Modeling planetary interiors

To understand the internal structure of a planet or moon, we need to relate its mass ($M$) and moment of inertia ($I$) to the size and density of internal layers. These properties can also be inferred from gravitational field measurements, specifically using the coefficients $J_2$ and the dimensionless parameter $\Lambda$.

Assume the planet/moon has:

• An iron core of radius $R_c$ and density $\rho_c$
• A rocky mantle extending from $R_c$ to $R_m$ with density $\rho_m$
• An outer shell (e.g. ice) extending from $R_m$ to $R_g$ (total radius) with density $\rho_i$

We assume constant density in each layer.

The mass of the object is the sum of the masses of its spherical layers:

$$ M = \frac{4\pi}{3} \left[ \rho_c R_c^3 + \rho_m (R_m^3 - R_c^3) + \rho_i (R_g^3 - R_m^3) \right] \tag{1} $$

This equation connects the mass $M$ to the layer boundaries $R_c$ and $R_m$.

Each spherical shell contributes to the moment of inertia. The total moment of inertia is:

$$ I = \frac{8\pi}{15} \left[ \rho_c R_c^5 + \rho_m (R_m^5 - R_c^5) + \rho_i (R_g^5 - R_m^5) \right] \tag{2} $$

These two equations (1) and (2) form a solvable system for the unknown radii $R_c$ and $R_m$.

In practice, we don’t always know $I$ directly. However, we can estimate it using measurements of:

• $J_2$ — the second zonal harmonic of the planet’s gravitational field, related to its shape
• $\Lambda$ — the ratio of centrifugal to gravitational force at the equator

The gravitational flattening $J_2$ is related to the difference in moments of inertia around the polar and equatorial axes:

$$ J_2 = \frac{I_p - I_e}{M R_g^2} \tag{3} $$

The parameter $\Lambda$ is defined as:

$$ \Lambda = \frac{\omega^2 R_g^3}{G M} \tag{4} $$

Where:

• $\omega$ = angular rotation rate
• $R_g$ = equatorial radius
• $G$ = gravitational constant
• $M$ = mass

Using these, the moment of inertia factor $\frac{I}{M R_g^2}$ can be approximated by:

$$ \frac{I}{M R_g^2} \simeq \frac{\frac{2}{3} J_2}{J_2 + \frac{1}{3} \Lambda} \tag{5} $$

This equation provides a practical way to infer $I$ from observable quantities, even without directly measuring internal mass distribution.

• Measure or estimate $J_2$, $\omega$, $M$, and $R_g$ from spacecraft data.
• Use Equation (5) to compute $\frac{I}{M R_g^2}$ and thus $I$.
• Plug $M$ and $I$ into Equations (1) and (2) and solve for $R_c$ and $R_m$.

By combining gravitational field measurements and simple physical models, we can infer the internal structure of distant planetary bodies—even when we can’t look inside them.