Lagrange point

In celestial mechanics, the Lagrange points /ləˈɡrɑːn/ (also Lagrangian points, L-points, or libration points) are orbital points near two large co-orbiting bodies. At the Lagrange points the gravitational forces of the two large bodies cancel out in such a way that a small object placed in orbit there is in equilibrium in at least two directions relative to the center of mass of the large bodies.

Smaller objects (green) at the Lagrange points are in equilibrium. At any other point, the gravitational forces 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

There are five such points, labeled 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 centers of the two large bodies, while L4 and L5 each act as the third vertex of an equilateral triangle formed with the centers 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.

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.[1] 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.[2][3]

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.[4]

Lagrange points

The five Lagrange points are labeled and defined as follows:

L1 point

The L1 point lies on the line defined by the two large masses M1 and M2, and between them. It is the point where the gravitational attraction of M2 partially cancels that of M1. 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.[5]

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.

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 from the center of Sun than Earth is. This placement occurs because the Sun is also affected by Earth's gravity and so orbits around the two bodies' barycenter, 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 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 barycenter 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 centers 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.

Point Stability

The triangular points (L4 and L5) are stable equilibria, provided that the ratio of M1/M2 is greater than 24.96.[note 1][6] 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.

Natural objects at Lagrange 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 Langrage points of other orbital systems:

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.

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) anticlockwise, and the effective potential in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).[13]
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.[14]

L1

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

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:

We may also write this as:

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 that of the larger body. We may also write:

where ρ1 and ρ2 are the average densities of the two bodies and and 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:

L2

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

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:

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:

with parameters M1,2 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:[15]

L4 and L5

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 need not have negligible mass.) The general triangular configuration was discovered by Lagrange in work 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:

where r is the distance from the large body M1 and sgn(x) is the sign function of x. The terms in this function represent respectively: force from M1; force from M2; and centrifugal force. The points L3, L1, L2 occur where the acceleration is zero — see chart at right.

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[note 1] times the mass of the secondary body (e.g. the Moon).[16][17] The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth[18]). 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)[17] curves the trajectory into a path around (rather than away from) the point.[17][19]

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 (×109m) L1 (×109m) 1 − L1/SMA (%) L2 (×109m) L2/SMA − 1 (%) L3 (×109m) 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.[20] 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).

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,[21] 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 is due to be positioned at L2.

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[22] 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 7-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 manned mission to near-Earth asteroids). In 2010, spacecraft transfer trajectories to Sun–Earth L3 were studied and several designs were considered.[23]

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 half-way manned 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,[24] and would be "an ideal location" for a propellant depot as part of the proposed depot-based space transportation architecture.[25]

Sun–Venus

Scientists at the B612 Foundation were[26] 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.[27]

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.[28] The idea is that this would protect the planet's atmosphere from the Sun's radiation and solar winds.

Lagrangian spacecraft and missions

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).[29]

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.[30] 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[31] and was a centerpiece in his 2006 film An Inconvenient Truth.[32]

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.

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.

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.[37]
  • 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.

Past and current missions

Color key:
  Unflown or planned mission   Mission en route or in progress (including mission extensions)   Mission at Lagrangian point completed successfully (or partially successfully)
MissionLagrangian pointAgencyDescription
International Sun–Earth Explorer 3 (ISEE-3)Sun–Earth L1NASA Launched in 1978, it was the first spacecraft to be put into orbit around a libration point, where it operated for four years in a halo orbit about the L1 Sun–Earth point. After the original mission ended, it was commanded to leave L1 in September 1982 in order to investigate comets and the Sun.[38] Now in a heliocentric orbit, an unsuccessful attempt to return to halo orbit was made in 2014 when it made a flyby of the Earth–Moon system.[39][40]
Advanced Composition Explorer (ACE)Sun–Earth L1NASALaunched 1997. Has fuel to orbit near L1 until 2024. Operational as of 2019.[41]
Deep Space Climate Observatory (DSCOVR)Sun–Earth L1NASALaunched on 11 February 2015. Planned successor of the Advanced Composition Explorer (ACE) satellite. In safe mode as of 2019, but is planned to reboot.[42]
LISA Pathfinder (LPF)Sun–Earth L1ESA, NASALaunched one day behind revised schedule (planned for the 100th anniversary of the publication of Einstein's General Theory of Relativity), on 3 December 2015. Arrived at L1 on 22 January 2016.[43] LISA Pathfinder was deactivated on 30 June 2017.[44]
Solar and Heliospheric Observatory (SOHO)Sun–Earth L1ESA, NASAOrbiting near L1 since 1996. Operational as of 2020.[45]
WINDSun–Earth L1NASAArrived at L1 in 2004 with fuel for 60 years. Operational as of 2019.[46]
Wilkinson Microwave Anisotropy Probe (WMAP)Sun–Earth L2NASAArrived at L2 in 2001. Mission ended 2010,[47] then sent to solar orbit outside L2.[48]
Herschel Space TelescopeSun–Earth L2ESA Arrived at L2 July 2009. Ceased operation on 29 April 2013; will be moved to a heliocentric orbit.[49][50]
Planck Space ObservatorySun–Earth L2ESA Arrived at L2 July 2009. Mission ended on 23 October 2013; Planck has been moved to a heliocentric parking orbit.[51]
Chang'e 2Sun–Earth L2CNSA Arrived in August 2011 after completing a lunar mission before departing en route to asteroid 4179 Toutatis in April 2012.[36]
ARTEMIS mission extension of THEMISEarth–Moon L1 and L2NASAMission consists of two spacecraft, which were the first spacecraft to reach Earth–Moon Lagrangian points. Both moved through Earth–Moon Lagrangian points, and are now in lunar orbit.[52][53]
WINDSun–Earth L2NASAArrived at L2 in November 2003 and departed April 2004.
Gaia Space ObservatorySun–Earth L2ESALaunched 19 December 2013.[54] Operational as of 2020.[55]
Chang'e 5-T1 Service ModuleEarth–Moon L2CNSALaunched on 23 October 2014, arrived at L2 halo orbit on 13 January 2015.[37]
QueqiaoEarth–Moon L2CNSALaunched on 21 May 2018, arrived at L2 halo orbit on June 14.[56]
Spektr-RGSun–Earth L2IKI RAN
DLR
Launched 13 July 2019. Roentgen and Gamma space observatory. En route to L2 point.

Future and proposed missions

MissionLagrangian pointAgencyDescription
"Lunar Far-Side Communication Satellites"Earth–Moon L2NASA Proposed in 1968 for communications on the far side of the Moon during the Apollo program,
mainly to enable an Apollo landing on the far side—neither the satellites nor the landing were ever realized.[57]
Space colonization and manufacturingEarth–Moon L4 or L5First proposed in 1974 by Gerard K. O'Neill[58] and subsequently advocated by the L5 Society.
EQUULEUSEarth–Moon L2University of Tokyo, JAXA 6U CubeSat, launch planned in 2019 as a secondary payload onboard SLS Artemis 1.[59]
James Webb Space Telescope (JWST)Sun–Earth L2NASA, ESA, CSA As of 2020, launch is planned for 2021.[60]
EuclidSun–Earth L2ESA, NASA As of 2013, launch is planned in 2020.[61]
Aditya-L1Sun–Earth L1ISROLaunch planned for 2021; it will be going to a point 1.5 million kilometers away from Earth, from where it will observe the Sun constantly and study the solar corona, the region around the Sun's surface.[62]
Demonstration and Experiment of Space Technology
for INterplanetary voYage (DESTINY)
Earth–Moon L2JAXA Candidate for JAXA's next "Competitively-Chosen Medium-Sized Focused Mission",
possible launch in the early 2020s.[63]
Exploration Gateway PlatformEarth–Moon L2[64]NASA Proposed in 2011.[65]
Nancy Grace Roman Space Telescope (WFIRST)Sun–Earth L2NASA, USDOE As of 2013, in a "pre-formulation" phase until at least early 2016; possible launch in the early 2020s.[66]
LiteBIRDSun–Earth L2[67]JAXA, NASA As of 2015, one of two finalists for JAXA's next "Strategic Large Mission"; would be launched in 2024 if selected.[68]
Planetary Transits and Oscillations of stars (PLATO)Sun–Earth L2ESAPlanned for launch in 2024 for an initial six-year mission.[69]
Space Infrared Telescope for Cosmology
and Astrophysics
(SPICA)
Sun–Earth L2JAXA, ESA, SRON As of 2015, awaiting approval from both Japanese and European side, launch proposed for 2025.[70]
Advanced Telescope for High Energy Astrophysics
(ATHENA)
Sun–Earth L2ESALaunch planned for 2028.[71]
Spektr-MSun–Earth L2RoscosmosPossible launch after 2027.[72]

See also

Notes

  1. Actually 25 + 621/224.9599357944

References

  1. One Year on Earth – Seen From 1 Million Miles on YouTube
  2. Koon, W. S.; Lo, M. W.; Marsden, J. E.; Ross, S. D. (2006). Dynamical Systems, the Three-Body Problem, and Space Mission Design. p. 9. Archived from the original on 2008-05-27. Retrieved 2008-06-09. (16MB)
  3. Euler, Leonhard (1765). De motu rectilineo trium corporum se mutuo attrahentium (PDF).
  4. Lagrange, Joseph-Louis (1867–92). "Tome 6, Chapitre II: Essai sur le problème des trois corps". Œuvres de Lagrange (in French). Gauthier-Villars. pp. 229–334.
  5. Cornish, Neil J. "The Lagrangian Points" (PDF). Archived from the original (PDF) on September 7, 2015. Retrieved 15 Dec 2015.
  6. "The Lagrange Points" (PDF)., Neil J. Cornish with input from Jeremy Goodman
  7. Choi 2011-07-27T17:06:00Z, Charles Q. "First Asteroid Companion of Earth Discovered at Last". Space.com.
  8. "NASA - NASA's Wise Mission Finds First Trojan Asteroid Sharing Earth's Orbit". www.nasa.gov.
  9. Slíz-Balogh, Judit; Barta, András; Horváth, Gábor (2018). "Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth-Moon Lagrange point L5 - Part I. Three-dimensional celestial mechanical modelling of dust cloud formation". Monthly Notices of the Royal Astronomical Society. 480 (4): 5550–5559. arXiv:1910.07466. Bibcode:2018MNRAS.480.5550S. doi:10.1093/mnras/sty2049.
  10. Slíz-Balogh, Judit; Barta, András; Horváth, Gábor (2019). "Celestial mechanics and polarization optics of the Kordylewski dust cloud in the Earth-Moon Lagrange point L5. Part II. Imaging polarimetric observation: new evidence for the existence of Kordylewski dust cloud". Monthly Notices of the Royal Astronomical Society. 482 (1): 762–770. arXiv:1910.07471. Bibcode:2019MNRAS.482..762S. doi:10.1093/mnras/sty2630.
  11. Freitas, Robert; Valdes, Francisco (1980). "A Search for Natural or Artificial Objects Located at the Earth–Moon Libration Points". Icarus. 42 (3): 442–447. Bibcode:1980Icar...42..442F. doi:10.1016/0019-1035(80)90106-2.
  12. "List Of Neptune Trojans". Minor Planet Center. Archived from the original on 2011-07-25. Retrieved 2010-10-27.
  13. Seidov, Zakir F. (March 1, 2004). "The Roche Problem: Some Analytics". The Astrophysical Journal. 603 (1): 283–284. arXiv:astro-ph/0311272. Bibcode:2004ApJ...603..283S. doi:10.1086/381315.
  14. "Lagrange Points" by Enrique Zeleny, Wolfram Demonstrations Project.
  15. "Widnall, Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem" (PDF).
  16. Fitzpatrick, Richard. "Stability of Lagrange Points". Newtonian Dynamics. University of Texas.
  17. Greenspan, Thomas (January 7, 2014). "Stability of the Lagrange Points, L4 and L5" (PDF).
  18. Pitjeva, E.V.; Standish, E.M. (2009-04-01). "Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit". Celestial Mechanics and Dynamical Astronomy. 103 (4): 365–372. Bibcode:2009CeMDA.103..365P. doi:10.1007/s10569-009-9203-8.
  19. Cacolici, Gianna Nicole, et al., "Stability of Lagrange Points: James Webb Space Telescope", University of Arizona. Retrieved 17 Sept. 2018.
  20. "ISEE-3/ICE". Solar System Exploration. NASA. Archived from the original on July 20, 2015. Retrieved August 8, 2015.
  21. Angular size of the Sun at 1 AU + 1.5 million kilometres: 31.6′, angular size of Earth at 1.5 million kilometres: 29.3′
  22. STEREO mission description by NASA, http://www.nasa.gov/mission_pages/stereo/main/index.html#.UuG0NxDb-kk
  23. Tantardini, Marco; Fantino, Elena; Ren, Yuan; Pergola, Pierpaolo; Gómez, Gerard; Masdemont, Josep J. (2010). "Spacecraft trajectories to the L3 point of the Sun–Earth three-body problem" (PDF). Celestial Mechanics and Dynamical Astronomy. 108 (3): 215–232. Bibcode:2010CeMDA.108..215T. doi:10.1007/s10569-010-9299-x.
  24. Jones, Andrew (2018-06-14). "Chang'e-4 relay satellite enters halo orbit around Earth-Moon L2, microsatellite in lunar orbit". SpaceNews.
  25. Zegler, Frank; Kutter, Bernard (2010-09-02). "Evolving to a Depot-Based Space Transportation Architecture" (PDF). AIAA SPACE 2010 Conference & Exposition. AIAA. p. 4. Archived from the original (PDF) on 2014-06-24. Retrieved 2011-01-25. L2 is in deep space far away from any planetary surface and hence the thermal, micrometeoroid, and atomic oxygen environments are vastly superior to those in LEO. Thermodynamic stasis and extended hardware life are far easier to obtain without these punishing conditions seen in LEO. L2 is not just a great gateway—it is a great place to store propellants. ... L2 is an ideal location to store propellants and cargos: it is close, high energy, and cold. More importantly, it allows the continuous onward movement of propellants from LEO depots, thus suppressing their size and effectively minimizing the near-Earth boiloff penalties.
  26. "B612 studying smallsat missions to search for near Earth objects". SpaceNews.com. June 20, 2017.
  27. "The Sentinel Mission". B612 Foundation. Archived from the original on 30 June 2012. Retrieved 1 February 2014.
  28. "NASA proposes a magnetic shield to protect Mars' atmosphere". phys.org.
  29. "ISEE-3 is in Safe Mode". Space College. 25 September 2014. "The ground stations listening to ISEE-3 have not been able to obtain a signal since Tuesday the 16th"
  30. US Department of Commerce, NOAA Satallites and Information Service. "NOAA's Satellite and Information Service (NESDIS)". Archived from the original on 2015-06-08.
  31. "At long last: Al Gore's satellite dream blasts off". USA TODAY. 7 February 2015.
  32. Mellow, Craig (August 2014). "Al Gore's Satellite". Air & Space/Smithsonian. Retrieved December 12, 2014.
  33. "Mission Complete! WMAP Fires Its Thrusters For The Last Time".
  34. Toobin, Adam (2013-06-19). "Herschel Space Telescope Shut Down For Good, ESA Announces". Huffington Post.
  35. Lakdawalla, Emily (14 June 2012). "Chang'e 2 has departed Earth's neighborhood for.....asteroid Toutatis!?". Retrieved 15 June 2012.
  36. Lakdawalla, Emily (15 June 2012). "Update on yesterday's post about Chang'e 2 going to Toutatis". Planetary Society. Retrieved 26 June 2012.
  37. "Chang'e 5 Test Mission Updates". Spaceflight 101. Retrieved 14 December 2014.
  38. "Solar System Exploration: ISEE-3/ICE". NASA. Archived from the original on 2011-06-14. Retrieved 2010-09-28.
  39. Lakdawalla, Emily (October 3, 2008). "It's Alive!". The Planetary Science Weblog.
  40. Chang, Kenneth (August 8, 2014). "Rudderless Craft to Get Glimpse of Home Before Sinking Into Space's Depths". The New York Times.
  41. "ACE MAG Spectrograms: 1 day starting 2019/3/15 (2019 074)". www.srl.caltech.edu. Retrieved 2020-02-12.
  42. "Software fix planned to restore DSCOVR". SpaceNews.com. 2019-10-01. Retrieved 2020-02-12.
  43. "LISA Pathfinder factsheet". ESA. 11 June 2012. Retrieved 26 June 2012.
  44. "LISA Pathfinder Will Concludee Trailblazing Mission". ESA Science and Technology. ESA. 20 June 2017. Retrieved 17 August 2017.
  45. "The Very Latest SOHO Images". sohowww.nascom.nasa.gov. Retrieved 2020-02-12.
  46. Tran, Lina (2019-10-31). "25 Years of Science in the Solar Wind". NASA. Retrieved 2020-02-12.
  47. "WMAP Facts". NASA. Retrieved 2013-03-18.
  48. http://map.gsfc.nasa.gov/news/events.html WMAP Ceases Communications
  49. "Herschel Factsheet". European Space Agency. 17 April 2009. Retrieved 2009-05-12.
  50. "Herschel space telescope finishes mission". BBC news. 29 April 2013.
  51. "Last command sent to ESA's Planck space telescope". European Space Agency. October 23, 2013. Retrieved October 23, 2013.
  52. Fox, Karen C. "First ARTEMIS Spacecraft Successfully Enters Lunar Orbit". The Sun-Earth Connection: Heliophysics. NASA.
  53. Hendrix, Susan. "Second ARTEMIS Spacecraft Successfully Enters Lunar Orbit". The Sun-Earth Connection: Heliophysics. NASA.
  54. "Worldwide launch schedule". Spaceflight Now. 27 November 2013. Archived from the original on 30 May 2010.
  55. "ESA Science & Technology - Fact Sheet". sci.esa.int. Retrieved 2020-02-12.
  56. Jones, Andrew (21 May 2018). "China launches Queqiao relay satellite to support Chang'e 4 lunar far side landing mission". GBTimes. Retrieved 22 May 2018.
  57. Schmid, P. E. (June 1968). "Lunar Far-Side Communication Satellites". NASA. Retrieved 2020-09-14.
  58. O'Neill, Gerard K. (September 1974). "The Colonization of Space". Physics Today. 27 (9): 32–40. Bibcode:1974PhT....27i..32O. doi:10.1063/1.3128863.
  59. "Extended Tisserand graph and multiple lunar swing-by design with Sun perturbation" (PDF). JAXA. 3 March 2016. Retrieved 2016-06-07.
  60. "Jim Bridenstine on Twitter". Twitter. Retrieved 2018-07-03.
  61. "NASA Officially Joins ESA's 'Dark Universe' Mission". JPL/NASA. 24 January 2013. Retrieved 12 April 2013.
  62. Aravind, Indulekha (15 November 2014). "After Mars Mission, what is Isro planning next?".
  63. "DESTINYについて " (in Japanese). Archived from the original on 2015-10-07. Retrieved 2015-10-23.
  64. "NASA teams evaluating ISS-built Exploration Platform roadmap". June 15, 2012.
  65. Bergin, Chris (December 2011). "Exploration Gateway Platform hosting Reusable Lunar Lander proposed". NASA Spaceflight.com. Retrieved 2011-12-05.
  66. Hertz, Paul (2013-06-04), NASA Astrophysics presentation to American Astronomical Society (PDF), archived from the original (PDF) on 2013-12-03, retrieved 2013-09-10
  67. Masahashi, Hazumi (1 September 2015). "LiteBIRD" (PDF). indico.cern. Retrieved 2015-10-23.
  68. "宇宙科学・探査分野工程表の取り組み状況について その3" (PDF) (in Japanese). Space Policy Committee. 13 October 2015. Retrieved 2015-10-23.
  69. "ESA selects planet-hunting PLATO mission". ESA. 19 February 2014. Retrieved 25 April 2016.
  70. Shibai, Hiroshi (2014-12-31), SPICA (PDF), retrieved 2015-02-24
  71. "ESA Science & Technology: Athena to study the hot and energetic Universe". ESA. 27 June 2014. Retrieved 23 August 2014.
  72. "В РАН рассказали, когда состоится запуск космического телескопа "Спектр-М"". РИА Новости. 2019-06-29. Retrieved 2019-07-13.
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