Solar System Stability and the Likelihood of Earthlike Planets

Stellar evolution predicts that the Sun will increase its radiation output over a time frame of hundreds of millions to billions of years. This poses an academic question concerning the long term fate of the Earth. If the Earth remains at its current orbital radius over the next one billion years then by the end of that time temperatures on the Earth's surface will become intolerable for the continuation of life. It has been suggested that "planetary engineering" might be able to prevent this by shoving the Earth into an orbit with a larger radius. However, it is argued here that this is not required as the three body interaction of the Sun, Jupiter and Earth will over time cause an outward drift in the Earth's orbit. Further, this drift exhibits 1/f behavior and is an indication of a degree of chaotic dynamics in the solar system. This drift on average compensates for the increased heating of the Sun. It is possible that this process has occurred through the existence of the solar system. This remarkable property of our solar system is used to examine what is known about extrasolar systems, and the question on the existence or frequency of occurrence of life on extrasolar planets.

This chapter engages in some rather advanced work with chaos theory in classical mechanics. For those unfamiliar with physics at this depth it is advised you skip over the detail and focus on the results

Korycansky, Laughlin, and Adams [14.1] reported that the orbit of the Earth could be modified through planetary encounters with comets whose orbits are directed into the solar system. It is proposed that energy associated with the orbit of a comet could be transferred to the Earth, and that the comet could then pick up this lost energy through a second orbital encounter with either Jupiter or Saturn. The motivation for this analysis is the fact that the radiation output of a main sequence star increases

during its stable existence. In one billion years the 10% increase in the solar radiation will push surface temperatures of the Earth to the boiling point of water, where the Earth's surface environment will shift into one similar seen on the planet Venus. Outside of possible thermophylic prokaryotes (Archeobacteria), the Earth will cease to be the biologically active planet it currently is.

The proposal advanced by Korycansky, Laughlin, and Adams is the Earth's orbit under repeated close encounters with a mass 1022g) on a highly elliptical orbit with a semi-major axis ~ 300 AU every 6000 years may be sufficiently shifted outward over a one billion year period to avoid the over heating of the planetary surface. The 10% increase in radiation output would be compensated by adjusting the orbit of the Earth so that its orbit would exist at 1.05 AU over a billion year period.

From a practical point of view it is highly problematic that such a program will ever be undertaken. The lifespan of more successful civilizations is 500-1000 years, the duration of nation states and empires is often shorter < 500 years, and the duration of the average mammalian species is 3 million years. Further, long term programs are at their longest around a century, as seen with cathedral building during the high middle ages in Europe and monolithic constructions during more ancient periods of history. The modern world engages in much more short term programs than did the ancient world. A long term program on a large scale undertaken in modern history was the 10 year program to put astronauts on the moon, which represents the longer time frames for programs in our age. Thus the long term fate of the Earth most likely does not rely upon humanity to devise such a long term planetary engineering scheme.

However, nature may provide the answer already. During the evolution of proto-life on a molecular level that the Sun would have heated the Earth by approximately 70% the current temperature. This would put the average temperature of the Earth some 3.5 billion years ago at around — 60°C if the Earth were at its current average distance from the Sun. Given that the Earth had a CO2 atmosphere up until 2.0 billion years ago the early temperature of the Earth would still be below freezing, ~ —30°C if we assume that the atmospheric pressure was comparable to today's. This has raised debates over the nature of how life evolved out of basic molecules. However, if the Earth were located at .83 AU at that time the temperature of the Earth's surface would be comparable to current average temperatures.

This drift in the average radius of the Earth is a process that is common to 3-body problems. In a 3-body problem it is expected in general that one of the masses will be expelled from the system. This is the so called resonance problem. For a 3-body problem with one huge mass, an intermediate mass and a small mass this process is likely to occur over a long period of time. It is illustrated below that the radial drift in the Earth's orbit and the stochasticity involved are an integral part of a maintaining the Earth at a radius that is appropriate for the continuation of life, presumably for the next 2.5 billion years.

Consider the simplified model of the solar system that includes only the Sun, Earth and Jupiter. The gravitational interaction is the classical Newtonian law of F = GMmr/r3. This is a three body problem that in general is not integrable. In this model some simplifications are imposed. The orbits of the Earth and Jupiter are considered to be on the same plane. Further, the small motion of the Sun is ignored. In addition, the initial orbits of Earth and Jupiter are assumed to be circular in order to avoid the ignore perturbation of perihelion advance. In this way perturbations that involve the radial position of the Earth can be exclusively examined. The problem is coded fairly simply and run for 48,000 Earth orbits. The interaction between Earth and Jupiter perturbs the solar motion of the Earth as a time varying harmonic oscillator force.

It is apparent the Earth wobbles about its Keplerian orbit by a significant percentage of the Earth's radius. On a small time scale this wobble appears comparatively regular. One can consider this wobble as due to a harmonic oscillator in a "first order analysis." The gravitational interaction between Earth and Jupiter is

First, as an approximation by a binomial expansion, the gravitational force between Jupiter and Earth is,

j where r are unit vectors. With F = me(rj — re) we find that the motion of the Earth's orbit away from the Keplerian orbit due to the motion of Jupiter is rj rj which is the differential equation for a harmonic oscillator with a driving force. The frequency of the oscillation from the solution to the unforced differential equation is then seen to be

which is a period of 16.7 years. This reflects the periodicity of Jupiter's orbit multiplied by a factor of a/2. The solution to the inhomogeneous differential equation effectively divides this by a/2 to give a period of 11.8 years, which is approximately the period of Jupiter's orbit. Since the orbit of Jupiter has a period that has an irrational ratio with the frequency of the Earth's orbit, this Earth's wobble motion irrationally winds around the tori that defines the energy surface in phase space. The solution to this wobbling is illustrated in Figure 14.2.

Now consider the long term evolution of the Earth's orbit. If the motion of the Earth remains on the energy surface the invariant tori is then preserved. This is the essence of the Kolmogorov, Arnold, Moser (KAM) theorem. However, if there is a drift in the motion of the Earth, the tori is then punctured and the motion of the planet can not be described by an equation that is an integrable solution to the equations of motion. The j

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Fig. 14.2. Perturbative wobble of Earth's orbit due to Jupiter.

Years 0 5 10 15 20

Fig. 14.2. Perturbative wobble of Earth's orbit due to Jupiter.

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Fig. 14.3. Radial drift of the Earth's orbit over 45,000 years.

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Fig. 14.3. Radial drift of the Earth's orbit over 45,000 years.

numerical result for the long term motion of the Earth is then illustrated on Figure 14.3.

This suggests that over a sufficiently long time period perturbations exhibit puncturing of the KAM surface. With these estimates it is possible to compute the mean drift of the Earth's orbital radius, and further find the average drift outward per year. The drift of the Earth's orbit is then ~ 4.2m away from the Sun annually. Over a billion year period this drift will amount to ~ 4.20 x 106 km for the data extrapolated in a linear manner. This would put the Earth at a distance of ~ 1.028 AU in a billion years. This means that for a "constant Sun" the flux of solar radiation would be reduced by ~ 94.6%. If a 10% heating of the Sun is assumed over the next billion years for the Earth at its current orbital radius the increased heating of the Earth's surface will increase by ~ 4.1%. Based upon this the catastrophic heating of the Earth would be pushed back to approximately ~ 2.5 billion years.

The heuristic inclusion of the inward pull of planet Venus the average pull is .21% of the outward pull of Jupiter. This would then reduce the possible biological future of the Earth to 2.05 billion years. One might further then consider the influence of the planet Saturn on the perturbation of the Earth, which might in some manner increase the future biological life of Earth. Of course at this point we are entering an arena of more and more uncertainty. The standard deviation of these data is ~ ±2.5 m motion annually. This puts the future age of the Earth within the range of 1.15 billion years or until the Sun finally ceases to function as a main sequence star in 5-6 billion years. In the outer range of possibility the solar heating of the Earth will decline by .96% per billion years which, means that the Earth will experience an overall cooling during its 5-6 billion years future.

If there is some overall trend in the climate of the Earth due to its radius from the Sun this should be observable from geological data. The history of climatic conditions on the Earth are relatively well understood during the Cenozoic period. During the earliest Paleocene and Eocene periods the Earth appears to have been comparatively warm, and the apparent absence of large ice sheets in the polar regions [14.2]. During the Oligocene period there appears to be the accumulation of ice on the Antarctic continent. It is most likely that these thermal or climatic fluctuations in Earth history are due to changes in the position of continents from tectonic motion and changes in the albedo of the Earth that had a positive feedback on the climate: the more ice sheets on the Earth the larger the planetary albedo and hence less solar radiation absorbed into the atmospheric and hydro-logical system of the planet. On a longer time frame it appears that since the Cambrian period that the average temperature of the planet has only varied between 12°-22°C, where currently the Earth's temperature is at the low end of this scale. This is a period of 400 million years [14.3]. If the Earth's radius from the Sun were constant it might be expected that there would be a 4% drift upward in the average temperature of the planet. This is not apparently observed.

It is then reasonable to conjecture that the average orbital radius of the Earth has not been constant in the past. If the average global temperature has been constant over the last 3.5 billion years then the orbital radius of the Earth was .83AU. This implies an average drift during this time period o

0 25 50 75 100 125 150 175 200 225 250

Units of frequences Unit frequency = 1.91 x10 sec

0 25 50 75 100 125 150 175 200 225 250

Units of frequences Unit frequency = 1.91 x10 sec

Fig. 14.4. Fourier transform of the radial drift in the Earth's orbit.

of 3.56 meters over each year period as measured currently. This is within these estimates of the current outward drift of the Earth's orbital radius.

This matter leads one into the issue of the stability of the solar system. This was found to be a problem early on in the development of classical mechanics. Newton in his Principia wrote on his considerable work on the three body problem of the Sun, Earth and moon. Therein Newton found that the problem of lunar motion appeared to defy capture by an analytic solution for all time. Newton even remarked that God may have to occasionally intervene to keep the entire solar system, which at that time was not known to contain Uranus, Neptune or Pluto (now demoted from the status of a planet), in a stable form. Later the problem of motion stability of planets was examined in 1773 by the mathematical mechanicians Laplace [14.4] and Lagrange [14.5]. This entailed that compound differential equations of motion of planets take into account all disturbances and interference, which amount to around 20000. This also required that resonances in orbital periods and cyclic inequalities in orbits be examined to determine whether the system is unstable or not. The effort by Laplace and Lagrange determined the stability of the solar System within a first order approximation. From a fundamental point of view this is an insufficient answer. It must also be mentioned that these efforts in theoretical mechanics were accompanied by the work of astronomers, opticians and instrument makers that permitted ever refined observations and records on planetary motion.

Fig. 14.5. Orbital drift of the Earth in phase space. The first diagram is a Poincare section and the second is the flow of these points. The third diagram is the log of drift frequencies.

Henri Poincare [14.6] and Lyapunov [14.7] took up the challenge with the derivation of a mathematically stringent and consistent theory of stability of motion. These developments involve the theory of irregular separatrices in phase space, Poincare sections in phase space, and the introduction of Lyapunov's exponent as a measure of how two orbits that are initially arbitrarily close in phase space will exponentially diverge in a finite time. Poincare won the Sweden prize offered for demonstrating solar system stability by illustrating how such stability can not be proven. In the 20th century the mathematician and mechanician V. I. Arnold [14.8] worked to solve the problem of solar system stability. Arnold worked to illustrate that regular orbits of planets in a solar system satisfy the conditions of the KAM theorem. Here with irrational ratios of orbital periods orbits in phase space exist on a torus with irrational winding, that are called the KAM surface.

The irrational winding on a torus is defined by the limit of a continued fraction expansion of various frequencies on regular tori that approach the irrational winding. The KAM surface in a stable theory represents the absolute limit of various tori with rational windings and are associated with integrable solutions. The existence of the KAM surface is seen as the stability of a system with incommensurate periodicities. However, due to overlapping of KAM surfaces or puncturing these integrability conditions can be seen to be less than universal. The breakdown of a KAM surface can lead to motion that is not dynamically predictable, or chaotic dynamics.

There are indications of chaos in the radial drift in the orbital radius of the Earth. First consider a Fourier transform of the time data for this drift. These datum are easily run through a fast Fourier transform. It is then apparent from the data that for frequencies < 3.5 x 10-11 sec-1 a 1/f type of behavior occurs. This indicates on a time scale of ~ 9000 years there is a noisy processes that are exhibited in the dynamics of the Earth's orbit. This means that over this time period and longer there is a net drift in the orbit with a change in the radius in the range G (15 km, 60 km). This 1/f behavior indicates that this drift is an irreversible process, or one that erases some of the information concerning the Earth's orbit every 9000 years. This lead to the conjecture this indicates that the solar system is over a sufficiently long time period not stable, and that the Earth will continue to drift outward until it might be ejected from the solar system. The sun will enter its red giant phase in 5-6 billion years. It is possible that the Earth orbit will have a radius of 1.14 AU during the Sun's red giant phase. So it is likely that the Earth will remain within the solar system while the Sun is a main sequence star. Yet if the Earth survives the red giant phase, it will likely end up in a complex orbital relationship with Jupiter, or be ejected from the solar system altogether within a ~ 1011 year time period.

This 1/f result is one indication that there is a stochastic process involved with the drift of the Earth's radius from the Sun. Such 1/f behavior is associated with noisy processes in electronics and is also existent in other systems. Another signature of chaos is the behavior of a trajectory as it intersects a Poincare section. This was included in the program, with the plane defined by y = 0 and py = 0 as the coordinate for the intersection of the Kepler orbit. The two graphs below illustrate the Poincare section for 1000 intersections of the orbit with the Poincare plane, and the second illustrates the first 100 intersections with the lines connected. This plot involves position and velocity which deviate from Kepler's law. It is apparent that there is utterly no apparent structure to the intersection of this orbit with the Poincare section. This gives further indication that the drift in the radius of the Earth is chaotic. It appears that for the initial conditions chosen that the dynamics of the drift in the Earth's orbit is determined by a punctured KAM surface.

The logarithm of the low frequency end of the behavior indicates a constant slope in a log-log plot. It is known that 1/f noise can be produced through intermittency. Intermittency is where a process that is regular or laminar will experience short periods of irregular behavior. For a scaling parameter small enough the system will exhibit laminar behavior. Then if this parameter is scaled to a sufficiently large value the intermittency is one possible route towards a general mechanism behind 1/f noise. Inter-mittency was investigated as one route to the understanding of stochastic behavior with the Lorentz equations [14.9]. Intermittency is further a property of iterated maps such as the logistics map.

It appears the orbit of the Earth drifts outward from the Sun significantly over a time period of ~ 107 years. Further it appears that this radial drift exhibits stochastic behavior over a large time scale, and where over a larger time scale the drift averages 4.2 km a year. This may be attributed to the near resonance condition that exists between the orbit of Earth and Jupiter where their orbital frequencies exhibit the ratio

/u>e = 11.8. There is the well known vanishing denominator problem for H(J, 9) = H0(J, 9) + eH1(J, 9) for J = (J1, J2) 9 = (6U 92). Here the generating function written according to the variable J'

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