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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:
- The low-mass
region m<1 MJ, where the body does not open a gap in a disk.
Ward(1998) proposed that de/dt<0, and Artymowicz (1993) in a series of
articles worked out the problem qualitatively and quantitatively. The
damping mechanism is due to so-called coorbital Lindblad resonances (LRs).
- The mid-mass region 1 MJ < m < mecc, where
the body does open a gap in the disk. Goldreich & Tremaine (1980) derived
de/dt<0, as a result of corotational resonances overpowering the external
LRs. Artymowicz (1992) confirmed that expectation in 2-D simulations of
proto-Jupiter, while Lin & Papaloizou (1993) reached a different conclusion
assuming a wide gap.
- The transition region where de/dt switches to positive valueas. Precise
mass mecc where de/dt=0 is unknown, and probably depends somewhat
on the nebula type, but
Artymowicz (1992) reported SPH simulations which suggested
that mecc lies between approximately
10 and 100 MJ, and argued that this
cross-over mass is in fact "close to" 10 MJ.
To illustrate the complex interaction between a protojupiter and the nebula
we show here unpublished PPM calculations by Artymowicz (1999). While oval
is the Roche lobe (circumplanetary region), all distances are in
units of star-planet distance. PPM=Piecewise parabolic method is
probably the most sophisticaded hydrodynamical code in wide use
in astrophysics today. On the same size of a grid, it offers
sharper representation of flow details than any other code (at the cost
of more computations). Disk-planet interactions computed with this
code confirm in the broad outline
the picture of eccentricity growth discussed here.
- High companion mass regime: secondary = brown dwarf or a star.
The theory of eccentricity pumping in disk-binary interaction has
been developed
by Artymowicz et al. (1991), and a series of papers by Lubow and Artymowicz
(1992-1997), as well as Lin & Papaloizou (1993).
The rates are high enough for a primordial eccentricity of any such body
to grow to the observed mid-eccentricity range e=0.15-0.7 in the
~few Myr time available during the existence of the protoplanetary disk.
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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