Chapter VI: The ubiquity of stellar oblateness as a driver of
6.3 Results & Discussion
time at 0.1 AU is given by
τν ≈ a2p ν
≈ a2p αh2Ω
≈ 2000 years, (6.11)
whereas the precession timescale from the stellar quadrupole is roughly 300 years. Given that the disk dissipates on a longer timescale than the precession timescale, the system might reduce its spin-orbit mis- alignment during disk dissipation as the stellar quadrupole begins to dominate over the disk’s quadrupole. More work is required in order to investigate this possibility. The timescales and physics governing disk dispersal are poorly understood, and so we leave this aspect of the problem as a caveat, to be returned to once better constraints become available.
-4 -3 .5 -3 -2 .5 -2 0
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
-4 -3 .5 -3 -2 .5 -2
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
-4 -3 .5 -3 -2 .5 -2
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
Kepler-51 Kepler-60 Kepler-18
log10(J2,0)
-3.5 -3 -2.5 -2 -3.5 -3 -2.5 -2
0 10 20 30 40 50 60 70 80 90
-4 -3.5 -3 -2.5 -2
stellar obliquity (degrees) -4
0 10 20 30 40 50 60 70 80 90
0 10 20 30 40 50 60 70 80 90
-4
3 2 1
unstable
Figure 6.3: The number of planets detectable in transit after 20 million years of simulation from an initially 3-planet configuration. The dotted line outlines the region where one or more planets were lost owing to dynamical instability.
An analytic formula was derived in Spalding and Batygin (2016), relating the mutual inclination excited owing to the stellar quadrupole, under the assumptions of circular orbits and low inclinations. For the 2-planet systems, we draw a solid black line that denotes this predicted boundary between coplanar and misaligned orbits.
As stated above, our primary goal was to delineate the ubiquity of stellar oblateness as an instability mechanism. To that end, we note that only Kepler-10 was immune to instability for all chosen parameters, with Kepler-36 remaining stable all but two times. All other systems were susceptible, at least for the upper range of J2,0. Accordingly, we conclude that the instability mechanism uncovered in Spalding and Batygin (2016) constitutes a viable pathway toward instability for low and high-multiplicity systems alike. In general terms, the range of J2,0
leading to instability is slightly smaller for the 3 and 4 planet systems than 2-planet systems; however, given our small sample size such a pattern is by no means statistically significant.
Eccentricities
If a single-transiting system is observed, it is difficult to infer whether there exist any non-transiting companions. Within the frame of our present investigation, a key signature of dynamical instability is the
-4 -3 .5 -3 -2 .5 -2 0
1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
-4 -3 .5 -3 -2 .5 -2
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0
Kepler-223
Kepler-79
-3.5 -3 -2.5 -2
-3.5 -3 -2.5 -2
0 10 20 30 40 50 60 70 80 90
-4
st el la r obl iqui ty (de gre es )
0
-4
10 20 30 40 50 60 70 80 90
unstable
4
3 2 1
log 10 (J 2,0 )
Figure 6.4: The number of planets detectable in transit after 20 million years of simulation from an initially 4-planet configuration. The dotted line outlines the region where one or more planets were lost owing to dynamical instability.
presence of significant eccentricity within the planetary orbits that remain subsequent to instability. Eccentricity therefore constitutes an observable signature of instability, however, for the shortest-period systems tidal effects are likely to have damped out any traces of pri- mordial eccentricity. The tidal circularization timescale is given by (Murray and Dermott,1999)
τe ≡ e eÛ
≈ 2 21
Q k2,pnp
mp M?
ap Rp
5
≈ 40 ap
0.1AU
132 Q/k2,p 1000
2REarth Rp
5
Gyr, (6.12) where k2,p is the planetary Love number and Qp is its tidal quality factor. Put another way, planets possessing 10 Earth masses and 2 Earth radii will circularize within a Gyr for semi-major axes below ap ∼ 0.05 AU. Those with semi-major axes exceeding ap ∼ 0.1 AU ought to possess eccentricities that are relatively unaffected by tides, though numerous other dynamical interactions are capable of exciting, or indeed damping, their eccentricities.
With the caveat regarding tidal circularization in mind, it is interest- ing to tabulate the orbital parameters of the planet that remains after dynamical instability within the four most unstable 2-planet exam- ples – K2-38, K epler-27, K epler-131, and Kepler-307. As can be seen from Table6.2, the mean eccentricity of the remaining planet is roughly ¯ei ≈ 0.3−0.4.
Cumulatively, we may propose the following observational signature.
First, consider a sample of single-transiting systems beyond 0.1 AU.
Suppose that they are composed of two populations: a fraction fin that have undergone dynamical instability and a fraction fin that have not.
The latter fraction did not pass a dynamical instability, and appear sin- gle owing to exhibiting mutual inclinations with unseen companions,
or alternatively were born single. According to our proposed mech- anism, the population that experienced instability should possess a mean eccentricity of ¯einst ≈ e¯i. If the mean across both populations is e¯ and the mean of the stable population is ¯est e¯inst, one can show that
fin = e¯−e¯st e¯inst −e¯st
≈ e¯
e¯inst, (6.13)
where the second equality assumes the stable population will exhibit eccentricities much lower than the unstable population.
Typically, theKeplerDichotomy is quoted as reflecting a roughly equal split between the large and small inclination systems, i.e., fin = 1/2 (Johansen et al., 2012; Ballard and Johnson, 2016). In order to reproduce this fraction with ¯eu,o ≈ 0.4, we would predict ¯eo ≈ 0.2. Of course, this is a very simplified picture, but outlines the feasibility of learning about the true underlying abundance of planets despite only observing the proportion that transit.
During the planets’ close encounters at high eccentricity, the semi- major axes of both planets are altered. The remaining planet generally experiences an increase in semi-major axis, indicating a gain in energy at the cost of the second planet, which usually ends up colliding with the central body. Related to this point, recall that the stellar radius was held fixed in the code, and J2 was forced to decay, leaving the star modestly larger than it would be in reality during the later stages of the simulation. Accordingly, there exists the possibility that in real systems, more energy would need to be transferred to collide with a smaller star, and larger semi-major axes and/or eccentricity influences might occur. In addition, depending upon the exact mechanism of in- stability, tides may “save” the inner planet by damping its eccentricity
before it enters a star-crossing trajectory. These details of the problem do not alter the general picture.
Our discussion in this section uses eccentricities and semi-major axes obtained from the mercury6 N-body code. However, we did not model collisions between planets, which is likely to influence the final eccentricity distribution. Accordingly, the eccentricities in reality may be smaller than we predict here owing to dissipative processes associated with the physics of merging. Though the quantitative nature of our predictions are subject to numerous uncertainties, the qualitative prediction is that a population of single-transiting systems ought to exhibit larger eccentricities than those possessing unseen companion planets.