Help Me Solve a Physics Homework Problem - Lagrange Points

[Meistro]

Hey a friend asked me for help on his physics homework, and I found this place and was wondering if you guys could help me out.

2: In 1772, the famed Italian-French mathematician Joseph Louis Lagrange was working on the infamous three-body problem when he discovered an intersting quirk in the results. If one mass is much smaller than the other two then there will exist points where this object can be stationary with respect to one of the two masses. These points are known as Lagrange points in his honor. In our treatment we could consider these points to be equilibrium points for a system. If we wanted to find Lagrange point for the Earth-Sun system located between the Earth and the Sun how far from the Earth is this point and what is the significance of the other solution? The mass of the Earth is 5.98 X 10^24 kg, the mass of the Sun is 1.991 x 10^30 kg and the radius of the Earth's orbit is 1.496 x 10^11 m. (solve using quadratic eq.)

 

[tony873004]

Google for Hill Sphere. Wikipedia has a good site. The Hill Sphere will give you the distance to the L1 and L2 points.

L3 is in Earth's orbit, on the opposite side of the Sun, so it is exactly 2 AU away.

L4 and L5 are 60 degrees ahead of and behind the Earth. The Earth, Sun, and L4 form an equilateral triangle, as do the Earth, Sun, and L5. So the Earth L5 distance is 1 AU. The Earth L4 distance is 1 AU. The Sun is also 1 AU from both these points.

Actually, one mass doesn't need to be smaller than the other two, at least for the L4 & L5. And I suspect the other 3 L points as well. The two orbiting masses combined need to be at most ~1/25 the mass of the large object. It's possible for an Earth-mass planet to be in Earth's L4 or L5 point.