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K. D. Kanagawa, H. Tanaka, T. Muto, T. Tanigawa, T. Takeuchi, Formation of a disc gap induced by a planet: effect of the deviation from Keplerian disc rotation, Monthly Notices of the Royal Astronomical Society, Volume 448, Issue 1, 21 March 2015, Pages 994–1006, https://doi.org/10.1093/mnras/stv025
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Abstract
The gap formation induced by a giant planet is important in the evolution of the planet and the protoplanetary disc. We examine the gap formation by a planet with a new formulation of one-dimensional viscous discs which takes into account the deviation from Keplerian disc rotation due to the steep gradient of the surface density. This formulation enables us to naturally include the Rayleigh stable condition for the disc rotation. It is found that the derivation from Keplerian disc rotation promotes the radial angular momentum transfer and makes the gap shallower than in the Keplerian case. For deep gaps, this shallowing effect becomes significant due to the Rayleigh condition. In our model, we also take into account the propagation of the density waves excited by the planet, which widens the range of the angular momentum deposition to the disc. The effect of the wave propagation makes the gap wider and shallower than the case with instantaneous wave damping. With these shallowing effects, our one-dimensional gap model is consistent with the recent hydrodynamic simulations.
1 INTRODUCTION
A planet in a protoplanetary disc gravitationally interacts with the disc and exerts a torque on it. The torque exerted by the planet dispels the surrounding gas and forms a disc gap along the orbit of the planet (Lin & Papaloizou 1979; Goldreich & Tremaine 1980). However, a gas flow into the gap is also caused by viscous diffusion and hence the gap depth is determined by the balance between the planetary torque and the viscous diffusion. Accordingly, only a large planet can create a deep gap (Lin & Papaloizou 1993; Takeuchi, Miyama & Lin 1996; Ward 1997; Rafikov 2002; Crida, Morbidelli & Masset 2006).
The gap formation strongly influences the evolution of both the planet and the protoplanetary disc in various ways. For example, a deep gap prevents disc gas from accreting on to the planet and slows down the planet growth (D'Angelo, Henning & Kley 2002; Bate et al. 2003; Tanigawa & Ikoma 2007), and also changes the planetary migration from the type I to the slower type II (Lin & Papaloizou 1986; Ward 1997). Furthermore, a sufficiently deep gap inhibits gas flow across the gap (Artymowicz & Lubow 1996; Kley 1999; Lubow, Seibert & Artymowicz 1999), which is a possible mechanism for forming an inner hole in the disc (Dodson-Robinson & Salyk 2011; Zhu et al. 2011).
Because of their importance, disc gaps induced by planets have been studied by many authors, using simple one-dimensional disc models (e.g. Takeuchi et al. 1996; Ward 1997; Crida et al. 2006; Lubow & D'Angelo 2006) and numerical hydrodynamic simulations (Artymowicz & Lubow 1994; Kley 1999; Varnière, Quillen & Frank 2004; Duffell & MacFadyen 2013; Fung, Shi & Chiang 2014). One-dimensional disc models predict an exponential dependence of the gap depth. That is, the minimum surface density at the gap bottom is proportional to exp [−A(Mp/M*)2], where Mp and M* are the masses of the planet and the central star, and A is a non-dimensional parameter (see also equation 38). On the other hand, recent high-resolution hydrodynamic simulations done by Duffell & MacFadyen (2013, hereafter DM13) show that the gap is much shallower for a massive planet than the prediction of one-dimensional models. According to their results, the minimum surface density at the gap is proportional to (Mp/M*)−2. Varnière et al. (2004) and Fung et al. (2014) obtained similar results from their hydrodynamic simulations. Its origin has not yet been clarified by the one-dimensional disc model. Fung et al. (2014) also estimated the gap depth with a ‘zero-dimensional’ analytic model, by simply assuming that the planetary gravitational torque is produced only at the gap bottom. Their simple model succeeds in explaining the dependence of the minimum surface density of ∝ (Mp/M*)−2. However, the zero-dimensional model does not give the radial profile of the surface density (or the width of the gap). It is not well understood what kind of profile accepts their assumption on the planetary torque. Further development of the one-dimensional gap model is required in order to clarify both the gap depth and width. Such a model enables us to connect the gaps observed in protoplanetary discs with the embedded planets.
One of the problems of the one-dimensional disc model is the assumption of the Keplerian rotational speed. The disc rotation deviates from the Keplerian speed due to a radial pressure gradient (Adachi, Hayashi & Nakazawa 1976). When a planet creates a deep gap, the steep surface density gradient increases the deviation of the disc rotation significantly, which affects the angular momentum transfer at the gap (see Sections 2.1 and 2.2). Furthermore, a large deviation of the disc rotation can also violate the Rayleigh stable condition for rotating discs (Chandrasekhar 1939). A violation of the Rayleigh condition promotes the angular momentum transfer and makes the surface density gradient shallower so that the Rayleigh condition is only marginally satisfied (Tanigawa & Ikoma 2007; Yang & Menou 2010). To examine such feedback on the surface density gradient, we should naturally include the deviation from the Keplerian disc rotation in the one-dimensional disc model.
Another simplification is in the wave propagation at the disc–planet interaction. The density waves excited by planets radially propagate in the disc and the angular momenta of the waves are deposited on the disc by damping. This angular momentum deposition is the direct cause of the gap formation. Most previous studies simply assume instantaneous damping of the density waves after their excitation (e.g. Ward 1997; Crida et al. 2006). If the wave propagation is taken into account, the angular momentum is deposited in a wider region of the disc, which increases the width of the gap (Takeuchi et al. 1996; Rafikov 2002). In a wide gap, the disc–planet interaction would be weak because the disc gas around the planet decreases over a wide region. Hence, we cannot neglect the effect of wave propagation on the gap formation.
In this paper, we re-examine the gap formation by a planet with the one-dimensional disc model, taking into account the deviation from Keplerian rotation and the effect of wave propagation. To include the deviation from Keplerian disc rotation, we modify the basic equations for one-dimensional accretion discs, detailed in the next section. The effect of the wave propagation is included using a simple model. In Section 3, we obtain estimates of gap depths for two simple cases. One estimate for a wide gap corresponds to the zero-dimensional model proposed by Fung et al. (2014). In Sections 4 and 5, we present numerical solutions of the gap without and with wave propagation, respectively. We find that the gap becomes shallow due to the effects of the deviation from Keplerian rotation, the violation of the Rayleigh condition and the wave propagation. With these shallowing effects, our results are consistent with the recent hydrodynamic simulations. In Section 6, we summarize and discuss our results.
2 MODEL AND BASIC EQUATIONS
We examine an axisymmetric gap in the disc surface density around a planet by using the one-dimensional model of viscous accretion discs. Although the Keplerian angular velocity is assumed in most previous studies, we take into account a deviation from Keplerian disc rotation in our one-dimensional model. The deviation cannot be neglected for a deep gap, as will be shown below. We also assume non-self-gravitating and geometrical thin discs. For simplicity, the planet is assumed to be in a circular orbit. We also adopt simple models for density wave excitation and damping to describe the gap formation.
2.1 Angular velocity of a protoplanetary disc with a gap
2.2 Basic equations describing a disc gap around a planet
To describe the deposition rate Λd, we consider the angular momentum transfer from the planet to the disc. This transfer process can be divided into two steps. First, the planet excites a density wave by the gravitational interaction with the disc (e.g. Goldreich & Tremaine 1980). Secondly, the density waves are gradually damped due to the disc viscosity or a non-linear effect (Takeuchi et al. 1996; Goodman & Rafikov 2001). As a result of the wave damping, the angular momenta of the waves are deposited on the disc. If instantaneous wave damping is assumed, the deposition rate Λd is determined only by the wave excitation. In Section 2.4.1, we will describe the deposition rate for the case with instantaneous wave damping. In Section 2.4.2, we will give a simple model of Λd for the case of gradual wave damping.
Equations (5)–(7) describe the time evolution of the three variables Σ, FM and FJ with the given mass source term SM, the angular momentum deposition rate from a planet Λd and the disc angular velocity Ω. Note that Ω depends on ∂Σ/∂R, as in equations (2) and (3).
Next, we consider the disc gap in a steady state (∂/∂t = 0). The time-scale for the formation of a steady gap is approximately equal to the diffusion time within the gap width, tdiff = h2/ν. For a nominal value of α (∼10−3), the diffusion time is roughly given by 103 Keplerian periods, which is shorter than the growth time of planets (105–7 yr; Kokubo & Ida 2000, 2002) or the lifetime of protoplanetary discs (106–7 yr; Haisch, Lada & Lada 2001). Hence, the assumption of a steady gap would be valid. In addition, we assume SM = 0 for simplicity. Although gas accretion on to the planet occurs for Mp ≳ 10 M⊕ (Mizuno 1980; Kanagawa & Fujimoto 2013), the assumption of SM = 0 would be valid if the accretion rate on to the planet is smaller than the radial disc accretion rate, FM.
2.3 Rayleigh condition
For a deep gap around a large planet, the derivative of the angular velocity deviates significantly from the Keplerian velocity, as shown in Section 2.1. A sufficiently large deviation in Ω violates the so-called Rayleigh stable condition of dj/dR ≥ 0 (see Chandrasekhar 1961). Such a steep gap is dynamically unstable, which would cause a strong angular momentum transfer, lessening the steepness of the gap. This would make the unstable region marginally stable (i.e. dj/dR = 0).
Actually, around a sufficiently large planet, equation (9) gives |$h_{\rm p}^2 \mathrm{d}^2\ln \Sigma /\mathrm{d} R^2 < -1$| in some radial regions. In such unstable regions, we have to use equation (14) instead of equation (9).
The Rossby wave instability may be important for the gap formation (e.g. Richard, Barge & Le Dizès 2013; Zhu, Stone & Rafikov 2013; Lin 2014). As well as the Rayleigh condition, the Rossby wave instability relates to the disc rotation (Li et al. 2000). Because it can occur before the Rayleigh condition is violated, however, the Rossby wave instability may suppress the surface density gradient more than the Rayleigh condition. For simplicity, we include only the Rayleigh condition in the present study. A further detail treatment including the Rossby wave instability should be done in future works.
2.4 Angular momentum deposition from a planet
In the disc–planet interaction, a planet excites density waves and the angular momenta of the waves are deposited on the disc through their damping. The angular momentum deposition rate Λd is determined by the later process. First, we will consider the deposition rate Λd in the case with instantaneous wave damping. In this case, the deposition rate is governed only by the wave excitation. Next, taking into account the wave propagation before damping, we will model the deposition rate in a simple form.
2.4.1 Case with instantaneous wave damping
Note that the WKB formula is derived for discs with no gap. Petrovich & Rafikov (2012) reported that the torque density is altered by the steep gradient of the surface density because of the shift of the Lindblad resonances. For simplicity, however, we ignore this effect in this paper. Hence, in our model, the excitation torque density Λex is simply proportional to the disc surface density at R, Σ(R), and is independent of the surface density gradient even for deep gaps. For a large planet with a mass of Mp/M* ≳ (hp/Rp)3; furthermore, the non-linear effect would not be negligible for wave excitation (Ward 1997; Miyoshi et al. 1999). This non-linear effect is also neglected in our simple model.
2.4.2 Case with wave propagation
When wave propagation is included, the angular momentum deposition occurs at a different site from the wave excitation and equation (18) is not valid. In this case, the angular momentum deposition is also governed by the damping of the waves. Although the wave damping has been examined in previous studies (e.g. Korycansky & Papaloizou 1996; Takeuchi et al. 1996; Goodman & Rafikov 2001), it is not clear yet how the density waves are damped in a disc with deep gaps. In the present study, therefore, we adopt a simple model of angular momentum deposition, described below.
In the case with wave propagation, we use equations (20) and (21) to obtain the gap structure with equation (9). It should be noted that T in equation (20) depends on the surface density distribution through the definition of equation (19), because Λex is proportional to Σ. These coupled equations are solved as follows. First, we obtain the surface density distribution with equation (9) for a given T. Next, we determine the corresponding mass of the planet from equation (19), using the obtained surface density.
2.5 Local approximation and non-dimensional equations
3 ESTIMATES OF GAP DEPTHS FOR SIMPLE SITUATIONS
3.1 Case of the Keplerian discs
For a very large K, the Rayleigh condition is violated and equations (37) and (38) are invalid. Tanigawa & Ikoma (2007) obtained the gap structure in Keplerian discs, including the Rayleigh condition. Their solution is described in Appendix A. In Appendix B, we also derive gap solutions in Keplerian discs, taking into account the wave propagation with the simple model of equations (20) and (21).
3.2 Case of the wide-limit gap
4 GAP STRUCTURE IN THE CASE WITH INSTANTANEOUS WAVE DAMPING
4.1 Linear solutions for shallow gaps
Here, we present the numerical solution of the gap in the case with instantaneous wave damping (i.e. λd = λex).
Fig. 1(a) shows y, which can be converted into the surface density s by equation (43). In these shallow gaps, the gap depth is almost the same as for the Keplerian case, though our model gives a smooth surface density distribution.
4.2 Non-linear solutions for deep gaps
Next, we consider deep gaps around relatively large planets. In this case, we numerically solve the non-linear equation (27) with the Rayleigh condition. We call the obtained non-linear solution the ‘exact’ solution.
At regions far from the planet, the surface density perturbation is rather small and the linear approximation is valid. Thus, we adopt a linear solution at |x| > 10. Note that this linear solution has different coefficients for the homogeneous terms from those in Section 4.1 (see Appendix C). The coefficients of the homogeneous solution are given to satisfy the boundary conditions of equation (34). At |x| ≤ 10, we integrate equation (27) with the fourth-order Runge–Kutta integrator. In the Rayleigh unstable region, the surface density is governed by the marginally stable condition (equation 29), instead of equation (27). Fig. 2 shows the surface density distributions of the exact solutions for K = 50 (a) and 200 (b). If we assume a disc with hp/Rp = 0.05 and α = 10−3, these cases correspond to Mp = 1/8MJ and 1/4MJ, respectively, where MJ is the mass of Jupiter. For comparisons, the Keplerian solution (equation 37) and the linear solution with equation (43) are also plotted. For K = 50, the linear solution almost agrees with the exact solution, while it is much deeper than the exact solution for K = 200. For K = 200, the Keplerian solution has a much smaller smin than the exact solution. Fig. 3 illustrates the angular velocities (a) and specific angular momenta (b) for the exact solutions for K = 50 and 200. Similar to the linear solution in Fig. 1, the shear motion is enhanced at |x| ≳ 1.4. This enhancement of the shear motion is also seen in Fig. 4. The enhancement promotes the angular momentum transfer and makes the surface density gradient less steep. For K = 200, the Rayleigh condition is violated. In the unstable region, the marginally stable condition further reduces the surface density gradient. This makes the gap much shallower than for the Keplerian solution, as seen in Fig. 2(b). We also plot the minimum surface densities, smin , of the wide-limit gap (equation 41) in Fig. 2. The wide-limit gap gives a much larger smin than the exact solution for K = 200. In the wide-limit gap, it is assumed that the density waves are excited only at the gap bottom with s ≃ smin . Fig. 5 shows the excitation torque density given by equation (18) for the exact solution with K = 200. This torque density indicates that the waves are excited mainly in the region with s > 10smin . Thus, the assumption of wave excitation at the gap bottom is not valid in this case. Since wave excitation with a larger s increases the one-sided torque, this can explain why the gap of the exact solution is much deeper than the wide-limit gap in Fig. 2. Note that this result for the wave excitation is obtained in the case of instantaneous wave damping. The effect of the wave propagation can change the gap width and the mode of wave excitation, as seen in the next section.
4.3 Effect of the Rayleigh condition
We further examine the effect of the Rayleigh condition on the gap structure. Fig. 6 shows the surface densities (a) and specific angular momenta (b) for the exact solution and the solution without the Rayleigh condition. The solution without the Rayleigh condition has unstable regions with dj/dx < 0 (i.e. 1.4 < |x| < 3.1). This comparison between these two solutions directly shows how the Rayleigh condition changes the gap structure. The Rayleigh condition increases smin by a factor of 6 for K = 200. This is because the marginal condition of d2ln s/dx2 ≥ −1 keeps the surface density gradient less steep and makes the gap shallow.
Fig. 7 shows νeff in the unstable region for K = 200. The effective viscosity is twice as large as the original value at x = 1.8. This enhancement of the effective viscosity causes the shallowing effect in Fig. 6(a).
In Fig. 6(a), we also plot the surface density distribution given by Tanigawa & Ikoma (2007, hereafter TI07), in which the Rayleigh condition is taken into account (for details, see Appendix A). Their model gives a shallower gap than our exact solution. This is because a very steep surface density gradient in the Keplerian solution is suppressed by the Rayleigh condition to a greater extent than in our model.
We also show that the Keplerian solution by TI07 does not satisfy the angular momentum conservation. The Keplerian solution without the Rayleigh condition (equation 37) is derived just from equation (35) (or equation 27), which is originated from equation (9). In this solution, thus, the angular momentum conservation is satisfied. However, when the Rayleigh condition is violated, the marginal stable condition (equation 29) is used instead of equation (37). Because of this, the surface density at the flat-bottom of TI07's solution does not satisfy equation (35) or the angular momentum conservation, either. This violation is resolved in our formulation because our exact solution always satisfies equation (27) outside the Rayleigh unstable region.3
4.4 Gap depth
Fig. 8 shows the minimum surface densities, smin , as a function of K for the exact solutions. For comparison, we also plot smin for the solutions without the Rayleigh condition and the Keplerian solutions. These solutions give deeper gaps than the exact solution, similar to the result of Section 4.2. It is found that the shallowing effect due to the Rayleigh condition becomes significant with an increase in K. This is because the Rayleigh condition is violated more strongly for large K.
In Fig. 8, on the other hand, the exact solution is much deeper than DM13's results and the wide-limit gap, though the latter two cases agree well with each other. The model of TI07 also gives much deeper gaps than DM13. These comparisons indicate that in the case with instantaneous wave damping, our exact solution cannot reproduce the hydrodynamic simulations of DM13. This difference in the gap depth from DM13 is likely to be due to the fact that the assumption of the wide-limit gap is not satisfied in the case with instantaneous wave damping (see Fig. 5). In the next section, we will see that the effect of wave propagation widens the gap and makes the assumption of the wide-limit gap valid.
5 EFFECT OF DENSITY WAVE PROPAGATION
In this section, we consider the effect of wave propagation. Wave propagation changes the radial distribution of the angular momentum deposition. A simple model of angular momentum deposition rate altered by wave propagation is described in Section 2.4.2. Using this simple model, we solve equation (27) with the Rayleigh condition in the similar way to the previous section. At the region far from the planet (i.e. |x| > 10), we use the linear solution to equation (C1) with g(x) = 0 in this case.
5.1 Gap structure for K = 200
Fig. 9 illustrates the surface densities (a) and the excitation torque densities (b) of the exact solutions in the case with wave propagation. The angular momenta of the excited waves are deposited around |x| = xd in our model. A large xd indicates a long propagation length between the excitation and the damping. The parameter K is set to 200. For an increasing xd, the gap becomes wider and shallower. The gap width is directly governed by the position of the angular momentum deposition. For xd = 3 and 4, the gap depths are consistent with the wide-limit gap (and also DM13). For xd = 4, the density waves are excited mainly at the bottom region with s ≃ smin , as seen in Fig. 9(b). Moreover, for xd = 3, a major part of the wave excitation occurs at the bottom. That is, the assumption of the wide-limit gap is almost satisfied for the solutions with xd = 3 and 4. This explains why the gap depths are consistent with the wide-limit gap for these large xd.
It is also valuable to compare the gap width with hydrodynamic simulations. DM13 performed a simulation for the case of Mp = 1/4MJ (2Msh in their notation), α = 10−3 and hp/Rp = 0.05. This case corresponds to K = 200. In this simulation, they found that the gap width is about 6hp, assuming that these gap edges are located at the position with Σ = (1/3)Σ0(Rp) (i.e. s = 1/3). If we adopt the same definition of the gap edge, the gap widths of our exact solutions with xd = 3 and 4 are 6.1hp and 7.7hp, respectively. Hence, if we take into account the wave propagation and adopt xd = 3–4, our exact solution can almost reproduce both of the gap width and depth of the hydrodynamic simulations by DM13, for K = 200.
It should be also noted that, for xd = 2, the wave excitation mainly occurs at |x| > xd (80 per cent of the excitation torques come from this region). However, the deposition site should be farther from the planet than the excitation site because the density waves propagate away from the planet. Thus, the case with xd = 2 does not represent a realistic wave propagation. From now on, we judge that our simple model for the wave propagation is valid if more than half of the one-sided torque arises from the excitation at |x| < xd. In the case with xd = 3 or 4, the excitation at |x| < xd contributes 55 per cent or 78 per cent of the one-sided torque, respectively. In Fig. 10, we check the effect of the width of the deposition site, wd, for xd = 3 and K = 200. It is found that the width wd has only a small influence on the gap structure. We show that the deviation from the Keplerian rotation is also important in the case with wave propagation. In Fig. 11, we plot the solution with the Keplerian rotation and our exact solution. The Keplerian solution is derived from equation (35) with the angular momentum deposition model (equations 20 and 21). When the Rayleigh condition is violated, the marginal stable condition (equation 29) is used. A detail derivation of this solution is described in Appendix B. In the Keplerian solution of Fig. 11, the Rayleigh condition is violated over the whole region of the angular momentum deposition. Then, the minimum surface density is given by equation (B3), which is much larger than our solution and equation (41). Because equation (B3) does not satisfy equation (35), the Keplerian solution does not satisfy the angular momentum conservation, as pointed out in Section 4.3. On the other hand, in the zero-dimensional analysis by Fung et al. (2014, or in equation 41), smin is estimated from a balance between the planetary torque and the viscous angular momentum flux (i.e. from the angular momentum conservation). Because of this difference, the Keplerian solution gives a much shallower gap than the estimation in equation (41). Note that because our exact solutions are given by equation (27), the balance between the planetary torque and the viscous angular momentum flux is always satisfied in our solutions. Hence, our solutions always satisfy the angular momentum conservation and gives a similar smin to equation (41) for a sufficiently wide gap.
5.2 Dependences of the gap depth and width on K
Fig. 12 shows the minimum surface densities smin as a function of the parameter K similar to Fig. 8 but the effect of the wave propagation is included in this figure. In this figure, we also show the dependence on the parameter xd while wd is fixed at hp since wd does not change the surface density distribution much (see Fig. 10). At K = 200, as also seen in Fig. 9, our exact solutions reproduce the gap depth of DM13 (or the wide-limit gap) for xd ≥ 3. At K = 1000, on the other hand, a larger xd (≥6) is required for agreement with DM13. That is, with an increase of K, a large xd is necessary for the values of the depth of the hydrodynamic simulations to agree. Note that the dashed lines in Fig. 12 represent the cases of unrealistic wave propagation, in which more than half of the one-sided torque is due to the excitation at |x| > xd, as for the case of xd = 2 in Fig. 9. At large K, a large xd is also required for realistic wave propagation.
We also show the Keplerian solution with the wave propagation of xd = 6 (see Appendix B). For K > 30, smin is given by equation (B3) and independent of K because of the Rayleigh condition, as seen in Fig. 9. This unrealistic result in the Keplerian solution is related with the violation of the angular momentum conservation, as pointed out in Section 4.3 (and see also Appendix B). To check which xd is preferable, we also compare the gap width with the hydrodynamic simulations. Fig. 13 shows the gap width of the exact solutions as a function of K. Similar to DM13, the gap edge is defined by the position with s = 1/3. In this definition, the gap width is roughly given by twice xdh for our exact solutions with K > 50. Note that this definition is useless for K < 50 because of shallow gaps with smin > 1/3. The dashed lines represent the cases of unrealistic wave propagation, similar to Fig. 12. The results of DM13 and Varnière et al. (2004) are also plotted in fig. 13. Varnière et al. (2004) also performed hydrodynamic simulations of gap formation for Mp/M* = 104–2 × 103, α = 6 × 10−2–6 × 10−5 and hp/Rp = 0.04 (i.e. K = 600–6 × 105). Their gap depths almost agree with DM13's relation. For K < 300, our exact solutions with xd = 3 and 4 agree with the results of DM13 and Varnière et al. (2004), respectively. For K > 300, on the other hand, the widths obtained by Varnière et al. (2004) are wider than those given by DM13. Our exact solution with xd = 6 agrees with the widths of Varnière et al. (2004), while widths of DM13 correspond to our solutions of unrealistic wave propagation. The preferable xd cannot be determined only by this comparison, and we still have a large uncertainty in the preferable value of xd.
The difference of widths between DM13 and Varnière et al. (2004) would be caused by different parameters in their simulations (e.g. the disc viscosity, spatial resolution and width of a computation domain). However, the origin of the difference is still unclear. Note that the results of Kley & Dirksen (2006) and Fung et al. (2014) may support the wide gap formation of Varnière et al. (2004). Kley & Dirksen (2006) also showed that the disc rotation has some eccentricity when the gap is extended to the 1:2 Lindblad resonance. The eccentric gaps are formed for a large K (≳104) (e.g. Kley & Dirksen 2006; Fung et al. 2014). Such wide gaps by massive giant planets are beyond the scope of our one-dimensional disc model adopting the local approximation.
In the above, we found that a larger xd is required for a larger K (i.e. a massive planet) in order to reproduce the minimum surface densities derived by DM13 and Varnière et al. (2004). It should be noted that the propagation distance is not proportional to the parameter xd. The propagating distance of waves is defined by the distance from the wave excitation site to the angular momentum deposition site (i.e. xd). Since the one-sided torque is radially distributed (see Fig 9b), the wave excitation site can be approximately given by the median of the distribution, i.e. the point within the half of the one-sided torque arises. Such an excitation site shifts away from the planet with an increase of K (see Fig. 12).4 D'Angelo & Lubow (2010) also showed this tendency that the peak of the excitation torque density shifts away from the planet as a deep gap is formed (fig. 15 in that paper), using hydrodynamic simulations. Hence, because of this shift of the excitation site, the propagating distance does not increase much as xd with K. Goodman & Rafikov (2001) showed that the propagating distance of waves decreases with an increase of the planet mass due to the non-linear wave damping. Hence, further studies are needed in order to confirm whether this results given by Goodman & Rafikov (2001) conflicts with ours because of the shift of the excitation site. Furthermore, the non-linear wave damping would be weakened by the steep surface density gradient at the gap edge, as pointed out by Petrovich & Rafikov (2012). In order to fix the parameter xd, such a wave damping effect in the gap should be taken into account in future work.
6 SUMMARY AND DISCUSSION
We re-examined the gap formation in viscous one-dimensional discs with a new formulation. In our formulation, we took into account the deviation from Keplerian disc rotation and included the Rayleigh stable condition, consistently. We also examined the effect of wave propagation. Our results are summarized as follows.
The deviation from the Keplerian disc rotation makes the gap shallow. This is because of the enhancement of the shear motion and the viscous angular momentum transfer at the gap edges (see Fig. 4).
For deep gaps, the deviation from the Keplerian disc rotation is so large that the Rayleigh stable condition is violated. An enhanced viscosity dissolves such unstable rotation and makes it marginally stable (see Fig. 7). This effect also makes the gap shallower (see Fig. 6).
To include the effect of wave propagation, we adopted a simple model where the position of the angular momentum deposition is parametrized by xd. A large xd indicates a long propagation length. The effect of wave propagation makes the gap wider and shallower (Fig. 9). In a wide gap, the waves are mainly excited at the flat bottom, which reduces the one-sided torque and the gap depth. For a sufficiently large xd, the gap depth of our exact solution agrees well with the wide-limit gap and with the results of hydrodynamic simulations. At K = 1000, our model requires xd ≥ 6 for the agreement (Fig. 12). In the case of instantaneous wave damping, on the other hand, our exact solution gives much deeper gaps than those of hydrostatic simulations.
To check the validity of the large xd, the gap width of our exact solution is compared with results of hydrodynamic simulations. For K = 1000, our exact solution with xd ≥ 6 has a gap width of 12hp, which is larger than those of DM13 (∼8hp). The gap widths of Varnière et al. (2004), on the other hand, are almost consistent with our exact solutions. Because of this uncertainty in the gap width of hydrodynamic simulations, it is difficult to fix the preferable xd by this comparison.
When the Rayleigh condition is taken into account, the deviation from the Keplerian rotation should also be included in order to keep the angular momentum conservation. The Keplerian solutions with the Rayleigh condition give much shallower gaps, as shown in Figs 8 and 12.
In future works, we need to determine the preferable value of xd. Previous studies (e.g. Korycansky & Papaloizou 1996; Takeuchi, Miyama & Lin 1996; Goodman & Rafikov 2001; Dong, Rafikov & Stone 2011) have investigated the wave propagation with no gap. As pointed out by Petrovich & Rafikov (2012), however, the gap structure can affect the wave damping. Since our result shows that the wave damping significantly affects both the gap depth and width, the wave damping should be treated accurately in both one-dimensional models and hydrodynamic simulations for gap formation.
Our simple model does not include the effect of the deviation from Keplerian disc rotation on the wave excitation. Petrovich & Rafikov (2012) showed that a steep surface density gradient modifies the excitation torque. Such an effect on the wave excitation should be included in future studies on the gap formation. Nevertheless, it is also considered that when the waves are mainly excited at the flat-bottom, such as for the wide-limit gap, the deviation of the disc rotation would not affect the wave excitation significantly.
We also neglect the non-linearity of wave excitation, whereas the non-linearity cannot be neglected for large planets as Mp/M* ≳ (hp/Rp)3. According to Miyoshi et al. (1999), the non-linearity makes the excitation torque small compared to the value for linear theory. This possibly leads to an additional shallowing effect. However, this effect would not significantly influence the gap depth since smin is scaled by only K in DM13's relation (equation 42).
The Rossby wave instability may be essential for the gap formation. In the present study, we included only the Rayleigh condition. A more detail investigation including both the Rayleigh condition and the Rossby wave instability should be done in future works.
We are grateful to Aurélien Crida, and Alessandro Morbidelli for their valuable comments. We also thank the anonymous referee for useful comments on the manuscript. KDK is supported by Grants-in-Aid for Scientific Research (26103701) from the MEXT of Japan. TT (Takayuki Tanigawa) is supported by Grants-in-Aid for Scientific Research (23740326 and 24103503) from the MEXT of Japan.
In the notation of Fung et al. (2014), K is given by q2/(α[h/r]5).
In the notation of DM13, K is given by |${\cal M}^{-1}(M_{\rm sh}/M_{\rm p})^{2} \alpha ^{-1}$|.
In Fig. 12, the exact solution have transition points from the realistic wave propagation (solid lines) to the unrealistic one (dashed lines) for each xd. At the transition point of K, the excitation site defined by the median is equation to xd. Fig. 12 shows that the excitation site moves away from the planet with an increasing K since the transition point of K increases with xd.