Lagrange points and regionally conserved quantities

The equilibrium points of the restricted three-body system are known as Lagrange points.^* (The word “restricted” means that one mass is so much smaller than the other two that it can be ignored when calculating the dynamics of the two more massive bodies.) The near-Earth Lagrange points of the Sun–Earth system, located approximately $1.5 × 10 6$ km from Earth, have been home to multiple satellites: SOHO, WMAP, Planck, and the James Webb Space Telescope (JWST).¹ The first of these is located near the sun-facing L₁ point for continuous solar observations. In their search for signals of cosmic origin, these other observatories have been parked near the L₂ point to take advantage of the clear view of deep space.² Dust clouds have been observed at the L₄ and L₅ points of the Earth–Moon system.³ Further afield, the Trojan asteroids cluster around Jupiter's L₄ and L₅ points. The L₃ point has no known natural or man-made significance.

While the pattern of five Lagrange points for the Sun–Earth system is relatively well-known, the configuration of these points in systems with significantly larger mass ratios, like the Pluto–Charon system with a mass ratio of about 0.12 or an equal-mass binary star system, is likely less familiar to readers. A symmetry argument for the latter readily establishes that the configuration for such a system must look quite different from those for stellar–planetary systems. Figure 1 compares the locations of the five Lagrange points for two scenarios, one for a system with very disparate masses and the other with equal masses. Beyond the qualitative aspect of the relative location of the Lagrange points, one might want to know how exactly NASA mission engineers determined that the JWST should be located at a distance of $1.5 × 10 6$ km from Earth?

The definitive text on the subject of Lagrange points is Theory of Orbits: the Restricted Problem of Three Bodies.⁴ Unfortunately, it and many other sources^5–10 analyze the motion in cartesian coordinates, which does not reflect the natural geometry of the problem. Where this problem appears in undergraduate physics textbooks, it receives very abbreviated treatment.^11–13 The number of articles aimed at the undergraduate and educator audiences are few and light on details.^14,15 Of these sources, only Ref. 4 provides approximate solutions to higher than first-order.

This work presents a thorough analysis of all equilibrium points using physically motivated arguments and polar coordinates. Many mathematical curiosities will be encountered on this journey, including the implications of Galois theory of quintic polynomials, expansions in fractional powers of a small parameter, symmetries, symmetry breaking, and a challenge to the local/global dichotomy of conserved quantities. While this work does not calculate the stability of the equilibrium points, it should be noted that stability can be calculated by extending the formalism presented here to second derivatives of the potential and including the effect of the Coriolis force. Such calculations show that only the L₄ and L₅ points are truly stable.

Section II presents preliminary aspects, including a definition of the coordinate system and a discussion of the effective potential. Building on this foundation, Sec. III analyzes the solutions for three cases. A discussion in Sec. IV summarizes the findings and offers some reflections on a few of the subtler points of the analysis, which is followed by concluding remarks in Sec. V.

The analytical goals of this work are formulas for the equilibrium points of a satellite that is gravitationally bound to two massive bodies. The two massive bodies, with masses m₁ and m₂, will be called the primaries, and the satellite, with mass m₃, will be called the tertiary.^† Subsections II A–II G define the coordinate system, the rotation frequency of the system, the effective potential, discuss the curious nature of angular momentum in this problem, present the azimuthal and radial components of the force, and introduce the dimensionless form that will be used in the final analysis.

Curiously, the sign of the inertial term in Eq. (6) is negative. One might be inclined to chalk this up to a typographical error, except that it yields correct solutions to the Lagrange point problem. The effective potential of Eq. (6) is also known as Jacobi's constant, a lesser-known conserved quantity that is found by integrating the radial force multiplied by $r ̇$ under the constraint of constant angular velocity.¹⁶ Further commentary on the nature of Eq. (6) follows in Sec. II F.

The only possible resolution to the aforementioned puzzle is that the angular momentum must be conserved in a small region around each equilibrium point but is not conserved on larger scales. This sense of local but not global conservation is rather opposite to the conventional understanding of conservation laws where local conservation (e.g., of charge) necessarily implies global conservation. The phrase “regionally conserved quantity” is introduced here to describe the peculiar nature of angular momentum conservation over a small region. This concept is similar to quasi-local conservation in general relativity where a locally flat but globally curved spacetime allows energy and momentum to be treated as both conserved and not-conserved depending on scale.^17–20 

In the vicinity of a Lagrange point, the azimuthal force is very weak given a small displacement from equilibrium. Because this force can be approximated as a linear function of the angular deviation over these scales, the first correction to the angular momentum will average to zero over an epicycle due to the fact that this force is an odd symmetric function. It follows that any net change to the angular momentum over longer time scales must enter as a second-order or higher effect and will, therefore, have zero derivative at the equilibrium point. It is in this sense that we can take the angular momentum to be conserved in a local region around each equilibrium point.

Equilibrium points of the system require $F θ = 0$⁠, which admits three cases: (A) θ = 0, (B) $θ = π$⁠, and (C) $r 1 ′ = r 2 ′$⁠. Case A yields Lagrange points L₁ and L₂, and case B yields Lagrange point L₃. The third case indicates that the position of the equilibrium point must lie along the bisector of the axis between the primaries, which describes Lagrange points L₄ and L₅.

Szebehely,⁴ Cornish,⁵ and Widnall et al.⁶ take an alternate approach and use $R = r 1 + r 2$ and $M = m 1 + m 2$⁠, which also satisfies $( r 1 + r 2 ) / R = ( m 1 + m 2 ) / M$⁠, where $μ * = m 2 / ( m 1 + m 2 )$ is the dimensionless maas ratio (note: while those texts also use the symbol μ, the “ $*$” is added here to differentiate this definition from the definition of μ used in this work). These two definitions of the mass ratio are related through $μ = μ * / ( 1 − μ * )$⁠. While the use of μ instead of $μ *$ is a departure from the standard of the literature, it is argued here that it has several advantages. First, there is the aesthetic quality that μ spans the range $0 < μ ≤ 1$ whereas $μ *$ spans the range $0 < μ * ≤ 1 / 2$⁠. Second, the first-order approximations of the radial force equation for the L₁ and L₂ points are much simpler when described in terms of μ. Third, Ref. 4 presents series solutions for the radial locations of L₁ and L₂ in terms of a quantity $μ * / ( 1 − μ * )$⁠, which is equal to μ and suggests that μ is in fact an optimal dimensionless form. Finally, and most importantly, the radial location of the L₂ point is non-monotonic when described in terms of $μ *$ and the L₄ and L₅ points decrease with increasing $μ *$⁠, both of which are rather counterintuitive and require significant care to correctly interpret. In contrast, the formulas for the L₂ point and the L₄ and L₅ points are increasing monotonic functions when described in terms of μ. Figure 3 depicts the pattern of solutions with increasing μ for all five Lagrange points. A detailed comparison of these normalization schemes, their associated solutions for the Lagrange points, and the transformations between them can be found in the supplementary material.²¹ 

The analysis is now reduced to a purely mathematical problem of finding the radial and angular coordinates that allow both components of the force to vanish simultaneously. Solutions are discovered analyzing the dimensionless part of Eq. (13) subject to the three cases defined in Sec. II E. For multiple reasons, it is helpful to begin by exploring first-order approximations to Eq. (13) before delving into the search for exact and higher-order solutions. The results of these analyses are summarized in Fig. 4, which shows how the radial coordinates of the Lagrange points vary with μ.

The Sun–Earth system has $( m E / 3 m S ) 1 / 3 ≈ 1 / 100$ to an accuracy of better than one part in a thousand. It follows that the distance of the L₁ and L₂ points from Earth is very close to $1 / 100 th$ of the Earth–Sun distance (about 150 × 10⁶ km), yielding the 1.5 × 10⁶ km value that was cited in the introduction. At the Sun–Earth L₂ point, the angle subtended by the Sun is approximately 9.3 mrad, whereas that of the Earth is approximately 8.5 mrad. It follows that the visible fraction of the Sun's area at L₂ is about $1 − ( 8.5 / 9.3 ) 2$⁠, or roughly 20% of the total, though the actual fraction of solar radiation received there is somewhat less than this due to solar limb darkening.²² 

An alternative to symbolic analysis is the discovery of solutions through a search procedure that calculates a finite table of solutions. The results of such a numerical search are presented as the solid color lines in Fig. 4. Many different numerical methods and tools can be used for such purposes. Two possible approaches are described here. Figure 4 was generated using a simple Python script with a nested loop structure. The first loop evolved μ through a specified set of values, and a secondary loop then scanned x values for each value of μ to find the roots of the relevant polynomials (presented subsequently) for the L₁, L₂, and L₃ Lagrange points. A simple and convenient way to explore the solution space can be done using graphing software, like Desmos (https://www.desmos.com/), with a first equation that specifies the range of μ values to be explored with a second equation defining the relevant polynomial. The roots can then be inspected by hand and a graph constructed from a table of solutions. Other tools, such as root-finding functions that are common in many programming languages, can be used to the same effect.

The following subsections present higher-order, quasi-analytic approximations for the L₁, L₂, and L₃ points, and derive exact solutions for the L₄ and L₅ points.

This case considers Lagrange points L₁ and L₂ that exist along the axis extending from m₁ through m₂. The search for analytic solutions proceeds best when using an alternate form of Eq. (13) that seeks zeros of the numerator after a common denominator has been extracted. The numerator becomes a fifth-order polynomial, which is described as $f ( x ; μ ) = ∑ n = 0 n = 5 f n x n$⁠, where the coefficients are given in Table I. Examples of the two forms of $f ( x ; μ )$⁠, calculated for the specific case of $μ = 0.5$⁠, are presented in Fig. 5. A single real-valued and positive root exists for each branch. Since all of the denominators of Eq. (13) are squared, and therefore positive, the sign of the effective radial force has been preserved in the transformations leading to $f ( x ; μ )$⁠. This means that the sign of the curves in Fig. 5 can be interpreted as the direction of the net radial force. This also illustrates the unstable nature of these equilibria since a positive displacement will lead to a positive force in both cases.

The exact values of the coefficients of the final terms are $− 220 / 81$ and $+ 212 / 81$⁠, respectively. These terms have been modified from their actual values to better match the solutions calculated through numerical analysis. These quasi-analytic forms work well over the entire range of μ and have average deviations, measured over the range $0 ≤ μ ≤ 1$⁠, from the numerical solutions of less than $10 − 2$⁠. Exact expansions out to sixth-order can be found in  Appendix B and derivations of these terms are presented in the supplementary material.

This pair of solutions possess interesting symmetry properties. The magnitudes of the coefficients of x₁ and x₂ are symmetric in the first three terms and then deviate starting at the fourth term, which begins with the introduction of the $μ 2$ term at that order in the expansion. In this light, the solutions to Eq. (20) possess qualities that may inspire thoughts of explicit symmetry breaking that is frequently encountered in quantum mechanics, as in Zeeman splitting.²³ 

Similar to the prior case, there are two sub-cases for $θ = π$⁠, depending on whether the radial location of m₃ is greater than or less than r₁, which are illustrated in Fig. 6 for the case of μ = 0.5. However, given that the linear approximations of Sec. III A established that there is only one viable branch for $θ = π$⁠, only the $x > μ$ branch is analyzed here. A full analysis of this case, including both branches, can be found in the supplementary material. Proceeding as for the prior case by changing variables from x to Δ, the fifth-order polynomial for the L₃ point is $g ( Δ ; μ ) = ∑ n = 0 n = 5 g n Δ n$ whose coefficients of are given in Table II.

The third case is defined by the condition that the tertiary be equidistant from the primaries and gives rise to the L₄ and L₅ points. By itself, this condition requires that the triangle formed by the locations of the masses be isosceles, with the tertiary at the point of symmetry. As will be shown, the additional constraint imposed by the requirement that the radial derivative of the effective potential vanish leads to the conclusion that the locations of the masses form an equilateral triangle for all values of μ.

Figure 7 presents Eq. (25) with μ = 0.5. Equation (26) admits two solutions, which produce the well-known angular positions of $± π / 3$ (⁠ $± 60 °$⁠) in the limit of μ = 0. Because θ is measured from the center of mass location, it is not obvious what geometry between the masses is implied by this solution for other values of μ. This is readily resolved by observing that the distance between the primaries is $1 + μ$ and, as established previously, the other two sides have lengths of $( x 2 + μ ) 1 / 2 = ( 1 + 2 μ + μ 2 ) 1 / 2 = 1 + μ$⁠, using Eq. (25). Therefore, the shape formed by the locations of the masses is always an equilateral triangle.

The foregoing analysis has shown how the positions of all Lagrange points can be calculated for arbitrary mass ratios. A relatively simple image of the solutions emerges when one considers the mass ratio as describing a fixed m₁, such as a star, with the mass of the second primary increasing from something infinitesimal in comparison (e.g., a small planet) to that of an equal mass binary star system. In this representation, all points except for the L₁ point move outward with increasing μ. The solutions presented here stand in contrast to other analyses which, using a different normalization scheme, generate solutions that are non-monotonic or decreasing functions of the mass ratio and whose interpretations require more care.

One of the main contributions of this work has been to clarify the meaning of the curious form of the effective potential described in Eq. (6), which is often used in discussions of the Lagrange points, either directly or indirectly through contour plots. While this form gives the correct equilibrium points upon taking a derivative with respect to r, it is a form that is rather devoid of physical meaning since it requires that the angular velocity be treated as a constant of the motion. This cannot be true since departures from stationary points result in epicyclic motion with variable angular velocity even in two-body systems.¹⁶ Rather than being mere convenience, it seems likely that the use of this unphysical form of the effective potential stems from the fact that contour plots of Eq. (6) conveniently illustrate the correct locations of the Lagrange points, in contrast to Eq. (7), which does not reveal the Lagrange points. This is because, fundamentally, the analysis of the Lagrange points is a force analysis, which means that illustrations of the solution space require a function that represents the spatial structure of the forces. While Eq. (7) is the correct starting place for this analysis, it does not lend itself to visualization of the solution space since the centrifugal term changes sign after taking the radial derivative of Eq. (7), which must be followed by a change of variables from L to $m 3 r 2 ω 0$⁠. Conversely, Eq. (6) preserves the overall structure of the forces and lends itself to visual representation. A comparison of contour plots generated in Eqs. (6) and (7) is presented in Sec. V of the supplementary material notes.

One might object that, despite being the more accurate physical representation, the loss of the ability to visualize the problem when using Eq. (7) is a loss of intuition and that Eq. (6) is justified on these grounds. That would be a good argument in favor of continued use of Eq. (6) if it were not for a third path. It is possible to maintain rigor by using Eq. (7) as a starting point for the analysis and presenting a visual illustration of the solution space through a contour plot of the norm of the total force, as is done in Fig. 8. The norm of the total force is a quantity that has minima (zeros) at the Lagrange points and therefore similarly motivates this analysis without any of the misleading aspects of Eq. (6). Of course, all such forms should be taken with a grain of salt since they lose their physical meaning outside of a small neighborhood around each of the Lagrange points where the angular momentum cannot be treated as a constant of the motion.

The primary goal of this work is a thorough analysis of the restricted three-body problem using the natural geometry described by polar coordinates, culminating in formulas that describe the locations of all Lagrange points. This problem was originally solved as special cases of the more general problem of harmonic motion in the three-body problem, first by Leonhard Euler in 1767 for the L₁, L₂, and L₃ points,^24,25 and subsequently by Joseph Louis Lagrange in 1772 for the L₄ and L₅ points.²⁶ It is humbling to revisit the studies of the past and see how much was done with so little, and it is remarkable that this problem, now over 250 years old, continues to provide fertile ground for new insights.

A number of interesting puzzles and mathematical curiosities were encountered throughout this work, among them an application of Galois theory of polynomials, symmetry and symmetry-breaking, and the role of normalization in shaping the solutions. A new contribution of this work is a set of quasi-analytic approximations for the radial locations of the L₁, L₂, and L₃ points that describe the solution space to high accuracy for all values of μ. Perhaps the most important aspect of this work is the clarification regarding the nature of angular momentum. The proper formulation of the problem requires that angular momentum be treated as an invariant near the equilibrium points even though it is not a locally conserved quantity. This observation naturally leads to the recognition that a third kind of conservation law, described here as regional conservation, is needed. Students of general relativity, in particular, may benefit from thinking of this aspect of the Lagrange point problem as they ponder the nature of energy conservation on local and cosmic scales.

The author has no conflicts to disclose.