Re: a surprising escape speed problem

[John Mallinckrodt <ajm@CSUPOMONA.EDU>]

> METHOD #2. Formal derivation using the conservation laws.
>
> System = earth + rocket. All velocities below are vector quantities
> so that we can explicitly test the directional advantage (if any) of
> launch.
>
> energy conservation => (v + v_orbit)^2 - v_esc^2 = M/m*(v'_orbit^2 -
> v_orbit^2)

The flaw here is in assuming that v_infinity = 0.  In fact, you'd
have to put in some *extra* effort to make the final velocity vanish
because you'd need to cancel out all that unnecessary orbital
velocity.  Orbital velocity, in this Sunless system, is an
*arbitrary* parameter and it can trick you into analyzing the problem
from the wrong reference frame.  The way to obtain minimal launch
speed is to let the final velocity of the probe equal the (arbitrary)
orbital velocity of the Earth.  And that, in turn, means that you
should simply make the launch velocity equal to the Earth escape
velocity.

Again, all of this only applies rigorously in the limit that the
radius of the Earth is negligible wrt the distance to the Sun.  This
same limit is required to justify the Barger and Olsson equation.  It
amounts to assuming that the potential well of the Earth is a mere
dimple on the potential well of the Sun (which it is!)

--
John Mallinckrodt        mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.