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Planet-disk interaction and the mass-eccentricity relation for planetary and stellar systems

The orbits of the smallest solid bodies forming in protoplanetary disk: meteoroids and small planetesimals up to several*100 km size are circularized by the aerodynamic gas drag (e.g., Weidenschilling 1977).

For more massive objects (planetary embrios) the dynamical friction (partial energy equipartition betweeen small and big bodies) may be very important as well (Stewart and Wetherill, 1988).

The eccentricity-mass relation can be understood in terms of disk-planet gravitational interaction:

Thus, the predicted regions of the mass-eccentricity plane which should be occupied according to the resonant interaction theory looks like this (keep in mind that the boundaries may not be sharp in reality!):

This prediction (since the theories pre-date the recent dicoveries!) should be compared with the current mass-eccentricity data:


The little dashes sticking to the right of the data points (which give the minimum masses in systems of unknown inclination) have length corresponding to the correction factor pi/2. Only about 1/4 of randomly oriented systems will have true masses larger than the ones indicated by the dashes. However, we now know (M. Mayor, talk at the Lisbon conference on extrasolar planets, 1998) that the orbits are not necessarily randomly distributed, and in fact several of the brown dwarf candidates are seen nearly pole-on. Therefore, the lines might be somewhat longer in reality.

The agreement of the theoretical predictions and the current restricted statistics of low-mass companions is impressive, except for a planet in 16 Cyg B system (empty quasi-square at e~0.7). This planet, however, is likely made eccentric by 16 Cyg A, its distant stellar companion, and hence could have been born on nearly-circular orbit like the rest of bodies in its mass range (Mazeh et al 1997, Holman et al 1997).


Addendum (Dec. 1999):


Many more exoplanets on primordial orbits unaffected by tides have been discovered by the end of the second millenium. The mass-eccentricity relation is found in the distribution of elements, although not as clear-cut as the above pictures based on the first few high-eccentricity planets. This is shown in the following table (Lubow and Artymowicz, in prep.). One system (16 Cyg B) has been omitted in view of the probable eccentricity generation by a companion star. Otherwise all the known planets including Jupiter were considered and, whenever known, the true masses rather than minimum masses used.

Mass-eccentricity correlation (Dec. 1999)

This table lists the number and mean eccentricity of companions (with orbital periods P > 10 days) in each of the 4 mass ratio bins (mass ratio = mass of a companion / mass of the host star). This is the only relevant quantity for most theories of eccentricity, not the value of mass in Jupiter masses. However, the mass ratio can be conveniently expressed in Jupiter units (mass ratio=0.001)
Type of companions Mass ratio [*0.001] # of comp. mean eccentricity
Low-mass giant planets 0-3.2 12 0.17 +- 0.04
Superplanets 3.2-10 9 0.31 +- 0.08
Brown Dwarfs 10-100 13 0.40 +- 0.05
Companion stars > 100 13 0.37 +- 0.02

We see that < e > for giant planets is ~ 1 sigma away from for superplanets, ~ 6 sigma away from for superplanets, and ~ 5 sigma away from for binary stars and brown dwarfs. There is thus little doubt about the existence of a trend.
On the other hand, the correlation isn't very tight, because in each mass bin we have a very large spread of individual eccentricities (larger than the uncertainty in mean ecc. in table). This can be easily understood as a result of varying conditions in different protoplanetary nebulae, most importantly varying viscosity. These differences can move the crossover mass ratio (mass above which eccentric instability of orbits occurs) between roughly 1 and 10 Jupiter mass ratio.


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