Gravitational Slingshot

Interplanetary space probes often make use of the "gravitational
slingshot" effect to propel them to high velocities.  For example,
Voyager 2 performed a close flyby of Saturn on the 27th of August in
1981, which had the effect of slinging it toward its flyby of Uranus
on the 30th of January in 1986.  Since gravity is a conservative 
force, it may seem strange that an object can achieve a net gain in
speed due to a close encounter with a large gravitating mass.   We 
might imagine that the speed it gains while approaching the planet 
would be lost when receeding from the planet.   However, this is not
the case, as we can see from simple consideration of the kinetic 
energy and momentum, which shows how a planet can transfer kinetic
energy to the spacecraft.

An extreme form of the maneuver would be to approach a planet 
head-on at a speed v while the planet is moving directly toward 
us at a speed U (both speeds defined relative to the "fixed" Solar
frame).  If we aim just right we can loop around behind the planet 
in an extremely eccentric hyperbolic orbit, making a virtual 
180-degree turn, as illustrated below.

                 

The net effect is almost as if we "bounced" off the front of the 
planet.  From the planet's perspective we approached at the speed 
U+v, and therefore we will also receed at the speed U+v relative to 
the planet, but the planet is still moving at (virtually) the speed
U, so we will be moving at speed 2U+v.  This is just like a very 
small billiard ball bouncing off a very large one.

To be a little more precise, conservation of kinetic energy and 
momentum before and after the interaction requires

           M U1^2  +  m v1^2   =   M U2^2  +  m v2^2

            M U1  -  m v1  =  M U2  +  m v2

Eliminate U2 and solve for v2 to give the result

               v2  =  ((1-q)v1 + 2U1)/(1+q)

where q = m/M.  Since q is virtually zero (the probe has negligible 
mass compared with the planet), this reduces to our previous estimate 
of v2 = v1 + 2U1.

Of course, most planetary fly-bys are not simple head-on reversals, 
but the same principles apply for any angle of interaction.  Let's 
take the planet's direction of motion as the x axis, and the 
perdendicular direction (in the orbital plane) as the y axis.  The
probe is initially moving with a speed v relative to the solar
reference frame, in a direction approaching the oncoming planet
at an angle theta.  Two views of this are shown below, one with
respect to the planet's rest frame, and the other with respect to
the solar reference frame.

       

By drawing a simple parallogram of speeds for the probe and planet 
intersecting at an arbitrary angle theta, and assuming we arrange 
for a hyperbolic orbit symmetrical about the x axis (with respect to
the planet's rest frame), the probe's initial velocity vector with
respect to the Sun's rest frame is 

    v1x = -v1 cos(theta)        v1y = v1 sin(theta)

and its final velocity vector is

   v2x = v1 cos(theta) + 2U       v2y = v1 sin(theta)

Thus its initial magnitude is v1, and its final magnitude is

  v2  =  (v1 + 2u) sqrt[ 1 - 4uv1(1-cos(theta))/(v1+2u)^2 ]

For example, suppose the initial speeds of the probe and the planet 
happen to be exactly the same (i.e., v1 = U).  In this case the above 
relation reduces to

              v2  =  v1 sqrt[5 + 4cos(theta)]

which confirms that when theta = 0 we have v2 = 3 v1, which is our 
head-on reversal case.  On the other hand, when theta=pi we have 
v2 = v1, which stands to reason, because in this case the probe and 
planet are going in the same direction at the same speed.  For a 
more realistic case, we can have the probe approach nearly 
perpendicular to the planet's path (i.e., theta = pi/2) and swing 
just behind it.  In that case the probe gets deflected in the 
direction of the planet's travel, at an angle given by the above 
formulas, and it's final speed is sqrt(5) = 2.23 times its original 
speed.

If the planets were point particles, then according to classical
physics it would be theoretically possible (in some rather contrived 
solar systems) for an object to acquire infinite speed in finite 
time by looping repeatedly around a set of planets.  Of course, in 
practice the *external* gravitational field of a planet would not 
be strong enough to "grab" the spaceship once it was travelling 
above a certain speed.  The limit is how fast you can loop around 
a planet without dipping into its atmosphere too deeply (let alone 
crashing into it).  Some NASA missions have repeatedly skimed the 
upper atmospheres of Venus and the Earth in their maneuvers (cross-
pollinating the environments?).

Come to think of it, if we (or someone else) ever found a star system 
consisting of multiple black holes orbiting each other, we might be 
able to apply this scheme to achieve relativistic speeds, by looping 
around from one to the other.  I suppose in this situation the 
achievable speed limit would depend on how close a spaceship could 
pass without be being destroyed by tidal forces.  Still, if the 
black holes were large enough, the tidal forces even at the event 
horizon would be tolerable, although it probably wouldn't be possible 
to have a controlable hyperbolic orbit pass closer than, say 3m.  
Also, stopping at our destination might be tricky.