The Astrophysical Journal December A Preprint typ eset using L T X style emulateap j E THE RESPONSE OF ACCRETION DISKS TO BENDING WAVES ANGULAR MOMENTUM TRANSPORT AND RESONANCES oline EJMLJ Terquem Car UCOLick Observatory University of California Santa Cruz CA Isaac Newton Institute for Mathematical Sciences University of Cambridge Clarkson Road Cambridge CB EH UK ctucolickorg The Astrophysical Journal December ABSTRACT We investigate the linear tidal p erturbation of a viscous Keplerian disk by a companion star orbiting We consider m p erturbations with o dd symmetry with resp ect to in a plane inclined to the disk midplane The resp onse frequency may b e either nite or vanishing These longwavelength the z p erturbations pro duce a welldened warp Since the resp onse of a viscous disk is not in phase with the p erturbing p otential a tidal torque is on the disk When the p erturb er rotates outside the disk this torque results in a decrease of the exerted disk angular momentum and thus in an increase of its accretion rate tal viscous stress acting on the background We show that this tidal torque is comparable to the horizon w when the p erturb ed velo cities in the disk are on the order of the sound sp eed If these velo cities o remain subsonic the tidal torque can exceed the horizontal viscous stress only if the viscous parameter which couples to the vertical shear is larger than the parameter coupled to the horizontal shear h v In protostellar disks b ending waves which are predominantly excited in the outer regions are found to propagate and transp ort a signicant fraction of the negative angular momentum they carry deep into the disk inner parts If the waves are reected at the center resonances o ccur when the frequency of the tidal waves is equal to that of some free normal global b ending mo de of the disk If such resonances exist tidal interactions may then b e imp ortant even when the binary separation is large Out of resonance the torque asso ciated with the secular p erturbation which is prop ortional to is generally much larger v than that asso ciated with the nite frequency p erturbations As long as the waves are damp ed b efore they reach the center the torque asso ciated with the nite frequency p erturbations do es not dep end on the viscosity in agreement with theoretical exp ectation These calculations are relevant to disks around y binary systems young stars and mayb e also to disks in Xra Subje ct headings accretion accretion disks binaries general hydro dynamics stars premain sequence waves INTRODUCTION Paczynski This has recently received Pringle observational conrmation b oth through the study of Tidal Interactions in PremainSequence Binary young binaries sp ectra Dutrey Guilloteau Simon Systems Coplanar versus Noncoplanar Systems Mathieu Fuller and direct imaging Ro d Jensen Thanks to recent optical images taken with the Hub dier et al ble Space Telescope accretion disks around lowmass The tidal eect of an orbiting b o dy on a dierentially premain sequence stars known as T Tauri are no rotating disk has b een well studied in the context of plane of theoretical sp eculations McCaughrean longer ob jects tary rings Goldreich Tremaine planet formation ODell As exp ected they app ear to b e common and interacting binary stars see Lin Papaloizou et al Furthermore around young stars Stauer and references therein In these studies the disk and orbit most T Tauri stars are observed to b e in multiple systems are usually taken to b e coplanar However there are obser see Mathieu and references therein Since the dis vational indications that the plane of the disk and that of tribution of the separation of premain sequence binaries the orbit may not necessarily b e aligned The most strik has a p eak around astronomical units which is smaller ing evidence for such noncoplanarity is given by HST and than the extent of the disks in such systems Edwards et adaptive optics images of the premain sequence binary it is exp ected that many circumstellar disks will al HK Tau Stap elfeldt et al The pro jected binary b e sub ject to tidal eects due to the inuence of binary separation in this system is AU the observed radius companions AU and the disk as of the circumsecondary disk is The theory of tidal interaction has already success p ect ratio at a radius of AU is ab out The authors fully satised many observational tests For example it suggest that the primary would b e lo cated at an angle of has b een predicted that angular momentum exchange b e at least ab ove the disk plane In the case of the hier tween orbital motion and disk rotation through tidal in archical triple system Ty Cra some mo dels suggest that teraction may truncate the disk see eg Papaloizou the orbit of the close sp ectroscopic binary and that of the ysique Observatoire de Grenoble Universite Joseph FourierCNRS Grenoble Cedex France On leave from Lab oratoire dAstroph TERQUEM studies Papaloizou Lin p erformed a full dimen wider binary are not in the same plane see Corp oron La sional linear analysis of m b ending waves in accretion grange Beust Bib o The Dawanas Also disks Comparing analytical results with numerical simu observations of the premain sequence binary T Tau have lations they showed that a vertical averaging approxima revealed that two bip olar outows of very dierent orien ohm Solf tion can b e used when the wavelength of the p erturbation tations originate from this system B Since it is unlikely that they are b oth ejected by the same is much larger than the disk semithickness Using the star each of them is more probably asso ciated with a dif WKB approximation they calculated the disp ersion re ferent memb er of the binary Furthermore jets are usually lation asso ciated with b ending waves taking into account thought to b e ejected p erp endicularly to the disk from selfgravity pressure and viscosity see also Masset Tag which they originate These observations then suggest ger for a study of thick disks They found that in that disks and orbit are not coplanar in this binary sys a non selfgravitating thin Keplerian disk in which is v tem More generally observations of molecular outows in small compared to the disk asp ect ratio b ending waves star forming regions commonly show several jets of dier propagate without disp ersion at a sp eed which is ab out ent orientations emanating from unresolved regions of ex half the sound sp eed Using these results Papaloizou tent as small as a few hundred astronomical units Davis Terquem calculated the m b ending wave re Mundt Eisloel sp onse of an inviscid disk with the rotation axis misaligned We note that the sp ectral energy distribution of a cir with a binary companions orbital rotation axis This cumstellar disk could b e signicantly aected by the lack work was extended by Larwo o d et al who p er of coplanarity in binary or multiple systems Terquem formed nonlinear calculations of the disk resp onse using Bertout This is b ecause repro cessing of radi smo othed particle hydro dynamics SPH This numerical ation from the central star by a disk dep ends crucially on metho d was further used by Larwo o d Papaloizou the disk geometry who analyzed the resp onse of a circumbinary disk sub ject In such noncoplanar systems b oth density and b ending to the tidal forcing of a binary with a xed noncoplanar waves are excited in the disk by the p erturbing companion circular orbit and by Larwo o d who considered the They are resp ectively of even and o dd symmetry with re tidal interaction o ccurring b etween a disk and a p erturb er sp ect to reection in the disk midplane Since much work following a parab olic tra jectory inclined with resp ect to has already b een carried out on density waves which are the disk the only waves excited in coplanar systems we shall fo cus The present work should b e contrasted with earlier here on b ending waves Because their wavelength is larger studies of purely viscous disks in which pressure and than that of density waves they are exp ected to propagate selfgravity were ignored The b ehavior of nonplanar and transp ort angular momentum further radially into the purely viscous accretion disk sub ject to external torques disk has b een considered previously by a numb er of authors Following the analysis of Bardeen Petterson Bending Waves these studies have b een develop ed in connection with Bending waves were rst studied by Hunter the precessing disks of Her X and SS for instance To omre in the context of the Galactic disk They eg Petterson Katz a b galactic warps calculated the disp ersion relation asso ciated with these eg SteimanCameron Durisen or warp ed disks waves taking into account selfgravity but ignoring pres around AGN and in Xray binaries Pringle How sure and viscosity Bending waves were further con ever the resp onse of a purely viscous disk is very dierent sidered in the context of galactic disks To omre from that of a viscous gaseous disk which is a b etter repre Sparke Sparke Casertano and planetary sentation of a protostellar disk In the former case evolu rings Shu Cuzzi Lissauer Shu In these tion o ccurs only through viscous diusion whereas in the studies the horizontal motions asso ciated with the ver latter case pressure eects manifesting themselves through tical displacement of the disk were neglected However b ending waves can control the disk evolution When these Papaloizou Pringle showed that these motions waves propagate ie when they are not damp ed