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Secular Excitation of Polar Neptune Orbits in Pure Disk-Planet Systems

Luke B. Handley, Konstantin Batygin

TL;DR

The paper addresses why Neptune-mass planets around Sun-like stars show a bimodal distribution of spin–orbit angles, particularly polar orbits at short periods. It proposes a self-contained mechanism operating during disk dispersal: photoevaporation opens a gap near $1$ au, the inner disk precesses rapidly under the outer disk while the outer disk erodes, and as the inner edge shrinks its precession slows until crossing a secular resonance with the planet’s $J_2$-driven precession, leading to adiabatic capture into a polar configuration. Using a decoupled disk–planet framework with softened Laplace–Lagrange coefficients, an extended Hamiltonian reduction to Henrard’s second fundamental model of resonance shows that Neptune-mass planets can be driven to $i_N\sim90^{\circ}$ for plausible disk parameters without requiring giant perturbers. The model aligns with observations of Neptune obliquities, offers falsifiable predictions for young systems, and highlights the imprints of disk dispersal on planetary architectures, suggesting that primordial disk processes play a significant role in shaping spin–orbit distributions.

Abstract

The stellar spin-orbit angles of Neptune-sized planets present a primordial yet puzzling view of the planetary formation epoch. The striking dichotomy of aligned and perpendicular orbital configurations are suggestive of obliquity excitation through secular resonance -- a process where the precession of a hot Neptune becomes locked onto a forcing frequency, and is slowly guided into a perpendicular state. Previous models of resonant capture have involved the presence of companion perturbers to the star-planet-disk system, but in most cases, such companions are not confirmed to be present. In this work, we present a mechanism for exciting Neptunes to polar orbits in systems without giant perturbers, where photo-evaporation is the self-contained mechanism. Photo-evaporation opens a gap in the protoplanetary disk at ~1 au, and the inner disk continues to viscously accrete onto the host star, precessing quickly due to the perturbation of the outer disk. As the inner disk shrinks, it precesses more slowly, and encounters a resonance with the J2 precession of the Neptune, quickly exciting it to a polar configuration. While likely not applicable to more massive planets which trigger back-reactions onto the disk, this mechanism reproduces the obliquities of small planets in multiple respects.

Secular Excitation of Polar Neptune Orbits in Pure Disk-Planet Systems

TL;DR

The paper addresses why Neptune-mass planets around Sun-like stars show a bimodal distribution of spin–orbit angles, particularly polar orbits at short periods. It proposes a self-contained mechanism operating during disk dispersal: photoevaporation opens a gap near au, the inner disk precesses rapidly under the outer disk while the outer disk erodes, and as the inner edge shrinks its precession slows until crossing a secular resonance with the planet’s -driven precession, leading to adiabatic capture into a polar configuration. Using a decoupled disk–planet framework with softened Laplace–Lagrange coefficients, an extended Hamiltonian reduction to Henrard’s second fundamental model of resonance shows that Neptune-mass planets can be driven to for plausible disk parameters without requiring giant perturbers. The model aligns with observations of Neptune obliquities, offers falsifiable predictions for young systems, and highlights the imprints of disk dispersal on planetary architectures, suggesting that primordial disk processes play a significant role in shaping spin–orbit distributions.

Abstract

The stellar spin-orbit angles of Neptune-sized planets present a primordial yet puzzling view of the planetary formation epoch. The striking dichotomy of aligned and perpendicular orbital configurations are suggestive of obliquity excitation through secular resonance -- a process where the precession of a hot Neptune becomes locked onto a forcing frequency, and is slowly guided into a perpendicular state. Previous models of resonant capture have involved the presence of companion perturbers to the star-planet-disk system, but in most cases, such companions are not confirmed to be present. In this work, we present a mechanism for exciting Neptunes to polar orbits in systems without giant perturbers, where photo-evaporation is the self-contained mechanism. Photo-evaporation opens a gap in the protoplanetary disk at ~1 au, and the inner disk continues to viscously accrete onto the host star, precessing quickly due to the perturbation of the outer disk. As the inner disk shrinks, it precesses more slowly, and encounters a resonance with the J2 precession of the Neptune, quickly exciting it to a polar configuration. While likely not applicable to more massive planets which trigger back-reactions onto the disk, this mechanism reproduces the obliquities of small planets in multiple respects.
Paper Structure (9 sections, 36 equations, 3 figures)

This paper contains 9 sections, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Dynamical Model
  3. The Gap Opened Disk
  4. Evolution of Short-Period Neptunes
  5. Discussion
  6. Observations of Neptune Obliquities
  7. Prospects for Validation
  8. Caveats and Room for Further Work
  9. Conclusion

Figures (3)

  • Figure 1: Top: time-evolution of the inner disk and planetary precession frequencies. Bottom: contours of the simplified Hamiltonian (Equation \ref{['eq:andoyer']}) as functions of the resonance proximity parameter $\delta$, plotted in the canonical cartesian coordinates (x, y) = ($\sqrt{2\Phi}\,\mathrm{cos}\,\phi$, $\sqrt{2\Phi}\,\mathrm{sin}\,\phi$). For $\delta\ll0$ (rapid precession of the inner disk), there is only a single equilibrium point around which all orbits circulate. As $\delta$ increases over time, that equilibrium shifts to higher actions (inclinations) establishing a libration region. When the precession rates of the Neptune and the inner disk are equal ($\delta=0$), the resonance is crossed, and two new equilibria are born at $\delta=3$. The unstable equilibrium lies on the contour which bounds the resonant region, the separatrix, which is plotted in black in the bottom-right two panels. During adiabatic capture, orbits of small action follow the leftward-migrating equilibrium (the crescent shape in the last panel) and remain trapped there for $\delta\gg0$, as $\nu_D$ approaches zero.
  • Figure 2: Integrations of the Neptune's Hamiltonian under several informative regimes with$i_\mathrm{in}=2\degree$ and an initial $i_N=1\degree$. 'Standard Model' indicates the expected outcome for realistic inner disk profiles (See Section \ref{['sec:Discussion']}) as well as our fiducial model with $a_X=0.1$ au. 'Early Saturation' indicates an integration with $a_X=0.08$ au that crosses the resonance adiabatically, but a large value of $\nu_N^{D,2}$ shifts the equilibrium to a sub-polar orbit (Equation \ref{['eq:saturation']}). 'Non-Adiabatic' corresponds to $a_X=0.15$ au such that the resonance was crossed, but the adiabatic criterion was not met. 'Resonance Missed' was integrated with $a_X=0.06$ au, and results in dynamics dominated by the inner disk such that the resonance is not crossed. Note that the qualitative differences in each case is primarily due to the asymptotic behavior of our power-law prescription of the surface density.
  • Figure 3: Observations of the stellar obliquity of planets from the TEPCAT database plotted as a function of planet-to-star mass ratio. Here, we focus on low mass hot Neptune planets (colored points) and giant planets (gray points), neglecting planets of masses which bridge the gap and may be difficult to distinguish. While hot ($\gtrsim6200$K) stars are differentiated for Neptune systems in this figure, a qualitative difference from cool hosts is not yet apparent.