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 they

are imp ortant in gaseous Keplerian disks where they are

do so on a timescale comparable to the sound crossing

nearly resonant These authors p ointed out that when the

time Since this is much shorter than the viscous diu

tly viscous so that its resp onse to verti disk is sucien

sion timescale communication through the disk o ccurs as

cal displacements is diusive rather than wavelike these

a result of the propagation of these waves Even when the

motions lead to the decay timescale of the warp b eing

waves are damp ed b efore reaching the disk b oundary a dif

v

Here is the Shakura times the viscous timescale t

fusion co ecient for warps that is much larger than that

v v

Sunyaev parameter that couples to the vertical

pro duced by the kinematic viscosity may o ccur b ecause of

shear and t is the disk viscous timescale asso ciated

pressure eects Papaloizou Pringle Selfgravity

v

to When is smaller than unity this timescale is

may also strongly mo dify the resp onse of viscous disks

v v

smaller than t which is the decay timescale of the warp

b ecause like pressure it allows b ending waves to propagate

v

obtained by previous authors

and its eect is global For a more detailed discussion of

In the context of protostellar disks Artymowicz

the resp ective eects of viscosity pressure and selfgravity

studied the orbital evolution of lowmass b o dies crossing see Papaloizou Terquem Lin

disks and Ostriker calculated the angular momen

d on the Disk by a Companion on an Torque Exerte

tum exchange b etween an inviscid disk and a p erturb er on

Inclined Orbit

a nearparab olic noncoplanar orbit outside the disk see

also Heller for numerical simulations The vertical There can b e no tidal torque exerted on an inviscid disk

structure of the disk was not taken into account in these with reective b oundaries that contains no corotation reso

ACCRETION DISKS AND BENDING WAVES

nance Goldreich Nicholson However if the disk zontal shear Numerical results are presented in x b oth

is viscous or has a nonreective b oundary the compan for uniform and nonuniform viscosity They indicate that

ion exerts a torque on the disk that leads to an exchange for parameters typical of protostellar disks b ending waves

of angular momentum b etween the disk rotation and the are able to propagate and transp ort a signicant fraction

orbital motion We can distinguish the following three of the negative angular momentum they carry deep into

comp onents of the torque the inner parts of the disk They also show that if the

The comp onent along the line of no des makes the waves can b e reected at the disks inner b oundary res

and disk precess at a rate that can b e uniform see x onances o ccur when the frequency of the tidal waves is

Kuijken Papaloizou Terquem Papaloizou et equal to that of some free normal global b ending mo de

al Although we will calculate in this pap er the pre of the disk Out of resonance if the disk and the or

cession frequency induced by the tidal torque we will not bital planes are not to o close to b eing p erp endicular the

discuss the disk precession in detail since this has already torque asso ciated with the zerofrequency p erturbing term

b een done in the pap ers cited ab ove is found to b e much larger than that asso ciated with the

The comp onent of the torque along the axis p er nite frequency terms As long as the waves are damp ed

p endicular to the line of no des in the disk plane in b efore they reach the center the torque asso ciated with

duces the evolution of the inclination angle of the disk the nite frequency p erturbations do es not dep end on the

plane with resp ect to that of the orbit This evolution viscosity in agreement with theoretical exp ectation Gol

which is not necessarily toward coplanarity o ccurs on the dreich Tremaine We summarize and discuss our

disks viscous timescale if the warp is linear Papaloizou results in x

Terquem Papaloizou et al Since it aects the

PERTURBING POTENTIAL

geometry of the system only on such a long timescale we

will not discuss this eect here We note that b oth these

We consider a binary system in which the primary has

comp onents of the tidal torque mo dify the direction of the

a mass M and the secondary has a mass M The binary

p s

disk angular momentum vector

We supp ose that the orbit is circular with separation D

The comp onent of the torque along the disks spin

primary is surrounded by a disk of radius R D and

axis do es not mo dify the direction but the mo dulus of the

of negligible mass M so that the orbital plane do es not

disks angular momentum vector thus changing the disks

precess and the secondary describ es a prograde Keplerian

accretion rate This eect is imp ortant since it is a source

orbit with angular velo city

of angular momentum transp ort in the disk and it will b e

r

the main sub ject of this pap er

G M M

s p

So far angular momentum exchange b etween a viscous

D

gaseous disk rotation and the orbital motion of a compan

ion on an inclined circular orbit has not b een considered

ab out the primary In the absence of the secondary star

It is the goal of this pap er to present an analysis of this

the disk is nearly Keplerian We adopt a nonrotating

pro cess In the case of an inviscid disk considered by Pa

y z centered on the pri Cartesian co ordinate system x

paloizou Terquem such angular momentum ex

mary star The z axis is chosen to b e normal to the initial

change o ccurs b ecause the waves are assumed to interact

disk midplane We shall also use the asso ciated cylindri

nonlinearly with the background ow b efore reaching the

cal p olar co ordinates r z The orbit of the secondary

disks inner b oundary Thus only ingoing waves propa

star is in a plane that has an initial inclination angle

gate through the disk This induces a lag b etween the disk

with resp ect to the x y plane

resp onse and the p erturbation which enables a torque to

We adopt an orientation of co ordinates and an origin of

b e exerted on the disk In a viscous disk the lag required

time such that the line of no des coincides with and the

for the torque to b e non zero is naturally pro duced by the

secondary is on the xaxis at time t We denote the

viscous shear stress

with D jDj p osition vector of the secondary star by D

due to the binary at The total gravitational p otential

the Paper Plan of

is given by a p oint with p osition vector r

The plan of the pap er is as follows In x we derive an

GM GM r GM D

expression for the p erturbing p otential In x we outline

s s p

the basic equations The mo del of the unp erturb ed disk

jrj jr Dj D

is describ ed in x The disk resp onse is presented in x

where G is the gravitational constant The rst dominant

where we give the linearized equations study the disk re

term is due to the primary while the remaining p erturbing

sp onse for b oth zero and nite p erturbing frequencies es

terms are due to the secondary The last indirect term ac

tablish the b oundary conditions and describ e the domain

counts for the acceleration of the origin of the co ordinate

of validity of this analysis In x we calculate the torque

system

exerted by the p erturb er on the disk and relate it to the

We are interested in disk warps that are excited by terms

disk viscosity by using the conservation of angular mo

in the p otential that are o dd in z and have azimuthal mo de

mentum We show that this tidal torque is comparable to

numb er m when a Fourier analysis in is carried out

the horizontal viscous stress acting on the background ow

The terms in the expansion of the p otential that are of the

when the p erturb ed velo cities in the disk are on the order

required form are given by

of the sound sp eed If these velo cities remain subsonic the

tidal torque can exceed the horizontal viscous stress only

if the viscous parameter that couples to the vertical

v

shear is larger than the parameter coupled to the hori h

TERQUEM

Z

n

n

sin

K n

n

0 0

n

r z t

r H

K

C

n

0 0 0

r z t sin d

nn

K n C r z

We consider a thin disk such that z r Since r D

n

and z D we then retain only the lowest order terms in r D

n

z

in the ab ove integral which b ecomes

nn

K

H

GM

s

0

r z cos sin sin t

Here is the Keplerian angular velo city given by

K

K

D

GM r H is the disk semithickness is the surface

p

mass density dened by

cos sin sin t sin sin

Z

H r

Because the principle of linear sup erp osition can b e

r r z dz

used the general problem may b e reduced to calculating

H r

the resp onse to a complex p otential of the form

and

0 i t

p

if r z e

n

C

Here the pattern frequency of the p erturb er is one of

n

p

n

or and f is an appropriate real amplitude

where denotes the gamma function

BASIC EQUATIONS

It then follows from the radial hydrostatic equilibrium

The resp onse of the disk is determined by the equation

equation that the angular velo city is given by

of continuity

n

n

K n

K

r v

r C

n

t

and the equation of motion

i h

n

K

r

v

F v r v rP r

In principle the surface density prole should b e de

t

termined selfconsistently as a result of viscous diusion

where P is the pressure is the mass density and v is

LyndenBell Pringle However since we ignore

the ow velo city We allow for the p ossible presence of a

the eect of viscosity on the unp erturb ed disk we are free

viscous force F but following Papaloizou Lin

to sp ecify arbitrary density proles for the equilibrium disk

we shall assume that it do es not aect the undistorted

mo del For we then take a function that is constant in

axisymmetric disk so that it op erates on the p erturb ed

the main b o dy of the disk with a tap er to make it vanish

ow only This approximation is justied b ecause the disk

at the outer b oundary

viscous timescale is much longer than the characteristic

timescales of the pro cesses we are interested in here in

r

r

particular b ending waves propagate with a velo city com

q

q

r

parable to the sound sp eed so that the waves crossing time

is much shorter than the viscous timescale

which approximates r min r with

For simplicity we adopt a p olytropic equation of state

N

p

r R

n

r

P K

is the p olytropic where K is the p olytropic constant and n

The parameters which represents the width of the tap er

index The sound sp eed is then given by c dP d

s

interior to the outer b oundary N q and p are constrained

such that the square of the epicyclic frequency which is

EQUILIBRIUM STRUCTURE OF THE DISK

dened by

We adopt the same mo del as in Papaloizou

Terquem Namely we supp ose that the equilibrium

d r

disk is axisymmetric and in a state of dierential rotation

r dr

such that in cylindrical co ordinates the ow velo city is

is p ositive everywhere in the disk For the numerical cal given by v r For a barotropic equation of state

culations we take N n R q and the angular velo city is a function of r alone

p For a thin disk under the inuence of a p oint mass in

The p olytropic constant K is determined by sp ecifying tegration of the vertical hydrostatic equilibrium equation

a maximum value of the relative semithickness H r in gives Papaloizou Terquem

ACCRETION DISKS AND BENDING WAVES

the disk and the ducial surface mass density is xed formulation we implicitly assume that the turbulent vis

by sp ecifying the total mass M of the disk cosity is not aected by the tidallypro duced velo cities

Figure shows the surface mass density the rela which is not necessarily true However since the wave

tive semithickness H r the ratio of the sound sp eed to length of the tidal waves we consider is much larger than

H and the ratio of the Keplerian frequency c the disk semithickness this formulation may b e correct

s K

the epicyclic frequency to the Keplerian frequency Papaloizou Lin have shown that when the ra

K

versus the radius r for the equilibrium disk mo dels de dial wavelength of the p erturbations is larger than the

scrib ed ab ove with M M and H r disk semithickness a vertical averaging approximation

p max

The epicyclic frequency departs from at the outer edge in whichg is assumed to b e indep endent of z can b e used

K

0 0 0

b eing of the disk b ecause of the abrupt decrease of there Each ofv andv are then prop ortional to z withv

r z

indep endent of z They have found some supp ort for the

THE DISK RESPONSE

pro cedure in a numerical study of the propagation of dis

turbances in disk mo dels that take the vertical structure

Linear Perturbations

fully into account

In this pap er we consider small p erturbations so that

Linearization of the equation of motion followed by mul

the basic equations and can b e linearized As the

tiplication by z and vertical averaging then gives the ve

relevant linearization of the equations has already b een

lo city p erturbations in the form

discussed by Papaloizou Lin we provide only an

0

abbreviated discussion here We denote Eulerian p ertur

v

r

g

p

bations with a prime Because the p erturbing p otential

z

0

is prop ortional to exp i t the and t de

p

g d

p endence of the induced p erturbations is the same We

g r

p p

p

dr

denote with a tilde the part of the Eulerian p erturbations

i

p v

that dep ends only on r and z For instance the Eule

0

0

0

rian p erturb ed radial velo city is denoted v and we have

v

r

i v

r

0 0

g

v r z exp i t v r z t

p

r r

z z

p

the physical meaning of which We dene a quantity g

0

will b e given at the end of this section through

v g

z

r

p

0

P

0

ir z g

where is a vertical average of the standard Shakura

v

Sunyaev parameter that couples to the vertical

shear It is dened by

and we also have g r z t g r z exp i t

p

It was rst shown by Papaloizou Pringle that

Z Z

1 1

horizontal motions induced in a near Keplerian tilted disk

dz z dz

v v

are resonantly driven The main eect of a small viscos

1 1

ity is then to act on the vertical shear in the horizon

As mentioned ab ove the main eect of viscosity is to

0 0

tal p erturb ed velo cities v z and v z Demianski

r

damp the resonantly driven horizontal velo cities There

Ivanov and Ivanov Illarionov have recently

fore the viscous terms which do not act directly on the

given the relativistic generalization of this eect Follow

resonance have b een neglected This is consistent with

ing Papaloizou Lin we then retain only the hor

keeping only the comp onents of the viscous force given by

izontal comp onents of the viscous force p erturbation

equation Again this is accurate only when

v

The same pro cedure applied to the continuity equation

gives

0

0

v

v

r 0 0

F F

v v

r

h i

z z z z

g d

r J i I

p p

dr

p

where is the kinematic viscosity that couples to the ver

v

0

0

tical shear This approximation is accurate when

v

v

v

r

r i

The numerical calculations have actually shown that

r r z z

when b ecomes larger than the terms that are ne

v

in which glected in this approximation b ecome comparable to the

terms that are retained For this reason we shall limit our

Z Z Z

1 1 1

z z

0 numerical calculations to values of smaller than

v

dz dz J z dz I

c c Here we have assumed that the disk is turbulent and

1 1 1

that dissipation can b e mo deled as a turbulent viscosity

We note that equation dep ends on the viscosity only

The most likely mechanism for pro ducing such a turbulent

through the p erturb ed velo cities

viscosity is the magnetorotational BalbusHawley instabil

To close the system of equations we need a relation

ity Balbus Hawley We note however that this

0 0

b etween P and For simplicity we assume that the instability may not op erate in all the parts of protostel

equation of state is the same for the p erturb ed and un lar disks Balbus Hawley Also it is not clear

p erturb ed ows Linearization of the equation of state that the tidal waves would b e damp ed by the turbulent

then gives viscosity in the way describ ed by equations In this

TERQUEM

where R and r is the disk inner b oundary Here

in

0 0

the disk has b een assumed to precess as a rigid b o dy This P c

s

is exp ected if the disk can communicate with itself ei

We now give a physical interpretation of g Equa

ther through wave propagation or viscous diusion self

tion can b e used to express g in terms of the vertical

gravity b eing ignored on a timescale less than the preces

Lagrangian displacement

z

sion p erio d As long as is smaller than the disk asp ect

v

ratio H r which is almost constant throughout most of

z

the disk b ending waves may propagate Papaloizou

g i

p

r

Pringle Papaloizou Lin The ability of

where again we have dened such that r z t

these waves to propagate throughout the disk during a pre

z z

j j cession time implies approximately that H r

r z exp i t

p

z p

This is the condition for near rigid b o dy precession in an

When the physical relative vertical displacement

p

inviscid disk Papaloizou Terquem The numer

is then given by

ical simulations of Larwo o d et al show that this

condition also holds in a viscous disk in which do es not

v

z

Re mg cos t Reg sin t I

exceed H r This indicates that in the absence of self

p p

r

H r which is likely to b e satised gravity as long as

v

in protostellar disks pressure is the factor that controls

Thus when Reg and I mg are the relative ver

p

the eciency of communication throughout the disk and

tical displacement along the y and xaxis resp ectively We

then the precessional b ehavior of the disk A more com

note that a constant value ofg corresp onds to a rigid tilt

plete discussion of the precession of warp ed disks which

includes selfgravity is given by Papaloizou et al

Zer oFrequency Response

The Coriolis force can b e taken into account by replac

From equation we have to consider the disk resp onse

0

ing with

to a secular p otential p erturbation with zero forcing fre

quency When and equations

p v

and reduce to the single secondorder ordinary

0

i r z sin

p

dierential equation forg Papaloizou Lin

0

dg iI d

For the zerofrequency case will represent the to

tal term and the real amplitude f will represent

dr dr r

i

v

f sin from now on

p

Here we have assumed that

With this new denition of f g is then given by

d

Re r

0

Z Z

dr

r i r

v

i

v

00 00 00 0

J r f r dr dr g r

which is certainly a reasonable approximation for

v

r r

in in

It can b e shown that with the term I in its present form

the xcomp onent of the torque is non zero This torque

where the quantity in the outer integral has to b e evalu

0

leads to the precession of the disk as in a gyroscop e By

ated at the radius r We have supp osed that g at

writing the equation of motion for in the frame

p

r r but we note that an arbitrary constant may b e

in

dened in x we have assumed that the resp onse of the

added tog This corresp onds to an arbitrarily small rigid

disk app ears steady in a nonrotating frame Because of the

tilt that we assume eliminated by the choice of co ordinate

precessional motion induced by the secular p erturbation

system

the resp onse is actually steady in a frame rotating with

The ab ove expression shows that dep ending on whether

the precession frequency The Coriolis force corresp ond

is larger or smaller than H r the imag

v

ing to this additional motion must then b e added in the

inary part of g is larger or smaller than its real part re

equation of motion the centrifugal force can b e neglected

sp ectively According to equation this corresp onds to

since it is of second order in the precession frequency

a vertical displacement pro duced by the secular resp onse

The magnitude of the Coriolis force in the frame in which

b eing mainly along the x or y axis resp ectively

the resp onse app ears steady has to b e such that the x

We nally comment that the ab ove treatment is accu

comp onent of the total torque involving b oth the Coriolis

rate only when j j is smaller than the maximum of

p

and the gravitational forces is zero

H r and If this is not the case then the

v

This requirement leads to the expression of the preces

precession frequency has to b e taken into account in the

sion frequency Papaloizou Terquem see also

p

resonant denominator i where it would

v

Kuijken in the context of galactic disks

b e the dominant term

M

s p

cos

FiniteFrequency Response

M M

s p

Z Z We now consider the resp onse generated in the disk by

R R

those terms in the p erturbing p otential with nite forcing

dr dr

the secondorder dierential frequency When

r r

in in p

ACCRETION DISKS AND BENDING WAVES

0

equation for g obtained from equations radius Using the expression of v this means that

z

and takes on the general form at these lo cations

0

d g dg

g v

r

B C g S A

r z

dr dr

z

p

with

and then by continuity

r

0

p

v g

A

r

D

z

p

at the inner b oundary

dl n

p

p

We note that for the largest values of we exp ect the

v

D r B

D dr

waves to b e damp ed b efore reaching the center of the disk

such that the results are indep endent of the inner b ound

ary condition For since there is no dissipation at

d D r

v

D

the b oundaries we exp ect no torque to b e exerted on the

dr D

disk and no net angular momentum ux to ow through

its b oundaries

S i I

p

where we have set D and D

p

alidity of This Analysis Domain of V

i By intro ducing the function

p v

In this analysis we have neglected the variation ofg with

such that h

z As noted ab ove Papaloizou Lin have shown

that this approximation is valid when the scale of variation

dh dA

B

ofg with radius is larger than the disk semithickness H

h dr A dr

ie when

equation can b e recast in the form

dg

dg d

g dr H

hA hC g hS

dr dr

When this is not satised disp ersive eects which have

the solutions of which are given by the Greens function

b een neglected here should b e taken into account

metho d

Since we also consider linear waves our analysis is valid

as long as the p erturb ed velo cities are smaller than the

Z

r

0 0 0

sound sp eed When they b ecome sup ersonic sho cks o ccur

g r g r g r s r dr

W

that damp the waves Nelson Papaloizou How

r

in

ever it is not correct to compare the Eulerian p erturb ed

Z

R

velo cities given by the equations and with

0 0 0

g r g r s r dr

the sound sp eed to know whether the waves are linear or

r

not This is b ecause the system should b e invariant un

der the addition of a rigid tilt ie a constant to g An

Here s h f r W is the constant

p

increase of g corresp onding to the addition of a rigid tilt

could make the Eulerian p erturb ed velo cities larger than

dg dg

g W hA g

the sound sp eed without the waves getting nonlinear In

dr dr

principle as p ointed out by Papaloizou Lin to

and g and g are the solutions of the homogeneous dier

get p erturb ed quantities that do not dep end on the ad

ential equation that satisfy the outer and inner b oundary

dition of a rigid tilt ie that dep end only on the radial

conditions resp ectively In the numerical calculations pre

gradient ofg we should use Lagrangian rather than Eu

sented b elow g and g are calculated directly from equa

lerian variables However since the Eulerian p erturb ed

tions and using the fourthorder Runge

radial velo city in spherical p olar co ordinates dep ends only

Kutta pro cedure with adaptative stepsize control given by

on the gradient of g in the limit of small p erturbing fre

Press et al Computation of the function h then

quency we can compare this quantity to the sound sp eed

allowsg to b e calculated from

to decide whether the motion is subsonic The criterion

that has to b e satised is then

Boundary Conditions

Equation has a regular singularity at the outer edge

dg

0 0

z r

r v z v

r R see Papaloizou Terquem Our outer

r z

dr

c

s

b oundary condition is thus that the solution b e regular

i

r z

p v

there

We take the inner b oundary to b e p erfectly reec

where we have assumed that and z r We

p

tive The radial velo city in spherical p olar co ordinates

have kept in the resonant denominator b ecause even

p

0 0

r v z v r z thus vanishes at the lo cations

though this term is small compared with unity it may still

r z

b e larger than r z such that r z r where r is the disk inner

in in

TERQUEM

net torque to b e exerted by the p erturb er This torque The WKB disp ersion relation for b ending waves gives

where k R is the wavenumb er Pa is transferred to the disk through dissipation of the waves c k

p s

paloizou Lin For nite p erturbing frequency we at the b oundary Because of the conservation of angular

can then rewrite the ab ove criterion in the form momentum the net torque is equal to the dierence of an

gular momentum ux through the disk b oundaries This

H

ux is constant indep endent of r inside the disk since

g max

v

there is not dissipation there

r

The situation is dierent when a corotation resonance

where g is the variation of the tilt angle across the disk

is present in the disk since this singularity provides a

and we have used the fact that H r H r

lo cation where angular momentum can b e absorb ed or

By H r we mean the asp ect ratio anywhere in the disk

emitted Goldreich Tremaine Goldreich Nichol

since this quantity do es not vary signicantly throughout

son

the disk For the condition we get is

p

When the disk is viscous its resp onse is not in phase

with the p erturb er A net torque is then exerted on the

H

g max

v

disk even if the b oundaries are reective and the angular

r

momentum ux inside the disk is not constant since the

As p ointed out in x dep ending on whether is larger

v

p erturb ed velo cities are viscously dissipated This situa

or smaller than the imaginary part ofg is larger

tion has b een investigated by Papaloizou Pringle

or smaller than its real part resp ectively Therefore the

and Papaloizou Lin in the context of coplanar or

ab ove criterion and equation tell us that the variation

bit Here we present the analysis corresp onding to a non

across the disk of the relative vertical displacement pro

coplanar orbit

duced by the secular p erturbation is limited by along

v We note that whenever the p erturb er rotates outside the

the xaxis and by H r along the y axis

disk the torque exerted on the disk is negative Through

We note that in an inviscid disk the zerofrequency p er

dissipation of the waves the disk then loses angular mo

turbation pro duces g j j R Papaloizou Ko

p mentum We rst derive an expression of the torque in

rycansky Terquem Since as we have p ointed

terms ofg and then relate it to using angular momen

v

out in x our analysis is valid only when j j R is

p tum conservation

smaller than the maximum of H r and the ab ove

v

criterion is always satised in an inviscid disk

Expression for the Torque

The most likely situation in a protostellar disk corre

The net torque exerted by the p erturb er on the disk is

sp onds to H r In that case when the p erturb ed

v

given by

velo cities are close to the sound sp eed the secular p ertur

Z

h i

bation pro duces a steady in the precessing frame tilt the

0 i t

p

Re e r T

variation of which across the disk is on the order of H r

V

or whichever term is the largest Sup erp osed on this

v

i h

steady tilt b ending waves pro duced by the nite frequency

0 i t

p

e Re r dV

terms propagate corresp onding to a tilt the variation of

which across the disk is on the order of H r This is ex

where the integration is over the volume V of the unp er

actly what was observed in the SPH simulations p erformed

turb ed disk Because of the p erio dicity the rstorder

by Larwo o d Papaloizou in which was larger

v

term is zero and the z comp onent of the torque is then

than H r We note that since in the case of protostellar

disks H r is likely to b e close to the variation of the

Z

vertical displacement across the disk due to the nonsecular

h i

0 i t 0 i t

p p

tidal p erturbations can b e as large as ab out a tenth of the

e T dV e Re Re

z

V

disk radius while the p erturbation remains linear

ANGULAR MOMENTUM EXCHANGE which is equivalent to

Z

Tidal p erturbation of a disk may lead to an exchange of

0 0

dV angular momentum b etween the disk and the p erturb er T I m

z

V

If the disk is inviscid and do es not contain any corotation

0

resonance the nature of the b oundaries would determine

Using equations and in the ab ove equation

whether such an exchange takes place or not see Lin

can b e replaced in terms ofg The expression of T then

z

Papaloizou for example Because of the conserva

involves J which in a Keplerian disk is equal to

tion of wave action in an inviscid disk waves excited at

We then get

the outer b oundary propagate through the disk with an

Z

R

increasing amplitude if the disk surface density increases

T I m g f r dr

z

inward or is uniform see Lightill for example It

r

in

is usually assumed that they b ecome non linear b efore

where f is the real amplitude dened by which has to reaching the center and are dissipated through interaction

b e mo died when in order to take into account the with the background ow Thus the inner b oundary can

p

disk precession see x In the numerical calculations b e taken to b e dissipative Papaloizou Terquem

presented b elow the torque T will b e computed from this Dissipation at the b oundary then intro duces a phase lag

z

expression b etween the p erturb er and the disk resp onse enabling a

ACCRETION DISKS AND BENDING WAVES

is the total angular momentum of the disk the If J

Z Z

D

  tidal eects we have considered would remove the angular

P dV B O dV

momentum content of the disk on a timescale

V V

p

where O is a selfadjoint linear op erator and B is a b ound

M GM R

J

p

D

ary term

t

d

T T

z z

Z

We shall compare this timescale with the viscous timescale

 

t r H R where we will take H r

h h

dS r P r n B

n

H r for our disk mo dels H r do es not vary by more

S

max

than in the main b o dy of the disk as indicated in

The integration is over the surface S of the disk and n

Fig Here is the viscous parameter that couples to

h

is the unit vector p erp endicular to this surface oriented

the horizontal shear

toward the disk exterior Since P and vanish at r R

We also dene the torque integral T r which is the

z

H and we consider mo dels for which r and z

in

torque exerted b etween the inner b oundary and the ra

we have B In addition since O is selfadjoint the

dius r

integral on the right hand side of equation is real

Z

r

The term involving P in equation is then zero

T r I m g f r dr

z

Using the fact that r z z r it can also

r

in

b e shown that the term involving r is zero

Since equation has no and tdep endence we can

Conservation Angular Momentum

now use the tilded variables again We dene such that

To get the angular momentum conservation equation



r z t r z exp i t We dene simi

p

which is of second order in the p erturb ed quantities we

0

larly The term involving in equation can b e writ

rst take the Lagrangian p erturbation of the equation of

ten as follows

motion and multiply by the unp erturb ed mass density

Z

To rst order this leads to LyndenBell Ostriker

0



r dV I m

V

t t t

Z Z

0 0 0



r I m dV dV I m

0 0

V V

rP F r r r

where an integration by parts has b een p erformed and the

linearized continuity equation

Here is the Lagrangian displacement vector denotes

the Lagrangian change op erator and we have used the re

0

r

lation b etween Lagrangian and Eulerian p erturbations of

to rst order in a quantity Q

see for example Tassoul has b een used The rst

term on the righthand side of equation is zero b e

0

Q Q r Q

cause vanishes at r R and z H and r

in

Equation then b ecomes

k with k b eing the unit vector in the The vector

z direction To derive equation we have used the

Z Z

fact that at equilibrium F

0 0 0

 

dV I m F dV I m We now take the scalar pro duct of equation with

V V

which is the complex conjugate of Since t i

p

and i all the terms on the left hand side of the

From equation we see that this is equivalent to

resulting equation are real We then take the imaginary

Z Z

part of this equation and integrate it over the volume V

R H

0



of the unp erturb ed disk This leads to

T I m F r dr dz

z

r H

in

Z

0

 

F P I m

We now use the relation b etween the Eulerian velo city

V

p erturbation and the Lagrangian displacement

0

 

r r r dV g

0

r v v v r

where we have dened the linear op erator P such that

t

P rP We note that if the terms on the

0

together with the expression of F the relation

left hand side of equation had non zero contribution

and the fact that H to transform equation

to equation it would b e related to the total time

into

derivative of the p erturb ed sp ecic angular momentum of

the disk

Using the p erturb ed equation of state in the form

P P n it can b e shown LyndenBell

Ostriker that

TERQUEM

NUMERICAL RESULTS

We compute the torque exerted on protostellar disks by Z

0 0

R

jv z j v z

r

a p erturb er with an orbit that is inclined to the plane of

dr T r

z v

p

the disk For a coplanar system one exp ects the circum

r

in

primary disk to b e truncated by tidal eects in such a

Z

0 0

R

way that its radius is not greater than ab out onethird of

z z v v

r

I m r

v

the separation of the system Papaloizou Pringle

r

p in

Larwo o d et al have shown that tidal truncation is

only marginally aected by the lack of coplanarity Thus

In the numerical calculations presented b elow the

R Since we consider binary in our mo dels we adopt D

torque will b e computed from equation and compared

mass ratio of unity M M the p erturbing p otential

s p

with the result of the ab ove equation

is small compared with the p otential of the central star

In a Keplerian disk Then when and

p

justifying the assumption of small p erturbation

equation is well approximated by

v

For computational convenience we normalize the units

0

0

such that M R and R Apart for

v

p K

v

r

i

the calculation of the epicyclic frequency the rotation law

z z

in the disk is taken to b e Keplerian the departure from

so that T can b e approximated by

z

the Keplerian law which o ccurs mainly at the disk outer

edge do es not exceed The inner b oundary of the

Z

R

0

v

and the outer b oundary is taken to b e disk is r

r

in

T dr r

z v

rather than in order to avoid a zero surface r

z

out

r

in

mass density at the outer edge

As exp ected the torque is negative Here we consider

We present here the results of some disk resp onse cal

linear p erturbations such that the p erturb ed velo cities re

culations for an inclination a p olytropic index

main smaller than the sound sp eed We have p ointed out

n and a disk mass M Results for an arbi

in x that since the system should b e invariant under

trary inclination prograde orbit or

the addition of a rigid tilt tog it is not in principle correct

retrograde orbit can b e obtained straightforwardly since

to compare the Eulerian p erturb ed velo cities to the sound

T and then t corresp onding to and is

z d p

sp eed to know whether the waves are linear or not How

prop ortional to cos sin and sin resp ectively

0

ever since the dominant term in the expression ofv

r

As shown by equation the precession frequency scales

is the term prop ortional to the gradient ofg we can com

with cos Since g is indep endent of M see equations

pare this velo city to the sound sp eed to decide whether

and expression shows that T M

z

the motion is subsonic Therefore we can get an estimate

The quantities t and t are indep endent of M

p d h

0

of the maximum value of the torque by setting jv j c

s

r

H and dropping the integration This leads to a

minimum value of t which is equal to t Only

d h h v

Uniform

v

if were larger than could t b e smaller than t

v h d h

In table we summarize the results obtained for dif

An other useful quantity for interpreting our results is

ferent values of the parameters which are the separa

the ux of vertical angular momentum in the radial direc

tion of the system D the viscous parameter and the

v tion at radius r

maximum value of the relative semithickness of the disk

H r The quantities we compute are the torque T

max z

Z Z

H

asso ciated with each of the p erturbing frequencies pro

0 0

ddz Re v v r Re J r

r

grade term retrograde term and the disk pre

H z

cession frequency and the ratio of the tidal timescale t

p d

to the viscous timescale t We have taken here

h h v

which can also b e written

The two values of the torque have b een computed using



0

0

equations and the result corresp onding to equa

v

v

r

J r r Re

tion b eing in parentheses The timescale t is com

d

z z

puted from equation with T b eing the sum of the

z

The torque T r exerted b etween the radii r and r re contributions from each frequency

z in

sults in a change of angular momentum of the disk Part We note that when either one of the criterions

of this angular momentum is advected through the b ound or is not satised cases indicated by a fo otnote it

aries this is J r J r J r since r and is always close to the disk outer edge This is b ecause the

in in

part is dissipated b etween these radii Thus the quantity density varies more rapidly there than in the other parts

T r J r is the amount of angular momentum dep osited of the disk Also the scale of variation ofg never b ecomes

z

in the disk b etween the inner b oundary and the radius r smaller than H by less than a factor of a few Similarly

As we shall see this quantity is negative which means when the radial velo city in spherical p olar co ordinates b e

that the disk loses angular momentum comes larger than the sound sp eed it is only by a factor of

In the numerical calculations presented b elow the a few We note that the results we present in this case can

torque integral will b e computed from equation and still b e used since the p erturb ed quantities can b e reduced

J r will b e computed directly from the velo cities using by cho osing appropriate scaling parameters like the initial

equations and inclination or the mass of the disk M

ACCRETION DISKS AND BENDING WAVES

In all cases the condition j j R max H r a semilog representation for mo dels plus

p v v

is satised see x The condition for near rigid b o dy other values of D and mo dels plus other val

v

j j R is also always satised precession H R ues of D and for and Before commenting on

p p

We note that there is a go o d agreement b etween the the resonances we note that the torque asso ciated with the

values of the torque given by equations and In nite frequency terms out of resonance do es not decrease

addition in the cases we computed we found this latter when the separation increases up to D This con

equation to b e very well approximated by rms the results of Papaloizou Terquem There

To simplify the discussion we note T T and T the are comp eting eects acting when the separation of the

z z z

torque asso ciated resp ectively with and system is increased On the one hand the wavelength of

p

Furthermore we separate the dep endence by writing the tidal waves b ecomes larger since it is inversely pro



T T cos sin and T T sin with T p ortional to its frequency which tends to increase the

 

z z

and T b eing indep endent of torque On the other hand the amplitude of the p erturb

Since part of this section is devoted to the tiltg we re ing p otential decreases which tends to reduce the torque

It call that g is related to the relative vertical displacement It seems that the rst eect is dominant for D

seems like the torque b egins to decrease with the separa through equation When Reg and I mg

p

tion when the wavelength b ecomes comparable to the disk are the relative vertical displacement along the y and x

radius which happ ens for D This is reasonable since axis resp ectively

the wavelength cannot increase further

FiniteFrequency Response

In Figure we can see resonances which o ccur when

Figure shows the real and imaginary parts ofg versus

the frequency of the tidal waves is equal to that of some

r for mo dels and gures for are

free normal global b ending mo de of the disk and cause

p p

similar We see that the wavelike character of the disk

the torque to b ecome very large even innite if there is

resp onse disapp ears when b ecomes larger than H r

no dissipation Papaloizou Lin In a resonance

v

which is in these mo dels This is in agreement with

the torque indeed increases when decreases in contrast

v

Papaloizou Pringle and Papaloizou Lin

to what is observed out of resonance It seems that there

and also with the relativistic generalization of Demian

is no longer any resonance for D This would not

ski Ivanov and Ivanov Illarionov who

b e surprising since the wavelength of the resp onse is then

have shown that in a near Keplerian disk selfgravitating

comparable to the disk radius A more detailed descrip

tion of the resonance which o ccurs at D is given in or not the longest wavelength disturbances undergo a

transition b etween wavelike and diusive b ehavior when

the App endix

H r We note however that when H r there

Since the resonances dep end on the sp ectrum of the

v v

are still some oscillations in the half outer part of the disk

free normal b ending mo des of the disk they are mo del

This is b ecause b ending waves have a long wavelength

dep endent In particular they o ccur only if the the waves

such that they can still p enetrate relatively far b efore b e

can b e at least partially reected at the inner b oundary

ing diused out

otherwise there is no free normal global b ending mo de in

The magnitude of Reg do es not vary with since it

the disk see also Hunter To omre for the case of a

v

is controlled mainly by the radial pressure force On the

purely selfgravitating disk

contrary the magnitude of I mg is much more sensitive

The values of the torque displayed in Figure corre

to the viscosity although it do es not vary monotonically

sp ond to However for this particular value of

with

the p erturb ed velo cities may b ecome larger than the

v

By comparing mo dels a and c and also mo dels c and

sound sp eed close to a resonance These values of the

c which dier only by the value of H r we have

torque should then b e scaled to b e used for other values

max

checked that the wavelength of the resp onse is prop ortional

of or for other disk parameters For the

v

to H r This is exp ected since b ending waves propagate

resonances are very weak indicating that the waves are

with a velo city which b eing half the sound sp eed is pro

almost completely dissipated b efore reaching the disk in

p ortional to H r and the wavelength is prop ortional to

ner b oundary

the wave velo city

We have compared our results with those obtained by

In all the cases we have computed except mo del f

Papaloizou Terquem for an inviscid disk in which

T T but T T in agreement with Papaloizou

the inner b oundary is dissipative Rememb ering that they

z z

Terquem In general the shorter the wavelength

had we found that in general we could repro

of the resp onse the smaller the torque and the wave cor

duce their results with This conrms that

v

resp onding to the retrograde term has a slightly longer

when the waves are dissipated b efore reaching

v

wavelength than that corresp onding to the prograde term

the disk inner b oundary



The dierence b etween T and T and T is still sig

Out of resonance the torque T asso ciated with the

z

nicant for the largest values of the separation computed

nite frequency terms is prop ortional to as long as this

v

here We exp ect all these quantities to converge toward

parameter is smaller than some critical value Then for



the same value when approaches which probably

the larger values of T is indep endent of the viscosity

v

z

This is in agreement with the theoretical exp ectation Gol means here R since the torque is controlled

v

by viscosity This is not satised even for mo del i which

dreich Tremaine and the work of Papaloizou

has R However we observe that the dier

Terquem We indeed exp ect the torque to b e inde

ence b etween these torques is reduced from a factor to a

p endent of whatever dissipation acts in the disk providing

factor less than when mo del i is run with

the waves are dissipated b efore reaching the disk inner

v

In Figure the net torque T is plotted versus D in z

TERQUEM

b oundary If the waves are reected on the disk inner Angular Momentum Dissipation as a Function of

b oundary the situation is more complicated b ecause the Radius

angular momentum carried by the waves has a dierent

Figures and show T r J r normalized to unity

sign dep ending on whether the waves are ingoing or out

z

versus r for various mo dels This represents the angu

going Therefore part of the angular momentum lost by

lar momentum dep osited in the disk b etween the inner

the disk while the waves propagate inwards is given back

b oundary and the radius r The fact that this quantity is

by the waves propagating outwards The net amount of

negative means that the disk loses angular momentum Of

angular momentum lost by the disk then dep ends on how

course jT r J r j should increase with r However we

eciently the waves are reected Using this argument

z

observe in Figure that for D and this func

we then nd that when H r and the waves

tion has a little hump close to the disk outer edge We

are damp ed b efore reaching the disk inner b oundary when

interpret it as b eing a numerical eect since its amplitude

is larger than and resp ectively

v

is found to dep end on the accuracy of the calculations

for H r this is a bit larger than the value found

We rst observe that b ending waves are able to trans

ab ove Such a dep endence of this critical value of with

v

p ort a signicant fraction of the negative angular momen

H r is exp ected Indeed the shorter the wavelength the

tum they carry deep into the inner parts of the disk For

more easily are the waves dissipated Since the wavelength

the parameters corresp onding to mo del b Figure shows

is prop ortional to H r see ab ove we then exp ect that

that the retrograde and prograde waves transp ort resp ec

for a given dissipation is more ecient for small values

v



tively and of the total negative angular momen

of H r For the same reason the net torque T increases

z

tum they dep osit into the disk at radii smaller than and

with H r see mo dels c and a and mo dels c and c

resp ectively In contrast only of the total negative

The coupling b etween the resp onse and the p erturbing p o

angular momentum carried by the zerofrequency p ertur

tential is indeed more ecient when the resp onse has a

bation is dep osited at radii smaller than We note

long wavelength and the integral in equation from

however that out of resonance the net amount of angular

which the torque is calculated is then larger

momentum dep osited at small radii by the p er

p

turbation is not necessarily smaller than that dep osited

ZeroFrequency Response

by the nite frequency p erturbations Also the fraction

Figure shows the real and imaginary parts ofg versus

of angular momentum dep osited close to the disk outer

r for mo dels and As exp ected we observe that

p

edge by the nite frequency resp onses esp ecially the ret

the zerofrequency resp onse is not wavelike

rograde wave is larger than that dep osited by the secular

From equation we see that as long as

v

resp onse

H r which is the case for Reg

v

It app ears that the secular resp onse gets dissipated very

is almost indep endent of the viscosity When

v

v

progressively throughout the disk whereas the prograde

dominates over and since it app ears in the ex

and retrograde waves are dissipated preferentially at some

pression of Reg with the minus sign Reg b ecomes neg

lo cations in the disk The p osition of these lo cations seems

ative The particular b ehavior of Reg in the outer parts

to dep end on the wavelength of these waves

of the disk is pro duced by the fact that the surface mass

The resonances do not aect the distribution of dissi

density drops to zero there thus increasing As

pation of angular momentum in the disk although they

exp ected from equation I mg is prop ortional to

v

obviously increase the absolute value of the actual amount

Papaloizou Korycansky Terquem computed

of angular momentum dep osited This is due to the fact

the real part ofg for an inviscid disk with an equilibrium

that the magnitude of the resp onse not its wavelength is

state similar to that we have set up here and for

p

aected by a resonance

and D We have checked that the results we get for

In Figure we have plotted T r J r versus r for

z

the smallest values of are in complete agreement with

v

and dierent values of We see clearly on

p v

theirs

these plots the transition that o ccurs when H r

v

Equation shows that T dep ends only on I mg not

z

When b ecomes larger than H r the wavelike resp onse

v

on Reg Since for the secular resp onse I mg T

v

z

of the disk disapp ears Thus the curves corresp onding to

is also prop ortional to This is b orne out by the numer

v

have a shap e similar to those corresp onding to

p

ical results Also equation indicates that g H

the secular resp onse

Thus if we vary H r while keeping constant we get

In general when is increased the fraction of nega

v

from equation that T H This is conrmed

z

tive angular momentum dep osited at small radii decreases

by mo dels c and a and mo dels c and c which dier

This is exp ected since the waves excited predominantly in

only by the value of H r This is in contrast to the

max

the outer parts of the disk get dissipated more easily at

nite frequency resp onse for which the torque increases

large radii when is large

v

with H r see ab ove

We observe in Figure a that there is hardly any dier



out of resonance Since T Figure indicates that T

z z

ence b etween the cases and This

v v

we also have T T T is always going to b e much larger



z

means that whatever the amount of angular momentum





than T for not to o close to typically for

z

transp orted by the wave the same fraction is dep osited b e

 

or tween r and r in b oth cases This probably indicates that

in

In contrast to the nite frequency resp onse the zero the wave is able to reach the disk inner b oundary for these

frequency resp onse is not wavelike and we then exp ect values of For larger for which the wave is completely

v v

T to decrease continuously with D This is indeed what dissipated b efore reaching the inner b oundary no angular

z

we observe in Figure momentum is dep osited at small radii so that the curves

ACCRETION DISKS AND BENDING WAVES

cation where the torque is exerted and part is dissipated lo ok dierent This indicates that for H r

max

lo cally through viscous interaction of the waves with the the waves do not reach the inner b oundary when is

v

background ow larger than For H r Figure b

max

We have shown that the tidal torque is comparable to the critical value of ab ove which the waves do not reach

v

the horizontal viscous stress acting on the background ow the inner b oundary is b etween and These

when the p erturb ed velo cities in the disk are on the order values are in agreement with those found ab ove Our cal

of the sound sp eed c If these velo cities remain subsonic culations also show that a larger fraction of the negative

s

the tidal torque can exceed the horizontal viscous stress angular momentum is dep osited at small radii when H r

only if the parameter which couples to the vertical is decreased However we have checked that this eect

v

shear is larger than the parameter which is coupled is less signicant when the distance interior to the outer

h

to the horizontal shear We note that so far there is no b oundary o ccupied by the tap er of the surface mass den

indication ab out whether these two parameters should b e sity is reduced along with the semithickness

the same or not Nelson Papaloizou have found

that when the p erturb ed velo cities exceed c sho cks re

Nonuniform

s

v

duce the amplitude of the p erturbation such that the disk

We have run mo del c with a nonuniform viscosity

moves back to a state where these velo cities are smaller

and x These mo dels cor x

v v

than c When sho cks o ccur the tidal torque exerted on

s

resp ond to a disk where resp ectively the inner and outer

the disk may b ecome larger than the horizontal viscous

parts are nonviscous

stress

Figure a shows the imaginary part of g versus r for

We have found that in protostellar disks b ending waves

corresp onding to these viscosity laws The uni

p

are able to propagate and transp ort a signicant fraction of

form case is also plotted for comparison The real part

v

the negative angular momentum they carry in the disk in

ofg is not shown since it do es not dep end on for these

v

ner parts This is due to their relatively large wavelength

small values of the viscosity see discussion ab ove We

Therefore tidal interactions in noncoplanar systems may

see that when the outer parts of the disk are nonviscous

not b e conned to the regions close to the disk outer edge

I mg in the inner parts is similar to the uniform case

v

where the waves are excited For the disk mo dels we have

while it has an almost constant value in the outer parts

set up which extends down to the stellar surface the value

When the inner parts of the disk are nonviscous I mg

of ab ove which the waves get dissipated b efore reaching

v

is almost zero there while it is signicant in the outer

the disk inner b oundary varies b etween and

parts This b ehavior can b e understo o d by rememb ering

for a disk asp ect ratio b etween and this critical

that I mg r is the integral from r to r of a function

in

value of increases with the disk semithickness We

v

prop ortional to see equation Since I mg varies

v

note that if we had set up an annulus rather than a disk

0

less through the disk than in the uniform casev given

v

r

the surface mass density would drop at the inner b ound

by equation is smaller This together with a smaller

ary Then for small the waves would have a tendency

v

leads to a smaller net torque given by equation

v

to b ecome nonlinear and then to dissipate b efore reaching

When there is not such a dierence b etween the

p

the inner b oundary Thus the b ehavior of b ending waves

uniform and nonuniform cases The torque is smaller

v

could b e similar in an annulus with small and in a disk

v

in the nonuniform case but I mg do es not change signif

with larger In the limit of small viscosity dissipation

v

icantly from one case to the other The reason is probably

could also o ccur as a result of parametric instabilities Pa

that I mg do es not vary monotonically with as shown

v

paloizou Terquem CF Gammie JJ Go o dman

in Figure

GI Ogilvie private communication the eect of

Figure b shows T r J r versus r for the same mo d

z

which may b e to lead to a larger eective However

v

els and We observe that as exp ected no angular

p

since the waves we consider here propagate in a nonuni

momentum is dep osited at radii where is very small

v

form medium it is not clear whether they would b e e

small radii for x and large radii for x

v v

ciently damp ed by these instabilities

It the disk mo del allows at least partial reection from

DISCUSSION

the center the tidal interaction b ecomes resonant when

In this pap er we have calculated the resp onse of a vis

the frequency of the tidal waves is equal to that of some

cous gaseous disk to tidal p erturbations with azimuthal

free normal global b ending mo de of the disk For the equi

mo de numb er m and o dd symmetry in z with resp ect

librium disk mo dels we have considered we have found

to the equatorial plane for b oth zero and nite p erturbing

that a particularly strong resonance o ccurs when the sep

frequencies We concentrated on these typ es of p erturba

aration is ab out times the disk radius The torque

tions b ecause they arise for inclined disks and they lead to

asso ciated with the nite frequency terms then increases

a longwavelength resp onse The eects of p erturbations

by many orders of magnitude However the resp onse may

with even symmetry which o ccur in the coplanar case

b e limited by sho cks and nonlinear eects The strength of

may b e sup erp osed on the eects of those with the o dd

the resonance is inversely prop ortional to In our calcu

v

symmetry

lations the range of separation for which the torque is sig

Since the resp onse of a viscous disk is not in phase with nicantly increased is rather large which means that the

the p erturbing p otential a tidal torque is exerted on the

eects of tidal interaction like disk truncation may o ccur

disk When the p erturb er rotates outside the disk this

even when the separation of the binary is large In addition

torque results in a decrease of the disk angular momen

to this strong resonance we have found a few other reso

tum Part of the negative angular momentum of the

nances with dierent strengths and widths From these

p erturbation is carried by tidal waves away from the lo

TERQUEM

calculations it app ears that the probability for the inter In the case of protostellar disks comparison b etween

action to b e resonant may b e signicant However these observations and theory is now b ecoming p ossible A non

results dep end on the particular equilibrium disk mo dels coplanar binary system HK Tau has b een observed for

we have set up the rst time very recently see x and subsequent

We note that the disk is exp ected to b e truncated such work will b e devoted to interpreting these observations

that the inner Lindblad resonance with the compan In the meantime we can comment briey on some obser

ion which could provide eective wave excitation is more vations related to Xray binaries

or less excluded from the disk There is evidence from the light curve of Xray binaries

Out of resonance we nd that the torque asso ciated such as Hercules X and SS that their asso ciated ac

with the zerofrequency p erturbing term is much larger cretion disks may b e in a state of precession in the tidal

than that asso ciated with the nite frequency terms Of eld of the binary companion Katz a b has in

course if the separation of the system is large enough dicated that the observed precession p erio ds of these two

at least times the disk radius in our mo dels the dif systems are consistent with the precession b eing induced

ference b etween nite frequency and secular terms disap by the tidal eld of the secondary

p ears When the secular resp onse is dominant the tidal In the case of SS it is interesting to note that an

dding motion with a p erio d half the or torque is prop ortional to in the limit This has additional no

v v

bital p erio d is observed see Margon and references the consequence that the ratio of the tidal timescale t

d

therein Katz et al suggested that this no dding time that would b e required for the tidal eects we have

motion is pro duced by the gravitational torque exerted by considered to remove the angular momentum content of

the companion In their mo del the no dding is viscously the disk to the disk viscous timescale t is prop ortional

h

transmitted from the outside where the torque is signi to

h v

cant to the interior where the jets are seen to resp ond to What observable eects would these tidal interactions

First as we mentioned the motion This implies an extremely large viscosity in pro duce in protostellar disks

in the intro duction they would lead to the precession the disk

of the jets that originate from these disks The preces Here we p oint out that the observed p erio d of this mo

sion timescale that we can infer from the observation of tion is consistent with the no dding b eing pro duced by

jets that are mo deled as precessing is consistent with this b ending waves As these waves propagate with the sound

motion b eing induced by tidal eects in binary systems sp eed and the disk is observed to b e very thick Mar

Terquem et al gon they could transmit the motion through the

Also as we have already p ointed out in x the sec disk on a timescale comparable to its dynamical timescale

ular p erturbation pro duces a tilt the variation of which For the waves to reach the disk interior would have to

v

can b e up to along the line of no des and H r along b e smaller than H r It is not clear that this condition

v

the direction p erp endicular to the line of no des Sup er is satised here The disk viscosity would then probably

p osed on this steady in the precessing frame tilt there have to b e smaller than that predicted by the mo del of

is a tilt pro duced by the nite frequency p erturbations Katz et al

the variation of which across the disk can b e up to H r We nally observe that a rapid communication time

Such asymmetries in the outer parts of protostellar disks through the disk also makes problems with the radiation

could b e observed Because the variations along the line of driven warping mo del prop osed by Maloney Begel

no des and along the p erp endicular to the line of no des de man to explain the precession of the disk in SS

p end on and H r resp ectively observations of warp ed In this mo del the communication b etween the dierent

v

protostellar disks have the p otential to give imp ortant in parts of the disk which is assumed to o ccur b ecause of

formation ab out the physics of these disks viscosity alone is on a timescale that is characteristic for

Protostellar disks are b elieved to b e rather thick ie mass to ow through the disk It is therefore very slow

H r Our calculations show that for such an asp ect since the viscous timescale must b e long for the warping

ratio the viscosity ab ove which b ending waves are damp ed instability to o ccur Pringle

b efore reaching the central star is Observa

v

tions seem to indicate that in protostellar disks is on

h

It is a pleasure to thank John Papaloizou for his ad

the order of If is smaller than or compa

v

vice encouragement and many valuable suggestions and

rable to resonances may o ccur as describ ed ab ove

discussions I acknowledge Steve Balbus Doug Lin Jim

providing the disk inner edge allows some reection of the

Pringle and Michel Tagger for very useful comments on an

waves In that case we could observe truncated disks with

earlier draft of this pap er and John Larwo o d for p oint

sharp edges even when the binary separation is large If

ing out the no dding motion in SS I also thank

is larger than but close to b ending waves are still

v

an anonymous referee whose comments help ed to improve

able to propagate throughout most of the disk In addi

the quality of this pap er This work was supp orted by the

tion given that protostellar disks are rather thick these

Re Center for Star Formation Studies at NASAAmes

waves propagate fast they cross the disk on a time com

search Center and the University of California at Berkeley

parable to the sound crossing time In that case it may

and Santa Cruz I am grateful to the Isaac Newton Insti

b e p ossible to observe some timedep endent phenomena

tute at Cambridge University for supp ort during the nal

with a frequency equal to twice the orbital frequency

stages of this work

APPENDIX

We describ e here in more detail the resonance that o ccurs at D since it is particularly strong For and p

ACCRETION DISKS AND BENDING WAVES

resonance o ccurs at D and D resp ectively We observe that for the values of considered here

v

the range of D for which the torque is signicantly increased is rather large We see that the strength of the resonance is

prop ortional to This is exp ected since the strength of the resonance is inversely prop ortional to the damping factor

v

which itself is prop ortional to when dissipation is due to viscosity We also note that the integral of the torque over the

v

range of frequencies in this resonance is indep endent of This can b e understo o d in the following way For

v p

the same argument would apply for the torque at or close to resonance can b e written

p

A

v

T

z

v

where R is the resonant frequency and A is an amplitude which do es not dep end on We have calculated

v

that A For the values of considered we have checked that the integral of the torque from to

v

do es not dep end very much on whether is taken to b e the range of frequencies in the resonance or for which

expression is valid the latter b eing smaller We then approximate the integral I of the torque over the resonance by

using so that

I A Ar ctan

v

For or the range of frequencies for which is valid is large compared with so that I A

v v

indep endent of For although is only a few times this relation is still valid within a factor less

v v v

than due to the rapid convergence of the function Ar ctan

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J AJ Lightill J W aves in Fluids Cambridge Cambridge Univ

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Terquem C Bertout C AA

Terquem C Bertout C MNRAS LyndenBell D Ostriker J P MNRAS

Terquem C Eisloel J Papaloizou J C B Nelson R P LyndenBell D Pringle J E MNRAS

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To omre A in IAU Symp Internal Kinematics and Dy Margon B ARAA

namics of Galaxies ed E Athanassoula Dordrecht Reidel Masset F Tagger M AA

TERQUEM

Table Torque and Corresp onding Tidal Evolution Timescale

H

Lab el D T t t

v z p

d h

r

max

b a

a   

b a

b   

a c

  c 

c a

d   

a   

b    

c   

d    

e   

a

a   

a

b   

a

c   

a

d   

a

e   

a

f   

a

g   

a

h   

a

i   

a   

b   

c   

d   

e   

f   

g   

h   

i   

NoteListed are the separation of the system D the viscous parameter the maximum value of H r the torque T asso ciated with each

v z

of the p erturb ed frequencies and the disk precession frequency and the ratio of the tidal timescale t to the viscous timescale

p

d

t

h

a

Cases whereat the outer edge of the disk g varies on a scale smaller than H

b

Cases whereat the outer edge of the disk the radial spherical velo city is larger than the sound sp eed

c

Cases whereat the outer edge of the disk g varies on a scale smaller than H and the radial spherical velo city is larger than the sound sp eed

ACCRETION DISKS AND BENDING WAVES

.1

.003

.08

.002 H(r)/r .06

.001

.04

0 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1

.05 1

.04 .8

.03

.6 .02

.01 .4

0

0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1

r r

Fig Surface mass density top left relative semithickness top right ratio of the sound sp eed to the Keplerian

frequency bottom left and ratio of the epicyclic frequency to the Keplerian frequency bottom right vs r for the

equilibrium disk mo dels The parameters are n M M and H r

p max

TERQUEM

.002

a) Real part of g

0

-.002 Re(g) -.004

-.006

-.008 0 .2 .4 .6 .8 1 r

b) Imaginary part of g

.0005 Im(g)

0

-.0005

0 .2 .4 .6 .8 1

r

Fig Real panel a and imaginary panel b parts of g vs r for mo dels and The dierent curves

p

corresp ond to solid line dotted line short dashed line long dashed line and

v

dotshort dashed line The parameters are D and H r The wavelike character of the resp onse

max

disapp ears when H r v

ACCRETION DISKS AND BENDING WAVES

3 4 5 6 7 8 9 10 11 D

3 4 5 6 7 8 9 10 11

D

Fig Net torque jT j vs D in a semilog representation for H r The panels a and b corresp ond to

z max

mo dels b and and mo dels c and resp ectively The dierent curves corresp ond to

v v p

solid line dotted line and dashed line These plots show several resonances

TERQUEM

.002

0

-.002 Re(g)

-.004 a) Real part of g

-.006

0 .2 .4 .6 .8 1 r

0

-.05 Im(g)

-.1

b) Imaginary part of g

-.15 0 .2 .4 .6 .8 1

r

Fig Real panel a and imaginary panel b parts ofg vs r for mo dels and The dierent curves corresp ond

p

to solid line dotted line short dashed line long dashed line and dotshort

v

dashed line The parameters are D and H r The zerofrequency resp onse is not wavelike max

ACCRETION DISKS AND BENDING WAVES

0

-.2

-.4

-.6

-.8

-1

0 .2 .4 .6 .8 1

r

Fig T r J r angular momentum dissipated b etween r and r normalized to unity vs r for D

z in

mo del b and solid line dotted line and dashed line p

TERQUEM

0

-.2

-.4

-.6

-.8

-1

0 .2 .4 .6 .8 1 r

0

-.2

0.1 -.4

-.6

-.8

-1

0 .2 .4 .6 .8 1

r

Fig T r J r angular momentum dissipated b etween r and r normalized to unity vs r for The

z in p

panel a corresp onds to mo dels ie D H r and solid line dotted line short

max

dashed line and long dashed line The panel b corresp onds to mo dels ie D H r and

max

dotted line short dashed line long dashed line and dotshort dashed line solid line

ACCRETION DISKS AND BENDING WAVES

0

-.01 Im(g)

-.02

a) Imaginary part of g

-.03 0 .2 .4 .6 .8 1 r

0

-.2

-.4

-.6

-.8

-1

0 .2 .4 .6 .8 1

r

Fig Imaginary part ofg panel a and T r J r angular momentum dissipated b etween r and r normalized

z in

to unity panel b vs r for and a nonuniform The dierent curves corresp ond to solid line

p

x dotted line and x dashed line The parameters are D and H r max