courses:ast401:4.1
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| ====== Modeling planetary interiors ====== | ====== Modeling planetary interiors ====== | ||
| - | ===== Deriving the Mass and Moment of Inertia for a Layered Planet Using $J_2$ and $\Lambda$ ===== | ||
| - | |||
| 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, | 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, | ||
| - | ==== 1. A Simple Three-Layer Model ==== | + | ===== - A Simple Three-Layer Model ===== |
| Assume the planet/moon has: | Assume the planet/moon has: | ||
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| We assume constant density in each layer. | We assume constant density in each layer. | ||
| - | ==== 2. Total Mass Equation ==== | + | ===== - Total Mass Equation |
| The mass of the object is the sum of the masses of its spherical layers: | The mass of the object is the sum of the masses of its spherical layers: | ||
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| This equation connects the mass $M$ to the layer boundaries $R_c$ and $R_m$. | This equation connects the mass $M$ to the layer boundaries $R_c$ and $R_m$. | ||
| - | ==== 3. Moment of Inertia Equation ==== | + | ===== - Moment of Inertia Equation |
| Each spherical shell contributes to the moment of inertia. The total moment of inertia is: | Each spherical shell contributes to the moment of inertia. The total moment of inertia is: | ||
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| These two equations (1) and (2) form a solvable system for the unknown radii $R_c$ and $R_m$. | These two equations (1) and (2) form a solvable system for the unknown radii $R_c$ and $R_m$. | ||
| - | ==== 4. Estimating Moment of Inertia Using $J_2$ and $\Lambda$ ==== | + | ==== - 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: | In practice, we don’t always know $I$ directly. However, we can estimate it using measurements of: | ||
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| This equation provides a practical way to **infer $I$ from observable quantities**, | This equation provides a practical way to **infer $I$ from observable quantities**, | ||
| - | ==== 5. Final Steps to Model the Interior ==== | + | ==== - Final Steps to Model the Interior ==== |
| * Measure or estimate $J_2$, $\omega$, $M$, and $R_g$ from spacecraft data. | * Measure or estimate $J_2$, $\omega$, $M$, and $R_g$ from spacecraft data. | ||
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| * Plug $M$ and $I$ into Equations (1) and (2) and solve for $R_c$ and $R_m$. | * Plug $M$ and $I$ into Equations (1) and (2) and solve for $R_c$ and $R_m$. | ||
| - | ==== Summary ==== | + | ==== - Summary ==== |
| ^ Quantity ^ Equation ^ Description ^ | ^ Quantity ^ Equation ^ Description ^ | ||
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| 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. | 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. | ||
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| + | ===== - Python ===== | ||
| + | < | ||
| + | <script src=" | ||
| + | </ | ||
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