Why can Kepler’s third law be applied to estimate the Galaxy’s mass using the Sun’s orbit?

Question

BlackTom AI · Accepted Answer

Kepler’s third law relates orbital period P and semi‑major axis a by P^2 = (4π^2/GM) a^3 when a single central mass M dominates the gravity.

1) Relevant knowledge points and concepts
- Kepler’s third law in Newtonian form: P^2 = (4π^2/GM) a^3 for a small body orbiting a dominant central mass M.
- Newton’s law of gravitation and centripetal force for circular orbits: m v^2/r = G M(r) m/r^2.
- Newton’s shell theorem: for a spherically symmetric mass distribution, the gravitational effect on an external particle depends only on the mass inside its radius (as if concentrated at the center).
- Galaxies are not perfect point masses, but if the mass inside the Sun’s orbit is the dominant contributor and roughly symmetric, Kepler’s law can be used to estimate the enclosed mass.

2) Step‑by‑step reasoning
- For the Sun orbiting the Galactic center, treat the Sun as a test particle orbiting under gravity of mass Menc inside its radius.
- Use centripetal = gravitational force to connect orbital motion to enclosed mass.

3) Complete calculations and intermediate steps
- Centripetal balance: m v^2 / r = G Menc m / r^2.
- Cancel m and simplify: v^2 = G Menc / r.
- Solve for Menc: Menc = v^2 r / G.
- Express period P = 2π r / v. Then v = 2π r / P.
- Substitute into Menc: Menc = ( (2π r / P)^2 r ) / G = (4π^2 / G) r^3 / P^2.
- Rearranged: P^2 = (4π^2 / G Menc) r^3, the Kepler form.

4) Why the correct answer is correct
Kepler’s law applies because the Sun orbits outside most of the mass that contributes to its gravitational attraction in a way that can be approximated by the mass interior to its orbit acting as if concentrated at the center (shell theorem / effective central mass). This lets us use the Kepler/Newton relation to estimate the Galaxy’s enclosed mass.

5) Why the other options are incorrect
- “Galaxy rotates as a solid object”: false—galactic rotation is differential; different radii have different orbital velocities.
- “Sun does not move relative to the Galaxy”: false—the Sun orbits the Galactic center.
- “Sun is the most massive object in the Galaxy”: false—the Galaxy’s mass is dominated by stars, gas, and dark matter, not the Sun.

6) Conclusion
Therefore the correct reason is that the mass inside the Sun’s orbit can be treated as concentrated at the center, which matches the provided answer.

Why can Kepler’s third law be applied to estimate the Galaxy’s mass using the Sun’s orbit?Single choice

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Why can Kepler’s third law be applied to estimate the Galaxy’s mass using the Sun’s orbit?Single choice

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