At its core, Newton’s law of gravity is simply a mutual attraction of masses. But when more than two bodies attract each other their motion can become wondrously complex. So complex that even the motion of three masses is often unpredictable. But there are also some amazing effects of gravitational dynamics, such as the ability to orbit a point where there is no mass.

The Lagrange points of two masses orbiting each other.

Joseph-Louis Lagrange first studied the complexities of gravitational motion in the late 1700s. He had no way to calculate the motion of a particular object, but instead focused on the gravitational potential of large bodies in motion, such as the Sun-Earth or Earth-Moon systems. He found that the gravitational pull of objects and their motions about each other could create balance points, now known as Lagrange points. These balance points move with the motion of the masses, but relative to each mass they don’t appear to move. So if you could put an object exactly at a Lagrange point, the gravitational pull of the other masses and their orbital motion would make it appear as if the object were stationary.

For three of these points, any slight deviation from their exact position would cause a small body to fly off in a strange orbit. But two of them (known as L4 and L5) are stable balance points. This means if a body deviates slightly from the position the effects of gravity and motion will work to bring it back towards the point. It’s similar to the effect of a mass hanging on a string. If you give it a little push it will wobble around its original position.

The L4 and L5 points for the Sun-Jupiter system have captured asteroids over time, and are now known as the Trojans. Another group of asteroids known as the Hilda family tend to cluster near Jupiter’s L3 point. The Sun-Earth L4 and L5 points contain some interplanetary dust. The L4 point also contains a small asteroid, which is Earth’s only known trojan. Mars has four Lagrange asteroids. Lagrange points have also become important for astronomical missions. Several satellites such as Planck and Gaia are located near the Sun-Earth L2 point. The SOHO satellite orbits near Sun-Earth L1. Satellites at L3 could provide a way to communicate continuously with missions to Mars and the outer solar system.

It’s all a part of Lagrange’s gravitational dance.

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In my Intermediate Mechanics class, we’ve been talking about gravitational potential.  Gravitational potential is a nice mathematical way to describe the effects of gravity on an object.

You can get an idea of how gravitational potential is related to gravity by imaging a ball on field of rolling hills.  The potential at a given point is the height of the ground.  The gravitational force can be determined by calculating how the ground varies.  If ground is perfectly flat in some region, then a ball placed there would remain at rest, so there is no gravitational force.  If the ground decreases in height as you travel eastward, then a ball placed in that region would roll eastward, thus there is a gravitational force in the eastward direction.

Gravitational potential can also be useful in visualizing gravity.  In the figure below, I’ve plotted a contour plot of the gravitational potential around two orbiting bodies. This contour plot is similar to a topological map where each dotted line represents a uniform potential.  I’ve also plotted dots where the potential is flat. These are points where the gravitational and rotational forces just cancel out, so that the effective force is zero.

You might imagine that there would be just one such point.  It would lie between the two masses right where the pull from one mass is balanced by the pull from the other mass.  While that is one of the points, it turns out that there are 4 others. Together the 5 points are known as Lagrange points.  Two of the points (the ones in the red region on either side of the masses) are actually quite stable, and they can even capture small masses.  The trojan asteroids that lead and trail Jupiter are an example of such captured objects.

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