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From: FIRTH@TL-20B.ARPA
Newsgroups: net.space
Subject: Getting stuff into Orbit
Message-ID: <8512261825.AA16515@s1-b.arpa>
Date: Thu, 26-Dec-85 13:26:02 EST
Article-I.D.: s1-b.8512261825.AA16515
Posted: Thu Dec 26 13:26:02 1985
Date-Received: Sat, 28-Dec-85 01:19:55 EST
Sender: daemon@ucbvax.BERKELEY.EDU
Organization: The ARPA Internet
Lines: 67

There has been some discussion recently about how to get mass into
stable Earth orbit, using some form of ground-based accelerator.

If we consider only one mass at a time, then of course it isn't possible.
When the mass leaves the accelerator, it is in free fall, and so its
trajectory must return to that point - which presumably is quite close
to the ground and so well within the atmosphere.  Lagrange strikes again!

One could also consider a mass that had aerodynamic lift, but again the
answer is the same: it will return to the point where ballistic motion
began; and if that was deep enough in the atmosphere to provide some lift
on the way out, it will surely introduce drag on the way back in.

The only way is to give the body some more impulse once it is outside the
atmosphere, by means of rockets, an orbiting grabber, or whatever.

However, the trick is possible if you consider TWO masses at a time.  Suppose
we launch a mass into orbit from a ground-based accelerator.  Just to play with
some numbers, suppose the orbit is an ellipse with major axis 20000 miles
and minor axis 16000 miles.  Perigee is then 4000 miles from the center of
the Earth, ie grazing contact, and apogee is of course 16000 miles from the
center or 12000 miles above the surface.

Now launch TWO such masses, so that their orbits are in the same plane, but
pointing away from the Earth in opposite directions.  That is, the two
ellipses share a common focus (the earth's center) and all three foci are
in a straight line.  These orbits intersect at a distance of ~6400 miles out,
ie an altitude of ~2400 miles.

Launch the two masses so that they meet at the point of intersection, one
inbound and one outbound (and they had both better be on their first orbits,
of course).  Let the masses be equal.  Then, they meet when travelling at
the same speed (but in different directions), and with the same energy.
Somehow, get then to join into one bigger mass.  The combined mass will
then be travelling in a new orbit, whose major axis is perpendicular to
the major axes of the old orbits, and with perigee the same 6400 miles.
Apogee of course will be ~13600 miles.  The new mass is comfortably outside
the atmosphere, and all propulsion was done on the ground.

[Here follows the boring math.  Given an ellipse with semimajor axis a and
 semiminor axis b, then the interfocal distance 2f is given by

	f = sqrt(a^2-b^2)

 and the peri- and ap- distances by a-f and a+f.

 Two such ellipses with a common focus, and the three foci in a straight
 line, intersect twice; the points of intersection being symmetrically
 placed about the common focus so that the line joining them passes through
 the common focus and is perpendicular to the major axes. The triangle formed
 by the common focus, either other focus, and either point of intersection is
 a right triangle.  If the intersection point is at distance x from the common
 focus and y (>x) from the other focus, then

	x^2 + 4f^2 = y^2	(recall that 2f is interfocal distance)
 and
	x + y = 2a		(the ellipse invariant)

 whence we may calculate x and y.

 Finally, recall that the energy of an elliptic orbit is -1/2a, independent
 of the minor axis (multiplied by GMm, of course, but we can ignore all that)
]

Robert Firth
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