Formation of Close-in Super-Earths by Giant Impacts:
Effects of Initial Eccentricities and Inclinations of Protoplanets
MATSUMOTO, Yuji

KOKUBO, Eiichiro

Nowadays, dozens super-Earths, which are composedrock or ice, are already found in orbits that are close to
their stars (e.g., [1]). Recent observations have reveled
the orbital eccentricity (e) and orbital inclination (i)distributions of super-Earths [2,3]. We considered how
the eccentricities and inclinations of close-in superEarths are determined in the giant impact stage. Theseevolve through gravitational scattering and collisions
between protoplanets in gas-free environments. The
eccentricities are increased by scattering between
protoplanets [4], and they are damped by collisions [5].
The collisions tend to occur at the point where theirorbits initially crossing since the small scattering crosssection of protoplanets in the close-in region. The orbitalcrossing initially occurs around the apocenter of theinner protoplanet and the pericenter of the outer planet.
As a result, the velocity of merged body is about equal tothe Kepler velocity.
We studied the effects of the initial e and i ofprotoplanets on the final orbits of planets formed bygiant impacts [6]. We performed N-body simulationsof the protoplanet accretion over 500 times. Numericalcomputations were carried out on the PC cluster at theCenter for Computational Astrophysics, NAOJ.
We found that the final eccentricities do not depend
on initial eccentricities, when orbits of protoplanets areinitially not crossed. While the eccentricities gradually
increase with each collision if initial e is small, they
gradually decrease when initial e is large. The finalplanets become similar due to these relaxations.
On the other hand, the initial inclinations are
not relaxed. When the initial i is small, the velocity
dispersions normal to the disk midplane keep small
through scattering, and the final planets have small i. If
the initial i is large, collisions do not occur immediately
after orbital crossing, since they can collide only inparts of their orbits. Protoplanets experience more
scattering until a collision, which make i larger. This is
why the merged protoplanets have larger i. And, final
inclinations of planets depend on initial inclinations.
The eccentricities In large i cases are larger those insmall i cases due to more scattering. We found that theinclinations of observed planets are consistent with ourresults, if protoplanets initially have ~ 10 −3 –10 −2 radinclinations (Figure 1, [6]).

(d)(Academia Sinica)		

(NAOJ)

10-1

 i M [rad]				

10-2

10-3-4-3-2-1
010 10 10 10 102
 i ini 1/2[rad]Figure 1: This figure is from the panel (d) of Figure 9 in [6]. The
mass-weighted inclination (iM) is shown as a functionof the initial inclination dispersion of protoplanets(i 2ini1/2). The error bars indicate the standard deviation.
The filled symbols are the results of models that initial
number of protoplanets is Nini = 16 (models N16), theopen symbols are those of models N8, and crosses
are those of models N32. The initial eccentricity
dispersions are different between red circles and blue
triangles. The solid and dashed lines are the fits by theleast-squares-fit method for models. The dotted lines
shows the mode of the mutual inclination [2,3].

References[1] Mullally, F., et al.: 2015, ApJS, 217, 31.
[2] Fabrycky, D. C., et al.: 2014, ApJ, 790, 146.
[3] Xie, J. W., et al.: 2016, P. Natl. Acad. Sci. USA, 113, 11431.
[4] Kokubo, E., Ida, S.: 2002, ApJ, 581, 666.
[5] Matsumoto, Y., Nagasawa, M., Ida, S.: 2015, ApJ, 810, 106 .
[6] Matsumoto, Y., Kokubo, E.: 2017, AJ, 154, 27.

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