Abstract
Sedna is the first inner Oort cloud object to be discovered. Its dynamical origin remains unclear, and a possible mechanism is considered here. We investigate the parameter space of a hypothetical solar companion which could adiabatically detach the perihelion of a Neptune-dominated TNO with a Sedna-like semimajor axis. Demanding that the TNO’s maximum value of osculating perihelion exceed Sedna’s observed value of 76 AU, we find that the companion’s mass and orbital parameters (m c , a c , q c , Q c , i c ) are restricted to
during the epoch of strongest perturbations. The ecliptic inclination of the companion should be in the range \(45{\deg}\lessapprox i_c\lessapprox 135{\deg}\) if the TNO is to retain a small inclination while its perihelion is increased. We also consider the circumstances where the minimum value of osculating perihelion would pass the object to the dynamical dominance of Saturn and Jupiter, if allowed. It has previously been argued that an overpopulated band of outer Oort cloud comets with an anomalous distribution of orbital elements could be produced by a solar companion with present parameter values
If the same hypothetical object is responsible for both observations, then it is likely recorded in the IRAS and possibly the 2MASS databases.
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Acknowledgements
The authors gratefully acknowledge informative exchanges with Rodney Gomes. J.J.L. received support from NASA Planetary Geology and Geophysics Grant 344-30-50-01.
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Appendix A: Dynamics
Appendix A: Dynamics
We approximate the companion orbit as an invariant ellipse of mass and orbital parameters m c , a c , q c , Q c , i c having orbit normal \(\hat{{\bf n}}_{\bf c}\). The heliocentric companion position is denoted by r c while the heliocentric Sedna position is r. The barycentric solar location is
Newton’s equations of motion for the STNO are then
where \({\bf g}_{\odot,{\bf p,c}}\) are the gravitational fields at the STNO’s location due to the Sun, the planets and the companion, respectively.
Further, we approximate the planetary perturbations by treating the planets as circular rings. In the limits r p r r c , we expand both the planetary and companion interactions. Thus
Combining these results, we obtain
where \({\cal I}_p\equiv\sum_p\mu_p r_p^2\) and \(\mu_{\circ}\equiv\mu_\odot +\sum_p \mu_p\).
We then construct the equations of motion for the scaled angular momentum vector and the eccentricity vector,
which yields:
Expressing the positions of the STNO and the companion in vector form
we sequentially perform secular averages over the short (STNO) orbital period and the long (companion) orbital period to obtain:
where
These analytic forms are obtained using Mathematica (Wolfram Research, 2003)).
We see in Eq. (A9) that the secular planetary interaction produces orbit normal precession around \({\bf \hat{n}_p}\), while a similar term in the secular companion interaction produces orbit normal precession around \({\bf \hat{n}_c}\). It is the term \(\propto ({\bf \hat{n}_c}\cdot{\bf e}) {\bf \hat{n}_c} \times {\bf e}\) that dominates the nutation of h and the changes in perihelion distances for large-eccentricity STNO. The analysis reproduces a well-known result: In the secular approximation, planetary perturbations alone do not change e (Goldreich, 1965).
The secularly averaged equations depend on the companion elements through the quantities \({\bf \hat{n}_c}\) and τ c . There are several symmetries evident in the equations, such as their invariance when \({\bf \hat{n}_c}\rightarrow - {\bf \hat{n}_c}\), i.e., i c → π −i c , and their independence of the companion perihelion direction, \({\bf \hat{e}_c}\).
Orienting our axes as shown in Figure 1, we see that the companion can be characterized by two parameters, γ c and i c , assumed to be constant here. Of course a wide-binary companion orbit is subject to perturbations from passing stars and the galactic tide. Therefore, these parameters essentially describe the epoch when companion interactions with the STNO are strongest, i.e., when γ c is largest. The galactic tide will change e c and i c , but changes are small for \(a_c\lessapprox\) 10,000 AU. Osculations proceed through \(>rapprox\) one half-cycle in 4.6 Gy when \(a_c>rapprox\)20,000 AU.
The STNO orbit is characterized by a secularly constant semimajor axis, a, and four variable elements i, ω, ω and e. The six coupled equations for the components of e and h are restricted by the two conserved quantities, h·e=0 and h 2+e 2=1, which serve as checks on our numerical solutions.
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Matese, J.J., Whitmire, D.P. & Lissauer, J.J. A Widebinary Solar Companion as a Possible Origin of Sedna-like Objects. Earth Moon Planet 97, 459–470 (2005). https://doi.org/10.1007/s11038-006-9078-6
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DOI: https://doi.org/10.1007/s11038-006-9078-6