Lagrange Points and You Robert Bond November 6, 2013
No one can make the argument that mankind has been merely a passive spectator on our now nearly 200,000 year old orbital journey around our sun and through our galaxy. As long as humans have been able to document their feelings about what is truly out there they have done so, not just simply in the form of writing down their thoughts and ideas, but also in the form of ancient cave paintings and Neolithic monuments of what could be observed in the sky, and more recently, mathematical theories of how it all works. In this paper we will be discussing one of these more proactive human beings, more specifically, one of his contributions to mankinds endless thirst for knowledge of the universe. His name was Joseph-Louis Lagrange and his theories and work on what is more commonly referred to as The Three Body Problem laid the foundation for the concept of what we now call, Lagrange Points. Joseph-Louis Lagrange was born on January 25, 1736, in Turin, Italy to a then wealthy father Guiseppe Francesco Lagrange who held office as a treasurer of the Sardinian military, however, Guiseppe would go on to lose a substantial amount of his wealth and social status due to bad property speculations. Joseph-Louise s mother, Maria Theresa Gros, was the only daughter of a wealthy and successful physician in Combiano, Italy, located thirteen kilometers from Turin. Joseph Lagrange would go on to cite the losses of his father as a positive influence on his success as a mathematician alluding to the idea that if he had been raised in an excessively wealthy household he would have never been so inclined to study mathematics so rigorously. Lagrange was thrusted onto the international mathematics stage at a young age of nineteen when he wrote a letter in 1755 to another famous mathematician 1
of the time and still famous to this day, Leonhard Euler. In the letter, Lagrange discussed with Euler what he called Calculus of Variations that Lagrange used to solve a problem that Euler had been working on for an extended period of time. Although Euler was on the precipice of arriving to a solution of the problem himself at the time, after receiving the letter from Lagrange, Euler decided that he would not pursue to publish his work on the isoperimetrical problem and would instead let Lagrange publish his own works and findings. Euler was so impressed that not only did he let Lagrange take credit for the findings but upon Eulers retirement as director of the Berlin Academy of Science, he recommended Lagrange for the job. Seventeen years later, Lagrange would go on to publish Essai Sur le Probleme des Trois Corps or Essay On the Three Body Problem. In this essay, Lagrange explains his theory of what we now call Lagrange Points, which are points in space that when you have the famous Three Body Problem the third body is being acted upon not only by the first body but the second body as well, which will cause the third body to revolve around the first body at the same rate that the second body revolves around the first. Lagrange s theory was that not only do the second and third body revolve around the first at the same rate, but that they will stay approximately the same distance away from each other. More specifically, if you have a three body system of the Sun, the Earth, and a Satellite, a Lagrange Point will be a point in space that if you were to place the Satellite at that point, the Satellite will orbit the Sun a relative distance away from the Earth at the same rate that the Earth is orbiting the Sun. Essentially this will cause the Satellite to remain the same distance away from the 2
Earth. Lagrange theorized that in a three body system there would be five points in space where Lagrange Points would exist. Lagrange was right. The force on a body at a Lagrange point can be found by solving the following equation, F = GM 1m r r 1 3 ( r r 1 ) GM 2m r r 2 3 ( r r 2 ) where r 1 and r 2 are functions of time and m 1 and m 2 are orbiting each other. then, by solving the equation, F (t) = m d2 r 1 (t) dt 2 you will acquire the relative positions of where the third body will remain a constant distance away from the first and second bodies. In a general description, if you were looking down an orthogonal plane at a two dimensional model of the Earth, Sun, Satellite, system, you would be able to acquire five Lagrange Points. One point in our two dimensional model will lie along a straight line from the Sun to the Earth, passed the Earth some distance. At this point, the gravity of the Sun pulls on the Satellite but so does the gravity of the Earth keeping the Satellite the same distance away from the Earth relatively constantly, this point is approximately 1.5 million kilometers from Earth and over 150 million kilometers from the Sun. Again, observing our three body system from an orthogonal plane looking down at a two dimensional model, another Lagrange point will lie again on a straight line from the Sun to the Earth but this time within the orbit of the Earth around the Sun instead of outside 3
of it. At this point, the gravity of the Sun will act on the Satellite pulling it towards the Sun however the Satellite will be so much closer to the Earth relative to its distance from the Sun that the gravity of the Earth will be pulling the Satellite in the opposite direction of the Sun so that the Satellite will remain at a relatively constant distance from Earth while the Satellite and the Earth orbit the Sun in our three body system. Staying with the idea of a top down view of a two dimensional model and a straight line from the Sun to the Earth, there is a third Lagrange point that will lie on a straight line from the Earth, through the Sun to a point in space that the gravity of the Earth and the gravity of the Sun effectively both pull on the Satellite as to make the Satellite orbit on the opposite side of the Sun relative to the Earth. Last but not least the fourth and fifth points looking down at our two dimensional model would revolve around the Earth just ahead Earth s orbit and just trailing Earth s orbit so that the Satellite would again be pulled in two different directions keeping it a relative distance from the Earth but still orbiting the Sun at the same speed as the Earth. Although most of the theories discussed in Lagrange s essay were just that, theories, in our more modern world of space exploration, the idea of Lagrange Points are much more applicable and useful. In fact, some of them are already in use by NASA. The L1 point which is our point that lies on the straight line of our two dimensional model from the Sun to the Earth is the point inside the Earth s orbit, with the Sun pulling it one way and the Earth pulling it the opposite direction. This point is approximately 1.5 million kilometers from Earth and was the destination of NASA s Solar and Heliospheric Observatory project which is now used to track solar storms and also was responsible for helping NASA scientists 4
determine the structure of the interior of the sun, how and why the corona of the sun reaches such high temperatures, around 1,000,000 degrees Celsius, and where exactly solar winds come from. SOHO is also an integral part of NASA s early warning system for solar storms, capable of giving almost three days advance notice of possibly dangerous and electrically disruptive solar storms. Another mission of NASA s, who s destination was a Lagrange Point, was flown to L2. Using our two dimensional model L2 lies on our straight line from the Sun to the Earth approximately 1.5 million kilometers outside the Earths orbit. L2 is home to the famous WMAP, or Wilkenson Microwave Anisotropy Probe. The WMAP is credited with having taken the most intricate and detailed pictures of our visible universe to date and is widely heralded as one of the most successful and influential space missions of 21st century. In fact, since the results of the WMAP mission, scientists and physicists have observed the universe in a completely different way than before. The contributions of the WMAP mission is somewhat difficult for one person alone to wrap their head around just exactly what it all means. In general it is widely accepted in the scientific community that data from the WMAP mission has confirmed theories that the universe is approximately 13.7 billion years old, and not only that but the WMAP mission has even been able to give us an idea of the curvature of space, nearly flat! Last but certainly not least, the WMAP mission added nearly all of the necessary evidence that was sought after to prove the now widely accepted theory of the big bang! In conclusion, the idea that Lagrange proposed almost 250 years ago has been a huge 5
part of our understanding of the universe. Without these virtual parking spots around our Earth for our satellites in space, these missions would all have been extremely more complicated and expensive and may not have even been attempted. Since these Lagrangian points are essentially that, parking spots, one can only wonder, are we the only ones using them? 6
Works Cited Ball, W W. R. A Short Account of the History of Mathematics. New York: Main Street, 2001. Print. Chuss, Dr. David T. Wilkinson Microwave Anisotropy Probe. N.p., 7 Jan. 2013. Web. 18 Oct. 2013. map.gfsc.nasa.gov. Cornish, Neil J. The Lagrange Points. Physics at MSU. N.p., n.d. Web. 18 Oct. 2013. Lagrange, J L. Ausdehnung Des Lagrange schen Behandlung Des Dreikorper-Problems (essai Sur Le Probleme Des Trois Corps) Auf Das Vierkorper-Problem. Vor Dr. A. Seydler. N.p., 1885. Print. 7