Question

41 review your understanding
key question: whats the relationship between the sun, the planets that orbit it, and the time they take to complete an orbit?
at the beginning of this chapter, you were asked to discuss your thoughts on various relationships between the planets and the sun. you should now be able to provide a more precise discussion on those relationships

1. (a) what is the relationship between a planets orbital period and its distance from the sun?

(b) why does this relationship occur?

1. the earth orbits the sun at 1au. what might happen to the earths orbit if the sun:

(a) suddenly doubled in mass:
(b) suddenly halved in mass:

1. comets have very eccentric orbits. at which points in their orbits are they moving the fastest and the slowest?

Explanation:

Step1: Recall Kepler's third law

The square of a planet's orbital period ($T$) is proportional to the cube of its average distance ($r$) from the Sun, $T^{2}\propto r^{3}$.

Step2: Explain the reason for the relationship

Gravitational force provides centripetal - force. According to Newton's law of universal gravitation and centripetal - force formula, $F = G\frac{Mm}{r^{2}}=m\frac{v^{2}}{r}$, and $v=\frac{2\pi r}{T}$. Combining these equations gives the relationship.

Step3: Analyze the effect of Sun's mass change on Earth's orbit (a)

If the Sun's mass ($M$) doubles, from $F = G\frac{Mm}{r^{2}}=m\frac{v^{2}}{r}$, the gravitational force increases. To maintain the balance, the Earth would move to a smaller orbit or increase its orbital speed. In a stable - state, the Earth's orbit would become smaller.

Step4: Analyze the effect of Sun's mass change on Earth's orbit (b)

If the Sun's mass halves, the gravitational force decreases. The Earth would move to a larger orbit or decrease its orbital speed. In a stable - state, the Earth's orbit would become larger.

Step5: Recall Kepler's second law for comets

Kepler's second law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. Comets move fastest at perihelion (closest to the Sun) and slowest at aphelion (farthest from the Sun).

Answer:

1. (a) The square of a planet's orbital period is proportional to the cube of its average distance from the Sun ($T^{2}\propto r^{3}$).

(b) This relationship occurs because gravitational force provides centripetal force. By combining Newton's law of universal gravitation and centripetal - force equations, we can derive this relationship.

1. (a) The Earth's orbit would become smaller.

(b) The Earth's orbit would become larger.

1. Comets move fastest at perihelion and slowest at aphelion.