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Illustration 12.2: Orbits and Planetary Mass
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When we consider the elliptical orbits of the planets (Kepler's first law), we assume that the Sun is stationary at one focus of the ellipse. Why does this happen? The mass of the Sun must be much, much greater than the mass of the planets in order for the motion of the Sun to be ignored. But how large does the mass of the Sun need to be in order to achieve this idealized planetary motion? For reference, the Sun is about 1000 times heavier than Jupiter (the most massive planet) and about 100 million (108) times more massive than the least massive planet, Pluto. Restart.
As you vary the mass ratio in the animation, the mass of the system changes such that the product of the masses, m1*m2, remains the same. Therefore as you change the mass ratio, the force will remain the same for the same separation between the masses.
The 1000:1 Mass animation closely resembles the Sun and Jupiter system (the distance is given in astronomical units [A.U.] and the time is given in 108 seconds). The green circle is like the Sun, while the red circle is like Jupiter. The force of attraction due to gravity is shown by the blue arrows (not shown to scale), and the relative kinetic energies are shown as a function of time on the graph (note that for this animation the unit for the kinetic energy is not given since we are comparing the relative amount for each object). Also note that the eccentricity of the orbit e = 0.048, the perihelion and aphelion distances, and the planet's period closely match those of Jupiter.
In the 100:1 Mass animation does the "Sun" remain motionless? What about the 10:1 Mass animation? The 2:1 Mass animation? The 1:1 Mass animation? What do you think this means for planetary dynamics in our solar system?
For elliptical orbits, the force due to gravity changes magnitude since the separation changes. But at every instant, the forces of gravitational attraction (the force of the green circle due to the red circle and the force of the red circle due to the green circle) are always the same. This is Newton's third law. It is not too surprising that the law of universal gravitation (described by Newton) contains the third law (also described by Newton).
At the same time, what happens to the kinetic energy of the system as a function of time? It too changes. But why? As the separation between the "Sun" and the "planet" changes, the gravitational potential energy of the system changes too. While the kinetic energy of the system changes, the sum of the kinetic energy and the potential energy of the system must—and does—remain a constant throughout the motion of the objects.