8. Centripetal Forces & Gravitation

Io and Ganymede are two of Jupiter's four Galilean moons. Io orbits at an average distance of 422,000km in 1.77 days. What is Ganymede's average orbital distance (in km), if it takes 4 times longer to orbit Jupiter?

(I) The asteroid Icarus, though only a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from the Sun?

(III) The comet Hale–Bopp has an orbital period of 2400 years.

(a) What is its mean distance from the Sun?

(I) Neptune is an average distance of 4.5 x 10⁹ km from the Sun. Estimate the length of the Neptunian year using the fact that the Earth is 1.50 x 10⁸ km from the Sun on average.

(II) A science fiction writer imagines a planet that has three times the mass of the Earth orbiting a star that has twice the mass of our Sun. The writer wants the planet’s “year” to be 10% longer than Earth’s orbital period. At what mean distance should she place the planet from its star?

(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.

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(III) The comet Hale–Bopp has an orbital period of 2400 years.

(c) What is the ratio of the speed at the closest point to the speed at the farthest point?

(III) The comet Hale–Bopp has an orbital period of 2400 years.

(b) At its closest approach, the comet is about 1.0 astronomical unit from the Sun ( 1 AU = distance from Earth to the Sun). What is the farthest distance?

(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. Compare your results to the values in Table 6–3. <IMAGE>

The Near Earth Asteroid Rendezvous (NEAR) spacecraft, after traveling 2.1 billion km, orbited the asteroid Eros with an orbital radius of about 20 km. Eros is roughly 40km x 6km x 6km. Assume Eros has a density (mass/volume) of about 2.3 x 10³ kg/m³ .

(c) Estimate the orbital period of NEAR around Eros, as if Eros were a sphere.

In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we’ll approximate it as a 50-km-diameter circular orbit with a period of 11 days. (b) What is the mass of the comet?

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. (d) How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface?