Introduction
From planetary to exo-planetary science
This idea was motivated by the coplanar arrangement of the planetary orbits and has been spectacularly confirmed by relatively recent images from the Hubble Space Telescope. But while our Jupiter "attacked" and moved out again (due to interactions with Saturn), the hot Jupiters continued their inward march to the inner edge of the disk.
The hot Jupiter debate
In the above expression, ˆρ is the radial unit vector in the plane of the disk. In other words, when the star is tilted, more of the disk is working.
Early excitation of spin-orbit misalignments in close-
Introduction
Furthermore, the statistical analysis of Crida and Batygin (2014) has shown that the expected spin-orbit distortion distribution of the disc rotation model is fully consistent with the observed one. Given the aforementioned successes of the disc rotation mechanism in resolving the discrepancy between disc-driven migration and.
Model
1 In agreement with the discussion above, the back-reaction of the star to the disk is not taken into account. In addition to the physical properties of the disk, we also need to prescribe its dynamic behavior to complete the problem specification.
Results
The post-resonant encounter stellar inclination of the star (measured in a frame coplanar with the binary orbit) as a function of the disk binary inclination is shown as a purple curve. The physical evolutions of the star and the disk are assumed to proceed as described in section 2.1.
Discussion
2Physically speaking, the movement of the disc material relative to the background field creates a flow within the disc. 1Physically speaking, the motion of the disc material relative to the background field creates a current within the disc. The y-axis refers to the misalignment β? between the axis of rotation of the star and the initial plane of the 6 planets.
Alignment of protostars and circumstellar disks
Introduction
As the layers fall inward and join the growing star/disc system, the angular momentum vector of the system must vary in direction (as well as magnitude). The collapse of a turbulent region also produces different directions for the angular momentum vectors of the forming star/disc systems as the collapse progresses. Numerical simulations of this process (Bate, Lodato and Pringle, 2010; Fielding et al., 2015) show that the angular momentum vectors of the discs change as different cloud layers fall inward.
Model Description
Perhaps most importantly, modulation of the stellar spin axis can arise from the star's magnetic disc. To complete the specification of the problem, we need to characterize the properties of the disk. To obtain the rate of precession of the stellar spin axis in the disc frame, we can assume that the disc remains stationary at β = 0 (this assumption will be removed later), which means that Zdisk = 0.
Numerical Simulations
Without the dissipative term, equation 3.19 describes the gravitationally conservative precession of the stellar spin axis around a time-varying disc angular momentum vector. To complete the problem specification, we describe the time evolution of a 1 M star as. Furthermore, the mass of the circumstellar disk (Mdisk) increases proportionally to that of the star so that.
Results & Discussion
In such a case the current induced is radial and in the plane of the disk. This induced field, B,i is represented as a fraction (also called a pitch angle), of the component of the stellar dipole field perpendicular to the disc's surface. This induced field, B,i is represented as a fraction (also called a pitch angle), of the component of the stellar dipole field perpendicular to the disc's surface.
Magnetic origins of the stellar mass-obliquity cor-
Introduction
Previous attempts to explain the trend (Winn, Fabrycky, et al., 2010; . Lai, 2012) have focused primarily on the path of forced migration. Furthermore, the presence of multi-transient fault systems (Huber et al., 2013) has yet to be given a practical explanation within the violent framework. In particular, recent observations (Gregory, Donati, et al., 2012) have revealed that low-mass T-Tauri stars possess dipole field strengths of the order of ~1 kG, an order of magnitude larger than their counterparts in their higher mass (~ 0.1 kg).
Model
Gain in such a frame is equivalent to subtracting νt from the argument of ascending node of the disk. When the star and disc are misaligned, this bulge results in gravitational torques that force a precession of the stellar spin pole around the disc plane (analogous to a top spinning on a flat table). This induced field, Bφ,i is represented as a fraction (also called a pitch angle), γ of the component of the star dipole field perpendicular to the disk's surface.
Results
In other words, the magnetic fields of higher mass stars are dynamically insignificant within the lifetime of the disk, consistent with the timescale analysis cited above. An important aspect of the dynamics is that the star's orientation does not converge to the disk plane, but rather to 30 degrees, i.e. the binary plane, which can be explained qualitatively as follows. The picture is very similar over 30, 45 and 60 degrees, that is, the magnetic fields of low-mass stars are able to realign them with the binary plane within the lifetime of the disk for a wide range of corners.
Discussion
Inclusion of this aspect increases the stability of the disk-locked equilibrium proposed elsewhere (Koenigl, 1991; Mohanty and Shu, 2008). In all simulations, we correct that the axis of rotation of the star is parallel to the z-axis. The solid lines illustrate the major semi-axis of the inner (red) and outer (blue) planets, while the dotted lines indicate the apocenter (top) and pericenter (bottom) of the orbits.
Spin-orbit misalignment as a driver of the Kepler
Introduction
For example, by comparing the transit durations of co-transiting planets, Fabrycky, Lissauer et al. 2014) inferred generally low mutual inclinations within close-packed Kepler systems. Further investigations have shown a similar trend among stars that host planets with lower mass and multiple transits (Huber et al., 2013; Mazeh et al., 2015). In this way, we naturally take into account the slightly larger observed size of males (Johansen et al., 2012).
Analytical Theory
An object with a mass of 10 Earths would have to orbit above ~100 AU to have the angular momentum of a star. By incorporating the above assumptions, we can now write the Hamiltonian (H) that governs the dynamical evolution of planetary orbits. Figure (5.1) shows that a mismatch of the order of twice the star skew can be easily induced for reasonable values of J2.
Numerical Analysis
By geometrical arguments, the potential for such misalignments to take one of the planets out of transit depends on the R?/a ratio. For our numerical runs, we choose 10 values of the stellar inclination and 11 of the initial stars J2 = J2,0 (i.e., the dimming immediately after the disc is dissipated). The choice of forτ is essentially arbitrary, provided that J2 decays over many precessional timescales, due to the adiabatic nature of the dynamics.
Results
We showed that for any given two-planet system, there is a resonant J2 if the inner planet has more angular momentum than the outer one. However, the picture becomes much more complicated in a multiplanet system, where each planet introduces two additional secular modes (one for eccentricity and one for inclination; Murray and Dermott 1999 ), increasing the density of resonances in Fourier space. 10−2.4, which coincides approximately with the onset of instability in the low-inclination directions (Figure 5.2), but not exactly, for the reasons mentioned above.
Discussion
A consequence of primordially generated spin-orbit offsets is that the stellar bias would be present at the end of the birth disc's life, leaving the planetary orbital architecture subject to the quadrupole moment of their young, rapidly rotating and extended host star. For each subpopulation, we illustrate the fraction of systems as a function of the number of planets observed in transit. Using Bayes' theorem, with a uniform prior, we generated a binomial distribution for hot stars and cool stars separately, illustrating the likelihood of the data, given an assumption about what fraction of systems are singletons.
Conclusions
Here, we investigate the ubiquity of stellar obliquity-driven instabilities within lower-multiplicity systems. We choose to parameterize this contraction by assuming that the radius of the star is fixed at R. The instability roughly corresponds to the time when the pericenter of the outer planet coincides with the apocenter of the inner planet.
The ubiquity of stellar oblateness as a driver of
Introduction
In contrast, extrasolar planetary systems are replete with examples of planets orbiting significantly closer than Mercury (Batalha et al., 2013). A particular method to determine mutual trends has been to compare the relative numbers of multi-transit systems with single-transit systems (Lissauer, Ragozzine, et al., 2011; Johansen et al., 2012; Tremaine and Dong, 2012; Ballard and Johnson, 2016). On the contrary, a fraction (up to 50%; Johansen et al. 2012; Ballard and Johnson 2016) of the systems either have large mutual inclinations and therefore detect only one planet at a time in transit, or else this fraction of stars hosts only one planet. .
Methods
The planets move under the influence of their own mutual gravity, along with the gravity of the host star. This approximation is valid under the condition that the angular momentum of the star is significantly larger than the planetary orbit. Given that the mutual inclination will change over time, which could cause pairs of planets to move in and out of mutual transit, we calculate the average number of transits over the last 10 time steps of the integration (spanning . ~ 105 years).
Results & Discussion
As can be seen from Table 6.2, the average eccentricity of the remaining planet is approximately ¯ei ≈ 0.3−0.4. The remaining planet generally experiences an increase in the semi-major axis, indicating a gain in energy at the cost of the second planet, which usually ends up colliding with the central body. Moreover, depending on the exact mechanism of instability, tides can "save" the inner planet by moderating its eccentricity.
Mechanism of instability
To deduce which resonance enters the system, we illustrate the evolution of the argument $1 − $2 in Figure 6.6. Resonances do not exist at low inclinations in K2-38 due to the low angular momentum of the inner body relative to the outer body. The planet-planet-induced precession cannot overcome the greater influence of the star-quadrupole at shorter orbital periods.
Conclusions
To conclude our discussion of the instability itself, we illustrate why the aforementioned eccentricity growth leads to instability. T Tauri stars rotate with periods varying between about 1-10 days, with the median of the distribution lying close to 3-5 days (Bouvier, 2013). We obtained a qualitative understanding of the instability mechanism, namely that the values of $Û of both planets can be brought close to each other by means of quadrupole-driven tendencies.
A secular resonant origin for the loneliness of hot
Introduction
Analytical Theory
Results & Discussion
Summary
