The James Webb Space Telescope, the largest and most powerful of its kind in history, has delivered images of unprecedented resolution and sensitivity since its launch in late 2021. Its observing instruments operate in the spectrum infrared and require temperatures of -364 degrees Fahrenheit or colder to function properly. This means that they must be protected both from direct radiation from the Sun and from that reflected by the Earth and the Moon, for which the telescope is equipped with a large sunshield.
To be effective, the James Webb must follow an orbit in which, at any given point, the Sun, Earth and Moon are all in the same direction, which the sunshade will face. This orbit cannot be around the Earth, like that of the Hubble telescope. Instead, a more sophisticated alternative had to be devised, with the help of dynamical systems theory.
The L2 point of the Earth-Sun system fulfills these criteria. This point is 1.5 million kilometers from Earth – about four times the distance between Earth and the Moon – in the opposite direction from the Sun. Its location stems from the work of 18th-century mathematicians Leonhard Euler and Joseph-Louis Lagrange on the three-body problem. They studied a system of two masses moving in circular orbits around their center of mass, and a very small third mass (an asteroid or an artificial satellite) which moves under the gravitational pull of the first two.
Euler and Lagrange proved that there are five equilibrium positions for the third mass: three of them (L1, L2 and L3) stay on the line that connects the masses, while the other two (L4 and L5) form with them an equilateral triangle. Also, since the five positions move with the two masses, the configuration does not change. Therefore, if the masses are the Earth and the Sun and a satellite is placed at the point L2, it will move around the Sun accompanying the Earth, remaining on the line that joins the Sun and the Earth, as shown the image.
The L2 orbit has an important characteristic: it is unstable. This means that over time, any object in it will change its course and leave. Near L2, there are other less unstable orbits, which also meet the mission requirements and, in addition, allow the telescope to avoid the shadow of the Earth and the Moon. This is an added advantage because the James Webb works with electrical energy obtained through solar panels. They are known as Halo orbits and revolve around L2, accompanying it on its journey around the Sun. One of them was chosen for the telescope. As it is unstable, trajectory correction maneuvers are necessary to keep the James Webb in orbit.
An advantage of this instability is that it is known that there is a (reduced) number of transfer paths which come close to it. Placing the James Webb on one of these trajectories will bring it closer to Halo’s orbit over time. The combination of all these transfer paths forms what is called in mathematics a differentiable manifold.
Calculation by computer-implemented algorithms shows that part of it is close to Earth; perfect for sending the James Webb to its destination. Thus, the Ariane 5 rocket used for the launch from Kourou, French Guiana, did not leave the telescope in orbit around the Earth – as it would be customary – but propelled it to this close transfer path. Until reaching Halo’s final orbit, the trip was very quick at first, taking about three days to pass the Moon’s orbit, and slower as it got closer to its destination, which has reached approximately 30 days after launch.
Angel Jorba is professor of applied mathematics at the University of Barcelona.
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