I got to thinking that maybe someone would like a better explanation of
the trajectory options available to Kranz, so here is one view of
Direct Abort...

TLI is not a commitment to go to the Moon. It is only a commitment to
go *toward* the Moon.

To understand why, consider this scenario: Shoot a basketball in a
high arch to land near the center of the court. At any point along
that arch, you can impart a delta-v (a change in velocity) on that ball
so that it goes directly to the target at the center of the court, no
longer traveling to the apex of the original arch (its apogee). This
change in basketball arch trajectory is analogous to a Direct Abort for
an Apollo spacecraft headed for the Moon. Both are trajectories within
the gravity field of the Earth.

To simplify the situation, let's take the Moon out of the picture for
the moment. In this case, you can see that all the Trans-Lunar
Injection burn is doing for you is taking you out of your circular Low
Earth parking orbit, and pumping your trajectory way up into a highly
elliptical orbit that will top out at lunar distance (where the Moon
used to be). This is like the high arching basketball shot. The first
half of the ellipse acts as the Hohmann Transfer. With nothing out
there, the spacecraft will just go way up, and then way back down. But
the stack has plenty of energy to change its direction. While it is
still going out, if you decide that you want to bring it back quickly
you can simply turn it around and point it toward the Earth and do a
burn that takes you directly back to Earth. This Direct Abort burn is
like swatting the basketball mid-flight so that its path goes directly
toward the target at the center of the court, never getting anywhere
close to the height of its original trajectory (never making it out to
"lunar" distance).

That is the simplest way I can describe the basics of a Direct Abort.


If anyone would like a more exacting analogy, we can place the Moon
back in the picture. Imagine that hanging at some height above the
basketball court is a spherical magnet. Imagine that the ball is
metallic so that it is attracted to the magnet. (Magnetic attraction
follows the inverse square law similar to gravity.) Now when you shoot
the ball in a high arch it gets attracted to the magnet. The Earth
never stops pulling on the ball, but when the ball gets close enough to
the magnet it is that force that dominates, opening the possibility of
the magnet capturing the ball. This brings the analogy to a three-body
problem where the Earth-basketball-magnet corresponds to the real-world
Apollo situation of Earth-ApolloSpacecraft-Moon. And the basketball
shooter is like the S-IVB that imparts the energy for the high-arching
trajectory.


To be even more complete, the spherical magnet would not be hanging
stationary over the court. It would be moving steadily at a fairly
high speed along some kind of track. So the actual problem of getting
the basketball captured by the magnet is not one of slowing the ball
down, but rather speeding the ball up to more closely match the
magnet's speed. Notice that the reason for a spacecraft falling back
toward Earth after completing the half-ellipse outbound trajectory of a
Hohmann Transfer is that it does not have enough speed for a circular
orbit like the Moon's. From the point of view of the Moon, the
spacecraft appears to be slowing down during the Lunar Orbit Insertion
burn. But in an inertial reference frame it becomes clear that the
spacecraft is actually speeding up. Notice that the LOI burn happens
behind the Moon with the thrust vector pointing in the same direction
as the Moon's orbit.

For one more step toward completeness...

Imagine now that the court is not rectangular and stationary, but
circular and rotating. So not only is it a moving target that you're
shooting for, but your position itself is moving when you take the
shot.


More and more complexity could be piled onto the analogy, but the basic
point regarding the Apollo 13 abort trajectory decision is that if you
want to get them home quickly, you can turn them around and come
directly home. You can swat the basketball directly back down to the
court.

I've seen little to support Kranz's decision to take the extra days
going all the way out to the Moon before bringing them home, especially
given the near pristine condition that I see in that hi-res photo of
the SM. Had Lovell&Co *not* made it back due to a consumables shortage
or some cold soaked parachute failure or such, I am certain that the
mishap board would have cited Kranz's trajectory decision as the fatal
mistake.


~ CT