3 Modeling and results

In order to study whether or not resonance trapping of particles is a plausible mechanism to confine solid material able to form primordial comets we consider the epoch when the masses of proto-Jupiter and proto-Saturn reach the present values by accreting the gases of the solar nebula. During this stage, drag-induced orbital decay of material from the outer solar nebula provides continuous replenishment of solid particles. We have carried out a series of numerical integrations of coplanar orbits of solid metric particles in the gravitational field of the Sun, proto-Jupiter and proto-Saturn, using the Bulirsch-Stoer method (Bulirsch & Stoer 1966). We investigate the response of the particles, in a standard protoplanetary (see, e.g., Cuzzi et al. 1993) disk with radius $R_{\rm D} = 100$ AU and mass $M_{\rm D} = 0.001~M_{\odot}$ (as Jupiter and Saturn are being formed). Our parallel direct integration code models gravitational and viscous forces, including self-gravitation (de la Fuente Marcos & de la Fuente Marcos 1998, 2001). We decide to focus our attention in meter-sized particles following Weidenschilling (1994). In his work he argues that cometary and Chiron-like objects could form by accretion of primary nuclei or building blocks, tens of metres in size. This theory is consistent with the observed sizes and structure of comet nuclei as well as outbursts and splitting events. On the other hand, photometric observations of the debris left by the disintegration of C/1999 S4 LINEAR suggest that the primordial building blocks of this comet were indeed metric in size, therefore metric (or submetric) particles are the most relevant as regards comet formation. In the classical core-instability scenario of giant planet formation (see, for example, Mizuno 1980), when the proto-giant planets were already formed, the planetesimals have grown to kilometre-sized bodies, therefore there is little room to comet formation following the building blocks model pointed out above. However, in the framework of the capture-in-vortex mechanism (see, for example, Barge & Sommeria 1995; Godon & Livio 2000; de la Fuente Marcos & Barge 2001; de la Fuente Marcos et al. 2002) the birth of the Giant planets could take less than a million years as well as allow for coexistence of metric particles and proto-giant planets. In the following, we will assume this non-standard scenario.

In our calculations, the initially circular coplanar orbits of the test particles are perturbed by the Sun, proto-Jupiter and proto-Saturn but do not themselves exert any gravitational forces. The initial heliocentric distance of the dust particle population is 20 AU. For the protoplanets we adopted their present eccentricities. Our models are followed for about 7 Myr as recent determinations constrained by D/H measurements in the Solar System (Hersant et al. 2001) suggest that the epochs of the formation of Jupiter and Saturn cannot be lower than 0.7 and 5.7 Myr, respectively, after the formation of the Sun. Instead of considering a continuous growth model for the giant planets we restrict our calculations to several static combinations for the masses of the growing protoplanets: $M_{\rm pJ}$, $M_{\rm pS} = (M_{\rm J}/10,~0)$, ( $M_{\rm J},~ 0)$, ( $M_{\rm J}/10$, $M_{\rm S}/100$), ($M_{\rm J}$, $M_{\rm S}/10$), ($M_{\rm J}$, $M_{\rm S}$), where $M_{\rm pJ}$ is the mass of proto-Jupiter, $M_{\rm pS}$ is the mass of proto-Saturn and $M_{\rm J}$ and $M_{\rm S}$ are the masses of Jupiter (0.000956 $M_{\odot}$) and Saturn (0.000286 $M_{\odot}$), respectively. Our qualitative results do not strongly depend on the choice of protoplanetary masses but the quantitative results (order of resonances) depend slightly on these values (see Table 1).

 

 

Table 1: Average orbital elements of the trapped bodies for the numerical experiments performed.

Size	$M_{\rm pJ}^{\star}$	$M_{\rm pS}^{\star\star}$	$a^{\star\star\star}$	e
metric	0.1	0.0	7.2	[0.03, 0.09]
metric	1.0	0.0	8.7	[0.05, 0.12]
	1.0	0.0	11.4	[0.10, 0.16]
submetric	0.1	0.01	6.8	[0.03, 0.07]
metric	0.1	0.01	16.1	[0.10, 0.15]
metric	1.0	0.1	12.5	[0.05, 0.12]
centimetric	1.0	1.0	12.5	[0.08, 0.16]
decimetric-metric	1.0	1.0	13.3	[0.04, 0.12]
metric	1.0	1.0	16.1	[0.10, 0.16]
	1.0	1.0	17.1	[0.22, 0.23]
	1.0	1.0	17.5	[0.14, 0.16]
	1.0	1.0	17.9	[0.10, 0.12]
	1.0	1.0	18.8	[0.12, 0.16]

$^{\star}$ proto-Jupiter mass (in $M_{\rm J}$).
$^{\star\star}$ proto-Saturn mass (in $M_{\rm S}$).
$^{\star\star\star}$ semi-major axis (in AU).

Figure 1: Resonant trapping of a 5 m particle by proto-Jupiter ($M_{\rm J}$/10) and proto-Saturn ($M_{\rm S}$/100).

Figure 2: Resonant trapping of a 5 m particle by Jupiter and proto-Saturn ($M_{\rm S}$/10).

Our first calculation considers a proto-Jupiter with $M_{\rm pJ} = 0.1 M_{\rm J}$ orbiting around the Sun with its current values of semi-major axis and eccentricity. Metric particles are trapped in outer resonances with semi-major axis 7.2 AU and eccentricity changing in the range [0.03, 0.09] with a period of about 15 000 yr. Increasing the mass of the growing proto-Jupiter shifts the non-gravitational resonance outwards. For a $M_{\rm pJ} = M_{\rm J}$ resonant capture for metric particles appears at 8.7 AU. The eccentricity is now changing in the interval [0.05, 0.12] with a period of a few 10³ yr. The exact values depend on the size of the particle: larger radius implies higher eccentricity and longer period. Besides the 8.7 AU resonance we find another one (see Table 1 and Fig. 3) at about 11.4 AU with higher eccentricity, span 0.10-0.16. This outer resonance is stronger as shown in Fig. 4 (lower panel). Figure 4 provides another characteristic of gas-induced resonance trapping, size-selective capture. The outermost resonance is rather size-selective as the innermost shows clear size gaps.

Figure 3: Numerical simulation of the evolution of 500 metric particles under the influence of the Sun, gas drag and Jupiter (or Jupiter+Saturn). Positions of the bodies (in heliocentric Cartesian coordinates) during the trapping. Cumulative points for a time interval of 2 Myr are plotted in order to improve contrast. (upper panel) Solid bodies subject to Sun attraction, nebular gas drag and perturbation from Jupiter. In this case the two resonances found do not overlap. (lower panel) Saturn also included. Now the resonances overlap. See text for further explanations.

The inclusion of a growing proto-Saturn changes the results dramatically. If $M_{\rm pJ} = 0.1 M_{\rm J}$ and $M_{\rm pS} = 0.01 M_{\rm S}$, particles of about 1 m or less are not captured by the outer resonances beyond Saturn but cross the Saturnian orbit and stop at 6.8 AU from the Sun. However, particles of about 5 m are trapped in an outer resonance at 16 AU with the eccentricity in the range [0.100, 0.153] in a time scale of 12 500 yr (see Fig. 1). This is the vicinity of the 2:1 resonance with Saturn. Increasing the proto-Jovian mass $M_{\rm pJ} = M_{\rm J}$ and the proto-Saturn mass, $M_{\rm pS} = 0.1 M_{\rm S}$, shifts this non-gravitational resonance inwards, 12.5 AU (Fig. 2). This is the 3:2 resonance with Saturn. When we consider current values for the masses of both Jupiter and Saturn, the resonant configuration also changes. Centimetric particles are trapped at the 3:2 resonance but decimetric grains are trapped at 13.3 AU and metric particles at the 2:1 resonance. Metric particles (see Figs. 4 and 3) are also trapped by other resonances (see Table 1), in fact the two resonances that appear in the Jupiter-alone case have been shifted outwards. Again, the size-selective nature of the trapping mechanism is clearly seen on the plots. The innermost resonance only includes particles of about 21 m and several size gaps appear in Fig. 4.

In summary, our results show that, depending on the masses of the protoplanets considered and the particle's radii, solid material is trapped in resonances. Our calculations suggest an evolutionary scenario for the dynamics of the solid material in this region with differential capture as a function of the size. Larger particles are trapped in the outermost resonances. On the other hand, particles with sizes of order of 100 m or larger are not affected by drag forces. The location of our resonances is for the nebula model described in (e.g.) Cuzzi et al. (1993), different nebula models produce different quantitative results.