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Satellites, black holes, exoplanets: when science extends beyond our planet

Satellites: why are ‘Lagrange points’ so important?

Paul Ramond, Post-doctoral Fellow in Astrophysics at Université Paris Dauphine-PSL
On June 1st, 2022 |
5 min reading time
Paul Ramond modifiée
Paul Ramond
Post-doctoral Fellow in Astrophysics at Université Paris Dauphine-PSL
Key takeaways
  • The JWST satellite, launched on 25th December 2021, recently reached its anchor point in orbit around the sun, known as the L2 Lagrange point.
  • Lagrange points are based on a mathematical conundrum known as the ‘three-body problem’, which involves, for example, two celestial bodies orbiting the sun. This orbit is the first Lagrange point.
  • The co-rotating frame of reference, which reduces the satellite’s trajectory to a single point, allows us to find the other two Lagrange points - L2 and L3 - on the same axis.
  • But there are actually more than three Lagrange points. It was Joseph Louis Lagrange who demonstrated that there are five. However, the two other points are not in the same reference frame as the first ones.

The James Webb Space Tele­scope has arrived safe­ly at its des­ti­na­tion – the famous Lagrange point L2 locat­ed in the inter­plan­e­tary void of our Solar sys­tem, some 1.5 mil­lion kilo­me­tres from Earth. But why there? What is so spe­cial about Lagrange points, of which there are sev­er­al, that space agen­cies have been send­ing satel­lites to for the past 50 years?

The three-body problem

Since Isaac New­ton for­mu­lat­ed the laws of mechan­ics and grav­i­ta­tion at the end of the 17th  Cen­tu­ry, the “three-body prob­lem”, which con­sists of deter­min­ing the motion of three celes­tial objects under their mutu­al grav­i­ta­tion­al attrac­tions, has been one of the most fruit­ful prob­lems of clas­si­cal physics. Attempts to solve it have led to count­less the­o­ret­i­cal advances, which in turn have enabled many rev­o­lu­tion­ary prac­ti­cal appli­ca­tions. Even though we have a much bet­ter under­stand­ing of the prob­lem and its dif­fi­cul­ty, in par­tic­u­lar thanks to the work of Hen­ri Poin­caré at the end of the 19th Cen­tu­ry, it is not, to this day, com­plete­ly solved.

Lagrange points nat­u­ral­ly appear as spe­cial solu­tions of the so-called ‘restrict­ed’ three-body prob­lem, in which one of the bod­ies is very small com­pared to the oth­er two. This is the case for the motion of a satel­lite around the Sun and a planet.

The balance of forces

Let us first con­sid­er the orbit of the Earth around the Sun, which is a near-per­fect cir­cle. The Earth makes a com­plete orbit around the Sun in one year, a peri­od known as its rev­o­lu­tion. Let us now imag­ine plac­ing a satel­lite in a cir­cu­lar orbit around the Sun with the the same rev­o­lu­tion (exact­ly one year) and such that it always lies on the Earth-Sun axis. By adjust­ing its dis­tance from the Sun in just the right way, the force of attrac­tion of the Sun will exact­ly com­pen­sate for that of the Earth and the satel­lite will be ‘in equilibrium’.

How­ev­er, the satel­lite is also sub­ject to a cen­trifu­gal force (the one that pulls us out­wards in a mer­ry-go-round). This force adds to the grav­i­ta­tion­al attrac­tions, but does not change the big pic­ture: we can always adjust the posi­tion of the satel­lite on the Sun-Earth axis so that the three forces can­cel one anoth­er, as shown in the fig­ure below.

The orbit of the Earth (green) and that of the satel­lite (grey) around the Sun (orange) in the cen­tre. The attrac­tive forces of the Earth and the Sun are shown, as well as the cen­trifu­gal force expe­ri­enced by the satel­lite, in purple.

The co-turning reference frame

The orbit that we have just con­struct­ed for the satel­lite is pre­cise­ly what astronomers call a ‘Lagrange point’. But why Lagrange ‘point’ and not ‘orbit’? The rea­son lies in anoth­er way of look­ing at our prob­lem. Indeed, since the satel­lite and the Earth orbit the Sun in a year, we can imag­ine orbit­ing with them, and take this rev­o­lu­tion into account in the graph­ic rep­re­sen­ta­tion of the prob­lem. The satel­lite’s tra­jec­to­ry is then reduced to a point, placed between the Sun and the Earth, which is also fixed in this so-called ‘co-turn­ing’ ref­er­ence frame. This point, which in fact rep­re­sents an orbit, is the Lagrange point L1.

The first three

The co-turn­ing ref­er­ence frame is very use­ful and allows us to uncov­er four addi­tion­al Lagrange points, by sim­ply look­ing for oth­er spots where all three forces (Earth and Sun’s attrac­tion and the cen­trifu­gal force) can­cel out. Two of these points, called L2 and L3, are also on the Sun-Earth axis, as shown in the fig­ure below.

The orbit of the Earth (green) and the Lagrange points (black) around the Sun (orange) at the cen­tre, in the co-rotat­ing ref­er­ence frame. The attrac­tive and cen­trifu­gal forces expe­ri­enced by a satel­lite in each of them are shown. Here the dis­tances and sizes are extreme­ly exag­ger­at­ed (see the box for the real values).

The Lagrange point L2 lies on the oth­er side of the Earth from L1, and L3 on the oth­er side of the Sun. It is clear from the fig­ure that the three forces expe­ri­enced by a satel­lite at these points can­cel out: this is why they are also called ‘equi­lib­ri­um points’. Togeth­er with L1, the points L2 and L3 cor­re­spond to three exact solu­tions of the three-body prob­lem deter­mined by Leonard Euler and pub­lished in 1765. Locat­ed at the oppo­site end of the Sun from us, the point L3 has been the sub­ject of many flights-of-fan­cy since ancient times. One of these is a hypo­thet­i­cal ‘anti-Earth’, which would exist at L3 but which would be exact­ly hid­den from us by the Sun.

Artificial satellites

On the prac­ti­cal side of things, it is main­ly the L1 and L2 Lagrange points that have inter­est­ed space agen­cies. Indeed, satel­lites are reg­u­lar­ly sent there for spe­cif­ic sci­en­tif­ic mis­sions. The first time was in 1978 with the ISEE (Inter­na­tion­al Sun-Earth Explor­er) pro­gramme, whose mod­ules were placed in orbit around the L1 Lagrange point of the Sun-Earth sys­tem (locat­ed at 1.5 mil­lion km, that is, at less than 1% of the Earth-Sun dis­tance). Since 2018, the L1 point in the Earth-Moon sys­tem (326,000 km from Earth, 15% of the Earth-Moon dis­tance) has also been home to the Chi­nese relay satel­lite Que­qiao, which com­mu­ni­cates with the Chang’e 4 lunar probe on the far side of the Moon.

The point L2 (also locat­ed about 1.5 mil­lion km from the Earth) of the Sun-Earth sys­tem is home to some of the most extra­or­di­nary space mis­sions. Among the most recent is the Planck satel­lite1, launched in 2009 to mea­sure the old­est light in the Uni­verse (known as the cos­mic microwave back­ground) with extreme pre­ci­sion. Anoth­er is the LISA Pathfind­er satel­lite, which was launched in 2015 and which aimed to demon­strate how mature the tech­nol­o­gy need­ed for the future LISA space grav­i­ty inter­fer­om­e­ter is. More recent­ly, the Gaia mis­sion2, which cat­a­logues the posi­tion, speed and light of more than a bil­lion celes­tial objects (stars, but also aster­oids, galax­ies, etc.), has also found itself at L2. The lat­est satel­lite to reach L2 is the James Webb Space Tele­scope3, launched on 25 Decem­ber 2021. Suc­ces­sor to the famous Hub­ble Space Tele­scope, its mis­sion is to study near­by exo­plan­ets and explore the far­thest reach­es of the Uni­verse to observe the very first galaxies.

It may sound crowd­ed up there and you might ask your­self: don’t all those satel­lites at the L2 point run the risk of col­lid­ing with each oth­er? After all, there is only so much space at a Lagrange ‘point’! For­tu­nate­ly, these satel­lites are not sent to the exact posi­tion of the Lagrange points, but in orbit around them, in an area that is huge com­pared to the scale of a satel­lite: sev­er­al hun­dred thou­sand kilo­me­tres in diam­e­ter. The satel­lites are thus in orbit around the Lagrange point, but the small num­ber of adjust­ments required to keep them there are neg­li­gi­ble com­pared to what it would take to keep them in such an orbit else­where in the Solar sys­tem. This is the main advan­tage of Lagrange points: grav­i­ta­tion and mechan­ics ‘nat­u­ral­ly’ take care of a satellite’s motion so that astronomers can devote them­selves to the beau­ti­ful sci­ence that these mis­sions allow!

The last two

It was Joseph-Louis Lagrange who showed in 1772 that two oth­er points, L4 and L5, exist. These are not on the Sun-Earth axis, but locat­ed at equal dis­tance from both, so as to make an equi­lat­er­al tri­an­gle. At these points, the two grav­i­ta­tion­al forces and the cen­trifu­gal force, although not aligned, still can­cel out exact­ly, as shown in the fig­ure below. In prac­tice, the satel­lite is there­fore almost on the Earth­’s orbit, 60° ahead of (L4) or behind (L5) it.

Natural objects

Since Lagrange points are asso­ci­at­ed with equi­lib­ri­um posi­tions, it would not be sur­pris­ing to find nat­ur­al objects (such as small aster­oids) at these loca­tions in the Solar sys­tem. In the case of the Sun-Jupiter sys­tem, about 10 000 aster­oids have been observed to date around the L4 and L5 Lagrange points. Called Tro­jan aster­oids, some of these even have nat­ur­al satel­lites of their own, such as the largest called ‘(624) Hec­tor’ and its aster­oidal moon “Sca­man­drios”.

These thou­sands of objects are proof that the Lagrangian sta­bil­i­ty regions are real and not just the­o­ret­i­cal. In fact, most plan­ets of the Solar sys­tem have Tro­jan aster­oids, that is,  small rocky bod­ies locat­ed near the L4 and L5 loca­tions of the cor­re­spond­ing Sun-plan­et sys­tem. How­ev­er, one mys­tery remains, as no Tro­jans have ever been observed for the Earth-Sat­urn sys­tem. It is sus­pect­ed that grav­i­ta­tion­al dis­tur­bances cre­at­ed by Jupiter pre­vent aster­oids from stay­ing there too long4. Yet, iron­i­cal­ly, two of Sat­urn’s nat­ur­al satel­lites, Tethys and Dionne, have Tro­jans themselves!

4Influ­ence of the coor­bital res­o­nance on the rota­tion of the Tro­jan satel­lites of Sat­urn, Philippe Robu­tel, Nico­las Ram­baux & Maryame El Moutamid Celes­tial Mechan­ics and Dynam­i­cal Astron­o­my vol­ume 113, pages1–22 (2012)

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