Lunar space elevator

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https://en.wikipedia.org/wiki/Lunar_space_elevator 

 

Diagram showing equatorial and polar Lunar space elevators running past L₁. An L₂ elevator would mirror this arrangement on the Lunar far side, and cargo dropped from its end would be flung outward into the solar system.

A lunar space elevator or lunar spacelift is a proposed transportation system for moving a mechanical climbing vehicle up and down a ribbon-shaped tethered cable that is set between the surface of the Moon "at the bottom" and a docking port suspended tens of thousands of kilometers above in space at the top.

It is similar in concept to the better known Earth-based space elevator idea, but since the Moon's surface gravity is much lower than the Earth's, the engineering requirements for constructing a lunar elevator system can be met using materials and technology already available. For a lunar elevator, the cable or tether extends considerably farther out from the lunar surface into space than one that would be used in an Earth-based system. However, the main function of a space elevator system is the same in either case; both allow for a reusable, controlled means of transporting payloads of cargo, or possibly people, between a base station at the bottom of a gravity well and a docking port in outer space.

A lunar elevator could significantly reduce the costs and improve reliability of soft-landing equipment on the lunar surface. For example, it would permit the use of mass-efficient (high specific impulse), low thrust drives such as ion drives which otherwise cannot land on the Moon. Since the docking port would be connected to the cable in a microgravity environment, these and other drives can reach the cable from low Earth orbit (LEO) with minimal launched fuel from Earth. With conventional rockets, the fuel needed to reach the lunar surface from LEO is many times the landed mass, thus the elevator can reduce launch costs for payloads bound for the lunar surface by a similar factor.

Location

There are two points in space where an elevator's docking port could maintain a stable, lunar-synchronous position: the Earth-Moon Lagrange points L₁ and L₂. The 0.055 eccentricity of the lunar orbit means that these points are not fixed relative to the lunar surface : the L₁ is 56,315 km +/- 3,183 km away from the Earth-facing side of the Moon (at the lunar equator) and L₂ is 62,851 km +/- 3,539 km from the center of the Moon's far side, in the opposite direction. At these points, the effect of the Moon's gravity and the effect of the centrifugal force resulting from the elevator system's synchronous, rigid body rotation cancel each other out. The Lagrangian points L₁ and L₂ are points of unstable gravitational equilibrium, meaning that small inertial adjustments will be needed to ensure any object positioned there can remain stationary relative to the lunar surface.

Both of these positions are substantially farther up than the 36,000 km from Earth to geostationary orbit. Furthermore, the weight of the limb of the cable system extending down to the Moon would have to be balanced by the cable extending further up, and the Moon's slow rotation means the upper limb would have to be much longer than for an Earth-based system, or be topped by a much more massive counterweight. To suspend a kilogram of cable or payload just above the surface of the Moon would require 1,000 kg of counterweight, 26,000 km beyond L₁. (A smaller counterweight on a longer cable, e.g., 100 kg at a distance of 230,000 km — more than halfway to Earth — would have the same balancing effect.) Without the Earth's gravity to attract it, an L₂ cable's lowest kilogram would require 1,000 kg of counterweight at a distance of 120,000 km from the Moon. The average Earth-Moon distance is 384,400 km.

The anchor point of a space elevator is normally considered to be at the equator. However, there are several possible cases to be made for locating a lunar base at one of the Moon's poles; a base on a peak of eternal light could take advantage of near-continuous solar power, for example, or small quantities of water and other volatiles may be trapped in permanently shaded crater bottoms. A space elevator could be anchored near a lunar pole, though not directly at it. A tramway could be used to bring the cable the rest of the way to the pole, with the Moon's low gravity allowing much taller support towers and wider spans between them than would be possible on Earth.

Fabrication

Because of the Moon's lower gravity and lack of atmosphere, a lunar elevator would have less stringent requirements for the tensile strength of the material making up its cable than an Earth-tethered cable. An Earth-based elevator would require high strength-to-weight materials that are theoretically possible, but not yet fabricated in practice (e.g., carbon nanotubes). A lunar elevator, however, could be constructed using commercially available mass-produced high-strength para-aramid fibres (such as Kevlar and M5) or ultra-high-molecular-weight polyethylene fibre.

Compared to an Earth space elevator, there would be fewer geographic and political restrictions on the location of the surface connection. The connection point of a lunar elevator would not necessarily have to be directly under its center of gravity, and could even be near the poles, where evidence suggests there might be frozen water in deep craters that never see sunlight; if so, this might be collected and converted into rocket fuel.

Cross-section profile

Space elevator designs for Earth typically have a taper of the tether that provides a uniform stress profile rather than a uniform cross-section. Because the strength requirement of a lunar space elevator is much lower than that of an Earth space elevator, a uniform cross-section is possible for the lunar space elevator. The study done for NASA's Institute of Advanced Concepts states "Current composites have characteristic heights of a few hundred kilometers, which would require taper ratios of about 6 for Mars, 4 for the Moon, and about 6000 for the Earth. The mass of the Moon is small enough that a uniform cross-section lunar space elevator could be constructed, without any taper at all." A uniform cross-section could make it possible for a lunar space elevator to be built in a double-tether pulley configuration. This configuration would greatly simplify repairs of a space elevator compared to a tapered elevator configuration. However a pulley configuration would require a strut at the counterweight hundreds of kilometers long to separate the up-tether from the down-tether and keep them from tangling. A pulley configuration might also allow the system capacity to be gradually expanded by stitching new tether material on at the Lagrange point as the tether rotated.

History

The idea of space elevators has been around since 1960 when Yuri Artsutanov wrote a Sunday supplement to Pravda on how to build such a structure and the utility of geosynchronous orbit. His article however, was not known in the West. Then in 1966, John Isaacs, a leader of a group of American Oceanographers at Scripps Institute, published an article in Science about the concept of using thin wires hanging from a geostationary satellite. In that concept, the wires were to be thin (thin wires/tethers are now understood to be more susceptible to micrometeoroid damage). Like Artsutanov, Isaacs’ article also was not well known to the aerospace community.

In 1972, James Cline submitted a paper to NASA describing a "mooncable" concept similar to a lunar elevator. NASA responded negatively to the idea citing technical risk and lack of funds.

In 1975, Jerome Pearson independently came up with the Space elevator concept and published it in Acta Astronautica. That made the aerospace community at large aware of the space elevator for the first time. His article inspired Sir Arthur Clarke to write the novel The Fountains of Paradise (published in 1979, almost simultaneously with Charles Sheffield's novel on the same topic, The Web Between the Worlds). In 1978 Pearson extended his theory to the moon and changed to using the Lagrangian points instead of having it in geostationary orbit.

In 1977, some papers of Soviet space pioneer Friedrich Zander were posthumously published, revealing that he conceived of a lunar space tower in 1910.

In 2005 Jerome Pearson completed a study for NASA Institute of Advanced Concepts which showed the concept is technically feasible within the prevailing state of the art using existing commercially available materials.

In October 2011 on the LiftPort website Michael Laine announced that LiftPort is pursuing a Lunar space elevator as an interim goal before attempting a terrestrial elevator. At the 2011 Annual Meeting of the Lunar Exploration Analysis Group (LEAG), LiftPort CTO Marshall Eubanks presented a paper on the prototype Lunar Elevator co-authored by Laine. In August 2012, Liftport announced that the project may actually start near 2020. In April 2019, LiftPort CEO Michael Laine reported no progress beyond the lunar elevator company's conceptualized design.

Materials

Unlike earth-anchored space elevators, the materials for lunar space elevators will not require a lot of strength. Lunar elevators can be made with materials available today. Carbon nanotubes aren’t required to build the structure. This would make it possible to build the elevator much sooner, since available carbon nanotube materials in sufficient quantities are still years away.

One material that has great potential is M5 fiber. This is a synthetic fiber that is lighter than Kevlar or Spectra. According to Pearson, Levin, Oldson, and Wykes in their article The Lunar Space Elevator, an M5 ribbon 30 mm wide and 0.023 mm thick, would be able to support 2000 kg on the lunar surface (2005). It would also be able to hold 100 cargo vehicles, each with a mass of 580 kg, evenly spaced along the length of the elevator. Other materials that could be used are T1000G carbon fiber, Spectra 200, Dyneema (used on the YES2 spacecraft), or Zylon. All of these materials have breaking lengths of several hundred kilometers under 1g.

Potential lunar elevator materials

Material	Density ρ kg/m3	Stress Limit σ GPa	Breaking height (h = σ/ρg, km)
Single-wall carbon nanotubes (laboratory measurements)	2266	50	2200
Toray Carbon fiber (T1000G)	1810	6.4	361
Aramid, Ltd. polybenzoxazole fiber (Zylon PBO)	1560	5.8	379
Honeywell extended chain polyethylene fiber (Spectra 2000)	970	3.0	316
Magellan honeycomb polymer M5 (with planned values)	1700	5.7(9.5)	342(570)
DuPont Aramid fiber (Kevlar 49)	1440	3.6	255
Glass fibre (Ref Specific strength)	2600	3.4	133

The materials will be used to manufacture the ribbon-shaped, tethered cable which will connect from the L₁ or L₂ balance points to the surface of the moon. The climbing vehicles which will travel the length of these cables in a finished elevator system will not move very fast, thus simplifying some of the challenges of transferring cargo and maintaining structural integrity of the system. However, any small objects suspended in space for extended periods of time, like the tethered cables would be, are vulnerable to damage by micrometeoroids, so one possible method of improving their survivability would be to design a "multi-ribbon" system instead of just a single-tethered cable. Such a system would have interconnections at regular intervals, so that if one section of ribbon is damaged, parallel sections could carry the load until robotic vehicles could arrive to replace the severed ribbon. The interconnections would be spaced about 100 km apart, which is small enough to allow a robotic climber to carry the mass of the replacement 100 km of ribbon.

Climbing vehicles

One method of getting materials needed from the moon into orbit would be the use of robotic climbing vehicles. These vehicles would consist of two large wheels pressing against the ribbons of the elevator to provide enough friction for lift. The climbers could be set for horizontal or vertical ribbons.

The wheels would be driven by electric motors, which would obtain their power from solar energy or beamed energy. The power required to climb the ribbon would depend upon the lunar gravity field, which drops off the first few percent of the distance to L₁. The power that a climber would require to traverse the ribbon drops in proportion to proximity to the L₁ point. If a 540 kg climber traveled at a velocity of fifteen meters per second, by the time it was seven percent of the way to the L₁ point, the required power would drop to less than a hundred watts, versus 10 kilowatts at the surface.

One problem with using a solar powered vehicle is the lack of sunlight during some parts of the trip. For half of every month, the solar arrays on the lower part of the ribbon would be in the shade. One way to fix this problem would be to launch the vehicle at the base with a certain velocity then at the peak of the trajectory, attach it to the ribbon.

Possible uses

Materials from Earth may be sent into orbit and then down to the Moon to be used by lunar bases and installations.

Former U.S. President George W. Bush, in an address about his Vision for Space Exploration, suggested that the Moon may serve as a cost-effective construction, launching and fueling site for future space exploration missions. As President Bush noted, "(Lunar) soil contains raw materials that might be harvested and processed into rocket fuel or breathable air." For example, the proposed Ares V heavy-lift rocket system could cost-effectively deliver raw materials from Earth to a docking station, (connected to the lunar elevator as a counterweight,) where future spacecraft could be built and launched, while extracted lunar resources could be shipped up from a base on the Moon's surface, near the elevator's anchoring point. If the elevator was connected somehow to a lunar base built near the Moon's north pole, then workers could also mine the water ice which is known to exist there, providing an ample source of readily accessible water for the crew at the elevator's docking station. Also, since the total energy needed for transit between the Moon and Mars is considerably less than for between Earth and Mars, this concept could lower some of the engineering obstacles to sending humans to Mars.

The lunar elevator could also be used to transport supplies and materials from the surface of the moon into the Earth's orbit and vice versa. According to Jerome Pearson, many of the Moon's material resources can be extracted and sent into Earth orbit more easily than if they were launched from the Earth's surface. For example, lunar regolith itself could be used as massive material to shield space stations or crewed spacecraft on long missions from solar flares, Van Allen radiation, and other kinds of cosmic radiation. The Moon's naturally occurring metals and minerals could be mined and used for construction. Lunar deposits of silicon, which could be used to build solar panels for massive satellite solar power stations, seem particularly promising.

One disadvantage of the lunar elevator is that the speed of the climbing vehicles may be too slow to efficiently serve as a human transportation system. In contrast to an Earth-based elevator, the longer distance from the docking station to the lunar surface would mean that any "elevator car" would need to be able to sustain a crew for several days, even weeks, before it reached its destination.

Lagrange point

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Lagrange_point 

Small-mass objects (green) at the Lagrange points are in equilibrium with one massive body (blue) orbiting another (yellow). At all other points, the gravitational forces due to the two massive bodies are non-equilibria.

 

Lagrange points in the Sun–Earth system (not to scale). A small object at L4 or L5 will hold its relative position. A small object at L1, L2, or L3 will hold its relative position until deflected slightly radially, after which it will diverge from its original position.

 

An example of a spacecraft at Sun–Earth L2
  WMAP ·   Earth

In celestial mechanics, the Lagrange points /ləˈɡrɑːndʒ/ (also Lagrangian points, L-points, or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem in which two bodies are very much more massive than the third.

Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other. This can make Lagrange points an excellent location for satellites, as few orbit corrections are needed to maintain the desired orbit. Small objects placed in orbit at Lagrange points are in equilibrium in at least two directions relative to the center of mass of the large bodies.

There are five such points, labelled L₁ to L₅, all in the orbital plane of the two large bodies, for each given combination of two orbital bodies. For instance, there are five Lagrangian points L₁ to L₅ for the Sun–Earth system, and in a similar way there are five different Lagrangian points for the Earth–Moon system. L₁, L₂, and L₃ are on the line through the centres of the two large bodies, while L₄ and L₅ each act as the third vertex of an equilateral triangle formed with the centres of the two large bodies. L₄ and L₅ are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.

The L₄ and L₅ points are stable points and have a tendency to pull objects into them. Several planets have trojan asteroids near their L₄ and L₅ points with respect to the Sun. Jupiter has more than a million of these trojans. Artificial satellites have been placed at L₁ and L₂ with respect to the Sun and Earth, and with respect to the Earth and the Moon. The Lagrangian points have been proposed for uses in space exploration.

History

The three collinear Lagrange points (L₁, L₂, L₃) were discovered by Leonhard Euler a few years before Joseph-Louis Lagrange discovered the remaining two.

In 1772, Lagrange published an "Essay on the three-body problem". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special constant-pattern solutions, the collinear and the equilateral, for any three masses, with circular orbits.

Lagrange points

The five Lagrange points are labelled and defined as follows:

L₁ point

The L₁ point lies on the line defined between the two large masses M₁ and M₂. It is the point where the gravitational attraction of M₂ and that of M₁ combine to produce an equilibrium. An object that orbits the Sun more closely than Earth would normally have a shorter orbital period than Earth, but that ignores the effect of Earth's own gravitational pull. If the object is directly between Earth and the Sun, then Earth's gravity counteracts some of the Sun's pull on the object, and therefore increases the orbital period of the object. The closer to Earth the object is, the greater this effect is. At the L₁ point, the orbital period of the object becomes exactly equal to Earth's orbital period. L₁ is about 1.5 million kilometers from Earth, or 0.01 au, 1/100th the distance to the Sun.

L₂ point

The L₂ point lies on the line through the two large masses, beyond the smaller of the two. Here, the gravitational forces of the two large masses balance the centrifugal effect on a body at L₂. On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than that of Earth. The extra pull of Earth's gravity decreases the orbital period of the object, and at the L₂ point that orbital period becomes equal to Earth's. Like L₁, L₂ is about 1.5 million kilometers or 0.01 au from Earth.

A notable example of an artificial satellite at L2 is the James Webb Space Telescope, designed to operate near the Earth–Sun L₂. See other spacecraft at Sun–Earth L₂.

L₃ point

The L₃ point lies on the line defined by the two large masses, beyond the larger of the two. Within the Sun–Earth system, the L₃ point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly closer to the center of the Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies' barycentre, which is well inside the body of the Sun. An object at Earth's distance from the Sun would have an orbital period of one year if only the Sun's gravity is considered. But an object on the opposite side of the Sun from Earth and directly in line with both "feels" Earth's gravity adding slightly to the Sun's and therefore must orbit a little farther from the barycentre of Earth and Sun in order to have the same 1-year period. It is at the L₃ point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycentre at one focus of its orbit.

L₄ and L₅ points

Gravitational accelerations at L₄

The L₄ and L₅ points lie at the third corners of the two equilateral triangles in the plane of orbit whose common base is the line between the centres of the two masses, such that the point lies behind (L₅) or ahead (L₄) of the smaller mass with regard to its orbit around the larger mass.

Stability

The triangular points (L₄ and L₅) are stable equilibria, provided that the ratio of M₁/M₂ is greater than 24.96. This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. When a body at these points is perturbed, it moves away from the point, but the factor opposite of that which is increased or decreased by the perturbation (either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable, kidney bean-shaped orbit around the point (as seen in the corotating frame of reference).

The points L₁, L₂, and L₃ are positions of unstable equilibrium. Any object orbiting at L₁, L₂, or L₃ will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ station keeping in order to maintain their position.

Put Simply:

A Lagrange (Lagrangian) Point is one of five infinitely small points where, in relation to two mutually orbiting bodies, the forces due to the gravity of each body in opposition to each other and to the centrifugal force, mutually exactly oppose each other.

Lagrange Point 1 (L₁) is found along a line drawn between the centres of gravity of the two objects, at a distance from one and the other which is proportional to their masses.

L₂ is a point along the same line extended beyond the smaller body to a distance where the same counterbalance of forces holds true.

L₃ is a corresponding point along that line extended beyond the larger body.

L₄ and L₅ are found at the opposite vertices of the equilateral triangles whose common base is the line drawn between the centers of gravity the of the larger and smaller masses.

Infinitely small items placed at these points will, in the absence of random instabilities, stay there. If displaced, they will, without any other force, move away at ever increasing speeds. However, at L₄ and L₅, the Coriolis force will act upon an item to tend to return it to that point.

Natural objects at Lagrange points

Main article: List of objects at Lagrangian points

Due to the natural stability of L₄ and L₅, it is common for natural objects to be found orbiting in those Lagrange points of planetary systems. Objects that inhabit those points are generically referred to as 'trojans' or 'trojan asteroids'. The name derives from the names that were given to asteroids discovered orbiting at the Sun–Jupiter L₄ and L₅ points, which were taken from mythological characters appearing in Homer's Iliad, an epic poem set during the Trojan War. Asteroids at the L₄ point, ahead of Jupiter, are named after Greek characters in the Iliad and referred to as the "Greek camp". Those at the L₅ point are named after Trojan characters and referred to as the "Trojan camp". Both camps are considered to be types of trojan bodies.

As the Sun and Jupiter are the two most massive objects in the Solar System, there are more Sun-Jupiter trojans than for any other pair of bodies. However, smaller numbers of objects are known at the Lagrange points of other orbital systems:

• The Sun–Earth L₄ and L₅ points contain interplanetary dust and at least two asteroids, 2010 TK7 and 2020 XL5.
• The Earth–Moon L₄ and L₅ points contain concentrations of interplanetary dust, known as Kordylewski clouds. Stability at these specific points is greatly complicated by solar gravitational influence.
• The Sun–Neptune L₄ and L₅ points contain several dozen known objects, the Neptune trojans.
• Mars has four accepted Mars trojans: 5261 Eureka, 1999 UJ7, 1998 VF31, and 2007 NS2.
• Saturn's moon Tethys has two smaller moons of Saturn in its L₄ and L₅ points, Telesto and Calypso. Another Saturn moon, Dione also has two Lagrangian co-orbitals, Helene at its L₄ point and Polydeuces at L₅. The moons wander azimuthally about the Lagrangian points, with Polydeuces describing the largest deviations, moving up to 32° away from the Saturn–Dione L₅ point.
• One version of the giant impact hypothesis postulates that an object named Theia formed at the Sun–Earth L₄ or L₅ point and crashed into Earth after its orbit destabilized, forming the Moon.
• In binary stars, the Roche lobe has its apex located at L₁; if one of the stars expands past its Roche lobe, then it will lose matter to its companion star, known as Roche lobe overflow.

Objects which are on horseshoe orbits are sometimes erroneously described as trojans, but do not occupy Lagrange points. Known objects on horseshoe orbits include 3753 Cruithne with Earth, and Saturn's moons Epimetheus and Janus.

Physical and mathematical details

A contour plot of the effective potential due to gravity and the centrifugal force of a two-body system in a rotating frame of reference. The arrows indicate the gradients of the potential around the five Lagrange points—downhill toward them (red) or away from them (blue). Counterintuitively, the L₄ and L₅ points are the high points of the potential. At the points themselves these forces are balanced.

 

Visualisation of the relationship between the Lagrangian points (red) of a planet (blue) orbiting a star (yellow) counterclockwise, and the effective potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).
Click for animation.

Lagrangian points are the constant-pattern solutions of the restricted three-body problem. For example, given two massive bodies in orbits around their common barycenter, there are five positions in space where a third body, of comparatively negligible mass, could be placed so as to maintain its position relative to the two massive bodies. As seen in a rotating reference frame that matches the angular velocity of the two co-orbiting bodies, the gravitational fields of two massive bodies combined providing the centripetal force at the Lagrangian points, allowing the smaller third body to be relatively stationary with respect to the first two.

L₁

The location of L₁ is the solution to the following equation, gravitation providing the centripetal force:

$${\displaystyle {\frac {M_{1}}{(R-r)^{2}}}-{\frac {M_{2}}{r^{2}}}=\left({\frac {M_{1}}{M_{1}+M_{2}}}R-r\right){\frac {M_{1}+M_{2}}{R^{3}}}}$$

where r is the distance of the L₁ point from the smaller object, R is the distance between the two main objects, and M₁ and M₂ are the masses of the large and small object, respectively. (The quantity in parentheses on the right is the distance of L₁ from the center of mass.) Solving this for r involves solving a quintic function, but if the mass of the smaller object (M₂) is much smaller than the mass of the larger object (M₁) then L₁ and L₂ are at approximately equal distances r from the smaller object, equal to the radius of the Hill sphere, given by:

$${\displaystyle r\approx R{\sqrt[{3}]{\frac {M_{2}}{3M_{1}}}}}$$

We may also write this as:

$${\displaystyle {\frac {M_{2}}{r^{3}}}\approx 3{\frac {M_{1}}{R^{3}}}}$$

Since the tidal effect of a body is proportional to its mass divided by the distance cubed, this means that the tidal effect of the smaller body at the L₁ or at the L₂ point is about three times of that body. We may also write:

$${\displaystyle \rho _{2}(d_{2}/r)^{3}\approx 3\rho _{1}(d_{1}/R)^{3}}$$

where ρ₁ and ρ₂ are the average densities of the two bodies and ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$ are their diameters. The ratio of diameter to distance gives the angle subtended by the body, showing that viewed from these two Lagrange points, the apparent sizes of the two bodies will be similar, especially if the density of the smaller one is about thrice that of the larger, as in the case of the earth and the sun.

This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M₂ in the absence of M₁, is that of M₂ around M₁, divided by √3 ≈ 1.73:

$${\displaystyle T_{s,M_{2}}(r)={\frac {T_{M_{2},M_{1}}(R)}{\sqrt {3}}}.}$$

L₂

The Lagrangian L₂ point for the Sun–Earth system.

The location of L₂ is the solution to the following equation, gravitation providing the centripetal force:

$${\displaystyle {\frac {M_{1}}{(R+r)^{2}}}+{\frac {M_{2}}{r^{2}}}=\left({\frac {M_{1}}{M_{1}+M_{2}}}R+r\right){\frac {M_{1}+M_{2}}{R^{3}}}}$$

with parameters defined as for the L₁ case. Again, if the mass of the smaller object (M₂) is much smaller than the mass of the larger object (M₁) then L₂ is at approximately the radius of the Hill sphere, given by:

$${\displaystyle r\approx R{\sqrt[{3}]{\frac {M_{2}}{3M_{1}}}}}$$

The same remarks about tidal influence and apparent size apply as for the L₁ point. For example, the angular radius of the sun as viewed from L₂ is arcsin(695.5×10³/151.1×10⁶) ≈ 0.264°, whereas that of the earth is arcsin(6371/1.5×10⁶) ≈ 0.242°. Looking toward the sun from L₂ one sees an annular eclipse. It is necessary for a spacecraft, like Gaia, to follow a Lissajous orbit or a halo orbit around L₂ in order for its solar panels to get full sun.

L₃

The location of L₃ is the solution to the following equation, gravitation providing the centripetal force:

$${\displaystyle {\frac {M_{1}}{\left(R-r\right)^{2}}}+{\frac {M_{2}}{\left(2R-r\right)^{2}}}=\left({\frac {M_{2}}{M_{1}+M_{2}}}R+R-r\right){\frac {M_{1}+M_{2}}{R^{3}}}}$$

with parameters M₁, M₂ and R defined as for the L₁ and L₂ cases, and r now indicates the distance of L₃ from the position of the smaller object, if it were rotated 180 degrees about the larger object, while positive r implying L₃ is closer to the larger object than the smaller object. If the mass of the smaller object (M₂) is much smaller than the mass of the larger object (M₁) then:

$${\displaystyle r\approx R{\frac {7M_{2}}{12M_{1}}}}$$

L₄ and L₅

Further information: Trojan (celestial body)

The reason these points are in balance is that, at L₄ and L₅, the distances to the two masses are equal. Accordingly, the gravitational forces from the two massive bodies are in the same ratio as the masses of the two bodies, and so the resultant force acts through the barycenter of the system; additionally, the geometry of the triangle ensures that the resultant acceleration is to the distance from the barycenter in the same ratio as for the two massive bodies. The barycenter being both the center of mass and center of rotation of the three-body system, this resultant force is exactly that required to keep the smaller body at the Lagrange point in orbital equilibrium with the other two larger bodies of the system (indeed, the third body needs to have negligible mass). The general triangular configuration was discovered by Lagrange working on the three-body problem.

Net radial acceleration of a point orbiting along the Earth–Moon line

Radial acceleration

The radial acceleration a of an object in orbit at a point along the line passing through both bodies is given by:

$${\displaystyle a=-{\frac {GM_{1}}{r^{2}}}\operatorname {sgn}(r)+{\frac {GM_{2}}{(R-r)^{2}}}\operatorname {sgn}(R-r)+{\frac {G{\bigl (}(M_{1}+M_{2})r-M_{2}R{\bigr )}}{R^{3}}}}$$

where r is the distance from the large body M₁, R is the distance between the two main objects and sgn(x) is the sign function of x. The terms in this function represent respectively: force from M₁; force from M₂; and centripetal force. The points L₃, L₁, L₂ occur where the acceleration is zero — see chart at right. Positive acceleration is acceleration towards the right of the chart and negative acceleration is towards the left; that is why acceleration has opposite signs on opposite sides of the gravity wells.

Stability

Although the L₁, L₂, and L₃ points are nominally unstable, there are quasi-stable periodic orbits called halo orbits around these points in a three-body system. A full n-body dynamical system such as the Solar System does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following Lissajous-curve trajectories. These quasi-periodic Lissajous orbits are what most of Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of station keeping keeps a spacecraft in a desired Lissajous orbit for a long time.

For Sun–Earth-L₁ missions, it is preferable for the spacecraft to be in a large-amplitude (100,000–200,000 km or 62,000–124,000 mi) Lissajous orbit around L₁ than to stay at L₁, because the line between Sun and Earth has increased solar interference on Earth–spacecraft communications. Similarly, a large-amplitude Lissajous orbit around L₂ keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.

The L₄ and L₅ points are stable provided that the mass of the primary body (e.g. the Earth) is at least 25 times the mass of the secondary body (e.g. the Moon), and the mass of the secondary is at least 10 times that of the tertiary (e.g. the satellite). The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth). Although the L₄ and L₅ points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable. The reason for the stability is a second-order effect: as a body moves away from the exact Lagrange position, Coriolis acceleration (which depends on the velocity of an orbiting object and cannot be modeled as a contour map) curves the trajectory into a path around (rather than away from) the point. Because the source of stability is the Coriolis force, the resulting orbits can be stable, but generally are not planar, but "three-dimensional": they lie on a warped surface intersecting the ecliptic plane. The kidney-shaped orbits typically shown nested around L₄ and L₅ are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.

Solar System values

Sun-planet Lagrange points to scale (click for clearer points)

This table lists sample values of L₁, L₂, and L₃ within the Solar System. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. Distances are measured from the larger body's center of mass with L₃ showing a negative location. The percentage columns show how the distances compare to the semimajor axis. E.g. for the Moon, L₁ is located 326400 km from Earth's center, which is 84.9% of the Earth–Moon distance or 15.1% in front of the Moon; L₂ is located 448900 km from Earth's center, which is 116.8% of the Earth–Moon distance or 16.8% beyond the Moon; and L₃ is located −381700 km from Earth's center, which is 99.3% of the Earth–Moon distance or 0.7084% in front of the Moon's 'negative' position.

Lagrangian points in Solar System

Body pair	Semimajor axis, SMA (×109 m)	L1 (×109 m)	1 − L1/SMA (%)	L2 (×109 m)	L2/SMA − 1 (%)	L3 (×109 m)	1 + L3/SMA (%)
Earth–Moon	0.3844	0.32639	15.09	0.4489	16.78	−0.38168	0.7084
Sun–Mercury	57.909	57.689	0.3806	58.13	0.3815	−57.909	0.000009683
Sun–Venus	108.21	107.2	0.9315	109.22	0.9373	−108.21	0.0001428
Sun–Earth	149.6	148.11	0.997	151.1	1.004	−149.6	0.0001752
Sun–Mars	227.94	226.86	0.4748	229.03	0.4763	−227.94	0.00001882
Sun–Jupiter	778.34	726.45	6.667	832.65	6.978	−777.91	0.05563
Sun–Saturn	1426.7	1362.5	4.496	1492.8	4.635	−1426.4	0.01667
Sun–Uranus	2870.7	2801.1	2.421	2941.3	2.461	−2870.6	0.002546
Sun–Neptune	4498.4	4383.4	2.557	4615.4	2.602	−4498.3	0.003004

Spaceflight applications

Sun–Earth

The satellite ACE in an orbit around Sun–Earth L₁

Sun–Earth L₁ is suited for making observations of the Sun–Earth system. Objects here are never shadowed by Earth or the Moon and, if observing Earth, always view the sunlit hemisphere. The first mission of this type was the 1978 International Sun Earth Explorer 3 (ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances. Since June 2015, DSCOVR has orbited the L₁ point. Conversely it is also useful for space-based solar telescopes, because it provides an uninterrupted view of the Sun and any space weather (including the solar wind and coronal mass ejections) reaches L₁ up to an hour before Earth. Solar and heliospheric missions currently located around L₁ include the Solar and Heliospheric Observatory, Wind, and the Advanced Composition Explorer. Planned missions include the Interstellar Mapping and Acceleration Probe (IMAP) and the NEO Surveyor.

Sun–Earth L₂ is a good spot for space-based observatories. Because an object around L₂ will maintain the same relative position with respect to the Sun and Earth, shielding and calibration are much simpler. It is, however, slightly beyond the reach of Earth's umbra, so solar radiation is not completely blocked at L₂. Spacecraft generally orbit around L₂, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L₂, the Sun, Earth and Moon are relatively close together in the sky; this means that a large sunshade with the telescope on the dark-side can allow the telescope to cool passively to around 50 K – this is especially helpful for infrared astronomy and observations of the cosmic microwave background. The James Webb Space Telescope has been positioned at L₂ (12/25/21).

Sun–Earth L₃ was a popular place to put a "Counter-Earth" in pulp science fiction and comic books. Once space-based observation became possible via satellites and probes, it was shown to hold no such object. The Sun–Earth L₃ is unstable and could not contain a natural object, large or small, for very long. This is because the gravitational forces of the other planets are stronger than that of Earth (Venus, for example, comes within 0.3 AU of this L₃ every 20 months).

A spacecraft orbiting near Sun–Earth L₃ would be able to closely monitor the evolution of active sunspot regions before they rotate into a geoeffective position, so that a seven-day early warning could be issued by the NOAA Space Weather Prediction Center. Moreover, a satellite near Sun–Earth L₃ would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed mission to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L₃ were studied and several designs were considered.

Missions to Lagrangian points generally orbit the points rather than occupy them directly.

Another interesting and useful property of the collinear Lagrangian points and their associated Lissajous orbits is that they serve as "gateways" to control the chaotic trajectories of the Interplanetary Transport Network.

Earth–Moon

Earth–Moon L₁ allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable space station intended to help transport cargo and personnel to the Moon and back.

Earth–Moon L₂ has been used for a communications satellite covering the Moon's far side, for example, Queqiao, launched in 2018, and would be an ideal location for a propellant depot as part of the proposed depot-based space transportation architecture.

Sun–Venus

Scientists at the B612 Foundation were planning to use Venus's L₃ point to position their planned Sentinel telescope, which aimed to look back towards Earth's orbit and compile a catalogue of near-Earth asteroids.

Sun–Mars

In 2017, the idea of positioning a magnetic dipole shield at the Sun–Mars L₁ point for use as an artificial magnetosphere for Mars was discussed at a NASA conference. The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.

Lagrangian spacecraft and missions

Main article: List of objects at Lagrangian points

Spacecraft at Sun–Earth L₁

International Sun Earth Explorer 3 (ISEE-3) began its mission at the Sun–Earth L₁ before leaving to intercept a comet in 1982. The Sun–Earth L₁ is also the point to which the Reboot ISEE-3 mission was attempting to return the craft as the first phase of a recovery mission (as of September 25, 2014 all efforts have failed and contact was lost).

Solar and Heliospheric Observatory (SOHO) is stationed in a halo orbit at L₁, and the Advanced Composition Explorer (ACE) in a Lissajous orbit. WIND is also at L₁. Currently slated for launch in late 2024, the Interstellar Mapping and Acceleration Probe will be placed near L₁.

Deep Space Climate Observatory (DSCOVR), launched on 11 February 2015, began orbiting L₁ on 8 June 2015 to study the solar wind and its effects on Earth. DSCOVR is unofficially known as GORESAT, because it carries a camera always oriented to Earth and capturing full-frame photos of the planet similar to the Blue Marble. This concept was proposed by then-Vice President of the United States Al Gore in 1998 and was a centerpiece in his 2006 film An Inconvenient Truth.

LISA Pathfinder (LPF) was launched on 3 December 2015, and arrived at L₁ on 22 January 2016, where, among other experiments, it tested the technology needed by (e)LISA to detect gravitational waves. LISA Pathfinder used an instrument consisting of two small gold alloy cubes.

After ferrying lunar samples back to Earth, the transport module of Chang'e 5 was sent to L1 with its remaining fuel as part of the Chinese Lunar Exploration Program on 16 December 2020 where it is permanently stationed to conduct limited Earth-Sun observations.

Spacecraft at Sun–Earth L₂

Spacecraft at the Sun–Earth L₂ point are in a Lissajous orbit until decommissioned, when they are sent into a heliocentric graveyard orbit.

• 1 October 2001 – October 2010: Wilkinson Microwave Anisotropy Probe
• November 2003 – April 2004: WIND, then it returned to Earth orbit before going to L₁ where it still remains
• July 2009 – 29 April 2013: Herschel Space Telescope
• 3 July 2009 – 21 October 2013: Planck Space Observatory
• 25 August 2011 – April 2012: Chang'e 2, from where it travelled to 4179 Toutatis and then into deep space
• January 2014: Gaia Space Observatory
• 2019: Spektr-RG X-Ray Observatory
• 2021: James Webb Space Telescope uses a halo orbit
• 2022: Euclid Space Telescope
• 2024: Nancy Grace Roman Space Telescope (WFIRST) will use a halo orbit
• 2031: Advanced Telescope for High-Energy Astrophysics (ATHENA) will use a halo orbit

Spacecraft at Earth–Moon L₂

• Chang'e 5-T1 experimental spacecraft DFH-3A "service module" was sent to the Earth-Moon L₂ lunar Lissajous orbit on 13 January 2015, where it used the remaining 800 kg of fuel to test maneuvers key to future lunar missions.
• Queqiao entered orbit around the Earth–Moon L₂ on 14 June 2018. It serves as a relay satellite for the Chang'e 4 lunar far-side lander, which cannot communicate directly with Earth.