Re: Planetary Motion Computation

> "joax" <mlacdc@ecenet.com> wrote in message
> news:3B5EF1E7.F8C1D580@ecenet.com...
>> We thought at first that we would get some kind of grand equation that would say:
>> If Jupiter is here, and the Sun is here, and Uranus and Neptune are here, and the
>> Earth and Venus are here and here, then HERE is where Mars has to be.  That did
>> not happen.   I guess I just want to caution people that want to make assumptions
>> about planetary positions and perturbations that it is not at all simple to do.
>> That there are so many interdependent feed-back mechanisms at work that a simple
>> deduction based on a few variables is not going to be accurate at all.
>> 
>> You might like to look at Section 4 of the Pioneer paper, much discussed in
>> this group. It is entitled "BASIC THEORY OF SPACECRAFT NAVIGATION". The paper
>> is available at:
>> 
>> http://www.arxiv.org/abs/gr-qc/0104064

Several years ago there was an article in Scientific American about
gravitational computations and simulations of the solar system.  The
results were interesting.

Since the three body (and up) problem with Newtonian mechanics does not
have a (known) closed form solution, these systems are simulated on
computers using numerical integration.  Numerical integration is a very
interesting math problem and avoiding the introduction of errors is a
very complicated issue, not solved simply by using a double precision
number as opposed to a single precision.  The order of operations in
computing the equation can have catastrophic consequences on the round
off errors introduced into the results.

A numerical simulation was set up and the whole solar system was
simulated for an extended period.  As simulation time progressed, orbits
varied slightly, but nothing out of the ordinary.  However, as the
simulation time became very long (i.e. millions of years) what happened
was the entire solar system became unstable, the planetary orbits became
"chaotic" and usually one of the planets was ejected from the solar
system.

This effect happened in spite of the time direction (of course, because
it is time symmetric) and independent of changes in the mathematical
algorithm.  The conclusion was that orbital mechanics has a chaotic
element IN THE LONG TERM, or in other words it was the famous "butterfly
effect" as applied to gravitationally bound systems.  Small variations
in initial conditions result in widely varying results after a
significant time.

Now, a simulation that runs for a time of several million years is quite
long.  It is obviously several million orbits of the earth, tens of
millions of orbits of Mercury and many tens of thousands of orbits of
Pluto.  The biggest problem with the simulation results is that they
were unable to make the system stable over the course of time frames
comparable to the estimated lifetime of our solar system.  While we do
not have a lot of specific data that is 5 billion years old, we have
empirical evidence that the solar system has been stable (specifically
at least the earth) because conditions for the evolution of life are
quite sensitive to the position of the earth's orbit, and life evolved
over a billions of years.

It is one thing to model the position of a spacecraft over tens of years
(and yes, it is an impressive accomplishment), it is another to state
that the same equation is capable of explaining the entire orbital
mechanics of the solar system over billions of years.

Since we have evidence that the solar system is, in fact, stable (look
in the mirror, you are here) we have to assume that any observed
instability in our computations is either a result of the inaccuracies
of the computation or in the incompleteness of the explanation of why
the solar system is stable to begin with.  In the article referred to,
the suggestion was that it was not the math per se because regardless of
what they did, the result was the same.

The point is that secondary and tertiary effects do not have to be
particularly big to matter.  The solar system is something like 5
billion years old, and has had the time to work all of this out.  The
integral will take into account small effects FOREVER.

The gravitational constant (coefficient, or whatever) was originally
determined with the Cavendish balance and since then has been refined
with more accurate experiments and with throwing several thousand pound
spacecraft at planets and measuring the results.

From that value of the gravitational constant, we then *compute* the
mass of the planet and other planets.  Any idea that we have a clear and
unambiguous measure of the weight (mass) of the earth is just
ridiculous, what we have is a reasonably self consistent system of
equations and no clear unambiguous data that contradicts it.

But this does not mean it is "right" it just means that we can do some
relatively impressive engineering calculations with the equations.

It has been stated that the repulsive force does not obey the inverse
square law.  So what? Neither does the strong force.  This has not
stopped us from discovering it or making atomic bombs.

It should also be pointed out that we have made experiments that seems
to constantly flirt with the existence of a repulsive component of
gravity.  The existence of a "fifth force" has been the subject of
legitimate (i.e. by university professors) debate for quite some time. 
The problem is that the experimental design has always had errors
greater to or comparable to the magnitude of the supposed result and
this has allowed the evidence to be discarded.  After all, if you are
going to grind the Hubble telescope mirror to 1/10 wave, you are not
going to measure this with a wooden ruler.  If somebody comes along and
says they used a wooden ruler and determined that the mirror was off,
you would naturally discount them, even if it eventually proved they
were right after all.

Our scientific advance has always been closely tied to our ability to
measure stuff.  Any significant improvement in measurement accuracy has
resulted in the need for new theories because the universe has always
been full of surprises.  The thread on the Pioneer anomaly is a good
example.  The above mentioned paper shows how sufficiently accurate
measurements have shown that there is an unaccounted force in distant
spacecraft.  Just out of curiosity, what would happen if this force was
applied to a whole planet for 5 billion years?  Is the force related to
the mass of the spacecraft, the surface area or the composition?  Is the
sun attracting the spacecraft or is interstellar space "repulsing" it? 
It is an interesting set of questions.

The idea that the planets are stable because of Newtonian mechanics is
clearly wrong.  Relativistic corrections help make the simulations more
accurate and allow very precise short term spacecraft trajectories and
the development of GPS.  But this does not mean that the universe
actually works this way and that therefore there is nothing more to
discover.

Like a repulsive gravitational force that is not inverse square.

The Small Kahuna