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Equation 7 · Part 10 · How We Know What Is Inside a Planet

=

U(r,θ)=GMr[1−∑n=2∞Jn(Rr)nPn(cos⁡θ)]U(r,\theta) = \frac{GM}{r}\left[1 - \sum_{n=2}^{\infty} J_{n}\left(\frac{R}{r}\right)^{n} P_{n}(\cos\theta)\right]
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

A spacecraft in orbit is a gravimeter. Its along-track velocity is perturbed by mass anomalies beneath it, and Doppler tracking of the radio link records those perturbations. Expanded outside the body, the potential from a rotationally symmetric mass distribution takes the form U(r,θ)=GMr[1−∑n=2∞Jn(Rr)nPn(cos⁡θ)]U(r,\theta) = \frac{GM}{r}\left[1 - \sum_{n=2}^{\infty} J_{n}\left(\frac{R}{r}\right)^{n} P_{n}(\cos\theta)\right]. and the factor (R/r)^{n} is the whole epistemology of gravity-based interior work. Each successive degree n decays faster with altitude, so high-degree terms are measurable only from low orbit and are dominated by shallow structure. Low-degree terms reach deep but are few. A gravity field is therefore a filter that attenuates depth information in a known, unavoidable way.

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the article section

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