In this paper, we discuss some problems concerning the formation of Uranus and Neptune. We find that the adoption of reduced Hill spheres as the region of close interaction between planetesimals introduces an enhancement of the gravitational cross-sections in previous numerical simulations. We also discuss a way to make possible the formation of Uranus and Neptune on time-scales shorter than the age of the Solar system.Facultad de Ciencias Astronómicas y Geofísica

Brunini, Adrián

Melita, Mario Daniel

Monthly Notices of the Royal Astronomical Society

English

El Servicio de Difusión de la Creación Intelectual

Mon. Not. R. Astron. Soc. 330, 184-186 (2002)
On the accretion of Uranus and Neptune
A. Brunini*  and M. D. Melita
*E-mail: abrunini@fcaglp.unlp.edu.ar
Observatorio Astronomico, Univer sidad Nacional de La Plata, Paseo del Bosque S/N, (B1900FWA) La Plata, Argentina
Accepted 2001 October 16. Received 2001 June 25; in original form 2001 February 12
ABSTRACT
In this paper, we discuss some problems concerning the formation of Uranus and Neptune. 
We find that the adoption of reduced Hill spheres as the region of close interaction between 
planetesimals introduces an enhancement of the gravitational cross-sections in previous 
numerical simulations. We also discuss a way to make possible the formation of Uranus and 
Neptune on time-scales shorter than the age of the Solar system.
Key words: accretion, accretion discs - planets and satellites: formation - planets and 
satellites: individual: Uranus - planets and satellites: individual: Neptune - Solar system: 
formation.
1 INTRODUCTION
In the 1960s, Safronov (1972) concluded that, owing to the low 
surface density of solid material in the outer solar nebula, the time­
scale of formation of Uranus and Neptune would be longer than the 
age of the Solar system. This conclusion was based on calculations of 
planetesimal accretion in the particle-in-a-box approximation, where 
the modem concept of multi-planet accretion is not present. 
Moreover, the distant interactions between the planetesimals were 
not included. Thus the mass evolution of the system was that of 
orderly growth of the bodies. When a more accurate model was 
produced (see, for example, Wetherill & Stewart 1989 and Greenberg 
et al. 1996), it was noticed that the more massive body would 
decouple rapidly from the distribution, exhibiting a runaway growth. 
This effect is mainly due to the dynamical friction experienced by the 
biggest body. Dynamical friction, which tends to produce 
equipartition of energy, reduces the relative velocities between the 
large objects and the smaller ones, producing an enhancement of 
the gravitational cross-section of the runaway body. The end-state of 
the runaway growth phase leaves most of the total mass of the system 
contained in a few big planetesimals (or protoplanets), which orbit 
embedded in a swarm of smaller objects. This stage is usually taken 
as the initial condition for numerical simulations of the formation of 
the outer planets by multi-planet accretion (Fernández & Ip 1984, 
1996; Brunini & Fernandez 1999). Indeed, a number of features in 
the outer Solar system have been called upon as observational 
evidence of the existence of such a stage (Stem 1991).
Numerical simulations of the macro-accretion process of Uranus 
and Neptune were first carried out by Fernández & Ip (1984, 1996). 
Their simulations incorporated in a natural fashion the multi-planet 
accretion process. However, it was based on Opik’s formalism for 
planetesimal-planetesimal interactions (Ópik 1951) (thus keeping 
Safronov’s particle-in-a-box approximation).
A serious limitation of Opik’s formalism is that it allows bodies 
to collide if they are on mutual crossing orbits only. This is not a 
serious limitation whenever relative velocities are high, so radial 
exclusions may be large. In this case planetary embryos grow at a 
moderate rate, because at large relative velocities the gravitational 
focusing factor is small. However, the most important regime of 
accretion, the runaway growth, occurs during the first stages of the 
process, when relative velocities are small enough to make 
dynamical friction operative (Wetherill & Stewart 1989; Ida 1990; 
Ida & Makino 1991; Greenberg et al. 1996).
In view of the previously mentioned limitations of the Opik 
formalism to explain relevant features of the accretion of Uranus 
and Neptune, Brunini & Fernández (1999) investigated this 
problem by means of a more complete numerical modelling of the 
accretion process, based on the numerical integration of the 
equations of motion that naturally overcomes the limitations 
mentioned above. The main results of Fernández & Ip (1984) were 
confirmed. As a consequence of the runaway growth, in these new 
simulations the time-scale of formation of Uranus and Neptune was 
reduced to only some 107yr. It was also found that the orbital 
evolution of the outer planetary system was governed by the 
exchange of orbital angular momentum between the protoplanets 
and the planetesimals, and the resonant coupling between the 
protoplanetary system.
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2 PROBLEMS
In Fernández & Ip (1984) as in Brunini & Fernández (1999), close 
encounters were treated neglecting the central gravitational 
attraction of the Sun. This approximation tends to overestimate 
the gravitational cross-sections during low relative velocity 
encounters by a factor of up to 4 (Ida 1990). The fact that 
Fernández & Ip (1984) have considered cross-sections multiplied 
by a factor of up to 4, in an attempt to simulate the presence of 
primordial atmospheres surrounding the protoplanets, does not
© 2002 RAS
On the accretion of Uranus and Neptune 185
mean that the factor of multiplication of the gravitational cross­
section was 4X4= 16, because in those simulations all the 
encounters were at a relative velocity regime where the two-body 
approximation is good.
However, as a way to simulate the effects of the presence of a 
primordial solar nebula on the heliocentric dynamics of the 
planetesimal, Fernández & Ip (1984) and Brunini & Fernández 
(1999) have considered the interaction of the planetesimals only 
where the interacting bodies were within a certain fraction of the 
mutual Hill radius. It seemed to be a simple way to damp the 
gravitational warming rate of the system, which would be a natural 
consequence of the presence of a gaseous medium. This strategy 
introduces a large and non-systematic overestimation of the rate of 
accretion.
To understand this fact, we must consider first that the 
planetesimal-planetesimal interaction was accounted for only within 
this reduced sphere of action. Therefore, from the border of the 
mutual Hill sphere up to this reduced sphere of action, the 
acceleration of the planetesimals due to the gravity of the planet was 
neglected. This tends to underestimate the relative velocity of the 
encounters and hence to overestimate the gravitational cross-sections.
When including the solar perturbations during close encounters, 
not predicting the accretion events by means of a two-body 
approximation, and considering the radius of the Hill sphere as the 
distance of close interactions, we noted that the accretion process 
was much less efficient. In fact, during some preliminary runs 
(using the same initial conditions as in Brunini & Fernández 1999), 
planets as massive as Uranus and Neptune were never formed 
(Melita & Brunini 1999). The rate of pairwise collisions was 
reduced by a factor of 6-10. Therefore the dynamical excitation of 
the system soon precludes the start of runaway growth of big 
planetesimals.
section, suggest that, in the outer Solar system, the formation of 
solid cores of those magnitudes is unlikely on any time-scale, at 
least without the inclusion of the effect of the primordial nebular 
gas. Radial migration of the planetary embryos embedded in 
the protoplanetary gaseous nebula, as well as the damping in the 
relative velocities of planetesimals due to the presence of the 
primordial gas may favour accretion at large distances from 
the central star (Papaloizou & Larwood 2000). Nevertheless, these 
are problems which are beyond the scope of the present paper.
4 STEVENSON’S PLANETOIDS
An alternative scenario was proposed by Stevenson (1982, 1984). 
He argues that, in the outer Solar system, solid bodies were rich in 
condensed ices. The gravitational energy released in a collision of 
planetesimal pairs is enough to vaporize the ices. Although the low 
temperatures at this distances from the Sun favour a rapid re­
condensation, a fraction of these vapours could mix with the 
primordial nebula, substantially enhancing its mean molecular 
density. A dense gas can be much more easily trapped by a low- 
mass planetesimal. Some simplifying arguments show that a Mars­
sized planetoid is able to acquire a large and dense envelope 
(Lissauer et al. 1994) of densities as high as 10-5gcm-3 at the 
distance of Neptune.
In this scenario, Uranus and Neptune accreted through 
successive pairwise collisions between those large planetoids. It 
is difficult to anticipate the fraction of gas that is retained during an 
energetic event such as a collision between two of those objects. 
Nevertheless, the sum of the gas present in all of the planetoid 
envelopes would be orders of magnitude larger than the combined 
gas in the actual atmospheres of Uranus and Neptune, so there is a 
certain tolerance for gas loss.
Another very interesting feature of this scenario is the fact that 
the time-scales of formation of Uranus and Neptune could be 
longer than the time-scale of complete dispersal of the primordial 
gas in the solar nebula, because the gaseous component is carried 
by the protoplanets.
We have performed a series of numerical experiments to 
determine the effects of these extended atmospheres in the 
dynamics of close encounters between these gaseous planetoids 
and planetesimals of different sizes.
We have adopted the model of a planetoid described by Lissauer 
et al. (1994). A solid core of mean density lgcm-3 and mass 
103 *26g is considered, orbiting the Sun at the distance of Uranus (an 
intermediate distance between Saturn and Neptune) on a circular 
orbit. We have assumed that these planetoids were surrounded by a 
dense, uniform and compact atmosphere, extended up to 10 
planetary radii. Beyond this distance, the mean density reaches 
nebular conditions. We have explored gaseous densities for the 
compact atmospheres ranging from 10-5 to 10-7 gem-3 [this last 
value is two orders of magnitude less than the densities considered 
by Lissauer et al. (1994)]. Planetesimals of radius 100-1000km 
were launched from the border of the Hill sphere, with a relative 
velocity (shown in Fig. 1 as a fraction of the escape velocity, given 
by Ve = v-GM',//?ii. Rh being the radius of the Hills sphere of the 
planet) such that during the first pericentric passage they pass 
through the planetary atmosphere. For each relative velocity V'o the 
impact parameter b was computed as
3 THE STANDARD MODEL
There is now a generalized consensus that the mainly gaseous
Jupiter and Saturn had to form before the dispersal of hydrogen and
helium from the primordial solar nebula, which is thought to have
occurred before lOMyr. This is supported by recent radio CO
observations (Zuckerman, Forveille & Kastner 1995) and the
observation of ‘naked T Tauri’ stars with ages of approximately 
1 Myr (Walter et al. 1988), suggesting that the molecular gas
surrounding young solar-type stars tends to dissipate rapidly, over
time-scales no longer than a few Myr. Because of the possible
presence of a non-negligible amount of hydrogen and helium in
Uranus and Neptune, of the order of 1.6 and 1.1 M® respectively.
Pollack et al. (1996) suggested that their time-scales of formation
could not have been much longer than the time-scale of dissipation
of the gaseous component of the nebula, perhaps no longer than a
few X 107yr. This conclusion was based on Mizuno’s core
instability theory of the accretion of gas by giant planets (Mizuno 
1980). In this theory, once solid cores of the order of 10-20 M©
have accreted (Bodenheimer & Pollack 1986; Pollack et al. 1996),
the primordial nebular gas enclosed within the Hill sphere of
the planet is compressed on to the surface of the planetary core.
The region left by this gas is occupied by fresh gas falling from the
surrounding nebula. In addition, the Hill sphere is continuously
expanding because of the increment in the planetary mass as a
consequence of the rapid accretion of gas. This process generates a 
self-sustained hydrodynamical instability by which the planet is 
able to accrete large amounts of gas on very short time-scales.
Our preliminary results, briefly described in the previous
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© 2002 RAS, MNRAS 330, 184—186
186 A. Brunini and M. D. Melita
h----------r—'\'X w\ \\ \\ \\
\
r=1000km
T
r=100 km
\ I
9
8
7
6
5
4
3
2
1
0 0.5 1 1.5 2 2.5 3
Figure 2. Relative enhancement of the cross-section due to the presence of 
an extended envelope, for a planetesimal of 100 km radius and different 
values of the density of the gas. The solid lines correspond to orbits with 
apocentre distance less than the Hill radius of the planet after the first 
passage through the atmosphere. Dotted lines correspond to direct 
collisions on the planet.
Figure 1. Relative enhancement of the cross-section owing to the presence 
of an extended envelope of density p ~ IO-’ gem1 for planetesimals of 
different radii. So is the gravitational cross-section in the absence of gas. 
The solid lines correspond to orbits with apocentre distance less than the 
Hill radius of the planet after the first passage through the atmosphere. 
Dotted lines correspond to direct collisions on the planet.
where R,nin is a distance ranging from the geometrical radius of the 
planet to a certain fraction of the radius of the extended atmosphere 
(we used 9 planetary radii as the upper limit). In addition to a direct 
collision on to the planetary surface, we have also considered 
accretion when, after the first passage through the planetary 
atmosphere, the planetesimal becomes permanently bonded to the 
planet (i.e. when its apocentric distance becomes less than the Hill 
radius). Each encounter was treated as a two-body problem, but gas 
drag was included. The code is described in full detail by Brunini, 
Giordano & Orellana (1996).
Fig. 1 displays the factor of enhancement of the effective cross­
section of the planet for a dense atmosphere and different 
planetesimal radii. It can be observed that, in this model, the cross­
sections are strongly enhanced even for the less favourable case of 
r = 1000 km.
Fig. 2 shows the same as in Fig. 1, but for the same planetesimal 
radii (r = 100 km) and different atmospheric densities. We can see 
that small planetesimals are efficiently trapped by the effect of gas, 
even in the low-density case.
For very low relative velocities, all the curves shrink around 
S/Sq ~ 9 (S is the gravitational cross-section in the presence of gas, 
whereas So is the same quantity when gas is not present). This is 
easily explained as follows. Gas drag is effective in reducing the 
apocentric distance. If after passing through the atmosphere the 
apocentric distance is smaller than the Hill radius, the planetesimal 
becomes bonded to the planet. So, below a sufficiently small 
relative velocity, all the planetesimals are trapped. This means that 
below a certain relative velocity all the planetesimals with impact 
parameter given by equation (1), with rmm = 9 planetary radii (the 
upper limit that we used), were considered as potential impactors. 
Using equation (1) to compare with the impact parameter for 
collision without gas (i.e. with = 1 planetary radius), we get 
S/So ~ 9.
As, during the phase of runaway growth in the presence of a 
gaseous medium, the encounters are at very low relative velocities, 
the presence of compact atmospheres surrounding the biggest 
planetesimals could in principle provide a way to enhance the
gravitational cross-sections by a factor high enough to solve the 
problem of the formation of Uranus and Neptune on time-scales 
shorter than the age of the Solar system.
REFERENCES
Bodenheimer R. Pollack J. B.. 1986. Icarus. 67. 391
Brunini A.. Fernández J. A.. 1999. Planet. Space Sei., 47. 591
Brunini A.. Giordano C. M.. Orellana R. B.. 1996. A&A, 314, 977
Fernández J. A.. Ip W. H.. 1984. Icarus. 58. 109
Fernández J. A.. Ip W. H.. 1996. Planet. Space Sei., 44. 431
Greenberg R.. Bottke W.. Carussi A.. Valsecchi G. B.. 1996. Icarus. 94. 98 
Ida S.. 1990. Icarus. 88. 129
Ida S.. Makino J.. 1991. Icarus. 98. 28
Lissauer J. J.. Pollack J. B.. Wetherill G. W.. Stevenson D. J.. 1994. in 
Cruikshank D. R. ed.. Neptune and Triton. Univ. Arizona Press. Tucson, 
p. 37
Melita M. D.. Brunini A.. 1999. in Svoreñ J.. Pittich E. M.. Rickman H.. 
eds. Proc. IAU Colloq. 173, Evolution and Source Region of Asteroids 
and Comets. Astron. Inst. Slovak Acad. Sei., p. 37
Mizuno H.. 1980. Prog. Theor. Phys., 64. 544
Ópik E. J.. 1951. Proc. R. Irish Acad., 54A, 165
Papaloizou J. C. B.. Larwood J. D.. 2000, MNRAS. 315, 823
Pollack J. B.. Hubickyj O.. Bodenheimer P. Lissaueer J. J.. Podolak M.. 
Greenzweig Y.. 1996. Icarus. 124, 62
Safronov V. S.. 1972. Evolution of the protoplanetary cloud and the 
formation of the Earth and the planets. Israel Program of Science 
Translation. Jerusalem
Stern S. A.. 1991. Icarus. 90. 271
Stevenson D. J.. 1982. Planet. Space Sei., 30. 755
Stevenson D. J.. 1984. Lunar Planet. Sei., XV. 822
Walter F. M.. Brown A.. Mathieu R. D.. Myers P. C.. Vrba F. J.. 1988. AJ. 
96. 297
Wetherill G. W.. Stewart G. R.. 1989. Icarus. 77. 330
Zuckerman B.. Forveille T. Kastner J. H.. 1995. Nat. 373, 494
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On the accretion of Uranus and Neptune

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