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The asteroid belt between Mars and Jupiter consists of many fragments (which some space
Chapter 5, Problem 64P(choose chapter or problem)
Problem 64P
The asteroid belt between Mars and Jupiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destroyed). (a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun? (b) Can we use these data to deduce the mass of this planet?
Questions & Answers
QUESTION:
Problem 64P
The asteroid belt between Mars and Jupiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destroyed). (a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun? (b) Can we use these data to deduce the mass of this planet?
ANSWER:
Solution 64P
Part (a)
Step 1 of 3:
Consider a hypothetical planet made the asteroid belt between the Mars and Jupiter. The center of mass of the asteroid belt is about three times greater than that of the Sun and the Earth. We are going to find the period of the hypothetical planet to orbit the Sun.
The center of mass of the planet aP = 3 aE
The time period of the Earth TE = 1 year
Where aE is the center of mass of the Earth from the Sun.
From the Kepler’s law of period,
∝
Where T is the period orbit the Sun