Name:
___________________
Math 122 Lab 7: Lagrange
Points
Introduction
One
of the most important uses of vector calculus (both historically and in modern
times) is the computation of the orbits of objects in space. In this lab, you
look at a special case of the notorious three-body problem. You have heard that
the earth’s orbit is an ellipse with the sun at one focus. Actually, this would
be correct if there were no other objects in the solar system (i.e., only two
bodies, the earth and sun, interacting). The true orbit of any of the planets
is more complicated than that and, in fact, is thought to be unpredictable and
unstable! However, the instability is on the order of billions of years, so we
don’t have much to worry about.
What to Turn In
Turn
in this worksheet with everything filled in. This will be graded for accuracy
and understanding, so write good explanations when you are asked to explain
something. If you need more space, turn in additional pages typed or neatly
written. Due Friday at 8:30. Late papers lose 4 points per day late.
The Five
Lagrange Points
In general, it is impossible
to find equations for the motion of three or more objects interacting through
gravity, because the orbits are too complicated. Imagine looking at a
simplified solar system from “overhead” as in the picture below, showing the
(large) sun, (medium) earth and five locations for a (small) spacecraft.
If we keep the sun and earth
in the same place, most objects such as the moon will appear to move. There are
five orbits (labeled L1-L5) where an object in orbit around the sun maintains a
constant position relative to the
earth and sun. These are called the Lagrange points. Our goal here is to find
an equation for L1, the point on a direct line between earth and sun.
Circular
Orbits
We will need one important fact from the physics of
circular orbits. If an object has position function r(t) = <Rcos(ct),Rsin(ct)>, its path is a circle with
center at the origin. What is its radius? What role does c play in the object’s orbit?
If the object’s mass is m, show that F = ma becomes F
= - mRc2 <cos(ct),sin(ct)> and find the magnitude of the
force vector.
Solving for
the Lagrange Points
Now, we solve for the L1 Lagrange point. In the
diagram, the sun is a distance r from
the earth, and the L1 point is a distance R
from the earth.
By 
while the pull of the
earth (with mass m) is 
. Then the combined
force on the object has magnitude
If we assume that the L1 object is in a circular
orbit around the sun with radius 
, then the magnitude of the force is given by 
for some constant c. Setting these equal to each other
gives
Modify this equation by dividing out the mass s.
Now, we also assume that the earth is in a circular
orbit around the sun. The corresponding equation for the earth is
for some constant d. Divide by mr and
rewrite this equation here.
The constants c
and d in our equations depend on
the speeds of the orbits of the L1 point and earth, respectively. If the L1
point is to stay in line with the earth, the speeds must be such that both
complete one rotation in the same amount of time. For this to happen, we must
have c = d. In the equation at the bottom of the previous page, replace 
with the expression
just found for 
. Then divide both sides by GM.
The rest of the derivation is done for you below. Naming
gives the equation
Multiplying both sides of the equation by 
gives
and renaming 
gives the equation
Finally, multiplying both sides of this equation by 
gives
which is the equation that you’ll use to find L1. For
the earth and sun, the masses are such that 
. Use your calculator to approximate the value of z.
Fill in the blank: the L1 point is about ______ % of
the way from the earth to the sun.
Next, we solve for the L2 Lagrange point. First,
redraw the picture from page 3 showing the sun, earth, L2, r and R.
In the equation for the gravitational force, explain
why each of the three negative signs in the L1 equation becomes positive to
give the equation
The same series of steps as before gives this
equation for z:
Using the same y
as before, solve this to obtain z and
write the value here.
Which one is closer to the earth, L1 or L2?
For L3, the derivation is easiest if we define R to be the distance from L3 to the sun.
Then the equation for is given by
Solve this for z
and write the value here.
How does the distance of L3 from the sun compare to
the distance of the earth from the sun?
Conclusion
Now that you have worked
with the equations, think about the significance of the Lagrange points. These
are places where (theoretically) the combined effect of the sun’s and earth’s
gravities is to pull the object along at a constant relative position. Explain
why L1 would be a great location for a solar observatory. (In fact, NASA has
its
Explain why L2 is a great
place for an observatory of deep space. (NASA is sending its Microwave Anisotropy Probe there.)
Explain why L3 is a popular
place for science fiction writers to locate Planet X, a place where (unknown to
us) intelligent life exists.
Solving for the Lagrange points L4 and L5 is
complicated, but it is an interesting fact that L4, the earth and the sun are
vertices of an equilateral triangle. So, L4 and L5 have the exact orbits of the
earth, but are 60 degrees ahead and 60 degrees behind, respectively. The
Lagrange points are not all stable. In particular, only L4 and L5 are stable,
while L1, L2 and L3 are unstable. This means that an object exactly at L1 will stay there, but an
object near L1 will wander away. Since the NASA solar observatory near L1 is not
exactly at L1, a small amount of fuel
must be used to keep it near L1. By contrast, an observatory placed near L4
will stay near L4 forever.
Lagrange points exist for
other systems of large, medium and small objects. In the sun-Jupiter system,
the L4 and L5 points are the locations of numerous asteroids, called Trojan
asteroids. Explain why a large number of asteroids might gather there.
In the earth-moon system, explain why L4 and L5 are
good locations for space stations.