[math-fun] GR: Solar system's future could be bumpy

FYI -- Mercury's orbit was perturbed by just .38 millimeters in the various simulations!! I heard a discussion of this article in which it was stated that general relativity actually provides a bit more stability for the solar system than simulations not including GR. Unfortunately, I don't have access to the article to see if it discusses this GR v. non-GR issue. It makes sense that GR helps to stabilize the solar system, since the ability of GR to "bleed off" information (e.g., perterbations/initial conditions) using gravitational waves provides a mechanism analogous to using heat radiation to stabilize simple mechanical systems. --- http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html Letter Nature 459, 817-819 (11 June 2009) | doi:10.1038/nature08096; Received 17 February 2009; Accepted 22 April 2009 Existence of collisional trajectories of Mercury, Mars and Venus with the Earth J. Laskar1 & M. Gastineau1 Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France Correspondence to: J. Laskar1 Correspondence and requests for materials should be addressed to J.L. (Email: laskar@imcce.fr). Abstract It has been established that, owing to the proximity of a resonance with Jupiter, Mercury's eccentricity can be pumped to values large enough to allow collision with Venus within 5 Gyr (refs 1–3). This conclusion, however, was established either with averaged equations 1, 2 that are not appropriate near the collisions or with non-relativistic models in which the resonance effect is greatly enhanced by a decrease of the perihelion velocity of Mercury 2, 3. In these previous studies, the Earth's orbit was essentially unaffected. Here we report numerical simulations of the evolution of the Solar System over 5 Gyr, including contributions from the Moon and general relativity. In a set of 2,501 orbits with initial conditions that are in agreement with our present knowledge of the parameters of the Solar System, we found, as in previous studies 2, that one per cent of the solutions lead to a large increase in Mercury's eccentricity­an increase large enough to allow collisions with Venus or the Sun. More surprisingly, in one of these high-eccentricity solutions, a subsequent decrease in Mercury's eccentricity induces a transfer of angular momentum from the giant planets that destabilizes all the terrestrial planets 3.34 Gyr from now, with possible collisions of Mercury, Mars or Venus with the Earth. Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France Correspondence to: J. Laskar1 Correspondence and requests for materials should be addressed to J.L. (Email: laskar@imcce.fr). --- http://www.usnews.com/articles/science/2009/06/11/solar-systems-future-could... Solar System's Future Could Be Bumpy Study calculates the odds that two planets collide or one crashes into sun in the next 5 billion years Posted June 11, 2009 By Sid Perkins, Science News It’s happened before, and it could happen again: Planets in the inner solar system may collide if gravitational interactions substantially disturb now-stable orbits, a new study suggests. Scientists and mathematicians have long known that the equations describing the orbital motions of any group of three or more objects can’t be solved exactly. Even with the most powerful computers, it is hopeless to try to determine precisely what planetary orbits will look like more than a few million years in the future, says Jacques Laskar of the Paris Observatory. The results of any single simulation offer nothing more than one possible outcome, but running a large number of simulations can provide insights into overall probabilities, he notes. Now, a large-scale study by Laskar and observatory colleague Mickael Gastineau, which appears in the June 11 [2009] Nature, provides a possibly frightening glimpse into the solar system’s future. It’s a future in which, literally, worlds collide. The computer model used by Laskar and Gastineau accounts for the gravitational interactions among the moon, the eight major planets and Pluto, as well as any effects of general relativity. For the new study, the researchers started with the best known information about the position and orbital velocity of each of the 10 bodies, and marched simulations forward in nine-day steps for the next 5 billion years, the projected life of the sun. To obtain a large sample of possible futures, Laskar and Gastineau ran 2,501 scenarios. The only difference between runs was the size of Mercury’s orbit. In the second and each subsequent simulation, the size differed by only .38 millimeters. For the first few million years, planetary orbits hardly differed from one run to the next. Eventually, though, variations began to accumulate. Most of the detrimental changes resulted from Jupiter’s gravitational tug on Mercury, the inner planet with the most lopsided orbit, Laskar says. In one simulation, Mercury collides with Venus about 1.76 billion years from now. In another, those planets come within 6,500 kilometers of a collision about 4.9 billion years in the future. In three other simulations, Mercury’s orbit becomes so lopsided that it falls into the sun. In yet another scenario, Mars swings within 800 kilometers of Earth about 3.34 billion years from now. During such a close approach, tidal flexing within Mars’ crust could break that planet apart and fragments could strike Earth ­ a generally disastrous result, Laskar notes. Using the Earth-Mars close encounter as a starting point, Laskar and Gastineau then ran another 201 simulations to assess the possibility that the near miss could lead to a direct collision. In 196 of those computer runs, one planet in the inner solar system either slammed into another or fell into the sun. Of those scenarios, Earth was struck 48 times: once by Mercury, 18 times by Venus and 29 times by Mars. Despite the dire outcomes of a few of these scenarios, the team’s findings hint that the odds are overwhelmingly in Earth’s favor. “I prefer to see the glass as 99 percent full and only 1 percent empty,” says Gregory Laughlin, an astronomer at the University of California, Santa Cruz. “It’s encouraging to think that the solar system is here for the duration,” he notes. ---- http://www.sciencenews.org/view/generic/id/44568/title/Solar_systems_future_... Solar system's future could be bumpy Study calculates the odds that two planets collide or one crashes into sun in the next 5 billion years By Sid Perkins July 4th, 2009; Vol.176 #1 (p. 9) PLANETARY BUMPER CARS. Chaotic perturbations of orbits, largely due to the gravitational effects of Jupiter, could send planets in the inner solar system crashing into one another (such as Mars and Earth, illustrated here) sometime in the next 5 billion years, a new study suggests. It’s happened before, and it could happen again: Planets in the inner solar system may collide if gravitational interactions substantially disturb now-stable orbits, a new study suggests. Scientists and mathematicians have long known that the equations describing the orbital motions of any group of three or more objects can’t be solved exactly. Even with the most powerful computers, it is hopeless to try to determine precisely what planetary orbits will look like more than a few million years in the future, says Jacques Laskar of the Paris Observatory. The results of any single simulation offer nothing more than one possible outcome, but running a large number of simulations can provide insights into overall probabilities, he notes. Now, a large-scale study by Laskar and observatory colleague Mickael Gastineau, which appears in the June 11 Nature, provides a possibly frightening glimpse into the solar system’s future. It’s a future in which, literally, worlds collide. The computer model used by Laskar and Gastineau accounts for the gravitational interactions among the moon, the eight major planets and Pluto, as well as any effects of general relativity. For the new study, the researchers started with the best known information about the position and orbital velocity of each of the 10 bodies, and marched simulations forward in nine-day steps for the next 5 billion years, the projected life of the sun. To obtain a large sample of possible futures, Laskar and Gastineau ran 2,501 scenarios. The only difference between runs was the size of Mercury’s orbit. In the second and each subsequent simulation, the size differed by only .38 millimeters. For the first few million years, planetary orbits hardly differed from one run to the next. Eventually, though, variations began to accumulate. Most of the detrimental changes resulted from Jupiter’s gravitational tug on Mercury, the inner planet with the most lopsided orbit, Laskar says. In one simulation, Mercury collides with Venus about 1.76 billion years from now. In another, those planets come within 6,500 kilometers of a collision about 4.9 billion years in the future. In three other simulations, Mercury’s orbit becomes so lopsided that it falls into the sun. In yet another scenario, Mars swings within 800 kilometers of Earth about 3.34 billion years from now. During such a close approach, tidal flexing within Mars’ crust could break that planet apart and fragments could strike Earth ­ a generally disastrous result, Laskar notes. Using the Earth-Mars close encounter as a starting point, Laskar and Gastineau then ran another 201 simulations to assess the possibility that the near miss could lead to a direct collision. In 196 of those computer runs, one planet in the inner solar system either slammed into another or fell into the sun. Of those scenarios, Earth was struck 48 times: once by Mercury, 18 times by Venus and 29 times by Mars. Despite the dire outcomes of a few of these scenarios, the team’s findings hint that the odds are overwhelmingly in Earth’s favor. “I prefer to see the glass as 99 percent full and only 1 percent empty,” says Gregory Laughlin, an astronomer at the University of California, Santa Cruz. “It’s encouraging to think that the solar system is here for the duration,” he notes.