Re: two body problem and collision time
- From: "Greg Neill" <gneillREM@xxxxxxxxxxxxxxx>
- Date: Sat, 9 Feb 2008 02:38:01 -0500
"Timo A. Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote in message
news:Pine.WNT.4.64.0802091355140.596@xxxxxxxxx
On Tue, 29 Jan 2008, Greg Neill wrote:
Consider the case of two equal masses coorbiting in
circular motion. Both describe circles about the
common center of mass. But if you choose one of
the masses as the origin instead, the other mass
*still* describes a circle about the new origin.
It doesn't matter if the masses are equal or not,
or if the orbits are circular or not. This is why,
for example, we perceive the Sun describing an
elliptical orbit about the Earth from an Earth-
centered frame.
No obvious proof occurs to me, and I haven't had time to grind out the
algebra. For a circular orbit, we have a special case, so it isn't
obvious to me that this generalises to elliptical orbits. Do you know
of a simple, hopefully forehead-smackingly simple, proof?
Maybe not forhead smacking. I can't recall where I
first read this factoid, so if there's a particularly
obvious demonstration I can't recall it. I do know
that I've plotted the results for the requisite change
of frames and the plots come out as ellipses.
Here's a go at a not too nasty proof.
Take the equation for the radius of an ellipse
with latus rectum p and eccentricity e and
construct the position vector for the center of
mass frame (one focus):
r(q) = p/(1 + e*cos(q)) r is the radius
R1(q) = r(q)*[cos(q) + i*sin(q)] R1 is the vector
The other body will orbit in an ellipse with the
same eccentricity but different latus rectum as
scaled by the mass ratio, and it will orbit the
same center of mass. One difference is that its
radius vector will be always in the diametrically
opposite direction as that of the first body. That
is, you can draw a straight line from one body
through the origin to the other body. So its
vector is given by:
R2(q) = -k*R1(q)
where k is the scaling factor.
Let's form the position vector from one body to the
other. We can choose either body as the new frame.
Here's one choice:
R(q) = R1(q) - R2(q)
= R1(q) + k*R1(q)
= (1 + k)*R1(q)
But this is simply R1(q), an ellipse, scaled by a
factor of (1-k).
.
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