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Space Forum / Astronomy / April 2005Fundamental and observational allowed ranges of planetary systems
Can anyone suggest good sources on the Internet on what are various multibody systems like and how they compare with each other? I find contradicting assertions on various points. 1) Hill "sphere" or "radius". It characterizes three-body systems where two bigger ones have very different masses. How far can a satellite orbit a planet without being removed by perturbations from the primary? There are expressions of this as a distance, expressed through masses of the primary and secondary and the distance between them. This seems stupid! The orbital period of course also scales with cube root of mass, so it seems the best to compare the orbital periods. So what are the allowed orbital periods for a tertiary body? Some analytical arguments claim that the limit is a third of secondary orbital period, others that it is a square-root-three ratio that is critical. And is there, in fact, a Hill "sphere", or does the stability of distant tertiaries depend on the eccentricity, or inclination, or perhaps even tbe sense of orbit? Observationally, Jupiter has satellites with periods as long as 982 says, which is nearly quarter of the secondary period. These have great eccentricities and also great inclinations - but closer to coplanar than perpendicular. Interestingly, all distant satellites are retrograde - the furthest prograde satellite has period of 456 days, less than a ninth of the secondary period. 2) Lagrange points. As for the equilaterar triangle, I have seen the proof that it only is stable if the ratio between primary and secondary masses is bigger than 26. Are there any constraints on the mass of the tertiary body? Other than not being, in fact, a secondary with a mass too great compared to that of the primary. 3) Titius-Bode law. Can anyone point to some places where the spacings of orbital periods are compared for Solar System, the systems of Jupiter, Saturn and Uranus, and known extrastellar planetary systems and multiple - above double - stars, and eccentrities and inclinations also given, and comparisons of masses? > chornedsnorkack@hushmail.com wrote: > Can anyone suggest good sources on the Internet on what are various [quoted text clipped - 10 lines] > stupid! The orbital period of course also scales with cube root of > mass, so it seems the best to compare the orbital periods. What is the definition of "Hill Sphere"? I thought it was where (in terms of the satellites distance from the planet) the gravitational force between satellite and planet equals the gravitational force between satellite and star (Sun). This seemed to make good sense to me. I read about the "Hill SPhere" recently in Astronomy Magazine. I didn't work out equations though but it would be: F = G*Ms*m/rs^2 = G*Mp*m/rh^2 ==> rh = rs * SQRT(Mp/Ms) ... where Mp = planet mass Ms = star mass rh = the Hill radius (distance between satellite and planet) rs = distance between satellite and star (which we approximate as the distance between planet and star since rh << rs one assumes). Did I get the equation right? One could assume other definitions and make other approximations and arrive at other "more accurate" equations of course I'm sure ... > So what are the allowed orbital periods for a tertiary body? Some > analytical arguments claim that the limit is a third of secondary [quoted text clipped - 4 lines] > distant tertiaries depend on the eccentricity, or inclination, or > perhaps even tbe sense of orbit? If the gravitational force between the satellite and star is greater than that between the satellite and planet then the satellite is "bound" more to the star than planet and yes, "Hill Sphere" makes alot of sense. > Observationally, Jupiter has satellites with periods as long as 982 > says, which is nearly quarter of the secondary period. These have great [quoted text clipped - 22 lines] > double - stars, and eccentrities and inclinations also given, and > comparisons of masses? Why not do this yourself (for Jupiter, Saturn, Uranus & Neptune)? You'd have to use a different scale factor though than the AU (distance between Sun and Earth). There might be some interesting results pop out. I seem to remember reading somewhere that the orbits of the planets about a pulsar did look like they "obeyed" Bode's Law. Sorry, I don't remember the reference ... > > chornedsnorkack@hushmail.com wrote: > > Can anyone suggest good sources on the Internet on what are various [quoted text clipped - 23 lines] > rh = the Hill radius (distance between satellite and planet) > rs = distance between satellite and star (which we approximate as the > distance between planet and star since rh << rs one assumes). > > Did I get the equation right? One could assume other definitions and > make other approximations and arrive at other "more accurate" equations > of course I'm sure ... I prefer other definitions. After all, it seems to me that if the gravitational forces due to the primary on the secondary and tertiary are nearly equal at all times, so that the attraction between secondary and tertiary is stronger than the difference, it seems secondary and tertiary would be bound. > > So what are the allowed orbital periods for a tertiary body? Some > > analytical arguments claim that the limit is a third of secondary [quoted text clipped - 41 lines] > I seem to remember reading somewhere that the orbits of the planets > about a pulsar did look like they "obeyed" Bode's Law. Sorry, I don't > remember the reference ... Well, let us try examples: Solar system gas giants: Taking Jupiter's period as 100 %, the other periods are Saturn 248,4 % (close to 5:2 resonance but not exact - are they resonant at all?), Uranus 708,3 %, Neptune 1389,5 %. The other ratios would be Uranus/Saturn 285,2 %, Neptune/Saturn 559,4 %, Neptune/Uranus 196,2 %. The mass ratios, with Jupiter as 100 %, are Saturn 29,9 %, Uranus 4,6 %, Neptune 5,4 %. The next lighter planet, Earth, is about 0,3 % and the Sun has 1050 times the mass of Jupiter. The inner rocky planets: Taking the period of Earth as 100 %, the orbital periods are Mercury 24,1 %, Venus 61,5 % (close to 8/13 - is it a resonance?), Mars 188,1 %. The other ratios are Venus/Mercury 255 %, Mars/Mercury 780%, Mars/Venus 306%. Masses Earth 100 %, Mercury 5,5 %, Venus 81,5 %, Mars 10,7 %. The Galileian satellites of Jupiter: Taking the period of Ganymedes as 100 %, Io and Europa are said to be resonant, with Europa's period 50 % and for Io, 25 %. However, checking the periods from http://nssdc.gsfc.nasa.gov/planetary/factsheet/joviansatfact.html gives the period ratio between Europa (given as 3,551181 days) and Ganymede (7,154553) to be 201,5 % instead of exactly 200 %. The ratio between Io (1,769138) and Europa is 200,73 %. Can anyone explain the movement of the inner three Galileian satellites? In any case, the period of Callisto is 233,26 % that of Ganymede. Suspiciously close to 7/3. Is it also a resonance? The mass ratios are, taking Ganymede as 100 %, Io 60,3 %, Europa 32,4 %, Callisto 72,6 %. No other satellite of Jupiter has 0,007 %. I wonder what is the significance of 1/2 resonance? The asteroid belt has a prominent Kirkwood gap near this resonance with Jupiter. How can the Galilean satellites be in 1/2 orbital resonances? chornedsnork wrote: > ...close to 5:2 resonance... close to 8/13... Suspiciously close to 7/3... I think you may be overdoing it here. First and formost, *any* rational number can be written as a ratio of integers - resonances are only important for very low integer ratios, like 1:2 or 2:3, etc. Given enough periods in the solar system, I'm certain you can find almost any ratio you want. And if you want to uncover resonances, try Saturn's system (Mimas-Tethys, Enceladus-Dione, , Titan-Hyperion, all the "lagrange" satellites as well as the co-orbitals). > Can anyone explain the movement of the inner three > Galileian satellites?... How can the Galilean satellites > be in 1/2 orbital resonances [when the Kirkwood 1/2 > resonance is empirically unstable]? That's a very, very good question. It has to do with tidal evolution of the orbits and the liberation of the line of the apsides of the mutual orbits. Very roughly, tidal effects with Jupiter are expanding Io's orbit, which then falls into a resonant relationship with Europa. At this point if the orbits are not perfectly circular, the resonance relationship isn't completely symmetric about the line of the apsides, resulting in a liberation of the orbit about that line, and resonant reactions now expand Europa's orbit in lock-step with Io. Once the pair expands the orbits enough, they fall into a resonant relationship with Ganymede, and now you have three in lock-step. To be honest, I can't claim I can fully understand it myself (I do when I read it; if you can find a copy, look at Lewis, "Physics & Chemistry of the Solar System", pg255), as I admit I've not yet tried to teach this in my 100-level class (& if I can't teach it to a 100-level, I really don't consider it understood). Consulate a good book on orbital mechanics and resonances. And never, *ever* ask about the "nu-dot-6" resonance <grin>. The more you learn, the more you realize of bloody little you actually know... or at least that's been my experience.  SignatureBrian Davis > chornedsnork wrote: > [quoted text clipped - 40 lines] > more you learn, the more you realize of bloody little you > actually know... or at least that's been my experience. What is a nu-dot-6 resonance? I'm creating space strategy so some realistic planetary orbits are necessary. My current biggest problem is, I had hard time to discover Gravitational constant. > What is a nu-dot-6 resonance? Ugh, that will teach me to open my big electronic mouth. The "nu-dot-6 resonance" refers to a specific term in a rather length perturbation equation. It has to do with how some main belt asteroids get ejected into Earth-crossing trajectories (IMS). I brought it up as an example of a resonance I *completely* do not understand. > I'm creating space strategy so some > realistic planetary orbits are necessary. > My current biggest problem is, I had > hard time to discover Gravitational constant. Then you can certainly ignore such odd resonances as mentioned above. SignatureBrian Davis >> What is a nu-dot-6 resonance? > [quoted text clipped - 4 lines] > (IMS). I brought it up as an example of a resonance I > *completely* do not understand. Interesting. Something simple like this? http://adsabs.harvard.edu/abs/1985Metic..20....1W >> I'm creating space strategy so some >> realistic planetary orbits are necessary. >> My current biggest problem is, I had >> hard time to discover Gravitational constant. Some citiations from articles about the problem. "One of nature's venerable constants - gravity - may not be the same for every type of particle in the universe, suggest new calculations." "Recently the value of G has been called into question by new measurements from respected research teams in Germany, New Zealand, and Russia. The new values disagree wildly. For example, a team from the German Institute of Standards led by W. Michaelis obtained a value for G that is 0.6% larger than the accepted value; a group from the University of Wuppertal in Germany led by Hinrich Meyer found a value that is 0.06% lower, and Mark Fitzgerald and collaborators at Measurement Standards Laboratory of New Zealand measured a value that is 0.1% lower" > Then you can certainly ignore such odd resonances as > mentioned > above. I would need to do resonances when I'd be removing wrong planets from simulation, and modify theirs parameters. > chornedsnork wrote: > [quoted text clipped - 6 lines] > enough periods in the solar system, I'm certain you can find almost any > ratio you want. Well, 2:5 and 3:7 are definitely relevant resonances, since they create Kirkwood gaps: http://ssd.jpl.nasa.gov/a_histo.html > And if you want to uncover resonances, try Saturn's system > (Mimas-Tethys, Enceladus-Dione, , Titan-Hyperion, all the "lagrange" > satellites as well as the co-orbitals). Yes, let us do it. Taking Rhea as 100 % periods: Dione 60,6 %; Tethys 41,8 %; Enceladus 30,3 %; Mimas 20,8 %; Titan 353 %; Hyperion 471 %; Iapetus 1756 %. with ratios Dione:Enceladus 199,8 %; Tethys:Mimas 200,4 %; Titan:Hyperion 74,95 %. masses: Dione 47,8 %; Tethys 26,9 %; Enceladus 3,7 %; Mimas 1,7 %; Titan 5870 %; Hyperion 0,5 %; Iapetus 85,7 %. > > Can anyone explain the movement of the inner three > > Galileian satellites?... How can the Galilean satellites [quoted text clipped - 9 lines] > resulting in a liberation of the orbit about that line, and resonant > reactions now expand Europa's orbit in lock-step with Io. But why is this relationship stable? > Once the pair > expands the orbits enough, they fall into a resonant relationship with [quoted text clipped - 8 lines] > more you learn, the more you realize of bloody little you actually > know... or at least that's been my experience. > What is the definition of "Hill Sphere"? Operationally, it is termed the "zero velocity curve", and has to do with the relative velocities of two objects in the presence of a third (and in a rotating reference frame). > I thought it was where... the gravitational force > between satellite and planet equals the gravitational > force between satellite and star (Sun). Nope. As a stability criterion, that ignores the fact that the planet & satellite essentially "share" an orbit (the planet-satellite relative velocity is far less than the Sun-planet relative velocity, for instance). The next step up would be the sphere of influence, (based on the ratio of the perturbation and direct attractions of the teritary to the primary & secondary). And the best guess we've got (well, that I've got anyway) is the Hill radius (1/3 a_hill actually). SignatureBrian Davis > Nope. As a stability criterion, that ignores the fact that the > planet & satellite essentially "share" an orbit (the planet-satellite > relative velocity is far less than the Sun-planet relative velocity, > for instance). It's also worth mentioning that the Sun attracts the Moon more strongly than the Earth attracts the Moon. Signaturehttp://hertzlinger.blogspot.com Chornedsnork wrote: > Can anyone suggest good sources on the Internet... I find > contradicting assertions on various points. In that case, first & formost, beyond the internet or reasonably intelligent sounding post (hopefully including this one) - Go To The Printed Sources!! That said... > 1) Hill "sphere" or "radius"... How far can a satellite orbit a > planet without being removed by perturbations from the primary? Well, empirically, satellite orbits seem to be stable out to about 1/3 a_hill (retrograde orbits, again empirically, are a little better, stable out to 1/2 a_hill). Here I'm using a_hill as: a_hill = a (m/3M)^(1/3) where a = distance between the secondary (planet) and primary (star) m,M = masses of the secondary & primary, respectively. > There are expressions of this as a distance... This seems > stupid! It may seem stupid, but it's how Hill and others worked out the problem when dealing with the Earth's Moon. If you don't want it that way, do the math. Heck, I'll do it here. Instead of the size of the Hill sphere, the orbital period of an object on the Hill sphere is: P_hill = P / 3^(1/2) where P = orbital period of the secondary (planet) around the primary (star) > Some analytical arguments claim that the limit is a third > of secondary orbital period... Interesting. Do you have a reference for this? > ...others that it is a square-root-three ratio that is critical. Well, that's what I got in the derivation above but for the Hill radius, not the stability issue. As far as references, IMS I got the "1/3 a_hill" limit from the book "Satellites" published by U of Az Press (Burns was the editor). It's a very good starting point for this stuff (although at a upper undergraduate to graduate level, but it's really not bad; review articles mostly). > does the stability of distant tertiaries depend on the > eccentricity, or inclination, or perhaps even tbe sense > of orbit? Yep, also the presence or absence of resonances, other bodies, etc. There are some generalities that can be sussed out (retrograde orbits tend to be more stable due to shorter synodic periods with respect to the system, for instance), but in general you are creeping up on the general three-body problem (worse, the "real solar system" is far from a three body system), not a solvable problem in closed form. > Observationally, Jupiter has satellites with periods as long > as 982 [days?]... I missed that: the longest period I know of for a bound object is Sinope, at 758 days. What object are you refering to? > These have great eccentricities and also great > inclinations... And are almost certainly captured objects. The high eccentricity & inclination is relic of the capture orbits - that far out, with such a long dynamical time and low tidal interactions, circularization of the orbit takes long than the age of the solar system. > the furthest prograde satellite has period of 456 days... Dang, I'm obviously way behind on the curve of "new Jovian satellites". Sorry 'bout that. > 2) Lagrange points. > [quoted text clipped - 3 lines] > > Are there any constraints on the mass of the tertiary body? If you've seen the actual proof, you've probably seen a reasonable derivation of the Lagrange solutions. This assumes the third body has a negligable mass. In other words, it is assumed the third body has no perturbing influence on the two larger objects in the problem, and their orbits can be treated as true two-body problems. I don't think we have any measured masses for the Lagrange point satellites in the Saturn system, but given their sizes and the measured mass of the secondary you can estimate the mass ratio of secondary to teritary as about 3e-6 as the largest example we've got empirically (Tethys & Calypso, assuming Calypso is around 2000 kg/m^3). > 3) Titius-Bode law. > > Can anyone point to some places where the spacings > of orbital periods are compared for Solar System, the > systems of Jupiter, Saturn and Uranus, and known > extrastellar planetary systems [etc]... No, but you could certainly do it yourself. While Bode's law is seriously flawed, an expanding spacing for the planets makes sense... again, in terms of the Hill sphere. For each object accreting in the solar nebula, the outer ones would have far larger Hill spheres (from which they might accrete materials), so outer accreting planets sweep out larger annuli (?) of the solar nebula. SignatureBrian Davis JRS: In article <1114013451.060407.124380@f14g2000cwb.googlegroups.com> , dated Wed, 20 Apr 2005 09:10:51, seen in news:sci.astro, brdavis@iusb.edu posted : >Chornedsnork wrote: >> 1) Hill "sphere" or "radius"... How far can a satellite orbit a >> planet without being removed by perturbations from the primary? [quoted text clipped - 7 lines] > a = distance between the secondary (planet) and primary (star) > m,M = masses of the secondary & primary, respectively. Is it known why the name is attached to that particular value, when gravity gradient equality is where r^3/m = R^3/M and stability is at about a_hill/3? Does that value itself have physical significance? Obviously the Hill radius, whatever its physical significance, can define a sphere. But does the physical significance apply to the sphere, or just to the coplanar circle? >P_hill = P / 3^(1/2) FWIW, confirmed. Since mass is proportional to density and to cube of size, and since the densities of relevant Solar System bodies are within the range 600 to 6000, a factor of 10, so the cube root varies by no more than a factor of 1.5 each way from 12.5, it follows that : if one finds oneself somewhere near a secondary, one can make a preliminary estimate of the possibility of being in a stable orbit around it by comparing the angular diameters of primary and secondary. Signature© John Stockton, Surrey, UK. ?@merlyn.demon.co.uk Turnpike v4.00 MIME. © Web <URL:http://www.merlyn.demon.co.uk/> - FAQqish topics, acronyms & links; Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc. No Encoding. Quotes before replies. Snip well. Write clearly. Don't Mail News. John Stockton wrote: > Is it known why the name is attached to that particular > value, when gravity gradient equality is where > r^3/m = R^3/M and stability is at about a_hill/3? Does > that value itself have physical significance? You know, I honestly don't know - but I'll try to dig it up. I've run across Hill's radius in numerous situations, but I've not gone back to Hill's original work on the subject. > Obviously the Hill radius, whatever its physical > significance, can define a sphere. But does the > physical significance apply to the sphere, or just > to the coplanar circle? I strongly suspect it is derived for the case of three coplaner bodies, as a conservative estimate for the effects (i.e.- "if I'm safe inside a_hill in the co-planner case, I should certainly be constrained in the non-coplanner one"), but again I really don't know. And I can't find any orbital mechanics books here at work (ironicly enough, a university physics dept; go figure). i'll see what I can dig up tonight at home. SignatureBrian Davis > There are expressions of this as a distance, expressed through masses > of the primary and secondary and the distance between them. This seems [quoted text clipped - 5 lines] > orbital period, others that it is a square-root-three ratio that is > critical. It's expressed through the masses, because the Hill radius is a characteristic of the gravitational fields, and gravitational fields are created by mass. If you want it in terms of orbital period, that's certainly just as easy to come up with, just plug in the Hill radius to the equation which determines the orbital period. > 2) Lagrange points. > > As for the equilaterar triangle, I have seen the proof that it only is > stable if the ratio between primary and secondary masses is bigger than > 26. Only L4 and L5 are stable under this criterion; L1 through L3 are never stable. > 3) Titius-Bode law. > [quoted text clipped - 3 lines] > double - stars, and eccentrities and inclinations also given, and > comparisons of masses? I've seen numerous places on the Web over the years; just use Google. There's nothing particularly meaningful to the Titius-Bode law; all it really does is say that the spacings of the orbits are spaced out in a roughly exponential manner, for varying degrees of "roughly." You can always tweak things a little bit to make them fit better, and with most "fits" the question is really whether or not anything meaningful is underlying the law, or whether the cleverness is in simply in the one doing the tweaking. SignatureErik Max Francis && max@alcyone.com && http://www.alcyone.com/max/ San Jose, CA, USA && 37 20 N 121 53 W && AIM erikmaxfrancis The great artist is the simplifier. -- Henri Amiel >There's nothing particularly meaningful to the Titius-Bode law; all it >really does is say that the spacings of the orbits are spaced out in a [quoted text clipped - 3 lines] >underlying the law, or whether the cleverness is in simply in the one >doing the tweaking. Interestingly, the four large moons of Jupiter follow a Bode-like law with different parameters (each moon's orbital period is twice that of the previous one) and Uranus' five largest moons follow yet another different Bode-like law. http://www.floridastars.org/9605cohe.html has details. It's possible that some process causes planets to tend to form with some sort of regular spacing, but not limited to just one particular pattern. > Interestingly, the four large moons of Jupiter follow a Bode-like law > with different parameters (each moon's orbital period is twice that of [quoted text clipped - 3 lines] > form with some sort of regular spacing, but not limited to just one > particular pattern. Well there's a hidden bias there, because the smaller the number of data points, the easier it is to find a fit. You can find a perfect exponential fit with any three orbits. Finding a fairly good one with four or five doesn't really say anything profound, other than that the orbits tend to be spaced out further apart as you move away from the central body. That doesn't exactly seem at all surprising, since the protoplanetary disk thinned out far away from the center, too. Fitting to an exponential equation of the form R = A B^n where n is the orbit number doesn't give all that _great_ fits for the Galilean satellites of Jupiter or Uranus' big five, either: max@oxygen:~/projects/stella% ./stella.py < jupiterGalilean.st A = 4.131e+02, B = 1.642e+00 n name | actual computed error 1 Io | 4.22e+02 4.13e+02 (-2.0%) 2 Europa | 6.71e+02 6.78e+02 (+1.1%) 3 Ganymede | 1.07e+03 1.11e+03 (+4.0%) 4 Callisto | 1.88e+03 1.83e+03 (-3.0%) (2.5%) average max@oxygen:~/projects/stella% ./stella.py < uranus.st A = 1.293e+02, B = 1.468e+00 n name | actual computed error 1 Miranda | 1.29e+02 1.29e+02 (-0.1%) 2 Ariel | 1.91e+02 1.90e+02 (-0.7%) 3 Umbriel | 2.66e+02 2.78e+02 (+4.6%) 4 Titania | 4.36e+02 4.09e+02 (-6.2%) 5 Oberon | 5.84e+02 6.00e+02 (+2.8%) (2.9%) average SignatureErik Max Francis && max@alcyone.com && http://www.alcyone.com/max/ San Jose, CA, USA && 37 20 N 121 53 W && AIM erikmaxfrancis The color is red / Under my shoe -- Neneh Cherry |
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