Lagrangian Points, Satellites, and Space Exploration

Source: NASA

Let’s take a look at the Sun and the Earth. Our planet orbits the Sun. What does this mean? Well, following Newton’s law of universal gravitation, the Sun and the Earth exert a gravitational force over each other. So, it follows that there is some point in space between these two bodies where the gravitational force of the Sun on the Earth will cancel that of the Earth on the Sun. Simple… or is it? There is another force at work: centrifugal. The Sun and Earth orbit a common center of gravity. Any object moving in a circle will experience centrifugal force, i.e. the force that pushes outward. If you’ve ever heard of an amusement park ride called the Gravitron, well, that’s centrifugal force at work. The opposite of centrifugal force is centripetal force, i.e. the force that pushes inwards, and this works to maintain an object’s circular path. To visualize these two forces at work, imagine yourself tying a piece of string to a rock. Now, start swinging that string like a lasso. Centrifugal force is pushing the rock outwards. At the same time, centripetal force is pushing back down the string towards the center of motion (you). Swing the string hard enough and centrifugal force wins the battle against centripetal force (and the tension of the string, which we’ll assume is negligible to keep the example simple), and your rock shoots off tangentially to its former circular path.

In the previous scenario, gravity isn’t really an issue, primarily because you and the rock do not exert a significant force on each other with respect to the Earth beneath you. But, when we’re talking about space and orbits, gravity becomes a big deal. The eighteenth-century French-Italian mathematician Joseph Louis Lagrange made an interesting discovery when tinkering with the interplay between gravity and centrifugal/centripetal forces in orbital systems: there exists positions in space where the gravitational pull of the two masses precisely equals the centripetal force required to rotate with them. Specifically, 5 such points exists, and, theoretically, an object sent to any of these points of Lagrange will remain stationary.

The figure at the top illustrates the 5 Lagrangian points of the Sun-Earth system. Some of these points have already been put to use by astronomers. Point 1 is where you will find the Solar and Heliospheric Observatory, or SOHO. Its position allows it to enjoy an uninterrupted view of the Sun at all times and collect information about the solar wind an hour before it reaches the Earth. At point 2 lies the Wilkinson Microwave Anisotropy Probe, used by astronomers to observe the Cosmic Microwave Background (CMB) and calculate the age of the universe (Weintraub, 2011). Point 2 will also serve as the placement site for the James Webb Space Telescope.

Outside of satellite destinations, popular astronomer Neil deGrasse Tyson envisions another use for these points: “… imagine fuel stations at every Lagrangian point in the solar system, where travelers fill up their rocket gas tanks en route to visit friends and relatives elsewhere among the planets. This travel model, however futuristic is reads, is not entirely farfetched. Note that without fueling stations scattered liberally across the United States, your automobile would require the proportions of the Saturn V rocket to drive coast to coast: most of your vehicle’s size and mass would be fuel, used primarily to transport the yet-to-be-consumed fuel during your cross-country trip. We don’t travel this way on Earth. Perhaps the time is overdue when we no longer travel that way through space.” (Tyson, 2007). It’s an interesting thought, to say the least, and, for me, it illustrates how cool physics can be. When you’re not being tested on it.

To learn more:

Death by Black Hole and Other Cosmic Quandaries, Neil DeGrasse Tyson

http://www.nasa.gov/missions/solarsystem/f-lagrange.html

http://map.gsfc.nasa.gov/mission/observatory_l2.html

http://math.ucr.edu/home/baez/lagrange.html

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