The Lagrangian Point: Staying in Synch with Earth

What is a Lagrangian Point ? Notebook Entry: There are Five Rotating Positions in a Solar Orbit where the combined Gravitational Force of the Sun and the Earth provides the exact Centripetal Force needed for a small Satellite to orbit at the same rate as the Earth. (Let’s ignore details like planetary orbits are actually elliptical and the Sun and Earth both rotate around a third point called their barycenter.) This works for Earth-Moon system, too!

Sun-Earth Lagrangian Point #2 - SEL2 Lissajous Orbit of L2 “Top” View “Side” View

Earth Moon JWST Separation L + 39 min Main Engine Cut-Off 1 L + 13 min Trajectory correction maneuver L + 8 hrs Sunshield deploys L + 2 days Telescope deploys L + 4 days L2 orbit Achieved L days Transit of JWST to SEL2

Open the simulation from the URL on your worksheet: galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/kepler6.htm … and complete tasks 1 through 5. Remember the Balance of Forces? Gravitational Force produced by a the Sun on the Earth: Centripetal Force needed for an object rotate around a point: (6.67)( )* (1.50) 2 (10 11 ) 2 Lets just look at the acceleration produced by the Sun’s gravity and see if it matches the acceleration of Earth’s orbit… G = 6.67x m 3 /kg-s 2 m S = 1.99 x kg m E = 5.97 x kg r S = 1.50 x m r L2 = 1.50 x 10 9 m v E = 2.98 x 10 4 m/s (2.98) 2 (10 4 ) 2 (1.50) (10 11 ) a G = 5.90x10 -3 m/s 2 a C = 5.92x10 -3 m/s 2 (1.99)(10 30 ) Why are these different? Then we’ll look at the math when a Satellite is added…

Velocity needed for a farther away object to have the same period as Earth: Centripetal Acceleration needed for an object rotate around a point: Gravitational Acceleration produced by a the Sun on the Earth: Balance of Forces on Satellite at L2? (2.98x10 4 m/s) (150x10 9 m) (1.50x10 9 m) (150x10 9 m) 1 (3.01) 2 (10 4 ) 2 (150x10 9 m x10 9 m) (6.67x ) (1.99x10 30 ) (150x x10 9 ) 2 (6.67x ) (5.97x10 24 ) (1.50x10 9 ) 2 + a G = 5.96x10 -3 m/s 2 a C = 5.98x10 -3 m/s 2 Are these close enough? Notebook: If a satellite is 1.5 million kilometers outside Earth’s orbit, the 2 nd Lagrangian Point, L2, the combined gravity of the Sun and the Earth will “pull” the satellite along fast enough in its longer orbit to “Stay in Synch” with the Earth as revolves in its shorter orbit at a slower speed.

Complete the Graphical Exercise on your Worksheet Objects: Sun, Earth, L1, L2, r S (with double ended arrow), r L2 (with double ended arrow) Velocity Vectors: Earth, v E, Satellite, v L2 (larger magnitude single arrow is longer) Acceleration Vectors: Earth, a E, Satellite, a L2 (larger magnitude arrow is longer)

Complete the Outline & Essay on your Worksheet When the James Webb Space Telescope is launched in 2018, your college roommate doesn’t know what a Lagrangian Point is! Your new roommate asks, “What’s so special about this L2 point?” and now you can tell him (with a little reflection, perhaps). Then, your roommate asks, “Why does the satellite remain opposite to the Sun in relation to the Earth at the L2 point?” In the space provided on the Worksheet, here is your Writing Assignment: 10) Compose a Detailed Outline or Draw a Detailed Concept Map of your answer to both questions. 11) Write a one page essay answering these questions using the following key terms; Sun, Earth, Gravitational Forces, Centripetal Force, Acceleration, Tangential Velocity, Orbital Period, and Balance.