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*From*: Leigh Palmer <palmer@sfu.ca>*Date*: Wed, 19 Feb 1997 18:12:51 -0800

I am confused by a question brought to me by one of my students who has

been reading about the SOHO spacevehicle. It has been inserted in an orbit

about the Sun at a Lagrange point ( defined as a point where Sun's Gravity

and Earth's Gravity fields balance). The press release I just read from the

net say it should be stable there fro 20 years due to wonderful work of

insertion.

There is a mistake in the release. SOHO (and another solar observer, WIND)

is not orbiting at the inner Lagrange point; it is in a hal;o orbit about

that point. The inner Lagrange point is in line with the Sun, which makes

for communications difficulties. You can see the gory details at

http://sohowww.nascom.nasa.gov/operations/commissioning/. You should click

on the first two pictures to see the transfer orbit in detail.

Here I have trouble: Is the net force at the Lagrange point zero? If so,

how does it manage to orbit at all? Is it not an unstable equilibrium?

The net force at the L1 point is indeed zero in the rotating frame of

reference. The gravitational forces of Earth and the Sun are just balanced

by the centrifugal force in that accelerated frame. While that is an

entirely appropriate way to look at the problem, there are religious

objections to using centrifugal force in high school pedagogy, and this

problem can, of course, also be viewed in the inertial frame fixed with

respect to the center of the Sun. in that frame the gravitational force

exerted on the satelite by the Sun is reduced slightly by the attraction

of Earth. The satellite orbits the Sun with the same period as the Earth,

however, because its orbit is smaller than Earth's.

Joseph Lagrange investigated the motions in what is called the restricted

three body problem. In this problem two massive bodies are set in circular

orbits about one another. The motion of a third, much less massive, body

is then analyzed as it moves in the gravitational field of the larger

bodies, not affecting that motion. Since the larger bodies always remain

the same distance from one another this analysis can be conveniently

carried out in the frame of reference which rotates with the larger bodies.

In that frame the large bodies are at rest and the small body moves in the

combined field of their gravity and the centrifugal potential. It is not

too difficult to show that there are five points in that potential field

where the potential has a zero gradient, and thus no net force acts on the

small body. (As I recall the equation generated is a quintic equation

which has five roots.) Two of those points (designated L4 and L5) are

points of stable equilibrium in the rotating frame. The other three are

points of unstable equilibrium.

The Earth and Sun approximate the setup for the restricted three body

problem. SOHO is the smaller mass in this case. SOHO is in an orbit which,

when viewed from Earth, appears to orbit the Sun in a "halo" centered on

L1.

Because L1 is a point of unstable equilibrium SOHO must be nerfed back to

its halo orbit from time to time by vernier rockets. What the press

release should have said is that there is enough rocket fuel aboard SOHO

to permit it to be kept in that orbit for twenty years.

For a more complicated transfer orbit leading to a halo result see

http://www-spof.gsfc.nasa.gov/cgi-bin/gif_walk?plot_type=orbit_wind

This orbit uses the Moon for gravitational boosts. (Our oldest son is

working on data returns from an instrument aboard this craft.)

I suspect that there is something rather rudimentary that I am overlooking

and I also know that real world Physics is more complex that high school

physics, but...I could use some help.

This is really very much easier to understand in the rotating frame of

reference. It is too bad that centrifugal force has been excluded from

the armamentarium of high school physics. It would be far better to forbid

the use of the term "centripetal force". I see numerous free body diagrams

drawn by students on which they place a vector confidently labelled as

the centripetal force! When one adds this to the other forces acting on

the body the result doesn't seem to work out somehow. They eventually get

straightened out (by second year) and become comfortable with accelerated

frames and centrifugal force and potential.

Leigh

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