This expectation has the form of a partition function or Euclideanized path integral. Here the dynamics given by f (Fi, Oj,) enters into the path integral, and the stochastic term acts as a diffusive term in this path integral. This indicates the function is explicitly evaluated using renormalization group techniques. This partition function is analogous to that in the Ising model [14.10], but here instead of a set of spins that exist in space there are stochastic kicks that exist in time. These stochastic kicks are assumed to be on average the same, which pertains for the orbit outside of a strong resonance condition.
This path integral can be demonstrated to be similar to the Ising model. For the variation in the stochastic variable 5£j = j — £j_i the product of any two variations vanish S£ii 5£j ~ 0, so that
for i = j the sum of these stochastic variables is
This means there exist additional "endpoint terms" which do not conform to the Ising type of construction. However, for a large enough n this error should be minimal. The expectation is approximately
for ¡3 = 1/sa2. ¡3 is analogous to the Boltzmann factor, but is here a constant is fixed by time the iterated map is run on a computer. As the partition function is invariant under rescaling of 3, an approximate renor-malization of the coupling constant is
Here 3 is a pseudotemperature that gives the time scale for the system. A correlation time or length scale exists that is t « 1/log(tanh 3). (14.21)
For large 3 this correlation time scale will exhibit a divergence. Physically this indicates that for large 3, or long run times, the correlation length becomes very large and that the planet (Earth) may then end up in a region that is exponentially far removed. This suggests that the Earth will after a time either be expelled from the solar system, absorbed into Jupiter, or its orbit may be highly perturbed so that it collides with the future white dwarf remnant of the Sun. It is impossible to predict which of these ultimate fates are in store for the Earth.
As a cautionary note, such renormalization procedures and decimations are not exact. For a repeated series of decimations the error due to the removal of the endpoint terms may cause difficulties if they are large. However, here it is assumed that for a sufficiently large Ising chain these contributions are negligible if the magnitude of each £3 is within some reasonable bounds so and £n are not significant contributors to the over all chain.
Physically this suggests that the correlation length is then related to a Lyapunov exponent. The Lyapunov exponent for an iterated map is
For an iterated map with a coupling strength r the interval r — r,, for r, the parameter for instability, the exponent is then
where 7 = log 2/ log S is a critical exponent. By adjusting r and S so that
this defines the boundary between stable and unstable behavior. If the parameter r is identified as the correlation length (time), thinking again of this parameter as analogous to temperature and T, as the critical point, the connection between a Lyapunov exponent and the correlation length is apparent.
Based upon the above Fourier transform and the onset of 1/f noise the time for criticality is on the order of 2.09 x 10-11 sec or 6600 years. A criticality for a putative planet at 1AU an approximate time interval of drift due to extrasolar gas giant planets exists. It is t' ~ (M/M3)((r3- — R)/(r — R))2t, where M is the mass of a known extrasolar gas giant planet, M3 is the mass of Jupiter, rj is the orbital radius of Jupiter, R = 1AU is orbital radius of the Earth, and r is the average orbital radius of an extrasolar gas giant planet under consideration. Similarly the parameter 3 for stochastic behavior is also scaled. This may also be applied to consider the relative influences of Venus and Saturn on the stochastic motion of Earth. This approximate formula gives different correlation times for these planets. The space correlation function for Jupiter doubles at 3 times the rate as a similar function for Venus, and 1.6 times that for Saturn. This leads to an estimate for 1/f behavior due to Venus occurs on a scale of ~ 20,000 years and for Saturn on a scale of ~ 10, 500 years. So the influence of these planets on the stochastic motion of the Earth is considerably less.
This discussion indicates that the Earth is in a position that may be relatively unique. The evolution of the solar system may be such that the Earth maintains a fairly constant input of solar radiation during the duration of the Sun as a main sequence star. This likely is important for the evolution of life. The earliest microfossils of life are 3.5 billion years old. The oxygen revolution that started 1.5-2.0 billion years ago and the Cambrian explosion of life 600 million years ago may have occurred under global temperature conditions that are not significantly different than those today. This suggests that the biosphere is not simply an emergent process unique to the planet Earth, but to the configuration and evolution of the entire solar system.
Speculations about life on other planets have been popular since Scar-parelli claimed to see canalli on the surface of Mars. While life has not been completely ruled out elsewhere in the solar system, it is now almost certain that no planet in the solar system has anything similar to the vast biosphere on Earth. Real information has largely pushed speculation on the existence of life onto extrasolar planets. It is reasonable to argue the star about which another bio-active planet orbits should be similar to the Sun, a G class main sequence star. A more luminous and massive star would exhaust it fuel too quickly, and a smaller star would require the planet to be much closer to the star where its rotation would end up tidally locked with the planetary orbit around the star. Yet main sequence stars evolve by burning hydrogen in a shell that migrates outward from a core of helium, and the star heats up over time. A planet that starts out in the right position for the evolution of unicellular life would then heat up within a 1 — 2 billion years and cease to be biologically active without the right gravitational environment established by gas giants. So it poses a question on what types of solar systems might sustain a bio-planet, and what is their frequency of occurrence.
Of course this question is framed in the context of biology as we understand it. This is carbon based biology with the requirements of water in the liquid phase, which should occur under temperatures and pressures comparable to those on the Earth's surface. Speculations of silicon based or alternative chemically based life are not considered here as there exists no information concerning them.
Since 1995 the observation of extrasolar planets has become nearly routine [14.11]. Most of the planets observed are generally rather massive gas giants. So far terrestrial planets are too small to be adequately detected. Yet a survey of these extra solar systems is rather disappointing. Many stars have gas giants in close orbits with periods on the order of 5-20 days to around 1 Earth year. This likely precludes the very existence of terrestrial planets. One possible candidate so far to possess an Earth-like planet with a solar system environment similar to our own is Epsilon Eridani. A gas giant with a mass of .86 times that of Jupiter and an orbital period of 6.85 years has been found. e Eridani is a K2 class star with a mass around .7 that of the Sun [14.12]. This would then put the gas giant at approximately the right position relative to a potential bio-planet situation closed to this star ~ 5.0 x 107km. However, tidal locking of such a planet's rotation may prevent any biologically activity on such a planet. Further, the larger perturbation on the orbit of a terrestrial planet due to a gas giant may make the prolonged occurrence of life impossible. So as a candidate for the existence of life e Eridani is probably marginal at best. The recently discovered extra solar system HD 72659 is another potential candidate. The star is classified as at G0 V star with a slightly smaller mass than the sun with 14% less in heavier elements. This star is 167 light years away, which puts it at the extreme range of a possible interstellar probe.
It appears questionable whether there exists an Earth-like planet around any of these stars. It might be best to look for an alternative prospect for the occurrence of a biologically active planet. Ironically a possible answer may lie with the Star Wars series of movies. They posit the existence of two biologically active planets that orbit gas giants. Of course gas giants tend to have magnetic fields that trap a considerable amount of high energy charged particle radiation, and a putative terrestrial planet would require a strong magnetic field to shield it. HD 177830 contains a gas giant at an orbital radius comparable to the Earth's orbit. However, this is a K0 star, which means that conditions at this distance from the star are likely to be quite cold. HD 28185 also possesses a gas giant at this radius, and as it is a G5 star temperatures conditions may be right for a possible biologically active moon around this gas giant. However, there is no other gas giant that would perturb the orbit of this identified gas giant to compensate for the increased heating of the star over its lifetime. However, as this orbit is probably comparative stationary this means that the time window for the evolution of life on such a putative moon is narrower than the time that Earth has possessed life.
The stars u Andromadae and HD 82943 might be a candidate. If a terrestrial planet orbits u Andromadae b it might be positioned properly for the existence of life. However, as this is an F class star this distance may be outside the boundaries for an a biologically active planet. Temperatures on a terrestrial planetary moon around this gas giant may simply be too high. Yet for a system similar to this one the two other gas giants may act to keep this second gas giant positioned within a radius appropriate for the sustenance of life on a terrestrial planet through the evolution of u Andromadae. It is doubtful that u Andromadae b exists in an orbit that is sufficiently stable. HD 82943 has two gas giants b and c, which interact very strongly with each other. Estimates their Lyapunov exponents to that of Earth AE for these gas giants around e Andromadae and HD 82943 are 6.3AE and 7.1AE respectively. The measurements of e Eridani are very noisy as this star exhibits variations which make the measurement of this gas giant somewhat problematic. Yet it is possible that a second gas giant exists with a smaller radius around epsilon Eridani with a terrestrial planet. However, the prospects for this type of configuration appears less than likely.
A general analysis of the ratio of the drift times and Lyapunov exponents for putative terrestrial planets around each of the extrasolar systems with that of the Earth is now discussed. The relevant parameter are a ratio of these exponents for each putative terrestrial planet with an orbital radius of 1AU with those of the Earth. Current extrasolar planetary results are given in appendix I. The Lyapunov exponents for the orbit of a putative terrestrial planet are in most cases considerably larger than that of the Earth. In many cases the drift rate is 2 orders of magnitude larger than that for the Earth. These analyzes only involve the radial shift of a putative terrestrial planet, and does not include the role of the eccentricity in the orbits of these orbits. In most cases these eccentricities are larger than the orbits of planets in our solar system. This will most likely reduce the prospects for stable orbits of terrestrial planets that could contain life. As such the prospect for a biologically active planet around any of these stars is highly problematic.
A Bayesian estimate for the occurrence of an Earth-like planet may be found at this point. This analysis can only realistically be done to estimate the number of potential orbits comparable to that of the Earth which may exist. Situations, such as with HD 28185 are difficult to study. It is not clear whether a terrestrial planet can form around a gas giant situated ~ 1 AU around the star. Let A be the potential occurrence of a terrestrial planet in an orbit comparable to the orbit of Earth. A indicates that a stable orbit of such a planet occurs. Let B be the occurrence of gas giants that permit the stable orbit of an Earth-like planet. Further, A and B are the complementary occurrences. Then P(A|B) is the probable occurrence of A given B, P(A) is the probable occurrence of A, and is the Bayesian prior estimate, P(B|A) is the probability of B given A. Bayes theorem gives the equation
To start with, a Bayesian prior estimate is used, which states that out of 145 solar systems examined, only our solar system has an Earth-like planet, P(A) = yl/145 ~ 0.083. This is admittedly a Bayesian prior fraught with uncertainty, but is currently the best possible. A much larger data set would be required, and would be bolstered if more extrasolar systems are found which have the proper orbital dynamics of gas giants that can permit the existence of an Earth-like planet. So far outside of our solar system there are only two possible candidates. Yet for the purpose of being conservative these are not included in the Bayseian prior. To find P(B\A) we put the numbers of putative planets with certain range of Lyapunov exponents into bins. A histogram of Lyapunov exponents relative to that of the Earth is presented in Figure 14.6.
To derive a probability distribution for these now sum the number of stars between the Lyapunov exponents in the range .9XEarth-1.1XEarth this gives a probability of 0.0138. Further we must then factor in the fact that many of these correspond to inward drift. Ignoring the divergent results only 0.27 of these correspond to outward drift. This is then used to obtain P(B\A) ~ 0.0037. Further, since P(B\A) < P(B\A) ~ 1. we may then approximate the Bayes equation as
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