Lagrange

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The author of this page will appreciate comments, corrections and imagery related to the subject. Please contact Anatoly Zak.


Acknowledgement:

The author would like to thank Igor Rozenberg for his help in preparing this section.

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Strategy

By 2011, NASA and its partners in the International Space Station program formulated several scenarios for the future of manned space flight. Credit: NASA


Cruise

The Russian PTK NP spacecraft could be designed to reach the L points. Copyright © 2012 Anatoly Zak


Lagrange

A cargo supply vehicle based on the Block DM upper stage was proposed by a Russian firm RKK Energia to resupply a manned outpost at the L1 or L2 points in the Earth-Moon system. Copyright © 2012 Anatoly Zak


Lagrange

A US depiction of a station in at a Lagrangian point featuring key Russian components. Observers in Russia found this architecture highly improbable. For example, the Russian government had so far flatly rejected all proposals for modifying the Soyuz spacecraft for deep-space missions. Also, there were no plans to build another version of the service module. Credit: Boeing/NASA


transfer

Transfer of the gateway complex from the low Earth orbit to a Lagrangian point with the help of electric propulsion system. Credit: NASA


Orion

The Orion MPCV spacecraft shown docked to the "gateway" facility in the L2 point of the Earth-Moon system Unless the spacecraft orbited the L2 point in a "halo" orbit, the Earth would be visible from that region of space. Credit: NASA


L2

The Russian Spektr-RG observatory was designed to watch the Universe from the L2 position located "behind" the Earth relative to the Sun as illustrated in this graphic. Credit: Keldysh IPM RAN ballistic center


Gaia

A Russian rocket was scheduled to send Europe's Gaia astronomy satellite 1.5 million kilometers away from Earth. Copyright © 2009 Anatoly Zak


Spektr RG

Spektr-RG would be the first Russian spacecraft to reach a Sun-Earth Lagrangian point. Credit: NPO Lavochkin


 

 

At the end of the first decade of the 21st century, the partners involved into the International Space Station, ISS, project faced a difficult dilemma: where to go next in space with the limited funding available to space agencies on both sides of the Atlantic. Although the ISS was given the green light to operate until at least 2020, decisions about the future would have to be made well in advance, particularly, if space agencies wanted to join forces in an effort to expand human space flight beyond the Earth orbit. In the absence of a bold commitment to go to the Moon, Mars or asteroids, space planners in the US and Russia considered sending missions to the so-called Lagrange points, which could serve as staging hubs for deep-space exploration, if such projects ever became affordable.


In 1772, the French mathematician Joseph-Louis Lagrange established that at certain locations in space gravitational forces and the orbital motion between two celestial bodies would balance each other. (579) A total of five such points were predicted to exist between a planetary body and its moon. Such locations became known as Lagrangian, Libration or L points. The closest to Earth Lagrangian points were those existing at the gravitational equilibrium between the Earth and the Moon.

Dark bodies, alien spaceships and space settlements

With the advent of the Space Age, Lagrangian points first attracted the attention of astronomers, sci-fi writers and futurists. Some astronomers suggested that Lagrangian regions acted as gravitational magnets could trap yet to be discovered planetary bodies or even remnants of ancient alien spaceships. At least two observations during the 1960s and 1970s, including one from the Skylab space station, provided evidence that light-absorbing clouds of dust might have accumulated at the L4 and L5 points in the Earth-Moon system, even though other attempts failed to confirm their existence. Around the same time, scientists involved in the search for extraterrestrial intelligence under the SETI project considered the Lagrangian regions as convenient parking spots for the deployment of antennas trained to listen for alien civilizations. SETI researchers also saw these regions as possible junk yards for hypothetical alien spacecraft, which might have explored the Solar System and been cast away in gravitationally stable orbits eons ago. (585) Finally, a famous space colony proposed by Gerard K. O'Neill during the 1970s, would also be located in the L5 Earth-Moon point, making it convenient to receive its construction materials from the Moon.

Lagrange points in space flight

In the 1961 sci-fi novel A Fall of Moondust by Arthur C. Clarke, a space outpost inhabited by a lone astronaut serves as a relay station for the trapped lunar tourists. Clarke, who is credited with the first proposal of a geostationary satellite, placed his imaginary outpost into a Lagrangian point not just to make it sound more exotic. In October 1967, Robert W. Farquhar, an aeronautical engineer at NASA's Electronics Research Center, published a paper proposing a relay satellite circling the L2 point behind the Moon in an orbit with a radius of 3,500 kilometers. Because this orbit is always visible from Earth as a circle around the Moon's disk, such a satellite could provide round-the-clock communications with the invisible far side of the Moon. (582) The spacecraft would need two weeks to complete each orbit.

Half a decade later, Farquhar pushed his idea further. As Apollo astronauts reached the Moon in 1969, a group of experts appointed by the president brainstormed future tasks for the US space program. By the summer, this so-called Space Task Group came up with an "Integrated Program Plan" for lunar exploration extending into the 1980s and farther into the future. This long-term strategy called for a manned station in a 110-kilometer polar orbit around the Moon. It would serve as a transport hub for operations on the lunar surface. However in June 1972, Farquhar, then at NASA's Goddard Space Flight Center, argued that such a station would be better off orbiting around the L2 point, instead of the low lunar orbit. The L2 hub would still fulfill a duty of the way station to the Moon and, additionally, it could serve as a command and control center for manned expeditions and robotic systems deployed anywhere on the lunar surface. (In order to "talk" to astronauts working on the near side of the Moon, the L2 outpost would use ground stations on Earth.)

Farquhar stressed that the communications capability of the station in the L2 point would be especially important for the astronomical observatory, which at the time was widely expected to be built in the equatorial region on the far side of the Moon. In contrast, the command center in the polar lunar orbit would lose direct contact with the Earth every time it went behind the Moon. (583) Ironically, the problem could be resolved with an unmanned relay satellite placed in the L2 point.

Comparison of L points and low-Earth orbit as a gateway to the Moon

Farquhar also argued that transport operations from the Earth to the Moon via L2-based outpost would be more efficient that those via the low lunar orbit. In both cases, it was assumed that three reusable vehicles would service the transport chain from the Earth to the Moon. In the first leg of the trip, the cargo and passengers would be carried to the low Earth orbit, where they would switch to the reusable tug shuttling between the Earth orbit and the vicinity of the Moon. There, the cargo and passengers would be picked up by a specialized lunar lander.

The most fuel-efficient trajectory from the Earth orbit to the L2 point would swing around the Moon. During such a journey, the spacecraft would make two engine firings -- one in the vicinity of the Moon to prevent too much acceleration and the second -- to ease itself into the L2 point. In total, such mission would spend less propellant than the one shuttling between the Earth orbit and the ideal lunar orbit. (80) However, the flight to L2 would take longer (around 140 hours - almost six days) than to the lunar orbit (three days), which, in case of a manned mission, would require more resources for the life-support system.

For the final leg of the trip, a lunar lander departing either from L1 or from L2 points would also need more propellant and time to reach the surface than the same vehicle heading to the Moon from its low orbit. However, the Lagrange-based lander would always need a nearly same amount of propellant to reach any destination on the surface. From L-points, a lander could depart to the Moon at practically any time and land at any location without limitations. (583) (584) In contrast, the lander based in the lunar orbit, would be limited to 14-day windows or would have to conduct propellant-hungry orbital plane adjustments to reach a desirable landing site on the Moon.

With all their advantages, Libration points presented their own problems in flight dynamics, first of all, more demands for flight accuracy during rendezvous and docking operations than in traditional orbits. Additionally, the L1 point located between the Earth and the Moon and the L2 point located 65,000 kilometers from the Moon are considered unstable, requiring propellant to keep the spacecraft either hovering in one place like a hummingbird or in a "halo orbit" circling around both points. (581) However the low lunar orbit is not stable either and the outpost there would crash into the Moon if left without a re-boost capability. In both cases, low-thrust electric engines (commonly used to keep satellites in geostationary orbit for many years with only tiny use of propellant) could perform station-keeping functions.

The jury is still out

In 2007, a group of European engineers presented the results of a study on the possible roles of Earth-Moon Lagrangian points for manned lunar exploration. The study ruled out the L2 point, as offering no advantages for transport operations and complicating overall missions. The study also favored the lunar orbit over both L1 and L2 points for the location of the transport hub, because it would cut travel time and provide more safety in case of emergency onboard manned vehicles. Lunar missions via L points would need more acceleration, hence propellant, (a total Delta V budget of around 10.5 kilometers per second) compared to missions via lunar orbit demanding only 9.7 kilometers per second, the study concluded.

At the same time, the authors of the study stressed that L1 and L2 would demand less propellant to maintain the outpost's orbit, enable global access to the lunar surface and provide better communications with the Earth. Even more importantly, the L2 point would far outcompete the lunar orbit for the role of a staging area for deep-space missions, such as expeditions to Mars. (584)

Sun-Earth Lagrange points

The Lagrangian points in the Sun-Earth system also became destinations for space flight. Particularly, the L2 position located 1.5 million kilometers behind the Earth relative to the Sun attracted interest thanks to its potential for astronomical observations unimpeded by the blinding sunlight in the shadow of the Earth. In turn, the L1 point would be convenient for spacecraft watching the Sun. On August 12, 1978, NASA launched the ISEE-3 probe, which was directed into the L1 point between the Earth and the Sun, some 1.6 million kilometers from its home planet. By November 21, the spacecraft entered a so-called halo orbit around the L1 point, which would take it as high as 150,000 kilometers above the Earth-Sun line and then the same distance below it. It would take ISEE-3 half a year to complete each orbit. This halo orbit would keep the spacecraft away from the disc of the Sun when viewed from Earth, thus enabling ground antennas to avoid "blinding" radio waves emitted by the Sun. (80) In the following years, astronomy satellites such as WMAP, Plank and Herschel were all based at the L2 point in the Earth-Sun system. (589)

In June 2011, China's Chang'e-2 spacecraft headed to the L2 point after completing its studies of the Moon surface from its orbit. The probe entered a halo orbit around L2 following August.

A Russian rocket will get its first try to fly in the direction of the Sun-Earth Lagrange point in 2013. A Soyuz launcher will send Europe's Gaia astronomy satellite from Kourou, French Guiana, to the L2 point in the Earth-Sun system on a five-year mission to catalog about one billion stars as faint as the 20th magnitude.

Around a year later, the Spektr-RG X-ray observatory is expected to become the first Russian (or Soviet) spacecraft reaching any of the Lagrange points. As of 2012, it was promised to fly as early as 2014, heading to an orbit around the L2 point in the Earth-Sun system. Around 2018, it would be followed by the James Web Space Telescope, JWST -- NASA's flagman space observatory, designed to replace the Hubble Space Telescope. The Russian equivalent of JWST, known as Spektr-M, was also expected to orbit the L2 point.

Lagrange points as springboard into deep space

At the beginning of the 21st century, or some 240 years after their discovery, Lagrangian points in the Earth-Moon system appeared for the first time on the short list of destinations for manned spacecraft. In its official strategy made public in 2011, NASA said that "initially, exploring the vast expanse of space surrounding the Earth and Moon, including the Lagrange points, will establish a human presence outside of LEO (low Earth orbit) as we prepare for more complex missions beyond the Earth’s gravitational influence... Exploring at Lagrange points could provide unique perspectives of the Moon, Sun, and Earth. The Lagrange point on the far side of the Earth-Moon system, called Earth-Moon L2 (EM L2), provides a “radio silence” zone for astronomical observations. Journeys to EM L2 would take humans farther than they have ever been from Earth. (587)

According to preliminary plans published in 2011-2012, the man-tended outpost would serve as a way station to the Moon, then to asteroids and, ultimately, to Mars. The alternative scenario called for the Lagrange-based station to serve as a spring-board to go to asteroids first, followed by lunar exploration, but it would also culminate with an expedition to Mars.

In 2011, the Boeing company, NASA's primary contractor in the ISS project, formulated the following rational for placing the so-called "gateway" into either the L1 or L2 points in the Earth-Moon tandem:

  • Primary destination for initial flights beyond LEO:
    • Provides a habitat destination for the Orion spacecraft and Soyuz for medium-duration stays
    • Enables early characterization of environment outside radiation belts
  • "Local" control of lunar robot assets:
    • Allows the use of tele-presence robots
    • Development of remotely controlled ISRU capabilities critical for Mars exploration
  • Gateway for a mission to a near-Earth asteroid:
    • Enables assembly, test and checkout of asteroid spacecraft prior to departure
    • Enables lowest mass mission spacecraft which will shorten trip times to and from the asteroid
  • Base for re-usable Lunar lander:
    • Allows re-use of expensive lunar lander assets
    • Enables much more flexible mission operations for lunar access and “anytime return”
  • Gateway for a human mission to Mars:
    • Enables assembly, test and checkout of Mars spacecraft prior to departure
    • Enables lowest mass mission spacecraft which will shorten trip times to and from NEA
    • Safe orbit for nuclear tug assets; (586)

Obviously, such justifications could only be sound if lunar robots, asteroid probes, reusable lunar landers, nuclear tugs and human missions to Mars really existed or were developed in parallel and came online by the time the "gateway" was ready for launch. Unfortunately, neither the NASA budget could afford any of these components, nor was an international agreement on the supply of these assets by other partners was available at the time. As a result, the "gateway" risked to appear on the scene without anything to go through it.

Boeing also proposed a service station in the L2 point behind the Earth relative to the Sun, where it could repair and refuel "telescope assets". However yet again, neither NASA's James Web Space Telescope, nor the Russian Spektr-RG X-ray observatory, which were both in development at the time, had design features enabling future astronauts to service or refuel them.

The authors of the Boeing study pictured the "gateway" relying heavily on Russian hardware, including the Zvezda service module from the ISS project and the Soyuz spacecraft. However critics pointed out that none of these vehicles were anywhere near the capability to function beyond low Earth orbit and, in case of Soyuz, to return to Earth from deep space. Despite numerous speculations in the West and very preliminary studies of the "lunar" Soyuz conducted during 2000s, the Russian space agency, Roskosmos, consistently rejected all such proposals in favor of a new-generation manned spacecraft, PTK-NP.

Immediate goals

Without a large-scale international program and a long-term commitment toward a particular destination in space, NASA attempted to justify an outpost in the Lagrange points as an experience-building exercise.

"A facility at an Earth-Moon Lagrange point provides a stepping stone for certifying technologies and staging future deep-space exploration missions while building experience and confidence in radiation countermeasures, high-reliability ECLSS (life-support systems), telerobotics, and more. Exploring cis-lunar space (Earth-Moon region) will give NASA the opportunity to develop tools and operational techniques to support decades of future exploration, while remaining in relative proximity to Earth."

"A long-duration habitation capability could support a Lagrange gateway at EM L1 or L2. This facility would provide a destination for exploration systems, like the (Orion) MPCV spacecraft; a staging point for future missions to many destinations; a crewed, non-terrestrial laboratory for NEA or Mars sample return; and a hub for accessing or servicing systems throughout cis-lunar space."

"The ISS is an invaluable resource for researching and testing exploration capabilities in space, and it may inspire future space station concepts. As NASA looks to explore beyond LEO, the agency is considering how a facility in cis-lunar space, potentially stationed at an Earth-Moon Lagrange point, could support research, testing, and astronomical observation, as well as provide a staging point for exploration missions. Such a facility, also known as a Lagrange gateway, would build upon ISS hardware and experience, and would serve as an initial in-space habitat, providing a basis for future long-duration habitation developments," a NASA-endorsed document said. (587)

Russian plans

In Russia, engineers also conducted some preliminary studies on the possibility of conducting missions to the Lagrange points. The nation's key contractor for manned space flight, RKK Energia, cited such advantages of a base in the Lagrange region between the Earth and the Moon, as the absence of harmful influence from the Earth's radiation belts and the modest demand for propulsion capability to escape the Earth's orbit. Echoing similar NASA studies, RKK Energia argued that a base in a Lagrange point would enable to test basic technologies for deep-space missions and provide all-time, global access to the Moon and back. The company stressed that the project would require to develop new techniques for flight control, dynamics and navigation in the vicinity of Lagrange points. (580)

With unmatched experience in delivering crews and supplies to the Earth-orbiting space stations, RKK Energia was well equipped to develop both manned and unmanned transport systems for a "gateway" positioned in the Lagrange point. The best candidate for carrying crews to Lagrangian points would be the next-generation spacecraft, PTK NP. Although by 2012, the main goal of the project was to develop a vehicle for transporting crews to the lunar orbit, the same ship would have the capability to reach the L1 or L2 points in the Earth-Moon system with minimal modifications.

Russian engineers also evaluated various options for delivering cargo and propellant to the Lagrangian points, which would be absolutely critical for the operation of a transport hub at such locations. In 2008, RKK Energia studied the possibility of launching the Progress cargo ships to the lunar orbit apparently onboard a US-built booster. Again, the similar transport system could be used to reach Lagrangian points.

In 2011, RKK Energia also published the preliminary evaluation of a cargo vehicle, which would rely on the off-the-shelf combination of the Proton rocket and the Block-D upper stage to carry cargo to the Lagrangian points. (588)

Future uncertain (Editor's note)

Under any circumstances, missions to Lagrangian regions could only be justified as a stepping stone within a wider strategy to reach long-term destinations in space, such as the Moon, the asteroids or Mars. Without a simultaneous effort to reach these goals, stand-alone missions into "empty space" would be very hard to sell to taxpayers, likely exposing the manned space program to the same kind of criticism which has been leveled against the International Space Station over the past two decades. Whether the human outpost is built in the low Earth orbit or set up further in space, it could hardly provide a rational for the continuous manned space flight by merely being a destination for the transport spacecraft. However, if it was developed as a component of a larger program, such as a permanent lunar base or an expedition to Mars, it could become a critical springboard into the Solar System.

Given the financial realities of the day, it is unlikely that any of the space-faring nations would be able to afford such an ambitious manned space program. As a result, a broad international agreement, which would share responsibilities among its signatories, including but not limited to the US, Russia and Europe, would be a much more important achievement in the short term than a decision to build another gateway in space with only a vague understanding of where this gateway ultimately leads.

(To be continued)

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APPENDIX

Comparison of various aspects of the mission in the the low lunar orbit versus the Lagrangian points:

Station in the low lunar orbit, LLO Station in the low lunar orbit, LLO
Station in the L2 point
Flight from the low-Earth orbit relative to the Moon position Constrained
Constrained
Flight from the low-Earth orbit relative to nodal regression of the Earth parking orbit Constrained
Constrained
Flight from the low-Earth orbit constrained by the orientation of the lunar orbit relative to the Earth-Moon line Constrained
Not constrained
Flight time from Earth orbit Shorter
Longer
Remote sensing of the Moon More effective (?)
Less effective (?)
Orbit stability, station keeping More costly
Less costly
Communications with the lunar surface Relay system needed
Direct view of the lunar far side
Direct contact with Earth Very limited
Unlimited*

*When placed into a halo orbit around the L2 point

 

Lagrangian (Libration) points, their properties and possible roles in space exploration:

Earth-Moon
Location
Properties
Application, role
L1 (EML)
58,000 kilometers from the Moon, 325,000 kilometers from the Earth
Unstable, requires propellant for station keeping
Transport hub to the Moon
L2
64,514 kilometers behind the Moon
Unstable, requires propellant for station keeping
Global communications support on the Moon
L3
Behind the Earth on the Earth-Moon line
Unstable, requires propellant for station keeping
-
L4
circles 60 degrees ahead of the Moon in lunar orbit
Stable, no orbit correction needed
-
L5
circles 60 degrees trailing the Moon in lunar orbit
Stable, no orbit correction needed
-
Earth-Sun
-
-
-
L1 (ESL)
Earth-Sun line, 1.49 million kilometers from Earth toward the Sun
-
Solar astronomy, Earth observation
L2
Earth-Sun line, 1.50 million kilometers behind Earth
-
Astronomy

 

Propulsion budgets (Delta V) for reaching Lagrangian points and comparable destinations:

Origin and destination
Flight time
Required Delta V
Flight from L1 to Mars or asteroids
-
up to 1,000 meters per second
Flight from L1 to lunar orbit
-
up to 800 meters per second
Flight from L1 to L2
-
10 meters per second
Flight from L1 to the lunar surface
-
2,600 meters per second
Flight from L2 to the lunar surface
-
2,600 meters per second
Flight from L2 to the lunar surface
-
2,600 meters per second
Flight from Earth-Moon L1 to the Earth-Sun Lagrange point
-
400 meters per second
Manned flights between Earth orbit and L1 point
-
3,800 meters per second
Cargo flights between Earth orbit and L1 point with the use of electric engines
-
7,000 meters per second
Flight between the Earth orbit and lunar orbit
-
4,400 meters per second
Flight between the lunar orbit and lunar surface
-
1,900 meters per second
Flight from Earth orbit to L2 via the Moon swing by (583)
9 days
3,470 meters per second
Flight from Earth orbit to L2 (direct) (583)
4 days
4,250 meters per second

Source: RKK Energia, August 2012

 

Propulsion budget for the Russian PTK NP spacecraft on missions to the Moon via low lunar orbit or the L1 point (estimates circa 2011):

Maneuver
Lunar orbit
L1 point
Escape from the low Earth orbit toward the Moon
3,220 meters per second
-
Escape from the low Earth orbit toward the L1 (Lagrangian) point (Earth-Moon)
-
3,125 meters per second
Entering 100-kilometer circular lunar orbit (any inclination)
1,300 meters per second
-
Entering the L1 point (including plane change)
-
820 meters per second
Landing at any location on the lunar surface from the 100-kilometer circular lunar orbit
2,100 meters per second
-
Landing at any location on the lunar surface from the L1 point
-
2,885 meters per second
Launch from the Moon to a 100-kilometer circular lunar orbit
1,760 meter per second
-
Launch from the Moon to the L1
-
2,560 meter per second
Two docking operations (arrival to the Moon, departure from the Moon)
-
100 meters per second
Escape from the lunar orbit toward Earth with a return trip not exceeding five days
1,305 meters per second
-
Escape from the L1 toward Earth with a a reentry at a 5-degree angle at an altitude of 100 kilometers
-
820 meters per second
Return trajectory correction
-
75 meters per second
Total
9,685 meters per second
10,385 meters per second

Page author: Anatoly Zak; Last update: July 12, 2014

Page editor: Alain Chabot; Last edit: October 12, 2012

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