Presenting Gravity Law proportion as Newton always used it. (no G)

 Here's a possible outline for such a class activity:

1. Review Newton's Law (Proportionally): Start by stating the law of gravitation as a proportionality: F ~ m1m2/r². Emphasize the dimensionless nature of the proportion.

2. Centripetal Force (Proportionally): Introduce centripetal force, also as a proportionality: F_c ~ mv²/r.

3. Stable Orbits: Explain that stable orbits represent a balance between these proportions: m1m2/r² ~ mv²/r.

4. Kepler's Third Law (Proportionally): Derive the proportional form of Kepler's Third Law: M ~ r³/T².

5. Introduce "Luna Mass": Define "1 luna mass" as the unit of mass. Explain that we'll be expressing all masses relative to the Moon.

6. Earth's Mass in Luna Masses: Provide the Earth-Moon mass ratio (approximately 81.3) and show how to express Earth's mass in luna masses.

7. Calculate the Sun's Mass (in Luna Masses): Guide students through the calculation of the Sun's mass in luna masses, using the proportional relationship derived earlier and the known orbital parameters of the Earth and Moon.

8. Challenge Problems: Present additional problems, such as:

   • Calculating the mass of Jupiter (in luna masses) given its orbital period and distance from the Sun.

   • Calculating the mass of a hypothetical planet orbiting a different star, given its orbital parameters and the star's mass (expressed relative to the Sun).

   • Discussing how uncertainties in the orbital measurements would affect the calculated masses.

9. Connect to G (Later): After students have worked through these exercises, you can then introduce G as a scaling factor that allows us to work with standard units (kg, m, s, N) and calculate absolute forces. This way, G is presented as a practical tool, not a fundamental mystery.

This activity would be a fantastic way to deepen students' understanding of gravity, empower them with proportional reasoning skills, and demystify the gravitational constant. It would be a valuable addition to any introductory physics or astronomy curriculum.