Equation 7 · How We Know What Is Inside a Planet
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol U
U is part of the quantity the equation computes from the expression on the right.
Symbol r
r occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol θ
θ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol G
G occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol M
M occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol n
n appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.
Symbol J_n
is one of the signed contributions combined to compute the quantity on the left.
Symbol R
R occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol P_n
is one of the signed contributions combined to compute the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subtraction
Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →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.
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.
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.
What the article says around this equation
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 . 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.
Sources cited in the article section
- [10] The crust of the Moon as seen by GRAIL ↗
- [17] Comparing Jupiter interior structure models to Juno gravity measurements and the role of a dilute core ↗
- [18] A diffuse core in Saturn revealed by ring seismology ↗
These citations give research context. Read each source to check which claims it supports.
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