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Space Forum / Astronomy / August 2004Hill Sphere versus Laplace Sphere
Please tell me which one is the real SOI (sphere of influence). Apparently there are two contenders for the title "sphere of influence" : that region of space around an astronomical body within which it can hold satellites in stable orbits around itself. Because the body in question is always(?) itself a satellite of a bigger body, the SOI is calculated by considering the two-body system of body #1 and its primary. Contender # 1 : The Hill sphere is named for George William Hill (1838-1914). It is determined by the Hill radius : r_H = D_sp·(M_s / (3·M_p))^(1/3), where r_H is measured from the center of the body (satellite), D_sp is the center-to-center distance between the satellite and the primary, M_s is the mass of the satellite and M_p is the mass of the primary. http://en.wikipedia.org/wiki/Hill_sphere Apparently it also is determined by L1 and L2, the Lagrange points closest to the satellite. http://stars5.netfirms.com/lagrangian.htm Contender # 2 : Don't bother looking up "Laplace sphere", it's my own coinage. Apparently the first person to calculate a SOI was French astronomer Pierre Simon de Laplace (1749-1827). It is determined by the following radius : r_L = D_sp·(M_s / M_p)^(2/5), where r_L is measured from the center of the satellite, and D_sp, M_s and M_p are defined as above. http://www.go.ednet.ns.ca/~larry/orbits/gravasst/gravasst.html Sincere insights are greatly appreciated. Jake >Please tell me which one is the real SOI (sphere of influence). > [quoted text clipped - 40 lines] > >Jake For some insight, I'd try http://www.ima.umn.edu/talks/workshops/10-29-11-2.2001/koon/tour.pdf Hill's formulation of the problem sounds to me more like it addresses the issue at hand - can the body escape - than does the simplistic quesiton of "where are the forces equal". I'd like to particularly draw your attention to the plots of the "allowed region" on page 12 of the pdf document I mentioned. The figure labelled case 2 should be the case where the body is no longer considered "bound" to the moon. While one can see that that the orbit of the body reaches the L1 point in order to escape (this is the point where the forces balance), one can also see that the orbit isn't exactly circular at this point anymore. So I think the answer to your question may simply lie in what is being measured. Do we measure the distance fromt the moon to the L1 point to determine the "region of influence", or do we more conservatively draw a sphere around the moon and insist that the whole sphere remain inside the "white" energetically allowed region of figure 2, when the body has "just enough" energy to escape the moon? I unfortunately haven't gone through Hill's analysis thoroughly enough to be positive of exactly how he's actually defined the Hill sphere, though I suspect he probably did what I described above. Anyway, I thought that this post might help you out more than it hurt, apologizies in advance if I turn out to be totally wrong :-). In addition http://scienceworld.wolfram.com/physics/JacobiIntegral.html might be helpful as another source / name for the conserved quantity that is being plotted and used to determine the "allowed regions" in the figures I mentioned. >>Don't bother looking up "Laplace sphere", it's my own coinage. >>Apparently the first person to calculate a SOI was French astronomer [quoted text clipped - 5 lines] >>where r_L is measured from the center of the satellite, and D_sp, M_s >>and M_p are defined as above. I've been looking into this some more, and I think much of the problem may lie right here, in the definition of r_L. There is no closed form solution for the location of the Lagrange points, so I suspect that r_L here is not what I thought it was supposed to be, the location of the L1 Lagrange point. There's an exact calculator for the Lagrange points at John Stockton's web hompage http://www.merlyn.demon.co.uk/gravity3.htm#L13 An approximate solution value for small mass ratios is given at http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf which puts the lagrange point at x = (alpha/3)^(1/3)*R, and the satellite at x=(1-alpha)*R alpha = mass of satellite/(mass of satellite+mass of primary) R = distance from satellite to primary Note that there is no 2/5 exponent here, so it can't be the same as your expression. The Hill equation appears to have the right exponent. > >>Don't bother looking up "Laplace sphere", it's my own coinage. > >>Apparently the first person to calculate a SOI was French astronomer [quoted text clipped - 31 lines] > your expression. The Hill equation appears to have the right > exponent. Well that's precisely the question I have. Nobody who accepts Laplace's formulation accepts the inner Lagrange points as the limits of orbital stability. The question is, who's right? I wish I knew how Laplace came up with his setup, because that would make it a much easier call. Maybe a better question would be, how do we measure orbital stability in a way that doesn't involve perturbation theory? >Well that's precisely the question I have. Nobody who accepts Laplace's >formulation accepts the inner Lagrange points as the limits of orbital >stability. Hmmm, I see Lagrange, Hill, and Poincare have done stuff on the 3 body problem, but Laplace? You did make me draw some of my own graphs to make sure I understood this - this has something that I've always meant to do anyway, you've just helped me along.... Unfortunately, there's no convenient way to post these graphs. Consider the Jacobi intergal from http://scienceworld.wolfram.com/physics/JacobiIntegral.html J(x,y)= y^2+x^2+2*(1-u)/((x+u)^2+y^2)^(1/2)+2*u/((x-1+u)^2+y^2)^(1/2)-xdot^2-ydot^2 You'll have to go to http://scienceworld.wolfram.com/physics/RestrictedThree-BodyProblem.html to see the expansion for the expressions for r1 and r2 that give the above usable expression for calculating J J is an invariant of the three body problem, it won't change For specificity, we will take the mass ratio u = .01, that's m2/(m1+m2)=.01, and actually draw some graphs of this function. Because it's essentially the negative of an energy, the region that a body can reach is expressed by the region J > (some constant). J reaches it's absolute maximum when xdot and ydot are zero. So we will plot these regions limited by "zero velocity" xdot=ydot=0. That's the absolute maximum that J can reach, it's very optimistic but obviously a limiting case. With u=.01, we find that the L1 point occurs at approximately xL1val := 1-(.01/3)^(1/3) = .8506198418. This isn't exactly right but it's close enough for our purposes. We evaluate the function J at x=xL1val,y=0, and find it's critical value is about 3.17, slightly less So we draw a nice plot of J>3.17 implicitplot(J1>3.17,x=.7..1.2,y=-.2 .. (.2),filled=true,grid=[100,100]); and we get a nice oval shaped curve, that the body can't get out of around the x=.99 point where the body is. (Too bad I can't post it). When we take J>3.16, we see that the body can escape through the L1 point. when we take J>3.15, we see that the body can escape through L2 (the outer lagrange point) as well as L1 (the inner one) Note that the website says that the Jacobi intergal is the proportional to the Hamiltonian in rotating coordinates. This is true, but they've done a sneaky change of variables - H should really be a function H(x,y,px,py) in this case H=px^2/2m + py^2/2m + w*(px*y-y*px)+V(x,y) where V is the potential energy. Explaining why H is like this is a bit tricky :-(. Hopefully you are somewhat familiar with Hamiltonian mechanics? (cross fingers). You can get xdot and ydot from Hamilton's equation's = xdot = d/dpx(H), ydot=d/dpy(H) (it's really a partial derivative). This analysis is very conservative, because of the drawing of zero velocity surfaces. Of course, in a real system, there will be more than three bodies, so the system will be perturbed in ways that aren't accounted for. > >Well that's precisely the question I have. Nobody who accepts Laplace's > >formulation accepts the inner Lagrange points as the limits of orbital > >stability. > Hmmm, I see Lagrange, Hill, and Poincare have done stuff on the 3 body > problem, but Laplace? Well, from the first page I cited (which I admit isn't very good): "Laplace derived the following for the Radius of the Sphere of Influence ..." http://www.go.ednet.ns.ca/~larry/orbits/gravasst/gravasst.html There's also this page: "Hill Sphere: named after George William Hill ... It is interesting to note that the properties of such sphere were earlier considered by Pierre Simon de Laplace (1749-1827)!" The article writer seems unaware that the SOI Laplace described had a different definition, and therefore a different formulation, than Hill's sphere. http://ksiezyce.republika.pl/encyclopaedia/h_en.html "What's important here is the concept of sphere of influence, developed by Laplace in the late 18th century. Comparing the size of the gravitational forces is too simplistic. One need to look at the ratios of the central forces from each body (sun,earth) to the perturbations from the other body." http://mathquest.com/discuss/sci.math/a/m/179986/179997 "One has to compare the ratios of the gravitational forces to the perturbing effect of the other body. That's what Laplace realized and led to the definition of Sphere of Influence (SOI). The earth's SOI has a radius of about 140 earth radii. All bodies within it are orbiting the earth because the sun's perturbation in proportion to the earth's gravitational force is less than the earth's perturbation in proportion to the sun's force. The moon is well within the SOI at 60 earth radii out. Any book on orbital mechanics will explain this." http://mathquest.com/discuss/sci.math/a/m/179986/180009 Then there's this very interesting exchange: http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&selm=DoH2I4.HGJ%25spenford%40 zoo.toronto.edu Unfortunately, the best page I ever saw on Laplace's SOI has been taken down: http://www.ae.utexas.edu/~lord/voyager2.html Well, I uploaded some of the plots I made with a brief explanatory text to http://www.geocities.com/syzygy303/ so that you and/or another interested reader can see, graphically, how the Jacobi intergal function constrains a "third body" to stay near a satellite, and what happens as this constraint is gradually relaxed. It looks like "The Hill Sphere" is the correct terminiology for this region (a region of no escape as long as there are no "outside" pertubations). The term "Hill Radius" appears to differ in usage from "The Hill Sphere" (there's a factor of (1/3)^(1/3) missing), and the term "sphere of influence" apparently means yet a third thing. Sorry I don't know more (anything, really) about the Laplace SOI, it's not in any of my textbooks, which are really more oriented towards physics than astronomy. I should note that in spite of the name, the Hill Sphere doesn't appear to be particularly spherical in the plots - it looks more like an egg.... > >Please tell me which one is the real SOI (sphere of influence). > > [quoted text clipped - 48 lines] > the issue at hand - can the body escape - than does the simplistic > quesiton of "where are the forces equal". Just to clear up any confusion, there's nothing simple about either Hill's or Laplace's SOI. Neither simply equates the gravities. That result would be r_E = D_sp·(M_s / M_p)^(1/2) One problem I have is that I get a different result than Hill did for the L1 distance from the secondary body (the body of interest), even though I understand the logic. I get the same result that Laplace did for his SOI radius, even though I *don't* understand his logic. Very frustrating. > I'd like to particularly > draw your attention to the plots of the "allowed region" > on page 12 of the pdf document I mentioned. The figure labelled case > 2 should be the case where the body is no longer considered > "bound" to the moon. Comets aren't true moons of Jupiter. Unless I completely misinterpreted the pdf, those plots were of allowable comet trajectories, not permanent orbits. I define the SOI as that region around a body in which a smaller object set into orbit will remain so forever, barring any major disruptions from another body coming from outside the SOI. > While one can see that that the orbit of the > body reaches the L1 point in order to escape (this is the point where [quoted text clipped - 20 lines] > that is being plotted and used to determine the "allowed regions" in > the figures I mentioned. (about the L1 point) The equation to solve for the position of L1 that I get is eq1 := -x+(1-u)/(x+u)^2-u/(x-1+u)^2 = 0 where u=m2/(m1+m2) and m1+m2=G=w=1. Or alternately m1, the primary, has mass 1-u and is located at x=-u m2, the secondary, has mass u and is located at 1-u (so the distance d between bodies is also 1). Note that (1-u)*-u + u*(1-u) = 0, so the center of mass is at the origin. Also note that since G=d=1, w=G*(m1+m2)/d^3 = 1 Solving it is another matter. I was able to duplicate the result from the paper I quoted and even add some terms when I _assumed_ that the solution was 1-k1*u^(1/3), but I don't see how to justify this assumption. w^2*x is the "centrifugal force", it points to the right, and w=1 so w^2*x = x. (1-u)/(x+u)^2 is the force towards the primary, it points to the left u/(x-1+u)^2 is the force towards the moon, it points to the right You say "Laplace", are you sure you don't mean "Lagrange"? > Comets aren't true moons of Jupiter. Unless I completely misinterpreted > the pdf, those plots were of allowable comet trajectories, not permanent [quoted text clipped - 3 lines] > set into orbit will remain so forever, barring any major disruptions > from another body coming from outside the SOI. This is not a usefull definition. First of all, I've not met anybody that uses this definition. 'set into orbit' is not a very concise. You should at least specify the energy and the eccentricity of the orbit. Orbits with the same energy but with different eccentricities may behave very differently. This of course applies to orbits with different energies also. Finally, it is almost impossible to prove that an orbit is stable. It can only be done in the most simplistic cases like the two body or the circular restricted three body problem. (There are indications that Mercury's orbit may not be stable. I would recommend the article "Large scale chaos in the solar system" in Astron. Astrophys. 287, L9-L12 by Laskar 1994.)  SignatureØystein Olsen, oystein.olsen_at_astro.uio.no, http://folk.uio.no/oeysteio Institute of Theoretical Astrophysics, http://www.astro.uio.no University of Oslo, Norway > Please tell me which one is the real SOI (sphere of influence). > > Apparently there are two contenders for the title "sphere of influence" > : that region of space around an astronomical body within which it can > hold satellites in stable orbits around itself. This is not a common definition of the the SOI. (I've never seen this usage of SOI in any text-books or articles.) A more common definition would be a surface centered on the secondary, within which it is more convenient(*) to analyze the orbit as centered on the secondary and perturbed by the primary. Contender two is the usual value for the SOI. (*) Consider an object to first be in an orbit around the primary and then to be in an orbit around the secondary. Let 'Ap' and 'As' be the accelerations due to the primary and secondary. Let 'Ps' and 'Pp' be the perturbing accelerations due to the secondary and primary. The SOI is then defined by the surface: Ps/Ap = Pp/As Roughly speaking, within the SOI the orbit is more similar to a conic section around the secondary than an orbit around the primary. > Contender # 1 : > > r_H = D_sp·(M_s / (3·M_p))^(1/3), > Contender # 2 : > > r_L = D_sp·(M_s / M_p)^(2/5), SignatureØystein Olsen, oystein.olsen_at_astro.uio.no, http://folk.uio.no/oeysteio Institute of Theoretical Astrophysics, http://www.astro.uio.no University of Oslo, Norway > > Please tell me which one is the real SOI (sphere of influence). > > [quoted text clipped - 4 lines] > This is not a common definition of the the SOI. (I've never seen this usage > of SOI in any text-books or articles.) Maybe this one is better: "The "sphere of influence" is the volume within which a mass can hold satellites permanently bound to itself." http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&selm=94339.011616LABBEY%40git vm1.gatech.edu > A more common definition would be a > surface centered on the secondary, within which it is more convenient(*) to [quoted text clipped - 11 lines] > Roughly speaking, within the SOI the orbit is more similar to a conic > section around the secondary than an orbit around the primary. Yes, this is what Laplace conjectured. My question is, why? What led him to consider the *ratios* as the important thing to consider, as opposed to, say, the differences? I can't follow the line of reasoning here that led to his conclusion. > > Contender # 1 : > > [quoted text clipped - 8 lines] > Institute of Theoretical Astrophysics, http://www.astro.uio.no > University of Oslo, Norway > > Apparently there are two contenders for the title "sphere of influence" > > : that region of space around an astronomical body within which it can [quoted text clipped - 5 lines] > Maybe this one is better: "The "sphere of influence" is the volume > within which a mass can hold satellites permanently bound to itself." This is basically the Hill-surface or Hill-sphere. The volume is (of course) related to the concept of Hill-stability. This definition of SOI is sensible, but it is not very common. I checked through several text-books at our library too, none of them uses this definition. d*(ms/mp)^(2/5) is used as the SOI in all of them. These two definitions will in most cases have reasonable similar values, so I would recommend using the most common meaning of SOI if only to avoid as many misunderstandings as possible. > The SOI is >> then defined by the surface: [quoted text clipped - 8 lines] > opposed to, say, the differences? I can't follow the line of reasoning > here that led to his conclusion. Basically, you're interested in effect of |Ps| on |Ap| compared to |Pb| on |As|. But what does the two differences really tell you? If the numbers are of very different magnitudes, you are basically comparing apples and oranges. In some way you must define what you mean by a perturbation on a two-body orbit. If we consider the Moon to orbit the Sun, then the deviation from a two-body orbit will be larger in absolute units than the deviation from a two-body orbit around the Earth. Is this entirely relevant? In most cases the answer is no. It's better to look at the ratios or the relative sizes of the perturbations. Let 'o' be an orbital element and 'Do' be the change in the orbital element due to the perturbations. How fast the series-expansions converge in an analytical solution will depend on the fraction 'Do/o' and not simply 'Do'. This is one reason to use the ratios and not simply the differences. Let's take a simple example. In one drink i mix in 1 unit of alcohol and 10 units of water. In the second drink I use 5 units of alcohol but 100 units of water. The differences are 9 and 95 units, but the fractions are 0.1 and 0.05. If you're only served 11 units, the first drink is clearly preferable:-) SignatureØystein Olsen, oystein.olsen_at_astro.uio.no, http://folk.uio.no/oeysteio Institute of Theoretical Astrophysics, http://www.astro.uio.no University of Oslo, Norway > > > Apparently there are two contenders for the title "sphere of influence" > > > : that region of space around an astronomical body within which it can [quoted text clipped - 13 lines] > have reasonable similar values, so I would recommend using the most common > meaning of SOI if only to avoid as many misunderstandings as possible. One big problem I have with the Hill radius is that I can't derive it! First I assumed that the relative acceleration of L1 to the Earth was zero. This gave the result: r_test = D_sp·(M_s / (2·M_p))^(1/3) , not the Hill radius. Second, I assumed the angular velocity of L1 was the same as Earth's. I got the same result: not the Hill radius. Grrrrrrr! > > The SOI is > >> then defined by the surface: [quoted text clipped - 14 lines] > oranges. In some way you must define what you mean by a perturbation on a > two-body orbit. I tried equating the relative accelerations, something I never did before, for some reason. I got r_test = D_sp·(M_s / (2·M_p))^(1/2) Plugging in the numbers for the Earth-Sun system, this doesn't even reach halfway to the moon. So obviously that isn't the right answer. > If we consider the Moon to orbit the Sun, then the deviation from a > two-body orbit will be larger in absolute units than the deviation from a [quoted text clipped - 11 lines] > 0.05. If you're only served 11 units, the first drink is clearly > preferable:-) Yeah, but force vectors aren't drinks and I still don't get it. :-( At least I can derive Laplace's radius from his conjecture. I guess I'll have to settle for that. :-\ > -- > Øystein Olsen, oystein.olsen_at_astro.uio.no, http://folk.uio.no/oeysteio > Institute of Theoretical Astrophysics, http://www.astro.uio.no > University of Oslo, Norway > One big problem I have with the Hill radius is that I can't derive it! > First I assumed that the relative acceleration of L1 to the Earth was [quoted text clipped - 4 lines] > Second, I assumed the angular velocity of L1 was the same as Earth's. I > got the same result: not the Hill radius. Grrrrrrr! This is how it is derived in "Solar System Dynamics" by C.D. Murray and S.F. Dermott 1) Start with the circular restricted three-body problem. The bodies mu1 and mu2 are located one the x-axis at -mu2 and mu1 respectively. 2) For simplicity set the mean motion, n to 1. 3) Assume small mass ratios. i.e. mu1 >> mu2. Replace mu1 with 1. Keep mu2 where it's multiplied with a fraction, but let x + mu2 -> x. 4) Do the transform x -> 1 + x 5) Assume the motion to be close to mu2. Then x, y, and r2 are small quantities of order mu2^(1/3). Neglect higher power of mu2. 6) Use r1 ~ sqrt(1 + 2x) 7) Remove small quantities 8) Let the radial force vanish and solve for r2. r2 is the Hill sphere. > Yeah, but force vectors aren't drinks and I still don't get it. :-( > > At least I can derive Laplace's radius from his conjecture. I guess I'll > have to settle for that. :-\ The important thing in both analytical and numerical studies are sizes of the perturbations relative to the orbit. This is why ratios are used. SignatureØystein Olsen, oystein.olsen_at_astro.uio.no, http://folk.uio.no/oeysteio Institute of Theoretical Astrophysics, http://www.astro.uio.no University of Oslo, Norway JRS: In article <41106255.E27CE3A1@NOSPAMhobonet.com>, dated Tue, 3 Aug 2004 21:13:09, seen in news:sci.astro, Jake <jakmal@NOSPAMhobonet.com> posted : >I tried equating the relative accelerations, something I never did >before, for some reason. I got [quoted text clipped - 3 lines] >Plugging in the numbers for the Earth-Sun system, this doesn't even >reach halfway to the moon. So obviously that isn't the right answer. The gravitational field from the Sun affects the Earth and the Moon equally, since they are at the same average distance. It is the gradient of the solar field, possibly aided by higher-order terms, that tends to break the Moon out of orbit. The Sun's gravity gradient (GM/R^3) is matched by the Earth's (Gm/r^3) when r = R(m/M)^1/3; with M = 2E30 kg, m = 6E24 kg, R = 150E6 km, this gives r = 1.2E6 km. That is a factor of 1.44 higher than the oft-quoted Hill expression should give; evidently well within the outfield. 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; some Astro stuff via astro.htm, gravity0.htm; quotes.htm; pascal.htm; &c, &c. No Encoding. Quotes before replies. Snip well. Write clearly. 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