Effects of Magnetic Braking and Tidal Friction on Hot Jupiters

Tidal friction is thought to be important in determining the long-term spin-orbit evolution of short-period extrasolar planetary systems. Using a simple model of the orbit-averaged effects of tidal friction Eggleton, Kiseleva & Hut (1998), we analyse the effects of the inclusion of stellar magnetic braking on the evolution of such systems. A phase-plane analysis of a simplified system of equations, including only the stellar tide together with a model of the braking torque proposed by Verbunt & Zwaan (1981), is presented. The inclusion of stellar magnetic braking is found to be extremely important in determining the secular evolution of such systems, and its neglect results in a very different orbital history. We then show the results of numerical integrations of the full tidal evolution equations, using the misaligned spin and orbit of the XO-3 system as an example, to study the accuracy of simple timescale estimates of tidal evolution. We find that it is essential to consider coupled evolution of the orbit and the stellar spin in order to model the behaviour accurately. In addition, we find that for typical Hot Jupiters the stellar spin-orbit alignment timescale is of the same order as the inspiral time, which tells us that if a planet is observed to be aligned, then it probably formed coplanar. This reinforces the importance of Rossiter-McLaughlin effect observations in determining the degree of spin-orbit alignment in transiting systems.

💡 Research Summary

The paper investigates how stellar magnetic braking influences the long‑term orbital and spin evolution of close‑in giant exoplanets, commonly known as hot Jupiters. Using the orbit‑averaged tidal friction formalism of Eggleton, Kiseleva & Hut (1998), the authors augment the tidal equations with a magnetic‑braking torque of the form proposed by Verbunt & Zwaan (1981). After nondimensionalising the stellar spin (Ω) and orbital mean motion (n) to variables ˜Ω and ˜n, the coupled evolution is described by two ordinary differential equations containing a single dimensionless parameter A that measures the strength of magnetic braking. For a Sun‑like star hosting a Jupiter‑mass planet, A≈100, indicating that magnetic braking dominates the stellar spin evolution.

A phase‑plane analysis in the (˜Ω, ˜n) plane shows two qualitatively different regimes. When A=0 (no magnetic braking), the system evolves toward the corotation line ˜Ω=˜n, which can be a stable equilibrium if the total angular momentum satisfies Hut’s (1980) condition. In contrast, with realistic magnetic braking (A≈100), the star’s spin rapidly declines, moving the system below corotation (˜Ω<˜n). The tidal torque then extracts orbital angular momentum, causing the planet to spiral inward. The authors demonstrate that, without magnetic braking, many initial conditions would settle into a long‑lived equilibrium, whereas with braking every bound orbit eventually decays in a finite time.

To test the analytic insights, the authors perform full numerical integrations of the coupled tidal‑spin equations for the misaligned XO‑3 system (λ≈70°, m_p≈12.5 M_J, e≈0.29, P≈3.2 d, host star ≈1.3 M_⊙). Simple timescale estimates (τ_a for orbital decay, τ_i for spin‑orbit alignment) often assume a constant stellar spin, which is invalid when magnetic braking is strong. Their integrations reveal that for typical tidal quality factors Q′≈10⁶ the alignment and inspiral times are of order a few Myr, far shorter than the observed system age (2–3 Gyr). To retain the present inclination and survive for gigayears, the stellar Q′ must exceed ≈10¹⁰, implying very weak tidal dissipation in the host star. This conclusion is supported by independent calculations using the Ogilvie & Lin (2007) formalism, which predict Q′>10¹⁰ for the relevant tidal frequencies.

The paper therefore reaches three main conclusions: (1) Stellar magnetic braking is essential for realistic modeling of hot‑Jupiter evolution unless the star is already rotating much slower than the orbit; neglecting it leads to qualitatively wrong predictions. (2) Coupled tidal‑spin evolution can be substantially faster (or slower) than simple exponential‑decay estimates, so full integration of the equations is required for accurate timescales. (3) For typical hot Jupiters, the spin‑orbit alignment timescale τ_i is comparable to the orbital decay timescale τ_a, meaning that observed alignment likely reflects primordial coplanarity rather than tidal realignment. Consequently, Rossiter‑McLaughlin measurements remain a powerful diagnostic for distinguishing formation pathways such as planet‑planet scattering, Kozai‑Lidov migration, or smooth disk‑driven migration.