Constraints to Uranus' Great Collision. IV. The Origin of Prospero
Gabriela Parisi, Giovanni Carraro, Michele Maris, Adrian Brunini
aa r X i v : . [ a s t r o - ph ] J a n Astronomy&Astrophysicsmanuscript no. gparisi c (cid:13)
ESO 2018October 30, 2018
Constraints to Uranus’ Great Collision IV
The Origin of Prospero
M. Gabriela Parisi , , ,⋆ , Giovanni Carraro , , Michele Maris , and Adrian Brunini Instituto Argentino de Radioastronom´ıa (IAR), C.C. N o
5, 1894 Villa Elisa, Argentinae-mail: [email protected] Departamento de Astronom´ıa, Universidad de Chile, Casilla 36-D, Santiago, Chilee-mail: [email protected] Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, Argentinae-mail: gparisi,[email protected] European Southern Observatory (ESO), Alonso de Cordova 3107, Vitacura, Santiago, Chilee-mail: [email protected] Dipartamento di Atronomia, Universit´a di Padova, Vicolo Osservatorio 2, I-35122 Padova, Italye-mail: [email protected] INAF, Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italye-mail: [email protected]
Received June 2007 ; accepted January 2008
ABSTRACT
Context.
It is widely accepted that the large obliquity of Uranus is the result of a great tangential collision (GC) with an Earth sizeproto-planet at the end of the accretion process. The impulse imparted by the GC had a ff ected the Uranian satellite system. Veryrecently, nine irregular satellites (irregulars) have been discovered around Uranus. Their orbital and physical properties, in particularthose of the irregular Prospero, set constraints on the GC scenario. Aims.
We attempt to set constraints on the GC scenario as the cause of Uranus’ obliquity as well as on the mechanisms able to giveorigin to the Uranian irregulars.
Methods. Di ff erent capture mechanisms for irregulars operate at di ff erent stages on the giant planets formation process. The mech-anisms able to capture the uranian irregulars before and after the GC are analysed. Assuming that they were captured before the GC,we calculate the orbital transfer of the nine irregulars by the impulse imparted by the GC. If their orbital transfer results dynamicallyimplausible, they should have originated after the GC. We then investigate and discuss the dissipative mechanisms able to operatelater. Results.
Very few transfers exist for five of the irregulars, which makes their existence before the GC hardly expected. In particularProspero could not exist at the time of the GC. Di ff erent capture mechanisms for Prospero after the GC are investigated. Gas dragby Uranus’envelope and pull-down capture are not plausible mechanisms. Capture of Prospero through a collisionless interactionseems to be di ffi cult. The GC itself provides a mechanism of permanent capture. However, the capture of Prospero by the GC is a lowprobable event. Catastrophic collisions could be a possible mechanism for the birth of Prospero and the other irregulars after the GC.Orbital and physical clusterings should then be expected. Conclusions.
Either Prospero had to originate after the GC or the GC did not occur. In the former case, the mechanism for theorigin of Prospero after the GC remains an open question. An observing program able to look for dynamical and physical families ismandatory. In the latter case, another theory to account for Uranus’ obliquity and the formation of the Uranian regular satellites onthe equatorial plane of the planet would be needed.
Key words.
Planets and satellites: general – Planets and satellites: formation– Solar System: general– Solar System: formation
1. Introduction
Very recently, rich systems of irregular satellites (hereafter irregulars) of the giant planets have been discovered. Enabled by theuse of large-format digital images on ground-based telescopes, new observational data have increased the known population ofJovian irregulars to 55 (Sheppard et al. 2003), the Saturnian population to 38 (Gladman et al. 2001, Sheppard et al. 2005a, 2006a)and the Neptunian population to 7 (Holman et al. 2004, Sheppard et al. 2006b). The Uranian system is of particular interest since apopulation of 9 irregulars (named Caliban, Sycorax, Prospero, Setebos, Stephano, Trinculo, S / / / Send o ff print requests to : M. Gabriela Parisi ⋆ Member of the Carrera del Investigador Cient´ıfico, Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina M. Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IV
Irregulars of giant planets are characterized by eccentric, highly tilted with respect of the parent planet equatorial plane, and insome case retrograde, orbits. These objects cannot have formed by circumplanetary accretion as the regular satellites but they arelikely products of an early capture of primordial objects from heliocentric orbits, probably in association with planet formation itself(Jewitt & Sheppard 2005). It is possible for an object circling about the sun to be temporarily trapped by a planet. In terms of theclassical three-body problem this type of capture can occur when the object passes through the interior Lagrangian point, L , with avery low relative velocity. But, without any other mechanism, such a capture is not permanent and the objects will eventually returnto a solar orbit after several or several hundred orbital periods. To turn a temporary capture into a permanent one requires a sourceof orbital energy dissipation and that particles could remain inside the Hill sphere long enough for the capture to be e ff ective.Although currently giant planets have no e ffi cient mechanism of energy dissipation for permanent capture, at their formationepoch several mechanisms may have operated: 1) gas drag in the solar nebula or in an extended, primordial planetary atmosphere orin a circumplanetary disk (Pollack et al.1979, Cuk & Burns 2003), 2) pull-down capture caused by the mass growth and / or orbitalexpansion of the planet which expands its Hill sphere (Brunini 1995, Heppenheimer & Porco 1977), 3) collisionless interactions between a massive planetary satellite and guest bodies (Tsui 1999) or between the planet and a binary object (Agnor & Hamilton2006), and 4) collisional interaction between two planetesimals passing near the planet or between a planetesimal and a regularsatellite. This last mechanism, the so called break-up process, leads to the formation of dynamical groupings (e.g. Colombo &Franklin 1971, Nesvorny et al. 2004). After a break-up the resulting fragments of each progenitor would form a population ofirregulars with similar surface composition, i.e. similar colors, and irregular shapes, i.e. light-curves of wide amplitude. Significantfluctuations in the light-curves of Caliban (Maris et al. 2001) and Prospero (Maris et al. 2007a) and the time dependence observedin the spectrum of Sycorax (Romon et al. 2001) suggest the idea of a break-up process for the origin of these bodies.Several theories to account for the large obliquity of Uranus have been proposed. Kubo-Oka & Nakazawa (1995), investigatedthe tidal evolution of satellite orbits and examined the possibility that the orbital decay of a retrograde satellite leads to the largeobliquity of Uranus, but the large mass required for the hypothetical satellite makes this possibility very implausible. An asymmetricinfall or torques from nearby mass concentrations during the collapse of the molecular cloud core leading to the formation of theSolar System, could twist the total angular momentum vector of the planetary system. This twist could generate the obliquities ofthe outer planets (Tremaine 1991). This model has the disadvantages that the outer planets must form before the infall is completeand that the conditions for the event that would produce the twist are rather strict. The model itself is di ffi cult to be quantitativelytested. Tsiganis et al. (2005) proposed that the current orbital architecture of the outer Solar System could have been produced froman initially compact configuration with Jupiter and Saturn crossing the 2:1 orbital resonance by divergent migration. The crossingled to close encounters among the giant planets, producing large orbital eccentricities and inclinations which were subsequentlydamped to the current value by gravitational interactions with planetesimals. The obliquity changes due to the change in the orbitalinclinations. Since the inclinations are damped by planetesimals interactions on timescales much shorter than the timescales forprecession due to the torques from the Sun, especially for Uranus and Neptune, the obliquity returns to small values if it is smallbefore the encounters (Hoi et al. 2007).Large stochastic impacts at the last stage of the planetary formation process have been proposed as the possible cause of theplanetary obliquities (e.g. Safronov 1969). The large obliquity of Uranus (98 ◦ ) is usually attributed to a great tangential collision(GC) between the planet and an Earth-size planetesimal occurred at the end of the epoch of accretion (e.g., Parisi & Brunini 1997,Korycansky et al. 1990). The collision imparts an impulse to Uranus and allows preexisting satellites of the planet to change theirorbits. Irregulars on orbits with too large semimajor axis escape from the system (Parisi & Brunini 1997), while irregulars witha smaller semimajor axis may be pushed to outer or inner orbits acquiring greater or lower eccentricities depending on the initialorbital elements, the geometry of the impact and the satellite position at the moment of impact. The orbits excited by this perturbationmust be consistent with the present orbital configuration of the Uranian irregulars (BP02).In an attempt to clarify the origin of Uranus obliquity and of its irregulars, we are using in this study the most updated informationon their orbital and physical properties.In Section 2, we improve the model developed in BP02 for the five Uranian irregulars known at that epoch and extend our studyto the new four Uranian irregulars recently discovered by Kavelaars et al. (2004) and Sheppard et al. (2005b). The origin of theseobjects after the GC is discussed in Section 3, where several mechanisms for the origin of Prospero are investigated. The discussionof the results and the conclusions are presented in Section 4.
2. Transfer of the irregulars to their current orbits:
Assuming the GC scenario, the transfers of the nine known irregulars to their current orbits are computed following the proceduredeveloped in BP02 for the five irregulars known in 2002. We present improved calculations using a more realistic code to computethe evolution of the irregulars current orbital eccentricities.If the large obliquity of Uranus has been the result of a giant tangential impact, the orbits of preexisting satellites changed dueto the impulse imparted to the planet by the collision. The angular momentum and impulse transfer to the Uranian system at impactwere modeled using the Uranus present day rotational and orbital properties as imput parameters (BP02).Just before the GC, the square of the orbital velocity ν of a preexisting satellite of negligible mass is given by: ν = Gm U r − a ! , (1) . Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IV 3 r being the position of the satellite on its orbit at the moment of the GC, a its orbital semiaxis and m U the mass of Uranus beforethe impact. The impactor mass is m i and G is the gravitational constant. After the GC, the satellite is transferred to another orbitwith semiaxis a acquiring the following square of the velocity: ν = G ( m U + m i ) r − a ! . (2)We set ν = A ν e and ν = B (1 + m i / m U ) ν e , where A and B are arbitrary coe ffi cients (0 < A ≤ B > ν e being the escapevelocity at r before the GC.The semiaxis of the satellite orbit before ( a ) and after ( a ) the GC verify the following simple relations: a = r − A ) , a = r − B ) . (3)If A < B then a < a . In the special case of B =
1, the orbits are unbound from the system. If A > B then a > a , the initialorbit is transferred to an inner orbit. When A = B , the orbital semiaxis remains unchanged ( a = a ).The position r of the satellite on its orbit at the epoch of the impact may be expressed in the following form: r = G m U ( ∆ V )2 " B ′ − A √ A cos Ψ ± √ ( B ′ − A ) + A cos 2 Ψ , (4)with B ′ = B (1 + m i / m U ). Since stochastic processes can only take place at very late stages in the history of planetary accretion (e.g.Lissauer & Safronov 1991), the GC is assumed to occur at the end of Uranus formation (e.g. Korycansky et al. 1990). The mass ofUranus after the GC, ( m i + m U ), is taken as Uranus’ present mass. Ψ is the angle between ν and the orbital velocity change impartedto Uranus ∆ V . An analytical expression for ∆ V is derived in BP02 assuming that the impact is inelastic (Korycansky et al. 1990) asa function of m i , the impact parameter of the collision b , the present rotation angular velocity of Uranus Ω , the spin angular velocitywhich Uranus would have today if the collision had not occurred Ω , and α which is the angle between Ω and Ω : ∆ V = R U b " Ω + Ω (1 + m i m U ) (1 + m i m U ) − ΩΩ cos α (1 + m i m U )(1 + m i m U ) / , (5) R U being the present equatorial radius of Uranus. A collision with the core itself was necessary in order to impart the requiredadditional mass and angular momentum (Korycansky et al. 1990). Since b is an unknown quantity, we take its most probable value: b = (2 / R C , where R C is the core radius of Uranus at the moment of collision assumed to be 1.8 × km (Korycansky et al. 1990,Bodenheimer & Pollack 1986). The results have a smooth dependence with the impactor mass which allows us to take m i ∼ m ⊕ (Parisi & Brunini 1997).The minimum eccentricity of the orbits before the collision is given by: e min = − A ) − i f A ≤ . , e min = − − A ) i f A > . , (6)while the minimum eccentricity of the orbits after the collision is: e min = − B ) − i f B ≤ . , e min = − − B ) i f B > . . (7)The minimum possible value of ∆ V ( ∆ V min ) is obtained from Eq. (5) for an initial period T =
20 hrs ( T = π/ Ω ) and α = o (BP02). Therefore, although simulations of solid accretion produce in general random spin orientations (e.g. Chambers 2001), wefurther assume T =
20 hrs and α = o in order to set a maximum bound on Eq. (4). Upper bounds in a ( a M ) and a ( a M ) areobtained from Eq. (3) through Eqs. (4) and (5) taking ∆ V =∆ V min with Ψ= o , i.e., assuming the impact in the direction oppositeto the orbital motion of the satellites and taking the positive sign of the square root in Eq. (4): a M = G m U ( B ′ − A )2( ∆ V min )2 (1 − A )( √ B ′ − √ A ) , a M = G m U ( B ′ − A )2( ∆ V min )2 (1 − B )( √ B ′ − √ A ) . (8)For each A, we calculate the value of B (B = B ′ / (1 + m i / m U )) corresponding to the transfer to a M = a , where a is the presentorbital semiaxis of each one of the Uranian irregulars shown in the second column of Table 1 in units of R U . From Eq. (7), thisvalue of B provides the minimum possible value of e min , e m , that the orbit of each irregular may acquire at impact for each initialcondition A and every initial condition for T , α , Ψ and m i , i.e., if a transfer of a given orbit (A,B) is not possible for Ψ= o , T =
20 hrs and α = o , the same transfer (A,B) is not possible for any other incident direction of the impactor and for any other valueof T and α either.Since the orbits of the irregulars are time dependent, the orbital evolution of the five Uranian irregulars known in 2002 wascomputed in PB02 by numerical integration of the equations of the elliptical restricted three body problem formed by the Sun,Uranus and the satellite. In this paper, we present the orbital evolution of the nine known Uranian irregulars for 10 yrs using the M. Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IVSatellite r s [ km ] a [ R U ] e mean a i [ R U ] e i ( ∆ a ) / a i ) ( ∆ e / e i )Caliban 49 283 0.191 287.5 0.1973 1.602 x 10 − − Sycorax 95 482 0.541 485 0.5436 6.307 x 10 − − Prospero 15 645 0.432 648 0.4342 4.585 x 10 − − Setebos 15 694 0.581 701.8 0.5853 1.123 x 10 − − Stephano 10 314 0.251 333 0.2781 6.058 x 10 − − − − Francisco 6 169 0.142 280.6 0.3593 0.6604 1.530Margaret 5.5 579 0.633 649 0.6705 0.1209 5.917 x 10 − Table 1.
Present parameters of the Uranian irregulars and orbital damping due to gas drag exerted by Uranus extended envelope. r s and a are the present physical radius and the present orbital semiaxis of the irregulars. e mean is their calculated mean eccentricitytabulated in Table 2. a i and e i are the orbital semiaxis and eccentricity just after the GC, while ( ∆ a ) / a i ) and ( ∆ e / e i ) are the dampingof these orbital elements since the epoch of the GC until the contraction of Uranus envelope. Satellite e mean e max e min Caliban 0.191 0.315 0.072Sycorax 0.514 0.594 0.438Prospero 0.432 0.571 0.305Setebos 0.581 0.704 0.463Stephano 0.251 0.381 0.121Trinculo 0.218 0.237 0.200Ferdinand 0.660 0.970 0.393Francisco 0.142 0.187 0.093Margaret 0.633 0.854 0.430
Table 2.
Variation of the eccentricity of the Uranian irregulars due to Solar and giant planet perturbations over a period of 10 yrs.symplectic integrator of Wisdom & Holman (1991), where the perturbations of the Sun, Jupiter, Saturn and Neptune are included.The mean ( e mean ), maximum ( e max ) and minimum ( e min ) eccentricities are shown in Table 2 for all the known Uranian irregulars.The transfer of each satellite from its original orbit to the present one is possible only for those values of (A,B) which satisfythe condition e m < e max . A satellite did not exist before the impact if it has no transfer and the satellites with the widest range oftransfers are those with the highest probability of existing before the impact.The transfers within a range of 20 R U around each present satellite semiaxis ( a M = a ± R U ; a taken from Table 1) for all theUranian irregulars are shown in Fig.1. There are few transfers for Setebos, Ferdinand and Margaret. This makes the existence ofthese satellites before collision little probable. The only transfers for Trinculo and Prospero are close to the pericenter of an eccentricinitial outer orbit ( e m > e m > e m for Trinculo isin the range [0.16-0.23], very close to e max (0.237). This result gives a very low probability for the existence of Trinculo before theGC. For Prospero e m is in the range [0.52-0.57], e m ∼ e max (0.571). Therefore this satellite could not exist before the GC. If thepresent large obliquity of Uranus was caused by a large impact at the end of its formation, Prospero had to originate after the event.Relating the origin of the outer Uranian system to a common formation process, all the Uranian irregulars probably were originatedafter the GC. The possible post-GC origin of Prospero and the other Uranian irregulars is discussed in the following section.
3. Origin of Prospero after the Great Collision
In this section, we analyze the possibility that Prospero have been captured after the GC. We investigate the possible dissipativemechanisms able to produce its permanent capture taking into account that the giant impact is assumed to have occurred at latestages in the planetary accretion process.
Bodenheimer & Pollack (1986) and Pollack et al. (1996) studied the formation of the giant planets by accretion of solids and gas.In their model, the so called core instability scenario, when the mass of the core of the planet has grown enough a gaseous envelopebegins to form around it. For Uranus, its envelope extended until its accretion radius which was ∼ R U at the end of Uranus’formation (Bodenheimer & Pollack 1986). The formation of Uranus is completed when there is no more nebular gas to accrete.Otherwise, gas accretion by proto-Uranus would have continued towards the runaway gas accretion phase and the planet wouldhave now a massive gaseous envelope. Bodenheimer & Pollack (1986) obtained that after the end of accretion, the radius of theenvelope of proto-Uranus remained almost constant ( ∼ R U ) over a time scale of 10 yrs and then contracted rapidly to ∼ R U in 10 yrs. The final contraction to the present-day planetary radius occurred on a slower timescale of 10 yrs.Korycansky et al. (1990) carried out hydrodynamical calculations of the GC for a large set of initial conditions at the endof accretion. They found a sharp transition between the cases where almost all the mass of the envelope of Uranus remainedafter the impact and those where it was almost entirely dispersed by the impact. This implies that the impact should have not . Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IV 5 Fig. 1.
The transfers capable of producing the present orbits of the Uranian Irregulars. A (B) is the square of the ratio of the satellite’sspeed just before (after) the impact to the escape velocity at the satellite’s location just before (after) the impact. e m ( e m ) is theminimum eccentricity of the orbits before (after) collision. The full-black line A = B divides the upper region (the current orbitsarise from inner orbits) and the lower region (the current orbits arise from outer orbits). The value of e max tabulated in Table 2 isshown on the dashed line for comparation with e m . Trinculo: empty down triangles, Caliban: full circles, Sycorax: empty rhombus,Ferdinand: empty squares, Francisco: full down triangles, Margaret: empty up triangles, Prospero: full up triangles, Setebos: emptyhexagons, Stephano: empty circles.dispersed the envelope, as there would have been no nebular gas to re-accrete on the planet. They showed that the envelope reactshydrodynamically at impact and it expands outward. After the shock the gas falls back on the core over a timescale of few hoursbeing the final result a readjustment instead of a catastrophic transformation. The timescale for this hydrodynamical process is muchshorter than the orbital period of the irregulars which is of the order of years. We may then assume that the GC did not change theenvelope density profile. M. Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IV
The extended envelope of Uranus could be in principle, a source of gas allowing the capture of Prospero and the other irregularsafter the GC. Assuming that the GC did not change the envelope profile, we fit from Fig.1 of Korycansky et al. (1990) the densityprofile of Uranus’ gaseous envelope before the GC, ρ g = c te R − g cm − with c te = and R being measured in cm. It gives anebular density of ∼ × − g cm − at the boundary of 500 R U in agreement with the minimum mass nebula model.As a first approximation, we compute the ratio of gas mass traversed by a body of density ρ s and radius r s in a characteristicorbital period P, to the mass of the body. Assuming for the body a circular orbit of radius R, we calculate the so-called β parameter(Pollack et al. 1979): β = P τ = π R ρ g π r s πρ s r s = πρ g R ρ s r s , (9)where τ is the characteristic timescale for changing any of the orbital parameters. For the permanent capture to occur β cannot bevery small, β ≥ c te = ) for an object the size of Prospero ( ρ s = − ) and at Prospero’s pericenter (R = R U ), β ∼ × − , which is toosmall to a ff ect the orbit of Prospero.Following BP02, we now investigate with more detail the possible e ff ect of gas drag on the Uranian irregulars after the GC dueto Uranus’ extended envelope before its contraction to its present state. Following the procedure of Adachi et al. (1976), we obtainthe time variations of the eccentricity e and semiaxis a of each Uranian irregular. The drag force per unit mass is expressed in theform: F = − C ρ g v rel , C = C D π r s m , (10)where v rel is the relative velocity of the satellite with respect to the gas. In computing the satellite mass m , a satellite mean density ρ s of 1.5 g cm − is taken for all the satellites (http: // ssd.jpl.nasa.gov / ?sat phys par). The drag coe ffi cient C D is ∼ r s is eachsatellite radius taken for each satellite from Table 1.Assuming that the orbital elements are constant within one Keplerian period (the variations of a and e are very small), weconsider the rates of change of the elements averaged over one period, that is: h dadt i = − C π ( G ( m i + m U ) a ) Z π ρ g ( e + + e cos θ ) (1 + e cos θ ) d θ h dedt i = − C π (1 − e ) G ( m i + m U ) a ! Z π ρ g ( e + cos θ )( e + + e cos θ ) (1 + e cos θ ) d θ. (11)Since after the end of accretion the gas density in the outer regions of the envelope contracts rapidly, we have integrated Eqs.(11)back in time on 10 yrs for the 9 Uranian irregulars. We have taken v rel as the satellite orbital velocity since we have assumed anull gas velocity. This assumption maximizes the orbital damping for retrograde satellites which allows us to set upper bounds inthe damping e ff ect for the orbital eccentricity and semiaxis of all the retrograde irregulars. The mean eccentricity e mean and theactual semiaxis a from Table 1 were taken as the initial conditions for the integrations. We take ρ g = c te R − g cm − with c te = and R = a (1 − e ) / (1 + e cos θ ), a being measured in cm. The orbital damping is shown in Table 1, where a i and e i are theinitial semiaxis and eccentricity just after the GC at the end of accretion and ∆ a = ( a i − a ) and ∆ e = ( e i − e mean ), are the damping inthe orbital semiaxis and eccentricity. Stephano, Trinculo and Margaret had experienced little orbital evolution while Francisco hadsu ff ered a large orbital damping (see Table 1). The permitted transfers of Fig.1 would increase for Francisco since the condition e m < e i should be then satisfied. However, the orbital evolution of the larger satellites, in particular of Prospero, results negligible. Evenincreasing the nebula density by a factor of 10 ( c te = ) the damping of the orbital elements of Prospero is too small with ∆ a / a i = ∆ e / e i = ff erent size. The fragments move away oneanother since drag forces vary inversely with size and act to separate them. The average pressure on the forward hemisphere of anon-rotating, spherical body as it moves through the gas with relative velocity v rel is approximately equal to the dynamic pressure, p dyn = ρ g v rel (Pollack et al.1979). The body will fragment into pieces if p dyn ≥ Q, where Q is the compressive strength. Valuesof Q on the order of 3 × dyne cm − are needed to shatter strong (e.g. rock / ice) targets ( which is 10 times lower than the valueadopted for asteroids), while compressive strength on the order of 3 × dyne cm − are appropriate for relatively weak (snow-like)targets (Farinella & Davis 1996, Stern 1996). For a body on circular orbit at the present pericenter of Prospero (R = R U ), where v rel is the circular speed around Uranus at R and taking c te = for ρ g , p dyn ∼ − , and at Sycorax’s pericenter p dyn ∼ − . In both cases p dyn << Q and Prospero could not have been originated by the dynamical rupture of a parent object.A collision may fracture the parent body but if the energy at impact is not su ffi cient to disperse the fragments, drag forces mayact to separate them against their mutual attraction. The relative importance of these e ff ects is measured by the ratio ǫ j of the dragforce on a given fragment j to the gravitational force acting on j by the other fragments i (Pollack et al.1979): ǫ j = X i , i , j F D j m j F Gi j m j , (12) . Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IV 7 where F Gi j is the gravitational force between the particles i and j, and F D j is the drag force on the particle j of radius r j : F D j = C D ρ g π r j v πρ s r j , (13)The fragment j is dispersed by the gas if ǫ j > ff ect, wecomputed Eq.(12) for j = Prospero at Prospero’s pericenter taking into account the gravitational attraction due to another fragment ofequal size, and at Sycorax pericenter taking into account the gravitational force of Sycorax on Prospero. In the first case, we obtain ǫ j = − and in the second case at Sycorax pericenter ǫ j = ff ered noorbital evolution due to gas drag and could not have been capture by Uranus’envelope after the GC. Within the GC scenario, runaway of the cores of the planets occurred during the first stages of accretion but stopped for eachembryo after it reached a size of about 1000 Km. At 10-35 AU the final mass distribution contained several hundreds of Mars-size(or larger) bodies dominating the mass of the residual disk. Beaug´e et al. (2002), investigated the e ff ects of the post-formationplanetary migration on satellites orbits. They obtained that if the large-body component (composed of Mars-size bodies) dominatedthe mass of the residual disk, the presently accepted change in the orbit of Uranus of ∼ ∼ ffi cient instabilities to eject allthe Uranian irregulars. Pull-down capture caused by the orbital expansion of the planet could then not be a plausible mechanism forthe origin of Prospero and the other irregulars. Pull-down capture caused by the mass growth of the planet after the GC would notbe possible given the impact is assumed to have occurred at the end of the accretion process when there was no more mass to beaccreted by the planet. Within the framework of the restricted three-body problem, a capture is always followed by an escape. To end up with a long termcapture, the satellite has to dissipate energy in a short time. The entrance energy ∆ E within the gravitational field of the planet is(Tsui 1999): ∆ E = − . µ / (1 − δ ) GM ⊙ a p , (14) µ = M p / M ⊙ and δ <<
1, where M p and a p are the mass and orbital semiaxis of the planet .Tsui (1999), suggested a permanent capture mechanism where a guest satellite encounters some existing inner orbit massiveplanetary satellite causing its velocity vector to be deflected keeping the irregular in orbit around the planet. In this way, thee ff ective two-body potential would be about twice the entrance energy ∆ E of the guest satellite. The radius R of the orbit of theguest satellite after deflection is then given by: R = . − δ µ / a p . (15)In the case of Uranus, for a minimum entrance energy of δ =
0, the minimum permanent orbital radius of the guest satellite is R = R U . This value of R is much larger than the present semiaxis of Prospero (see Table 1), making the capture of Prospero bythis mechanism implausible.The fact that binaries have recently been discovered in nearly all the solar system’s small-body reservoirs suggests that binary-planet gravitational encounters could bring a possible mechanism for irregulars capture (Agnor & Hamilton 2006). One possibleoutcome of gravitational encounters between a binary system and a planet is an exchange reaction, where one member of the binaryis expelled and the other remains bound to the planet. Tsui (1999) extended the scenario of large angle satellite-satellite scattering tothe formation of the Pluto-Charon pair assuming that Pluto was a satellite of Neptune and that Charon was a guest satellite. ThroughEqs.(14) and (15), the conditions for the escape of the pair was found. Following their scenario, let us consider the hypothesis thatProspero was a member of a guest binary entering Uranus’ field, with energy density ∆ E bin , above the minimum density given byEq.(14) and δ =
0. A close encounter with Uranus could result in disruption of the binary, leading to the ejection of one memberand capture of the other. The minimum semiaxis R is given by Eq.(15). However, even this scenario seems to be unlikely since thesemiaxis of Prospero is smaller than 955 R U . Collisional interactions between two planetesimals passing near the planet or between a planetesimal and a regular satellite, the socalled break-up process, leads to the formation of dynamical groupings (e.g. Colombo & Franklin 1971, Nesvorny et al. 2004). Theresulting fragments of each progenitor body after a break-up will form a population of irregulars expected to have similar surfacecomposition, i.e. similar colors, and irregular shapes, i.e. large temporal variations in the light curve as these irregular bodies rotate.
M. Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IV
The critical rotation period ( T c ) at which centripetal acceleration equals gravitational acceleration for a rotating spherical objectis: T c = π G ρ ob ! / , (16)where G is the gravitational constant and ρ ob is the density of the object. With ρ ob = − , T c = ff er centripetal deformation into aspherical shapes. For a given density and specific angular momentum(H), the nature of the deformation depends on the strength of the object. In the limiting case of a strengthless (fluid) body, theequilibrium shapes have been well studied (Chandrasekhar 1987). For H ≤ GM R sphe ) / ], where M (kg) is themass of the object and R sphe is the radius of an equal-volume sphere, the equilibrium shapes are the oblate Maclaurin spheroids.For 0.304 ≤ H ≤ > m s , the energyrequired at impact to result in a break-up is given by:12 m s v col ≥ m s S + Gm s γ R ms , (17)where v col is the collision speed, S is the impact strength, R ms is the radius of the target and γ is a parameter which specifies thefraction of collisional kinetic energy that goes into fragment kinetic energy and is estimated to be ∼ v col = v e + v in f , where v e is the scape speed at the target surface and v in f is the typical approach velocityof the two objects at a distance large compared with the Hill sphere of the target. For two bodies colliding in the Kuiper disk, v in f isgiven by (Lissauer & Stewart 1993): v in f = v k e + i ! , (18)where v k is the keplerian velocity, e is the mean orbital eccentricity and i the mean orbital inclination of the KBOs. We take h e i = h i i (Stern 1996). Eq.(17) was computed using Eq.(18) for values of e in the range [0.01-0.1] and orbital semiaxes of the KBOs in therange [30-60] AU. In computing the mass of the target m s , we consider the radius R ms of the KBOs in the range [10- 500] km anddensities between [0.5 - 2] g cm − . Impact strengths in the range [3 × -3 × ] erg cm − were taken. We obtain that targets with R ms ≤
210 km su ff er disruption for all the values of these parameters in the Kuiper Disk. Prospero has a radius of just only 15 km. IfProspero originated from the Kuiper Belt, it would have been more likely a collisional fragment rather than primary body. Prosperowould preserve an irregular shape after disruption since it is such small object that is unable to turn spherical because its gravitycannot overcome material strength. Prospero could have been captured during the break-up event if the two KBOs collided withinUranus’ Hill sphere, which could be possible for a minimum orbital eccentricity of the original KBOs of 0.37.We also consider the case in which the target is a satellite of Uranus which collides with a KBO that enters the Hill sphere ofthe planet. Eq.(17) remains valid but the following expression of v in f is considered: v in f = v ip − v sp , (19)where v ip is the velocity of the KB0 with respect to Uranus and v sp is the satellite orbital velocity, at the epoch of th event. We assumethat the satellite orbital velocity is circular. In order to get bounds in the relative velocity, we take two values of v in f , v in f = v ip ± v sp .For v ip , we assume that the KBO describes a hyperbolic orbit around Uranus during the approach giving: v in f = G ( m i + m U ) a s + GM ⊙ a kb , (20)where we assumed GM ⊙ / a kb as the relative velocity between the KBO and Uranus far from the encounter, ( m i + m U ) is the presentmass of Uranus and a s the orbital semiaxis of the satellite. We calculate Eq.(17) using Eqs.(19) and (20) for a kb in the range [20,60]AU and a s [100 -700] R U for the same values of S, R ms and densities we have used for the collisions among KBOs. We obtain that . Gabriela Parisi et al.: Constraints to Uranus’ Great Collision IV 9 for all the possible parameters, any Uranus’ satellite with radius R ms ≤ ff ers disruption if it collides with a KBO largerthan 10 km. This process would lead to the formation of two clusters of irregulars, one associated to the preexisting satellite andthe other to the primary KBO. This process has the disadvantage that it is unlikely that the preexisting satellite were formed from acircumplanetary disk as regular satellites given the large orbital semiaxis required for this object.Break-up processes predict orbital clustering. However, no obvious dynamical groupings are observed at the irregulars of Uranus.A further intensive search of more faint irregulars around Uranus is needed in order to look for dynamical and physical families. We now turn to the question of whether the GC itself could have provided a capture mechanism (BP02). Since all the transfers with A > B ′ lead to a more bound orbit, this process might transform a temporary capture into a permanent one (see Section 2 and Fig.1).Moreover, a permanent capture could even occur from an heliocentric orbit (transfers with A = N in heliocentric orbits at the time of the GC, at distances from Uranus lessthan or equal to 300 R U . Assume that the GC occurred when Uranus was almost fully formed, meaning that its feeding zone wasalready depleted of primordial planetesimals. We assume that the objects passing near Uranus at that time were mainly escapeesfrom the Kuiper belt. Using the impact rate onto Uranus and the distributions of velocities and diameters given by Levison et al.(2000), and assuming that the mass in the transneptunian region at the end of the Solar System formation was 10 times its presentmass, a back-of-the-envelope calculation gives one object of diameter D ≥
20 km passing at a distance R ≤ R U from Uranusevery 6 yrs at the end of accretion (BP02). The typical crossing time T C among protoplanets in the outer Solar System is larger thanone millon years (Zhou et al. 2007). The number of objects passing near Uranus during a timescale T C is then 167000, which givesa probability of 6 × − for the capture of an object at about 300 R U by the GC. This low rate of incoming objects, makes thepossibility of the capture of all the irregulars from heliocentric orbits di ffi cult. Even the capture of a single object, Prospero (notethat Prospero could not have an orbit bound to the planet before the GC), turns out to be low probable. Since temporary capture canlengthen the time which a passing body can spend near the planet, a more plausible situation arises if we assume that the GC couldproduce the permanent capture of one or more parent objects which were orbiting temporarily around Uranus being the presentirregulars the result of a collisional break-up occurring after the GC.
4. Discussion and Conclusions
It is usually believed that the large obliquity of Uranus is the result of a great tangential collision (GC) with an Earth-sized proto-planet at the end of the accretion process. We have calculated the transfer of angular momentum and impulse at impact and haveshown that the GC had strongly a ff ected the orbits of Uranian satellites. We calculate the transfer of the orbits of the nine knownUranian irregulars by the GC. Very few transfers exist for five of the nine irregulars, making their existence before the GC hardlyexpected. In particular, Prospero could not exist at the time of the GC. Then, either Prospero had to originate after the GC or theGC did not occur, in which case another theory able to explain Uranus’ obliquity and the formation of the Uranian regular satelliteswould be needed. It is usually believed that the regular satellites of Uranus have accreted from material placed into orbit by the GC(Stevenson et al. 1986).Within the GC scenario, several possible mechanisms for the capture of Prospero after the GC were investigated. If the Uranianirregulars belong to individual captures and relating the origin of the outer uranian system to a common formation process, gasdrag by Uranus’ envelope and pull-down capture seem to be implausible. Three-body gravitational encounters might be a sourceof permanent capture. However, we found that the minimum permanent orbital radius of a guest satellite of Uranus is ∼ R U while the current semiaxis of Prospero is 645 R U . The GC itself could provide a mechanism of permanent capture and the captureof Prospero could have occurred from a heliocentric orbit as is required within the GC scenario, but due to the low rate of incomingobjects it turns out to be di ffi cult. Break-up processes could be the mechanism for the origin of Prospero and the other irregularsin the frame of di ff erent scenarios. Prospero might be a fragment of a primary KBO fractured by a collision with another KBO.The fragment could have been captured by Uranus if the two KBOs had a minimum orbital eccentricity of 0.37. Prospero could bea secondary member of a collisional family originated by the collision between another satellite of Uranus and a KBO where theparent satellite of Prospero could have been captured by any mechanism before or after the GC. This process has the disadvantagethat it is unlikely that the preexisting satellite were formed from a circumplanetary disk as regular satellites given the large orbitalsemiaxis required for this object. Since collisional scenarios require in general high collision rates, perhaps the irregulars wereoriginally much more numerous than now. Then, Prospero and also the other irregulars might be the result of mutual collisionsamong hypothetical preexisting irregulars (Nesvorny et al. 2003, 2007) which could have been captured by any other mechanismbefore the GC.The knowledge of the size and shape distribution of irregulars is important to know their relation to the precursor Kuiper Beltpopulation. It could bring valuable clues to investigate if they are collisional fragments from break-up processes occuring at theKuiper Belt and thus has nothing to do with how they were individually captured later by the planet, or if they are collisionalfragments produced during or after the capture event (Nesvorny et al. 2003, 2007). The di ff erential size distribution of the Uranianirregulars approximates a power law with an exponent q = in laboratory impact experiments. They found that while the asteroids and laboratory impact fragments show similar distributionof axis ratio ( h b / a i ∼ h b / a i ∼ h b / a i ∼ / a of 0.8. The knowledge of the size and shapedistribution of irregulars would shed light in the size and shape distribution of small KBOs as well as on the irregulars capturemechanism.Colors are an important diagnostic tool in attempting to unveil the physical status and the origin of the Uranian irregulars.In particular it would be interesting to assess whether it is possible to define subclasses of irregulars just looking at colors, andcomparing colors of these bodies with colors of minor bodies in the outer Solar System. Avalilable literature data show a dispersionin the published values larger than quoted errors for each Uranian irregular (Maris et al. 2007a, and references therein). We haveconcluded in Maris et al. (2007a), that the Uranian irregulars are slightly red but they are not as red as the reddest KBOs.An intensive search for fainter irregulars and a long term program of observations able to recover in a self consistent mannerlight-curves, colors and phase e ff ects informations is mandatory. Acknowledgements.
MGP research was supported by Instituto Argentino de Radioastronom´ıa, IAR-CONICET, Argentina and by Centro de Astrof´ısica, Fondo deInvestigaci´on avanzado en Areas Prioritarias, FONDAP number 15010003, Chile. MM acknowledges FONDAP for finantial support during a visit to Universidadde Chile. Part of the work of MM has been supported by INAF FFO-
Fondo Ricerca Libera - 2006. AB research was supported by IALP-CONICET. We appreciatethe useful suggestions by the Reviewer, which have helped us to greatly improve this paper.
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