Dipolar Magnetic Moment of the Bodies of the Solar System and the Hot Jupiters

The planets magnetic field has been explained based on the dynamo theory, which presents as many difficulties in mathematical terms as well as in predictions. It proves to be extremely difficult to calculate the dipolar magnetic moment of the extrasolar planets using the dynamo theory. The aim is to find an empirical relationship (justifying using first principles) between the planetary magnetic moment, the mass of the planet, its rotation period and the electrical conductivity of its most conductive layer. Then this is applied to Hot Jupiters. Using all the magnetic planetary bodies of the solar system and tracing a graph of the dipolar magnetic moment versus body mass parameter, the rotation period and electrical conductivity of the internal conductive layer is obtained. An empirical, functional relation was constructed, which was adjusted to a power law curve in order to fit the data. Once this empirical relation has been defined, it is theoretically justified and applied to the calculation of the dipolar magnetic moment of the extra solar planets known as Hot Jupiters. Almost all data calculated is interpolated, bestowing confidence in terms of their validity. The value for the dipolar magnetic moment, obtained for the exoplanet Osiris (HD209458b), helps understand the way in which the atmosphere of a planet with an intense magnetic field can be eroded by stellar wind. The relationship observed also helps understand why Venus and Mars do not present any magnetic field.

💡 Research Summary

The paper proposes an empirical formula for estimating the dipolar magnetic moment (M) of planets based on three readily observable or inferable planetary properties: total mass (m), rotation period (P), and electrical conductivity (σ) of the most conductive internal layer. The authors begin by compiling data for the seven magnetic bodies in the Solar System – the four giant planets (Jupiter, Saturn, Uranus, Neptune), Earth, Mercury, and the moon Ganymede – including measured dipole moments, masses, rotation periods, and assumed conductivities (metallic hydrogen for the giants, iron‑nickel alloy for terrestrial cores, and seawater or iron‑nickel for Ganymede).

Separate log‑log analyses reveal strong power‑law correlations between M and each individual variable: M ∝ m^β (β≈1.2, R²≈0.98), M ∝ P^−δ (δ≈1, R²≈0.97), and a qualitative trend that higher σ yields larger M. By combining the three variables into a single dimensionless parameter X = m·σ/P, the authors find that the data collapse onto another power law, M = k·X^α, with α close to, but slightly above, unity (≈1.1). They note that removing Jupiter from the fit drives α even nearer to 1, suggesting that Jupiter’s magnetic field is unusually strong relative to the simple scaling.

To provide a physical rationale, the paper models the planet’s interior as a conducting sphere carrying a net current I = σ·ε·A, where ε is an electromotive force generated by rotation (ω = 2π/P). Using the scalar magnetic potential of a dipole and equating it to that of a distant current loop, they derive M ≈ I·A·R, which after substituting I, the planetary volume (V ≈ 4πR³/3), and the relation m = ρV, yields the empirical form M ∝ (m·σ/P)^α. The exponent α deviates from 1 because of simplifications (e.g., neglect of fluid dynamics, magnetic diffusion, and detailed interior stratification).

The authors then apply the scaling law to “Hot Jupiters,” a class of close‑in gas giants whose rotation is assumed to be tidally locked (P equal to orbital period). Using measured masses and orbital periods for 31 Hot Jupiters and adopting σ ≈ 2 × 10⁵ S m⁻¹ (metallic hydrogen), they compute X and thus M for each planet. For HD 209458b (Osiris), the predicted dipole moment is 1.26 × 10²⁶ A·m², about 5.5 times Saturn’s and one‑sixth Jupiter’s. The derived magnetic moments allow estimation of surface field strengths and magnetopause stand‑off distances (R_ss) when planetary radii are known.

The paper also discusses non‑magnetic bodies such as Venus and Mars. Plugging their masses, long rotation periods, and low σ (silicate conductivity ≈1 S m⁻¹) into the formula yields magnetic moments far below observational upper limits, offering a qualitative explanation for their weak or absent fields. Ganymede’s position on the scaling plot depends sensitively on the assumed σ; using seawater conductivity places it off the main trend, while iron‑nickel conductivity aligns it, supporting the hypothesis of a metallic core.

In conclusion, the study presents a simple, data‑driven relationship that can quickly estimate planetary magnetic moments, especially for exoplanets where direct magnetic measurements are impossible. However, the approach rests on several strong assumptions: (1) a single bulk conductivity adequately represents complex, layered interiors; (2) rotation period alone captures the dynamo’s kinetic energy input, ignoring convective vigor, magnetic Reynolds number, and compositional effects; (3) the limited sample size (seven Solar System bodies) may not capture the full diversity of planetary interiors. Consequently, while the scaling law offers a useful first‑order tool, its quantitative predictions should be treated with caution. Future work incorporating high‑pressure conductivity experiments, detailed interior structure models, and full magnetohydrodynamic simulations will be essential to refine or replace the empirical relation with a physically robust theory.