← Mathematical compendium

Published equation contexts

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]

Why this formula appears here

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.

Read the full article-specific guide →

Read the representative guide

rr

Symbol r

r occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Read this term in its guide →
θ\theta

Symbol θ

θ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

Read this term in its guide →
nn

Symbol n

n appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

Read this term in its guide →
n=2n=2

Starting index or lower bound: n=2

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

Read this term in its guide →
∞\infty

Ending index or upper bound: infty

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

Read this term in its guide →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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]

Equation 7 · Planetary Science

How We Know What Is Inside a Planet

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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.

Equation guide → · Article →