Re: [Phys-l] Rocket Science

[John Denker <jsd@av8n.com>]

Bob Sciamanda wrote:
> Fowles equation 6.47 gives an orbit equation convenient for calculating a 
> new orbit after a forward thrust from a circular orbit:
>
> R(Theta) = Ro[(Vo/Vc)^2] / [1 + { (Vo/Vc)^2 - 1}Cos(Theta)],          [1]
>
> where Vo is the speed at Theta = 0, and the orbit is a circle with radius Ro 
> if Vo = Vc.

OK.

It is useful to define /eps/ such that Vo = Vc(1+eps).  Then equation [1] can be
expanded to leading order as

   R(theta) = Ro(1 + 2 eps - 2 eps cos(theta))

which checks out because:
 -- R(theta) = Ro when theta=0
 -- The new orbit is (2 eps) higher on average, which agrees with what we
  get from KE + virial arguments.

========

To finish the task, we need to know dt/d(theta).  It is tempting to write
R(theta)/Vo by dimensional analysis, but alas there is more to physics than
dimensional analysis.  [I tried and failed to do this problem in my head;
this is the point where I dropped the ball.]  The correct argument, using
conservation of momentum, throws another factor of R(theta)/Ro in there.
At the end of the task we get

   t(new)
  -------  =  1 + 3 eps + higher-order terms
   t(old)               + terms that vanish when averaged over a whole orbit

That's the same "3" that several people have obtained by other means.

> I think we need more info than JD supplied. 

Several people have solved the problem using just the supplied info.

>  We do not know Vo or Vc ; we only know Delta V = Vo-Vc.

Do you really not know Vc for a "nice, circular, low-earth orbit" ???

There is an interesting pedagogical point here.  If I had simply asked
what is the velocity of a nice, circular, low-earth orbit, presumably
everybody could have come up with the answer.  But if I present a story
problem which _implicitly_ requires folks to figure out Vc, suddenly it
becomes a monstrous barrier to solving the problem.  This suggests that
the curriculum has not nearly enough story problems (or not nearly enough
story problems _of the right kind_).  A related tirade can be found at:
  http://www.av8n.com/physics/ill-posed.htm

==========================================================

A spreadsheet to do the rendezvous calculation gnumerically can be found at
  http://www.av8n.com/physics/rendezvous.xls

The numerical methods used here are somewhat fancier than what ye olde
standard-issue introductory-level student would be expected to come up
with.  The main thing is that I use _balanced_ steps.  For instance,
when stepping the velocity from row 7 to row 8, I use the gravitational
force not at the row-7 position or the row-8 position, but rather the
_average_ of those two positions.  Similarly, when stepping the position
to row 8, I step from the row 6 (not row 7) position, using the row 7
(not 6 or 8) velocity.  As a general rule, when stepping you want to use
the !average! derivative, and using the derivative at the midpoint of the
step is typically a better estimate of the average derivative -- usually
muuuch better than using the derivative at either end of the step.

If you don't use balanced steps, your simulation will produce orbits with
unacceptable amounts of precession, or orbits that spiral off to never-never
land ... unless you use an ultra-small stepsize (dt), which causes problems
of its own:  it is a waste of computer time, and greatly increases your risk
of roundoff errors.

I can code balanced steps in my sleep, 'cuz I've done it a bazillion times.
You might find it worthwhile to remember this trick.  Even if you don't
remember the details, at least remember that not all steppers are created
equal.  Small changes in the numerical methods can lead to big improvements.

Another hint:  There are lots of things you can do to check your numerics.
 *) Always check that any result you care about is independent of stepsize.
 *) In the special case of Keplerian orbits, the following additional checks
  apply:
   -- Make sure the orbit is circular, if the initial conditions should have
    produced a circular orbit.
   -- Make sure all orbits are closed ... no spiraling or precessing.  Run
    off several periods to make sure.
   -- Make sure the orbital period is 2pi in the appropriate units.
   -- Check that the energy is constant all along the orbit.
   -- Check that the angular momentum is constant all along the orbit.
 *) More generally, when you're doing a physics problem, there are almost
  always some sort of sum rules or conservation laws that you can use to
  check your numerics.

==========

Another amusing bit of physics:  As you can see in the "Delta R" column in the
spreadsheet, the unwise intercept trajectory actually does begin by moving toward
the other spacecraft, in the very short term.  There is some kind of correspondence
principle involved here;  In the very short term, the two craft don't know or
care about orbital dynamics.

This has practical consequences for docking.  If you start out sufficiently close
that you can complete the docking maneuver before you rack up any appreciable
fraction of a radian of orbit, you won't be harmed by using unsophisticated
distance=rate*time thinking.