A small body follows an elliptical orbit with the central star at one focus. Initially, its farthest point from the star is 2 times as far from the star as its closest point.
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最遠点での速さは、最接近点での速さの何倍か？
半径1.0と2.0（×10¹¹ m）の円軌道。外側の速さは内側の何倍か？
The equations explain the phenomenon
1We write the gravitational force.F=G M mr²……
2We derive the rate of change of angular momentum from the equation of motion.d(𝐫 × m𝐯)dt=𝐫 × 𝐅……
3We derive conservation of angular momentum for a force directed toward the center.𝐫 × m𝐯=Constant……
4We find the ratio of the speeds at the closest and farthest points.vfarvnear=rnearrfar……
We now consider circular orbits, moving from the closest and farthest points of an ellipse to an orbit with a constant radius.
5We derive the speed in a circular orbit.v=√(G Mr)……
6We derive the potential energy.U=−G M mr……
7We apply conservation of energy.½ m v²−G M mr=E……
Further study: Earth–Moon Motion and the Laws of Orbits
8We apply the result at Earth's surface.g=G MR²……
9We check whether the Moon is also falling.aMoon=g (Rr)²……
10We derive the relationship between period and radius.T²=(4π²G M) r³……
11We derive the escape velocity.v=√(2 G MR)……
12We solve numerically to examine the shapes of orbits.m d²𝐫dt²=−G M m 𝐫r³……
What this chapter has shown
As the body approaches the Sun, its potential energy decreases and its kinetic energy increases, so it speeds up. Angular momentum is also conserved. Since rv has the same value at the closest and farthest points, we can determine the ratio of the speeds at those points.
0.500 timesThe ratio of the initial speeds
For 2 circular orbits of different radii, doubling the radius reduces the speed to about 0.71 of its original value.
In everyday lifeEarth is closest to the Sun in early January, when it moves at about 30.3 km/s. At its farthest point in early July, its speed is about 29.3 km/s.
Detailed summary
Substituting the gravitational force GMm/r² into the equation of motion yields all 3 of Kepler's laws. Because the force acts toward the center, angular momentum and areal velocity remain constant, and the speeds at the nearest and farthest points of an elliptical orbit are inversely proportional to their distances from the Sun. The force balance for a circular orbit gives T² ∝ r³, and solving the equation of motion numerically produces an elliptical orbit with the Sun at one focus. Both the force mg at Earth's surface and the Moon's motion are manifestations of 1 universal force.