News and Views
Comments about the article in Nature: The solar System's extended shelf life
Following is a discussion about this "NEWS & VIEWS" article in Nature Vol 459 11 June 2009 by Gregory Laughlin
For a copy select:The solar System's extended shelf life
 The text in italics is copied from the article
 Immediate followed by some comments
In the last paragraph I explain my own opinion.
The solar System's extended shelf life
In the first half of the header of the document we read:

Simulations show that orbital chaos can lead to collisions between Earth and the inner Planets.

I have a two problems with that sentence.
 Orbital chaos is not the cause that are collisions possible between the planets in the Solar system. Collisions are a fact of live in the future but also in the past.
 The issue is that we can not accurately simulate the future (and past) of the solar system.
It should be mentioned that here is meant: Simulations using Newton's Law.
In the second half of the Header we read:

But Einstein's tweaks to Newton's theory of gravity render these ruinous outcomes unlikely in the next few billion years

This sentences raises two questions:
 Why the word "ruinous" ? If there are collisions between the planets, let they happen.
 When you read the next of the document it becomes clear that the authors have performed two types of simulations:
 One set of simulations with Newton and one set with Einstein.

The results are that using Einstein there are much less collisions between the planets. This should be explained in detail why we can rather accurate simulate the positions of the planets in the Next 1000 years and because the difference in that prediction using Einstein is so small.
i.e. the 43 arc sec per century angle in Mercury.
Next we read in the text:

Their work shows that the orbits of the terrestial planets have a roughly
99% chance of maintaining their current wellordered clockwork for the next 5 billion years etc

This sentence gives the same information as discussed above.
Next we read:

The constant interplay of gravitational attractions between planets acts to degrade their
repetitive predictable motions

The importance of Newton's Law by introducing forces (gravitation) is that we can predict the movement of the planets compared with the time before Newton.
You can not make any statement about a physical system, neither that a system is predictable or repetitive.
The fact that we can not accurate simulate the movement is because the issue is much more complex.

Over time, a system of orbits can become increasingly disordered and planets
might fling each other out into space or into their parent star or collide with each other.

It is wrong to call this phenomana disorder. It is normal
The issue is that in principle als those things can happen. The question which of all those phenomena will actual happen.

By the 1850s it was recognized that the higherorder terms in the planetary "disturbing function" could not be neglected,
and consideration of these terms revived the question of orbital stability.

The issue is not stability but predictability.
More terms mean better predictability.

In 1889, Henri Poincare demonstrated that even a threebody problem cannot be solved by analytic integration,
thereby eliminating any possibility than an analytic solution for the entire future motion of the eight planets could be found.

What this means accordingly to HP is that we cannot express the position of the planets with any function of t. Let this be.

Poincare's work anticipated the nowfamiliar concept of dynamical chaos and the sensitive dependence of nonlinear systems on initial conditions.

See Later

Orbital predictions etc demonstrated that the planetary orbits will indeed become chaotic,
with typical Lyapunov times  the time required for chaos to significantly degrade the predictability of a system  of the order of 5 million years.

The only thing that you can say is that the predictability of the system decreases the longer you perform your simulation.
The reasons are:
 the accuracy of your observations,
 the model used (Newton versus Einstein)
 the accuracy of your PC both hardware and software i.e. implementation
 constraints (The size of the objects involved)
It is true, if your model is Newton's Law, that the outcome of two close sets of initial conditions will divergence.
That means that the positions at a certain moment t wil differ.
This becomes the most obvious the larger the range of distances that are involved. Specific between close encounters.
What this means for the Solar System depents on the objects involved.
 If the objects are only planets than the predictability is high.
 If the objects include planets and asteroids (even if they are considered round) the predictability is low.

Statements regarding stability of the Solar System must therefore be expressed in terms of probabilities.

Why ? What is the definition of stability ?
Stability is not the issue.
Is the solar system now stable ? Was it stable 2 billion years ago ?. Will it be stable 2 billion years from now ?
IMO the answer on all those questions is Yes
IMO the most important event is when two planets collide (Or a planet collides with the Sun). The chalenge is to predict that moment as accurate as possible.

They report highly detailed numerical simulations of the evolution of the whole Solar System using the most accurate available planetary ephemerides
(a table of the precise positions and velocities of the planets at a specific time)

From a pure scientific point of view this is a wrong method.
 Starting point should be a set of observations. Those observations are the positions of the Sun and Planets over a long period of time.
 The next step is to select an algorithm which decribes the movements of the planets and which can be used to calculate the positions of the planets at any moment tx.
 Together with the algorithm you establish the parameters involved
When the algorithm used is Newton's Law those parameters are the positions, velocities and masses of each of the planets and the Sun.
 Next you make an estimate of the initial values of these parameters required at a certain moment t0
For the initial values of positions and velocities the ephemerides can be used.
 Using the algorithm, the initial values and the observations you improve the values of those parameters.
 When the error between the calculated positions and the observations is the smallest the first phase is finished. The result is a final set of parameters at a certain moment t0
 The second phase is to calculate the future behaviour of the solar system based on those final parameters and the selected algorithm.
The important point is that each algorithm in effect requires it own set of initial values.
That means the initial values are different using Newton's Law compared to Einstein's

The simulations indicate that Mercury poses the greatest risk to the present order.

I expect the greatest risk to the predictability of the solar system i.e. that it will collide with an other planet or the Sun.

But the odds of Mercury entering a secular resonance are greatly reduced bij the small modifications that Einsteins theory of relativity
imparts to the planetary motions.
Famously 43 arceconds per century of Mercury's total precession is due to the effect of general relativity.

The total precession is 573 arcseconds per century of which 531 can be explained by Newton's Law.
The task of simulation the solar system using the Newton versus Einstein is totally different.
Using Einstein is much more complicated.
As I stated above using Einstein requires a different set of initial parameters including the masses of the planets and the Sun.
My estimate will be that the outcome of an simulation over a range of 1 billion years can be quite different, including the time that Mercury will cease to exist.

This correction effectively detunes the MercuryJupiter interaction and
decreases the chance that resonance will occur in the next 5 billion years to roughly 1%

Newton's Law is a very powerfull tool (may be not perfect) to simulate the behaviour of the planets in the Solar system.
The whole question is to what extend that Law can be used in order to make a prediction of the state (the positions) of the planets based on
current observations over a period of 1 million years
IMO very little because much more is involved. The issues are accuracy. Accuracy of the observations and accuracy in computer calculations one of which be the step size.
Over a period of 100 Million years ? IMO less.
Over a period of 5 Billion years ? IMO nothing
The same with Einstein's method ie General Relativity
Letters
Comments about the article in Nature: Existence of collisional trajectories of Mercury, Mars and Venus with the Earth
By J Lasker & Mgastineau

Owing to the chaotic behaviour of the Solar System, the distance between two initial close orbital solutions increases by a factor of ten every ten million years.

It is true from a mathematical point of view that if you perform a simulation of the Solar System using Newton's Law that and with two sets of initial conditions that the two outcomes will be different (will divergence).
This divergence will be larger the further away you want to look into the future (and is ofcourse also a function of the difference between the two sets of initial conditions)
But that does not mean that our Solar System is chaotic.

It is thus hopeless to search for a precise solution for the motion of the Solar System over 5 Gyr

IMO the fact that there is no precise solution I think that does not worry most people. This is also not the issue.
The issue is to try to predict the state of the Solar System over 5 Gyr as accurate as possible.
This is not easy.
The four reasons are: The observations, the calcultion model or method used (Newton versus Einstein), accuracy and constraints.
The accuracy of your observations and the accuracy of your computer.
From a mathematical point of view it is possible to find a precise solution for a system consisting of two objects. But does that mean that you can predict the state of such a system 5 Gyr in the future ?. The answer is no for the same four reasons.

The most precise longterm solutions of the Solar System are not valid over more than a few ten million of years.

What does the word valid means here ?
In fact the most precise solution (calculation) of the Solar System over 10 Myr has a lot of scientific value, because it should answer the question if we still have 8 planets.

A numerical integration etc over 5 Gyr can thus only be considered as a random sample of its possible evolution.

The whole question is does it make any sense to try susch long periods when the outcome over a period of 100 Myr already is questionable.

the step size is 2.5*0.01 years (which is equivalent to 788400 seconds) unless the eccentricity of the planets increases beyond 0.4, in which case the step size is increased.

IMO this step size is too large. What you have to do is try different step sizes for the whole period and observe if the outcomes are the same. They should.


Reflection
 In the Nature article a step size of 788400 seconds is used.
To test that I have performed 3 simulations of the solar systems (including 7 planets) with different step sizes of: 50 sec, 100 sec and 1000 seconds written in Visual Basic
For the results see here: Planet simulation in 3D
The Ephemerides are from the "Explenatory supplement to the astronomical almanac" Edited by P.K. Seidelmann page 304
What the results show is that only the 50 second step size can be considerd correct.
This raises serious doubts about the step size of 788400 seconds used in the Nature article.
 The idea behind the simulation is to observe the forward movement of the planet Mercury.
 For the results of the planet Mercury over a longer period (step size 50) see here: Mercury simulation in 3D .
 For the results of all the 7 planets See here: Planet simulation in 3D
 For the results of all the 7 planets over a period of 210000 years See here: Planet simulation in 3D
 If you want to do a simulation over 5 Gyr you must take the movement of the Sun through our Galaxy into account.
If you don't, the simulation is not real.
 If you take the movement of the Sun through our Galaxy into account you have to explain the details using both Newton's Law and Einstein.
In both cases you have to demonstrate the forward movement of the planet Mercury. (i.e. the famous angles of 531 and 573 arc second per century)
A specific issue is here accuracy because you have to into account very large distances (the distance of the Sun versus the centre of Our Galaxy) compared to very small distances (The distance of Mercury versus the Sun).
 To observe the results of a simulation through our Galaxy using Newton's Law see here: Galaxy simulation in 3D .
Newton's Law assumes that gravity acts instantaneous.
The simulation uses Newton Law but also takes speed of gravity propagation into account for four different values: 0 (instantaneous), 30000000, 90000000 and 100000000 km/sec.
 What the simulation shows is that the average distance of planet Mercury increases when gravity propagation is included.
This implies that under those conditions the solar system is less stable compared with clasic Newton's Law which assumes that gravity acts instantaneous.
I expect the same should be true using Einstein. Secondly the simulation shows how important the mathematics behind each simulation is.
This is much more important than the claim that the solar system is chaotic.
 One of the first persons to calulate the speed of gravity was Paul Gerber.
See: (1) Gerber's Gravity ,(2)
Effect of the Earth’s TimeRetarded Gravitational Field on Spacecraft Flybys and (3)
Die räumliche und zeitliche Ausbreitung
der Gravitation. Paul Gerber .
My understanding of Paul Gerber's mathematics is that he start with the Sun at rest.
The problem is that introducing with a Sun at rest in my simulations (including time retardation) there is no difference in the forward movement of Mercury when gravity acts instantaneous compared with a speed of gravity equal to c.
There is only a difference between the two when the speed of the Sun is not equal to zero.
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Created: 16 June 2009
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