====== 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 $\Lambda$. ===== - A Simple Three-Layer Model ===== 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. ===== - Total Mass Equation ===== 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 $R_m$. ===== - Moment of Inertia Equation ===== 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 $R_m$. ==== - Estimating Moment of Inertia Using $J_2$ and $\Lambda$ ==== 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. ==== - Final Steps to Model the Interior ==== * 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 $R_m$. ==== - Summary ==== ^ Quantity ^ Equation ^ Description ^ | Mass from interior model | $M = \frac{4\pi}{3} [\dots]$ | Adds up volume × density of layers | | Moment of inertia from model | $I = \frac{8\pi}{15} [\dots]$ | Integrates $r^2$ weighted mass for each layer | | $J_2$ | $J_2 = \frac{I_p - I_e}{M R_g^2}$ | Measures deviation from spherical symmetry | | $\Lambda$ | $\Lambda = \frac{\omega^2 R_g^3}{G M}$ | Measures centrifugal force’s effect | | $I$ from $J_2$ and $\Lambda$ | $\frac{I}{M R_g^2} \simeq \frac{\frac{2}{3} J_2}{J_2 + \frac{1}{3} \Lambda}$ | Useful approximation to get $I$ from observations | 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. ===== - Python =====