The orbital celestial equations of motion are a set of nonlinear ordinary differential equations with initial values.
Nonlinear ordinary differential equations are analyzed using bifurcation theory.
The hallmark of this analysis is the sensitive dependence on initial conditions.
'As Poincare experimented, he was relieved to discover that in most of
the situations, the possible orbits varied only slightly from the initial
2-body orbit, and were still stable, but what occurred during further
experimentation was a shock. Poincare discovered that even in some of the
smallest approximations some orbits behaved in an erratic unstable manner. His
calculations showed that even a minute gravitational pull from a third body
might cause a planet to wobble and fly out of orbit all together.'
Even measuring initial conditions of the system to an arbitrarily high, but finite accuracy, we will not be able to describe the system dynamics "at any time in the past or future". To predict the future of a chaotic system for arbitrarily long times, one would need to know the initial conditions with infinite accuracy, and this is by no means possible.
The Hamiltonian formulation of this set of nonlinear ordinary differential equations (mechanical system without friction) leads to the famous KAM theory.
Two of the greatest Soviet mathematicians of the 20th century, A.N. Kolmogorov and V.I. Arnold asked the following question: to what extent the geometric structure of the quasi-periodic dynamics of a Hamiltonian system persists under small perturbations that destroy the toroidal symmetry?
This led to the famous KAM theory (Kolmogorov-Arnold-Moser); however, it is valid for "sufficiently" small perturbations.
In reality, the perturbations in the solar system are far too large to apply KAM theory: the conditions of the KAM theorem are too strict to apply to a realistic solar system.
So, the mathematicians have to rely on computing Lyapunov exponents, in order to try to predict any region of instability/chaos.
Even in this case, the measured Lyapunov exponent may have no relation to the true exponent: great care has to be taken in computing such quantities.
In 1989, J. Laskar proudly announced that the exponential divergence time for the solar system is 5 million years.
However, again, this calculation DOES NOT take into account the sensitivity of the results due to uncertainties of the knowledge of true masses and the INITIAL CONDITIONS of the planets.
Jack Wisdom (MIT): It is not possible to exclude the possibility that the orbit of the Earth will suddenly exhibit similar wild excursions in eccentricity.
The exponential divergence of chaotic trajectories precludes long-term prediction given the limited knowledge of the state of our solar system.
Lyapunov exponents and symplectic integration.
Let d(t) be the distance between two solutions, with d(0) being their initial separation. Then d(t) increases approximately as d(0)e
λt in a chaotic system, where λ is the Lyapunov exponent. The inverse of the Lyapunov exponent, 1/λ, is called the Lyapunov time, and measures how long it takes two nearby solutions to diverge by a factor of e.
Since the solar system is not integrable, and experiences unpredictable small perturbations, it cannot lie permanently on a KAM torus, and is thus chaotic.
Sussman and Wisdom's 1992 integration of the entire solar system displayed a disturbing dependence on the timestep of the integration (measurement of the Lyapunov time).
Thus, different researchers who draw their initial conditions from the same ephemeris at different times can find vastly different Lyapunov timescales.
Wayne Hayes, UC Irvine
To show the importance and the dependence on the sensitivity of the initial conditions of the set of differential equations, an error as small as 15 meters in measuring the position of the Earth today would make it impossible to predict where the Earth would be in its orbit in just over 100 million years' time.
A difference in the initial position of 1 cm grows to ∼1 AU (= 1.496 x 10^11 m) after 90–150 million years.
Let us take a closer look the chaotic dynamics of planetary formation; thus, a clear indication that the initial conditions cannot be predicted with accuracy (as we have seen, a mere 15 meters difference in the data will have catastrophic consequences upon the calculations).
OFFICIAL SCIENCE INFORMATION
Four stages of planetary formation
Initial stage: condensation and growth of grains in the hot nebular disk
Early stage: growth of grains to kilometer-sized planetesimals
Middle stage: agglomeration of planetesimals
Late stage: protoplanets
For the crucial stages, the initial and early stages, prediction becomes practically impossible.
As if this wasn't enough, we have absolute proof that in the age of modern man planet Earth underwent sudden pole shifts (heliocentrical version), thus making null and void any integration of the solar system/Lyapunov exponents calculations which do not take into account such variations of the system's parameters:
http://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1635693#msg1635693http://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1546053#msg1546053Let me show what sensitive dependence on initial conditions means, using one of the most famous examples: the Lorenz attractor butterfly effect.
In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.Here is the set of Lorentz equations:
Now, the set of differential equations which describes the planetary orbits is much more complicated than this.
Numerical expressions for precession formulae and mean elements for the Moon and the planets: http://adsabs.harvard.edu/abs/1994A&A...282..663SLaskar's calculations are useless without specifying the initial conditions and parameters of the equations:
https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1774581#msg1774581The best that modern astronomy can claim is a 300 year limit on the stability of the solar system, using Newcomb's series:
https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1935048#msg1935048And that hypothesis does not take into account the fact that the initial conditions cannot be specified at all for the orbital equations in question.