# Lagrange Points: The Stable Regions of Space

Figure 1: Lagrange Points placements with regard to the celestial bodies in space

Credit: solarsystem.nasa.gov

Abstract

Lagrange points are the points of equilibrium and stability for any small third body with respect to two large celestial bodies in space.This article will give an introduction to Lagrange points as a universal concept and then cover achievements at the Sun-Earth system’s Lagrange points so far. It also includes the scope of future missions at these locations of not only the Sun-Earth system but others too. These points are significant for space exploration and as future space destinations. Mathematically, coordinates of Langrange points of two large body systems can be calculated easily. Formulas to calculate the coordinates for each and every Lagrange point with the exact numbers for the Sun-Earth and Earth-Moon system are computed.

Figure 2: Gravitational waves around the lagrange points due to the celestial bodies

Credit: solarsystem.nasa.gov

Introduction

Every object in space has a gravitational force of its own which it forces on other bodies around it. The combined gravitational forces of two large bodies creates a point of equilibrium in the space where both equals the centripetal force required for a small object to move with them, forming an area of completely stable environment, with no gravitational forces to influence it. These regions formed due to two spherical bodies are known as Lagrange points or L points or Liberal points. Each system of a pair of celestial bodies has five such points that can exist, named from L1 to L5 . They are named after Joseph-Louis Legrange, an Italian- French mathematician who discovered them in the 18th century and mentioned them in his paper called the “Three Body Problem”. Anything placed at these points of equilibrium will be stable and in constant pattern with respect to these pairs of large bodies. [1][2]&[3]

As shown, the first three points L1, L2 and L3 are in line with the centre of the two bodies and are meta-stable. While lagrange points, L4 and L5 form the apex of the [1] equilateral triangles with the large bodies as the other two vertices of it are more stable. If we take the system of Sun and Earth, then the L4 and L5 are at an angle of 60 degrees from the earth, with L4 leading the orbit of the Earth around the Sun and L5 following it.

Lagrange Point 1 (L1)

The L1 point lies between the two bodies of the system in line with their centres.[1][2] For the Sun-Earth system, it lies at a distance of 1.5 million kilometers from the Earth. L1 has the privilege of giving an uninterrupted view of the Sun. At any point of time, [3] multiple astronomical and earth observatories can be there at L1 point, so a number of

missions have been planned at L1 in the past and future too. This is one of the most utilized Langrange points.

Past and current missions at L1:

The first mission to the Sun-Earth L1 was the International Sun-Earth Explorer-3 in 1978. The objective was to study the interaction between the Earth magnetic field and solar wind however contact with this probe was lost in Sep, 2014. NASA’s Genesis probe collected particles of solar wind for the analysis on Earth from 2001 to 2004 at L1. At present, since December 1995, Solar and Heliospheric Observatory (SOHO) has been there to make observations of the Sun. There are three more probes at present at L1 point other than SOHO. They are ACE (Advanced composition Explorer) to study matter comprising energetic particles from the solar wind, the interplanetary medium, and other sources. Wind (spacecraft) to study radio waves, complete plasma in the solar wind and in the Earth’s magnetosphere. Also, NASA’s Deep Space Climate Observatory (DSCOVR) was launched at L1 in February, 2015 which is important for timely and accurate space weather alerts on Earth so as to safeguard power grids, aviation, GPS and telecommunications on the earth. European Space Agency’ s LISA pathfinder mission was also sent to the L1 in 2015, in order to demonstrate the underlying technologies for a future space-based gravitational wave observatory.This was deactivated in 2017 but it has paved the way for such an observatory which is planned to be launched in 2034. [6]

Future missions at L1:

The NEOcam (Near Earth Object) surveyor mission is planned to be launched in 2026 by NASA which is a space based infrared telescope to observe the sky and look for potentially hazardous asteroids. The Aditya-1 mission from ISRO is also set to launch in mid-2022, and with the objective to provide observations of the Sun’s photosphere, chromosphere and corona. An idea proposed by researchers to slow down climate change from L1 is by using a single large sunshade/ giant fresnel lens that can distort the light from the Sun, if manufacturing and launching possible. [6] For the L1 point between Sun and Mars an idea proposed by Jim Green the Chief Scientist at NASA, is that we can position an artificial magnetosphere shield at L1 that would block the solar wind that constantly tears away the Martian Atmosphere. This would allow traces of gases to form a thin atmosphere to Mars, thickening it up further over time. The greenhouse gases will also start to warm the atmosphere causing the trapped water to also melt and get converted into water vapour. This could replenish about 1/7th of Mars’s oceans.

Similar concept can be used to make the planet of Venus habitable over the centuries just like Mars, i.e. by placing a Sun shade between the Sun and Venus at L1, though 4 times its size to block the Sun radiations.

Significance of L1:

The Earth-Moon L1 point can be used to place multiple satellites and observatories which can study the lunar surfaces and the Earth as it will rotate about its axis making it possible to observe the whole of the Earth every day. It is also recognized as the perfect spot to make the space elevator that has been in the works and ideas for a really long time. [6]

Coordinates for the Lagrange Point 1 (L1) for any two large bodies can be calculated as explained by Neil J. Cornish of Wilkinson Microwave Anistropy Probe Team as:

where,

Distance between the two masses

= ,

and are the masses of the two bodies

For the Sun-Earth system and , therefore giving a distance of that can approximately be written as from the Earth.

For the Earth-Moon system and , therefore giving a distance of that can approximately be written as from the Moon.

Lagrange Point 2 (L2)

The L2 point is located behind the smaller of the two bodies, in line with their centre.[1] For the Sun-Earth system, L2 is 1.5 million kilometers behind the Earth from the point of view of the Sun. It is the most convenient place to observe the larger universe for its clear view without any disturbances as the satellites there do not have to orbit the Earth and can communicate effectively with Earth [3][7]

Past and Current mission to Sun-Earth L2:

The first spacecraft that was sent to L2 was NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) in June 2001 which spent a decade and mapped the cosmic microwave background radiation. After that, NASA’s WIND satellite was also sent to L2 for half an year to study Earth’s magnetosphere, after which it was sent to L1 through the Earth’s orbit to study the impact of the Sun’s magnetosphere. In May 2009, ESA’s Herschel Space Observatory was sent which was equipped with an infrared telescope 1.5 times larger than that of the Hubble Telescope. It tracked the presence of water in our milky way galaxy. It ran out of fuel in 2013 and joined WMAP in order to orbit the Sun. Planck Space Telescope was also launched in May 2009 to replace WMAP making detailed observations of the cosmic microwave background radiation. The spacecraft was soon shut down in 2013 and sent into the heliocentric orbit. These missions were soon followed by ESA’s Gaia Space Observatory in 2014 to chart a 3D map of billions of stars in the Milky Way Galaxy. It is still orbiting in the L2 region along with Russia and Germany’s Spectra RG launched in July 2019, which is an x-ray observatory designed to map out the high energy regions of the universe.[7]

Future missions at L2:

To achieve the human aspiration of knowing and mapping the universe till the edges, L2 is a significant location. Hence many ambitious missions are planned at this point. The most awaited is NASA’s James Webb Space telescope to be launched by the end of 2021. It is a tennis court size space telescope optimized to study infrared spectrum wavelengths and is designed to study the observable universe till its edge to see the first stars and galaxies that were formed after the big bang. It would soon be followed by the ESA’s Euclid Space Telescope in 2022 to study the visible and near-infrared wavelengths deep in the universe. It will also map out the speed at which the galaxies in space are moving away from us to a distance of about 10 billion years in the past. The Nancy Grace Roman Space Telescope (WFIRST) is planned to be sent by 2024, in order to observe a wide view of the sky and map out dark energy. WFIRST will continue the work of NASA’s Kepler Telescope to find extrasolar planets using gravitational lensing and onboard chronographs. ESA’s Advanced Telescope for High Energy Astrophysics (ATHENA) is also set to launch in 2031 and is an X-Ray observatory mission designed to map hot gas structures, determine their physical properties, and search for supermassive black holes. Another mission planned by ESA is Planetary Transits [7] and Oscillations of Stars (PLATO) to be launched in 2026 in order to detect terrestrial exoplanets up to the habitable zone of solar-type stars and characterisation of their bulk properties needed to determine their habitability.[8]

Significance of L2:

The Earth-Moon L2 is the best spot for communicating and exploring the far undiscovered side of the Moon and as a jumping point for the space missions further into the solar system. Till now it has been used by China’s Chang’e 5-T1, launched in January 2015 in order to maneuver upon the future lunar missions. Queqiao was also sent to Moon-Earth L2 in June 2018 to form communication with Chang’e 4 lander on the far-side of the lunar surface, which couldn’t communicate directly with Earth. [7]

Coordinates for the Lagrange Point 2 can be given as:

where,

Distance between the two masses

= ,

and are the masses of the two bodies

For the Sun-Earth system and , therefore giving a distance of that can approximately be written as from the Earth.

For the Earth-Moon system and , therefore giving a distance of that can approximately be written as from the Moon.

Lagrange Point 3 (L3)

The L3 point lies beyond the larger of the two bodies, in line with their centres . In the [1] Sun-Earth system, this region is completely behind the Sun. Space organisations have not used the spot for any of their missions yet or plan to use it in the future too as we can never have direct communication with any object at the L3 point.

But, L3 has been a popular topic amongst science fiction writers who say that there might be a secret planet hidden in that region. However this point has been refuted by the scientists with support of two reasons. First, for anything to exist at L3, it needs to have negligible mass compared to the other bodies in the system. Second, if we take a hypothetical situation in which an Earth-size planet exists at L3, it would not be able to remain there for more than hundred and fifty years. One idea that does have some practical use for us is that the far side of the Sun would be a good site to do ongoing observations of the sun from an angle we can’t observe from the Earth. This would give us a 3D map of the entire Sun, as proposed by a 2009 paper, ‘Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem’. But, NASA’s twin stereo satellites [9] orbiting the Sun in the Earth’s orbit, already gives us a complete 3D view of the Sun, therefore making the above mentioned mission unnecessary.[7]

Significance of L3:

L3 of the Sun-Earth system is an unexplored region as this is behind the Sun and is of interest amongst science fiction writers only.

The coordinates for the Lagrange Point 3 (L3) can be given as:

where,

Distance between the two masses

= ,

and are the masses of the two bodies

For the Sun-Earth system and , therefore giving a distance of that can approximately be written as from the Earth.

For the Earth-Moon system and , therefore giving a distance of that can approximately be written as from the Moon.

Lagrange Point 4 (L4) and Lagrange Point 5 (L5)

L4 and L5 point form the apex of the equilateral triangles with the large bodies as the vertices of it. For the Sun-Earth system, the L4 and L5 are at an angle of 60 degrees from the earth, with L4 leading the orbit of the Earth around the Sun and L5 following it. Unlike L1, L2 and L3 points, a spacecraft at L4 and L5 is completely stable. It is due to this stability that natural objects are also found here. These natural objects that linger around the L4 and L5 points are generally referred to as Trojans by the scientists. [1][2] Sun and Jupiter are the biggest bodies in our Solar System therefore they have the largest number of trojan asteroids in their L4 and L5 points. The Sun-Neptune system also has dozens of asteroids in its L4 and L5 points followed by the Sun-Mars system, which has around four trojan asteroids in their Lagrange points. Several Moons of Saturn are also said to have multiple Trojan companions with them. The Sun-Earth system mainly consists of interplanetary dust but also one Trojan Asteroid named 2010 TK7. The Earth-Moon system has interplanetary dust present in its L4 and L5 points [10]

called Kordylewski clouds. It is the stability and presence of Trojans which makes L4 point of not only Earth-Sun system interesting but of other systems too as these Trojans can be a good resource for space related aspirations of humans. Though there have been no missions until now, a lot is planned for the future.

Future missions at L4:

One of such missions is named Cubesat, proposed at the 46th Lunar and Planetary Science Conference by astronomers. In this mission, a satellite will be sent to one of the Sun-Earth L4 or L5 points in order to search for additional Trojan asteroids. These Trojans can further be taken as potential candidates for future robotic orbiters and landers.

The Lucy Mission is also planned by NASA which will spend 12 years visiting 7 different asteroids, one in the asteroid belt and 4 in the L4 region of the Sun-Jupiter system and 2 in the L5 region of the same system.

Another mission proposed by NASA is Astrodynamical Space Test of Relativity using Optical Devices (ASTROD). This would be a gigantic Gravitational Wave Interferometer which would not only use the L4 and L5 points of the Sun-Earth system but also the L3 point. It would also use the LISA observatory to detect gravitational waves with more sensitivity than it was ever possible. Due to its huge sensitivity it will be able to detect collisions between supermassive black holes, intermediate black holes, compact binary stars and even the gravitational wave vibrations left over from the Big Bang. [10]

Significance of L4, L5 point

In the world of Science fiction the L4 and L5 points have been given a lot of thought for a possibility of building space colonies in them. The most credible proposals for the same have been presented by the Princeton Physicist Gerard O’Neill through his O’Neill Cylinder Space Settlement in the 1970s. Since the space colony will be placed outside the magnetosphere of the Earth, therefore the inhabitants inside would require protection against the cosmic radiations, this can be done by increasing the mass and we can obtain such a huge amount of metal through space mining, whether it be mining the Moon or an Asteroid. Another problem that can occur in such a system is the axis of its rotation, which was solved by O’Neill and that was by building the station in pairs, rotating in opposite directions therefore balancing out the forces. One of the biggest reasons this idea is not possible is that it would require a very large amount, almost 4.5 billion tonnes of rocks, steel, machinery etc. Therefore by the end of everything including making, assembling, launching and controlling it would cost at least \$ 450 trillion. The only way we’ll be able to make such a large spaceship is by developing completely space-based mining and manufacturing. [10][11]

The coordinates for the Lagrange Point 4 (L4) and Lagrange Point 5 (L5) can be given as:

where,

Distance between the two masses

and are the masses of the two bodies

For the Sun-Earth system , and , therefore giving the L4 and L5 coordinates as in opposite directions from the Earth and Sun.

For the Earth-Moon system , and , therefore giving the L4 and L5 coordinates as in opposite directions from the Moon and Earth.

Conclusion

Knowledge about Lagrange points has been a stepping stone for human’s rendezvous with space exploration. L1 and L2 points of the Sun-Earth system have been the most utilised so far and even though there has not been much use for the Earth-Moon L1 and L2 points at present, but the plans in future for them are promising. L4 and L5 points are very stable and even natural objects known as trojans are settled at these locations. It is due to this stability that the future missions conceived for L4, L5 are exceptionally ambitious and technologically possible.Therefore, possibilities are endless, all that’s required is time, money and constant advances in the field of technology to put these ideas into action.

References

[1] Howell, Elizabeth. “Lagrange Points: Parking Places in Space”. Space.com. August 22, 2017. Accessed August 26, 2021
https://www.space.com/30302-lagrange-points.html

[2] NASA/WMAP Science Team. “What is a Lagrange Point?”. NASA Solar System Exploration. June 23, 2020. Accessed August 26, 2021
https://solarsystem.nasa.gov/resources/754/what-is-a-lagrange-point/

[3] European Space Agency. “What are Lagrange points?”. Accessed August 26, 2021
https://www.esa.int/Enabling_Support/Operations/What_are_Lagrange_points

[4] Cornish, Neill J. “The Lagrange Points”. NASA WMAP Education and Outreach. 1998. Accessed August 27, 2021 https://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf

[5] Curtis, Howard D. “The Two-Body Problem. Orbital Mechanics for Engineering Students”. Sciencedirect.com. Chapter-2(2010)
https://doi.org/10.1016/B978-0-12-374778-5.00002-7

[6] “A Tour Of The Lagrange Points. Part 1 – Past And Future Missions To L1”. Filmed August 31, 2019. Youtube Video. 13:56. Posted by “Fraser Cain”. Accessed August 28, 2021

[7] “A Tour Of The Lagrange Points. Part 2 – Space Telescopes At L2 And Nothing At L3”. Filmed September 4, 2019. Youtube Video. 17:00. Posted by “Fraser Cain”. Accessed August 28, 2021

[8] eoPortal Directory. “PLATO (PLAnetary Transits and Oscillation of stars)”. Accessed August 28, 2021 https://directory.eoportal.org/web/eoportal/satellite-missions/p/plato

[9] Tantardini, Marco, Elena Fantino, Yuan Ren, Pierpaolo Pergola, Gerard Gómez, and Josep J. Masdemont. “Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem.” Celestial Mechanics and Dynamical Astronomy 108, no. 3 (2010): 215-232.https://link.springer.com/article/10.1007/s10569-010-9299-x

[10] “A Tour Of The Lagrange Points. Part 3 – Trojans and Space Colonies at L4/L5”. Filmed September 11, 2019. Youtube video. 11:50. Posted by “Fraser Cain”. Accessed August 30, 2021

[11] O^ NEILL, G. E. R. A. R. D. “The colonization of space.” In Space Manufacturing Facilities, p. 2041. 1974.
https://doi.org/10.2514/6.1975-2041

Image Credits

[1] Figure 1: Lagrange Points placements with regard to the celestial bodies in space

Credit: solarsystem.nasa.gov, NASA/WMAP Science Team, June 23, 2020

https://solarsystem.nasa.gov/resources/754/what-is-a-lagrange-point/

[2] Figure 2: Gravitational waves around the lagrange points due to the celestial bodies Credit: solarsystem.nasa.gov, NASA/WMAP Science Team, June 23, 2020

https://solarsystem.nasa.gov/resources/754/what-is-a-lagrange-point/