arxiv: 2605.10868 · v1 · submitted 2026-05-11 · ⚛️ physics.ed-ph

Recognition: 2 theorem links

· Lean Theorem

The Solar System as a lab for the Law of Universal Gravitation

Mauricio Mendivelso-Villaquir\'an

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:00 UTC · model grok-4.3

classification ⚛️ physics.ed-ph

keywords Newton's law of gravitationphysics educationdeductive reasoningsolar system datainverse square laworbital periodsKepler's lawsclassroom laboratory

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The pith

Students can deduce the inverse-square form of the law of gravity from online data on planetary orbits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the mathematical dependence in Newton's law of universal gravitation need not be presented as a given historical fact. Instead, students can use publicly available internet data on planetary distances and orbital periods to perform a deductive analysis that reveals the inverse-square relationship. This turns the solar system into a classroom laboratory where the form of the law emerges from the data rather than from authority. A sympathetic reader would see value in replacing passive acceptance of an equation with an active process of scientific reasoning accessible at the middle- or high-school level.

Core claim

By gathering orbital data for the planets from reliable online sources and examining how orbital period relates to distance, it is possible to deduce that the gravitational attraction must decrease with the square of the distance in order to account for the observed motions.

What carries the argument

Collection and direct comparison of planetary semi-major axes and orbital periods drawn from public internet databases to test the functional dependence of gravitational force on distance.

If this is right

The law of gravitation appears in the curriculum as a result derived from observation rather than an axiom supplied by Newton.
Classroom activities can shift from memorizing an equation to carrying out a data-based deduction.
The same approach supplies a concrete example of how scientific laws are established from measurements.
Educators gain a ready-made exercise that uses only free, existing data sources.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method could be extended to other inverse-power laws or to motion under different central forces using the same public datasets.
Students might next test whether the deduced law still holds for moons, comets, or artificial satellites using additional online records.
The exercise makes explicit the difference between knowing a formula and knowing how the formula was obtained from evidence.

Load-bearing premise

Publicly available internet data on planetary orbits is accurate, complete, and free of hidden assumptions that would prevent students from uniquely recovering the inverse-square form without prior knowledge of the law.

What would settle it

A student following the paper's recommended online sources and analysis steps obtains an exponent for the distance dependence that is not uniquely equal to minus two, or finds that multiple different exponents fit the data equally well.

Figures

Figures reproduced from arXiv: 2605.10868 by Mauricio Mendivelso-Villaquir\'an.

**Figure 1.** Figure 1: The Solar System rotation curve. In addition to the Sun-centered system, gravitational systems such as Jupiter, Saturn, Neptune, and Uranus can be analyzed, as their satellites are included in the NASA datasets. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Examples of rotation curves of other systems. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Linear adjustment of v 2R versus mass of central body. After that process, the obtained outcome for the slope is Γ = 6.54×10−11 m3kg−1 s −2 . This suggests that the constant in (2) is proportional to the mass of the central body. Consequently, v 2 = Γ·M R and, due to the centripetal nature of gravitational attraction4 : F = Γ·M ·m R2 This is the same expression found in (1). The slope turns out to be the g… view at source ↗

read the original abstract

The Law of Universal Gravitation is part of middle and high school's general physics and astronomy curricula. This topic is included in the most popular physics textbooks available as a fact whose origin remains in the detailed work of Sir Isaac Newton 300 years ago. Consequently, its mathematical form is presented as an equation without any deductive process. Nevertheless, deduction of the mathematical form of this law is an opportunity to discuss how a deductive process can be performed using the data available on the Internet from reliable sources.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a classroom exercise for deriving the inverse-square law from online orbital tables, but the tables already embed the target relation so the deduction is not independent.

read the letter

The main takeaway is that this paper offers a teaching activity where students pull planetary periods and distances from public web sources and work toward the 1/r^{2} form. It frames this as a way to show the deductive origin of the law rather than just stating the equation. That goal is reasonable for middle or high school physics classes that want to connect data to theory without heavy math upfront. The paper does a clean job of pointing to accessible sources like NASA tables and keeping the focus on process over result. For educators, that could be a practical hook to discuss how laws get their shape from observations. The soft spot is the data itself. Standard tables list values that were selected or computed under Newtonian assumptions including the inverse-square law and Kepler's third law. Using those numbers to recover T^{2} proportional to a^{3} is mostly recovering what the models already put in. The abstract gives no sign of starting from raw angular positions or ranging data that would be independent of the force law. Without that step or a clear discussion of the assumptions, the exercise risks becoming verification dressed as deduction. The full manuscript might add worked examples or error checks that change this, but nothing in the description shows it. This is for physics teachers who want a data-oriented supplement to the usual textbook presentation of gravity. A reader planning lessons on the history of the law or looking for internet-based activities might pick up an idea or two. It is not aimed at researchers and does not claim new results. I would not send it for peer review. It is a modest educational note that can stand as a preprint without formal refereeing, though a light edit for clarity on the data assumptions would make it more useful to its intended audience.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes an educational activity in which middle- and high-school students use publicly available internet data on planetary orbital periods and semi-major axes to deduce the inverse-square form of Newton's law of universal gravitation, framing this as a deductive process rather than an accepted fact.

Significance. If the derivation can be shown to proceed from model-independent data without circular embedding of Kepler's third law, the activity would usefully illustrate scientific deduction in the classroom. The absence of any concrete steps, data tables, or error analysis in the provided text prevents assessment of whether this goal is achieved.

major comments (2)

[Abstract] Abstract: the claim that students can 'uniquely deduce' the inverse-square form from internet data is not supported by any derivation steps, tabulated values, or discussion of how periods and distances are converted to forces; without these the central educational claim cannot be evaluated.
[Data sources] Data sources (implied throughout): standard tables (NASA, Wikipedia) list periods and semi-major axes already consistent with T²/a³ = constant because those values are selected or computed under Newtonian inverse-square assumptions; this embeds the target relation and converts the exercise into verification rather than independent deduction.

minor comments (1)

The manuscript should add a dedicated section with explicit data sources, the exact sequence of calculations, and a sample error analysis to allow replication.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which identify key areas for improving the clarity and rigor of the proposed educational activity. We address each major comment below and will revise the manuscript to incorporate additional details and discussion.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that students can 'uniquely deduce' the inverse-square form from internet data is not supported by any derivation steps, tabulated values, or discussion of how periods and distances are converted to forces; without these the central educational claim cannot be evaluated.

Authors: We agree that the manuscript lacks explicit derivation steps, sample data, and calculations, which limits evaluation of the central claim. The current text emphasizes the conceptual opportunity for deduction but does not include practical implementation details. In revision, we will add a new section with concrete steps: selecting NASA-sourced orbital periods and semi-major axes for the planets, computing centripetal accelerations as proxies for gravitational force (F ∝ 4π²a / T²), tabulating results, and showing via log-log plots or ratios that the force scales as 1/a². Example tables and error considerations will also be included to support the educational claim. revision: yes
Referee: [Data sources] Data sources (implied throughout): standard tables (NASA, Wikipedia) list periods and semi-major axes already consistent with T²/a³ = constant because those values are selected or computed under Newtonian inverse-square assumptions; this embeds the target relation and converts the exercise into verification rather than independent deduction.

Authors: We acknowledge the potential for perceived circularity when using modern tabulated values. Periods are observationally determined, while semi-major axes derive from geometric and ranging methods independent of the gravitational force law form. The deduction proceeds by combining these data with the centripetal force requirement for assumed circular orbits. To mitigate concerns, the revised manuscript will include explicit discussion of data provenance, reference Newton's use of Kepler's empirically derived third law (from pre-Newtonian observations), and note that the activity illustrates the logical steps of scientific inference rather than pure verification. We will also suggest supplementary use of historical datasets where feasible. revision: partial

Circularity Check

1 steps flagged

Orbital data tables already encode Kepler's third law derived from inverse-square gravity

specific steps

fitted input called prediction [Abstract]
"deduction of the mathematical form of this law is an opportunity to discuss how a deductive process can be performed using the data available on the Internet from reliable sources."

The 'reliable sources' supply periods and semi-major axes that are not model-independent measurements but values already consistent with Kepler's third law (T²/a³ = constant). Kepler's law is itself derived from the inverse-square force law; therefore the input data already contains the functional dependence the paper claims to deduce, making the output equivalent to the input by construction.

full rationale

The paper presents a deductive process to recover the inverse-square form of Newton's law from publicly available internet data on planetary periods and semi-major axes. Standard sources (NASA, Wikipedia) publish values already selected or computed to satisfy T² ∝ a³, which is a direct mathematical consequence of central inverse-square force under the assumptions of negligible planet mass and two-body reduction. The derivation therefore starts from inputs that presuppose the target functional form, reducing the claimed deduction to a verification exercise rather than an independent recovery from raw observations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters, axioms beyond standard orbital mechanics, or invented entities; it relies on existing planetary data and the assumption that orbital relations can be used to back out the force law.

axioms (1)

domain assumption Planetary orbital data from public sources accurately reflect Keplerian motion without significant perturbations or measurement error for the purpose of this exercise.
The deduction process depends on treating the downloaded distances and periods as reliable inputs that encode the underlying gravitational dependence.

pith-pipeline@v0.9.0 · 5368 in / 1167 out tokens · 71270 ms · 2026-05-12T04:00:54.404787+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Once the curve has been obtained, a function must be adjusted... v ∝ R^{-1/2} ... v²R = constant ... Γ = 6.54×10^{-11} ... F = Γ·M·m / R²
IndisputableMonolith/Patterns/Gravity.lean zero-parameter gravity certificate unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the slope turns out to be the gravitational constant G

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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