THE LAGRANGE POINTS ================= John Pazmino NYSkies Astronomy Inc firstname.lastname@example.org www.nyskies.org 2001 October 3 initial 2001 October 27 current
Introduction ---------- A series of chats about the SOHO solar probe, the Trojan asteroids, and space colonies at assorted meetings of the Amateur Astronomers Association in February 2001 led to the question: Just what are the Lagrange points in a planet's orbit? Curiously, most AAA members have heard of these points and can recite, likely by rote, the casual explanations given in textbooks and NASA promos. But they lack the fuller details about them. Uh, there's a good reason for this shortfall of understanding. The theory of the Lagrange points is quite hairy. At the same time it is utterly fascinating! It's also now increasingly Important for home astronomers to appreciate with the prospect of actually seeing a Trojan asteroid in the new larger home scopes and following news about the space probes parked at these points. I do spend some time on actually computing the locations of these points because this procedure either is glossed over in literature aimed at home astronomers or the treatment is presented as overwhelmingly convoluted. It isn't, thanks to home computers, and I show simple equations that naturally lead to these points. And, in the same blow, they lead to an other fascinating feature of orbits, the Roche lobes.
History ----- The story goes back to the late 1600s with Isaac Newton. He sussed out the theory of gravity among the bodies within the solar system. He even proved that comets, until then total mysteries, moved by gravitational influence from the Sun. He found that for a pair of masses, like Sun and planet, his theory could totally describe the motions with absolute certainty. He could write down explicit formulae for the place and speed of the planet at any moment in the past or future and know they were correct. This case is today called the 'two-body problem'. In fact, it is more of an exercise for a freshman physics student to work out the orbit of a comet using this model. You can do so, too, given a sci/tech calculette and a jug of wine. When Newton focused attention on the motion of the Moon, he found that a two-body model gave lousy results. The Moon deviates from the nice pure ellipse because there is a third body around to distort the Moon's motion. This is, of course, the Sun. As hard as Newton tried, he could find no definitive solution for gravity among three bodies. He related that the only time he ever had a headache was when struggling with this three-body problem.
The three-body problem -------------------- Given an arbitrary set of three masses in an arbitrary configuration, it is not possible to write out formulae that will determine the future or past motion, as in the two-body case. Because of this characteristic of the three-body problem -- and it IS a 'problem'! -- mathematicians and astronomers ever since Newton's day have devoted years and decades to it. In 1889 Henri Poincare' showed, as winner of a math contest set up by King Olaf II of Sweden, that there really is NO general method of working out the motion of an arbitrary triplet of bodies. In doing so he demonstrated that Newton physics, so touted as perfectly ordered and determinative, embeds chaotic behavior of celestial motion. In this regard, he founded the modern discipline of chaos, which was rediscovered quite a full century later in the 1980s. Today, when we work with a three-body case, like the Sun-Earth- Moon system, we impose simplifying conditions on the masses. For the Earth and Moon we put the Sun very far from the two and elaborate the motions for that geometry. Such conditions on a three-body case leads to the theme of the 'restricted three-body problem'.
Josef-Louis Lagrange ------------------ Josef-Louis Lagrange was a mathematician in the mid 1700s living in Turin, in present-day Italy. He made his mark with seminal work in differential equations and number theory. He also excelled in formalizing mechanics and dynamics. He won honors from his native city, Berlin, and Paris. He eventually worked in Berlin and then Paris. The Paris Academy of Sciences had contests on mathematical problems, which Lagrange often entered and several times won. In 1772 he shared the prize with Euler on solving the three-body problem, which was important in mathematics as well as astronomy. In his paper 'On the problem of three bodies' Lagrange demonstrated that in a particular three-body situation there are five magic points in space around the masses. The case was of two considerable masses which generate gravity fields around them and a third of infinitesimal mass and no sensible field. In this article I call the bodies Sun, planet, and particle. The particle is typicly an asteroid or spacecraft. These five points are disposed around the Sun and planet such that a particle placed at any of them will stay there indefinitely. It would orbit the Sun with the same period as the planet, so its elongation from the Sun seen at the planet remains fixed and its synodic period is infinite. The synodic period of a body is the interval between successive occurrences of a given elongation from the Sun. Normally this is months or years because the body does circulate thru the zodiac and overtakes the Sun repeatedly. If the body is at a Lagrange point, it keeps the same elongation all the time and never scores a 'successive occurrence' of it. It occurs after an 'infinite' time.
Further restrictions ------------------ It is common at least for home astronomers to add a further restriction to the Lagrange three-body case. The two larger bodies are in circular orbital motion. This is closely achieved in the solar system, where the planets enjoy nearly circular orbits only modestly excentric off of the Sun. The math for an elliptical orbit, where the distance between the Sun and planet oscillates from perihelion to aphelion, is just too much for the home astronomer. Additionally, I'm restricting the motion to the orbital plane of the two large bodies. There is a 3D extension, which is really necessary for asteroids because of their large inclination to the ecliptic, but I leave that out for this work. So everything here deals with circular orbits with the third body, the particle, moving in the same plane as the other two.
Trojan asteroids -------------- In October 1906 Wolf during routine searches discovered asteroid 588, which he named Achilles. He soon found it paced Jupiter but stayed roughly 60 degrees ahead of that planet in the zodiac. In November 1906 Kopff found an other peculiar asteroid, 617 Patroclus; in December 1907 he bagged 624 Hektor (sometimes spelled Hector). These, too, paced Jupiter. Patroclus is some 60 degrees behind the planet while Hektor is near Achilles ahead of Jupiter. 1908 saw 659 Nestor join Achilles and Hektor. All four happened to be named for warriors of the Trojan Wars, chronicled by Homer. The gentleman's agreement was made to name all future Jupiter-pacing asteroids after other figures from Troy. Moreover, the asteroids leading the planet will be named for Greeks and those lagging are the Trojans. The exceptions are among the first of these so-called 'Trojan' asteroids! Hektor is a 'spy' among the Greeks; Patroclus, among the Trojans. For most of the 20th century the Trojan asteroids remained a curious feature for asteroid historians. Their number increased slowly. At the end of 1909 there were 4 Trojans; 1919, 6; 1929, 6; 1939, 12; 1949, 13; 1959, 15. 1969, 16. Despite their low number, the Trojans enjoyed extensive study for their orbital behavior. Starting in the 1970s as a spinoff of the space exploration projects and the commissioning of larger more sensitive observational equipment the ranks of Trojans exploded. I lost count but by the end of 1979 there were about 60 Trojans; 1989, about 400; 1999, hold your hat, 2,000! Thus almost ALL the Trojan asteroids of Jupiter were discovered in the last decade of the 20th century! The 2,000 are disposed into about 1,200 Greeks and about 800 Trojans. There were myriads of participants in the Trojan Wars, so don't worry about running out of names. The name 'Trojan' for this type of asteroid alludes to the Trojan Wars, not to the side its namesake was on. Remember, the Greeks won, so they write the history. Most asteroid specialists say the imbalance of members between the leading and lagging groups is just a chance of discovery. They have no theory to predict more asteroids in the one or the other. It wasn't long before celestial dynamicists recognized the Trojan groups as sitting near two of Lagrange's special points. These two are near the planet's orbit forming an equilateral triangle with Jupiter and the Sun. They are, from the example with Jupiter and the geometry, called 'Trojan' or 'triangular' points. Due to tidal forces from neighboring planets, the Trojan asteroids wander away from the exact 60-degree marks but always slip back toward them. They 'balance' or 'teeter' on these points, so the points are also known as 'libration' points. Yet another term is plain 'L' point, after Lagrange's initial.
Prediscovery of a Trojan ---------------------- Wolf properly is honored for recognizing the first Trojan asteroid. However, there is an asteroid found in 1904 where the observer did not recognize the object as a Trojan! Edward Barnard at Yerkes Observatory was looking for Phoebe, a new moon of Saturn discovered by Pickering in 1896. Its orbit was still loose and Barnard, with Pickering's help, tried to collect more datapoints. On 12 September 1904 he saw what he thought was Phoebe but it moved contrary to how a Saturn moon would move. He then treated it as an asteroid but lost track of it after that night's work. Its modern designation is 1904-RD. At that time Saturn was some 73 degrees west of Jupiter and in front of the region around Jupiter's lagging triangular point where its Trojans can roam. On 26 September 1999 -- 95 years later! -- Gareth Williams, at Minor Planet Center, linked records of several asteroids into a chain belonging to a one and the same body. He tied, for example, 1999-RM11, 1996-HJ22, and 1978-VH6. All these are the same asteroid lost after discovery. And on each new finding of this same asteroid it got an allnew designation. The orbit for this object allowed Williams to track down asteroids in 1988, 1987, 1985, 1975, and, ta-TAH!, 1904. Because the newer asteroids were known to be Jupiter Trojans, so must be 1904-RD! By the way, the greater number of new asteroids are lost within days of their confirmation and it is quite normal that many new asteroids, getting their own designations, are really recoveries of these lost, and sometimes forgotten, asteroids. More intriguing is that Barnard, when he wrote about his find in 1907 -- with three Trojan asteroids on the books -- did not claim his own asteroid as the true first Trojan!
Observing Jupiter's Trojans ------------------------- The Trojan asteroids of Jupiter are way too faint for a small instrument to see, but they may be visible in the larger home telescopes and can be captured by CCD systems. The Trojans are at best 14th to 15th magnitude when they are at opposition. If you already can see with your faculties Pluto, you are ready to go for the brighter Trojan asteroids. Besides requiring some aperture to discern, these asteroids require detailed finder charts. Because they are farther out than the main belt of asteroids, their motion thru the stars is slower and, therefore, not as obvious. The Trojan asteroids are not in quiet orbits like the larger main belt members. They suffer perturbations from the other planets that push them to and fro around the triangular points. You need addiurnate orbital elements, obtainable from the Minor Planet Center's website at 'www-cfa.harvard.edu/iau/mpc.html'. The nine brightest Trojans are 624 Hektor, 911 Agamemnon, 1143 Odysseus, 3451 Mentor, 617 Patroclus, 3317 Paris, 1437 Diomedes, 1172 Aeneas, and 2797 Teucer. 588 Achilles and 659 Nestor are probably too dim for home astronomy equipment.
Numbering ------- Lagrange himself did not assign numbers or other nomenclature to the points. He merely marked them on his diagrams. Somewhere in the early days of the space age the numbering scheme arose. The leading triangular point is numbered L4; the lagging, L5. Everyone agrees with this numbering. The other three points are very inconsistently labeled among authors, causing rank confusion and misapprehensions. There happens to be no agreement, but the most sensible one is that used within the orbital dynamics crowd. All three other Lagrange points sit on the Sun-planet line; they are sometimes referred to as the 'colinear' points. L1 is in inferior conjunction as seen from the planet. L2 is in opposition, and L3 is in superior conjunction. One mixup comes by shifting the eye to the Sun. From the Sun's eye, L2 becomes L3 and L3 is now L2; the labels are swopped. An other mixup comes from looking at the Earth-Moon system. Articles on space missions often don't make it clear which set of Lagrange points they're referring to. Some authors label these colinear points sequentially left to right, or vice versa, on their diagrams. So L1 may be at either superior conjunction or opposition. For this reason it is most unwise to merely state the L-number. Do deliberately describe the actual location of the point, in words or on a chart. I use the convention here, based on the Sun and a planet as seen by the planet's eye. The planet revolves counterclockwise around the Sun.
+L4 / \ / \ / \ revolution / \ _ / \ planet |\ +L3 O Sun + o + | \ L1 L2 | \ / / (L5 symmetrical with L4)
I give here a tabula to compare the Lagrange points within the Sun- Earth system with those of the Earth-Moon system.
+-------+-----------+------------+ | point | Sun-Earth | Earth-Moon | +-------+-----------+------------+ | eye | on Earth | on Moon | | L1 | btwn S & E| btwn E & M| | L2 | opp to S | opp to E | | L3 | beyond S | beyond E | | L4 | leads E | leads M | | L5 | lags E | lags N | +-------+-----------+------------+
For the record, O'Neill, when in the 1960s he proposed a human habitat in one of the Trojan points of the Earth-Moon system, chose the lagging point L5. No particular reason; it was a coin flip. From this came the name of the now-defunct L5 Society, which advocated a migration of excess human population into space.
Other Trojan bodies ----------------- A spinoff of the new observational techniques, like supersize scopes and planetary probes, was the discovery of 'Trojans' around planets other than Jupiter. Mars has two Trojan asteroids. The first was found by Levy & Holt in 1990 at Mars's L5 point. It was later named 5261 Eureka. The second one, as yet unnumbered and unnamed, is 1998-VF31, also at the Mars L5 point. Saturn's moons Dione and Tethys have Trojan attendants! All were discovered in 1980 from Earth observatories, not, as some authors assert, from the Voyager missions. Accompanying Dione is Helene at L5; Tethys hosts Telesto in L4 and Calypso in L5. In 1995 we found a temporary cloud at the L4 of Enceladus. It soon faded, likely due to evaporation into invisible gas. Even Earth itself has a Trojan, asteroid 3753 Cruithne! Its behavior is, erm, sturrange, involving BOTH triangular points AND Earth's L3 point on the far side of the Sun. I have a separate section for this fellow. Altho these other bodies are not true Trojans -- they are not at Jupiter's triangular points -- they are commonly called Trojans anyway. So it is more or less all right to refer to a body at ANY L4 or L5 point a Trojan body.
Kordylewski clouds ---------------- The prospect of Earth having supernumeral natural satellites is a recurring theme. Several serious efforts were made over the ages to find such other moons. When in the early 1950s manmade satellites were feasible, it was presumed that they would be tracked by radar. They would be the inert final stage of a rocket with no radio beacon. If there were natural moons, likely large meteorites, they would be detectable by radar, too, and be a grand annoyance for the manmade satellite project. A special camera was constructed at Lowell Observatory to map out these meteorites by tracking at their anticipated speeds. Despite spurious reports of success, the searches were all negative. No natural moons of Earth were found. The camera, on the other hand, was a godsend for tracking the early manmade moons, particularly those of the Soviet Union. This machine evolved into the fabled Baker-Nunn camera in 1957. At Krakow Observatory the search for secondary moons was focused on the Moon's L4 and L5 points for being where moonlets could congregate and be easier to spot. Telescopic searches were fruitless. In 1956 Kordylewski and Wilkowsky figured that such meteorites, gathered around L4 and L5, would be best seen by eye. An ordinary telescope would dilute them out of sight. In September 1956 Kordylewski did spot, at the L4 and L5 places of the Moon's orbit, vanishingly faint patches dimmer then the zodiacal band. Altho this discovery was debunked as some Cold War propoganda, it was confirmed by subsequent searches by spaceprobes. These clouds are spread over a roughly two-degree diameter as seen from Earth and librate around the L4 and L5 points. They seem to be secularly variable in prominence, likely due to continual depletion and refilling as the meteorites are nudged around. So far, I could find no confirmed report of a home astronomer positively seeing or imaging the Kordylewski clouds.
Lagrange points for spacecraft ---------------------------- In the late 20th century the Lagrange points, all five of them, took on major importance as possible stations for spacecraft. ACE, SOHO, Triana (Al Gore's satellite), and WIND hover near Earth's L1 point. Note that L1 seen from Earth is smack in front of the Sun. A craft at this point would suffer intolerable radio interference from the Sun. To get away from this interference, probes 'at' L1 really circulate in an orbit around L1 a couple degrees away from the Sun. A fantasy project to combat global warming is to position gigantic solar shades at Earth's L1! These would be manoeuvered to block more or less of incoming sunlight and balance out the warming trend. The Moon's L2 point, behind the Moon as seen from Earth, is home for the MAP mission. It is under consideration for the Next Generation Space Telescope, successor of the Hubble Space Telescope. So far no space mission considers use of Earth's L3, the one in superior conjunction behind the Sun. Science fiction writers love L3 as the place for a contraterrestrial world always hidden from us in the Sun's glare. L4 and L5 in the Sun-Earth system are potential sites for the remote end of a ultralongbase interferometry. The Earth-Moon L4 and L5 could in the far off future be home to humongous depots to house displaced Earthlings.
The WIND spacecraft ----------------- WIND was launched in 1994 to study solar particle, radiation, and magnetic flux. It cruises around the Earth-Moon region out to about 150 Earth radii (the Moon is about 60 radii away). From time to time WIND is sent to Earth's L1 point for a couple months, around which it gyrates to avoid radio interference from the Sun. On 19 February 2001 the Spacewatch project found an asteroid scudding past Earth only 100 Earth radii away! What's more, it was backtracked to a near miss of the Moon in August 2000! It was duly designated 2001-DO47. Suddenly, while under close monitoring, 2001-DO47 on 23 February 2001 jumped onto an altogether other orbit! Celestial dynamicists went crazy. It's as if the asteroid turned on a rocket. It did. Frenzied inquiries revealed that 2001-DO47, a name forever frozen into the record books, is the WIND craft being moved from one orbit to an other and will repeatedly in the future hover at Earth's L1. The Minor Planet Center, thru its office that monitors near-Earth asteroids, in late February 2001 started to include ephemerides of spacecraft near Earth in manoeuverable orbits. One bad scare was enough.
Features of the Lagrange points ----------------------------- The cardinal feature of a Lagrange point is that a particle placed at one has the same orbital period, or angular velocity, as the planet. That's what really is meant by 'the particle stays still relative to the planet'. In other words, the total gravity force at a Lagrange point must be the same as that at the planet and be directed toward the center of mass of the Sun-planet system. This is not the same as requiring that the net gravity force at the particle be merely the same strength, with no regard to direction, as that at the planet! Nor it is the place where the Sun and planet gravity forces cancel out. This is the zero-point often used in science fiction stories, like Verne's cannon shot to the Moon. A moment's reflection will demonstrate that there is only one zone where a zero-point can exist. That's between Sun and planet. Every where else the forces can not cancel and there is always a net nonzero force on the particle. Altho the erroneous explanation is commonly applied for L1, it dismally fails for L2 thru L5.
Locating the Lagrange points -------------------------- L1, L2, L3 are the colinear points. They are accurately on the line of the Sun and planet. L4 and L5, are exactly 60 degrees, as measured from the Sun, ahead of or behind the planet. The 60 degree angle is exact; L4 and L5 do form an equilateral triangle with Sun and planet. Note that L4 and L5 are NOT on the planet's orbit! They happen to sit there ONLY because we premised circular motion! And the planets DO have nearly circular orbits. L4 and L5 are at the triangular places, which will fall inside or outside the nearby section of the planet's orbit.
Turned off by math ---------------- It seems that the usual treatment for calculating these points is unnecessarily complicated. It's as if the author, not knowing what to do, copied off some stew of equations to make his paper look impressive. Or maybe just to dazzle the reader. I found something very very odd about the explanations of Lagrange points. Textbooks show that dropdead sexy picture of contours for gravity energy surrounding two masses. This is particulary common in illustrations of close or interacting binary stars. Certain contours embrace one or the other of the masses. Some embrace both. And a certain set are closed loops triangular to the masses. So far so good. Guess what? On these drawings the Lagrange points are usually plotted! L1 is between the masses, where two contours cross. L2 and L3 are outside the masses, also where two other contours cross. L5 and L6 are eithin the contour loops to either side of the masses. Hmmm. Let's look at the standard derivation of Lagrange points. Yikes! A stew of sines of angles, sines of sums of angles, arrows pointing this way, arrows pointing that way, dotted lines, dashed lines, Greek symbols on this point, Latin letters on that point. For someone knowing geometry and trigonometry and vectors this is good gumbo. Most home astronomers do not understand such math. This mathwork seems to have nothing to do with the pretty contour drawings. Why? If you do understand gravity, you may recognize that the contour map is a plot of the gravitational energy around the bodies. The Lagrange points are critical places in that field where the energy is a relative minimum, or 'low point'. If you imagine this plot as an elevation map, the L1, L2, L3 sit on saddles, like those between two adjacent hills. L4 and L5 are swales or shallow depressions. Hmmm.
Easy way of calculation --------------------- I'm hugely puzzled why the easy method is not promoted more vigorously among home astronomers. The method uses no massive math and no vectors. It's all arithmetic and simple algebra. This simplicity comes with a price; heavy repetition. The computations must be done over and over and over again for many many points all around the Sun and planet. You will have to do these calcs for smaller parts of the field near the Lagrange points. You'll perhaps have to cut finer slices of the field for detailed calcs. Not all that long ago I would have told you this work takes a summer of weekends, even with a calculette and that jug of wine. But can you think of any device that just loves to do repetitive tasks, and do them very quickly? Yes, your computer fitted with a general purpose programming language. I used BASIC, but whichever one you were brought up with will do just as well so long as it does have the usual algebraic operations built in. Besides a knowledge of BASIC programming, you need the BASIC language program. Included with earlier machines, this seems to be left out of the present day models. You may have to fetch GWBASIC, the BASIC once part of a home computer's equipage, from your favorite software website.
Center of mass ------------ Lo here the diagram.
+ A /| \ / | \ / | \ / | \ / | \ / | \ S O---+-----------------------o P C
S, P, A are Sun, planet, and asteroid. S and P also stand for the mass of Sun and planet. C is the center of mass (COM) of Sun and planet. The asteroid is so small it does not distort the gravity field of the other bodies. I use a double letter to name a line segment, like 'SC' for 'line from S to C'. The place along SP for the COM is the proportion
SC = SP*P/(S+P) PC = SP*S/(S+P)
Please verify that the sum of these two segments is in deed the whole line SP. Try it; just algebraicly add the above two equations and shuffle it a bit. It's a good and wise strategy to build in to any complex mathwork checks like this. You'll catch blunders quickly before they swell out of control -- commonly without you knowing it! - - and you end up with nonsense answers. Now some authors posit the center of mass is at the Sun. This happens to work very well within the solar system because the mass of planet is so very tiny compared to the Sun. The COM is buried deep inside of the Sun. Even for Jupiter, whose mass is quite 1/1000 of the Sun, the COM is still within the inner corona. In the diagram I just put the COM well away from the Sun for clarity's sake.
Rotation about center of mass --------------------------- The rotation of a body in the solar system is measured from the COM, not the Sun. Recalling the cardinal feature of a Lagrange point, we now must emphasize that the angular velocity or, inversely, the orbital period is measured from the center of mass. The angular velocity of a point is, by definition,
(Angular Velocity) = (Tangential Velocity)/(Radius from COM)
AngVel = TangVel/RadCOM = TangVel/CA
Now this is the SAME for Sun, planet, and particle. Only the proportion of TangVel and RadCOM vary among them. So we can avoid the messy handling of three velocities and three distances by taking this one common constant parameter AngVel. Note that it is a pure number divided by unit time, radians/second. The orbital period is the inverse of this AngVel, expanded to the full circuit of the orbit. (There are two pi radians in a circle.)
Period = Circumference/(Tangential Velocity) = (2*pi*Radius)/TangVel = (2*pi)*(Radius/TangVel) = (2*pi)/AngVel
which seems tame enough. The numerator is the circumference of the orbit; the denominator, the tangential speed within the orbit.
Gravity energy function --------------------- A point, like A in the above diagram, senses an energy due to gravity that is the net sum of that from the two bodies and that of the rotation of the entire system. I see some authors harp on the gravity part of the energy, probably because this looks a lot like Newton's law, but essentially all miss out on the rotational part. Why? Likely because in all the diagrams, it is so easy to think the three bodies are standing still in space. They are in mutual rotation! We are looking at the diagram WHILE ROTATING WITH IT, like hiding in the rafters of a merry-go-round spying on the riders on the deck. Such a vantage point is called a 'corotating frame'. Failure to appreciate this feature of the usual representations of Lagrange points, and other rotating systems in astronomy, can lead to dreadful and awful misunderstandings of how such systems behave. What makes this energy function so wonderful is that it works with regular arithmetic. Energy is a scalar, or 'regular' number with direction associated with it. So at any point A in the field around Sun S and planet P we have immediately by the definition of potential and kinetic energy
(Potential energy from Sun) = -gamma*(Sun mass)/(Sun-ast dist) (Potential energy from planet) = -gamma*(pla mass)/(pla-ast dist)
PotErgSun = -gamma*S/SA PotErgPla = -gamma*P/PA
gamma is the gravitational or Newton constant, 6.672e-11 n.m2/Kg2.
(Kinetic energy of asteroid) = (1/2)*(Tangential Velocity)^2
KinErgAst = (1/2)*TangVel^2 = (1/2)*(AngVel*CA)^2
This last is the trick of using the common angular velocity in stead of worrying about the tangential velocity for every point we examine. The total energy at the given point is the potential energy plus the kinetic energy
(Total energy at asteroid) = PotErgSun + PotErgPla + KinErgAst
TotErgAst = PotErgSun + PotErgPla + KinErgAst = -gamma*S/SA + -gamma*P/PA + (1/2)*(AngVel*CA)^2
See how this resembles the total energy of a particle in the two- body situation, like a satellite in orbit around a planet? The math is hardly more troublesome. You do have to work out SA, PA, and CA for each point, but that's really an exercise of the Pythagoras rule of triangles. You remember that, no? hypotenuse^2 = xside^2 + yside^2. That is, to get SA, you do
xside = (xvalue of A) - (xvalue of S) yside = (yvalue of A) - (yvalue of S)
SA = sqrt(xside^2 + yside^2)
Note that the coords of S and P are fixed thruout the calcs by the premise of the plane circular orbits. You only to play with those of A, which you move around to map out the energy field. By coding the whole works into BASIC, this chore can be automated.
Mapping the energy field ---------------------- I recommend a brute-force tactic. In the stead of analyticly hunting for where the Lagrange points are, just grid up the entire zone around the Sun and planet and calculate TotErgAst for each. For the Sun and Earth you can take a 3 AU grid, with 1/100 AU cells, centered on the Sun. First calculate SC, or PC, for the location of the COM. For each cell's coordinates, work out SA, PA, and CA. Send these into the energy equation. Then plug in the known values of S, P, and gamma to get TotErgAst. Plot this value at the cell's coordinates. You can exploit symmetry bu working thru the space on one side of the Sun-planet axis. The other side is merely mirrored across that axis. You will actually get, in a cruder form perhaps, the same sort of lobal layout shown in the textbooks! You'll see that there is a minimum point of the energy function outside the bodies; these are L2 and L3. The minimum between the two is L1. The ones at the sides are L4 and L5. For a smallscale plot and for Earth being so small compared to the Sun, you'll probably not get L1 and L2. Run the program again centered on Earth with smaller cell size and a zoomed in plot. Continue this iterative process until you get a good fix on all the Lagrange points. The actual place is that coordinate of the cell with the local minimum value of TotErgAst.
L1, L2, L3 unstable points ------------------------ The colinear, points L1, L2, and L3 are unstable. A particle placed there will not stay put for long. Any little disturbance, like tidal action of an other planet, will shove the particle off the point, much like a ball pushed off of a hilltop. More accurately, these points are like saddles. If you imagine a stereotypical Western saddle, the front and rear curl upward to support the rider. The sides curl downward to fit the horse's back. From the eye of the planet, the saddle is sideways as if the horse were crossing in front of you. A ball in the middle, where the rider's own [snip] would rest, can be nudged fore or aft and will then roll back to the middle. That ball nudged sideways rolls off of the saddle. It is impossible to make the ball stay put in the middle of the saddle under random displacements, like those induced by the horse walking. Moreover, for L1 and L2 being close to the planet, any moons of that planet will repeatedly upset the body, so in real life there is no fixed position for these points. The Earth-Sun-Moon system is an extreme example. The Moon passes near enough to Earth's L1 and L2 to displace them out from under a particle resident there. Likewise, the Moon's L2 and L3 points are moved around by the influence of the Sun. In fact, a craft stationed at, say, Earth's L1, must be fitted with manoeuvering rockets. These continually nudge the craft back to L1 after being shoved away by the planet's moons. For this reason, such L1 probes like SOHO have on board a large stock of fuel to last out the lifetime of the mission.
L4 and L5 stable points --------------------- The triangular points L4 and L5 are stable. A particle put there will slide back toward it under a disturbance. A planet passing in an adjacent orbit may budge the body off of L4. When the planet moves on and its tidal pull subsides, the body wanders back to L4 much like the ball rolling back to the bottom of a shallow bowl. In fact, it is almost impossible to sit exactly at the geometric point itself due to perturbances from the other planets. It is easiest to place the body near L4 or L5 and let it swirl around it in a loose and lazy orbit. This is like the ball rolling in an 'orbit' around the bottom of the bowl. This orbit never repeats and is not at all simple to describe. It resembles soewhat the swirlly patterns of an oscilloscope, called a Lissajou figure, with random signals applied to the two axes.
Tidal pull of other planets ------------------------- When I explored the effect of planets in adjacent orbits on a given planet's L4 and L5, an amazing fact emerged. I then found that it was common knowledge already among celestial dynamicists, so I'm not famous yet. For all of the planets except Jupiter, the disturbing gravity at L4 and L5 from an adjacent planet is LARGER than the primary field of the planet that created these points! That is, a body in L4 and L5 WILL suffer substantial displacements and can NOT sit quietly on the exact geometrical point. This accounts for the paucity of Trojans around any planet other than Jupiter. They were long ago kicked away too far to roll back, like a ball pushed so hard it overtops the rim of the bowl. So in the real solar system, or within the Earth-Moon system, Lagrange points are not absolutely stable at all. They may be a particle's home for years, centuries, millennia, but sooner or later they'll be upset by the action of the neighboring planets.
Size of Lagrange wells -------------------- In the bowl analogy, the L4 and L5 points are surrounded by a larger or smaller bowl within which the particle can rattle around and not be expelled from its home point. As long as it does not run 'over the rim' it's safe. How big is the bowl or well? I tried what I thought would be the two extreme cases. I first worked out the well for a planet of infinitesimal mass, like an asteroid. The well actually is broad and long, enclosing the entire orbit of the planet and merging all around! In fact, there is no 'well' because I reduced the three-body problem to the trivial case of two-bodies. Thus a particle can be placed ANYWHERE along the planet's orbit and still pace the planet! This is what goes on within the main belt of asteroids and with Janus and Epimetheus, one pair of coorbital moons of Saturn. I next tried with the planet of the same mass as the Sun. Uh-oh. Something really wrong here. The 'well' shrank to nothing, becoming just an other point on the surrounding contours. Any slight displacement will roll the particle off the point and out of synchronous motion with the planet. Further reading and experimentation revealed that when the planet is more than 1/25 of the Sun's mass the wells around L4 and L5 close up and are unstable. When the planet is slightly less than 1/25 of the Sun's mass, the well opens up as a very tiny dimple hugging L4/L5. This gradually enlarges and elongates and broadens with diminishing mass of the planet. When I got to planets of solar system mass, the Lagrange wells are really quite ample, occupying vast zones of the planet's orbit.
Three-dimensional wells --------------------- Virtually all treatises deal with a planar case where the wells are flat microbe-looking patches around the Lagrange points. In fact, the wells are three-dimensional volumes enclosing the points. For the Trojan asteroids of Jupiter, the volume is surprisingly huge, several AU along the orbit and a full AU north and south of the orbit! One vigorous thread of research is the stability of such swarms in the longterm. Do asteroids leave the nest and new ones come into it? How steep an inclination can still allow membership in the Lagrange club? Could a spaceprobe exploit the width of a Lagrange well to study high latitude solar activity?
Roche lobes --------- Edouard Roche discovered in 1849 that there is a zone around the Sun and planet, like a 'sphere of influence', within which a particle is tied to just the one or the other body. It will remain in an orbit about the body, within its Roche lobe as we call it now, and the other body can not capture it. There will be disturbances from the other mass, yes, but the particle is trapped gravitationally to the one in whose Roche lobe it sits. Recall the notion of a well surrounding the Lagrange point, the bowl or saddle. If we trace out the closer rims of these wells all around the Sun we get the frontier of the Sun's Roche lobe. Similarly for the planet. The lobe of each passes thru L1 between Sun and planet. Now look at your field plot again. See where the energy is a relative MAXIMUM? There is a 'ridgeline' all around Sun and planet. This is the inner or closer edges of the wells around the Lagrange points! Within the solar system the Roche lobes are important in capture studies. The reason the Moon can stay attached to Earth as a satellite is that it orbits within Earth's Roche lobe. Altho the Sun distorts the Moon's motion, it can not cause the Moon to leave Earth. With Earth's L1 being about 1.5 million kilometers away, the Roche lobe is a roughly round zone 1.5 million kilometers out from Earth. This is Earth's 'sphere of influence'.
Binary stars ---------- In binary star work Roche lobes are everything. Some binaries are so close that they stand but a couple of their own diameters apart. They tidally distort each other. If they are small enough, their matter is entirely within their own Roche lobes. If one star grows in size, as it will when it becomes a red giant, it may actually expand to fill its Roche lobe! At this moment all hell breaks loose for the star. Matter that touches the frontier of the lobe spills out, quite like water overtopping a levee. It is expelled from the binary system or it may spiral around and fall into the companion star. Such action is a mechanism for streamers and jets in close binary stars. A most strange case is when the star fills its lobe and material touches L1. L1 is shared with the companion and the two lobes join here. L1 becomes a floodgate for material from the large lobe-filling star to the other smaller one! Accretion discs and circumstellar rings are formed by flooding thru L1.
Leonid meteor shower ------------------ Until the mid 1990s it was presumed that the strong Leonid meteors seen near the nodal crossing of the parent comet Tempel-Tuttle were spit out during that passage. The meteors were newly released into the streamtube along Tempel-Tuttle's orbit. Comet Tempel-Tuttle hits its descending node, almost at Earth's orbit, every 33.2 years, and for the most part the Leonid shower was much stronger in the nodal years. Leonid storms came in 1833, 1866, and 1966; weaker, but still enhanced, displays came in 1899 and 1932. Study of reports before 1800 evidence Leonid displays pretty much at 33 year intervals, with s few 'duds'. Attempts to reconstruct the historical displays by this prevailing model were erraticly successful. The 1966 storm could not be replicated by this model at all, leaving some serious doubt about a storm in the late 1990s. In the mid 1990s new understanding of comets was in hand to revise the simple model of a comet shrouded by newly issued meteors. A comet nucleus is covered with dust and grit. As the comet rounds perihelion and is heated, this dust lofts up and can escape from the nucleus's surface. In addition, differential gravity from the Sun across the nucleus raises the dust by tides. To appreciate this method, note that in the peculiar case of the Leonids, comet Tempel-Tuttle passes its descending node and its perihelion only a weekish apart. That's why in meteor litterature we read both 'perihelion' and 'node' when discussing the Leonids. It is the nodal crossing that dumps meteors into the Earth's orbit. Perihelion passing generates the new meteors around the comet. The nucleus, being so small, has a tiny Roche lobe only a few hundred meters in radius. With approach to the Sun's stronger gravity field near perihelion, this Roche lobe contracts, likely to below the depth of the lofted dust and grit. The particles now drifting outside the Roche lobe can drift away from the comet into independent orbits of their own. Because the particles were not expelled or jetted out, their speeds are still very nearly that of the comet and so their orbits remain close to the comet's orbit. It takes several years for this dust to move away from the comet and form a filament or thread of dust parallel to the comet's orbit. Such dust threads were first observed in infrared radiation by the IRAS satellite in 1984, but the image resolution was too low to make out the individual strands. It is the intersection of Earth with these filaments that causes the mass Leonid storms, not mere proximity to comet Tempel-Tuttle. Hence, the meteors seen in a given year are not new meteors but ones cast off from a previous revolution. The storm witnessed in November 1999 came from the filament released on the 1899 return. Filaments from other returns were too far from the Earth to throw down meteors. The dud years, like 1997 just before Tempel-Tuttle's arrival at the node, result from all the filaments missing the Earth, regardless of how close Earth was to the comet itself.
Common misconceptions ------------------- Most misunderstandings about the Lagrange points and Roche lobes come from science fiction or, sadly, a dumbing down of science fact. A Lagrange point is a fixed place in or near the planet's orbit. It perpetually paces the planet for ever and can be plotted on a map permanently like a geographic place. The Lagrange point does wander from overlaying energy fields of the other planets. A truly stable point obtains only for the restricted circular three-body problem. A Lagrange point is a gravity pit. A particle -- even large other planets! -- will drop into it and get stuck there for ever. The hero spaceship has a good map; the villan's does not. Hero skirts past L5 to one side. Villan, in chase a little off the line, plows into L5 and falls in like it's an elephant trap. It's really a local minimum of energy with very gentle 'walls'. Any ordinary spaceship can fly thru it without so much as a bump. So can comets and high-excentricity asteroids. The Lagrange point is an exact place. Step off of it a few meters and you're swept away into your own independent orbit around the Sun. Perhaps it's the size of a manhole cover or phonebooth? Because of the small size and extreme military advantage, sci-fi wars are fought over who gets possession of the points. The conqueror builds his battleship there and pummels the planet from it. The point is merely the lowest energy of a lower 'elevation' well or depression or swale. In the Earth-Moon system, the Kordylewski clouds are within a well at least two degrees diameter as seen from Earth. This is already four times the size of the whole Moon itself. a lot of real estate. SOHO and others are in neat little 'halo orbits' around L1. The NASA chart shows the craft lazily winding round and round L1 snapping picture after picture of the Sun. SOHO and the other probes must have onboard rockets and enough fuel to last out the mission. There are continual burns to nudge the craft closer to L1 as it is urged away by the pull of the Moon and other planets, mainly Venus. One of the grander goofs relate to Roche lobes. These were forcibly put before the home astronomer in the 1970s. That's when interacting binary stars were first observable in detail. The lobes are depicted as bottles or sacks enclosing each star. The vessels copulate thru their mutual L1. The lobes are just mathematical surfaces, like an isobar or isotherm on a weather map. Matter can freely pass thru it at will, as by stellar expulsions or heaving. The region near an interacting binary system is more likely to have a storm of material swirling all around oblivious to any Roche lobe. Routinely the flow of matter around the stars is claimed to be driven only by gravity, so its motion is conservative (reversible with no loss of energy or momentum) and the lobes are static constructs. With the generally strongly elliptical orbits, presence of magnetic fields and stellar wind, differential rotation of the stars, and viscosity from ambient dust, plus some other factors, the concept of a Roche lobe is vastly more involved.
Earth's Trojan companion ---------------------- This asteroid was found in 1986 as 1986-TO but not immediately named. It wasn't until 1997 that its weird motion was realized and it was named 3753 Cruithne. 'Cruithne' is a Celtic word pronounced somewhat like 'kroo-EEth-nigh'. However, if you use a continental sounding, 'kroo-Ith-nay' you'll be understood. It is tough to describe adequately the motion of this weird fellow in words or ASCII diagrams! I'll try, but you may get very dizzy.
. . . . . . + L4. . . . . . / \ . . . / . \ . . . / . .\ . . / \ . . / \ . . / \ . + L3 + Sum + Earth . . \ / . . \ / . . \ / . . \ . ./ . . \ . / . . . . . . \ / . . + L5. . . . . .
In the Sun-Earth system L3 is rather closely on the Earth's orbit in superior conjunction. L4 and L5 are at the 60 degree marks. L1 and L2 play no part in this scenario; the diagram leaves them out. Also for the Earth the wells of L4 and L5 are large, almost blending with the well of L3. It takes very little effort to shove Cruithne out of the L4 well into L3's. Or from L3's into L4's. In 1900, by simulations of its motion, Cruithne was near its closest point to Earth teetering on L4. Since then it gently slided around the orbit toward L3, not yet hopping over the 'rim' between L4 and L3. It takes 385 years to make the complete lap all the way to L5. In 2285 it'll be closest to Earth on the lagging side near L5. On this lap, L4 thru L3 to L5 it is just inside the Earth's orbit with a period of 364 days. Its conjunctions and oppositions relative to the Sun are caused by the ENTIRE ENSEMBLE revolving around the Sun. The motion is hardly so simple as this. 3753 executes an elongated loop AROUND the central place extending for some thirty heliocentric degrees along the Earth's orbit. So it does some weird gyrations in the sky, making a fancy Lissajou pattern among the stars. It does NOT circulate around the zodiac like a regular asteroid. In 1950, for example, Cruithne moved in a Lissajou loop almost entirely within Pisces and Cetus! This loop was centered roughly on beta Ceti (Diphda) and was some 25 degrees across. This loop gradually expanded in size and its center drifted southeast. This loop spanned only a limited range of ecliptic longitudes, from about 350 degrees to about 15 degrees. Cruithne's high inclination made it range from a couple degrees north to some 25 degrees south of the ecliptic. Watch out! The center of this loop migrates among the Lagrange points and thru the stars in 770 years, round trip. The very asteroid circles this center tracing out the loop among the stars in 364 days. The 3D path in space is a sort of spiral around the Sun! In 1995 the loop enlarged so much that it touched and crossed the south ecliptic pole. At this moment, and still now in 2001, Cruithne can runs thru all degrees of the ecliptic, altho with almost all large southern latitudes. In 2000 the loop was huge and still enlarging slowly, it now is centered in Horologium. (You thought this constellation would never have any importance?) The path takes it thru Scorpius, Ophiuchus, Serpens, Aquila, Aquarius, Pisces, Cetus, Eridanus, Lepus, Canis Major, Puppis, Carina, Centaurus, Lupus, thru Scorpius to repeat the cycle in 364 days. Rounding L5 it wanders back thru L3, on the outside of Earth's orbit with a 366 day period, until it reaches L4 again. The whole round trip takes 770 years. What a ride! If you want to play with 3753 here are its elements, taken from JPL's DASTCOM service:
Name = 1986-TO = Cruithne = 3753 Elements Epoch = JD 2,451,800.5 = 2000 September 13 0h UT Semimajoraxis = 0.9977426 AU Excentricity = .5147552 Inclination = 19.80836 degree Longitude of Ascending Node = 126.33434 degree Argument of Perihelion = 43.69687 degree Perihelion Date = JD 2,451,916.1610631 = 2001 January 6.6610632 Perihelion Distance = 0.484149408 AU Absolute Magnitude H = 15.1 Magnitude Coefficient G = 0.15
Cruithne is, unfortunately, beyond small home telescopes, being but 5 kilometers across and never brighter than 15-1/2 magnitude.
Simulation of Lagrange points --------------------------- To model the behavior of Lagrange points and Roche lobes, you need a competent gravity simulator. There are two kinds. One is a general model of gravity aimed more at physicists where you arbitrarily assign masses and distances to the bodies. The other is for astronomers and includes the solar system properties. I used the latter in the form of Dance of the Planets. This venerable DOS program is the best overall program that moves planets and other bodies with true gravity. A comet, for instance, will deviate from its initial path and an asteroid will wander off course when other planets tug on it. It allows insertion of a new body by either its orbital elements, like for Cruithne above, or by raw XYZ vectors of position and velocity. Because the gravitational influences of the other planets is small but accumulative, a simulation must run for many centuries of scaletime. Yes, I really really did leave the computer running overnight or on weekends to do these runs! You can not use a 'trolley track' program or one that employs only a two-body model of gravity. You'll do nicely for a comet's current apparition or an asteroid with addiurnate orbital elements. Trying to run the model for many hundreds of years will get you utterly stupid results. On the Web there are a few Java programs or applets, within sites dealing with gravity and celestial mechanics. Some are preloaded with examples. Others you have to hack at the code to initialize them. These run thru your Web browser or thru the website itself. There are a couple simulators in DOS/WIN executables issued as part of a college class in celestial dynamics. These can be fetched from the school's or professor's wensite. I gave up on listing websites. They mutate so often and many links lead to deadends. Search for 'libration point', 'Trojan asteroid', 'Lagrange point', and similar.
Lagrange points beyond the solar system ------------------------------------- SInce the 1960s, as an outgrowth of improved observation and theory, we found application of Lagrange points elsewhere in the universe. I sketch out these examples without details. I already discussed the importance of Lagrange points for binary stars in the sections on Roche lobes and misconceptions. More recently, in the 1990s, studies done of globular clusters in the Milky Way's halo show that they can lose their stars thru their L1. This is the Lagrange point between the cluster and the galactic center. A fascinating application comes from studies of atomic structure! The electrons in an atom behave sort of like a miniature solar system. Mapping of the electron distribution around the nucleus shows Lagrange effects.
Hearty thanks ----------- This paper started out as a couple pages to answer some small questions. It grew as I learned of the paucity of good tuition about Lagrange points and Roche lobes among home astronomers. In its gestation this paper passed thru many reviews, notably by members of the AAA Recent Astronomy Seminar. Special thanks for comments, clarifications, and corrections, presented on marked up copies of the article go to Shyly Amarasinghe, Bruce Kamiat, Bernie Kleinman, Charlie Ridgway, and Stewart Rorer. In addition to hardcopies distributed at the Seminar in late May 2001, the paper is hung in the file area of the NYSkies maillist in Yahoogroups.com.