How Can the I‑Love Relation Reveal Secrets of Planets and Their Moons?

In the realm of stars and planets, two quantities seem to speak different languages: the moment of inertia, which tells how hard it is to spin an object, and tidal deformability, which measures how much it gets squeezed by another body’s gravity. Recent calculations have shown that, for ideal polytropic stars, these two properties are linked by an almost universal relation, nicknamed the “I‑Love” relation.

Under Newtonian gravity, the I‑Love relation varies by less than 1 % across different polytropic indices, while in General Relativity the variation rises to about 10 %. This means the connection is more stable and predictable in the Newtonian regime.

But the discovery doesn’t stop at stars. The authors compared the theoretical formula with real data from eight planets and several moons in our Solar System. The results match surprisingly well, suggesting the same law can describe smaller bodies too.

Why does this matter? If we know an exoplanet’s mass and one of the two quantities (moment of inertia or tidal deformability), the I‑Love relation lets us estimate the other. In practice, we could gauge how “flexible” a distant planet is without directly observing it, opening new ways to probe its interior structure.

This finding shows once again how seemingly simple laws can connect diverse astronomical phenomena, from a star’s heartbeat to the dance of moons around their planets.

The I‑Love relation originates from the interaction between two fundamental parameters of self‑gravitating bodies. The moment of inertia I is defined as I = ∫_0^R ρ(r) r^4 dr, where ρ(r) is the radial density and R the body’s radius. The tidal deformability λ, on the other hand, is linked to the quadrupolar response of the body to an external gravitational field: λ = (2/3) k_2 R^5 / G, with k_2 the Love number and G the gravitational constant. For a polytropic EOS, P = K ρ^{1+1/n}, the internal structure is described by a Lane‑Emden equation of order n. Solving this equation numerically both in Newtonian gravity and in GR (using the Tolman‑Oppenheimer‑Volkoff equations) yields the functions I(M) and λ(M) for various n. Normalizing by the mass M allows plotting λ(I) in a dimensionless form, revealing a curve almost independent of n. The 10 % discrepancy in GR stems from spacetime curvature, which modifies the internal pressure distribution and consequently the k_2 values. Applying the relation to Solar System planets requires assuming an approximate polytropic model for their interiors, typically n≈1 for rocky bodies and n≈0.5 for gas giants. Observational I data come from node precession measurements or flyby missions, while λ is inferred from orbital variations due to tidal effects. The good agreement observed suggests that, despite simplifications, the I‑Love relation captures the fundamental elastic properties of self‑gravitating bodies, making it a powerful tool for characterizing exoplanets, especially when direct observations are limited.