Energetics of star-planet magnetic interactions: Novel insights from 3D modelling

Abstract

Star-planet magnetic interactions (SPMI) occurring in the sub-Alfvenic regime can, in principle, induce stellar chromospheric hotspots. Currently, estimates of the power generated by SPMI primarily rely on analytical scaling laws that relate stellar and planetary parameters to the interaction energetics. The existing scaling laws published in the literature so far do not agree with each other by at least an order of magnitude. Our aim is to quantify an absolute upper limit on the power that a planet can channel back to its host star during such interactions, which in turn lead to the formation of stellar hotspots. By performing a series of 3D MHD simulations with varied parameters known to influence the energetics of SPMI, we derive a numerically supported scaling law that can be used to reliably estimate the energy channeled from the planet back to the star. Our results suggest that existing analytical scaling laws may not fully capture the power transferred from the planet to the star through SPMI. The scaling law derived from our numerical simulations appears to provide a more comprehensive estimate, reflecting dependencies on common stellar and planetary parameters also considered in earlier models. Moreover, our findings indicate that power generation involves not only the planetary obstacle itself but also the extended magnetic structure of the Alfven wings interacting with the streaming stellar wind. This study suggests that care should be taken when applying analogies directly from jovian sub-Alfvenic interactions to SPMI, as the underlying physical conditions (specifically the value of the Alfvenic Mach number) may not be directly comparable. Our numerically derived scaling law offers a potentially improved approach for estimating SPMI power, capturing some of the interaction's complexities exclusive to SPMI.

Figures (14)

• Figure 1: Schematic from a simulation depicting a close-in exoplanet orbiting its host star. The central red sphere represents the star, while the smaller white sphere indicates the exoplanet. White tubes illustrate magnetic field lines connecting the planet and star, with the surrounding sheath representing the isosurface of one of the Alfvén wings. Translucent hemispheres show the surfaces over which inward Poynting flux was integrated (see main text for details).
• Figure 2: Panel (a) displays the quantity $\vec{s} \cdot \vec{\hat{c}}_A$ on a spherical surface of radius $2.5R_{\star}$ centered on the star. Panel (b) shows the radial component of the Poynting vector, $s_r$, on the same spherical surface. The black contours in this panel indicate lines of constant $(\vec{s} \cdot \vec{\hat{c}}_A) = 1\ \text{W m}^{-2}$.
• Figure 3: Surface-integrated quantities $S_r$ and $S \cdot \hat{c}_A$ shown as functions of radius for the system with an orbital radius of $5 R_{\star}$. The surface integration is performed over spherical shells of varying radii centered on the star, following the approach illustrated in figure \ref{['fig:3d_schematic']} and using the formulations in equations \ref{['eq:sr_integral']} and \ref{['eq:sca_integral']}. The translucent portion of the solid red ($S \cdot \hat{c}_A$) curve corresponds to integration radii where the chosen surfaces are suboptimal for evaluating the integral (see explanation in appendix \ref{['sec:surface_int_truncated']}). Consequently, the solid portion of the red curve highlights the range over which the integrated quantity is considered most reliable. The dotted and dashed black horizontal lines represent the corresponding analytical predictions from the Saur and Lanza models, as given by equations \ref{['eq:Saur_power']} and \ref{['eq:Lanza_power']}, respectively.
• Figure 4: Comparison of SPMI power as a function of planetary magnetic field strength at two orbital radii. Panel (a) shows the results for $R_{\text{orb}} = 3.9,R_{\star}$, and panel (b) for $R_{\text{orb}} = 5.9,R_{\star}$. The scatter points indicate simulation results; the solid lines represent power-law fits. The blue markers and lines denote $S_r$ from simulations, the red squares and lines show the predictions from the Saur model, and the green hexagons and lines correspond to the Lanza model.
• Figure 5: New scaling law for the power directed toward the star via SPMIs, as described by equation \ref{['eq:scaling_law']}. The data points are drawn from all simulation setups listed in Table \ref{['tab:simulation_setups']}. The blue diamonds indicate the default cases, while the red and green circles represent cases with varying planetary magnetic field strengths. For clarity, each point is labeled with the corresponding setup name from Table \ref{['tab:simulation_setups']}.