Satellites: why are ‘Lagrange points’ so important?

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?

Since Isaac New­ton for­mu­lated the laws of mech­an­ics and grav­it­a­tion at the end of the 17^th  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 19^th 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.

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 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 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 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.

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.

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!