explicitly, and the viscos- q GMi cs = RT/µ. We assume a vanishing physical bulk viscosity ity and radiation are treated implicitly. The formalism for ζ, but consider it in the artificial viscosity coefficient (see application to thin disks in (r − ϕ) geometry has been de- Kley 1999). scribed in detail in (Kley 1998, 1999). Here this scheme is extended to allow for a Dual-Grid system (see 3.2) and a more detailed energy balance Eq. (9). 2.4. Radiative Balance As the calculations have shown, the system behaves ex- The influence of radiative and viscous effects on the disk tremely dynamically with very strong, moving gradients temperature are treated in a local balance. The balance at the inner edge of the disk. To model such a situation, equation for the internal and radiative energy considering the viscosity has to be treated implicitly to ensure numeri- only these two effects reads cal stability. The coupling of ur and uϕ requires a solution 4 method similar to that in Kley (1989). For stability pur- ∂ ΣcvT + aT = Qdiss − Qrad (9) poses an additional implicit artificial viscosity has to be ∂t included in the computations (Kley 1999). where Qdiss and Qrad denote the viscous dissipation and This code has been employed successfully in related radiative losses, respectively. For the viscous losses we use simulations dealing with embedded planets in disks (Kley the vertically averaged expression 1999, 2000). The numerical details of the present simula- 1 tions are presented in Günther (2001). Q = u · ∇T = T r T2 (10) diss 2Σν For the radiative transport 3.2. Dual-Grid technique

∂Fz Qrad = −∇ · F 0 − (11) The Dual-Grid technique employed here combines two de- ∂z sirable features of thin disk computations: a) An ideally where F 0 is the flux vector in the z = 0 plane. Here suited coordinate system (r − ϕ) in the outer bulk parts we consider only the losses in the vertical direction, i.e. of the disk which ensures conservation of angular momen- F 0 = 0, a standard approximation in accretion disk the- tum, and b) Coverage of the region near the central binary ory. Integration over the vertical direction yields star allowing the determination of the mass flow onto the 4 stars. Qrad =2Frad =2σBT (12) eff In the main part the domain is covered by a cylindrical with the local effective (surface) temperature Teff which is grid. On this (outer) grid the equations are formulated in related to the midplane temperature T through polar coordinates, and the solution technique guarantees −1/4 exact conservation of global angular momentum. The con- 1/4 1 3 Teff = τ T = + κΣ T (13) servation property is absolutely crucial in obtaining phys- 2 4 ically reliable results for long term disk evolution (Kley using an interpolated opacity κ(ρ0,T ) adapted from Lin & 1998). Papaloizou (1985). Note that this formulation of radiative However, such a grid cannot be extended to the very transfer is a very good approximation only in the optically center and leaves a hole in the middle. For typical com- thick regime. putations this does not pose serious problems, but in this 4 Richard Günther & Wilhelm Kley: Circumbinary Disk evolution

Fig. 1. Grid system of a typical calculation. Shown is the grid structure of the Cartesian and cylindrical grid at the center of the computational domain. Circumstellar disks of the system can be seen at the top and bottom edges of the picture. Fig. 2. Radial shock in first (infalling) phase right after crossing the grid boundaries. Surface density is color coded, the grid structure is shown. case we are dealing with eccentric binary systems in the middle. Hence, the stars or their circumstellar disks may a complete coverage of the physical space in between the easily cross the inner grid boundary causing strong irreg- two stars clearly outweighs this disadvantage. ularities. In any case a central hole does not allow for an accurate determination of the individual mass accre- 3.3. Tests tion rates onto or a possible overflow between the stars. Therefore, using a Cartesian grid centered on the origin We ran several tests to check the accuracy and numerical enables an accurate calculation of the mass flows in the stability of the grid overlap region, including an infalling vicinity of the stars. Applying this technique, a high lo- shock and a Sedov like explosion. For these tests no viscos- cal resolution can be obtained in the center as well. An ity nor radiative effects are applied. All tests show that the example of such a grid structure is displayed in Fig. 1. grid overlap technique is sufficient for handling problems Related numerical schemes using a nested grid tech- with radial symmetric features such as accretion disks. nique, though not in different coordinate systems, have The shock experiment initial setup is a step function been applied in astrophysical simulations by a number of of density and temperature with the floor starting at r0 authors (eg. Ruffert 1992; Yorke et al. 1993). (about 10 grid cells outside of the inner boundary of the The necessary equations (in Cartesian and cylindri- cylindrical grid) and covering the whole Cartesian grid: cal coordinates) are then integrated, independently and Σ(r)=Σ +Σ Θ(r − r ) . (14) in parallel, on the two grids, and after each timestep the min 0 0 necessary information in the overlap region has to be ex- In Fig. 2 you can see the grid setup and the shock in the changed. Restrictions are imposed on the time step for initial infalling phase just after crossing the grid bound- stability reasons, the Courant-Friedrichs-Lewy (CFL) con- aries. The shock front remains radially symmetric and no dition must be fulfilled during each integration, on each reflections are visible. After the shock compresses into a grid. single point it switches to the outflowing phase. You can Test calculations using a single Cartesian grid covering see the shock front propagating outwards just after cross- the whole domain have shown that the non-conservation of ing the grid boundaries in Fig. 3. Apart from unavoidable angular momentum yields serious deviations from known grid effects near the center the shock front remains nicely analytical results in particular for longer integration times. circular and again neither disturbances nor reflections are As the grids possess a different geometry, exact con- visible at the grid interface. servation of momenta and energy cannot be guaranteed in the conversion/interpolation process which uses linear 4. General model design interpolation to exchange density, temperature and velocity. Exchanging density, internal energy and momentum The main goal of this study is the investigation of the was also implemented and leads to more correct conserva- characteristic features of the hydrodynamic flow of the tion of momenta and energy. However, the advantage of circumbinary disk. Richard Günther & Wilhelm Kley: Circumbinary Disk evolution 5

the whole resolved stellar core (up to 11x11 grid-cells) for the tight systems, making it a locally conﬁned process. Two methods for deciding whether a grid cell is tar- geted for accreting a part of its mass were implemented. Accreting a constant fraction f per time of the mass exceeding a deﬁned minimum density Σmin leads to a very smooth accretion process. The accreted mass per iteration is computed by

Macc = min(1.0,f∆t) max(0.0, Σij−Σmin)∆Volij (15) i,jX∈R Limiting the density in the accretion region R to an ar- bitrary value Σmax results in a more realistic structure of the circumstellar disks and to more correct spectral energy distributions of the circumstellar disks.

Macc = max(0.0, Σij − Σmax) ∆Volij (16) i,jX∈R This accretion process is not smooth, but consists of accretion events because exceeding the maximum density is Fig. 3. Radial shock in second (explosion) phase right af- a discrete process and not distributed evenly over time. ter crossing the grid boundaries again. Surface density is Nevertheless the time-averaged accretion rates are the color coded, the imaged domain is the same as in Fig. 2. same for both methods. The inferred accretion rates in any event compare fa- vorably with the average disk accretion rate, as estimated from the radial mass fluxes through different surfaces, and From now on, we refer to non-dimensional units. All the evolution of the total mass found in the circumstellar the lengths are expressed in units of the semi-major axis disks. a of the binary. This is constant because the time scale on which the binary orbital elements change due to mass and angular momentum transfer is much longer than the 4.2. Spectra generation dynamical time scales considered here (see Artymowicz & Spectral energy distributions are generated decoupled Lubow 2001). Masses are in units of the solar mass and from the dynamic evolution of the disk as a post- time is given in units of the binary’s orbital period. processing step. Here the model from Adams et al. (1988) The whole azimuthal range of the disk is taken into ac- is extended to work on data generated by our numerical count by considering a computational domain represented simulation. The flux at the observer is obtained by inte- by 2 π × [0, rout], where typically rout = 10 in units of grating over the disk surface assuming local black-body the binary separation. The cylindrical grid covers a re- radiation gion 2 π × [rin, rout] with rin = 0.1 represented by up to cos i Rd 2π 256 × 256 meshpoints, and the central Cartesian grid has F = B [T (r, ϕ)] 1 − e−τ(r,ϕ) rdϕ dr(17) ν D2 ν up to 128 × 128 meshpoints covering 2 rin × 2 rin with an Zr0 Z0 additional small overlap. where i is the inclination of the disk, D the distance to the observer and τ the line-of-sight optical depth through the disk. τ can be obtained using the opacity κ, τ(r, ϕ)= 4.1. Accretion onto the Stars κΣ(r,ϕ) cos i . T is the surface disk temperature which has to As will be seen, matter flows from the disk through the be estimated using the optical depth of the disk obtained gap and circulates the stars in small circumstellar disks. from an opacity table. From those it can be accreted onto the stars. This stellar The integration is limited to regions outside of the stel- accretion is accounted for by removing some mass from lar cores as temperatures and densities from inside the a region close to the stars. This removal is performed on cores can be nothing but wrong. The star emission is ac- both grids dependent on the star positions. The removed counted for by adding a black-body spectrum from an mass is not added to the dynamical mass of the stars, but appropriate effective star temperature. is just monitored. The region R out of which matter is removed is given as 4.3. Initial and boundary conditions a fraction (0.2) of the Roche lobe size for the wide systems and as the stellar cores for the tight spectroscopic systems. The initial density distribution is proportional to r−d, The accretion process typically involves only a few grid where d is either set to 0.5 which is the equilibrium den- cells on each side of the stars for the wide systems and sity distribution for an accretion disk around a central star 6 Richard Günther & Wilhelm Kley: Circumbinary Disk evolution

Σ Σ -1 1.4 (r) = 0r Initial gap profile 1.2

0.8 Σ

0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5 3 3.5 4 r

−1 Fig. 4. Initial gap profile composed out of Σ0r and the gap function Eq. (18). The figure shows a gap at rgap =1 Fig. 5. Evolution of an equal mass binary after 90 or- with Σ0 = 1. bital periods. Color encoding is logarithmic surface density. Both the inner part of the circumbinary disk and the having a constant kinematic viscosity, or given by a fit of circumstellar disks are visible. the generated spectral energy distribution to the one observed. However, we superimpose to this an axisymmetric 5.1. Equal Mass Binary gap around the COM of the two stars, given by an approximate gap radius rgap (Artymowicz & Lubow 1994) The equal mass binary on a circular orbit serves as a test- and a gap function ing platform for numerical stability and accuracy. In a coordinate frame corotating with the binary one expects af- r−r −1 − gap ter some transient time a quasi-stationary situation, which f := e 0.1rgap +1 . (18) gap is symmetric with respect to the line connecting the stars. As the viscosity is solved implicitly by solving a large Figure 4 shows a typical surface density at t = 0. The matrix system iteratively, exact symmetry cannot be pre- initial velocity field is that of a Keplerian disk orbiting served numerically. Also usual density contrasts are of the the center of mass of the two stars. order of 103 for numerical floor to circumstellar disks and The gravitational potential in Eq. (4) is phased-in from circumstellar disks to the circumbinary disk. Hence, smoothly from an initial central potential for the total this test serves as an excellent example for estimating the mass, M1 +M2, during an initialization phase lasting typ- ability to preserve symmetry over long time scales. ically ten orbital periods. This allows the system to adjust In Fig. 5 you can see a circumbinary disk after 90 orbits smoothly from the semi-stationary setup of a single cen- evolution time in almost perfect quasi-stationary state. tral star to the binary star situation. The deviation from line-symmetry is small. Only with very For the boundary conditions, periodicity is imposed at high resolution simulations we observe spiral features in ϕ = 0 and ϕ = 2 π. We set outflowing boundary con- the outer regions of the disk, while spiral arms originating ditions at the outer radial border (rout), i.e. the radial from the inner gap of the circumbinary disk flowing onto velocity gradient at Rmax is zero, and truncated to posi- the circumstellar disks can be observed even for low reso- tive values. Angular velocity there equals the unperturbed lutions. Those spiral arms feeding the circumstellar disks initial Keplerian value. remain stable for the whole period of a non-eccentric bi- For single grid calculations with a cylindrical grid we nary. use zero gradient boundary conditions for both velocity The accretion rates for both accretion mechanisms components and the density at the inner border. onto the two stars are nearly identical and do not show any significant variations for different numerical parameters. Thus, our implementation of the accretion mechanism is 5. Results robust. Changing model parameters such as Σ0, gap width, Before discussing the results concerning particular astro- or disk size we obtain characteristic variations of the re- nomical objects, we would like to present the details of a sulting spectral energy distribution of an evolved disk. specific test case of an equal mass binary star on a circu- Figures 6-8 show dependencies of these parameters on the lar orbit. Such a system is well suited to check important spectral energy distribution that are used to fit parameters properties (symmetries, stability) of the numerical method of the spectroscopic systems to match observed spectra. used. Also general properties of accretion disks in binary systems can be derived from such a test case. Richard Günther & Wilhelm Kley: Circumbinary Disk evolution 7

1e+35 AK Scorpii DQ Tauri no initial gap d d d d 1 AU P 13 .60933 ± 0 .00040 15 .8043 ± 0 .0024 2.5 AU e . ± . . ± . 1e+34 5 AU 0 469 0 004 0 556 0 018 10 AU q 0.987 ± 0.007 0.97 ± 0.15 ] ◦ ◦ -1 i 63 23 1e+33 a 0.16 AU 0.135 AU [erg s ν T F ∗ 6500 K 4000 K ν

2 1e+32 d R∗ 2.2R⊙ 1.785 R⊙ π 4 M∗ 1.5 M⊙ 1.3 M⊙ 1e+31 d 152 pc 140 pc Md 0.005 M⊙ 0.001 M⊙ Rd 1e+30 40 AU 50 AU 1e+11 1e+12 1e+13 1e+14 1e+15 ν [Hz] Table 1. Parameters for AK Sco and DQ Tau.

Fig. 6. Dependency of the gap width on the SED.

1e+35 2.5 AU extension 5 AU extension 25 AU extension 1e+34 50 AU extension 100 AU extension ] -1 1e+33 [erg s ν F ν

2 1e+32 d π 4

1e+31

1e+30 1e+11 1e+12 1e+13 1e+14 1e+15 ν [Hz]

Fig. 7. Dependency of the disk size on the SED. Fig. 9. AK Sco circumbinary disk after 82.0 orbital pe- 1e+36 star emission riods in apastron. Color coding is log(Σ), the size of the 10-3 solar masses -2 stars reﬂects the actual stellar radii, the length scales are 1e+35 10 solar masses 10-1 solar masses in AU 100 solar masses ]

-1 1e+34 vant information for the simulations has been summarized [erg s

ν 1e+33

F in Table 1, disk masses Md are integrated densities over ν 2

d the computational domain specified by the disk radii R . π 1e+32 d 4 Images showing the disk configuration in apastron and 1e+31 periastron phase are shown in Figs. 9 and 10. Generally, for eccentric systems we observe spiral fea- 1e+30 tures in the outer circumbinary disk only for very high 1e+11 1e+12 1e+13 1e+14 1e+15 resolution simulations, while again spiral arms feed the ν [Hz] circumstellar disks even with low resolutions. Opposed to the non-eccentric systems the spiral arms develop during Fig. 8. Dependency of Σ0 on the SED. apastron phase and are torn off together with the outer parts of the circumstellar disks after periastron phase for 5.2. Spectroscopic Systems high eccentric systems. This tearing off is caused by raising centrifugal forces acting on the circumstellar material. Hereafter we discuss models for the two spectroscopic For both systems we compared the accretion rates re- young binary stars DQ Tau and AK Sco which have a sulting from the simulation with accretion rates derived separation of only a fraction of an AU. The physical pa- from of observations as done by Hartigan et al. (1995) and rameter for the systems are taken from Andersen et al. Gullbring et al. (1998) (see Table 2). As no data was avail- (1989); Favata et al. (1998); Jensen & Mathieu (1997) for able for AK Sco, a comparison with observational data was AK Sco and Mathieu et al. (1997) for DQ Tau. The rele- not possible, but as both spectroscopic systems have sim- 8 Richard Günther & Wilhelm Kley: Circumbinary Disk evolution

1e-07 orbital phase accretion rate (primary) accretion rate (secondary) average accretion rate

1e-08 Mdot [Msol/yr]

30 31 32 33 34 35 t [orb]

Fig. 11. Time-dependent accretion rate of the AK Sco system. The orbital phase curve schematically shows the distance of the two stars. Fig. 10. DQ Tau circumbinary disk after 85.5 orbital periods in periastron. Color coding is log(Σ), the size of the 1e-07 stars reﬂects the actual stellar radii, the length scales are orbital phase accretion rate (primary) in AU. accretion rate (secondary) average accretion rate Hartigan Gullbring simulation DQ Tau 0.50 · 10−7 0.60 · 10−9 0.62 · 10−8 −8 AK Sco n.a. n.a. 0.83 · 10 1e-08 GG Tau 0.20 · 10−6 0.175 · 10−7 0.11 · 10−8 −6 −7 −7 UY Aur 0.25 · 10 0.656 · 10 0.5 · 10 Mdot [Msol/yr]

−1 Table 2. Accretion rates in M⊙yr derived from observational data from Hartigan et al. (1995) and Gullbring 1e-09 et al. (1998) compared to rates as resulting from our sim- 26 28 30 32 34 ulations. Our rates are for the primary component of the t [orb] binaries, for the other data we assume it is an averaged accretion rate for both stars. Fig. 12. Time-dependent accretion rate of the DQ Tau system. The orbital phase curve schematically shows the distance of the two stars. ilar system parameters, accretion rates of the same order of magnitude can be expected. As can be seen from Figs. 11 and 12 the accretion −9 −1 M˙ ≃ 7 · 10 M⊙yr the disk is very optically thin. In process happens synchronized with periastron phase of that regime the implemented radiative loss mechanism is the binaries. This is in agreement with numerical simu- not very accurate and thus these results are not meaning- lations from Artymowicz & Lubow (1996) and Rozyczka ful. & Laughlin (1997) and with observations which show ex- For the spectroscopic binaries we can generate spec- cess luminosity at these phases (Mathieu et al. 1997). tral energy distributions out of the simulation data which From our simulations we can derive an averaged ac- −8 −1 can be made to match the observed ones by manually ad- cretion rate of 0.83 · 10 M⊙yr for the primary, 0.82 · −8 −1 justing initial model parameters such as disk mass, gap 10 M⊙yr for the secondary of AK Sco and 0.62 · width and density profile d. For DQ Tau Fig. 13 shows 10−8 M yr−1 for the primary, 0.51 · 10−8 M yr−1 for the ⊙ ⊙ the spectral energy distribution composed out of the disk secondary of DQ Tau. These accretion rates are in good and the star, Fig. 14 shows the one for the AK Sco sys- agreement with the observational rates. tem. The effective stellar temperatures used in the model It is expected that the luminosity increases in pe- have been adjusted to match the observational data. They riastron because tidal compression of the disk leads to differ from the quoted values in the literature in Table 1. a temperature rise with subsequent luminosity increase. This may be due to difficulties in separating individual We tried to generate luminosity curves to show that in- stellar and disk contributions to the total flux in models deed excess luminosity is produced at periastron phase, and observations. Also occultation is not accounted for on but unfortunately those curves show both brightening adding the star spectrum to the disk. and darkening dependent on the implemented accretion mechanism. At the very low averaged accretion rates of Richard Günther & Wilhelm Kley: Circumbinary Disk evolution 9

UY Aurigae GG Tauri measurements stars emission (2450k) P 2074 yr 490 yr 1e+33 disk emission e . . stars and disk emission 0 13 0 25 q 0.63 0.77 ◦ ◦ ◦ ] i ± -1 1e+32 42 43 5 a 190 AU 67 AU [erg s ν

F T∗ 4000 K 3800 K ν 2

d 1e+31

π R∗ 2.6R⊙ 2.8R⊙ 4 M∗ 0.95 M⊙ 0.65 M⊙ 1e+30 d 140 pc 150 pc Md 1.1 M⊙ 0.15 M⊙ Rd 2100 AU 600 AU 1e+12 1e+13 1e+14 1e+15 ν [Hz] Table 3. Parameters for UY Aur and GG Tau.

Fig. 13. Spectral energy distribution of DQ Tau (d =2.0), crosses show observational data taken from Mathieu et al. (1997).

measurements stars emission (4450k) disk emission 1e+34 stars and disk emission ] -1 [erg s ν F ν 2 d π 4 1e+33

1e+13 1e+14 1e+15 ν [Hz] Fig. 15. UY Aur circumbinary disk after 20.5 orbital periods. Color coding is log(Σ), the length scales are in AU. Fig. 14. Spectral energy distribution of AK Sco (d =1.5), crosses show observational data taken from Jensen & Mathieu (1997). GG Tau is one order of magnitude less than the one for UY Aur. Accretion is reinforced at periastron phases as for the 5.3. Wider Systems spectroscopic systems, but events are an order of magnitude lower for GG Tau and nearly vanish for UY Aur due There are only two well known systems where the binary to the lower eccentricity of the systems. Also both systems and the disk have been imaged directly, GG Tau (Dutrey show more phase dependent accretion for the secondary et al. 1994; Guilloteau et al. 1999) and UY Aur (Close than for the primary. et al. 1998; Duvert et al. 1998). These systems have a In Figs. 15 and 16 you can see the inner parts of the much wider separation of tens to over 100 AU, a mass simulated circumbinary disks. The circumstellar disks re- ratio different from unity and show consequently a quite main intact over the whole period. It was not possible to different behavior. In Table 3 the system parameters used generate meaningful spectral energy distributions of the for the simulations are shown. disks as they are too cold for these wide systems. Also From our simulations we derive averaged accretion observational data is lacking for these systems. rates for both UY Aur and GG Tau that are comparable with the observed quantities in Table 2. In the −7 −1 6. Conclusion case of UY Aur we get 0.5 · 10 M⊙yr for the pri- −6 −1 mary and 0.25 · 10 M⊙yr for the secondary, respec- With the present calculations we have modeled the accre- tively. Simulations of GG Tau lead to accretion rates of tion flow onto the stars within a circumbinary disk in more −8 −1 −9 −1 0.11·10 M⊙yr for the primary and 0.33·10 M⊙yr detail using more realistic physics, much higher resolution for the secondary. As expected from the difference of the and a much longer time integration. Our main achieve- disk masses (0.15 M⊙ vs. 1.1 M⊙) the accretion rate for ments may be summarized as follows: 10 Richard Günther & Wilhelm Kley: Circumbinary Disk evolution

References Adams, F. C., Shu, F. H., & Lada, C. J. 1988, ApJ, 326, 865 Andersen, J., Lindgren, H., Hazen, M. L., & Mayor, M. 1989, A&A, 219, 142 Artymowicz, P., Clarke, C. J., Lubow, S. H., & Pringle, J. E. 1991, ApJ, 370, L35 Artymowicz, P. & Lubow, S. H. 1994, ApJ, 421, 651 —. 1996, ApJ, 467, L77 Artymowicz, P. & Lubow, S. H. 2001, in IAU Symposium, Vol. 200, 439 Bate, M. R. & Bonnell, I. A. 1997, MNRAS, 285, 33 Close, L. M., Dutrey, A., Roddier, F., et al. 1998, ApJ, 499, 883 Duquennoy, A. & Mayor, M. 1991, A&A, 248, 485 Dutrey, A., Guilloteau, S., & Simon, M. 1994, A&A, 286, Fig. 16. GG Tau circumbinary disk after 56.4 orbital pe- 149 riods. Color coding is log(Σ), the length scales are in AU. Duvert, G., Dutrey, A., Guilloteau, S., et al. 1998, A&A, 332, 867 Favata, F., Micela, G., Sciortino, S., & D’Antona, F. 1998, A&A, 335, 218 1. Development of a new Dual-Grid technique combin- Guilloteau, S., Dutrey, A., & Simon, M. 1999, A&A, 348, ing two coordinate systems, which is ideally suited for 570 modeling planar accretion disks including their central Gullbring, E., Hartmann, L., Briceno, C., & Calvet, N. objects. 1998, ApJ, 492, 323 2. Performing long-time integrations of binary stars and Günther, R. 2001, Dynamik und Entwicklung von circumbinary disks, including a detailed vertical energy zirkumbinären Scheiben (Diploma Thesis, University of balance. Tübingen,), 64 3. For close binaries with separations of only a fraction of Hartigan, P., Edwards, S., & Ghandour, L. 1995, ApJ, an AU we find a relatively large accretion rate crossing 452, 736 −8 −1 the gap of the order of 10 M⊙yr . Jensen, E. L. N. & Mathieu, R. D. 1997, AJ, 114, 301 4. For wide binaries (a = 10 − 200AU) the accretion rate Kley, W. 1989, A&A, 208, 98 is comparable to those of the close binaries only be- —. 1998, A&A, 338, L37 cause of the large disk masses of the two systems. —. 1999, MNRAS, 303, 696 5. For a suitable variation of the initial density distribu- —. 2000, MNRAS, 313, L47 tion Σ ∼ r−d of the disk, with d between 1.5 and 2 the Lin, D. N. C. & Papaloizou, J. 1985, in Protostars and spectral energy distributions of DQ Tau and AK Sco Planets II, 981 could be fitted well to the observational data. Mathieu, R. D., Ghez, A. M., Jensen, E. L. N., & Simon, M. 2000, Protostars and Planets IV, 703 Future models of this kind need to include a more detailed Mathieu, R. D., Stassun, K., Basri, G., et al. 1997, AJ, structure of the stars as well as a more detailed model of 113, 1841 the accretion process. The numerical resolution we have Rozyczka, M. & Laughlin, G. 1997, in ASP Conf. Ser. 121: reached by now is so fine that the individual stars in close IAU Colloq. 163: Accretion Phenomena and Related binaries are already resolved by usually about 100 grid- Outflows, 792 cells within low-res calculations. Ruffert, M. 1992, A&A, 265, 82 To obtain a detailed comparison with observed light Shakura, N. I. & Sunyaev, R. A. 1973, A&A, 24, 337 curves in particular the phase dependence a more elabo- Tassoul, J. 1978, Theory of rotating stars (Princeton rate model of the radiative loss of the very thin material Series in Astrophysics, Princeton: University Press, in the circumstellar disks is required. In that case 3d sim- 1978) ulations may be necessary which are beyond present day van Leer, B. 1977, Journal of Computational Physics, 23, computational resources. 276 Yorke, H. W., Bodenheimer, P., & Laughlin, G. 1993, ApJ, 411, 274 Acknowledgements. Part of this work was funded by the Zinnecker, H. & Mathieu, R. 2001, in IAU Symposium, Simulation German Science Foundation (DFG) under SFB 382 Vol. 200 physikalischer Prozesse auf Höchstleistungsrechnern. We would like to thank Dr. Hubert Klahr for many stimulating discus- sions during this project.