Page 22 - GALIET PHYSICS BLOSSOMS II+

Because P1 and P2 begin to feel each other’s gravitational pull more strongly as they approach PI (the point of intersection), they will accelerate slightly until PI. As P1 and P2 begin to pull apart past PI, they will decelerate slightly progressively past PI, at which point each will reassume their regular path and speed.
P1 crosses PI (the point of intersection) first while P2 continues to accelerate until it crosses PI, at which time it will slow down slightly due to the gravitational attraction of P2.
According to Kepler’s First Law, each planet moves around the sun in an elliptical (not circular) orbit, with the Sun at one focus. In our scenario, however, P1 moves around a roughly circular orbit, while P2 moves around a highly elliptical orbit around the same parent star.
However, a planet does not orbit the exact sun’s center by Newton’s Laws. Instead, the planet and the Sun orbit the mass’ common focus. For example, if we imagine a seesaw, the center of mass of two equal-mass bodies is midway between them. When the mass of one body increases, the center of mass moves towards it:
Figure 2.12
Because equal and gravitational forces act upon the Sun and the planet (Newton’s Third Law), the Sun must also move (Newton’s First Law) according to the planet’s gravitational pull. The Sun’s mass, however, is greater than any planet, hence, the planet-sun’s center of mass has to be very close to the Sun’s center just as the center of mass is nearer the two heavier masses in the drawing above, corroborating the accuracy of Kepler’s laws. In this way, Kepler’s first law becomes:
When planets orbit around the Sun, they do so in an ellipse, with the center of mass of the planet-Sun system at one focus.13
According to Kepler’s Second Law, the radius vector of the orbit (connecting the sun to the planet) sweeps out equal areas of the ellipse in equal times. However, this is not necessarily the case given that in P1 and P2 there is a slight gravitational wobble or going off the path slightly which suggests that the swept areas are not equal, but slightly smaller at PI. Of course, this is so infinitesimal small that it would not be noticeable to the naked eye!
According to Newton’s Second Law, when a force F acts on a body of mass m, it produces in it an acceleration a equal to the force divided by the mass. Thus, a = F/m, or F = ma. Consequently, the greater the force that acts on the object, or the smaller the mass of the object, the greater is its acceleration. If we pull two objects with the same force, the larger one will accelerate less; if we pull two identical objects with different forces, the one
12 Drawing and summary explanation of Kepler’s Second Law from class notes and from slides and from Chaisson, McMillan. Astronomy Today. 51-53.
13 Chaisson, McMillan. Astronomy Today. 51-53.
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