Re: [Phys-l] Rocket Science

[John Mallinckrodt <ajm@csupomona.edu>]

Just for the record, my own calculation was based on the so-called  
"vis-viva" equation and Kepler's third law:

     v^2 = k(2/r - 1/a)         and       a^3 ~ T^2

with r = ai so that vi^2 = k/ai and Ti = 90 minutes (ai, vi, and Ti  
being the initial values in the circular orbit). I took differentials  
and eliminated the proportionality constants to find

     2 (delta v)/vi = (delta a)/ai      and        (delta a)/ai =  
2/3 (delta T)/Ti

Eliminating (delta a)/ai it's easy to see that

extra lag ~ vi (delta T) = 3 (delta v) Ti = 3 (1 km/min) (~90  
minutes) ~ 270 km

Of course, this is ultimately the same thing JD has said more  
elegantly, but it may enjoy the virtue of being at least somewhat  
more straightforward.

Clearly, then, the way to accomplish the rendezvous at the end of one  
more orbit is for the trailing rocket to decrease its orbital speed  
by 20 km/hr -- 1/3 the amount that a naive application of d = vt  
would suggest one should increase the speed.

On Jul 26, 2006, at 10:03 AM, John Denker wrote:

> 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.

John "Slo" Mallinckrodt

Professor of Physics, Cal Poly Pomona
<http://www.csupomona.edu/~ajm>

and

Lead Guitarist, Out-Laws of Physics
<http://outlawsofphysics.com>