In-Class Exercise Solutions: Orbits and Motions

Astronomy 23/223 In-Class Exercise:

Solution: Orbits and Motions

1. Individually, make a list of all the controls in an automobile which can induce acceleration. Then form groups of 3 and discuss your individual lists.

Accelerator — obvious intent of the control
Brakes — decreasing speed is a negative acceleration
Steering wheel — turning is “directional” acceleration

2.

How could you determine the mass of Jupiter by making observations? Students who already took Astro 2/25: How could you determine the mass of Mercury?

Manipulation of Kepler’s laws, along with universal gravitation and centripetal acceleration show us that the period and semi-major axis of an orbit are related via the Gravitational constant and the mass of the central body. Since Jupiter has satellites, we only must observe the distance and period of the orbits (preferably for more than one object), and follow this procedure:
When we combine the concepts of Universal Gravitation and circular motion, we came up with the expression F_gravity = F_centripetal
F_gravity = G m₁ m₂ / r²
F_centripetal = m₂ V² /r
Let Jupiter be m₁ and the satellite be m₂. For circular motion the distance r is the semi-major axis a. The orbital velocity of the satellite can be described as distance/time, or circumference of the circular orbit divided by the orbital period:
V = 2 pi a /P

so setting the forces equal yields
G m₁ m₂ / a² = m₂ V² /a
note that the m₂ will cancel, so that circular orbital motion is independent of the mass of the orbiting body!
G m₁ / a² = ((2 pi a)²/P²)/a
which we rearrange to place all the a-terms on the right and all the P-terms on the left:
P² G m₁/(4 pi²) = a³
which should look startlingly like Kepler’s third law, but this time for Jupiter’s mass instead of the sun’s mass. To use a and P to solve for mass, manipulate once more so that
m₁ = m_Jup = a³ (4 pi²/G) / P²
and if you have observed P and a directly (given a decent amount of time), you can get the mass.

3.
If the Earth suddenly became half its current mass, how would this change the orbit about the Sun? What would happen to the orbit of the Moon? Discuss these questions in groups of 2 or 3, and at least one group will be asked to share their results.
As found above, the orbit of a body only depends on the mass of the central object (the larger mass), and the mass of the orbiting body is irrelevant, so halving the mass of the Earth would have NO effect whatsoever on its orbit about the sun.
Now, as for the Moon, that is more interesting. As above, there is a relationship between the mass times the period squared and the distance cubed. So, at its current distance, with a half-massive Earth, the moon’s orbital period would be sqrt(2) times its current value. If you were able somehow to keep the orbital period the same, then the distance between the Earth and the moon would have to decrease by a factor of 0.5^1.5.
Now, obviously we cannot simply change the mass of the Earth, but if we did, life would not be so simple, because although the moon would suddenly “feel” less attraction from the Earth, it would suddenly find itself in a situation with way too much momentum (it is moving along with its “normal” circular velocity), and many shenanigans would likely take place before things settled down.

4.
Can you find a place in the solar system where an object could remain at rest (as seen by a non-moving observer outside the solar system)? If so, where? If not, why not?
If you could place yourself at the true center-of-mass of the entire solar system, yes. You might suggest the center of the sun, but it is offset slightly by the pull of the planets. You might suggest some “stable” point between the Earth and the Sun, or between Jupiter and the Sun, where the pull of gravity is equal and opposite, but an outside observer would see even that “stable” point moving in orbit around the sun.

1.	Individually, make a list of all the controls in an automobile which can induce acceleration. Then form groups of 3 and discuss your individual lists. Accelerator — obvious intent of the control Brakes — decreasing speed is a negative acceleration Steering wheel — turning is “directional” acceleration

2.	How could you determine the mass of Jupiter by making observations? Students who already took Astro 2/25: How could you determine the mass of Mercury? Manipulation of Kepler’s laws, along with universal gravitation and centripetal acceleration show us that the period and semi-major axis of an orbit are related via the Gravitational constant and the mass of the central body. Since Jupiter has satellites, we only must observe the distance and period of the orbits (preferably for more than one object), and follow this procedure: When we combine the concepts of Universal Gravitation and circular motion, we came up with the expression F_gravity = F_centripetal F_gravity = G m₁ m₂ / r² F_centripetal = m₂ V² /r Let Jupiter be m₁ and the satellite be m₂. For circular motion the distance r is the semi-major axis a. The orbital velocity of the satellite can be described as distance/time, or circumference of the circular orbit divided by the orbital period: V = 2 pi a /P so setting the forces equal yields G m₁ m₂ / a² = m₂ V² /a note that the m₂ will cancel, so that circular orbital motion is independent of the mass of the orbiting body! G m₁ / a² = ((2 pi a)²/P²)/a which we rearrange to place all the a-terms on the right and all the P-terms on the left: P² G m₁/(4 pi²) = a³ which should look startlingly like Kepler’s third law, but this time for Jupiter’s mass instead of the sun’s mass. To use a and P to solve for mass, manipulate once more so that m₁ = m_Jup = a³ (4 pi²/G) / P² and if you have observed P and a directly (given a decent amount of time), you can get the mass.

3.	If the Earth suddenly became half its current mass, how would this change the orbit about the Sun? What would happen to the orbit of the Moon? Discuss these questions in groups of 2 or 3, and at least one group will be asked to share their results. As found above, the orbit of a body only depends on the mass of the central object (the larger mass), and the mass of the orbiting body is irrelevant, so halving the mass of the Earth would have NO effect whatsoever on its orbit about the sun. Now, as for the Moon, that is more interesting. As above, there is a relationship between the mass times the period squared and the distance cubed. So, at its current distance, with a half-massive Earth, the moon’s orbital period would be sqrt(2) times its current value. If you were able somehow to keep the orbital period the same, then the distance between the Earth and the moon would have to decrease by a factor of 0.5^1.5. Now, obviously we cannot simply change the mass of the Earth, but if we did, life would not be so simple, because although the moon would suddenly “feel” less attraction from the Earth, it would suddenly find itself in a situation with way too much momentum (it is moving along with its “normal” circular velocity), and many shenanigans would likely take place before things settled down.

4.	Can you find a place in the solar system where an object could remain at rest (as seen by a non-moving observer outside the solar system)? If so, where? If not, why not? If you could place yourself at the true center-of-mass of the entire solar system, yes. You might suggest the center of the sun, but it is offset slightly by the pull of the planets. You might suggest some “stable” point between the Earth and the Sun, or between Jupiter and the Sun, where the pull of gravity is equal and opposite, but an outside observer would see even that “stable” point moving in orbit around the sun.