10-4 10-3 10-2 10-1 10° 101 102 103 104 105 106 107 Diameter, cm

Fig. 5.24 Evolution of the size distribution of particles with a disk as a function of time. Initially, the grains have a diameter of 1 |m. The conditions in the protoplanetary disk are those in the primordial solar nebula, for a distance of 1 AU from the Sun (After Weidenschilling, 2000)

5.4.3 From Protoplanets to Planets

Kilometre-sized objects move in Keplerian orbits and are no longer susceptible to drag forces exerted by the surrounding gas. The orbits are almost circular and copla-nar, but the differential between the Keplerian velocities creates collisions and gravitational interactions. These interactions have the effect of increasing the eccentricity and inclination of the bodies, whereas collisions and friction with the gas tends to circularize the orbits. Because the relative velocities of the bodies are related to their velocities and their masses, the size distributions and the velocities evolve together and in a non-linear fashion.

The following section summarizes the discussion by de Pater and Lissauer (2001) on the growth rate of planetesimals. The Effect of Collisions

Collisions between solid bodies may result in accretion, fragmentation or inelastic rebound. The impact velocity at which two solid bodies collide is:

where v is the relative velocity of one body with respect to the other far from encounter, and ve is the escape velocity at the point of impact:

where m1 and m2 are the respective masses of the two bodies, and R1 and R2 are their respective radii. The rebound velocity is evi, with e < 1. Accretion may occur if evi is smaller than ve. For a 10-km rocky object, the escape velocity is about 6 m/s. This is larger than the typical relative velocities of planetesimals, so that a 10-km body is likely to accrete the surrounding planetesimals, whereas fragmentation will preferentially occur for very small planetesimals.

The mean growth of a planetary embryo of mass M is dM 2

where ps is the density of the planetesimals, and v is the average relative velocity between the embryo and the small bodies (assumed to be much smaller than the embryo). F is the gravitational enhancement factor, given by

in the 2-body approximation.

It is possible to express the growth rate of the embryo's radius as a function of the surface mass density op, the embryo density pP and the Keplerian orbital angular velocity n:

dt y n 4pp

This equation leads to a growth time of about 2 x 107y for the Earth, and more than 108 yr for Jupiter. In the latter case, we know that other more efficient factors have been involved, because Jupiter and Saturn must have formed within 107 years, before the T-Tauri phase and the dissipation of the gas. The Runaway Growth of Planetary Embryos

As the embryo grows, its escape velocity increases. If the relative velocity of the embryo versus the swarm of planetesimals remains small, the F factor may increase by large factors, leading to the runaway growth of the embryo. The embryo's feeding zone is limited to the annulus of planetesimals which the embryo may perturb gravi-tationally. Thus, rapid runaway will stop when the embryo has consumed the matter contained in the annulus. This mechanism thus leads to the formation of gaps inside protoplanetary disks. The size of the object that is formed depends on the material available within the gap that it creates.

The size of the gap depends primarily on the mass of the object that is forming. For a disk with no gas, the size of the gap is equal to a few times rH, where rH is the Hill radius, the latter being defined as the distance beyond which the gravitational force exerted by the star exceeds that of the protoplanet:

where a is the semi-major axis of the protoplanet's orbit, m is the mass of the proto-planet, and M the mass of the star. In the case of a disk with a high gas content that is strongly viscous, the width of the gap is a function of the Reynolds number where A is the width of the gap (Varniére et al., 2004; Beust, 2006).

Another situation occurs when the relative velocity of the swarm of planetesimals is comparable with or higher than the escape velocity of the embryo. In this case, the F factor remains close to unity and the evolutionary path of the planetesimals exhibits an orderly growth in the entire size distribution. This is why numerical simulations of planetary formation typically lead to two types of solutions:

• a burst of growth, which results in the rapid growth of one body, which sweeps up material from the surrounding space;

• orderly growth, resulting in several large bodies of similar mass, with a power distribution of smaller bodies.

In the latter case, the small bodies could have a very long lifetime. Such evolution could possibly explain the existence of debris disks around evolved stars (Table 5.1).


Acker, A., Astronomie-Astrophysique, Introduction, Dunod, Paris (2005) 134,137, 138,146, 149,151 Andre, J.-P. and Montmerle, T., 'From T-Tauri stars to protostars: circumstellar material and young stellar objects in the Rho-Ophiuchi cloud', Astrophys. J., 420, 837-862 (1994) 147 Artymowicz, P., 'Beta Pictoris and other solar systems', in From Dust to Terrestrial Planets, (eds) Benz, W., Kallenbach, R. and Lugmair, G.W., 69-86, Kluwer, Doordrecht (2000) 158, 159, 161 Aumann, H.H., Beichmann, C.A., Gillett, F.C., 'Discovery of a shell around Alpha Lyrae', Astrophys. J., 278, L23-L27 (1984) 139 Beckwith, S.V., Sargent, A.I., Chini, R.S. and Guesten, R., 'A survey for circumstellar disks around young stellar objects', Astron. J. 99, 924-945 (1990) 156 Bertout, C., Mondes lointains, Flammarion, Paris (2003) 139,143

Beust, H., 'Modelisation des disques de debris, in, Formation planetaire et exoplanetes', in Comptes-rendus de l'école thématique de Goutelas 2005, (eds) Halbwachs, J.-L., Egret, D. and Hameury, J.-M., 155-190, Observatoire de Strasbourg/SF2A (2006). 165 Beust, H. and Morbidelli, A., 'Mean motion resonances as a source of infalling comets toward Beta Pictoris', Icarus, 120, 358-370 (1996) 161

Cassen, P., 'Protostellar disks and planet formation', in extrasolar planets, Saas-Fee Advanced Course 31, (eds) Cassen, P., Guillot, T. and Quirrenbach, A., 369-444, Springer-Verlag, Heidelberg (2006) 135

Chiang, E.I. and Goldreich, P., 'Spectral energy distributions of T Tauri stars with passive circum-

stellar disks', Astrophys. J., 490, 368-376 (1997) 148,149 Choi, M., Evans, N.J. II, Gergersen, E.M. and Wang, Y., 'Modeling line profiles of protostellar collapse in B335 with the Monte Carlo method', Astrophys. J. 448, 742-747 (1995) 137 Cole, G.H.A. and Woolfson, M.W., Planetary Science, Institute of Physics Publishing, Bristol and Philadelphia (2002) 138

Cuzzi, J.N., Dobrovolski, A.R. and Champney, J.M., 'Particle-gas dynamics in the mid-plane of the solar nebula', Icarus, 106, 102-134 (1993) 162 Dutrey, A., Lecavelier des Etangs, A. and Augereau, J.-C., 'The observation of circumstellar disks: dust and gas components', in Comets II, (eds) Festou, M.C., Keller, U. and Weaver, H.A., 81-95, University of Arizona Press, Tucson (2004) 155 Elmegreen, B.G., Efremov, Y., Pudritz, R.E and Zinnecker, H., 'Observation and theory of star cluster formation' in Protostars and Planets IV, (eds) Mannings, V. et al., 179-215, University of Arizona Press, Tucson (2000) 135 Goldreich, P. and Ward, W.R., 'The formation of planetesimals', Astrophys. J., 183, 1051-1061 (1973) 162

Hartmann, L., Accretion Processes in Star Formation, Cambridge University Press, Cambridge (1998) 142

Heap, S.R., Lindler, D.J., Lanz, T.M. and Cornett, R.H. et al., 'Space telescope imaging spectrograph coronographic observations of Beta Pictoris', Astrophys. J., 539, 435-444 (2000) 159 Hersant, F., Gautier, D. and Hure, J., 'A two-dimensional model for the primordial nebula constrained by D/H measurements in the solar system: implications for the formation of giant planets', Astrophys. J., 554, 391 (2001) 154, 155 Johns-Krull, C.M., Valenti, J.A. and Koresko, C., 'Measuring the magnetic field of the classical

T-Tauri star BP Tau', Astrophys. J., 510, L41-44 (1999) 144 Larson, R.B., 'Star formation in groups', Mon. Not. Roy. Ast. Soc, 272, 213-220 (1995) 135 Larson, R.B., 'The physics of star formation', Reports on Prog. in Physics, 66, 1651-1697 (2003) 135,144, 145 Lecavelier des Etangs, A., Hobbs, L.M. and Vidal-Madjar, A. et al., 'Possible emission lines from the gaseous Beta Pictoris disk', Astron. Astrophys., 356, 691-694 (2000) 159, 160 Lequeux, J., The Interstellar Medium, Springer-Verlag, Heidelberg (2005) 134, 136 Lynden-Bell, D. and Pringle, J.E., 'The evolution of viscous disks and the origin of the nebular variables', Mon. Not. R. Astron. Soc., 168, 603-637 (1974) 137,142 Malfait, K., Waelkens, C. and Waters, L. et al., 'The spectrum of the young star HD100546 observed with the Infrared Space Observatory', Astron. Astrophys., 332, L25-L28 (1998) 156 Matsumoto, T. and Hanawa, T., 'Fragmentation of a molecular cloud core versus fragmentation of the massive protoplanetary disk in the main accretion phase', Astrophys. J., 595, 913-934 (2003) 144

Meyer, M.R., Hillenbrand, L.A. and Backman, D.E., et al., 'The formation and evolution of plan-netary systems: placing our solar system in context with Spitzer', Pub. Astron. Soc. Of the Pacific, 118, 1690-1710(2006) 139,157 Najita, J., 'Star formation', in Encyclopedia of Astronomy and Astrophysics, 3016-3027, Na-

ture/IoP Publishing, Bristol (2001) 135, 140, 141 Papaloizou, J.C. and Terquem, C., 'Critical protoplanetary core masses in protoplanetary disks and the formation of short-period giant planets', Astrophys. J., 521, 823-838 (1999) 153 Pfenniger, D. and Combes, F., 'Is dark matter in spiral galaxies cold gas? II. Fractal models and star non-formation', Astron. Astrophys., 285, 94-118 (1994) 135 Pfenniger, D., Combes, F. and Martinet, L. 'Is dark matter in spiral galaxies cold gas? I. Observational constraints and dynamical clues about galaxy evolution', Astron. Astrophys., 285, 79-93 (1994) 135

Safronov, V.S., Evolution of the Protoplanetary Cloud and Formation of the Earth and Planets (Nauka, Moscow, 1969; English translation: NASA TTF-677 (1972) 162

Schneider, G., Smith, B.A. and Becklin, E. et al., 'NICMOS imaging of the HR 4796A circumstel-

lar disk', Astrophys. J., 513, L127-L130 (1999) 158 Smith, B.A. and Terrile, R.J., 'A circumstellar disk around Beta Pictoris', Science, 226, 1421-1424 (1984) 139

Spitzer, L., Physical Processes in the Interstellar Medium, Wiley & Sons, New York (1978) 136 Telesco C.M., Fisher R.S., Pina R.K., et al., 'Deep 10 and 18 micron imaging of the HR 4796A circumstellar disk: Transient dust particles and tentative evidence for a brightness asymetry', Astrophys. J., 530, 329-341 (2000) 158 Varniere, P., Quillen, A.C. and Frank, A., 'The evolution of protoplanetary disk edges', Astrophys.

J., 612, 1152-1162 (2004) 165 Vidal-Madjar, A., Lagrange-Henri, A.-M. and Feldman, P.D., et al., 'HST-GHRS observations of Beta Pictoris: additional evidence for infalling comets', Astron. Astrophys., 290, 245-258 (1994) 159

Weidenschilling, S.J., 'The origin of comets in the solar nebula: a unified model', Icarus, 127, 290-306 (1997) 161

Weidenschilling, S.J., 'Formation of planetesimals and accretion of the terrestrial planets', in From Dust to Terrestrial Planets, (eds) Benz, W., Kallenbach, R. and Lugmair, G.W., 295-310, Kluwer, Doordrecht (2000) 162, 163 Weidenschilling, S.J. and Cuzzi, J.N., 'Formation of planetesimals in the solar nebula', in The Formation and Evolution of Planetary Systems, (eds) Weaver, H. and Danly, L., 1031-1060, Cambridge University Press, Cambridge (1993) 161 White, R.J., Greene, T.P. and Doppmann, G.W. et al. (2007), 'Stellar properties of embedded protostars', in Protostars and Planets V, (eds) Reipurth, V.B., Jewitt, D. and Keil, K., 117-132, University of Arizona Press, Tucson (2007) Wood, J.A. 'Pressure and temperature profiles in the solar nebula', in From Dust to Terrestrial

Planets, (eds) Benz, W., Kallenbach, R. and Lugmair, G.W., 87-96, Kluwer, Doordrecht (2000) 155 Wood, J.A. and Morfill, G., 'A review of solar nebular models', in Meteorites and the Early Solar System, (eds) Kerridge, J.F. andMathhews, M.S., 329-347, University of Arizona Press, Tucson (1988) 153,154

Was this article helpful?

0 0

Post a comment