On corotation torques, horseshoe drag and the possibility of sustained stalled or outward protoplanetary migration

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1 Mon. Not. R. Astron. Soc. 394, (9) doi:1.1111/j x On corotation torques, horseshoe drag and the ossibility of sustained stalled or outward rotolanetary migration S.-J. Paardekooer and J. C. B. Paaloizou DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 WA Acceted 9 January 15. Received 9 January 13; in original form 8 Setember 1 INTRODUCTION A lanet embedded in a gaseous disc is subject to a net torque that gives rise to orbital evolution. Since the discovery of the first hot Juiter (Mayor & Queloz 1995), lanet migration due to interaction with the rotolanetary disc has become a necessary ingredient of lanet formation theory. A considerable amount of analytical and numerical work has gone into understanding the disc lanet interactions leading to lanet migration (see Paaloizou et al. 7 for an overview). Three modes of migration can be distinguished, in most cases leading to migration towards the central star. High-mass lanets, for which the Hill shere is larger than the disc scaleheight, oen u dee gas in the disc, after which migration roceeds on a viscous time-scale (Lin & Paaloizou 1986). For standard disc arameters, S.Paardekooer@damt.cam.ac.uk ABSTRACT We study the torque on low-mass rotolanets on fixed circular orbits, embedded in a rotolanetary disc in the isothermal limit. We consider a wide range of surface density distributions including cases where the surface density increases smoothly outwards. We erform both linear disc resonse calculations and non-linear numerical simulations. We consider a large range of viscosities, including the inviscid limit, as well as a range of rotolanet mass ratios, with secial emhasis on the co-orbital region and the corotation torque acting between disc and rotolanet. For low-mass rotolanets and large viscosity, the corotation torque behaves as exected from linear theory. However, when the viscosity becomes small enough to enable horseshoe turns to occur, the linear corotation torque exists only temorarily after insertion of a lanet into the disc, being relaced by the horseshoe drag first discussed by Ward. This haens after a time that is equal to the horseshoe libration eriod reduced by a factor amounting to about twice the disc asect ratio. This torque scales with the radial gradient of secific vorticity, as does the linear torque, but we find it to be many times larger. If the viscosity is large enough for viscous diffusion across the co-orbital region to occur within a libration eriod, we find that the horseshoe drag may be sustained. If not, the corotation torque saturates leaving only the linear Lindblad torques. As the magnitude of the non-linear co-orbital torque (horseshoe drag) is always found to be larger than the linear torque, we find that the sign of the total torque may change even for mildly ositive surface density gradients. In combination with a kinematic viscosity large enough to kee the torque from saturating, strong sustained deviations from linear theory and outward or stalled migration may occur in such cases. Key words: lanets and satellites: formation lanetary systems: formation. lanets more massive than Juiter migrate in this Tye II regime (Crida & Morbidelli 7). Intermediate-mass lanets, comarable to Saturn, embedded in massive discs may undergo runaway or Tye III migration (Masset & Paaloizou 3), which may be directed inwards as well as outwards (Peliński, Artymowicz & Mellema 8). In this aer, we focus on low-mass lanets, with masses tyical u to a few times the mass of the Earth (M ). Low-mass lanets excite linear waves in the disc, the action of which leads to Tye I migration (Ward 1997). Linear, semianalytical calculations have resulted in a widely used torque formula (Tanaka, Takeuchi & Ward, hereafter TTW) for isothermal discs. The total torque on an embedded lanet can be decomosed in torques due to waves, which are generated at Lindblad resonances and roagate away from the lanet, and corotation torques, generated near the orbit of the lanet, where material on average corotates with the lanet (see Goldreich & Tremaine 1979). For low-mass lanets, the wave torque is usually thought to dominate the total C 9 The Authors. Journal comilation C 9 RAS

2 84 S.-J. Paardekooer and J. C. B. Paaloizou torque, and, being relatively insensitive to background gradients, leads to inward migration. Although Tye I migration was thought to be mathematically well understood, it nevertheless osed a serious roblem for lanet formation. Alying the torque formula from TTW (see also Korycansky & Pollack 1993, hereafter KP93) to lanets of a few M embedded in a tyical disc resulted in inward migration timescales that are much shorter than the lifetime of the disc. In other words, these lanets would all migrate to very small orbital radii, or even into the central star (Ward 1997). Since gaseous giant lanets are thought to form around a solid core that is in this mass range, Tye I migration theory essentially redicts that there should be no lanets at large radii. This has led to investigations on how to slow down or sto Tye I migration, for examle through the action of magnetic fields (Terquem 3; Nelson & Paaloizou 4) or shar surface density gradients (Masset et al. 6). More recently, efforts have been made to relax the isothermal assumtion that was made in revious works, and to roerly account for the energy budget of the disc. Since radiation is the dominant cooling agent, radiation-hydrodynamical simulations are needed. The outcome of these simulations was surrising: Paardekooer & Mellema (6a) found that the torque on the lanet was ositive, leading to outward migration, whenever the oacity of the disc at the location of the lanet was high enough to make cooling inefficient. In Paardekooer & Mellema (8), it was recognized that this ositive torque was a corotation effect, related to a radial entroy gradient in the unerturbed disc. Baruteau & Masset (8), through a linear analysis of the corotation torque, suggested that the linear corotation torque in the resence of an entroy gradient can be strong enough to overcome the negative wave torque. However, Paardekooer & Paaloizou (8) showed that this linear contribution is small, and that a genuinely non-linear effect is resonsible for the change of sign of the total torque. In this aer, we take a ste back and reanalyse the isothermal case, in a two-dimensional setu. The simlicity of this model allows us to erform numerical simulations with high enough resolution to investigate in detail non-linear effects on the corotation torque and to run them for long enough to study its ossible saturation. We will show that linear theory only alies for short times of the order of a few orbits when a rotolanet is inserted into a disc for which the viscosity is not too large, and there is a non-zero corotation torque. We also show that it is a non-linear effect associated with horseshoe bends (Ward 1991) that results in a dearture from linear theory. This dearture is found to result in torques that can have a much larger magnitude than the linear ones. As the torque scales with the gradient of secific vorticity, this may roduce notable effects that may even lead to stalled or outward migration, when this gradient is relatively large as, for examle, occurs when the surface density increases gently outwards. For these effects to be sustained, the viscosity must be large enough to revent torque saturation. Viscosities comarable to those often assumed in rotolanetary disc modelling are found to be large enough to enable these non-linear corotation torques to be sustained for rotolanets in the Earth mass range. The lan of this aer is as follows. We start in Section by reviewing the model used throughout this aer. In Section 3, we review and discuss linear corotation torque estimates as well as torque estimates based on the horseshoe drag exerienced by material executing horseshoe turns. These are shown to be distinct henomena with different deendencies on the hysical variables, the horseshoe drag being essentially non-linear even though the associated torque scales in the same way as the linear one. We go on to erform detailed linear calculations in Section 4, and then comare the linear and non-linear (horseshoe drag) torques in Section 5. In Section 6, we resent the results of numerical hydrodynamical simulations, torques are also comared to linear and horseshoe drag estimates confirming the view outlined above. The horseshoe drag is found to be significantly larger than the linear corotation torque and otentially a very imortant contributor to the total torque for disc surface densities that increase even mildly outwards. We go on to consider the long-term torque evolution and saturation as a function of viscosity finding corotation torque saturation in lowviscosity cases and sustained corotation torques when the viscosity is large enough to resuly angular momentum to the co-orbital region. We discuss our results further in Section 7. Finally we resent a short summary, together with some concluding remarks, in Section 8. In addition, we give an analysis and discussion of the time develoment of the linear corotation torque acting on a lanet after immersion into a disc, demonstrating that the characteristic time is the orbital eriod as was found in the numerical simulations. BASIC EQUATIONS The evolution of a gaseous disc is governed by the Navier Stokes equations. We will work in a cylindrical coordinate olar frame (r, ϕ), centred on the central star, and integrate the equations of motion vertically to obtain a two-dimensional roblem. We are then left with the continuity equation and two equations of motion: + v = (1) t v t + (v )v = + f visc, () where is the surface density, v = (v, r ) T is the velocity, is the vertically integrated ressure, is the gravitational otential and f visc reresents the viscous force. We use a locally isothermal equation of state, = c s, where the sound seed c s may be a rescribed function of radius. We take c s to be a ower law with index β, which makes β basically the ower-law index of the radial temerature rofile. Writing c s = H K,whereHis the vertical ressure scaleheight and K is the Kelerian angular velocity, we usually adot either a sound seed that gives rise to a constant asect ratio h = H/r (or, equivalently, β = 1) or a urely isothermal disc with constant c s (β = ). In the latter case, we usually quote the asect ratio at the location of the lanet, h = c s /(r ). The initial, or for linear calculations the background, surface density is taken to be a ower law with index α. Thus = (r/r ) α, where is the surface density at the location of the lanet. The gravitational otential is the sum of the otential due to the central star, the lanet s otential and an indirect term that arises due the acceleration of the coordinate frame centred on the star. For, we use a softened oint mass otential: = GM, (3) r rr cos(ϕ ϕ ) + r + b r where G is the gravitational constant, M is the mass of the lanet and (r, ϕ ) are the coordinates of the lanet. We will also use q to indicate the mass ratio M /M,whereM is the mass of the central star. The softening arameter b should be a sizeable fraction of h, in order to aroximate the result of aroriate vertical averaging of the otential. The exact form of the viscous terms can be found elsewhere (e.g. D Angelo, Henning & Kley ). We take the kinematic C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

3 viscosity ν to be a ower law in radius, such that the initial surface density rofile is a stationary solution. It is easy to see that the required viscosity law is ν r α 1/. This way we can neglect a global radial velocity field in the model, which greatly simlifies the analysis. Values quoted in the text refer to ν at the location of the lanet, and will be in units of r,where is the angular velocity of the lanet. 3 COROTATION TORQUES The wave (or Lindblad) torque exerted by a low-mass lanet on a gaseous disc is relatively well understood (Goldreich & Tremaine 1979; TTW). When the Hill shere of the lanet is much smaller than the scaleheight of the disc, the density waves that are excited at Lindblad resonances are linear and the resulting torque can be calculated by erforming a linear resonse calculation and then summing the contributions arising from individual Fourier comonents (TTW). This torque usually leads to inward migration, and has been thought to dominate in the Tye I regime of low-mass lanets. There are two aroaches that can be followed in order to obtain corotation torques. The first aroach, which we will refer to as the linear estimate, is based on a linear resonse calculation. The second aroach is a fundamentally different aroach based on a direct torque calculation made after making some assumtions about the trajectories of the disc fluid elements, which we refer to as a horseshoe drag calculation. The first aroach alies at early times after a rotolanet is inserted into a disc as, at least for a low-mass rotolanet, the resonse must first of all be linear. The second aroach alies at later times after the streamline attern has adjusted to the resence of the lanet. 3.1 Formula for the linear corotation torque for each azimuthal mode number In order to obtain linear estimates for corotation torques, the equations are linearized (see Section 4) and Fourier-decomosed in azimuth. The resulting single second-order equation (see Goldreich & Tremaine 1979) governing the disc resonse can be solved to find the corotation torque acting on the lanet. Using an aroximation scheme that assumes the erturbing otential varies on a scale significantly longer then the scaleheight, Goldreich & Tremaine (1979) find this torque given by Ɣ c,m = mπ m d /dr d dr ( ω ), (4) where m is the mth Fourier comonent of the lanet s otential, m is the azimuthal mode number and ω is the flow vorticity, being equal to / in a Kelerian disc. All quantities in equation (4) should be evaluated at corotation (r = r ), which makes the total torque roortional to the radial gradient in secific vorticity, or vortensity, in the unerturbed flow there. Of course unless the softening arameter is rather large, the erturbing otential does not vary slowly on the corotation circle and so (4) cannot be used as described above. However, it becomes valid if the erturbing otential m is relaced by the generalized otential obtained by adding the enthaly erturbation to it (see e.g. Lai & Zhang 6 and the Aendix). But then the linear resonse equations need to be solved in order to determine the enthaly erturbation in order to evaluate torques using the modified form of (4), which are nonetheless still roortional to the gradient of secific vorticity. Corotation torques and horseshoe drag The total linear corotation torque in the limit of zero softening To obtain the total torque, one needs to sum the contributions from all values of m (see Ward 1989). For a Kelerian disc with zero softening, the total torque has been calculated by finding the linear resonse numerically, to be given by (see TTW) ( ) 3 q Ɣ c,lin = 1.36 α h r 4. (5) For a three-dimensional disc, it is found that the same exression holds but with the numerical coefficient being relaced by.63. An imortant issue is the time required to develo the linear corotation torque. Being linear this should not deend on the mass of the rotolanet but only on intrinsic disc arameters. Furthermore, when a rotolanet is inserted into a disc, there has to be an initial linear hase and for sufficiently low rotolanet mass, the full linear resonse should develo. As it is somewhat involved, we relegate the discussion of the time develoment of the linear corotation torque to the Aendix. From this discussion, the characteristic develoment time is exected to be of the order of the orbital eriod. We now go on to consider the subsequent develoment of the corotational flow. 3.3 The horseshoe drag Ward (1991) derived a formula for the corotation torque using a fundamentally different aroach. The argument was based on the exected form of the gas streamlines. These can be classified into four grous as viewed in a reference frame corotating with the lanet (see Fig. 1). The first grou asses the rotolanet interior to the co-orbital region and the second grou asses it exterior to the co-orbital region. These grous extend to large distances from the rotolanet. The other two grous consist of material in the φ Γ Γ 1 I III IV II r Figure 1. Schematic overview of the horseshoe region, with the lanet denoted by the black circle. Solid grey lines indicate aroximate streamlines, with the searatrices in black. The searatices divide the region into four arts, labelled by roman numerals: I and II, the inner and the outer disc, resectively, and III and IV the leading and trailing art of the horseshoe region, resectively. Two horizontal-dashed lines indicate the boundaries of the torque calculation, Ɣ 1 and Ɣ. Note that in this figure, the location of Ɣ 1 and Ɣ is chosen arbitrarily. In the model, they are chosen to be far enough from the lanet so that the corotation torque is determined within. C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

4 86 S.-J. Paardekooer and J. C. B. Paaloizou corotation region on horseshoe orbits that execute turns close to the lanet. The third grou aroaches and leaves the rotolanet from its leading side, while the fourth does so from its trailing side. These grous are searated by two searatrices (see e.g. Masset, D Angelo & Kley 6 and Paardekooer & Paaloizou 8 for more discussion of this asect). For low-mass rotolanets, most of the corotation torque is roduced in a region close to the lanet with a length-scale exected to be the larger of br or H. It is imortant to note that it takes some time for the streamline structure described above to develo on this scale after insertion of a rotolanet into a disc. As this structure reresents a finite deviation from the initial form, this time deends on the mass of the rotolanet. The calculation of the horseshoe drag alies to the situation when this streamline structure has develoed on this scale. It is imortant to note that the time required is shorter than that required to develo it on a scale comarable to r which is when issues of torque saturation need to be considered. Following Ward (1991), we consider the torque roduced by material on streamlines undergoing horseshoe turns. We consider a region R interior to the two searatrices searating these from the first and fourth grous of streamlines that are bounded by two lines of constant ϕ, Ɣ 1 and Ɣ on the trailing and leading sides of the rotolanet, resectively (see Fig. 1). These boundaries are suosed to be sufficiently far from the rotolanet that the corotation torque is determined within. Assuming a steady state, this torque may be obtained by considering the conservation of angular momentum within R written in the form ( ) G Ɣ c,hs = r dϕ dr R ϕ [ ] Ɣ = (j j )( )r dr. (6) Ɣ 1 Here, j = rv ϕ is the secific angular momentum and j is j evaluated at the orbital radius of the rotolanet. We now follow Ward (1991) and assume that the streamline attern is symmetric on the leading and trailing sides such that a streamline entering on Ɣ can be identified with a corresonding streamline leaving on Ɣ 1 at the same radial location. But note that on account of the horseshoe turns, these streamlines will enter R at different radii. Since, in the barotroic case, otential vorticity or vortensity /ω is conserved along streamlines, and these streamlines are disconnected, this will differ on them. Accordingly we write the torque as Ɣ c,hs = Here, ( ) ω Ɣ ( ω ( = ω ) ω ) ω(j j )( )r dr. (7) is the vortensity difference on the corresonding streamlines which have been assumed to enter R at the same radial distance from the lanet on oosite sides and ω = / is the vorticity at the lanets orbital radius, and we have assumed the vortensity to be an even function of radial distance from the rotolanet. The above integral is easily done if one adots a first-order Taylor exansion for the quantities in brackets and integrates from r x s r to r + x s r, where the dimensionless width of the horseshoe region is x s. One obtains (Ward 1991) Ɣ c,hs = 3 ( ) 3 4 α x 4 s r 4. (9) (8) Note that as is aarent from the above discussion and that given in the Aendix and also the results of numerical simulations to be resented later, the horseshoe drag, given by equation (9), occurs as a non-linear effect that has no counterart in linear theory (see also Paardekooer & Paaloizou 8, in addition to Paardekooer & Paaloizou 9). We also note that numerical hydrodynamical calculations necessarily have b >, and, two-dimensional simulations, generally adot b h to account for three-dimensional effects in an aroximate way. We will see that this strongly affects the width of the horseshoe region x s. A detailed analysis of the horseshoe region is resented in Paardekooer & Paaloizou (9). Here, we adot a simle estimate for x s that has roved to be reasonable for smoothing lengths comarable to h (Paardekooer & Paaloizou 8): q x s = 3b. (1) In general, this deendence is to be exected when the searatrix streamline asses through the location of the lanet with b ossibly being relaced by h for small softening (see also Masset et al. 6). We will comare the horseshoe drag resulting from x s to the linear corotation torque in Section 5. Note that, since Ɣ c,hs is roortional to x 4 s, the horseshoe drag would then be roortional to q /h in the small softening case and q /b in the large softening case, just as Ɣ c,lin. This means that the deendence of the total torque on q, b and h would not be a good indication of linearity, since both the linear and the non-linear corotation torque scale in the same way. The only ways to distinguish these are through their magnitudes, and by their time evolution. The linear corotation torque is set u in aroximately an orbital time-scale (see the Aendix), similar to the wave torque. This means that, when the background state of the disc is stationary, any evolution in the torque after a few dynamical time-scales is due to non-linear effects. The finite width of the horseshoe region gives rise to a libration time-scale τ lib = 8π 1 3x, (11) s which is basically the time it takes for a fluid element at orbital radius r (1 + x s ) to comlete two orbits in a frame corotating with the lanet. Therefore, this is the time-scale on which the corotation torque will saturate due to hase mixing (see Ward 7), unless some form of viscosity oerates on smaller time-scales (Masset ). Note that saturation is a non-linear rocess (Ogilvie & Lubow 3), since it hinges on the finite width of the corotation region. We will see that there are basically three time-scales in the roblem, two of which are closely related to non-linearity. First, there is the orbital time-scale, on which all linear torques are set u. This is the shortest time-scale in the roblem. The longest time-scale is τ lib, on which saturation oerates. A third time-scale governs the develoment of the horseshoe drag, which we will see takes a fraction of a libration time (see also Paardekooer & Paaloizou 8), and lies in between the time-scale for the linear torque develoment and the saturation time-scale. 4 LINEAR CALCULATIONS Since we are interested in deartures from linear theory, it is necessary to first firmly establish what linear theory redicts. The twodimensional linear calculations of KP93 and TTW use a vanishingly small value for b, and these results are therefore not directly comarable to our hydrodynamical simulations. Our customized C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

5 Corotation torques and horseshoe drag 87 linear calculations, as briefly outlined below, with b comarable to h, can be directly comared to hydrodynamical simulations. 4 Re(W) 4.1 Governing equations Linearizing equations (1) and (), with f visc =, and assuming a Fourier decomosition such that erturbation quantities are the real art of a sum of terms ex(imϕ im t), m =, 1,..., one obtains the following system of equations (Goldreich & Tremaine 1979) for the corresonding Fourier coefficients: W Im(W) i σv m u m + dw m + d m + βw m =, (1) dr dr r i σu m + Bv m + imw m + im m =, (13) r r i σw m + dv m cs dr + (1 α) v m r + imu m =, (14) r where rimes denote erturbation quantities and the subscrit m, being the azimuthal mode number, indicates the mth Fourier coefficient. Here, the velocity erturbation is v m = (v m, u m ), W m = c s m/ is the enthaly erturbation, σ = m( ),B is the second Oort constant and m is the mth Fourier comonent of, being a real quantity. The term roortional to β is not resent in the equations of KP93, because they considered a strictly barotroic equation of state. Eliminating u m, we obtain a system of ordinary differential equations (KP93): dv m dr dw m dr = ( 1 α mb σ ( m + i ( = i σr σ ) c σ κ σ s ) v m m (W m + m ) σr ) v m r W m + im m (15) σr d m dr βw m, (16) r where κ = 4B is the square of the eicyclic frequency. We have solved equations (15) and (16) using a sixth-order Runge Kutta method and outgoing wave boundary conditions (see KP93). In Fig., we show the resulting W m for m = 1, for an isothermal, constant surface density disc with h = 1 3/ and a very low value of the softening arameter b = 1 4. The same case was shown in KP93 (their fig. ), and the results agree very well. The imaginary art of W m is directly related to the torque density: dɣ m dr = πm m Im(W m ), (17) cs and the total torque on the lanet Ɣ m can be found by integrating over the whole disc. Goldreich & Tremaine (1979) showed that the Lindblad torque is carried away by density waves, resulting in an angular momentum flux F m = mπr σ κ { Im(W m ) d [ Re(W m dr ) + ] m ] dim(w m ) }. (18) [ Re(W m ) + m dr Therefore, the Lindblad torque is given by F m = F m (r out ) F m (r in ), where r in and r out denote the inner and the outer radius r Figure. Real (solid) and imaginary (dotted) arts of the enthaly erturbation W, for an isothermal, constant surface density disc with h = 1 3/ and a softening arameter as used in KP93 (b = 1 4 ) lotted as a function of radius in units of r. of the disc, resectively. We can, therefore, calculate the corotation torque as (KP93) Ɣ c,m = Ɣ m F m, (19) and the total torques can be found by summation over all m. 4. Results We start by considering a disc with constant secific vorticity, which means that the corotation torque vanishes. This allows us to look in some more detail at the Lindblad torque alone. In Fig. 3, we show the total torque 1 for three different cases: strictly isothermal (dashed line), locally isothermal (h constant; solid line) and locally isothermal without the term roortional to β in equation (1) (dash dotted line). The latter case is similar to the one considered in KP93, and aroaches the result of KP93 for small softening. Similarly, the strictly isothermal case aroaches the two-dimensional result of TTW for small softening. For aroriate values of b h, differences between the strictly and locally isothermal models can be as large as 3 er cent. In the remainder of this aer, we will restrict ourselves to the strictly isothermal equation of state, since the horseshoe drag has been analysed for barotroic fluids only (Ward 1991). For this case, a smoothing length of b =.5, corresonding to.5 h, results in the total torque being equal to the three-dimensional calculations of TTW. We now introduce a corotation torque in the calculations by considering different values of α. In Fig. 4, we show the linear 1 In all figures, the torque is given in units of r 4 and is divided by q to make it indeendent of the mass of the lanet. In Paardekooer & Paaloizou (8), horseshoe drag was studied in adiabatic flows, where entroy (but not vortensity) is conserved. For a locally isothermal equation of state, neither of these is conserved, with the consequence that it is unclear what the horseshoe drag is in this case. C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

6 88 S.-J. Paardekooer and J. C. B. Paaloizou Γ/q b (h.1 ).1 1. TTW (3D) TTW (D) h const isothermal b (r ).1 KP93 h const isothermal h const (KP93) Figure 3. Total torque on the lanet, obtained from linear calculations, as a function of the softening arameter b. The surface density is roortional to r 3/ (α = 3/), which means that the corotation torque vanishes in the barotroic case. The disc is either fully isothermal with h =.5 (dashed line) or has a constant asect ratio h =.5 (solid line). Calculations similar to those of KP93 with constant h are indicated by the dash dotted line. Also shown are the results of linear calculations in two-dimensional by KP93 (for constant h) and the isothermal results of TTW, for two-dimensional as well as three-dimensional. Both two-dimensional studies used a very small (KP93) or vanishing (TTW) softening length. Filled circles denote the torques measured from hydrodynamical simulations for a q = lanet in a disc with constant h, and diamonds denote the results for isothermal simulations with h =.5. Γ/q b (h.1 ).1 1. α=1/ α=-1/ Γ Γ C Γ L TTW (3D) b (r ) Figure 4. Lindblad (dash dotted curves) and corotation (dashed curves) torques obtained from an isothermal (h =.5) linear calculation as a function of softening arameter b for two different density rofiles. Black curves: α = 1/, grey curves: α = 1/. The solid curves denote the total torques and the horizontal-dotted lines denote the three-dimensional results of TTW. Lindblad and corotation torques for α = 1/ and 1/. Note that in the latter case, the surface density gradient is ositive, which may be unrealistic excet for secial locations in the disc (see e.g. Masset et al. 6), but it serves as a good examle of a case with strong corotation torques. For α = 1/, the corotation torque is much smaller than the Lindblad torque, and is increased by a factor of 3 going from b = h to, similar to the Lindblad torque. Therefore, the corotation torque is a small fraction of the Lindblad torque for all values of b considered here. That is not the case for α = 1/, where for small softening arameters the corotation torque is almost the same as the Lindblad torque. The sign of the total torque will change for α< 1/, which is exected from the two-dimensional results of KP93 and TTW. For three-dimensional calculations, the situation is different (TTW), which is illustrated by the results in Fig. 4 for larger softening. For all values of α considered in Figs 3 and 4, we find that for a softening length of b =.5 h we can match our two-dimensional calculations with the three-dimensional work of TTW. 4.3 The limit of large softening We now relate our numerical results to the redictions made using the torque formula (4) given by Goldreich & Tremaine (1979). This is alicable to the situation where the softening arameter is significantly larger than the scaleheight, and we use it to evaluate the corotation torque in that limit. To do this, we adot the aroximation for m introduced by (Goldreich & Tremaine 198) for m > in the form π m = 1 cos(mϕ)dϕ GM K (mζ ), () π π r where K is the standard Bessel function and (r r ) ζ = + b. (1) r This is valid in the imortant domain of interest where m 1/b and r r < br. Summing the torques given by (4) over m, using () we obtain ( ) 3 8q Ɣ c,m = α 3 r 4 mk (mb), () m=1 m=1 where denotes the unerturbed surface density at the lanet s location. Noting that b is small and that the dominant contribution to the sum on the right-hand side comes from large m 1/b, we relace the sum by an integral, thus ( ) 1 mk (mb) xk (x) dx = 1 b m=1 Thus we obtain Ɣ c,m = Ɣ c,lin = m=1 ( ) 1. (3) b ( 3 ) 4q α 3b r 4. (4) We remark that Ward (199) obtained a corresonding exression for a vertically averaged otential that has the same scaling when b is relaced by h in the above equation. A comarison of the torques given by equations (5) and (4) indicates that the effect of softening should be significant for b > 1.4 h. But note that in addition to requiring b/h 1, (4) also formally requires b 1. This means, as we shall see below, that the limit where (4) alies requires rather small h <.5. C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

7 Corotation torques and horseshoe drag 89 1 model h=.1 h=.5 h=.5 8 b (h ) Γ C Γ HS Γ C 3D Γ C /q b (r ) Figure 5. Corotation torque, obtained from a linear calculation with α = 1/andh =.5 (solid line), h =.5 (dashed line) and h =.1 (dash dotted line). The dotted line gives the rediction of equation (4), valid for h b 1. We now directly comare linear calculations with the rediction of equation (4). The results are illustrated in Fig. 5 for three different values of h. Note that in the derivation of equation (4) it was assumed that b 1, so we exect the torque to be given by equation (4) for h b 1. From Fig. 5, we see that for h =.5 we can nicely reroduce equation (4) for.5 < b <.. However, as h increases there are increasing deviations from equation (4). For h =.5 and.1 < b <., the maximum and minimum deviations are by a factor of and 1.5, resectively, while for h =.1 the deviation always exceeds a factor of. 5 A COMPARISON OF THE EXPECTED HORSESHOE DRAG WITH THE LINEAR COROTATION TORQUE In this section, we comare corotation torques obtained from linear calculations to the exected horseshoe drag. We obtain the latter from equation (9) which requires an estimate of x s which we obtain from equation (1) which simulations have indicated gives a good estimate for b h. We shall obtain estimates for the horseshoe drag directly from simulations and make further comarisons below. In Fig. 6, we show the linear corotation torque (solid line), together with the horseshoe drag, obtained as indicated above (dashed line), for a disc with α = 1/, for different values of the smoothing arameter b. For reasonable values of b, the horseshoe drag is stronger than the linear corotation torque making the torque on the disc more negative. Thus the horseshoe drag on the lanet is ositive and larger than the linear corotation torque. For smoothing arameters. < b <.3, this non-linear torque is a factor of 3 larger than the linear torque. For smaller values of b, equation (1) redicts a value of x s that is too large and accordingly a horseshoe drag that is too large. In reality we exect both the horseshoe width and therefore the horseshoe drag to reach limiting values for small b, and these values should be larger than those we obtain here using values of b for which equation (1) alies. A more comlete discussion of these asects is given in Paardekooer & Paaloizou Γ/q b (r ) Figure 6. Corotation torque, obtained from a linear calculation with α = 1/ andh =.5 (solid line), together with the horseshoe orbit drag (dashed line) for our simle estimate of the width of the horseshoe region. Also shown is the three-dimensional result of TTW (dash dotted line). (9). Also shown in Fig. 6 is the corotation torque obtained for the corresonding three-dimensional disc (TTW) which has zero softening. For b =., we can match our two-dimensional linear calculation to the three-dimensional result of TTW. However, for the same softening the horseshoe drag is three times as large. For a large softening with b =.7h =.35, the horseshoe drag is equal to the three-dimensional unsoftened linear corotation torque. However, numerical results show that for this smoothing length, equation (1) actually slightly underestimates the true value of x s, with the consequence that the intersection of the horseshoe drag with the three-dimensional linear corotation torque occurs for aroximately b =.8h. Lacking a full three-dimensional model of the horseshoe region, it is difficult to say what value of b to choose so that two-dimensional results match three-dimensional results. However, numerical results suggest that b <. may be aroriate (Masset et al. 6). We come back to this issue in Section 7, but it is good to kee in mind that the effects discussed in the next sections may well be stronger in three-dimensional calculations. 6 HYDRODYNAMICAL SIMULATIONS In the revious section, we have established that horseshoe drag is otentially much stronger than the linear corotation torque. We now turn to numerical hydrodynamic simulations to show that indeed strong deviations from linear theory are encountered in ractice. We use the ROe solver for Disc Embedded Objects method (Paardekooer & Mellema 6b) in two satial dimensions, on a regular grid extending from r =.5r to 1.8r and which covers the whole π in azimuth. Since we want to resolve the horseshoe region for even the smallest lanets we consider, a relatively high resolution is required: 14 cells in the radial and 496 cells in the azimuthal direction, making the resolution at the location of the lanet aroximately.15r in both directions. Tests have shown C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

8 9 S.-J. Paardekooer and J. C. B. Paaloizou that this resolution is sufficient to cature the horseshoe dynamics for x s >.4. We include all disc material in the torque calculation (not excluding any material close to the lanet), and the lanet is introduced with its full mass at t =. For low-mass lanets, this does not affect the results. First, in Section 6.1, we study the develoment of the horseshoe drag and its relationshi to linear theory. Then, in Section 6., we study the long-term behaviour (saturation) of the corotation torque. 4 ν= ν=1-4 ν=1-3 Γ lin +Γ HS 6.1 Develoment of the non-linear torque We start by considering a lanet with a relatively high mass, q = , which corresonds to 4 M around a solar mass star. Although this is the lanet with the largest mass we consider, it has been exected to be well within the linear regime (Masset et al. 6). In Fig. 3, we show that for a disc with constant secific vorticity, we can reroduce the exected linear torque for all reasonable values of b. For very small values of b, one may exect a dearture from linear theory, since at this oint, an enveloe may form that is gravitationally bound to the lanet, which robably should be excluded from the torque calculation. We do not consider this regime here, since we exect to adot a value of b comarable to h in order to account for vertical averaging. The imortant oint is that we can match our linear calculations to the hydrodynamical simulations in the absence of corotation torques (linear as well as non-linear). This is further illustrated in Fig. 7, where we show the time evolution of the total torque on the lanet for different surface density rofiles. We see that the case with α = 3/ nicely falls on the corresonding linear result. Also note that this torque is set u in aroximately one orbital eriod. This is to be exected for both the Lindblad and the linear corotation torque (see the Aendix). Different surface density rofiles give remarkably different results. All cases excet α = 3/ show a dearture from linear theory, the sign of which is dictated by the vortensity gradient. This indicates that the corotation torque is enhanced with resect to its linear value. This enhancement takes aroximately orbits to develo Γ/q -5-1 α=5/ α=3/ α=1/ α=-1/ t (orbits) Figure 7. Total torque on the lanet for an isothermal, inviscid h =.5 disc with b =.3 and q = , for different surface density rofiles. The dotted lines indicate the linear torque for increasing α from to to bottom anels. Γ/q Γ lin -Γ C,lin +Γ HS Γ lin t (orbits) Figure 8. Total torque on the lanet in an isothermal h =.5 disc with α = 1/forb =.3 and q = for different magnitudes of the viscosity. The horizontal lines indicate, from bottom to to anels, the total linear torque, the Lindblad torque with the horseshoe torque (found using the rocedure outlined in Section 5) added and the total linear torque with the horseshoe torque added. after which the torques attain steady values for the remainder of the simulations. This time eriod, as we show below, can be understood as being a fraction of the libration time-scale, and is therefore related to a non-linear effect. Note that in all cases, linear theory is only valid at early times (less than about two orbits) during which as exected the linear corotation torque is set u on a dynamical timescale. At later times, it gets relaced by the non-linear horseshoe drag. We take a closer look at the α = 1/ case illustrated in Fig. 8. The horizontal-dotted lines indicate, from bottom to to anels, the linear torque, the linear Lindblad torque with the horseshoe drag added and the total linear torque with the horseshoe drag added. From the solid line, we see that the total torque can be understood as the linear torque, with the linear corotation torque relaced by the horseshoe drag. We have found this to be true for all values of α considered (see Fig. 9). It is also exected from the discussion giveninsection3. The magnitude of the non-linear torque deends on the magnitude of the viscosity in the disc. Recall that whenever we include viscosity in the model, we do not allow for large-scale mass flow with resect to the lanet (see also Section 7). It is easy to understand this deendence, since the horseshoe drag model hinges on vortensity conservation while the fluid executes a horseshoe turn. For strong enough viscosities, we can recover the linear torque, but note that the large values ν> 1 3 required corresond to a very large α viscosity arameter of α visc =.4. This is because only then can the viscosity directly affect the horseshoe turn. In Section 6., we will see that lower viscosities are in fact sufficient to kee the torque unsaturated. Returning to inviscid discs, we show in Fig. 9 the total torque on the disc for different values of α, confirming that the horseshoe drag indeed relaces the linear corotation torque. Note that we exect a torque reversal around α = 1, while the linear calculations redict this would only haen around α = 3. Also note that stee density C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

9 Corotation torques and horseshoe drag Γ L +Γ C,lin Γ L +Γ HS Γ L +Γ C +Γ HS -1 - Γ/q α Figure 9. Total torque on a q = lanet embedded in an isothermal h =.5 inviscid disc for b =.3 after 3 orbits. The solid line indicates the linear torque, and the dashed line shows the total linear torque lus the estimated horseshoe drag. The dotted line indicates the linear Lindblad torque lus the non-linear horseshoe drag. Symbols denote results obtained from hydrodynamical simulations for different values of the surface density ower-law index α. Γ/q q=1.6e-5 q=9.45e-6 q=6.34e-6 q=3.15e t (orbits) Figure 1. Total torque on a lanet in an isothermal h =.5 disc for b =.3 and α = 1/, for different values of q. The dotted line indicates the linear torque. rofiles result in an acceleration of inward migration with resect to the linear estimate. Linear theory redicts that the torque should scale as q. Since in our simle model, the horseshoe width scales as q 1/, the non-linear contribution to the torque (the horseshoe drag) also scales as q. Therefore, it could be misinterreted as a linear effect. However, the time-scale on which the horseshoe drag term is set u deends on the mass of the lanet. We will see below that it takes a fixed fraction Γ/q q=1.6e-5 q=9.45e-6 q=6.34e-6 q=3.15e t (τ lib ).15 Figure 11. Total torque on the lanet embedded in an isothermal h =.5 disc for b =.3 and α = 1/, for different values of q. The time is in units of τ lib, which scales as q 1/. of a libration time-scale. This means that although the full non-linear torque scales as q, this will not be the case at intermediate stages. This is illustrated in Fig. 1, where we show the time evolution of the torque for different mass ratios. For linear torques, all curves would fall on to of each other. This is indeed true at early times, when the horseshoe drag has not yet develoed. When the horseshoe drag takes over, however, lanets of different mass give different results; lower-mass lanets take longer to develo the non-linear torque. If we rescale the time axis to the libration time (see Fig. 11), the curves fall on to of each other again. This is because the timescale to set u the horseshoe drag is a fixed fraction of the libration time (aroximately 15 er cent, according to Fig. 11). One might exect that, as the scale of the region contributing the torque is of the order of H, the time required would be of the order of a factor of h smaller than the libration time. Thus the results resented here are consistent with this time being about h τ lib. This tye of henomenon was also discussed in Paardekooer & Paaloizou (8), where it was shown that the develoment of the non-linear torque is due to a high-density ridge resulting from material that has executed a horseshoe turn at constant entroy (relacing secific vorticity alicable to this case). 6. Long-term evolution Lindblad torques give rise to waves that carry away angular momentum. Therefore, the lanet can continue to exchange angular momentum with the disc at Lindblad resonances. However, there is no wave transort in the corotation region, and therefore in the absence of any other form of transort, only a finite amount of angular momentum exchange with the lanet can occur before the structure of the co-orbital region is significantly modified. In other words, the corotation region is a closed system unless there is some diffusive rocess oerating in the disc that can transort angular momentum there. In the absence of such diffusion, the corotation torque will saturate (Ogilvie & Lubow 3). C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96

10 9 S.-J. Paardekooer and J. C. B. Paaloizou ν= ν=1-7 ν= Γ/q t (orbits) Figure 1. Total torque on a lanet in an isothermal h =.5 disc for q = , b =. and α = 1/, for different values of the viscosity ν. Saturation is an inherently non-linear rocess, as it deends again on a finite width of the corotation region. The time-scale on which saturation oerates is the libration time-scale, and diffusion must oerate on a smaller time-scale in order to revent saturation. For a viscosity coefficient ν, one needs ν> x3 s 4 r (5) (see Masset, who considered this rocess for a disc with constant surface density). For a level of viscosity that has become standard in disc lanet interaction studies, ν = 1 5 r, all rotolanets considered here are of small enough mass that we exect the corotation torque to be unsaturated. We consider a lanet of q = , and use a softening arameter b =. to make the libration time-scale as short as ossible to ease the comutational burden. In Fig. 1, we show the long-term evolution of the torque for three different levels of viscosity. The inviscid case shows strong libration cycles, before the torque starts to settle to a value that is close to the linear Lindblad torque (Ɣ L 1q ;see Fig. 4). A small viscosity of ν = 1 7 gives less rominent libration cycles, and the torque settles at a value of aroximately 85. In this case, corotation torques are artially saturated. This is in quantitative agreement with the analysis of Masset (1), where it is argued that the horseshoe drag should be multilied by a factor of F(z) with z = x s (πν) 1/3 to account for saturation. Choosing the smallest otion for F given in Masset (1), which gave the best fit in Masset (), we find F =.114. This redicts a corotation torque of Ɣ C = 15, while the difference between the Lindblad torque and the total torque as measured from the simulations (the asymtotic difference between the dashed and solid curve in Fig. 1) is 15. For ν = 1 5, the libration cycles disaear, indicating that the torque remains unsaturated. There is basically no evolution of the torque once the non-linear contribution has been set u. This again agrees with the analysis of Masset (1), where for this value of ν we exect F close to unity. Note, however, that the maximum torque Γ/q - -4 ν= ν=1-5 b=.3 b= t (orbits) Figure 13. Total torque on a lanet in an isothermal h =.5 disc for q = and α = 1, for different values of softening arameter b and viscosity ν. that can be reached is reduced for this relatively high viscosity (see also Fig. 8). Finally, we show in Fig. 13 that, for a softening arameter b =., low-mass lanets will feel a sustained ositive torque when α = 1, which corresonds to a mildly ositive surface density gradient. Although a viscosity ν = 1 5 reduces the non-linear torque, making the torque negative for b =.3, it is necessary for the torque to be sustained. For a slightly smaller softening arameter, b =., we find sustained outward migration. Note that from Figs 3 and 4 we exect a smoothing around b =.5 to reroduce threedimensional effects. Full three-dimensional simulations, together with a three-dimensional understanding of horseshoe dynamics, are needed to see how this non-linear torque behaves in a threedimensional setting (see also Section 7). 7 DISCUSSION We have shown that non-linear effects commonly occur in the coorbital region, even for low-mass lanets. In inviscid discs, the corotation torque eventually always becomes non-linear, saturating after a few libration cycles, so that only the (linear) Lindblad torque survives. Viscosity can revent the torque from saturating, and for a large enough viscosity, which however decreases with rotolanet mass, the linear torque will be restored. In Fig. 14, we resent a schematic overview of the three ossibilities in the (log q, log ν) lane. Two lines define the boundaries between the three regions. The boundary between the region where the non-linear torque is maintained by viscosity and the saturated regime is given aroximately by the condition that viscous diffusion across the co-orbital region occurs in one libration cycle. The line searating the sustained horseshoe drag regime from the linear regime is exected to occur where viscous effects become large enough to disrut horseshoe turns. Note that the regime for which the corotation torque is linear occuies only a small fraction of the arameter sace. Also note that the borders between the different regions are not razor shar. In reality the non-linear torque can be artially saturated, or be reduced C 9 The Authors. Journal comilation C 9 RAS, MNRAS 394, 83 96