

Astronomy 23/223 InClass Exercise:
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 
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 semimajor 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 m_{1} m_{2} / r^{2} Let Jupiter be m_{1} and the satellite be m_{2}. For circular motion the distance r is the semimajor 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 m_{1} m_{2} / a^{2} = m_{2} V^{2} /a note that the m_{2} will cancel, so that circular orbital motion is independent of the mass of the orbiting body! G m_{1} / a^{2} = ((2 pi a)^{2}/P^{2})/a which we rearrange to place all the aterms on the right and all the Pterms on the left: P^{2} G m_{1}/(4 pi^{2}) = a^{3} 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 m_{1} = m_{Jup} = a^{3} (4 pi^{2}/G) / P^{2} 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 halfmassive 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 nonmoving observer outside the solar system)? If so, where? If not, why not? If you could place yourself at the true centerofmass 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. 
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