3 or More Body Interactions

This page introduces the concept of n-body interactions (with 3 or more bodies). This is a basic overview of the concept. You do not have to know this material in detail.

Main Idea

Problems involving two bodies that are interacting gravitationally are relatively easy to solve. Once there are more than two bodies, however, this is no longer the case. Generally, three or more body problems require numerical integrations to solve, and are quite complex. It becomes difficult to predict the motion of the bodies under the influence of multiple other gravitational forces, and the system most often becomes chaotic. There are, however, a number of cases where motion is not chaotic and that thus can be studied.

Restricted 3-Body Problem

In the restricted three body problem, we assume that the third body has a negligible mass, and that it moves under the influence of two other massive bodies. This simplifies calculations, as we can treat the two massive bodies as though they are in a simple two body problem to predict their motion. We assume that these two bodies orbit around their mutual center of mass, and that the third body, being of negligible mass, does not affect this motion.

Through this, we can also assume for calculations involving only the third body that the two bodies are in fact one point mass that is located at their mutual center of mass. This greatly simplifies calculations involving, say, a planet rotating around a stellar binary and like problems.

The restricted 3-body problem is useful for analyzing motion for many objects in the solar system, chiefly the Earth-Moon-Sun system, and other such systems involving moons. Because the moon is much less massive than the Earth, which is in turn much less massive than the Sun, we can treat that problem as a restricted 3-body problem.

An example of a stable solution to the 3-body problem.

Other Solutions

There are also a number of stable orbits associated with three-body problems. One such example is a stable figure eight orbit, as depicted here.

Examples

This is a simple example that corresponds to the restricted three body problem. More complex examples require computer simulations and integrators in order to solve.

The moon has a mass of [math]\displaystyle{ m_{moon} }[/math] kg. If it is located a distance [math]\displaystyle{ d_1 }[/math] m from the Earth (mass [math]\displaystyle{ m_E }[/math] kg) and a distance [math]\displaystyle{ d_2 }[/math] m from the Sun (mass [math]\displaystyle{ m_{Sun} }[/math] kg). What are the magnitudes of the forces of the Sun on the Moon and the Earth on the Moon?

Recall that the following equation applies to all bodies in this problem:

• [math]\displaystyle{ F_g=\frac{GM_1M_2}{r^2} }[/math]

Here, we can replace the relevant distances and masses into the equation relatively easily to come up with the following force magnitudes:

• [math]\displaystyle{ F_{Sun,Moon}=\frac{Gm_{moon}m_{Sun}}{d_2^2} }[/math] for the Sun on the Moon
• [math]\displaystyle{ F_{Earth,Moon} = \frac{Gm_{moon}m_E}{d_1^2} }[/math] for the Earth on the Moon

For a fun extra reference, the Sun's mass is about 300,000 times greater than Earth's mass, but the Sun-Moon distance is about 389 times greater than the Earth-Moon distance. Via a simple fraction, as shown below, the magnitude of the force that the Earth exerts on the Moon is thus much lesser than the magnitude of the force the Sun exerts on the Moon, despite the fact that the Moon is orbiting around the Earth.

• [math]\displaystyle{ \frac{F_{Earth}}{F_{Sun}}=\frac{m_E}{300000m_E}\frac{(389d_E)^2}{d_E^2}\approx{}\frac{150000}{300000}=0.5 }[/math]

Connectedness

My research involves an n-body integrator. Using these types of simulations to predict the motion of the stars and other celestial bodies is important in understanding how systems came to form as they are today. The idea that we can use integrators to predict the motion, or to run backwards in time to see where bodies used to be is extremely interesting. Our ability to create mathematical models of space is incredibly useful in furthering our understanding of space.

I am a physics major and, as such, 3-body problems come up quite a lot in my classes. Whether there be a problem involving a circum-stellar binary (a planet rotating around a binary system, where the orbit of the planet goes around both stars), or if there is just a problem involving our own Solar System, 3 (or more) body problems exist everywhere in nature.

History

The subject of three body gravitational interactions was first brought up by Isaac Newton in 1687 when Principia was published. Prior to this, attempts to predict the postion of the Moon in the Earth-Moon-Sun system were rather important for navigational purposes. Vespucci and Galileo had both, respectively, attempted to address this problem physically. This was difficult and kind of imprecise, however, due to the influence of the other planets and the Sun on the Moon's position.

D'Alembert and Clairaut would both later, in 1747, attempt to refine and further address the issue of multi-body systems. Both men would present competing analyses to the French Academy of Sciences. It was at this time that the phrase 'three body problem' would first be applied to discuss the issue.

Later attempts at definitively solving the n-body problem would lead to many different attempts at solutions and the beginning the foundations of chaos theory.