Section 9–7 Planetary motions

(Motion of a planet / Motion of Neptune, Jupiter, and Uranus / Computation time)

In this section, the three interesting points discussed are the motion of a planet, motions of Neptune, Jupiter, and Uranus, as well as the computation time for a numerical analysis.

1. The motion of a planet:

“The above analysis is very nice for the motion of an oscillating spring, but can we analyze the motion of a planet around the sun? (Feynman et al., 1963, section 9–7 Planetary motions).”

To be more precise, the Sun, Earth, and all planets in our solar system rotate around the barycenter (center of mass of all astronomical objects in the solar system). To achieve an approximation to the elliptical motion of a planet around the Sun, Feynman assumes that the Sun is infinitely heavy. Furthermore, we ignore how the Sun rotates about the center of our galaxy. Importantly, the Sun would move because of Newton’s third law of motion and gravitational forces on the Sun due to the planets. Essentially, we compute the motion of a planet orbiting around the Sun in an elliptical curve by using Newton’s second law of motion and Newton’s law of gravitation. In addition, we need to determine the “initial” position of a planet and how it is moving with a certain velocity.

To analyze the motion of a planet further, we have to calculate the components of the planet’s acceleration along two directions x and y. Feynman says that we shall suppose z is always zero because there is no force in the z-direction and there is no initial velocity v_z. Alternatively, one may assume that all planets in our solar system lie on the same plane. Physicists may explain that most of the mass of a solar system are on the same plane except for Pluto due to its unusual orbit. More importantly, Leapfrog method is used again in the simulation of a planet’s orbit as Feynman calculates the velocities v_x(0.05) and v_y(0.05) instead of the velocities v_x(0.00) and v_y(0.00).

2. The motion of Neptune, Jupiter, or Uranus:

“…let us see how we can calculate the motion of Neptune, Jupiter, Uranus, or any other planet (Feynman et al., 1963, section 9–7 Planetary motions).”

According to Feynman, we can calculate the force on planet number i, by using equations such as m_idv_ix/dt = − Gm_im_j(x_i−x_j)/r_ij³. Curiously, he specifies i = 1 to represent the Sun, i = 2, Mercury, i = 3 Venus, and so on. This is potentially misleading because the Sun is not a planet. Thus, he could have used the term “astronomical body i” instead of “planet number i.” Similarly, we can still use Leapfrog (midpoint) method to determine the motions of the n astronomical bodies including the Sun and planets. The positions of these astronomical bodies can be downloaded from NASA websites.

Unlike the previous two-body problem, this example has nine bodies (it seems that Pluto is excluded). Note that it is not even easy to solve a three-body problem. In general, a complete solution for the three-body problem would have the positions and velocities of the bodies for a period of time, provided three initial positions and initial velocities were determined. The motion of the three bodies is generally non-repeating and possibly “chaotic,” except in special cases. Interestingly, Feynman mentions in the next chapter that “the famous three-body problem, which so long challenged human powers of analysis.” The three-body problem is a special case of the n-body problem that is even more complicated.

3. Computation time:

“…we would need 4 × 10⁵ cycles to correspond to one revolution of a planet around the sun. That corresponds to a computation time of 130 seconds (Feynman et al., 1963, section 9–7 Planetary motions).”

Feynman explains that a very good computing machine may take a millionth of a second to do an addition. Furthermore, he estimates that it takes only two minutes to compute the motion of Jupiter around the Sun while including the perturbations of all the planets correct to one part in a billion. Currently, a computer can do multiple computations at the same time, and so the time needed for an addition is not a useful indicator of how fast the motions of planets can be computed. Specifically, the physical size of a processor, software program, and the speed of light are factors that determine the computation time needed. It should be worth mentioning that Feynman was involved in the “Connection Machine” project that connects 64000 processors for numerical computing (Hillis, 1989).

Feynman suggests that we can calculate motions of planets by using a computing machine to handle the arithmetic. He also claims that it is possible to achieve as high a degree of precision as we wish even for the tremendously complex motions of the planets. On the contrary, one may argue that there are dark matter particles (or unknown matters) in our solar system. Some physicists propose that dark matter have gravitational effects on motions of planets orbits despite difficulties in detecting the hypothetical matter (Gibney, 2017). However, the mass of astronomical bodies may not be sufficiently accurate and there could be a correction factor that accounts for “unknown matters” in our solar system in the first place.

Questions for discussion:

1. How would you compute the motion of a planet by using the Leapfrog method?

2. How would you compute motions of eight planets by downloading the initial positions of all planets?

3. What is the minimum computation time needed to simulate motions of planets in our solar system?

The moral of the lesson: we can compute complex motions of the planets to a relatively high degree of precision by using Leapfrog (midpoint) method.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Gibney, E. (2017). Dark-matter hunt fails to find the elusive particles. Nature News, 551(7679), 153.

3. Hillis, W. D. (1989). Richard Feynman and the Connection Machine. Physics Today, 42(2), 78-83.