Orbital mechanics wants to drive the trajectory into an ellipse.
I'm curious about that: Assuming the Shuttle is 100 m distant from the ISS, i.e., in a 100 m higher orbit, then this means that the Shuttle has to travel 2*PI*Δh (628 m) further each orbit than the ISS (also assuming both Shuttle and ISS are in a perfectly circular orbit).
So, my intuitive thinking says that the Shuttle would travel more slowly (Kepler's laws) than the ISS if it weren't using the RCS thrusters (other than to move it into the higher orbit to begin with).
Wrong initial conditions. The flyaround starts with the orbiter along the ISS velocity vector, not above or below ISS, and the initial velocity to start the flyaround is in the radial-up (away from Earth) direction.
But, how does the elliptical motion occur? Is this a motion of the Shuttle and ISS relative to each other? Do the Shuttle and ISS exert an infinitesimally small (but measurable) gravitational force on each other? Links/pointers to useful info related to this would be appreciated. Thanks!
The mutual gravitation of the two spacecraft is negligible. The apparent "force" causing the orbiter to want to move in an ellipse is an artifact of examining the relative motion in a rotating frame.
If you look at the motion in the inertial (non-rotating) frame, ISS is in a circular orbit and the orbiter is in a slightly elliptical equiperiod orbit. Since the orbiter's orbit has the same period, it would return to the same spot relative to ISS after each orbit.
If you solve the equations of motion in an ISS-centered rotating frame, you wind up with what are called Hill's equations or the Clohessy-Wiltshire equations:
http://academy.gsfc.nasa.gov/2003/interns/collange/module/Hill.htmThis example uses a different frame than JSC uses (RSW vs. LVLH) but will suffice for our purposes.
The flyaround is in-plane, so ignore the Z equation.
We are interested in unperturbed (i.e. where the orbiter would go if the crew took hands off the stick) so set fx and fy to zero and solve for x-double-dot and y-double-dot.
You wind up with an x-term (the "Rbar term"), an x-dot term and a y-dot term. The x-dot and y-dot terms cause the elliptical motion. They're essentially the Coriolis terms.
The analytical solution is also on that page. Take the x(t) and y(t) equations and solve them for these initial conditions:
omega = 1.15e-3 (orbital rate for ISS, mrad/s)
x0 = 0
y0 = 125 m
xdot0 = 0.3 m/s
ydot0 = 0
and you'll see the ellipse that would result if the crew went hands-off after starting the flyaround.