The formation of the Moon is still an open problem. One possible formation hypothesis is accretion from a circumterrestrial disk that was generated by a giant impact on the Earth. Direct N-body simulations show that a single large moon accretes from the disk in a time scale of a month to a year. We present a direct relationship between the size of the formed moon and the initial configuration of the disk, which gives an important clue to the Moon formation problem.

Many models have been proposed for formation of the Moon, but no one has succeeded to show the formation satisfactorily. The popular "Giant Impact" scenario states that a Mars-sized protoplanet hit the proto-Earth and generated a circumterrestrial debris disk from which the Moon accretes. This scenario has been favored since it may well account for the dynamical and geochemical characteristics of the Moon (large angular momentum of Earth-Moon system, depletion of volatiles and iron). Many hydrodynamic simulations (a smoothed particle method) have modeled the impact process. They calculated the impact between two large protoplanets with iron cores and silicate mantles and followed the orbital evolution of the debris after the impact for short time scales (~ a few orbital periods). It is found that an impact by a Mars-sized body usually results in formation of a circumterrestrial disk rather than direct formation of a clump. (This trend is most clear in recent simulations.) The disk mass is usually smaller than 2.5 ML where ML is the present lunar mass (= 0.0123 ME; ME is the Earth mass). Most of the disk material is distributed near or interior to the radius aR of the Roche limit (~ 2.9 RM where Re is the radius of the Earth) if the orbital angular momentum of the impact is 1-2 JEM where JEM is the angular momentum of the present Earth/Moon system. Within and near the Roche limit~ the tidal force of the Earth inhibits accretional growth.

The only published accretion calculation is Canup and Esposito with a gas dynamic approach. They approximated disk particles as particles in a box and tracked the evolution of the mass distribution function at individual regions of the disk, modeling velocity evolution, accretion, and rebounding of the disk particles. They showed that, in general, many small moonlets are formed initially rather than a single large moon and concluded that the simplest way to form the present-sized moon is to begin with at least a lunar mass of material outside the Roche limit. However, in gas dynamic calculations it is difficult to include non-local effects such as radial migration of the disk material and global interaction between formed moons and the disk. The importance of the radial diffusion out from the Roche limit has been pointed out through analytical argument9.

Here we perform direct N-body simulations, which automatically include non-local effects, to investigate global lunar accretion processes. The sequence of accretion of the moon from an impactgenerated disk may be as follows. Initially, the disk would likely be a hot, silicate vapor atmosphere/torus. Solid particles condense due to cooling of the disk, possibly after some radial migration. Subsequent collisions and fragmentation of the particles would damp initially large orbital eccentricities and inclinations of the particles to moderate values in a few orbital periods. Our simulations starts from this stage and follow the collisional evolution to a moon(s). On a longer time scale, a formed moon(s) gradually migrate outward bv tidal interaction with the Earth, sweeping remnants. We do not pursue such long time evolution here.

We present the results of 27 simulations with different initial disk conditions. We found that a single large moon, rather than multiple moons, is usually formed at similar distance from the proto-Earth in 100-1000 orbital periods (~ a month to a year). We also found that the final moon mass is mostly determined by a simple function of initial total mass and angular momentum of the disk. In order to estimate the final moon mass, we do not need to know the details of initial mass, size, and velocity distributions of the disk particles. The predicted moon mass from the disks obtained by the previous impact simulations might be as large as the present lunar mass in some cases. However, we cannot make a definitive conclusion at present, since the previous impact simulations did not provide enough data about the disk angular momenta. Improved simulations are needed to provide total mass and angular momentum of the disk. The combination of more refined N-body and impact simulations would clarify whether a giant impact could indeed have produced the Moon or not.

Model Description

We simulated the formation of a moon from a (three-dimensional) circumterrestrial debris disk initially consisting of 1000-2700 particles with mass m ~ l0e-5 - 10e-2 ML, assuming that solid particles had condensed and attained such sizes through accretion. (In the inner part of the disk, the particles may remain very small, since accretion becomes increasingly inhibited inside the Roche limit. Furthermore, the disk material may remain liquid due to longer cooling time in the inner part. We will comment on these effects later.)

We calculated disks with as many different initial conditions as possible, since we do not have enough knowledge about disk conditions after the vapor/liquid phase and initial collisional evolution The parameters we examined are summarized in Table l, where we show 19 runs of the 27 simulations for which we retained detailed output data. As shown below, the final outcome of accretion has only a weak dependence on the details of conditions of a starting disk. We scale the orbital radii by the Roche radius defined by aR = 2.456(pE/p)e1/3RE where (pE,/p) is the ratio of the internal density of the Earth to that of the disk particles. For disk particles with p = 3.34g/cm3 (the bulk density of the Moon), aR is located at about 2.9RE. Using aR, physical radii of disk particles with mass m are given by R = (1/2.456)(m/ME)e1/3 aR, independent of (pE/p).

Near the Roche radius, tidal forces of the proto-Earth affect w hether colliding particles rebound or accrete. Interior to about 0.8aR, tidal forces preclude accretion, w hile in the transitional zone 0.8-1.35aR, limited accretional growth can occur. Exterior to this zone, accretion is largely unaffected by tidal forces. This transitional zone will be referred to as the "Roche zone'`. \Ne adopt here the accretional criteria of Canup and Esposito which include this transition in addition to the impact velocity condition that for accretion, the calculated rebound velocity must be smaller than some critical value corresponding to (mutual) surface escape velocity. If the colliding bodies in our simulation satisfy the criteria, we produce a merged body, conserving momentum. If not, the bodies rebound with given restitution coefficients.
 

Figure 1. Snapshots of disk particles are plotted in geocentric cylindrical coordinates (r, z). (Particles at negative z are plotted at abs(z) ) The units of length and time are the Roche limit radius aR and kepler time TKep at aR(~ 7 hours). Solid and dotted circles are disk particles and the Earth. The sizes of the circles exprese physical sizes. The snapshots here are the result of run 4. The mean specific angular momentum Jdisk/Mdisk is initially 0.692(G ME aR)1/2. At t = 1500, the moon has the mass 0.40 ML, the semimajor axis 1.20 aR, the eccentricity 0.09, and the inclination (radian) 0.02. Second body's mass is only 0.025ML. The mass ejected from the system (MJ) and that hit the Earth are 0.026ML and 1.95ML, respectively.

 

Figure 2. The same snapshots as Fig. 1 for run 9 of a more extended disk (Jdisk/Mdisk = 0 813 At t = 1000, the largest moon mass is 0.71ML.

 

Figure 3. The same snapshots as Fig. 1 for run 13 of a very extended disk (Jdisk/Mdisk = 0.958(G ME aR)e1/2. In this case, two large moons are formed. At t = 1000, the largest moon has the mass 0.63ML and the semimajor axis 1.98aR, while the second one has 0.391ML and 0.93aR.

 

Figure 4. Final moon mass M is plotted on M/Mdisk - Jdisk/Mdisk plane. Smaller Jdisk/Mdisk cases are confined disks with smaller amax and larger q. Solid fine is the analytical estimate with MJ = 0 and dotted line is that with MJ = 0.051Mdisk. Squares are N-body results with en = 0.01 and triangles are with en = 0.5. The small squares and triangles are the cases where the mass of a second moon is larger than 30% of the largest one's mass. In these cases, we plotted the sum of the largest and the second moons' masses. Here we plotted all the resulte of the 27 simulations.

 
 

References

1. Boss, A. P. & Peale, S. J. Dynamical constraints on the origin of the Moon. In Origin of the moon. eds. W. K. Hartmann, R. J. Phillips, & Taylor, G. J., Lunar and Planetary Institute, Houston, pp. 59-102 (1984).

2. Hartmann, W. K. & Davis, D. R. Satellite-sized planetesimals and lunar origin Icarus 24, 504-515 (1975).

3. Cameron, A. G. W. & Ward, W. R. The origin of the Moon. Proc. Lunar Planet Sci. Conf. 7th, 120-122 (1976). 4. Benz, W., Slattery, W. L. & Cameron, A. G. W. The origin of the Moon and the single impact hypothesis I. Icarus 66, 515-535 (1986).

5. Benz, W., Cameron, A. G. W. & Melosh, H. J. The origin of the Moon and the single impact hypothesis III. Icarus 81, 113-131 (1989).

6. Cameron, A. G. W. & Benz, W. The origin of the Moon and the single impact hypothesis IV. Icarus 92, 204-216 (1991).

7. Cameron, A. G. W., The origin of the moon and the single impact hypothesis V. Icarus 126, 126-137 (1997).

8. Canup, R. M. & Esposito, L. W. Accretion of the Moon from an Impact-Generated Disk. Icarus 119, 427-446 (1996).

9. Ward, W. R. & Cameron, A. G. W. Disc evolution svithin the Roche limit. Proc. Lunar Planet Sci. Conf. 9th, 1205-1207 (1978).

10. Stevenson, D. J. Origin of the moon-the coliision hypothesis, Ann. Rev. Earth Planet Sci. 15, 271-315 (1987).

11. Makino, J. ~ Aarseth, S. J. On a Hermite integrator with Ahmad-Cohen scheme for gravitational many-body problema. Publ. Astron. Soc. Jpn. 44, 141-151 (1992).

12. Makino, J. A modified Aarseth code for GRAPE and vector processors. Publ. Astron. Soc. Jpn. 43, 859-876 (1991).

13. Canup, R. M. & Esposito, L. W. Accretion in the Roche zone: co-existence of rings and ringmoons. Icarus 113, 331-352 (1995).

14. Goldreich, P. and Tremaine, S. The d~vnamics of planetary rings. Ann. Rev. Astron. Astrophys. 20, 249-248 (1982).

15. Stewart, G. R. & Wetherill, G. W. Evolution of planetesimal velocities. Icarus 74, 542-553 (1988).

16. Safronov, V. S. Evolution of the protoplanetary cloud and formation of the earth and planeta. Nauka Press (1969).

17. Lin, D. N. C. & Papaloizou, J. C. B. On the tidal interaction bet~veen protostellar disks and companions. In Protostars and Planets III, eds. E. H. Levy and J. I. Lunine, Univ. of Arizona Press, pp.749-836 (1993).

Questions & Answers

Subject:   lunar question
   Date:   Thu, 16 Oct 1997 10:01:31 -0600
   From:   canup@sargon.colorado.edu (Robin Canup)
     To:   aparra1@pie.xtec.es (Tony Parra)

On the paper "Lunar accretion...", at page 2, I've found a statement I would like to understand better: "a former moon(s) gradually migrate outward by tidal interaction with the Earth, sweeping remnants". I know the fact since long time ago, but I'm not able to explain it.

Regards,
Tony

Hi Tony:

The moon (or anything else in Earth orbit for that matter) orbitally evolves due to "tidal interaction" with the Earth. The gravity of the moon raises tidal bulges on the near and far side of the Earth (these are the diurnal oceanic tides). The formation of these bulges represent an asymmetry in the gravitational potential of the Earth (in addition to the Earth's oblateness of course) which exerts a torque on an orbiting body. If the orbiting body is located within geosynchronous orbit (the orbit at which the orbital period=the rotational period or day of the Earth), then this torque causes a moon to evolve inward towards the Earth. For a moon outside geosync. orbit, the tidal torque causes angular momentum to be transferred from the Earth's rotation to the orbit of the moon, causing the Earth's spin to slow and the moon's orbit to expand. This is currently the case for our moon. Now the rate of this expansion is proportional to the mass of the orbiting moon (since a bigger moon produces a bigger tidal bulge on the Earth, which in turn provides a bigger torque). So an inner massive moon will overtake outer smaller moons as they all evolve due to tides: this is the origin of the sentence in the Nature paper that you referred to.

Hope this helped,

Robin.