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?单项选择题

类似问题

It takes the planet Jupiter 12 years to orbit the Sun once. What is the average distance from Jupiter to the Sun? (The distance from the Earth to the Sun is 1.5 × 1011 m.)

It takes the planet Jupiter 12 years to orbit the Sun once. What is the average distance from Jupiter to the Sun? (The distance from the Earth to the Sun is 1.5 × 1011 m.)

According to Kepler's 2nd law, during what month is Earth moving the fastest in its orbit?

Kepler's second law states that the line joining the Sun to a planet

Kepler's first law states that the orbits of the planets are

A comet moves in a highly elliptical orbit. Where does it spend most of its time?

Kepler's First Law of Planetary Motion states that a planet will orbit on a(n) _ with the Sun located at _.

Europa, a moon of Jupiter, has an orbital diameter of 1.34 x 109 m, and a period of 3.55 days. What is the mass of Jupiter?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单

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

类似问题

It takes the planet Jupiter 12 years to orbit the Sun once. What is the average distance from Jupiter to the Sun? (The distance from the Earth to the Sun is 1.5 × 1011 m.)

It takes the planet Jupiter 12 years to orbit the Sun once. What is the average distance from Jupiter to the Sun? (The distance from the Earth to the Sun is 1.5 × 1011 m.)

According to Kepler's 2nd law, during what month is Earth moving the fastest in its orbit?

Kepler's second law states that the line joining the Sun to a planet

Kepler's first law states that the orbits of the planets are

A comet moves in a highly elliptical orbit. Where does it spend most of its time?

Kepler's First Law of Planetary Motion states that a planet will orbit on a(n) _____ with the Sun located at _____.

Europa, a moon of Jupiter, has an orbital diameter of 1.34 x 109 m, and a period of 3.55 days. What is the mass of Jupiter?

更多留学生实用工具

希望你的学习变得更简单

Kepler's First Law of Planetary Motion states that a planet will orbit on a(n) _ with the Sun located at _.