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For exoplanet fans, this week has been an exciting one, with some amazing new discoveries being announced at the American Astronomical Society meeting in Austin, Texas – our galaxy is brimming with planets, probably billions, and the smallest known planets have been found (again), with one about the size of Mars. But that’s not all; it was also announced that Kepler has found not one but two more planets orbiting binary stars!
The two star systems are Kepler-34 and Kepler-35; they consist of double stars which orbit each other and are about 4,900 and 5,400 light-years from Earth. The two new planets, Kepler-34b and Kepler-35b, each orbit one of these pairs of stars and are both about the size of Saturn. Since they orbit fairly close to their stars, they are not in the habitable zones; Kepler 34-b completes an orbit in 289 days and Kepler-35b in 131 days. It’s more the fact that they orbit double stars that makes them so interesting.
This is now the third planet found in a binary star system. The first, Kepler-16b, was nicknamed Tatooine as it was reminiscent of the world orbiting two suns in the Star Wars films. Until recently, it was unknown if any such star systems had planetary companions. It was considered possible, although unlikely, and remained only a theory. But now, the view is that there may indeed be a lot of them out there, just as planets are now apparently common around single stars. That’s good news for planet-hunters, as most stars in our galaxy are binaries.
According to William Welsh of San Diego State University who participated in the study, “This work further establishes that such ‘two sun’ planets are not rare exceptions, but may in fact be common, with many millions existing in our galaxy. This discovery broadens the hunting ground for systems that could support life.”
Eric B. Ford, associate professor of astronomy at the University of Florida, stated: “We have long believed these kinds of planets to be possible, but they have been very difficult to detect for various technical reasons. With the discoveries of Kepler-16b, 34b and 35b, the Kepler mission has shown that the galaxy abounds with millions of planets orbiting two stars.”
The hope now is that Kepler will continue until 2016 to be able to further refine its findings so far. That will require a mission extension, but scientists involved are optimistic they will get it.
According to Ford, “Astronomers are practically begging NASA to extend the Kepler mission until 2016, so it can characterize the masses and orbits of Earth-size planets in the habitable zone. Kepler is revolutionizing so many fields, not just planetary science. It would be a shame not to maximize the scientific return of this great observatory. Hopefully common sense will prevail and the mission will continue.”
Yes, indeed.
The study was published January 11, 2012 in the journal Nature (payment or subscription required for access to full article).
See also PhysOrg.com for a good overview of the new findings.
The biggest problem remains in how these planets were formed in such a variable gravitational field. Against our knowledge of star formation and the conservation of angular momentum means that something is quite wrong in our basic theories in making stars. Influence like magnetic fields are likely the added quantity that leads towards some plausible explanation in regions where planets should not be able to exist.
As to the preponderance of exoplanets around other binary stars is yet to be proven, most;y because of the smallness of the data sample. Time, no doubt, will soon tell.
Variable gravitational field? May be, I don’t get it, but I thought that the two components of the binary star system excert a gravitational field exactly as a point mass located at the common center of mass, and that this field is not variable.
Yes,the planet will orbit a central axis but think about when the three elements are lined up as opposed to in a triangle. And one is orbiting in only 131 days which means it comes quite close to each sun in turn. Big time variabilities in gravitational attraction there, I would think. My goofy guess is that these planets form way out at double Pluto distances and then work their way in.
Well, I think about the triangle situation … thinking 😉 …, but it doesn’t change anything — Newtonian-wise.
Another thing is when the planets come very close to the stars. Then we would have similar relativistic effects as with our sun and the planet Mercury, of course: the perihelions of the orbits precess. But this would not be relevant, if your gess is correct, that the planets formed far away from the stars.
So, as far as I can tell, when the planets were formed, the gravitational field the binary stars excerted on the planets was constant.
Ever heard of perturbation theory, or even what the tides do to planetary positions or planetary orbits? Don’t these constitute and cause variable gravitational fields too?
Also note that I said “formation of planets” not planets in situ. Accretion disks would be ripped apart fairly quickly in binary systems.
So do planets form just around both stars and/or orbit around a common their centre of gravity during the accretion processes? If they do there must be other forces or actions at work.
You are right, perturbation is relevant here, and this indeed causes variations in the gravitational field of the system (but, sorry, you didn’t mention it in your first comment). And tides are only important rather close to the stars.
Whether accreation disks are rather stable or are ripped apart and how quick — is an open question, I think.
But, as a matter of fact, the gravitational field of binary stars is the same as the field of a point mass located at the common center of mass (as long as only Newton’s physics is relevant; see above). The accretion disk, gas, dusk, planets, etc. are exclusively influenced by this kind of gravitational field.
Because of this — you are right — something else must be there: electromagnetic fields; and e.g. collisions, friction, and radiation pressure work by means of such fields.
“….the gravitational field of binary stars is the same as the field of a point mass located at the common center of mass ”
I’m afraid it is not. Each star provides a separate gravitational field, that are then superimposed. Every additional object would superimpose it’s gravitational field aswell, and the resulting gravitational field is NOT that of a single gravitational field at the center of mass.
Ha, I missed that, despite reading your previous comment too.
Only in the case of large distances would a like charge (here, mass) superposition resemble a unary mass. You would need gravitational quadrupoles to generate gravitational waves to get an observable difference from far field effects.
Those aren’t the distances we discuss. (A least for the hard case, intermediate distanced binaries.)
Sorry to say this, but that “the gravitational field of binary stars” — more general: of several masses — “is the same as the field of a point mass located at the common center of mass” is basic Newtonian physics. I hope we don’t have to discuss these fundamentals on and on.
And, again, a different case is considering relativistic effects occuring near to a mass (like Mercury being near to our sun), especially near to a rotating mass (in some way or other “taking space with it”).
Because of the above, binary stars excert a gravitational field like a single point mass on planets — and on parts of the accretion disk — being, well, far away.
“…is basic Newtonian physics”
No it is not, you have missunderstood what has been said.
‘One’ object, like a star, have a gravitational field that basically is the same as one that originates from the center of mass.
‘Two’ objects orbit around the center of mass, but each have its own gravitational field that will be superimposed. They do NOT superimpose their gravitational fields into one superfield at the center of mass. This is the part that you have missunderstood.
More objects than 2 will introduce what is commonly termed perturbations, because of the complex apperance of all the superimposed fields. But all the fields are _individually_ superimposed. It becomes simpler if one of the masses dominate all the other, like in the solarsystem.
General relativity modifies the newtonian ‘law’ of gravity, one effect being that all the planets have slightly different perihelion advance (mercury has the largest difference), but the differences in the solarsystem are still very small – preturbations from all the masses in the solarsystem are a significantly larger effect, and those perturbations are because of the individually superimposed gravitational fields.
“…is basic Newtonian physics”
No it is not, you have missunderstood what has been said.
‘One’ object, like a star, have a gravitational field that basically is the same as one that originates from the center of mass.
‘Two’ objects orbit around the center of mass, but each have its own gravitational field that will be superimposed. They do NOT superimpose their gravitational fields into one superfield at the center of mass. This is the part that you have missunderstood.
More objects than 2 will introduce what is commonly termed perturbations, because of the complex apperance of all the superimposed fields. But all the fields are _individually_ superimposed. It becomes simpler if one of the masses dominate all the other, like in the solarsystem.
General relativity modifies the newtonian ‘law’ of gravity, one effect being that all the planets have slightly different perihelion advance (mercury has the largest difference), but the differences in the solarsystem are still very small – preturbations from all the masses in the solarsystem are a significantly larger effect, and those perturbations are because of the individually superimposed gravitational fields.
To be sure, I again looked up, what is written in a physics textbook I use since being at university — The Feynman Lectures on Physics, vol. 1, p. 12-9 — about the principle of superposition of fields, and, as far as only Newtonian physics is considered (!), for the gravitational field of several bodies it supports what I said.
So, in my job — simulation and visualization of physical systems — I will continue to just add all the force vectors to get the resultant force on a body. And this will be the same, if those masses were concentrated in one point mass at the suitable location. This in some cases results in a more performant simulation.
But I assume this will not convince you, so let us just agree to disagree and close this discussion.
No, i wont let you bail out on such flawed reason. And you siting papers and lectures does not impress me when clearly you have missunderstood this. If you do this for a job, you should be fired.
*Yes, fields can be superimposed.
*Yes, center of mass is maintained in all closed gravitatinal system.
*No, there is not a single gravitational field originating from the center of mass of multiple objects. And here is why:
Lets assume for sake of argument that there is such a field. Then we can send a probe to that region, and the closer we get the higher the gravity becomes. Eventually we would discover that right where the cente of mass is, there is a black hole where the gravity is so large that nothing can escape. Do you believe there is a black hole near the center of mass of our solarsystem??
What you have misstaken is a first order analogy of how orbital mechanics can be simplified and examplified. That analogy breaks down when you inspect it closer, because in reality gravitational fields always originate from a massive body, and superposing them requires that you add up the individual fields based on their relative positions.
I thought it through again, and, yes, you are right and I’m wrong. I’m sorry.
I inspected some pertinentsimulation projects of mine from the past, and, as it happens, I implemented them correctly.
I don’t know why I confused myself now (exhaustion?). And there is no need to fire me.
…reading this, well frankly, all it shows is that my original comment and its point was absolutely valid. From the above statements suggests you probably need to do a bit more reading on the subject.
(2 likes here, I doubt this. Do you have multiple avatars?)
Well, if you click on the “[#] Likes”, you can see who liked the dude.
The reason I Liked it was that I had just watched this SETI seminar by Kress on habitable planets and she discusses the basic disk formation, its stability and how different mechanisms aside from gravitation comes in. I recognized those latter two aspects (stability, additional mechanisms in disks outside of gravity) in Ivry’s comment, so it looked sound to me. I didn’t Like it because I have managed to follow your discussion precisely.
“Like”‘s are as all web opinion to be read informed by the mediocracity principle. If it can be explained by stupidity (or in this case lack of knowledge), it is most likely the case. =D
As I started in on the topic, but mind that I haven’t grasped your discussion yet, maybe Kress is helpful since that part of the seminar (Ch 2 starting in at 16:50) is a beginner’s overview. She discusses disk, and hence the timeframe of planetary formation, in terms of how to get rid of angular momentum. At the same time some mechanism, likely turbulence, has to keep the disk “fluffy” against gravitational collapse to a too thin annulus.
… okay, I have now read the thread, and it is too vague a subject for me to grasp. The experts have some handles on monary disk formation and evolution; this is what we surveyed in astrobiology class. Some of that can be applied to some binaries, either well separated or well clustered. The middle case looks like a nightmare.
So I am going to cheat. (O.o) I have no problem against finding planets in gravitationally long term stable orbits around any star formation, seeing how we find plenty of giant planets close to stars that have to migrate there. See Kress’ talk, btw.
This is very loose of course, because you need a disk for migration. But giants for quickly and that is how they manage to migrate in the first place. Presumably it takes some time for a binary disk to form disk structures.
If you guys can manage to find some mechanism for in situ formation in the hard case, kudos!
Sorry. I just made a general comment. I’ve also made no supposition nor expressed arbitrary personal notions.
Why should I have to defend myself, when the significant dichotomy between theory and observation is plainly obvious?
As the article says; “It was considered possible, although unlikely, and remained only a theory.”
My comment is reasonably pertinent to this question. I.e. Why it is believed planet formation in binaries is improbable or unlikely.
Makes perfect sense to me, but seems beyond the grasp of my fellow commenters here. Just saying.
I don’t think I have an answer to any of that, since as I said I have only the foggiest idea of how this applies. I have a ‘cheat’ that satisfies me until the day someone explains this subject of disk formation in binary systems to me.
Have you seen lcrowell’s comments? He says the same on the subject theory vs observation as you do here.
The reason I Liked it was that I had just watched this SETI seminar by Kress on habitable planets and she discusses the basic disk formation, its stability and how different mechanisms aside from gravitation comes in. I recognized those latter two aspects (stability, additional mechanisms in disks outside of gravity) in Ivry’s comment, so it looked sound to me. I didn’t Like it because I have managed to follow your discussion precisely.
“Like”‘s are as all web opinion to be read informed by the mediocracity principle. If it can be explained by stupidity (or in this case lack of knowledge), it is most likely the case. =D
As I started in on the topic, but mind that I haven’t grasped your discussion yet, maybe Kress is helpful since that part of the seminar (Ch 2 starting in at 16:50) is a beginner’s overview. She discusses disk, and hence the timeframe of planetary formation, in terms of how to get rid of angular momentum. At the same time some mechanism, likely turbulence, has to keep the disk “fluffy” against gravitational collapse to a too thin annulus.
… okay, I have now read the thread, and it is too vague a subject for me to grasp. The experts have some handles on monary disk formation and evolution; this is what we surveyed in astrobiology class. Some of that can be applied to some binaries, either well separated or well clustered. The middle case looks like a nightmare.
So I am going to cheat. (O.o) I have no problem against finding planets in gravitationally long term stable orbits around any star formation, seeing how we find plenty of giant planets close to stars that have to migrate there. See Kress’ talk, btw.
This is very loose of course, because you need a disk for migration. But giants for quickly and that is how they manage to migrate in the first place. Presumably it takes some time for a binary disk to form disk structures.
If you guys can manage to find some mechanism for in situ formation in the hard case, kudos!
The analogy of a gravitational field behaving like it comes from the center of the mass applies for spherical bodies (actually, it applies to rotationally symmetric bodies).
But the combinatorial of several orbiting masses does lead to distinct fluctuations in the combined gravitational fields, tides on earth is one direct example. Jupiter gravitational influense on earht is more when it is in opposition (closest to earth) and less when in conjunction with the sun (furthest away from earth). Same would apply to the starsystems aswell.
So basically, for a 2-body system, the analogy you describe works, but for 3-body or more systems, it doesnt quite. Thats one reason there is a direct analytical solution to 2-body dynamics, but none for N-body systems.
A bit of controversy has arisen over this. A pair of masses, same holds for a pair of charges, has a monopole term. That term dominates in the far field where the distance from the pair is much larger than the distance between the masses in the pair. As one approaches the pair the dipole term dominates. So from a great distance the two masses will behave approximately as a single mass at their common center of mass. However, close in the gravity field is dipolar.
This does say a bit about the motion of a third body or satellite. If it is orbiting at a distance much larger than the separation distance of the masses then its orbit will approximate the orbit of a body around a single mass. If the satellite is much closer then that orbit will be considerably more complicated, and indeed it is not integrable. The satellite in this case will exhibit chaotic dynamics which will cause the satellite to either collide with one of the stars or be ejected from the system.
LC
What is the distance between the two binary stars? Are we talking 50 AU (kuiper belt/pluto) or 500 AU, or 50,000 AU (Oort cloud – ~1/10th of a light year) distance between the pair?
If the distance is 50,000 AU, then Star 2 has no real influence on the inner/outer solar system of Star 1 (it simply scatters the outer dust cloud). In this scenario, Star 2 would also be quite dim (nothing like a second sun in the Tatooine example) to a person on the planets around Star 1. The movement of Star 2 should be very obvious to people on the planet. But unless it was a bloated giant or a bright high mass star, it would be a very bright star (that for some reason rotated ours). It wouldn’t be lighting up the sky.
Even at 50 AU, the naked eye would struggle to recognize a sun sized star as something more than a point source. It would be very, very bright for a star, but certainly nothing more than a bright yellow dot (as opposed to a pale blue dot). But at 50 AU, it would certainly impose a tremendous influence on planet formation. The probably of a planet forming in stable orbit would be nearly zero.
If the stars were close enough to appear like they do in the movies (visually appears to be even less than 10 AU)…..there is not a chance a planet could actually form in that environment. There’s even less than “not a chance” they could form in the habitable zone around it. None, zero, zilch. The probability of a planet being captured into a tight binary system is also quite low, but plausible.
That’s not to say that the two stars were always this close either. Binarys with planets more likely formed further away and simply moved inward after the planets formed.
Also one can envision a small tight binary around a somewhat distant third star, the 3rd star forms planets…the tight binary pair is moving inward, one star is eventually ejected and the other remains in stable orbit….this is a more realistic scenario for habitable planet around a tight binary. This certainly would not be the norm for a binary system.
The probability of something like Tatooine is very low. But there are so many stars that I’m sure Tatooine exists somewhere….by some mechanism however improbable
From the abstract there are good news and bad (?) news.
The good news first:
“The observed rate of circumbinary planets in our sample implies that more than ~1% of close binary stars have giant planets in nearly coplanar orbits, yielding a Galactic population of at least several million.”
So despite the difficulties of planet formation and long time stability, there is a healthy population of these.
At a guess binary systems will be good for constraining and testing results of protoplanetary disk and planet system evolution both, when people start to riddle more of the simpler and immediately interesting unary systems.
The bad (?) news:
“The planets experience large multi-periodic variations in incident stellar radiation arising from the orbital motion of the stars.”
I haven’t access to the paper right now, so I’m curious how much of a variation we are talking about here. What would it do to planets in the nominal habitable zone of binary stars (assuming planets fit there in the first place)?
If it isn’t enough to deep freeze or vaporize oceans, it may be a driver for some very interesting evolution. A good target for
Star WarsSETI search, perhaps.Our solar system may be one star up on both Kepler-34&35, since we may have formed with a hierarchical triple star composed of a close binary pair orbited by a smaller tertiary stellar component (TSC) inside the orbit of Mercury.
The TSC was resonance locked to the planets, causing the binary pair to decay and pushing itself and the planets outward in a event we’ll call ’planet inflation’. Originally, Venus and Uranus may have formed between Earth and Jupiter, but as the orbits inflated, first one then the other moved into the 2:1 (Kirkwood Gap) resonance with Jupiter and were tossed into their current relative orbits, accounting for their highly inclined axial tilts. Mars and 4 Vesta are small because they formed late in the orbits vacated by Venus and Uranus.
The binary pair merged in the primary luminous red nova (LRN) at about 4.5682 Ga, rolling the planetary accretion disk dust and gas out to the inner Oort cloud (IOC) which formed the IOC comet clusters and differentiated to form gneiss-schist-dolomite rocky cores (mantled gneiss domes).
The TSC orbit became eccentric in the primary LRN, forming a contact binary with the sun at aphelion, streaming plasma from the sun which volatile depleted the terrestrial planets, causing the ’planetary volatility trend’. Before volatile depletion, Venus and Earth may have been ice giants the size of Uranus and Neptune.
The TSC merged with the sun in a secondary LRN, blasting the volatile-enriched solar envelope out to the outer Oort cloud (OOC) which formed the OOC comet clusters which differentiated to form volatile-enriched granite-greenstone rocky cores. The LRNe also formed the short-lived-radionuclides of the early solar system which caused the granite-greenstone comet cores to melt, forming massive plutonic granite.
Schist-dolomite and greenstone are hydrothermal rock formed from diagenesis and lithification of the underlying gneiss and granite respectively.
Comet clusters underwent core collapse (similar to stellar globular clusters) by evaporating off the smaller comets, causing the larger comets to settle into the collapsed core and merge to form far-larger compound comets, such as the Sierra-Nevada (OOC) compound-comet core and the Appalachian-Basin-Province (IOC) compound-comet core.
Interesting, and certainly provocative. But too much catastrophism to my liking, which makes the chain of events highly unlikely I would think. (Layman here, but with an astrobiology interest.)
I will refer to Kress’ talk that I linked to in another comment for two reasons.
First she shows how the inner planets form naturally in place late in the game. When the disk has dispersed, planetesimals form planets on time scales ~ 10 – 100 by aggregating over protoplanets to planets. (In this scenario the rapidly formed Mars, ~ 4 Ma formation time, would be a lone protoplanet.)
The gap for the terrestrial formation described by Kress is set by the highly predictive Nice model that predicts so many features of the rest of the system. It contains one catastrophic 1:2 (standard Nice) or 2:3 (5th giant expulsion) resonance. And it constrains giant migrations nicely.
Second, she discuss volatile enrichment in detail. Analogously to the snow line for mainly water ices there is a “soot line” for PAH, which is the carbon compound that can go into asteroids.* Most asteroids contain more than enough carbon and water, too much actually, compared to Earth.
The only ones that are devoid of water are the enstatite chondrites, which she describes as “odd” IIRC. They are so reduced that they must have formed somewhere around the orbit of Mercury. Yet their oxygen isotopic composition correspond to the Earth-Moon system.
That can be fitted to Kress’ model, I think, since the terrestrials aggregate from, and scatter to, all over as they form by gravitational effects and concomitant collisional impacts. See her visuals of typical results. Adding asteroid volatiles to a dry Earth made enough oceans later as a late aggregation tail and/or late heavy impacts spike. (From the Nice model.) I believe anything out to ~ 3-5 AU is now claimed to be generally isotopically consistent with the Earth-Moon volatiles, but I haven’t checked it for a while, it is such a dynamic subject.
—————-
* Btw, the problem of how so much observed PAHs (~ 20 % of carbons, IIRC) forms finally seems on its way to a solution, in a news release just yesterday.
“To unravel the formation of naphthalene as the simplest representative of PAHs, University of Hawai?i at M?noa chemists Dorian S.N. Parker, Fangtong Zhang, Seol Kim, and Ralf I. Kaiser conducted gas phase crossed molecular beam experiments in their laboratory and presented that naphthalene can be formed as a consequence of a single collision event via a barrier-less and exoergic reaction between the phenyl radical and vinylacetylene involving a van-der-Waals complex and submerged barrier in the entrance channel. […]
Mebel’s computations showed that naphthalene is formed from the reaction of a single phenyl radical colliding with vinylacetylene. Most importantly, since the temperatures of cold molecular clouds are very low (10 K), the computations indicate that the reaction has no entrance barrier (‘activation energy’).
“These findings challenge conventional wisdom that PAH-formation only occurs at high temperatures such as in combustion systems and implies that low temperature chemistry can initiate the synthesis of the very first PAH in the interstellar medium,” said co-author Tielens.”
There is a bit of confusion here with regards to treating gravitational sources in Newtonian physics. A system with two masses in a mutual orbit has a dipole moment if the mutual orbit is circular. If the orbits are elliptical there is then a quadrupole moment. There is of course a monopole moment for the two body system, which if you measured the Newtonian gravity acceleration at a great distance would dominate. So a two body system appears approximately as a one body system at a great distance D which is much larger than the orbital radial distance R between the two bodies D >> R.
All of this is important if you put a satellite with a mass m <> 2?/?.
The classical mechanics of systems with three or more bodies can’t be solved in closed form. For any N body problem there are 3 equations for the center of mass, 3 for the momentum, 3 for the angular momentum and one for the energy. These are 10 constraints on the problem. An N-body problem has 6N degrees of freedom. For N = 2 this means the solution is given by a first integral with degree 2. For a three body problem this first integral has degree 8. This runs into the problem that Galois illustrated which is that any root system with degree 5 or greater generally has no algebraic roots. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for 8 solutions there is some eight order polynomial p_8(x) = ?_{n=1}^8(x – ?_n), with 8 distinct roots ?_n that are constant along the 8 solutions in this eight fold product. Since p_8(x) = p_5(x)p_3(x), a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding their roots, or a set of solutions that are algebraic. This means that any system of degree higher than four is not in general algebraic. At the root of the N-body problem Galois theory tells us there is no algebraic solution for N >= 3. Galois theory is a subject outside the scope of this book, but it is the basis of abstract algebra, groups and algebras used in physics.
This reasoning is the basis for a series of theorems by H. Bruns in 1887, which lead to Poincare’s proof that for n > 2 the n-body problem is not integrable. For this reason the stability of the solar system can’t be demonstrated in a closed form solution, but only demonstrated for some finite time by perturbation methods. Poincare won the Sweden prize for this with respect to the solar system. The stability of the solar system was an outstanding problem, and the King of Sweden raised a contest and award for a proof of the stability of the solar system. Previously Laplace had made an attempt at this early in the 19th century, but could not work it in completely. Poincare won the prize for showing the solar system is in fact not stable, but is only quasi-stable for time period much larger than the orbital periods of the planets.
It had been astrophysics “lore” that double star systems could not have planets, for it was thought the perturbing terms would cause planets to not be quasi-stable over long enough time periods. Data of late has thrown these ideas into trash heap. Tatooine type of planets are being found, which suggests otherwise.
LC
Well said. However, you said; “There is a bit of confusion here with regards to treating gravitational sources in Newtonian physics.”
True. But to importantly reiterate, Duncan Ivry made the supposition of a solitary gravitation point source. This was in response to my statement; “The biggest problem remains in how these planets were formed in such a variable gravitational field.”
The confusion here is with Duncan incorrect statements and is not with me or in what I’ve originally said. Please do tar me with this foolhardiness by Duncan Ivry (who is probably just attempting to look important)
I was not particularly pointing fingers. Actually, my attempts to post this were frustrated, and now I find that it got posted multiple times. I am going to try to rectify this.
LC
Can I nick that spartan expression, because it fits my situation below as well? “I was not particularly pointing fingers.” There.
SJSStar, you really are a nice person, aren’t you? Don’t make unfounded conclusions regarding my motivation, please.
Yes I am…. Thanks for saying so.
I remembered that, when you prompted me. [I am a swede, in case my bad case of bad english hasn’t clued everyone in.]
A very small nitpick is that it was a prize in Oscar II’s name for his 60th birthday, not in the name of Sweden but a “Swedish” prize I take it.
I had to google the rest though. The swedish mathematician Mittag-Leffler was turning around the abysmal status of the field in Sweden. He had instituted the still renowned “Acta Mathematica” series published by the Swedish Royal Academy of Science, and politically shrewdly proposed and later published the king’s prize.
Seems from above nitpick of association error it worked splendidly! =D
There is a bit of confusion here with regards to treating gravitational sources in Newtonian physics. A system with two masses in a mutual orbit has a dipole moment if the mutual orbit is circular. If the orbits are elliptical there is then a quadrupole moment. There is of course a monopole moment for the two body system, which if you measured the Newtonian gravity acceleration at a great distance would dominate. So a two body system appears approximately as a one body system at a great distance D which is much larger than the orbital radial distance R between the two bodies D >> R.
All of this is important if you put a satellite with a mass m <> 2?/?.
The classical mechanics of systems with three or more bodies can’t be solved in closed form. For any N body problem there are 3 equations for the center of mass, 3 for the momentum, 3 for the angular momentum and one for the energy. These are 10 constraints on the problem. An N-body problem has 6N degrees of freedom. For N = 2 this means the solution is given by a first integral with degree 2. For a three body problem this first integral has degree 8. This runs into the problem that Galois illustrated which is that any root system with degree 5 or greater generally has no algebraic roots. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for 8 solutions there is some eight order polynomial p_8(x) = ?_{n=1}^8(x – ?_n), with 8 distinct roots ?_n that are constant along the 8 solutions in this eight fold product. Since p_8(x) = p_5(x)p_3(x), a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding their roots, or a set of solutions that are algebraic. This means that any system of degree higher than four is not in general algebraic. At the root of the N-body problem Galois theory tells us there is no algebraic solution for N >= 3. Galois theory is a subject outside the scope of this book, but it is the basis of abstract algebra, groups and algebras used in physics.
This reasoning is the basis for a series of theorems by H. Bruns in 1887, which lead to Poincare’s proof that for n > 2 the n-body problem is not integrable. For this reason the stability of the solar system can’t be demonstrated in a closed form solution, but only demonstrated for some finite time by perturbation methods. Poincare won the Sweden prize for this with respect to the solar system. The stability of the solar system was an outstanding problem, and the King of Sweden raised a contest and award for a proof of the stability of the solar system. Previously Laplace had made an attempt at this early in the 19th century, but could not work it in completely. Poincare won the prize for showing the solar system is in fact not stable, but is only quasi-stable for time period much larger than the orbital periods of the planets.
It had been astrophysics “lore” that double star systems could not have planets, for it was thought the perturbing terms would cause planets to not be quasi-stable over long enough time periods. Data of late has thrown these ideas into trash heap. Tatooine type of planets are being found, which suggests otherwise.
LC
There is a bit of confusion here with regards to treating gravitational sources in Newtonian physics. A system with two masses in a mutual orbit has a dipole moment if the mutual orbit is circular. If the orbits are elliptical there is then a quadrupole moment. There is of course a monopole moment for the two body system, which if you measured the Newtonian gravity acceleration at a great distance would dominate. So a two body system appears approximately as a one body system at a great distance D which is much larger than the orbital radial distance R between the two bodies D >> R.
All of this is important if you put a satellite with a mass m <> 2?/?.
The classical mechanics of systems with three or more bodies can’t be solved in closed form. For any N body problem there are 3 equations for the center of mass, 3 for the momentum, 3 for the angular momentum and one for the energy. These are 10 constraints on the problem. An N-body problem has 6N degrees of freedom. For N = 2 this means the solution is given by a first integral with degree 2. For a three body problem this first integral has degree 8. This runs into the problem that Galois illustrated which is that any root system with degree 5 or greater generally has no algebraic roots. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for 8 solutions there is some eight order polynomial p_8(x) = ?_{n=1}^8(x – ?_n), with 8 distinct roots ?_n that are constant along the 8 solutions in this eight fold product. Since p_8(x) = p_5(x)p_3(x), a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding their roots, or a set of solutions that are algebraic. This means that any system of degree higher than four is not in general algebraic. At the root of the N-body problem Galois theory tells us there is no algebraic solution for N >= 3. Galois theory is a subject outside the scope of this book, but it is the basis of abstract algebra, groups and algebras used in physics.
This reasoning is the basis for a series of theorems by H. Bruns in 1887, which lead to Poincare’s proof that for n > 2 the n-body problem is not integrable. For this reason the stability of the solar system can’t be demonstrated in a closed form solution, but only demonstrated for some finite time by perturbation methods. Poincare won the Sweden prize for this with respect to the solar system. The stability of the solar system was an outstanding problem, and the King of Sweden raised a contest and award for a proof of the stability of the solar system. Previously Laplace had made an attempt at this early in the 19th century, but could not work it in completely. Poincare won the prize for showing the solar system is in fact not stable, but is only quasi-stable for time period much larger than the orbital periods of the planets.
It had been astrophysics “lore” that double star systems could not have planets, for it was thought the perturbing terms would cause planets to not be quasi-stable over long enough time periods. Data of late has thrown these ideas into trash heap. Tatooine type of planets are being found, which suggests otherwise.
LC
There is a bit of confusion here with regards to treating gravitational sources in Newtonian physics. A system with two masses in a mutual orbit has a dipole moment if the mutual orbit is circular. If the orbits are elliptical there is then a quadrupole moment. There is of course a monopole moment for the two body system, which if you measured the Newtonian gravity acceleration at a great distance would dominate. So a two body system appears approximately as a one body system at a great distance D which is much larger than the orbital radial distance R between the two bodies D >> R.
All of this is important if you put a satellite with a mass m <> 2?/?.
The classical mechanics of systems with three or more bodies can’t be solved in closed form. For any N body problem there are 3 equations for the center of mass, 3 for the momentum, 3 for the angular momentum and one for the energy. These are 10 constraints on the problem. An N-body problem has 6N degrees of freedom. For N = 2 this means the solution is given by a first integral with degree 2. For a three body problem this first integral has degree 8. This runs into the problem that Galois illustrated which is that any root system with degree 5 or greater generally has no algebraic roots. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for 8 solutions there is some eight order polynomial p_8(x) = ?_{n=1}^8(x – ?_n), with 8 distinct roots ?_n that are constant along the 8 solutions in this eight fold product. Since p_8(x) = p_5(x)p_3(x), a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding their roots, or a set of solutions that are algebraic. This means that any system of degree higher than four is not in general algebraic. At the root of the N-body problem Galois theory tells us there is no algebraic solution for N >= 3. Galois theory is a subject outside the scope of this book, but it is the basis of abstract algebra, groups and algebras used in physics.
This reasoning is the basis for a series of theorems by H. Bruns in 1887, which lead to Poincare’s proof that for n > 2 the n-body problem is not integrable. For this reason the stability of the solar system can’t be demonstrated in a closed form solution, but only demonstrated for some finite time by perturbation methods. Poincare won the Sweden prize for this with respect to the solar system. The stability of the solar system was an outstanding problem, and the King of Sweden raised a contest and award for a proof of the stability of the solar system. Previously Laplace had made an attempt at this early in the 19th century, but could not work it in completely. Poincare won the prize for showing the solar system is in fact not stable, but is only quasi-stable for time period much larger than the orbital periods of the planets.
It had been astrophysics “lore” that double star systems could not have planets, for it was thought the perturbing terms would cause planets to not be quasi-stable over long enough time periods. Data of late has thrown these ideas into trash heap. Tatooine type of planets are being found, which suggests otherwise.
LC
There is a bit of confusion here with regards to treating gravitational sources in Newtonian physics. A system with two masses in a mutual orbit has a dipole moment if the mutual orbit is circular. If the orbits are elliptical there is then a quadrupole moment. There is of course a monopole moment for the two body system, which if you measured the Newtonian gravity acceleration at a great distance would dominate. So a two body system appears approximately as a one body system at a great distance D which is much larger than the orbital radial distance R between the two bodies D >> R.
All of this is important if you put a satellite with a mass m <> 2?/?.
The classical mechanics of systems with three or more bodies can’t be solved in closed form. For any N body problem there are 3 equations for the center of mass, 3 for the momentum, 3 for the angular momentum and one for the energy. These are 10 constraints on the problem. An N-body problem has 6N degrees of freedom. For N = 2 this means the solution is given by a first integral with degree 2. For a three body problem this first integral has degree 8. This runs into the problem that Galois illustrated which is that any root system with degree 5 or greater generally has no algebraic roots. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for 8 solutions there is some eight order polynomial p_8(x) = ?_{n=1}^8(x – ?_n), with 8 distinct roots ?_n that are constant along the 8 solutions in this eight fold product. Since p_8(x) = p_5(x)p_3(x), a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding their roots, or a set of solutions that are algebraic. This means that any system of degree higher than four is not in general algebraic. At the root of the N-body problem Galois theory tells us there is no algebraic solution for N >= 3. Galois theory is a subject outside the scope of this book, but it is the basis of abstract algebra, groups and algebras used in physics.
This reasoning is the basis for a series of theorems by H. Bruns in 1887, which lead to Poincare’s proof that for n > 2 the n-body problem is not integrable. For this reason the stability of the solar system can’t be demonstrated in a closed form solution, but only demonstrated for some finite time by perturbation methods. Poincare won the Sweden prize for this with respect to the solar system. The stability of the solar system was an outstanding problem, and the King of Sweden raised a contest and award for a proof of the stability of the solar system. Previously Laplace had made an attempt at this early in the 19th century, but could not work it in completely. Poincare won the prize for showing the solar system is in fact not stable, but is only quasi-stable for time period much larger than the orbital periods of the planets.
It had been astrophysics “lore” that double star systems could not have planets, for it was thought the perturbing terms would cause planets to not be quasi-stable over long enough time periods. Data of late has thrown these ideas into trash heap. Tatooine type of planets are being found, which suggests otherwise.
LC
Copy of post below — deleted
Duncan, no worries. Ill reply to you here because the thread was going to deep.
We are all wrong some time. My concern was just that you seemed to ignore what was said, and i appologize if i turned harsch.