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By which I mean, the area around a Lagrange point where the linear-force approximation of the 3-body system defined at the Lagrange point itself is "good".

Background Consider the restricted 3-body problem, where there are two masses that affect each other and a test one which feels the gravity (but does not effect) the other two. In this setting, the sphere(oid) of influence of a smaller of the two main masses (ie planet) is the region of space where the dynamics of the test mass is better approximated by considering only the gravitational force from the planet than only that from the star.

If we restrict the star and planet to moving around each other in a circle, we find that there are 5 Lagrange points where the test particle can be placed at rest with respect to the two principle masses (ie, in a co-rotating frame) and feel no net force. This naturally invites us to expand the total 3-body forces (including things the fictitious centrifugal Coriolis forces from being in a rotating frame) acting on that test mass as a Taylor series about a Lagrange point. If we keep the linear terms (or, equivalently, the quadratic terms in the series expansion of the pseudo-potential), we get the linearized approximation of the forces at that Lagrange point. This approximation is hugely useful and has been extensively studied for centuries because it can be solved exactly (unlike the full 3-body problem) and so can be used to determine the stability and general structure of the dynamics near the Lagrange point.

My question is, how far out is this linear approximation good? Specifically, how far away can we place the test mass before this approximation of the forces acting on it is less good than the 2-body approximation of only considering the star's or only the planet's gravity? By analogy with how the Sphere of Influence of the planet is defined, I'm calling this region of space where the linearized forces are a better approximation than the alternatives the area of influence of a Lagrange point.

Not stability. Note that I'm not asking about the region of stability. That's a completely different question. L1, L2, L3 have no stable orbits, yet it's a mathematical certainty that as you get close to these points the linear approximation of the forces become arbitrarily good. Indeed, this linearized approximation around L2 predicts unstable orbits that are very similar to (if not exactly the same as) the halo orbits we see at L2.

Is this a meaningful thing to define? I think it ought to be, because basic calculus tells us that as you approach the Lagrange point the linearized approximation becomes better, and matches exactly at the point itself. Very far away, this same approximation is obviously terrible. Hence, there must be a cross over point.

Has this been done already? I'd assume so, but I couldn't find anything on the matter online or in a textbook on the matter. It seems like it would be useful though, as it could be used to improve the patched conics approximation (moving between conic orbits around masses, and linear dynamics around Lagrange points; all of which can be solved exactly without numerical integration).

How could this range of influence be calculated? My intuition is that you could do it by taking the 3rd derivative of the total potential at the Lagrange point, and look for the distance where this contribution is of the same size as the 2nd derivative. Aka, look for where the neglected terms in the approximation start to matter. I'm not sure that this would be accurate in practice however. There's also the problem that the potential around the Lagrange point includes a velocity-dependent term (from the Coriolis fictitious force), which might potentially mean that the region of influence would be velocity dependent? Does this render the whole concept meaningless? Or does it just mean that we have to think of it as being in phase space?

EDIT:

Seeing some interest but a lack of answers, I did some preliminary numerical exploration. Below are examples of which types of approximation has the smallest error compared to the real dynamics (yellow is the heavier body, green the smaller one, and blue the linearized equations of motion around the respective Lagrange point):

enter image description here enter image description here enter image description here

The top one has a mass ratio of ~1/10 (similar to the Pluto-Charon system), the middle one ~1/82 (Earth-Moon) and the bottom one ~1/1000 (Jupiter-Sun). Note that because I'm ignoring the z axis and eccentricity completely, these shouldn't be considered accurate of those particular systems, I just wanted to show the dependency on mass ratio. All the forces were computed in a co-rotating frame with the principle two body dynamics and so including centrifugal forces. Coriolis forces are not included because they are linear in velocity and so represented exactly in the linear approximations at the Lagrange points. Note that this doesn't exactly match the definition of Sphere of Influence as defined in places like here, but it wasn't obvious to me how to map that definition exactly to this context. For "smallest error" I calculated the norm of the difference between the exact force vector and the force vector for each approximation. The source code can be found as a Mathematica notebook here.

A few observations: Firstly, the regions aren't tiny. They do shrink with mass ratio (and L4/L5 shrink much faster than the rest, to the point of being invisible at this resolution for a 1:1000 mass ratio) but L1/L2 remain pretty large, comparable with the SoI of the smaller mass. Secondly, they're not at all spheroid. L1 and L2 seem to eat into the smaller masses SoI in a Pacman like fashion, while L3 is weirdly rectangular. Thirdly, just because the linear approximation around a Lagrange point is better than ignoring one mass or the other entirely, it doesn't make the approximation good, it might be that a full 3-body treatment is needed in some places to get anything vaguely accurate or useful.

All this aside, I'm no closer to computing any of these areas analytically. Or even establishing how their general size scales with mass ratio.

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    $\begingroup$ Some good posts on our sister sites on Lagrange points: physics.stackexchange.com/q/36092/123208 space.stackexchange.com/q/49002/38535 & links therein. $\endgroup$
    – PM 2Ring
    Feb 23 at 18:30
  • $\begingroup$ Thanks, I've had a skim through them, but I think they're asking a very different question. Stability is about whether the forces near the Lagrange point will tend to kick an object out or not. But that itself is working under the framework that the forces acting are the ones that come from a quadratic approximation of the true 3-body problem about that Lagrange point. This is a good approximation (arbitrarily good even) provided one is close enough to the Lagrange point. But I'm asking how far out this approximation can be made, regardless stability of orbits. $\endgroup$ Feb 24 at 0:48
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    $\begingroup$ Cross-posting is discouraged. Your question is fine on this site, and all the celestial mechanics experts on Physics.SE also visit here. $\endgroup$
    – PM 2Ring
    Feb 24 at 6:30
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    $\begingroup$ Instead of editing answers into the question, you could self-answer. If you have solved your problem yourself that should be celebrated! Your diagrams look interesting. One observation, the usual pictures of the pseudopotential in a rotating frame don't include the effect of the coriolis force, $\endgroup$
    – James K
    Feb 24 at 8:16
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    $\begingroup$ You may enjoy this closely related topic: Polar Orbits Around Binary Stars by Greg Egan. arxiv.org/abs/1510.05345 $\endgroup$
    – PM 2Ring
    Mar 1 at 18:33

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Right, so this is everything I've calculated on this topic, and as far as I'm able to go on this; at least, as far as I'm able without devoting serious mathematical machinery and computing power. This answer will serve as my own notes on the topic, so I'm being particularly thorough.

Problem Definition

Consider two masses in circular orbit around each other. We pick our units such that they have a separation of $1$ and with total mass $M_1 + M_2 = 1$ with $M_1 > M_2$. We define the mass parameter $\mu = M_2 / (M_1 + M_2)$, which is between $0$ and $1/2$, which acts as the sole parameter of the dynamical system. Throughout we'll be working in a co-rotating reference frame with the origin at the barycentre such that $M_1$ is always at $(-\mu, 0)$ and $M_2$ is at $(1-\mu, 0)$, and confining ourselves to the x-y plane. For reasons of mathematical simplicity (which also coincides with astronomical reality) we are mostly concerned with the situation where the two bodies have very different masses ($\mu << 1$) and will often expand our equations as a power series in $\mu$ in order to solve them.

In this set up, the total force vector (per unit mass, but this distinction will not be made from now on) acting on a test particle at point $\underline{r}$ is $$ F_{exact}(\underline{r}) = -\frac{(1-\mu) \underline{r}_1} {r_1^3} -\frac{(-\mu) \underline{r}_2}{r_2^3} + \underline{r} - 2 \hat{\underline{z}}\times\underline{v} $$ where $\underline{r}_{1,2}$ is the vector from $M_{1,2}$ to the point $\underline{r}$ respectively, and $\underline{v}$ is the velocity of the particle. The terms in that equation are the gravitational attraction from $M_1$, from $M_2$, the centrifugal pseudo force, and the Coriolis pseudo force respectively. We assume that this test particle is of negligible mass such that it does not perturb the orbits of the two main bodies.

It is well known that there are 5 points where $F_{exact} = 0$, known as the Lagrange points. Three are colinear: $L_1$ is between the two masses, $L_2$ beyond $M_2$, and $L_3$ on the opposite side of $M_1$. Finally, $L_{4,5}$ form an equilateral point on the plane with the two masses.

Approximate Forces

Because the three-body system (even in its simplified, restricted form) given above is non-integrable, it can be desirable to approximate the exact forces such that the problem becomes exactly solvable. A common way to do this is to ignore one or the other mass: $$ F_{M_1}(\underline{r}) = -\frac{(1-\mu) \underline{r}_1} {r_1^3} + \underline{r} - 2 \hat{\underline{z}}\times\underline{v}\\ F_{M_2}(\underline{r}) = -\frac{(-\mu) \underline{r}_2}{r_2^3} - 2 \hat{\underline{z}}\times\underline{v} $$ The first case corresponds to simply ignoring the smaller mass: transforming back to an inertial frame gives the standard Kepler problem for the test particle and $M_1$. For the second one we cancel out the attractive gravitational force with the centrifugal repulsion. Transforming from the rotating frame to one with fixed alignment but who's origin moves to stay centred on $M_2$ also reduces to the Kepler problem.

Both of these approximations are intuitively quite good if $r_i << r_j$, but become poor when the forces are close to cancelling out. This happens precisely at the Lagrange points. Hence, another approximation to consider is to expand $F_{exact}$ in the vicinity of the i'th Lagrange point (located at $\underline{R_i}$)

$$ F_{L_i}(\underline{r}) = 0 + \tilde{M} \underline{\Delta r} + O\big( \underline{\Delta r}^2 \big) - 2 \hat{\underline{z}}\times\underline{\Delta v} $$

where $\underline{\Delta r} = \underline{r}-\underline{R_i}$ is the displacement from the Lagrange point, essentially shifting the centre of the co-ordinate system, and $\underline{\Delta v} = \underline{v}$ is the (unchanged) velocity in this translated frame. $\tilde{M}$ is a $2\times2$ matrix whose entries depend on $\mu$ and which Lagrange point is in question: its exact values can be found be differentiating $F_{exact}$. Note that, as the force is precisely $0$ at the Lagrange point itself, if we neglect quadratic and higher order terms in the displacement from $\underline{R_i}$, we have dynamics that are linear in $\underline{\Delta r}$. This makes them integrable and so completely solvable - this is precisely the method used in determining which Lagrange points are stable and which are not.

'Best' Approximation

Let us define the Area of Influence of the Lagrange point $L_i$ to be the region where

$$ |F_{L_i} - F_{exact}| < |F_{M_1} - F_{exact}| \text{ and }\\ |F_{L_i} - F_{exact}| < |F_{M_2} - F_{exact}|$$

aka, where it is the 'best' approximation of the dynamics. Note that as the Coriolis force appears in exactly the same way in every expression for the force, it cancels out completely. The region defined this way thus depends only on position and not on general phase space. This definition is closely linked (though not quite identical) to both the definition for the Sphere of Influence and to the definition of the Hill Sphere

For illustrative purposes here, we'll define the Area of Influence of $M_2$ as the region where

$$ |F_{M_2} - F_{exact}| < |F_{M_1} - F_{exact}|$$

and similarly for $M_1$.

Solution

As shown in the (edited) question, it is possible to plot those areas via brute force numerical methods once the value of $\mu$ has been fixed.

Sphere of Influence plot where the arrows indicate the Lagrange points, the blue regions are their SoI, while the green one is the SoI of $M_2$ and yellow of $M_1$.

I have not been able to compute the exact boundary of the blue regions analytically. I'm even dubious whether an exact closed-form solution to this exist (just calculating the position of the colinear Lagrange points is already a quintic equation, and these are an input when it comes to calculating their SoI).

What I did do was find an approximation for the extent of the SoIs on the x-axis, as a series in $\mu$ that holds exactly as $\mu\to0$. The results for $L_1$ and $L_2$ are illustrated here

Force vs x-axis near L1 and L2

The exact force and it's various approximations in the vicinity of $M_2$. The black line is $F_{exact}$, the green one $F_{M_2}$, the orange one $F_{M_1}$, and the blue/purple ones $F_{L_{1,2}}$ respectively. Note that the latter lines are tangent to the exact force and touch it where it crosses the x-axis (the position of the Lagrange points) by construction. The vertical red lines are where I calculated (to lowest order in $\mu$) the boundaries of the SoI of the Lagrange points as defined above. That is, where either the green or orange line is as close to the black line as the blue/purple one.

Going outwards from $M_2$ we have the first boundary between the SoI of $M_2$ and of $L_{1,2}$ at a distance of $\mu^{1/2}$ from $M_2$. The position of the $L_1$ and $L_2$ themselves are a Hill radius away from $M_2$: $ (\mu/3)^{1/3}$. The boundary between the SoI of the Lagrange points and of $M_1$ are a further half Hill radius from themselves: $1/2 (\mu/3)^{1/3}$. For $L_3$ the results are simpler because they never interact with the SoI of $M_2$, here the boundary for the SoI of the Lagrange point is a symmetric $0.288... \mu^{1/2}$ along both direction.

I'll stress once again that these results are only for the x-axis and only to leading order in $\mu$. While the approximations for the boundary between the SoI of Lagrange points and $M_1$ holds pretty well even for largish $\mu \approx 1/100$, it converges much more slowly for the $M_2$ interface. Indeed, even for the $\mu=1/10000$ plotted we can see that the red line at the $M_2$-$L_2$ interface isn't quite where the purple and green lines cross. Also, because this only applies along the x-axis, it doesn't tell us anything about how the SoI extends in other directions. Nevertheless, based on the 2-d plot above, it's not an unreasonable approximation to treat them as roughly circular, with their extent in the y-direction being of a similar magnitude.

As the $L_{4,5}$ points are not on the x-axis, I could find no analytical results for them. Nonetheless, but playing around graphically with different values of $\mu$, I found what appears to be a decent heuristic for their SoI, as shown by:

Region plot for L4

The yellow background is the SoI of $M_1$ and the blue area the SoI of $L_4$. The dotted boundary is my heuristic for this: an ellipsoid with semi-major axis $5\mu/3$ and semi-minor axis $\mu/3$, rotated by $24.7...^{\circ}$. And symmetrically for $L_5$. Once again, this holds very well provided that $\mu$ is small enough, but discrepancies are clear once it gets above $1/100$.

To summarise the mixture of numerical and approximate analytical insights: the SoI of $L_{4,5}$ scales linearly with $\mu$, while for $L_3$ it scales as its square root; and for $L_{1,2}$ it scales with the cube root.

Application

The question remains of whether the sphere of influence of a Lagrange point, as defined here, is of any use. I propose that it might be a good way of 'upgrading' the patched conics approximation. In that method, orbits through a complicated system are approximated by only considering the dominant gravitational attraction and neglecting all others; resulting in a chain of Keplerian orbits as the test particle moves between the sphere of influence of various bodies of the solar system. The addition I suggest is to also consider the sphere of influence of the Lagrange points, chaining together the solutions of the linear dynamics in those regions with Keplerian orbits. This would be of negligible extra computational cost (because propagating a trajectory through a linear system comes down to a single matrix multiplication), and without the problems that numerically integrating the full 3-body dynamics brings. Importantly, it ought to represent the most important elements of 3-body dynamics, qualitatively if not precisely quantitatively. If I have the time, I'll put that claim to the test.

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Is there a meaningful way of defining the area of influence of Lagrange points?

No.

For several reasons actually!

The basic one that you already recognize is that there's nothing there, there's no special gravity field that comes from these particular places in empty space.

And you don't experience gravity from the other two bodies any differently there than you do anywhere else.

This is a dynamical problem. It's not like you're on a "normal" 3-body orbit until you get close to a Lagrange point when suddenly a new kind of influence starts to kick in, you have to consider the full trajectory.

You should proceed with the simulations discussed in comments, they will provide a lot of insight.

That said

If you focus on the concept of libration points which are Lagrange points with some semblance of stability (e.g. L4 and L5), you could think about how large of an area objects can move and still stay associated with those points.

The problem is, even in 2D, it's a 4D problem! The potential plots we see for stability are usually zero-velocity potentials the trajectories and stability regions are different for each non-zero initial velocity you might choose.

See for example

Definitely start simulating, it's really fun and instructive!

And feel free to post your result an an answer post!

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    $\begingroup$ Of course there aren't different forces around a Lagrange point, just like there aren't new or different forces that appear when you cross the boundary from the Sun's SoI to the Earth's. It's the 3-body problem everywhere. The whole concept of SoI is about which approximation is best: the 2-body problem with just the Sun or the one with just the Earth? My question is why not add the linearization of the forces around a Lagrange point to this list of approximations? I thought I explained that in the OP, can you say which bit you didn't understand so I can edit it to make it clearer to others? $\endgroup$ Feb 25 at 11:59
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    $\begingroup$ @ScienceSnake I'll give it some more thought. btw there are more orbital mechanics folks in Space Exploration SE as well. You might consider asking a follow-up question there and linking back to your question here. We don't ask the same question in two sites in Stack Exchange, but it's quite common to ask related or follow-up questions in the same or a different site. Check it out :-) $\endgroup$
    – uhoh
    Feb 25 at 14:52

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