Detailed Calculations of the Efficiency of Planetesimal Accretion in the Core-Accretion Model -II: The effect of Saturn

As part of our ongoing initiative on accurately calculating the accretion rate of planetesimals in the core-accretion model, we demonstrated in a recent article that when the calculations include the gravitational force of the Sun (the original core-accretion model did not include solar gravity), results change considerably [ApJ, 899:45]. In this paper, we have advanced our previous study by including the effect of Saturn. To maintain focus on the effect of this planet, and in order to be consistent with previous studies, we did not include the effect of the nebular gas. Results demonstrated that as expected, Saturn's perturbation decreases the rate of accretion by scattering many planetesimals out of Jupiter's accretion zone. It also increases the velocities with which planetesimals encounter the envelope, which in agreement with our previous findings, enhances their break-up due to the ram-pressure. Results also show that, because the effect of Saturn in scattering of planetesimals increases with its mass, this planet might not have played a significant role in the accretion of planetesimals by proto-Jupiter during the early stage of its growth. Finally, the late accretion of planetesimals, as obtained in our previous study, appears in our new results as well, implying that combined with the rapid in-fall of the gas, it can result in the mixing of material in the outer regions of the envelope which may explain the enhancement of the envelope's high-Z material.


INTRODUCTION
It is generally accepted that the collisional growth of km-sized planetesimals serves as a prelude to the formation of planetary systems. Depending on their radial distances from the central star, which determines their compositions, material strength, and impact velocities, the collision and accretion of these bodies can result in the formation of Moon-to Mars-sized planetary embryos in the inner part of a protoplanetary disk, and the cores of giant planets in regions beyond the disk's snowline. The core-accretion model of giant planet formation suggests that the latter is naturally followed by the accretion of gas from the nebula and results in the formation of gas-giant planets.
During the formation of a gas-giant, the contribution of planetesimals continues beyond the formation of the core. In the gas-accretion phase, and also during the time that the protoplanet's envelope collapses, many planetesimals enter the envelope and contribute to its metallicity and the luminosity of the planet by depositing their materials as they vaporize due to heating by the gas-drag.
While the physics of processes governing planetesimalenvelope interactions are independent of the characteristics of the system, their outcomes directly depend on the physical and dynamical properties of planetesimals and the state of the gas. For instance, as the envelope evolves and its density and temperature structures change, its response to encountered planetesimals varies which causes the rate of the accretion of these objects to vary as well.
These variations in the accretion rate of planetesimals strongly affect the formation of the planet as the outcome of the accretion determines the composition of the envelope and the onset of its collapse (Pollack et al. 1996;Iaroslavitz & Podolak 2007;Movshovitz et al. 2010;D' Angleo et al. 2014;Venturini et al. 2016;Lozovsky et al. 2017). It is therefore, imperative that any model of giant planet formation take these interactions into account and calculate their contributions accurately and self-consistently.
Because the strength of the interaction of planetesimals with the envelope, which determines the efficacy of their vaporization and mass-deposition, depends directly on their encounter velocities, to calculate the realistic amount of the mass that is deposited during the evolution of the envelope, the rate of the encounters of planetesimals and their encounter velocities must be obtained from the natural dynamical evolution of the system. A comprehensive model will also need to continue the integration of those planetesimals that enter the envelope, and calculate their mass-deposition in concert with the envelope's density and temperature variations.
While the above requirements have been known since the inception of the core-accretion model, realistic simulations have been possible only during the past few years. For almost three decades, limitations in computer technology did not allow for large scale simulations, forcing scientists to tune their approaches to the available computational resources by making simplifying assumptions. For instance, in the original work of Pollack et al. (1996), planetesimal contributions were calculated using a semianalytical approach. To avoid computational complexities, these authors assumed that all planetesimals entered the envelope with the same velocity and determined the rate of their encounters using a gravitational enhancement factor that was obtained from fitting to the results of the numerical simulations of Greenzweig & Lissauer (1990, 1992. Once inside the envelope, the orbit of a planetesimal and the amount of its mass-deposition were calculated using the methodology and the code developed by Podolak et al. (1988). The orbit was, however, integrated including only the effect of the proto-Jupiter (i.e., a two-body system) and external sources such as the gravitational forces of the Sun and additional (growing) giant planet(s) were not included. Pollack et al. (1996) considered a planetesimal to be fully accreted if its total energy became lower than a minimum value, or if it broke up due to ram-pressure. The mass-accretion rate was then determined by carrying out the above calculations for a series of impact parameters and calculating the protoplanets effective capture cross section and planetesimals flux.
Subsequent advances in computational techniques combined with the desire to achieve a shorter formation time for Jupiter prompted researchers to revisit some of the above processes. For instance, Inaba & Ikoma (2003) examined the full-accretion criteria by developing an analytical approximation to the time-evolution of the envelope as well as an analytical fit to the results of the N-body simulations of Ida & Makino (1993), Nakazawa et al. (1989), and Inaba et al. (2001). These authors ignored the effect of the Sun on the orbits of planetesimals inside the envelope and calculated the ablation of planetesimals using an ablation factor that did not vary with the envelope's temperature. Inaba & Ikoma (2003) argued that their simulations point to a larger rate of planetesimal encounters and, therefore, a shorter formation time for Jupiter. They also argued that based on their results, ram-pressure would not reach high values to break up planetesimals.
While the above calculations demonstrated the proof of the concept, their results were not fully realistic. In the work of Pollack et al. (1996), the encounter flux of planetesimals and their velocities had not been obtained from the natural dynamical evolution of the system, and the analytical approach of Inaba & Ikoma (2003) ignored the effect of the variation of the envelope's temperature on the ablation of planetesimals. Furthermore, neither of the above calculations included the gravitational perturbation of the Sun and other giant planets.
In a recent article (Podolak et al. 2020, hereafter, Paper-I), we revisited the planetesimal-envelope interaction and developed a comprehensive and self-consistent approach that would address the above shortcomings. Using a special purpose integrator (ESSTI, see section 2.4 for more details), we integrated the orbits of a large number of planetesimals starting from their original locations in the protoplanetary disk and calculated the ablation of those that entered the envelope using their encounter velocities obtained from their orbital evolution. Our calculations took into account the time-variations of the density and temperature of the envelope, and as the first step to address some of the shortcomings of the previous studies, they included the gravitational effect of the Sun as well.
In this study, we advance the calculations of Paper-I by including the gravitational perturbation of a planet in the orbit of Saturn. As demonstrated by Haghighipour & Scott (2012) and Haghighipour & Winter (2016), Saturn played a significant role in the formation of terrestrial planets and the structure of asteroid belt. It also con- tributed significantly to the origin of the parent bodies of the iron meteorites (Haghighipour & Scott 2012). Given the extent of the perturbation of this planet, it is certain that Saturn has also played a significant role in the orbital evolution of planetesimals in the region beyond 5 AU, especially those in the influence zone of Jupiter. It is expected that the gravitational perturbation of Saturn directly influenced the encounter rate of planetesimals as well as the velocity with which they entered Jupiter's envelope. In this paper, we calculate these effects and study their consequences on planetesimal-envelope interaction and the metalicity of the envelope. The rest of this paper is organized as follows. In Section 2, we explain our numerical approach and describe the details of our system and its initial set up. In this section, we also discuss the physics of planetesimal-envelope interaction and the way that they have been included in our calculations. We present the results in Section 3 and discuss the effect of Saturn in Section 4. We conclude our study in Section 5 by summarizing our findings and discussing their implications for the formation of gas-giant planets in our solar system and beyond.

The System
We consider a four-body system consisting of the Sun, Jupiter, Saturn and a planetesimal. The mass of the planetesimal is negligible and its orbit is between Jupiter and Saturn. Because our goal is to determine the amount of the mass that a planetesimal loses during its passage through the Jupiter's envelope, we include the gravitational force of Saturn as a constant perturbation, and treat Jupiter as an extended mass. For the purpose of this study, we do not consider a diffused core in Jupiter. Instead, we assume that the core is condensed and has a well-defined surface, separating heavy elements and H/He.
During the evolution of the system, the variations of 2.8 × 10 10 8.08 × 10 10 2.8 × 10 10 P 0 (dyn cm −2 ) 3.891 × 10 11 1.50 × 10 13 3.891 × 10 11 A (K) −2.1042 × 10 3 −2.4605 × 10 4 −2.1042 × 10 3 the radius, mass, and internal density distribution of Jupiter are included using the model A of Lozovsky et al. (2017). This model follows the growth of a planet in the core-accretion scenario for an assumed background surface density of solids of σ = 6 g cm −2 and a planetesimal radius of 100 km. Figure 1 shows the variations of the envelope's gas density in one of such models. The dotted line represents the gas density of 10 −6 g cm −3 , small and large dashed lines correspond to the gas densities of 10 −5 g cm −3 and 10 −4 g cm −3 , respectively, and the solid line shows a gas density of 10 −3 g cm −3 . As shown by the figure, when Jupiter grows, its envelope grows as well, both in radius and mass. This causes the density of the gas to vary with time at different elevations inside the envelope. In this model, at approximately 1.7 × 10 6 years, the envelope becomes unstable and the runaway gas-accretion ensues. To avoid complications due to the time evolution of the nebula, and for the mere purpose of maintaining focus on the perturbing effects of Saturn, we do not include the interaction of planetesimals with the nebular gas in our calculations. We also do not consider the appearance of a gap in the gaseous disk around the orbit of proto-Jupiter (Shibata & Ikoma 2019) and the migration of the planet due to its interaction with the nebula (Shibata et al. , 2022. While these effects are indeed important, they strongly depend on the model used for the disk gas distribution. They also depend on the subsequent evolution of the gas disk over the 3 Myr period we are considering here. Because, at present, there is no standard model for the structure and evolution of the disk, we prefer to focus on the dynamical effect of a growing Saturn, and put aside, for the present, the disk-model-dependent effects.

The Composition and Structure of Planetesimals
Planetesimals are considered to be perfectly spherical and have three sizes of 1, 10, and 100 km. Because during the interaction of a planetesimal with Jupiter's envelope, the efficiency by which it loses mass, and the amount of its mass-loss, in addition to its size, shape and velocity, vary with its internal composition as well, we consider three compositions for each size of the planetesimals: Pure ice. With a bulk density of 1.0 g cm −3 , these planetesimals represent the low-end of the density spectrum. Pure-ice planetesimals are volatile and evaporate easily as they are heated up by gas-drag while passing through the envelope.
Pure rock. These planetesimals represent the hardest material with the bulk density of 3.4 g cm −3 . They are not easily evaporated and their dynamics is less strongly affect by gas-drag. Rocky planetesimals also have high tensile strength which makes them stable against breakup due to ram-pressure.
Mix of ice plus rock. In reality, a planetesimal in the region between Jupiter and Saturn is made of a mix of ice and rock. When such a planetesimal is heated, its ice component vaporizes and as it escapes the planetesimal, it carries some of its rocky material with it. This rocky debris enhances the metalicity of the envelope and increases its dust/gas ratio. As the ice-vapor escapes the planetesimal, it also increases the planetesimal's porosity weakening its internal strength which in turn makes the planetesimal more susceptible to breakage due to ram-pressure. It is, therefore, imperative to study the response of such mixed planetesimals to their interactions with envelope as these bodies can have large contributions to the envelope's metalicity and the possible growth of the planet's core. In choosing the fraction of the ice and rock in our mix-composition planetesimals, we follow the trend observed in the bulk density of some of the large bodies in the outer solar system and consider planetesimals with 30% ice and 70% rock corresponding to a bulk density of 2.0 g cm −3 . This bulk density is similar to that of objects such as Pluto, Triton, and Titan,

Planetesimal-Envelope Interaction
When inside the envelope, in addition to the gravitational forces of the Sun, Saturn, Jupiter's core and the portion of the envelope that is interior to its orbit, the motion of a planetesimal is also affected by gas-drag. At this stage, the combined heating due to the gas-drag and the radiation received from the hot ambient gas, increases the temperature of the planetesimal causing it to lose mass. The rate of mass-loss depends on the material composition of the planetesimal and its response to heating. The latter is quantified by the critical temperature of the planetesimal material (T crit ) above which there is no phase change.
If, while the planetesimal passes through the envelope, its surface temperature (T p ) does not exceed (T crit ), the surface material can exist in two phases of solid and gas (vapor). At this state, the mass-loss will be primarily due to evaporation, In this equation, M p and R p are the mass and radius of the planetesimal, m p is the mass of the planetesimal molecule, and k is the Boltzmann's constant. The quantity P vap given by represents the upper limit of the pressure of the vapor surrounding the planetesimal. If the vapor pressure exceeds P vap , any excess vapor re-condenses onto the planetesimal surface until its pressure stays equal or drops below the above value. In equation (2), P 0 and A are constant quantities whose values depend on the composition of the planetesimal. Tables 1 shows the values of these quantities and other properties of the planetesimals used in our study.
Once the planetesimal's surface temperature reaches T crit , the vapor can have any pressure. We assume that at this state, any surface material is automatically turned into vapor and is blown away by the ambient gas that streams past of it. The mass-loss will then be due to the combined effects of evaporation and radiation heating/cooling, and is given by In this equation, ρ g and T g are the density and temperature of the envelope gas at the position of the planetesimal, v rel represents the magnitude of the velocity of the planetesimal relative to the gas (v rel > 0), σ is the Stefan-Boltzmann constant, C D is the drag coefficient and E 0 is a constant quantity whose value depends on the composition of the planetesimal (see Table 1). We refer the reader to the appendix of Paper-I and Podolak et al. (1988) for more details on the derivations of these equations.

Numerical Integrations
To accurately quantify the interaction of a planetesimal with the envelope, specifically, its mass-loss and size reduction, and to ensure that the velocity with which the planetesimal enters the envelope would be the natural outcome of its dynamical evolution, we integrated the orbit of the planetesimal starting from its initial position between Jupiter and Saturn.
To avoid complications due to the stiffness of the differential equations, we broke the integrations into two parts: inside and outside the envelope. When outside the envelope, we integrated the four-body system of Sun-Jupiter-planetesimal-Saturn using the N -body integrator MERCURY (Chambers 1999). At this stage, we included Saturn as a point mass and Jupiter as an extended object whose radius and mass varied with time according to the model A of Lozovsky et al. (2017). When the integrator indicated a collision between the planetesimal and the extended-mass Jupiter, N -body integrations were stopped and integrations were continued using our special purpose integrator, ESSTI (Explicit Solar System Trajectory Integrator).
ESSTI has been developed to integrate the orbit of a planetesimal inside the proto-Jupiter's envelope and outside, at any location in a protoplanetary disk. It has been designed to automatically include in the equations of motion, the physical processes associated with the location of the planetesimal. For instance, when a planetesimal enters the envelope, ESSTI adds to the equations of motion the drag force of the gas. It keeps the forces of the Sun and Saturn intact, but replaces the gravitational force of Jupiter with the gravitational force of the portion of the mass that is located between the orbit of the planetesimal and the center of mass of Jupiter. It accounts for the planetesimal's mass-loss by including the effects of the aerodynamical heating due to gas-drag, radiative heating due to the ambient gas, radiative cooling, and evaporation cooling. All these processes have been hard-wired in ESSTI with their corresponding analytical formulae (Table 2). We refer the reader to Paper-I for more details on this integrator.
In addition to accounting for mass-loss due to ablation, ESSTI also includes the effect of ram-pressure. The latter is due to the pressure difference across the planetesimal caused by the balk motion of the gas. For a planetesimal moving with a relative velocity v rel in a gas with a density ρ g , the ram-pressure on its front hemisphere is given by (Pollack et al. 1979) The planetesimal will fragment when the ram-pressure exceeds the object's compressive strength. If the selfgravity of the fragmented body cannot hold the fragments together, the body will break apart. The critical radius above which the self-gravity of the planetesimal will be strong enough to hold its fragments together is (Pollack et al. 1979) where ρ p is the bulk density of the planetesimal and G is the gravitational constant. ESSTI monitors all these processes until either the planetesimal has left the envelope or is fully absorbed (accreted). A planetesimal is considered fully absorbed if it collides with the Jupiter's core, breaks up due to ram-pressure, or loses more than 80% of the mass with which it entered the envelope, due to ablation. If the planetesimal leaves the envelope, any mass lost due to ablation is considered accreted and is added to the mass of Jupiter. At this stage, ESSTI integrations are stopped and the planetesimal's departing orbital elements are passed to MERCURY where N -body integrations are continued with the planetesimal's new mass. If the planetesimal returns to the envelope, the above process is repeated with its new mass, new radius, and the new state of Jupiter. Integrations are continued until either the planetesimal is fully accreted, collided with the Sun or Saturn, or is ejected from the system.

RESULTS AND ANALYSIS
We carried out a total of 14,076 four-body integrations corresponding to 1173 integrations for each line of size+composition presented in Table 3. In each system, we started Jupiter and Saturn in their current orbits and randomly chose the semimajor axis of the planetesimal from the range of 3.7 AU to 6.7 AU (corresponding to 4 Jupiter's Hill radii on either side of the planet). This range was chosen so that our initial conditions would be consistent with those in Paper I. The eccentricity of the planetesimal varied from 0 to 0.05. To avoid computational complications due to including Saturn as a growing body, we considered the planet in the orbit of Saturn to have been fully formed. However, to examine the dynamical and physical consequences of its perturbation, we ran integrations for three different values of its mass corresponding to 1/10 th , 1/3 rd and the full mass of Saturn. No inclination was considered for planetesimals, and the entire system was considered to be co-planar.
Integrations were carried out for 3×10 6 years, approximately twice the time for the protoplanetary envelope to collapse (see figure 1). The N -body integrations were carried out using MERCURY's hybrid routine with a time step of 10 days and ESSTI integrations were carried out using a 4 th order Runge-Kutta integrator. Trial runs carried out prior to the main inetegrations indicated 10 −12 as the optimal accuracy. We, therefore, set the accuracy of both integrators to this value. The ESSTI integrator has an adaptive time step capability that allows for adjusting this quentity for each integration for a given accuracy (10 −12 ).
As mentioned before, if a planetesimal entered the envelope, one of the following would occur. Either the planetesimal would leave the envelope and continue its orbit around the Sun after having lost energy and mass due to its interaction with the envelope, or it would continue its motion entirely in the envelope until it would collide with Jupiter's core, or it would vaporize completely while inside the envelope, or it would break up into smaller pieces due to ram-pressure. In the latter case, we assumed that these pieces gradually descended toward the center of mass of Jupiter until they either collided with Jupiter's core or fully vaporized. Table 3 shows the percentage of the occurrence of these outcomes for some combinations of the size and compositions of planetesimals. In the following, we explain these cases in more detail.

Pure-ice planetesimals
The most notable results shown by Table 3 are those of 1 km pure-ice planetesimals. Because these planetesimals are small and volatile, it is natural to expect their motion to be mainly affected by vaporization due to heating by gas-drag. However, as shown by columns 5 and 6, it is the ram-pressure that dominates their motion. Vaporization is in fact a rare occurrence with no vaporization occurring in systems where Saturn has its full mass, and only 1%-1.5% of the planetesimals vaporize when the planet in the orbit of Saturn has 1/3 and 1/10 of Saturn-mass (figure 2). As shown by columns 4 and 5, when these planetesimals enter the envelope, the majority of them breakup at large altitudes ( Figure 3) and their small pieces are rapidly (within almost one integration time step) vaporized such that none survives to collide with the Jupiter's core intact. 3.2. Mixture of ice and rock As mentioned earlier, when a mix-composition planetesimal is heated, its ice component vaporizes faster and carries some of its rocky grains with it. This rocky debris is then released into the envelope and enhances the envelope's metalicity. Because in this case, the planetesimal loses mass in the form of both ice and dust, the rate of its mass-loss is higher than other planetesimal types. Although this higher rate of mass-loss causes the planetesimal's surface-to-mass ratio to increase, which in turn makes the planetesimal more susceptible to be fully accreted through heating by the gas-drag, its larger bulk density (2 g cm −3 ) gives it more inertia making its rocky component more resistive to vaporization and breakage due to the ram-pressure. The 10 km mix-composition planetesimals, as intermediates between the 1 km and 100 km bodies, offer the best case to examine how these two competing effects play off against each other. Figure  4 shows this in more detail for 1/10 th of Saturn mass. The top panel in this figure shows the time and location of vaporization, and the bottom panel shows the same quantities for breakage due to the ram-pressure. As shown here, similar to the case of pure-ice planetesimals, ram-pressure has the dominant effect causing the mix-composition planetesimal to break up relatively high in the envelope. The vaporization, on the other hand, is rare and occurs mainly at the early stages of Jupiter's evolution.
3.3. Pure-rock planetesimals As expected, rocky planetesimals show strong resistance to vaporization. For instance, as shown by figure 5, 25% − 30% of 100 km pure-rock bodies that encountered the envelope, penetrated through and impacted the surface of Jupiter's core at early times and without losing much of their masses. However, and despite the latter, ram-pressure was still the dominent factor affecting the motion of these bodies. While this seems to be similar to the situation with icy and mix-composition planetesimals, it differs from those cases in the sense that the higher material strength of rock does not allow the ram-pressure to break up rocky planetesimals as easily. Figure 6 shows this for the 100 km rocky planetesimals that encountered the envelope. As shown here, breakups occur later and in deeper layers. This suggestes that pure-rock planetesimals may pass through the envelope several times and lose mass until their self-gravity becomes so weak that it cannot hold them together against the ram-pressure. Figure 7 shows this for a 100 km rocky planetesimal. The top panel in this figure shows the actual path of the body and the bottom panel shows the variations in its mass as a function of its distance from the center of mass of Jupiter. The black oval in the top panel shows the outer boundary of the envelope extending to approximately 3 × 10 10 cm from Jupiter's center of mass. The vertical axis in the bottom panel shows the instantaneous mass of the planetesimal relative to its initial mass (M 0 ). The red arrows on both panels show the location of the planetesimal and the direction of its motion. When approaching the envelope, the planetesimal is shown in red and the variations in its mass are presented by dashed lines. When moving away from the envelope, the planetesimal is in blue and its mass is shown by solid lines.
As shown here, at t = 0, the planetesimals is at approximately 7.5 × 10 8 cm from the center of Jupiter (the dotted line represening M/M 0 = 1 in the bottom panel). When it starts moving toward the envelope, it maintains its mass until it passes a critical distance at about 2×10 10 cm where it starts to lose mass due to evaporation. The ambient temperature at this region is approximately 2700 K meaning that, in addition to the gas drag, the hot envelope also contributes to the heating leading to mass loss. Although the planetesimal has lost mass and therefore, kinetic energy, it still retains enough energy to leave the envelope. However, its new kinetic energy is not large enough for escaping Jupiter's Hill sphere completely. As a result, the planetesimal returns to the envelope and repeats this process while losing energy and mass in each encounter until it can no longer leave the envelope. At this stage, either it breaks apart with some of its fragments hitting the surface of Jupiter's core (see column 4 of Table 3) or it is fully vaporized. As can be seen from the figure, the vaporization occurs almost entirely in the region where the temperature is above ∼2700K.
3.3.1. The case of 1 km pure-rock planetesimals As shown by Table 3, similar to the case of 100 km rocky bodies, 1 km pure-rock planetesimals, too, show large number of impacts with Jupiter's core. The main difference, however, is that because of the small sizes of these bodies, their motion is more complex. On the one hand, rock is much less volatile than ice, implying that a 1 km rocky planetesimal can survive longer in the envelope than a 1 km ice or mix-composition object. On the other hand, because of its small radius, the effects of gasdrag can be more significant causing the planetesimal to be more easily absorbed. Figure 8 shows the outcome of the interaction of two of such bodies with the envelope. The vertical axis on the left shows the temperature of the envelope and the one on the right shows the instantaneous and relative mass of the planetesimal. The black curve shows the envelope's temperature as a function of  Figure 3. Graphs of the break-up of 1 km pure-ice planetesimals due to ram-pressure for different values of Saturn's mass. From top to botom, panels corresponds to 1/10 th , 1/3 rd , and full Saturn mass. As shown here, break up happens at high altitudes in the envelope.  Figure 4. Graphs of the vaporization (top) and break-up (bottome) of 10 km mix-composition planetesimals for 1/10 th of Saturn mass. As shown here, vaporization is rare and ram-pressure breakup is the dominent process.
distance from the center of Jupiter at the time 1.66 Myr. The red curve shows an example of those planetesimals whose trajectories take them directly into the envelope.
In the case of this specific planetesimal, almost the entire mass-loss occurs in the region where the temperature is ∼ 3000 K. The green curve shows an example of those planetesimals that have grazing encounters with the envelope. In this case, the planetesimal shows similar behavior as the 100 km body: While it loses some mass and energy, it cannot escape Jupiter's Hill sphere and makes multiple returns until it is entirely absorbed. In both cases, the vaporized mass is deposited in the same region of the envelope that it is produced. It should be noted that although the deposition of these materials is in a very limited region, convective mixing is likely to spread them over a significantly larger region of the envelope. It is also important to emphasize that the scenarios presented in figures 7 and 8 can be affected by the presence of the nebular gas outside of the envelope especially that these effects are mass (and composition) dependent.

THE EFFECT OF SATURN
An analysis of the orbital dynamics of the planetesimals in our integrations indicates that, compared to the systems of Paper-I, more planetesimals in our systems were left un-accreted and/or were ejected from the system. Figure 9 shows this for the case of 100 km pure-rock planetesimals. Shown here are the final eccentricities of the surviving planetesimals (in red) compared to their initial eccentricities (in black) as a function of their initial semimajor axes. The top-left panel corresponds to the integrations in Paper-I and the three other panels show the results for our integrations with the mass of the planet in the orbit of Saturn equal to 1/10 (topright), 1/3 (bottom-left), and full Saturn-mass (bottom -right). As shown here, in integrations without Saturn, all planetesimals in the region between 5 AU and 5.5 AU were accreted by the growing proto-Jupiter. While there is a small number of scattered planetesimals, many remained in the system maintaining their original eccentricities (i.e., remain unaffected by the growing Jupiter). However, Once a planet is introduced in the orbit of Saturn, the perturbation of this planet causes the orbits of more planetesimals to become highly eccentric and less number of planetesimals in the vicinity of Jupiter to be  Distance from the Center of Jupiter (cm) Figure 8. Graphs of the mass-deposition of two 1 km rocky planetesimals in term of distance from the center of Jupiter. Note that the left vertical axis shows the envelope's temperature and the right vertical axis shows the planetesumal's relative mass. The black line represents the temperature of the envelope at 1.66 Myr. The red curve shows a planetesimal that penetrates through the envelope and is fully absorbed. The green lines show a planetesimal which undergoes multiple encounters until it is fully accreted. Note that in both cases, full accretion occurs at ∼ 3000k.
accreted. As expected, the scattering becomes stronger for larger values of Saturn's mass.
In addition to increasing their orbital eccentricities, interaction with Saturn also increases the velocities with which planetesimals encounter Jupiter's envelope. This increase in the encounter velocity, enhances the effects of gas-drag and ram-pressure, thereby, reducing the time of accretion. Figure 10 shows the tuning fork diagrams corresponding to the accretion time of fully captured planetesimals in term of their initial semimajor axis. Once again, the top-left panel shows the results for the case without Saturn, and the three subsequent panels represent results for the cases of 1/10 (top-right), 1/3 (bottomleft), and full Saturn-mass (bottom-right). As shown here, when the mass of Saturn increases, more planetesimals are captured at earlier times. However, irrespective of the mass of Saturn, accretion peaks at approximately 1.7 Myr for those planetesimals in the close vicinity of Jupiter. The latter is not unexpected as it corresponds to the time of the collapse of the envelope where the increase in the envelope gas density enhances the rate of accretion.

SUMMARY AND CONCLUDING REMARKS
We have studied the interactions of planetesimals with the gaseous envelope of the growing Jupiter in the coreaccretion model. Our study aims at calculating a more accurate mass accretion rate for the growing giant planet and advances our previous calculations (Paper-I) by including the gravitational perturbation of Saturn.
Using our special purpose integrator ESSTI, we tracked the orbits of planetesimals inside the envelope and calculated their mass-loss due to the heating by gas-drag and fragmentation caused by ram-pressure. Results confirmed our previous findings that, unlike the reports by Inaba & Ikoma (2003), in general, ram-pressure is the main process that affects the orbits of planetesimals when they first encounter the envelope. The high encounter velocities of these objects, especially when Saturn is included, triggers ram-pressure, causing the body to break up into small pieces and subsequently be accreted as its small fragments vaporize due ablation while descending in the envelope.
An analysis of the efficiency of the breakage and vaporization of planetesimals with different material compositions indicated that mix-composition planetesimals have the largest contribution to the metalicity of the envelope and even the growth of the planet's core. As these planetesimals disintegrate, their ice components vaporize almost immediately. Their rock components, however, dissolve gradually as they descend toward the center of the planet, enhancing the envelope's metalicity. Integrations show that some of these rocky fragments survive the descent and collide with the surface of the core contributing to its growth as well.
Our analysis also shows that the rate of collision of fragments with the core increases with Saturn's mass (Table 3). This is an expected result that has to do with the fact that a larger planet perturbs the orbits of planetesimals more strongly increasing their encounter velocities and enhancing the fragmentation effect of the ram-pressure. The latter results in producing more fragments with larger post-break up velocities.
Among the three types of compositions considered here, pure-rock planetesimals show a more interesting and complicated motion. The high density of these objects prevents them from being disintegrated when they first encounter the envelope. As a result, these planetesimals pass through the envelope multiple times los- ing mass to vaporization until their internal structure becomes weak enough for ram-pressure to break them apart. Results of integrations for 1-100 km-sized rocky bodies show that the full accretion occurs after approximately 5 × 10 5 years and almost entirely deep in the envelope where the temperature is higher than 2700-3000 K.
A comparison between the results presented here and those of Paper-I demonstrates that when Saturn is included, the rate of mass-accretion becomes smaller. The gravitational perturbation of Saturn increases the orbital eccentricities of many of the planetesimals, scattering them to the regions outside the influence zone of Jupiter, thereby reducing the number of bodies that encounter Jupiter's envelope. For those planetesimals that enter the envelope, the increase in orbital eccentricity appears as an increase in their encounter velocities. The latter enhances the efficiency of gas-drag and ram-pressure in dissolving the body, and shortens the time of its accretion. Because the perturbation of Saturn is a direct function of its mass, these findings imply that the rate of the accretion of planetesimals by Jupiter's envelope was higher when Saturn was small and decreased with time as Saturn became larger (and its gravitational perturbation became stronger).
That the rate of accretion decreases with Saturn's mass has a direct consequence on the final size of Jupiter's core. Results of Paper-I suggested that during the first 5 × 10 5 years, the rate of mass accretion was lower than that assumed by Lozovsky et al. (2017). When the perturbation of Saturn is included, that rate becomes even smaller. A smaller core, in turn, implies a smaller and less massive envelope meaning that the envelope considered here might not have been of a fully realistic size. A self-consistent model requires that the calculations, in addition to the effect of Saturn on the motion of plan- etesimals, to include the effect of Saturn's growth on Jupiter's envelope, as well. As mentioned in Section 2, to maintain focus on Saturn's effects, we did not consider the interaction of planetesimals with the nebular gas. This interaction, which appears in different forms, can affect the motion of planetesimals prior to their encounter with the envelope. For instance, as shown by Zhou & Lin (2007), Shiraishi & Ida (2008) and Shibata & Ikoma (2019), the drag force of the nebula will change the encounter velocities of planetesimals which may, at least partially, counter the perturbing effect of Saturn by preventing the orbital eccentricities of planetesimals from reaching very high values. The latter may slightly improve the accretion rate of these objects by reducing the number of scattered planetesimals and increasing the rate of their encounter with the envelope. Including this effect is the subject of future studies.
Finally, it is important to note that to ensure that our simulations would start from similar initial conditions as those in paper I, so that a comparison between the results of the two studies would be meaningful, we only included planetesimals with initial semimajor axes between 3.7 (AU) and 6.7 (AU). In paper I, this range was sufficient because only Jupiter was included. Saturn, however, will affect planetesimals further out, and some of these planetesimals may enter Jupiter's envelope. In future calculations, the initial distribution of planetesimals will be extended to farther distances.