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the discovery of a large number of short - period extrasolar planets has generated renewed interests in the study of tidally induced migration of protoplanets in disks .
the basis of migration is the gravitational interaction between a gaseous disk and an orbiting perturber , and has been analyzed well before extrasolar planets were actually discovered @xcite .
it has been suggested that this tidal interaction induces the embedded protoplanet to migrate under two different circumstances . in type  i migration ,
the protoplanet mass is small and the disk structure is weakly perturbed . in this limit
, the disk response can be analyzed with a quasi - linear approximation .
linear calculations have revealed an intrinsic imbalance of inner and outer disk torques @xcite , so the planet could migrate quite quickly inward . in type ii migration , the protoplanet mass is sufficiently large for it to open up a gap in the disk @xcite .
the protoplanet s orbital evolution is locked to the evolution of the disk , which is likely to be on the viscous timescale @xcite .
both types of migration have been basically confirmed by the non - linear numerical simulations .
the high mobility of small mass protoplanets or embryo cores of giant planets from type i migration has presented a challenge to the formation of giant planets .
the absence of deep oceans is contrary to the inference that any terrestrial planet in the solar system may be formed well outside the snow - line and migrated to their present location .
it is thus imperative to explore the cause which may have inhibited the type i migration .
one potential contributor to this process is the disk response to low mass planets , especially in the co - orbital region where only sketchy analytical arguments have been developed .
there are , however , two notable recent exceptions .
masset ( 2001 , 2002 ) has evaluated the co - orbital corotation torque and found that they can play a crucial role for the migration rate .
extensive numerical simulations of viscous disks have been performed to understand the torque dependence on the planet mass , sound speed , disk viscosity , and the torque saturation mechanism .
one generic behavior he found is that the total torque on the planet first shows an oscillation consistent with the librating fluid motion in the co - orbital region then it levels off to a constant due to the disk viscosity .
another study is by balmforth & korycansky ( 2001 ) , who used matched asymptotic expansions to explore the non - linear dynamics of the flows in the co - orbital region .
they investigated the role of re - arranging the potential vorticity profile as a response to the planet s driving and its role in the corotation torque saturation . for inviscid flows , they also noticed the formation of vortices which they attributed to the secondary instabilities in a forced critical layer problem ( the critical layer in this case is the corotation resonance ) .    in this _
letter _ , we present results on the non - linear dynamics of the co - orbital region , based on direct , high resolution numerical simulations (  2 ) .
our approach is quite similar to that of masset s except that we have focused on the inviscid limit simulations . as a consequence , the total torque on the planet exhibits some interesting new behavior , instead of leveling off to a constant as in the case of viscous disks .
some of the findings are similar to what balmforth & korycansky have found in the excitation of vortices (  3 ) .
we provide an interpretation of why these vortices are produced in  4 and discuss the implications of our results in  5 .
we assume that the protoplanetary disk is thin and can be described by the inviscid 2-d euler equations in a cylindrical @xmath0 plane with vertically integrated quantities .
two gravitating bodies , a central star and a protoplanet , are located at @xmath1 and @xmath2 respectively and the 2-d disk is modeled between @xmath3 .
the planet is on a fixed circular orbit at @xmath2 .
the self - gravity of the disk is not included .    the mass ratio between the planet and the central star is taken to be @xmath4 .
its hill ( roche ) radius is @xmath5 .
the disk is assumed to be isothermal with a constant temperature throughout the simulation region ( i.e. , it is attached to a thermal bath ) . the isothermal sound speed , scaled by the keplerian rotation speed @xmath6 at @xmath2 is @xmath7 where @xmath8 is the disk scale height .
we used a set of three sound speeds @xmath9 . in the simulations presented here ,
we choose an initial surface density profile with @xmath10 , in order that the ratio of vorticity to surface density ( potential vorticity ) has a flat radial profile @xmath11 ( note that @xmath12 has a small deviation from being a precise constant due to the finite pressure gradient which slightly modifies @xmath6 from its keplerian value . ) with this choice of initial condition , we avoid the generation of inflection points due to the rearrangement of @xmath12 distribution as it is carried by the stream lines .
we have also made other runs which have an initial potential vorticity gradient and we find that the results presented here also hold for them .
the detailed analysis of these results will be presented elsewhere .    since @xmath13 for all three different sound speeds , the planet is embedded .
the planet s gravitational potential is softened by an approximate 3-d treatment .
the key approach is to take the surface density at each cell spreading it out vertically based on a gaussian density profile with a scale height determined by the local temperature .
the gravitional force between the planet and the disk matter is calculated using this 3-d configuration but all the forces with the same @xmath0 positions are summed together .
this gives a correction factor @xmath14 to the gravitational forces , which effectively softens the gravitational potential .
the torque overestimation in a 2-d geometry was studied in @xcite which arises only from locations near the planet . as a test
if our key finding is affected by the 3-d treatment , we excluded several roche radii from the torque calculation and show that it is not the case .
although there are some accumulation of gas near the planet , we do not allow any disk gas to be accreted onto it .
this boundary condition is equivalent to that of sufficiently low - mass protoplanets with quasi hydrostatic envelopes ( pollack _ et al .
the planet s potential is switched on over 10 orbits in order for the disk to make an adiabatic adjustment .
( we have used other , longer turn - on times but the results do not depend on them . )    simulations are carried out using a 2-d code whose basic algorithm was based on fargo by @xcite and @xcite .
the co - rotating frame is used so that the positions of the central star and the planet are fixed at @xmath15 and @xmath16 ( acceleration due to frame rotation is also included , see @xcite ) .
runs are made using several radial and azimuthal grids to study the influence of resolution , and those presented here have @xmath17 .
outflow boundary conditions are used at both the inner and outer radial boundaries .
the simulations typically last several hundred orbits at @xmath2 .
one key difference between our simulations and previous studies is that our simulations are performed in the inviscid limit , i.e. , we do not explicitly include a viscosity term , though numerical viscosity is inevitable and is needed to handle for example shocks .
in addition , our simulations tend to be of higher resolution . with a radial @xmath18 ( some runs go up to @xmath19 ) and @xmath20 , the diameter of the hill sphere of a planet with @xmath21 is resolved by at least @xmath22 cells in each direction . using a nested - grid technique ,
@xcite have also made very high resolution simulations , though their simulations include an explicit viscosity .
our simulation results are generally in agreement with previous simulations for the planet on a fixed circular orbit ( see * ? ? ?
* ) in terms of the generic structures produced in the disk by the tidal interactions between the planet and its surrounding flows .
since the planet mass is relatively small , only density dips are produced without gaps ( at least during a few hundred orbits simulated here ) .
we define the co - orbital region here as @xmath23 .
the co - orbital region can be separated into several parts depending on their streamline behavior ( see * ? ? ?
* ) , including the horseshoe ( or librating ) region @xmath24 , the separatrix region @xmath25 , and the streaming region @xmath26 .
the horseshoe region can be further divided into real horseshoe streamlines enclosing all three libration points ( @xmath27 ) and tadpole like streamlines enclosing either @xmath28 or @xmath29 .
our key finding is shown in fig .
[ fig : torq_t ] which displays the total torque on the planet from the disk .
it has two very distinct phases : phase i : the total torque is negative and smooth , modulated by a period that is similar to the libration period near the l4 and l5 points ( @xmath30 where @xmath31 is one orbit at @xmath2 ) .
phase ii : the total torque shows very large amplitude and fast oscillations with a quasi - period of a few orbits .
we see such a behavior in all the runs with three different sound speeds , though the times when the `` phase '' transition occurs are different .
they are at @xmath32 and @xmath33 for @xmath34 and @xmath35 , respectively .
the obvious question is what causes such large amplitude and fast changes in torques .
closer inspection of the co - orbital region at times around @xmath36 ( using @xmath37 as an example ) reveals that vortices are generated in the separatrix region .
this development is shown in fig .
[ fig : kh ] , which depicts maps of potential vorticity @xmath12 through the times when the system undergoes its `` phase '' transition .
one can see that these vortices are anti - cyclones which are also blobs of higher densities ( not shown here ) .
being in the separatrix region , they experience very close encounters with the planet repeatedly , thus exerting strong `` impulses '' to the planet .
this results in the large amplitude and fast variations in the torque evolution .    to understand this phenomenon further , in fig . [
fig : pv ] , we plot the azimuthally averaged radial potential vorticity profile @xmath38 for @xmath37 at different times @xmath39 .
one sees that this profile deviates progressively away from the initially flat radial profile , developing two prominent structures at @xmath40 , with a minimum at @xmath41 and a maximum at @xmath42 . comparing with fig .
[ fig : kh ] , vortices apparently emerge from the minima .
the existence of the extrema ( or inflection points ) in the potential vorticity profile is usually regarded as a necessary condition for non - axisymmetric instabilities in rotating shear flows ( drazin & reid 1981 ) .
the progressive steepening of this profile , say , at @xmath43 until @xmath44 and the subsequent flattening ( see the curve at @xmath45 ) strongly suggest that there is a secondary instability which is excited around @xmath41 and vortices are the non - linear outcome of such an instability .
how could the potential vorticity profile develop such inflection points / structures from an initially flat and stable profile ?
as can be seen from the evolution equation for potential vorticity @xmath46 where @xmath47 is the lagrangian derivative , that @xmath12 is conserved along the streamlines in 2d ( korycansky & papaloizou 1996 ) .
thus , the initial uniform @xmath12 distribution should be preserved in an inviscid disk .
this conclusion , however , is only true provided that streamlines can be defined everywhere in a dissipationless flow .    disk flows in the co - orbital region around the planet , however , do not satisfy this condition ( except perhaps for flows within @xmath48 ) because of the two sets of spiral shocks produced by the planet .
these two sets of shocks emanate from close to the planet and cut through the whole disk .
flow lines which encounter these shocks are broken and the dissipation in shocks breaks the potential vorticity conservation . the consequence of the shock is equivalent to that of a strong viscous stress in the momentum equation , the curl of which , does not vanish .    to establish the role of shocks in breaking the potential vorticity conservation , we have performed three runs with different sound speeds @xmath49 . even though they only span a small range , but the effects are quite clear . in fig .
[ fig : pv3cs ] , we plot the azimuthally averaged potential vorticity profiles for these three runs at a time just before the emergence of vortices .
vortices first appear at the minimum of the `` valley '' in the profiles which moves outwards as @xmath50 increases .
this growth pattern is consistent with the fact that the starting location of the shocks is also moving away from the planet when the sound speed increases .
note that the cause for the emergence and growth of inflection points in the azimuthally averaged @xmath12 distribution presented here is different from that due to its re - arrangment along the horseshoe orbits ( balmforth and korycansky 2001 ) . in our calculation with a more general initial @xmath51 ( and @xmath12 ) distribution , we confirm that onset of the shearing instability occurs at an earlier epoch .
the appearance of the subsequently self - excited vortices is quite similar to the development of the kelvin - helmholtz ( kh ) instability observed at shear interfaces .
a detailed study on the properties of this secondary instability , such as its threshold and growth rate , will be presented in a forthcoming paper .
in this paper , we consider the tidal interaction between a protostellar disk and an embedded planet with a modest mass .
linear torque calculations suggest that such interaction leads to angular momentum transfer and inward migration of the planets . for planets with mass comparable to that of the earth , the inferred migration time scale ( ward 1997 )
is much shorter than the observationally determined depletion time scale of the disk @xcite .
this analytic result raises a problem for the formation of giant planets through the core accretion process ( pollack _ et al .
we reinvestigate this process in an attempt to resolve this paradox .
the embedded protoplanet induces a circulation flow pattern near its corotation region in the disk .
we show here that even in disks with initially uniform potential vorticities , inflection points may emerge and grow spontaneously as potential vorticity is generated near the spiral shock in the vicinity of the protoplanet .
these inflection points lead to the onset and growth of secondary instabilities .
the existence of secondary instabilities and their non - linear outcome as vortices indicate that the flows in the co - orbital region are perhaps more complicated than linear analyses have suggested .
these vortices should exert strong torques on the planet .
one limitation of our current study is that the planet is being artificially held on a fixed circular orbit .
such strong and rapidly varying torques from the vortices on the planet might cause the planet to `` oscillate '' and lose its phase coherence with the surrounding flow .
this feedback process should have important implications for the type i migration problem . allowing the planet to migrate radially
must be included in order to understand the planet s response to such torques .
these issues will be addressed in a forthcoming publication .
we wish to thank neil balmforth , darryl holm and edison liang for useful discussions .
this research was performed under the auspices of the department of energy .
it was supported by the laboratory directed research and development program at los alamos and by lanl / igpp .
we are also supported in part by nasa through nag5 - 11779 , nag5 - 9223 , and nsf through ast-9987417 .
we present global 2-d inviscid disk simulations with an embedded planet , emphasizing the non - linear dynamics in its co - orbital region . 
 we find that the potential vorticity of the flow in this region is not conserved due to the presence of two spiral shocks produced by the planet . as the system evolves , the potential vorticity profile develops extrema ( inflection points ) which eventually render the flow unstable . 
 vortices are produced in association with the potential vorticity minima . 
 born in the separatrix region , these vortices experience close - encounters with the planet , consequently exerting strong torques on the planet . 
 the existence of these vortices have important implications on understanding the migration rates of low mass planets .