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

This paper proposes using publicly available astronomical data from the Solar System to guide students through a deductive process for deriving the mathematical form of Newton's Law of Universal Gravitation, rather than presenting it merely as a historical fact.

Original authors: Mauricio Mendivelso-Villaquirán

Published 2026-05-12

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Mauricio Mendivelso-Villaquirán

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Solar System as a giant, cosmic laboratory where the rules of gravity are written in the stars. For centuries, students have been handed a "finished recipe" for gravity: a complex formula (Newton's Law of Universal Gravitation) that they are told to memorize and use, but rarely asked to figure out how it was discovered in the first place. It's like being given a cake and told, "This is how it tastes," without ever seeing the ingredients or the mixing process.

This paper proposes a new way to teach this concept. Instead of just memorizing the formula, the author suggests letting students act like detectives to re-discover the law using data that is freely available on the internet.

Here is how the paper breaks it down, using simple analogies:

1. The Problem: The "Magic Formula"

Currently, physics textbooks present gravity as a finished fact. They show the equation $F = G \frac{m_1 m_2}{r^2}$ and move on. The paper argues that this skips the most exciting part: the detective work. Students are left wondering, "How did Newton actually know this was true?" without a clear path to find out for themselves.

2. The Tool: The "Cosmic Speedometer" (Rotation Curves)

The author introduces a tool used by astronomers to study entire galaxies, called a rotation curve.

The Analogy: Imagine a carousel. If you stand on the edge, you have to run faster to keep up than someone standing near the center. A rotation curve is just a graph that plots how fast things are spinning against how far they are from the center.
The Application: The paper suggests using NASA's public data to plot the speed of planets (how fast they orbit the Sun) against their distance from the Sun. You can do this for our Solar System, or even for the moons of Jupiter and Saturn.

3. The Discovery: Finding the Pattern

When students plot this data on a graph, a clear pattern emerges. It's like finding a hidden code in the noise.

The graph shows that the speed of a planet is related to its distance in a very specific way: Speed squared multiplied by distance equals a constant number.
When the author tested this with different systems (Sun, Jupiter, Saturn), they found that this "constant number" wasn't actually constant across the whole universe—it changed depending on how heavy the central object was.
The Analogy: Think of it like a dance. The heavier the partner in the center (the Sun or Jupiter), the faster the dancers (planets or moons) have to spin to stay in the circle. The graph reveals a direct link between the "weight" of the central body and the speed of the orbit.

4. The Result: Deriving Gravity from Scratch

By connecting these dots, the students can mathematically derive the Law of Universal Gravitation themselves.

They start with the speed and distance data.
They realize the speed depends on the mass of the central body.
They calculate the slope of the line connecting these points, and voilà, that slope turns out to be the famous "G" (the gravitational constant).
The paper notes that the value they calculate is only about 2% different from the official, accepted value used by scientists today. This proves that you don't need a super-complex lab to find the law; you just need data and a graph.

5. The Classroom Experience: A Group Puzzle

The paper emphasizes that this isn't a solo activity. It's designed as a team project.

The Setup: Students work in groups, each analyzing a different planetary system (e.g., one group looks at Jupiter's moons, another at Earth's satellites).
The Process: They use free software (like Google Sheets or Excel) to fit the data to a curve.
The Learning: If a group gets a weird result, it's not a failure; it's a discussion point. Maybe they included a comet with a weird, stretched-out orbit that doesn't fit the "circular" pattern. This teaches them about data quality and scientific reasoning.
The Goal: By working together to build the formula from the ground up, students stop seeing gravity as a "magic rule" and start seeing it as a logical conclusion drawn from real-world evidence.

Summary

In short, this paper argues that we should stop treating the Law of Universal Gravitation as a mysterious gift from Isaac Newton and start treating it as a puzzle that students can solve today. By using the Solar System as a giant dataset and plotting the "speed vs. distance" of planets, students can mathematically reconstruct the law of gravity themselves, turning a dry textbook fact into an engaging, hands-on scientific discovery.

Technical Summary: The Solar System as a Lab for the Law of Universal Gravitation

Problem Statement
The Law of Universal Gravitation (LUG) is a staple of middle and high school physics curricula, yet it is frequently presented as a static mathematical fact ( $F = Gm_1m_2/r^2$ ) rather than a derived conclusion. A review of standard university, college, and AP/IB physics textbooks reveals a consistent pedagogical pattern: the law is introduced via its final equation or through a somewhat artificial derivation invoking Kepler's Third Law and circular orbits, often without a clear deductive process accessible to students. Consequently, learners are rarely given the opportunity to independently deduce the law from empirical data, leaving a gap between the historical discovery and the modern student's understanding. While historical experiments like the Cavendish balance exist, they are often delicate, assume the validity of LUG to calculate $G$ , and fail to address the fundamental question of how the law itself was originally deduced.

Methodology
The paper proposes an alternative, data-driven approach to deriving LUG using the "rotation curve" method, a tool typically employed by galactic astronomers (e.g., Vera Rubin) to study spiral galaxies. The methodology involves the following steps:

Data Acquisition: Students utilize publicly available, reliable datasets (specifically NASA Solar System Data) containing orbital velocities ( $v$ ) and mean orbital radii ( $R$ ) for planets and satellites.
Rotation Curve Generation: Using accessible software tools (e.g., Google Sheets, Desmos, GeoGebra, or specialized plotting tools), students plot $v$ against $R$ to generate a rotation curve for the Solar System and other gravitational systems (e.g., Jupiter's moons, Saturn's moons).
Function Fitting: Students fit a power-law function to the data points. The analysis reveals that the data follows a relationship of the form $v \propto R^{-0.5}$ , or $v^2R = \text{constant}$ .
Cross-System Analysis: This constant ( $v^2R$ ) is calculated for various central bodies (Sun, Jupiter, Saturn, Uranus, Neptune) and compared against their known masses.
Linear Regression: A log-log plot of the derived constant ( $v^2R$ ) versus the mass of the central body ( $M$ ) is constructed. A linear regression is applied to determine the slope of this relationship.
Derivation: The slope is identified as the gravitational constant ( $G$ ), leading to the deduction that $v^2 = GM/R$ . By equating this to the centripetal force equation ( $F = mv^2/R$ ), the Law of Universal Gravitation ( $F = GMm/R^2$ ) is derived.

Key Results

Empirical Verification: The rotation curves generated from NASA data for the Solar System and other planetary systems consistently exhibit Keplerian behavior, fitting a power law with an exponent very close to $-0.5$ (ranging from $-0.495$ to $-0.500$ across various software platforms).
Constant Determination: For the Solar System, the calculated value of $v^2R$ is approximately $1.2996 \times 10^{20} \, \text{m}^3\text{s}^{-2}$ .
Derivation of G: When plotting $v^2R$ against the mass of the central body, a linear relationship is observed. The slope of this regression yields a value of $\Gamma = 6.54 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2}$ .
Accuracy: The derived value for the gravitational constant differs from the accepted value by approximately 2%, a result attributed to the fitting process and data limitations.

Significance and Pedagogical Claims
The paper argues that this methodology offers a significant pedagogical shift by allowing students to reconstruct a foundational scientific law using modern computational tools and real-world data. Key claims regarding its significance include:

Demystification: It transforms LUG from an isolated axiom into a deduced result, illuminating the origins of the law and the scientific process of building knowledge from data.
STEM Integration: The activity embodies STEM principles by connecting historical breakthroughs (Newtonian physics) with contemporary technology (data analysis software, online repositories).
Collaborative Inquiry: The process is designed for collaborative work, where students analyze different systems, compare results, and engage in critical discussion regarding errors, unit consistency, and model fitting.
Accessibility: The activity is algebra-based and accessible to students from eighth grade onward. It can be adapted for Project-Based Learning (PBL) and does not strictly require internet access if offline tools are used, though online data is central to the primary method.
Error Analysis: The approach naturally incorporates the concept of scientific uncertainty. Deviations in fitted exponents or the final value of $G$ serve as opportunities for students to discuss measurement errors, data quality, and the limitations of assuming circular orbits for highly eccentric bodies.

The paper concludes that this rotation-curve approach provides a rigorous yet accessible entry point into gravitational analysis, enabling students to independently arrive at the Law of Universal Gravitation without relying solely on textbook assertions or complex historical experiments.