On the Early Evolution of Forming Jovian Planets II Analysis of Accretion and Gravitational

a r X i v :a s t r o -p h /0301620v 1 30 J a n 2003On the Early Evolution of Forming Jovian Planets II:Analysis of Accretion

and Gravitational Torques

Andrew F.Nelson 1

Max Planck Institut f¨u r Astronomie,K¨o nigstuhl 17,D-69117Heidelberg,Germany

Willy Benz

Physikalisches Institut,Universit¨a t Bern,Sidlerstrasse 5,CH-3012Bern,Switzerland ABSTRACT We continue our numerical study of the migration of an already formed proto-Jovian com-panion embedded in a circumstellar disk.We ?rst study the sensitivity of the planet’s migration to its mass accretion rate,and ?nd that the disk can supply a forming planet with mass at an essentially in?nite rate (～1M J /25yr)so that a gap could form very quickly via further dynam-ical interactions between the planet and remaining disk matter.The accreted matter has less orbital angular momentum than the planet and exerts an e?ective inward torque,so that inward migration is slightly accelerated.However,if a partial gap is formed prior to rapid accretion,the e?ective torque is small and its contribution to the migration is negligible.Although the disk can supply mass at a high rate,we show that mass accretion rates faster than ～10?4M J /yr are not physically reasonable in the limit of either a thin,circumplanetary disk or of a spherical envelope.Planet growth and ultimately survival are therefore limited to the planet’s ability to accept additional matter,not by the disk in which it https://www.sodocs.net/doc/5110925002.html,rge gravitational torques are produced both at Lindblad resonances and at corotation res-onances.We compare the torques in our simulations to analytic theories at Lindblad resonances and ?nd that common approximations to the theories predict torques that are a factor ～10or more larger than those obtained from the simulations.Accounting for the disk’s vertical structure (crudely modeled in our simulations and the theory with a gravitational softening parameter)and small shifts in resonance positions due to pressure gradients,to disk self gravity and to inclu-sion of non-WKB terms in the analysis (Artymowicz 1993a)can reduce the di?erence to a factor

～3?6,but do not account for the full discrepancy.Torques from the corotation resonances that are positive in sign,slowing the migration,contribute 20-30%or more of the net torque on the planet,but are not well resolved and vary from simulation to simulation.A more precise accounting of the three dimensional mass distribution and ?ow pattern near the planet will be required to accurately specify the torques from both types of resonances in the simulations.

We show that the assumption of linearity underlying theoretical analyses of the interactions

at Lindblad resonances is recovered in the simulations with planets with masses below 0.5M J ,but the assumption that interactions occur only at the resonances may be more di?cult to support.Angular momentum transfer occurs over a region of ?nite width near both Lindblad and corotation resonances.The shape of the disk’s response there (due e.g.to local variations in epicyclic frequency)varies from pattern to pattern,making the true position of the resonance less clear.We speculate that the ?nite width allows for overlap and mixing between resonances and may be responsible for the remainder of the di?erences between torques from theory and simulation,but whether accounting for such overlap in a theory will improve the agreement with the simulations is not clear.

Subject headings:Planets:Migration,Accretion Disks,Hydrodynamics,Numerical Simulations

In a companion paper(Nelson&Benz2002,hereafter Paper I),we began a study of the early stages of the migration of a Jovian planet through an accretion disk,using numerical simulations.That work was a study of several important systematic e?ects that a?ect the results of numerical simulations and an exploration of a very large parameter space of systems.The parameter study was designed to explore the range of possible systems likely to be encountered during the evolution of real circumstellar disks and to make a number of qualitative comparisons of our results to theory.In this work,we concentrate our study on two of the outstanding issues raised in that work.

The?rst issue concerns our?nding that a planet could not complete a transition to Type II migration, unless it was more massive than0.3M J,and that the migration rate was both very rapid and very sensitive to the details of the mass distribution within1–2Hill radii of the planet.The second issue concerns our ?nding that the migration rates did not follow the theoretically expected linear dependence on planet mass, but that the M pl2/3proportionality predicted for the gap width and based on the same physical model did hold.

We study these two issues through a detailed investigation of mass accretion and of the gravitational torques between the disk and planet,and their origin.In section1,we outline the initial conditions for an additional series of simulations,not discussed in Paper I(the‘gap’series of simulations),how we implement mass accretion in our code and the physical signi?cance we can derive from it.We also summarize the main theoretical results for the derivations of gravitational torques between the planet and disk.In section2,we look at accretion onto the planet,whether it can a?ect the migration rate,and given a lower limit on the accretion rate we discuss the likely morphology of the object.In section3,we will examine in detail four simulations from Paper I(the low mass and high resolution prototypes with and without disk self gravity) in order to explore in quantitative detail the di?erences between our hydrodynamical simulations and linear analyses.Our purpose will be to determine the extent to which linear theory may be applied to the problem, as well as where and how it may https://www.sodocs.net/doc/5110925002.html,stly,in section4,we compare our results to others in the literature.

1.The initial conditions and input physics

In this work we will make use of the‘low mass’and‘high resolution’prototypes,discussed in detail in Paper I.Brie?y summarized,these models consists of a circumstellar disk with mass M D=0.05M⊙,that extends from0.5to20AU.Mass is distributed according to an r?3/2power law and the temperature is de?ned by an r?1/2power law and the condition that the temperature at1AU is T=250K.We use an isothermal equation of state and the boundary conditions at the inner and outer edges of the grid are re?ecting.For the low mass prototype,the disk is resolved on a128×224cylindrical2grid(r×θ),while for the high resolution prototype we doubled the grid resolution in each dimension,to256×448.

The planet is set into an initially circular orbit at5.2AU and is thereafter in?uenced by gravitational forces from the1M⊙central star(?xed to the origin)and the disk.The gravitational softening required was de?ned to be equal to the size of the grid zone where the planet is found at each time.In order that the simulations would be identical,the high resolution prototype de?ned the softening to be twice the current cell size,so that the absolute values were identical.We have simulated two versions of these models,one with and the other without disk self gravity.All of our simulations studying accretion will include self gravity.

For our study of accretion we will use initial conditions and a physical model that are identical to our low mass prototype.We will perform two separate studies,one beginning from the low mass prototype, and the other with an already formed gap in the disk.We produce a gap from the unperturbed initial state by allowing the system to‘pre-evolve’for3000yr.In this pre-evolution,we begin with the low mass prototype,except that we assume that while the planet exerts gravitational forces on the disk,it is not in turn a?ected by the disk.Its orbit therefore remains?xed at5.2AU.During the pre-evolution,spiral patterns are generated by the planet/disk interaction and a gap structure forms in the disk.We show the unperturbed pro?le and the azimuth average of the mass distribution after3000yr in?gure1.While it is clear that the evolution has produced a signi?cant change in the pro?le,the gap region is not totally devoid of matter.We monitored the pro?le during the evolution and found no signi?cant changes during the last 1000yr of this pre-evolution.We therefore believe that the system is at or near a steady state.

After the simulation has completed its pre-evolution,we save the resulting state‘as is’and use it as an initial condition for multiple simulations in which we‘turn on’the disk-on-planet gravity and allow the planet to migrate.One additional modi?cation is required between the pre-evolution simulation and each of the models based on it.Turning on this additional gravity term requires that disk gravity be accounted for in the planet’s velocity.We assume that the its orbit is circular and calculate the angular velocity required for its orbit from the net gravitational forces due to the disk and star.

1.1.The mechanics of accretion from a grid,and what we can learn from it

In the acc and gap series’of simulations(see de?nitions in section2)we study mass accretion onto the forming planet.In these simulations,we allow mass to accrete onto the planet at a fraction,f,of the Bondi-Hoyle-Littleton(BHL)rate(see e.g.Shu1992):

˙M pl =4πρ∞f

(GM pl)2

R H 2

r iδr iδθj

where R is the distance of the zone center from the planet,r iδr iδθj de?nes the area of a grid zone and surface area of the Hill sphere,A,is taken as the sum over the areas of all grid zones for which R

The accretion of mass onto the planet will of course be accompanied by some amount of angular momentum,both as an orbital component(around the star)and a spin component.We add this momentum to the planet and subtract it from each grid zone,so that the net result of the accretion conserves total angular momentum.The mass accreted onto the planet will change its orbital angular momentum both by increasing the planet’s mass and by changing its trajectory.Mathematically,the change in orbital angular momentum of the planet due to accreting matter can be decomposed into three parts as

dJ z

dt rVθ+M pl

d

dt

,(3)

where r and Vθare the planet’s orbit radius and azimuthal velocity.The?rst term in this equation is due to the accretion of mass from the disk which is already co-moving with the planet.In other words,this change in angular momentum(‘torque’)has no net e?ect on the planet’s trajectory.The second term represents the dynamical e?ect of the mass accretion and can be compared to the gravitational torques on the planet due to the generation of spiral density structures in the disk.We discuss the importance of this e?ect in section 2.1.The third term in equation3is necessary to properly account for the fact that the spin of the planet will also be changed by the accretion.This is because mass accreted onto the planet will in general not have zero angular momentum,as calculated around the center of mass of the planet and accreted mass.

While a Bondi-like accretion rate is not strictly valid for the geometry and?ow pattern we consider here,we believe it represents an interesting limit.It is at or near the highest rate at which matter can be supplied to the planet by the disk and at or near the highest rate that the planet can accrete matter because it assumes that all matter within reach of the planet will be accreted and that none is able to build up into a long-lived envelope and oppose further infall.By including accretion or not we will be able to determine whether mass accreting onto the planet can bring with it enough orbital angular momentum(an e?ective torque),to signi?cantly change the orbit of the planet.We will also be able to explore the conditions in the circumplanetary environment.Under what conditions is it reasonable to expect a circumplanetary disk to form?Should we expect the environment to be a somewhat spherical envelope,or something still more complex?

1.2.Analytic torque formulae

In section3,we shall make detailed comparisons of the gravitational torques on the planet as produced in our simulations,to those predicted by theory.Here we will brie?y state the torque formulae to which we make those comparisons.Analytic methods typically proceed by linearizing the equations of motion and evaluating the gravitational torques of the disk and planet on each other as the sum contributions from a Fourier series of density perturbations in the disk.The density perturbations are generated at an inner or outer Lindblad resonance(ILR or OLR,respectively)or a corotation resonance(CR).

Goldreich&Tremaine(1979,1980,hereafter GT79,GT80)found that by far the largest portion of the torque exerted by a perturber is transmitted in the neighborhood of a resonance,and developed approxima-tions for these torques that were based upon this fact.Because the torques are transmitted to and from the

disk only at resonances,the torque on the planet can be expressed as the change in angular momentum?ux through the disk at that resonance.Under this approximation,they found that the angular momentum?ux generated by an m’th order spiral pattern at Lindblad resonances is

F LR=mπ2f c Σ(r)dr+2?

H(1+4ξ2) 2HK0(2H/3)+K1(2H/3)

1+ξ2andξ=m(c s/r?)p.

GT79also approximate the?ux due to interactions near corotation by

F CR=?mπ2

d?/dr d B

r c e?|qx|(7)

where B=κ2/4?is the Oort parameter,q=(rκ/c s)r c,x=(r?r c)/r c and r c is the corotation resonance location.In both equation4and7,the gravitational potential of the planet,φm,is one term in an in?nite series expansion in Laplace coe?cients,b m

1/2

.In our comparisons,we calculate the Laplace coe?cients using the method of Olvers(1967).

Equations4and7have been further re?ned by,e.g.Ward(1988);Artymowicz(1993a);Ward(1997) and Takeuchi et al.(1996).In particular for the LRs,the inclusion of non-WKB terms in the wave equation, and will result in a modi?ed resonant denominator

D?=D+ mc s

r 2?2πGΣ m

π

K0(m|1?α|)(10)

whereα=r/a pl,a pl is the semi-major axis of the planet and r is evaluated at the Lindblad resonances.For an exactly Keplerian rotation curve,the argument of the Bessel function is nearly constant(at2/3)for all m patterns at all Lindblad resonances.Both Ward(1988)and Korycansky&Pollack(1993)have de?ned a generalized Laplace coe?cient in very similar ways in order to avoid numerical singularities at the planet’s orbit https://www.sodocs.net/doc/5110925002.html,ing the form from Korycansky&Pollack(1993),it is

b m1/2(α)=2

(s2+p2α2?2αcos mθ)1/2

.(11)

where p=1,s=1+r20/a2pl and r0is the softening radius.The interpretation given to this modi?cation is to account in an approximate way for either a numerically required softening coe?cient(Korycansky& Pollack1993)or(in a slightly di?erent form)the vertical structure of the disk(Ward1988).In combination, the improvements of the theory represented in equations8,9and11are su?cient to negate the need for the cuto?function.In general,we will use only this form(without the cuto?function)in our comparison calculations but will also make comparisons to the form de?ned in equation10in order to investigate the changes in the torques they are responsible for.

Using these analyses,the torque of the disk on the planet will be the amount of?ux injected into or removed from the disk at a given resonance.For the Lindblad resonances,wave propagation is forbidden between the ILR and OLR and the?ux is zero there.Therefore,we can calculate the theoretical torque on the planet if we calculate the angular momentum?ux at the resonance,using equation4.For the corotation resonances,the torque on the planet may be calculated from equation7as the di?erence between the left and right limits as r approaches r c.

The torque,δT,exerted on any given ring,δr,in the disk is accounted for as a torque density,dT/dr, as is the torque actually transmitted to the disk matter.The?rst form will be a function of the amplitude and phase of a wave as it propagates through the disk and in the second will be a function of the dissipation in the disk as it decays.In the?rst case,studied by Ward(1986),the torque density will be characterized by an oscillating function of distance from the planet while in the second(Takeuchi et al.1996),the torque will instead be a decaying function of distance.Since our interest lies mainly in the torque of the spiral pattern on the planet,our torque densities will be characterized by the oscillatory form.Corotation torques on the other hand are not associated with waves.Torque densities on the disk for these resonances may be obtained directly from equation7by di?erentiation with respect to r.

1.3.Deriving torques from the simulations

In order to make comparisons to the torque formulae above,we must determine the amplitude and phase of each Fourier component of spiral density structures present in our simulations.As a function of radius,these amplitudes are

X m(r)=

1

Re(X m) .(13)

Calculation of the torque on the planet is a straightforward reconstruction of the density due to each pattern in each grid zone and a sum of the net torque over all grid zones.With a change of sign,these torques are also those exerted by the planet on the gas in the disk at each radius,and the summation can be separated into two partial sums to account for torques from radially inward or outward of the planet.

1.4.Connection of the numerical results to physical systems

In following discussion,we will show the torque magnitudes and torque densities exerted by the disk on the planet,either due to accretion or to gravitational interactions.As a point of reference in evaluating the?gures,recall that the present day Jupiter has a total orbital angular momentum of1.9×1050g cm2/s. Therefore a torque of1038g cm2/s2exerted for a time of103years will cause it to migrate inward from 5.2AU to5.0AU.A0.3M J planet would move inward to4.6AU under the same in?uence.When spread over a radial region of0.25AU,a torque of this magnitude yields a torque density ofδT/δr～7×1024g cm/s2.

2.The importance of accretion on planet migration

Many workers have shown in linear and nonlinear analyses that a planet embedded in a disk,especially before a gap has formed,is expected to have strong dynamical interactions with the disk matter within a few disk scale heights radially inward and outward of the planet.We found in Paper I that in fact the migration was dynamically very sensitive to the disk mass close to the planets-within one or a few Hill radii in both radius and azimuth coordinates.One additional interaction speci?c only to this region is mass accretion. In the following sections,we examine the importance of mass and momentum accretion onto the planet on its trajectory and the formation of a gap using simulations that allow accretion onto the planet at varying rates.We then inquire into the physical reasonability of those rates,in order to better constrain the possible morphology and growth processes of forming planets.The parameters de?ning each of the simulations in these experiments are summarized in table1.

2.1.Accretion

We saw in Paper I that planets with enough mass can open a deep gap in the disk via gravitational torque interactions and in so doing drastically slow their migration.Since the disk begins in a somewhat arti?cial condition(a1M J planet should have already either formed a gap or been accreted by the star)we cannot consider the rapid motion prior to gap formation typical behavior of the long term behavior of real systems with such massive planets.

Lower mass planets remain at issue.The migration of a planet less massive than0.3M J does not lead to a gap and is fast enough to move inward on a<<105yr time scale,far shorter than the expected disk lifetime of>106yr and the expected formation time scale for planets.The critical conclusion to note is that low mass planets can be in?uenced by gravitational torques from the disk strongly enough to migrate quickly through it,but cannot in?uence the disk strongly enough to form a deep gap and enter the much slower Type II migration phase.How then can such objects survive long enough to remain separate from the central star as the system moves into its main sequence lifetime?

Accreted matter changes the angular momentum of the planet due both to the increase of mass and because that matter does not in general have identical speci?c angular momentum to the planet when it is accreted.Additional matter accreting onto the planet continues to bring with it orbital angular momentum, exerting an e?ective torque on the planet’s trajectory.If forming planets are to survive for more than a few thousand years,then at least one of three conditions about this accreted matter must be true.First,if this angular momentum contribution is positive(and large enough),then it will provide a positive torque on the planet,counteracting the negative dynamical torques acting on it.Second,the planet could accrete all of the nearby disk matter,thus opening a gap and eliminating the large gravitational torques driving the migration.Third,mass accretion onto the planet could proceed faster than Type I migration,so that planets become massive enough,quickly enough to transition to Type II migration before migrating all the way inward to the stellar surface.Each of these possibilities can be true only if the disk can supply matter at a fast enough rate and the planet can accept matter at that rate.

We have explored the viability of these three conditions with a set of simulations that include accretion onto the planet,and use our low mass prototype simulation as a model.In order to test whether the evolution is merely an artifact of the fact that our initial condition is an unperturbed disk,we consider two cases. We consider an initial condition identical to our low mass prototype,and we consider an initial condition which begins with an already formed gap in the initial surface density pro?le around the planet,as shown in?gure1.These two series are designated acc and gap in table1,respectively.For each series,we vary the accretion rate by changing the assumed fraction,f,of the Bondi-Hoyle-Littleton rate(see eq.1)in di?erent simulations.

Is the accretion torque positive and large,satisfying our?rst condition?In?gures2and3,for the unperturbed and initial gap simulations respectively,we show the accretion torque as a function of time for a high accretion rate and for a low accretion rate simulation.For the two f=1models,the torque magnitude is initially large compared to the dynamical torques exerted on the planet by individual spiral patterns(see section3.1below),but within 500yr drops to near zero as the planet accretes additional mass.In both examples,the migration of the planet proceeds inward from5.2AU to4.8AU and stops.The low accretion rate models each have accretion torques which are far smaller than the dynamical torques on the planet.In the acc4model,the migration proceeds in near identical fashion to the low mass prototype simulation,while the gap4model,the planet remains at5.2AU for the duration of the simulation.The sign of the torque is always negative.Therefore it acts to increase the planet’s inward migration rate,and we conclude that a planet’s migration cannot be halted by the accretion of orbital angular momentum from the disk.Our?rst condition is not con?rmed.

Can a forming planet can accrete all of the nearby disk,thus forming a gap and satisfying our second condition?In?gure4,we show the mass of the forming planet as a function of time for di?erent fractions,f. For fractions,f≥0.01,the planet accretes between four and eight M J of mass from the disk in only1800yr. The initial accretion rates are as high as～1M J/25yr in the most extreme cases.The accretion occurs so quickly that the limiting factor in the accretion is not the BHL rate given by eq.1,but rather the fact that the disk simply cannot supply matter to the planet at the calculated BHL rate fraction.Instead,the rate is limited by the maximum amount of mass that we allow to be removed from a given zone per time step. The region inside the Hill sphere of the planet becomes completely drained and a gap is quickly generated in the disk as matter continues to pass into the planet’s Hill sphere.As found by Bryden et al.(1999)and Lubow,Siebert&Artymowicz(1999),we also?nd(by virtue of the fact that this region is the?rst to be emptied of matter)that matter accretes onto the planet?rst from the regions approximately2-4R H inside and outside the planet’s orbit radius.Matter in the‘horseshoe’region,at the planet’s orbit radius but with

a phase relative to the planet greater than±π/3,is able to remain unaccreted for longer periods,but is also lost within103yr when f>0.01.

The accretion rate decreases only when a wide and deep enough gap has begun to form in the disk. When such a gap begins to form,a relatively smaller amount of matter can come into close contact with the planet,and so become perturbed onto a planet intersecting orbit and accreted.The planet masses at which this turno?occurs in the gap simulations are a factor of～2below those reached in the acc series.While it is true that the‘?nal’masses for the planet at the end of each simulation are smaller by about a factor of two,the basic conclusion remains.These rates are su?cient to increase the planet’s mass to several Jupiter masses in only a few hundred years.

For a0.05M⊙disk,more than20M J are available to be accreted in the region±1AU from the planet, so that the planet would reach a mass of several tens of Jupiter masses rather than the～6?8M J that it reaches before the accretion slows.This means that the disk does not supply matter to the planet at a rate such that all of the nearby disk matter is accreted.Therefore our second condition is not con?rmed. However,planets this massive still do open a gap and make the transition to Type II migration via dynamical processes alone,as we saw in Paper I.This means that the disk is capable of supplying the planet with matter quickly enough to drive gap formation by dynamical processes,so that our third possibility may be con?rmed if the planet can accept matter at these rates.

Our?rst two conditions for survivability of Jovian planets(torques due to accretion and accretion of all available matter)could be de?nitely ruled out in this section.While accretion of additional mass can indirectly drive gap formation by dynamical processes,the accretion torques have little direct e?ect on halting or accelerating the migration,even for the most extreme simulations we performed.These conclusions are true for both the unperturbed and initial gap simulations and so are not an artifact of the initial condition. The third condition(accretion of enough matter to open a gap via dynamical processes)could not be con?rmed or ruled out directly.However,we can take the information that Jovian mass planets exist around the sun and other stars,along with the failure of our?rst two possibilities,as an indirect con?rmation.One signi?cant ambiguity remains however:how fast can mass accretion onto the planet actually proceed?

2.2.The morphology of the circumplanetary environment and its in?uence on the planet’s

accretion rate and?nal mass

Deriving limits on the planet’s accretion rate is important for both the survivability of the planet and determination of its?nal mass.In this section,we will set a lower limit on the accretion rate from the condition that it survive long enough to form a gap,and a(weak)upper limit by examining the morphology of the planet implied by the accretion.We will conclude that the planet’s envelope must undergo periodic dynamical instabilities during this phase of its evolution.The e?ciency of these instabilities in accelerating the mass accretion will in?uence the?nal mass of the planet.

We can de?ne a lower limit on the mass accretion rate of the forming planet from the results Paper I .We know that low mass planets migrate on a time scale of order0.5–1AU per thousand years,and can open a gap when they reach～0.3M J.This means that in order to survive long enough to open a gap and slow their migration rate,planets less massive than0.3M J must grow faster than～10?4M J/yr.The limit will be somewhat lower in disks less massive than the M D/M?=0.05ratio implicit in these calculations,but recall that we are considering a relatively early stage in the planet’s evolution when the disk may still be more massive than the nominal minimum mass solar nebula.

How much larger than this lower limit can the accretion rate be?The e?ciency of accretion will depend to some extent on the distribution and dynamic and thermodynamic conditions of mass very close to the planet.For the planet masses discussed in our study,any envelope that existed earlier has grown massive enough that gravitational contraction and/or ejection of some of the matter has begun(Bodenheimer& Pollack1986;Wuchterl1991).Eventually the forming planet will begin to develop a central core+disk structure.If we assume a disk structure has already formed,we can derive a self-consistency check of this assumption from the gravitational potential energy that must be radiated by the gas before it is accreted onto the planet and from a measure of the disk’s thickness.

For a given accretion rate,and independent of the speci?c form of dissipation in the disk,the central temperature for a steady state,internally heated accretion disk is given by

T(r)= 9GM pl˙M plτ

γk B r?1(15) where m pr is the mass of the proton,k B is Boltzmann’s constant,μis the average molecular weight of the gas andγis the ratio of speci?c heats.In order for the circumplanetary environment to be‘disky’,gas temperatures must be well below the limits de?ned by eq.15.

For what accretion rates are such temperatures obtained in the circumplanetary environment?In order to answer this question we require the values of the average molecular weight and the ratio of speci?c heats which,for conditions appropriate for circumplanetary disks,will beμ≈2.3andγ≈1.4.We also require a value for the optical depth,τ(=Σκ).We can make a very conservative estimate of the surface density,Σ,if we assume that the circumplanetary disk mass at given time is the mass of the present day Jovian moon system,which we will assume at the time of formation to consist of both gas and solids in solar nebula proportions.Then the surface density isΣ～10?20g/cm2for a disk extending outwards to the Hill radius.For a solar composition,the Rosseland opacity of this matter at typical nebular temperatures will beκ～2?4cm2/g(Pollack et al.1994),so that the optical depth isτ=Σκ～10?100.

For an optical depth ofτ=100,equations14and15yield curves as plotted in?gure5for a0.3M J planet. The condition that the disk have H/R=1is de?ned by a temperature of T≈200K at a distance of one Hill radius(0.24AU)from the planet and reaches the destruction temperature of silicate dust(T≈1200K)at a distance of0.04AU from the planet(～40?50R pl if R pl=2R J).For all accretion rates˙M pl>10?4M J/yr, the condition that the disk be thin is severely violated in all but the innermost parts of the circumplanetary disk.Lower mass planets will have curves with similar characteristics,but with lower temperature scales, through the M pl1/4and M pl1proportionalities in the temperature laws,eq.14and15,respectively.

Coupling this result to the lower limit on the accretion rate,we conclude that survival of the planet is inconsistent with the assumption of a circumplanetary disk at this stage of the planet’s evolution.For most of the time that it is growing,the planet must accrete matter faster than˙M pl>10?4M J/yr.This conclusion

places a strong constraint on the planet’s morphology.It must be characterized by an envelope or thick disk structure.

We can use this constraint to derive a weak upper limit to the accretion rate.The most liberal upper limit is that at which radiation pressure suppresses accretion of additional material.Assuming that the opacity source is ionized hydrogen atoms,then this rate is the Eddington accretion rate,which for an accreting surface at a radius～1?2R J,is～10?1M J/yr.However,the Eddington rate is probably not relevant for accreting Jovian planets in circumstellar disks,because the main source of opacity is not ionized hydrogen, but rather dust or exited states of complex molecules.If we instead assume that the main opacity source is dust and that the gas and dust remain well mixed,then the appropriate mass opacity is theκ～2?4g/cm2 value noted above rather than theκ～0.4g/cm2appropriate for Eddington accretion,and an analogue of the Eddington rate would be a factor ten smaller.

A much more restrictive limit than radiation pressure is the requirement to overcome the gas pressure of the planetary envelope composed of the recently accreted matter.This rate is available from an analysis of the Kelvin Helmholtz contraction of the envelope.Bryden et al.(2000)de?ne the maximum accretion rate allowed by Kelvin-Helmholtz contraction of the envelope as

˙M

=M KH0 M pl

KH

envelope and circumstellar disk is allowed,only～2?3%of the angular momentum remained in the planetary system,with the rest immediately returned to the solar nebula.Since their calculations were one dimensional,they could make no statements about the nature or mechanism for this transport however. Rotational instabilities are known to develop in morphologically similar entities such as rotating polytropes (Pickett et al.1996;Toman et al.1998).We speculate that the mass accretion rate in forming Jovian planets could be enhanced if large scale dynamical instabilities in the planet’s envelope could develop,due to this spin accretion and the processes that return it to the solar nebula.Clearly,much more work is required before a more?rm conclusion can be drawn on this point.

3.The dynamical interaction of the planet and disk

In Paper I and in section2we studied the planet and the disk somewhat qualitatively in terms of their actions on each other and the consequences for the evolution.We found that some of the major qualitative predictions of analytic theories were only partially recovered in our simulations.Here,we will attempt to make the analysis more quantitative in terms of the applicability of analytic formalisms to the system,and in what limits they break down.Because of the large and important di?erences in outcome,depending both on grid resolution and on whether disk self gravity was included or not,we will do a side by side analysis of four models from Paper I:the low mass(with our‘standard’resolution of128×224)and high resolution prototype models with and without disk self gravity.The standard resolution(in the low mass prototype)vs.high resolution comparisons will provide insight into the role of the mass distribution in the torque calculations,while the self gravitating vs.non self gravitating models provide insight into the physical model itself.In a sense,these comparisons will also touch on the di?erences between gap forming and non gap forming systems,since the two models with self gravity form gaps,while those without it do not.

In sections3.1and3.2,we show the torques from our simulations,?rst as a function of radius,then as a function of Fourier pattern number,m,and compare them to the analytic predictions.Then in section3.3 we examine the approximations made in deriving the analytic torque formulae and the approximations made in our numerical realization of the system.While very important for mathematical models of migration, this discussion will be rather detailed and of lesser interest to some of our audience.These readers may safely skip forward to section3.4,where we discuss the consequences of failures of various mathematical assumptions on the gravitational torques when they break down.

3.1.The torques exerted by the planet and disk on each other

In?gure6,we show plots of the gravitational torque density of the disk on the planet as a function of radius for the low mass and high resolution prototype simulations.The torques densities shown are those at the same time(300yr),as in the top panels of?gures4and6of Paper I,for which spiral patterns have fully developed and have had time to propagate through the entire disk,but before a deep gap has formed. These torques will be representative of those expected from Type I migration assumptions.

In each case,the torque density is very large near the planet(radially)and decays as a function of distance inwards and outwards from the planet,as the spiral patterns themselves decay.Qualitatively,the morphology present is consistent with the theoretical model that low order m patterns contribute little to the total torque,and that higher order m patterns,whose resonances fall closer to the planet,are excited and provide most of the net torque contribution.The sign of the torque oscillates so that at some radii it acts

to increase the disk’s angular momentum at the expense of the planet,while at other very nearby locations it acts in the opposite sense.The oscillations are stable in time relative to the planet’s position.The torque curves shown with the dotted line(omitting the contribution from inside the Hill sphere)shows that while the matter inside the Hill sphere makes some contribution to the total,it is not by itself the determining contribution to the torque from this radial region.

The contribution to the total due to the m=1,2and10spiral patterns are shown in?gures7and 8,for the same four models.In each case,a torque oscillating in sign originates at an LR and extends with decreasing amplitude in the direction away from the planet.Consistent with our resolution dependent numerical dissipation,the lowest order(longest wavelength)patterns propagate the furthest distances from the planet and produce torques over nearly the entire radial extent of the disk,while higher order patterns contribute to the torques only very close to the planet.The low order m patterns each display a large torque near the planet as well,clearly distinct from the oscillating torque pattern further away and presumably due to the corotation interaction.

The m=1pattern represents a special case pattern–it has no inner Lindblad resonance.Therefore we expect that no spiral structures should be generated there and no torque on the planet from interior to its orbit due to this pattern should exist.Indeed this is the case–the torques from the m=1pattern from well inside the planets orbit are near zero,while a decaying wave structure in the exterior disk is visible.The m≥2patterns do have both inner and outer Lindblad resonances and for the m=2pattern,the torque oscillates in sign and decays as a function of distance from the planet,while for the higher order symmetry m=10pattern only the?rst wave maximum can be observed.

In each case,the torques and their oscillations are larger in the non self gravitating disks than in the self gravitating versions.This is most likely a consequence of the di?erence in the migration rates and their e?ect on the forming gap.Although we have attempted to examine a point in time before substantial evolution has occurred,gap formation has begun to reduce the surface density near the planet,unfortunately causing the torques to be decreased by varying amounts in each simulation.While slightly visually disturbing,the di?erences will have few if any consequences for our comparisons below.

Each pattern also produces a large torque contribution from locations radially very close to the planet, near the corotation resonance locations and where waves generated from Lindblad resonances are forbidden. For all simulations except the high resolution non self gravitating version,the contribution is positive in sign(increasing the planet’s angular momentum)both inside and outside the planet’s orbit,but negative in sign(decreasing its angular momentum),slightly further inward.These torques are resolved on our grid in the sense that they are distributed over many radial rings of grid zones,however they di?er greatly in character between simulations at di?erent resolution.Therefore,we believe that their true character is not fully resolved by our simulations.

https://www.sodocs.net/doc/5110925002.html,parison to linear analyses

We have seen above that the torques on the planet can include some contribution from locations far from resonance locations,and torques from near both the corotation and Lindblad resonances.Cursory examination of?gures6–8suggest that some characteristics predicted by theory for those resonances may not be identically reproduced in the simulations(e.g.we would expect from equation7that the torque near CR would have the same sign over the entire radial range where it is exerted,but this is rarely the case). Nevertheless,we will proceed by identifying the torques from the region around each resonance location,

with the interaction predicted theoretically for that resonance.Separating these torques from each other in our simulations however,proves to be a challenging problem because for higher order m patterns,the Lindblad resonances are found progressively closer to the planet.

For the lowest m patterns,a clear separation exists between the CR and LR torques because their positions are well separated from each other.Therefore,for the purposes of our study,we shall de?ne the LR torques as those exerted in the region between r=∞(r=0)and the point half the distance between the Lindblad resonance and the corotation resonance for the OLR and ILR torques respectively.This allows us to include the portion of the torque which may be produced at some distance from the resonance itself,but nevertheless is clearly associated with it.We arbitrarily assign the di?erence between the total torque and the sum of the two LR contributions to the corotation resonance.For higher m patterns(above m=15), where all three resonance locations are very close together and it becomes impossible to distinguish between the torques from the LRs from the corotation torque,we shall assign the entire torque to the LRs.

3.2.1.Torques from near the Lindblad Resonances

The contributions to the torque from each Fourier component and due to the Lindblad resonances are shown3in?gure9and10for the models with and without disk self gravity.We also show the torques broken down to show the contribution from inside and outside the planet’s orbit.Di?erences exist,but in general, many qualitative features obtained for the di?erent runs are similar to each other.In each simulation,the dominant contribution to the torques comes from the spiral patterns with10

While the inner(positive)torques are quite similar to each other in all four simulations,the outer (negative)torques are larger in the non self gravitating disks than in the self gravitating versions,especially for the higher m patterns.This is consistent with the result that planet migration proceeds more rapidly in absence of disk self gravity.In the high resolution versions,there is more variation between the torques of individual patterns,probably because of the improved ability to distinguish them from the CR torques.

Also shown in?gures9and10are the torques predicted from equation4.To derive these torques,we use the azimuth averaged surface density and rotation curves of each simulation as inputs and include the generalized resonant denominator of equations8or9both to determine the resonance positions and in the torque formula itself.We also use the generalized Laplace coe?cients of equation11,with the same softening as is used in the simulations.Some qualitative features are reproduced well by the theoretical calculations. In each case,the torques are small for both small and large m,with maxima in both the net and inner and outer torque contributions near m=10.The positive net torques near m～5are also reproduced.

There are also several very serious di?erences between the torques from theory and from simulation.The most signi?cant is that in each case,the torques from theory are systematically larger by as much as a factor of six,than those from the simulations.Secondly,the torques in the non-self gravitating versions are larger than those in the corresponding self gravitating version.In three of four simulations,the torque distributions

from theory are smoother as functions of m in the sense that variations between nearby m patterns are small. The exception is the high resolution self gravitating model,which produces a large,negative net torque at m=8and a large increase in both the inner and outer torques near m=10,gradually decreasing as m increases.

3.2.2.Torques from near the Corotation Resonances

Figure11shows the total torques on the planet from the corotation resonances with m≤15.In each simulation,the most striking feature is that many patterns produce torques that are as large or larger than the net contribution from the LRs.The contributions for the patterns with m 3are particularly interesting because not only are they large,they are also positive,thus acting to slow the inward migration of the planet.For patterns with m>3,the torques are negative and again signi?cant in magnitude compared to the torques from the LRs.Only for patterns with m>10,do the torques decrease to near zero and only when disk self gravity is present.The net torque from the CRs with m≤15is about a third the value of the torque from the LRs.

After the simple observation that many of the CR torque components are large,we are confronted with the uncomfortable situation that large di?erences that exist between the results of each of the simulations, including di?erences between the simulations which di?er only in resolution.The most signi?cant di?erences are found in the magnitude of the torque from the m=1pattern.It is larger when self gravity is not included and it increases by nearly a factor of two when we increase from our standard to high resolution.

In both the standard and high resolution versions without self gravity,the torque from m=1was so much larger that it could not be displayed on the plot while also displaying features from m>1.The same was true in?gure8for the radial distribution of the torque.More important to note than its exact value is that the contribution is more than three times the largest one sided LR torque(i.e.ILR or OLR).It also has no counterbalancing torque of opposite sign as the LRs do,making its contribution very important in a determination of the migration rate.Further,for the high resolution simulations(but not the standard resolution models),the torque is large and positive when disk self gravity is included,but large and negative when it is suppressed.

Higher order m patterns also display di?erences.While the standard resolution simulations have near zero CR torques,both of the high resolution counterparts have large negative torques.In this case,both high resolution simulations produce negative torques for5

Because of the large and qualitative di?erences between the simulations,we are forced to conclude that we have not adequately resolved the e?ect that the CR torques will have.We have therefore not attempted to make a direct comparison of the CR torques with their analytic predictions,as we have for the LRs.While we cannot make reasonable comparisons with theory,we may still draw important physical conclusions from the large size of the CR https://www.sodocs.net/doc/5110925002.html,ly,that they may have a much larger e?ect on migration than previously realized.In order to constrain this possibility further,extremely high resolution global simulations of disks during the Type I migration stage must be performed.

3.3.Examination of the Assumptions and Approximations Made in Linear Analyses

What is the origin of the serious di?erences between theory and simulation,and between one simulation and another?In this section,we will examine the e?ects of the variation of the rotation curve on the torque and small shifts in the resonance locations may have on the calculated torques,and in the following section, their signi?cance for the torques.Such shifts may be global in nature,for example a shift in the initial state due to self gravity or pressure forces.Variation in the rotation curve may also be more local in nature,for example the changes in the rotation curve due to steep pressure gradients at the edges of the forming gap. Except for the discussion of linearity in section3.3.2,in the following sections we will concentrate speci?cally on our high resolution prototype simulation(with and without self gravity)so that these small shifts may be determined more precisely.

3.3.1.Validity of the resonance approximation

The wavelike behavior of the lowest order patterns is a consequence of the fact that at some radii, the spiral patterns have a di?erent phase relative to the planet than at other radii.The torque exerted on the planet by the disk matter at that radius is therefore positive or negative depending on this phase. The important points to note are that the torque is exerted at locations signi?cantly di?erent from the Lindblad resonance and that the sign of this local torque can be opposite that of the prevailing?ow of angular momentum.

How much do the torques exerted far from the resonances contribute to the total?This question is important because one simplifying assumption made in deriving equations4and7is that the disk and planet interact only at the resonances,rather than at some distance away from it.Errors in the comparison will enter if torques far from resonances contribute a signi?cant fraction of the total.

Based on inspection of?gures7and8,the simulation most strongly a?ected by torques far from the res-onance will be the non self gravitating,high resolution version because its torques have the largest amplitude oscillations everywhere.For this model,?gure12shows show the cumulative sum of the torque as a function of distance from the planet(both inward and outward)originating from the Lindblad resonances,Torques from corotation interactions are suppressed.While some errors are made due to incorrect separation of the CR and LR contributions,in every case the cumulative torque magnitude increases to its maximum value slightly outside or inside the exact resonance position for the OLR or ILR respectively.At greater distances, it undergoes decreasing amplitude oscillations around what becomes its?nal value at large distances from the planet.In every case the oscillations are smaller in amplitude than in the?rst1/2cycle of the wave. Therefore,the largest fraction of the net torque from a given resonance is indeed derived from near the resonance position itself.

The maximum cumulative torque is never obtained at the resonance,but rather slightly further away from the planet,after which the cumulative sum falls to a value as much as30–50%below its initial maximum. This is to be expected since the torque density waveform can be approximated as an Airy function(Ward 1986)whose?rst maximum is found about1/4cycle more distant from the planet than the resonance.The correspondence is also strengthened by the fact that the integral(i.e cumulative sum)of the Airy function also drops nearly40%from its initial maximum during the next1/2cycle of the wave.

A useful visual diagnostic for comparison between patterns is the position of the?rst maximum of the cumulative torque relative to the resonance https://www.sodocs.net/doc/5110925002.html,ing this measure,there is variation between one

pattern and another.For the m=2pattern for example,the ILR position is at approximately the half maximum,while the OLR position,even accounting for the positive o?set from misattributing a portion of the CR torque to the LR,is found nearly at the‘foot’of the wave.Variation similar to that shown are present in all other patterns as well.

We conclude that the resonance approximation of GT79is partially supported by our simulations for at least the Lindblad resonances,since most of the torque between the planet and the disk originates from the ?rst cycle of the wave.Our method of separating the CR and LR torques already constrains the CR torques to a narrow region,so we can make no statements about these contributions.The speci?c approximation that the conditions at the exact resonance position can be used to determine the torque may not be as well justi?ed,because the maximum cumulative sum may be found at varying distances from the exact resonance.

3.3.2.Linearity

In Paper I we found that the migration rates were nearly?at as a function of planet mass,varying by less than a factor of two over a factor20change in mass.In contrast,theory predicts that the rates will scale linearly with planet mass.The theory is based on the assumption that the perturbations are small,so that the models are in the linear regime.In this limit,we expect the perturbations to scale with the mass of the perturber,and through them also the migration rate.If instead perturbations are large,they will saturate–larger perturbers will not produce larger perturbations.

Are our migration rates?at because the perturbation amplitudes are saturated?In?gure13,we show the maximum pattern amplitudes as a function of mass(i.e.for the mas series of simulations–see Paper I ),obtained in the regions de?ned for each of the three resonances,each obtained at a time300yr after the beginning of the run.We again show the m=1,2and10patterns as typical representatives of the behavior of each of the other low and high order symmetry patterns.

The m=1and m=2pattern amplitudes may reach as high as40–50%.Even with these very large amplitudes,a close correlation between the relative amplitudes of each of the three resonances with each other is maintained.Although we have not attempted to?t linear functions to the data,a clear linear increase is present over most of the mass range.For example,the amplitude near the m=1OLR is～2%at 0.1M J and increases by a factor ten to～20%as the planet mass increase by a factor10to1.0M J,a perfect linear dependence.The direct proportionality continues even to the2M J simulation with an amplitude20 times that found at0.1M J.The maximum amplitude near corotation displays similar characteristics.For the m=2pattern,the maximum amplitudes near the ILR and CR increase in direct proportion to the planet mass,but the amplitude near the OLR saturates at～20?25%for simulations with planets above 1M J.

On the other hand,higher order patterns typi?ed by m=10produce much di?erent behavior.The amplitudes are not as large as for the m=1,2patterns,and they no longer increase in linear fashion over the whole range.Instead,above～0.75M J,the close correlation between the amplitudes for each of the three resonances is lost and the growth with planet mass appears to have saturated.Even below0.75M J the increase is no longer directly proportional to planet mass.For example,a2%perturbation increasing only a factor four with a factor7.5increase in planet mass.The mass at which saturation occurs is also similar to that for which the migration rate shown in?gure9of Paper I undergoes its only real change in its behavior,from a slowly growing rate,to a completely?at function of planet mass.We conclude that the reason for this change is the saturation of the pattern amplitudes for the patterns most strongly a?ecting

the migration.

At the low end of the plot of mass vs.migration rate from Paper I(?gure9),an extrapolation of the migration rates to zero planet mass yields a non-zero rate.Such a phenomenon would appear to be either inconsistent with the view that the torques are dominated by Lindblad resonance interactions in the linear regime,or with the view that gravitational torques are responsible for the migration(since a zero mass planet would not generate such torques).Neither alternative can be fully supported.Instead,we favor the view that the simulations of the lowest mass planets(below0.3?0.5M J)are in the linear regime,but that the true linear increase in the migration rate is not observed because of the positive torques due to low order m corotation resonances,and perhaps because of mixing between the torques from the CR and LRs of a given pattern(see discussion in section3.4.2below).More accurate quanti?cation of the relative strength of the two sources of torques(from the CR and the LRs),must be done in order to determine the true migration rate of a planet.

3.3.3.The epicyclic frequency

In the theory of circumstellar disks,perhaps the two most critical parameters describing the problem are the natural resonance frequency of the system(the epicyclic frequency,κ(r))and the driving frequency (the planet’s orbit frequency,?pl).The latter is very well determined since the planet is a single object. The former is more di?cult due to the important e?ects of pressure and self gravity on the rotation pro?le. Nevertheless,a widely implemented approximation in theory is that the di?erences from true Keplerian orbits are small,so that the identityκ(r)=?(r)is approximately held.To what extent does the epicyclic frequency deviate from equality with the orbital frequency?

In?gure14,we show the ratio of the epicyclic frequency at each location in the disk to the orbital frequency?for the high resolution prototype simulations,with and without self gravity.The variation ofκis as large as20%above and10%below the orbital frequency at each radius in the disk for the self gravitating disk,but only～10%in the non-self gravitating disk.In the latter case,no gap is able to form due to the very rapid migration of the planet.In the former,the largest variation occurs about2Hill radii inward and outward from the planet,which corresponds to the positions of local minima in the surface density distribution in the nascent gap.Variation>5%extends to a distance of2AU inwards and outwards of the planet and drops to a～2%o?set(due to the e?ects of pressure gradients and self gravity on the rotation curve)at locations further inward and outwards.Thus,although the equality between the two quantities is not exact,di?erences are larger than20%nowhere in the disk and are10%or less in most regions.

Di?erences of this magnitude remain important because as Ward(1997)and others have pointed out, the e?ect on the epicyclic frequency may be small but the e?ect on the gravitational torque su?ered by the planet can be very large.Modifying the epicyclic frequency at each orbit radius will cause two separate modi?cations of the torque.First,the resonance positions will deviate from their Keplerian locations,and this change in position will a?ect the value of the gravitational potentials calculated at the resonance position. Second,the resonant denominator(for the LRs)and the Oort constant,B,(for the CR)may change their values from that predicted in an unperturbed disk.Are the di?erences in the epicyclic frequency from the orbital frequency large enough to a?ect strongly the values of these quantities?

3.3.

4.The in?uence of the true rotation curve on the resonance positions

In?gure15,we show the resonance positions for all three resonances(ILR,CR and OLR)as a function of m,for the high resolution prototypes with and without self gravity.In only a few cases,do the true resonance positions correspond to the ideal Keplerian values.With self gravity,the CR positions are nearly coincident with the planet’s orbit,while without it,they shift inward.At the same time,the LR positions are shifted due to the additional terms in the de?nition of the resonant https://www.sodocs.net/doc/5110925002.html,rge m resonance positions follow closely the bu?er region(of size～2H/3)around the planet expected from the analysis of Artymowicz (1993a).Without self gravity the inner resonances shift inward by～1%,but the outer resonances remain largely una?ected because they are limited by the bu?er region.For smaller m,the positions follow the ideal Keplerian positions more closely,but remain modi?ed,especially for patterns m～10.For these patterns,the resonance positions fall near the edges of the forming gap,which means that they will be disproportionately a?ected by large pressure gradients there.

If we follow instead the GT79analysis and use the original de?nition of D of equation5,small deviations from the Keplerian values are present both above and below the Keplerian values including self gravity,but in general the correspondence is quite close.Without self gravity,each of the resonances are systematically shifted inward by about1%of the semi-major axis,which corresponds to about20%of the Hill radius.The inner resonances are systematically shifted further away from the planet,while the outer resonances are shifted closer.Neither example displays the bu?er zone expected from the use of equations8or9.

3.3.5.The in?uence of the true rotation curve on|rdD/dr|L.

The quantity,D,in equation4,de?nes the strength and shape of the response of the disk to the perturbation from the planet.In combination with the torque cuto?function(equation6),it determines the disk’s response to the planet.In the interests of mathematical tractability,it is usually approximated by the?rst term in a Taylor series as D≈(rdD/dr)r L x,where x=(r?r L)/r L,an approximation that in

turn is often further approximated4as rdD/dr|r L≈?3(1?m)?2r

L ≈±3m?2r

L

,which is equivalent to the

statement that the disk is Keplerian(i.e.κ=?).To what extent are these approximations valid after the evolution begins?

The quality of the approximation may be measured by the ratio of the approximate value to the‘real’value obtained numerically from our simulations.A high value of the ratio means that more torques are produced for that pattern in the simulation compared to what would be predicted using the approximation. In?gure16,we show this ratio for both the inner and outer Lindblad resonances.For the self gravitating disk simulation,the approximation varies by up to a factor～2.3above and below the numerically obtained value for patterns with m 15.Very sharp peaks exist in both the OLR and ILR ratios,coming at m=8 and m=10for the OLRs and ILRs respectively.For the ILRs,the peak is broader and extends to m=15. For lower m,the ratio drops below unity,indicating that the simulation produces less torque than expected. Above m=15and for both the ILR and OLR ratios,the approximation and the simulation produce very similar values,indicating that the approximation may be used without large errors in the torque.

In contrast,the non-self gravitating disk model shows di?erences only for the lowest m patterns.For m>10only relatively small deviations of order a few percent are present and no large peaks are present for

any pattern although,as in the self gravitating version,the ratio increases to as high as1.3and as low as 0.8for the lowest m patterns.

The patterns for which the approximation fails most severely(the ratio peaks)are those most sensitive to the structure of a gap.Their resonance positions tend to fall near the gap edges,both inside and outside the planet.Since the planet is nearer to the inner edge of the forming gap,more inner patterns,of higher order,are a?ected by its presence.For higher order patterns,where the resonance position falls well within the forming gap,the approximate form reproduces the simulation value to within a few percent.For the non-self gravitating simulation,the migration was so rapid that gap formation did not occur.The resonant denominator was therefore not strongly a?ected and the variations remained small.

3.3.6.The variation ofΣ/B

The strength of the CRs is proportional to the gradient ofΣ/B at the resonance,as indicated in equation 7.This quantity measures asymmetries in the circulation of matter as it travels on horseshoe orbits in the planet’s vicinity(Ward1991).For a Keplerian disk,the Oort constant reduces to B=?/4∝r?3/2.Since our assumed surface density is also a power law with the same proportionality,we expect that the value of Σ/B will be constant,and its gradient zero,making the CR torques insigni?cant.Clearly,torques from near the CR exist in our simulations.Is their origin due to the failure of this condition?

Figure17shows the ratio ofΣ/B to its initial value as a function of radius in the region near the CR. The slope of this quantity presents a fair measure of the variation(i.e.the gradient)inΣ/B itself since the initial value is constant.In the self gravitating simulation,Σ/B decreases more than a factor two below its initial value within～2R H both interior to and exterior to the planet,while remaining nearly unchanged at the exact CR.The non self gravitating version displays variation of some30%below its initial value and over a wider radial extent.The pattern is also o?set relative to that seen with self gravity.While CR appears near a local maximum ofΣ/B with self gravity,it appears near a local minimum without it.We speculate that this fact may simply be due to the very rapid migration in the latter case,so that the planet simply outruns the disk’s ability to keep up.

While at the exact CR,the slope may indeed be relatively small,we recall the physical phenomenon of circulation on horseshoe orbits that gives rise to the CR torques in the?rst place.In that context,the conditions over the radial extent of the horseshoe orbit become important,and using the exponential term in equation7as our guide,we may associate the width of the resonance with the disk scale height.In both simulations,theΣ/B varies over a radial scale of2–4disk scale heights and over this range,the size of variation is as large as(or larger than)a factor of half its magnitude and takes both positive and negative sign.We conclude that evolution modi?es the surface density and rotation pro?les of the disk enough to lead to very large gradients inΣ/B,and will therefore also lead to signi?cant CR torques.

3.4.The sensitivity of the torques to the various physical and numerical approximations

Figures14–17show that variations from the physical assumptions underlying the analytic derivations for the gravitational torques do occur.Each of the quantities probes a di?erent aspect of the interaction, so that we can regard them as acting independently and test the sensitivity of the torque to each in turn. Which,if any,of the variations are important for correct evaluation of the torque?Similarly,which,if any,

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恰恰相反，非常刺激。 https://www.sodocs.net/doc/5110925002.html, But onthecontrary, lazy. 却恰恰相反，懒洋洋的。 https://www.sodocs.net/doc/5110925002.html, Onthecontrary, I hate it! 恰恰相反，我不喜欢！ https://www.sodocs.net/doc/5110925002.html, Onthecontrary, the club gathers every month. 相反，俱乐部每个月都聚会。 https://www.sodocs.net/doc/5110925002.html, Onthecontrary, I'm going to work harder. 我反而将更努力工作。 https://www.sodocs.net/doc/5110925002.html, Onthecontrary, his demeanor is easy and nonchalant. 相反，他的举止轻松而无动于衷。 https://www.sodocs.net/doc/5110925002.html, Too much nutrition onthecontrary can not be absorbed through skin. 太过营养了反而皮肤吸收不了. https://www.sodocs.net/doc/5110925002.html, Onthecontrary, I would wish for it no other way. 正相反，我正希望这样 Provided by jukuu Onthecontrary most likely pathological. 反之很有可能是病理性的。 https://www.sodocs.net/doc/5110925002.html, Onthecontrary, it will appear clumsy. 反之，就会显得粗笨。 https://www.sodocs.net/doc/5110925002.html,

英语造句

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base on的例句

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