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This Week's Finds in Mathematical Physics (Week 222) 
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#73
Oct1206, 05:10 AM

P: n/a

Isolated quasars are a natural part of Halton Arp's "ejection"
scenario, although they are far more likely to be associated with a nearby galaxy. Since electromagnetism scales (conformal invariance of EM), such ejections could be similar in principle to the "coronal mass ejections" on the Sun, only on a far more vast scale. There are cases of quasars and clearly associated galaxies where the members have widely varying redshifts. A sensible interpretation would conclude that enormous electromagnetic fields are at work, and that some joint manifestation of gravity and electromagnetism is underway. See http://xxx.uniaugsburg.de/abs/astroph/?0101538 drl 


#74
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#75
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#76
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#77
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#78
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#79
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#80
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#81
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#82
Oct1206, 05:10 AM

P: n/a

In article <slrndlehja.qdn.robert@atdotde.iubremen.de>,
Robert C. Helling <helling@atdotde.de> wrote: >On Tue, 18 Oct 2005 12:57:32 +0000 (UTC), John Baez ><baez@math.removethis.ucr.andthis.edu> wrote: >> o Twotino  A twotino is a Kuiper belt object whose orbit is in 2:1 >> resonance with Neptune. These are rare compared to plutinos, and >> they're smaller, so they're stuck with boring names like 1996 TR66. >> There are also a couple of Kuiper belt objects in 4:3 and 5:3 >> resonances with Neptune. >Is there an easy way to see why these resonance orbits come about? Why >do three body systems with a large central object, an intermediate >planet and a small probe happen to get the probe in resonace with the >planet? Is this just "frequency locking happens in chaotic systems" >or is there an easy but more quantitative way to understand this? I'm shamefully ignorant of this, so ten minutes' research on the web was able to double my knowledge. I got ahold of this paper online: B. Garfinkel, On resonance in celestial mechanics: a survey, Celestial Mech. 28 (1982), 275290, http://adsabs.harvard.edu/cgibin/np...eMec..28..275G and while not easy to understand (I guess there's a huge body of work on this subject), it uses Hamiltonian perturbation theory and continued fractions to study resonance, and talks about a difference between "shallow" and "deep" resonances. It says that Laplace first explained the Great Inequality in the motion of Jupiter and Saturn by means of a 5:2 resonance, which is a "shallow resonance". I have no idea what the "Great Inequality" is, other than a strange name for this 5:2 resonance. But, I read elsewhere that: The dynamics of the SunJupiterSaturn system was recognized as problematic from the beginnings of perturbation theory. The problems are due to the socalled Great Inequality (GI), which is the JupiterSaturn 2:5 meanmotion nearcommensurability. This is from: F. Varadi, M. Ghil, and W. M. Kaula, The Great Inequality in a Planetary Hamiltonian Theory http://arxiv.org/abs/chaodyn/9311011 Somehow this shallow resonance is related to the continued fraction 1/(2 + 1/(2 + 1/(14 + 1/(2 + .... )))) which is close to 2/5. The PlutoNeptune resonance, on the other hand, is a "deep resonance" and related to the continued fraction 1/(2  1/(2 + 1/(10 + .... ))) which starts out close to 2/3. (Recall that plutinos go around the Sun about twice each time Neptune goes around thrice.) >Probably related: There are people doing numerical long term stability >analysis of the solar system. From what I know, they are not just >taking F=ma and Newton's law of gravity, replace dt by delta t and >then integrate but use much fancier spectral methods. Could somebody >please point me to an introduction into these methods? Here's a bit of stuff about that from "week107", perhaps not all that helpful, but still pretty interesting: ....................................................................... ... Later Jon Doyle, a computer scientist at M.I.T. who had been to my talk, invited me to a seminar at M.I.T. where I met Gerald Sussman, who with Jack Wisdom has run the best longterm simulations of the solar system, trying to settle the old question of whether the darn thing is stable! It turns out that the system is afflicted with chaos and can only be predicted with any certainty for about 4 million years... though their simulation went out to 100 million. Here are some fun facts: 1) They need to take general relativity into account even for the orbit of Jupiter, which precesses about one radian per billion years. 2) They take the asteroid belt into account only as modification of the sun's quadrupole moment (which they also use to model its oblateness). 3) The most worrisome thing about the whole simulation  the most complicated and unpredictable aspect of the whole solar system in terms of its gravitational effects on everything else  is the EarthMoon system, with its big tidal effects. 4) The sun loses one Earth mass per 100 million years due to radiation, and another quarter Earth mass due to solar wind. 5) The first planet to go is Mercury! In their simulations, it eventually picks up energy through a resonance and drifts away. For more, try: 4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system, Science, 257, 3 July 1992. Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion of Pluto is chaotic, Science, 241, 22 July 1988. James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom, The outer solar system for 200 million years, Astronomical Journal, 92, pp 176194, July 1986, reprinted in Lecture Notes in Physics #267  Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986. James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay Sussman, A digital orrery, in IEEE Transactions on Computers, C34, No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in Physics #267, Springer Verlag, 1986. 


#83
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


#84
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


#85
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


#86
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


#87
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


#88
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


#89
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


#90
Oct1206, 05:10 AM

P: n/a

Jonathan Thornburg  remove animal to reply wrote:
> Robert C. Helling <robert@hellingdell600.iuhb02.iubremen.de> wrote: > >>There are people doing numerical long term stability >>analysis of the solar system. From what I know, they are not just >>taking F=ma and Newton's law of gravity, replace dt by delta t and >>then integrate but use much fancier spectral methods. Could somebody >>please point me to an introduction into these methods? > > > I don't do this sort of work myself, but the buzzwords you want are > "symplectic ODE integrator". The basic idea is to use an ODE integration > scheme which conserves energy, angular momentum, and maybe other nice > things, up to floatingpoint roundoff error, rather than just up to > finite differencing error like a standard ODE integrator would do. > T. Fuse, Planetary Perturbations on the 2: 3 Mean Motion Resonance with Neptune, http://astronomy.nju.edu.cn/~xswan/r...PASJ54_493.pdf uses symplectic integration to study 2:3 resonances numerically. The thesis Timefrequency analysis based on wavelets for Hamiltonian systems by VelaArevalo, http://www.cds.caltech.edu/~luzvela/th2s.pdf contains in Chapter 4 interesting numerical information about chaos, resonances, and stability in the restricted 3body problem. Other interesting papers include: http://users.auth.gr/~hadjidem/Asymmetric1.pdf Symmetric and asymmetric librations in planetary and satellite systems at the 2/1 resonance astroph/0501004 Regimes of Stability and Scaling Relations for the Removal Time in the Asteroid Belt astroph/0203182 The Resonant Dynamical Evolution of Small Body Orbits Among Giant Planets http://cns.physics.gatech.edu/~luzve...enCQG_2004.pdf Time–frequency analysis of the restricted threebody problem: transport and resonance transitions http://www.astro.auth.gr/~varvogli/varv5.ps The “Third” Integral in the Restricted ThreeBody Problem Revisited Arnold Neumaier 


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