8. Centripetal Forces & Gravitation

The dwarf planet Pluto has an elliptical orbit with a semimajor axis of 5.91 × 10¹² m and eccentricity 0.249. During Pluto's orbit around the sun, what are its closest and farthest distances from the sun?

According to Kepler's Third Law, if a planet orbits a star in a year of length $T$, which of the following correctly relates the orbital period $T$ to the semi-major axis $a$ of its orbit (assuming the mass of the star is much greater than the planet's mass)?

According to Kepler's Third Law, what is the orbital period (in years) of a planet with a semimajor axis of $35au$ around the Sun?

According to Kepler's Third Law, which of the following planets (on average) takes the longest to orbit the Sun (has the longest orbital period)? $T^2=\frac{a^3}{k}$

According to Kepler's Third Law, what is the orbital period in Earth-years of a planet whose average distance from the Sun is $3$ astronomical units ($\mathrm{a.u.}$)?

Why is Newton's version of Kepler's Third Law so useful to astronomers?

According to Kepler's Third Law, what is the orbital period in Earth-years of a planet whose average distance from the Sun is $3$ astronomical units (AU)?

(III) The comet Hale–Bopp has an orbital period of 2400 years. What is the ratio of the speed at the closest point to the speed at the farthest point?

A satellite orbits the sun with a period of 1.0 day. What is the radius of its orbit?

According to Kepler's Third Law, which of the following correctly relates the square of the orbital period $P^2$ to the cube of the semi-major axis $a^3$ for a planet orbiting the Sun?

The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0 earth years. What are the asteroid's orbital radius and speed?

Large stars can explode as they finish burning their nuclear fuel, causing a supernova. The explosion blows away the outer layers of the star. According to Newton’s third law, the forces that push the outer layers away have reaction forces that are inwardly directed on the core of the star. These forces compress the core and can cause the core to undergo a gravitational collapse. The gravitational forces keep pulling all the matter together tighter and tighter, crushing atoms out of existence. Under these extreme conditions, a proton and an electron can be squeezed together to form a neutron. If the collapse is halted when the neutrons all come into contact with each other, the result is an object called a neutron star, an entire star consisting of solid nuclear matter. Many neutron stars rotate about their axis with a period of ≈ 1 s and, as they do so, send out a pulse of electromagnetic waves once a second. These stars were discovered in the 1960s and are called pulsars. How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface?

(II) Determine the mean distance from Jupiter for each of Jupiter’s principal moons, using Kepler’s third law. Use the mean distance of Io and the periods given in Table 6–3.

A new planet is discovered orbiting the star Vega in a circular orbit. The planet takes 55 earth years to complete one orbit around the star. Vega's mass is 2.1 times the sun's mass. What is the radius of the planet's orbit? Give your answer as a multiple of the radius of the earth's orbit.

(III) The comet Hale–Bopp has an orbital period of 2400 years. What is its mean distance from the Sun?