3728:
91:
3590:
1426:
4330:
4497:, despite the fact that the existence of a planetary body in this location had been understood as an impossibility once orbital mechanics and the perturbations of planets upon each other's orbits came to be understood, long before the Space Age; the influence of an Earth-sized body on other planets would not have gone undetected, nor would the fact that the foci of Earth's orbital ellipse would not have been in their expected places, due to the mass of the counter-Earth. The SunâEarth L
972:, 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 barycenter of Earth and Sun in order to have the same 1-year period. It is at the L
43:
3711:(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 L
3604:
137:
7744:
7755:
51:
6917:
4688:
2411:
993:
4674:
4315:
3530:
3131:
763:
similar to JWST. Each of the space observatories benefit from being far enough from Earth's shadow to utilize solar panels for power, from not needing much power or propellant for station-keeping, from not being subjected to the Earth's magnetospheric effects, and from having direct line-of-sight to
2252:
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
1065:
is greater than 24.96. This is the case for the SunâEarth system, the SunâJupiter system, and, by a smaller margin, the EarthâMoon 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
5827:
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.
4501:, however, is a weak saddle point and exponentially unstable with time constant of roughly 150 years. Moreover, it could not contain a natural object, large or small, for very long because the gravitational forces of the other planets are stronger than that of Earth (for example,
2614:
1671:
3314:
5836:
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
3585:
occur where the acceleration is zero â 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.
2222:
3697:
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), The Earth is over 81 times the mass of the Moon (the Moon is 1.23% of the mass of the Earth). Although the
2927:
967:
point exists on the opposite side of the Sun, a little outside Earth's orbit and slightly farther from 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'
1184:
As the Sun and
Jupiter are the two most massive objects in the Solar System, there are more known SunâJupiter trojans than for any other pair of bodies. However, smaller numbers of objects are known at the Lagrange points of other orbital systems:
3747:
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 (but see
3706:
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,
4440:, 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 â this is especially helpful for
1453:, could be placed so as to maintain its position relative to the two massive bodies. This occurs because the combined gravitational forces of the two massive bodies provide the exact centripetal force required to maintain the
2437:
1494:
2756:
1855:
4613:
Lagrangian points in the EarthâMoon system proposed as locations for their huge rotating space habitats. Both positions are also proposed for communication satellites covering the Moon alike communication satellites in
4352:
is suited for making observations of the SunâEarth 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
2398:
5236:"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"
2103:
3603:
3274:
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
913:. On the opposite side of Earth from the Sun, the orbital period of an object would normally be greater than Earth's. The extra pull of Earth's gravity decreases the object's orbital period, and at the L
2120:
976:
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.
5185:"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"
2823:
2030:
4662:
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.
1917:
909:
point lies on the line through the two large masses beyond the smaller of the two. Here, the combined gravitational forces of the two large masses balance the centrifugal force on a body at L
3233:
5797:
5563:
3295:
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
4520:
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
7460:
1957:
877:
would typically have a shorter orbital period than Earth, but that ignores the effect of Earth's gravitational pull. If the object is directly between Earth and the Sun, then
7001:
3525:{\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}}}}
4424:
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
1127:, 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 '
3126:{\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}}}}
1101:
will tend to fall out of orbit; it is therefore rare to find natural objects there, and spacecraft inhabiting these areas must employ a small but critical amount of
4521:
2623:
1722:
615:
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
881:
counteracts some of the Sun's pull on the object, increasing the object's orbital period. The closer to Earth the object is, the greater this effect is. At the L
66:
of a two-body system in a rotating frame of reference. The arrows indicate the downhill gradients of the potential around the five
Lagrange points, toward them (
5884:
2298:
4406:
2037:
3291:
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
5991:
5521:
5778:
4818:
4531:
would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed missions to
3664:
missions, it is preferable for the spacecraft to be in a large-amplitude (100,000â200,000 km or 62,000â124,000 mi) Lissajous orbit around L
1435:
7438:
7028:
5808:
532:
4551:
allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable
6794:
7018:
6953:
5570:
1178:
5666:
5330:
681:
points are stable points, meaning that objects can orbit them and that they have a tendency to pull objects into them. Several planets have
5995:
4957:
1170:
938:
101:
6854:
6061:
3756:
showing a negative direction. The percentage columns show the distance from the orbit compared to the semimajor axis. E.g. for the Moon, L
2609:{\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}}}}
1666:{\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}}}}
1429:
Visualisation of the relationship between the
Lagrange points (red) of a planet (blue) orbiting a star (yellow) counterclockwise, and the
304:
90:
4846:
2109:
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 L
3276:
2785:
1992:
969:
373:
1860:
7038:
4746:
4390:
4304:
1114:
816:
3194:
7058:
5030:
3645:
does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following
799:". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special
5706:
1015:
in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of (L
6849:
6729:
6148:
5761:
4717:
3653:
are what most of
Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of
720:
to study solar wind coming toward Earth from the Sun and to monitor Earth's climate, by taking images and sending them back. The
5653:
3792:
from Earth's center, which is 99.3% of the EarthâMoon distance or 0.7084% inside (Earthward) of the Moon's 'negative' position.
728:. This allows the satellite's large sunshield to protect the telescope from the light and heat of the Sun, Earth and Moon. The L
7213:
6814:
6568:
4524:
4354:
3673:
46:
Lagrange points in the SunâEarth system (not to scale). This view is from the north, so that Earth's orbit is counterclockwise.
7588:
7450:
7033:
6887:
6526:
6517:
6254:
4358:
1405:
are sometimes erroneously described as trojans, but do not occupy
Lagrange points. Known objects on horseshoe orbits include
713:
194:
1066:(either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable,
1351:
about the Lagrange points, with Polydeuces describing the largest deviations, moving up to 32° away from the SaturnâDione L
7784:
5921:
4707:
4402:
4319:
6103:
5939:
1699:
are the masses of the large and small object, respectively. The quantity in parentheses on the right is the distance of L
7779:
7154:
6834:
6304:
6027:
6004:
5012:
4425:
963:
point lies on the line defined by the two large masses, beyond the larger of the two. Within the SunâEarth system, the L
5695:
Angular size of the Sun at 1 AU + 1.5 million kilometres: 31.6â˛, angular size of Earth at 1.5 million kilometres: 29.3â˛
7566:
7402:
7149:
6996:
6779:
5415:(May 2007). "Equipotential Surfaces and Lagrangian Points in Nonsynchronous, Eccentric Binary and Planetary Systems".
4736:
4445:
1929:
700:
Some Lagrange points are being used for space exploration. Two important Lagrange points in the Sun-Earth system are L
638:, all in the orbital plane of the two large bodies. There are five Lagrange points for the SunâEarth system, and five
525:
458:
31:
6037:
1131:' or 'trojan asteroids'. The name derives from the names that were given to asteroids discovered orbiting at the Sunâ
4475:
of roughly 23 days. Satellites at these points will wander off in a few months unless course corrections are made.
7638:
7407:
7080:
6759:
6586:
4639:
4449:
4338:
3768:
from Earth's center, which is 84.9% of the EarthâMoon distance or 15.1% "in front of" (Earthwards from) the Moon; L
934:
721:
7561:
7443:
7181:
6946:
6897:
2217:{\displaystyle \rho _{2}\left({\frac {d_{2}}{r}}\right)^{3}\approx 3\rho _{1}\left({\frac {d_{1}}{R}}\right)^{3}}
1231:
5080:
4357:(ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances. Since June 2015,
3727:
3589:
7676:
7176:
7164:
7136:
6882:
6407:
4702:
4575:
1461:
453:
368:
7395:
5885:"TYCHO: Supporting Permanently Crewed Lunar Exploration with High-Speed Optical Communication from Everywhere"
5604:"Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit"
4329:
5705:
Tantardini, Marco; Fantino, Elena; Ren, Yuan; Pergola, Pierpaolo; GĂłmez, Gerard; Masdemont, Josep J. (2010).
7680:
7616:
7297:
7208:
7112:
7085:
6986:
6892:
6200:
4579:
3654:
3261:
1359:
1102:
708:, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Currently, an
624:
597:
324:
6051:
7728:
7690:
7124:
7043:
6981:
6754:
6356:
6276:
6264:
5965:
4378:
1086:
518:
241:
6117:
See the Lagrange Points and Halo Orbits subsection under the section on Geosynchronous Transfer Orbit in
7578:
7385:
7023:
6877:
6819:
6789:
6577:
6454:
6422:
6392:
6351:
6336:
6215:
6055:
5849:
5008:
3708:
789:
426:
261:
169:
5098:
3680:
keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.
1476:, allowing the smaller third body to remain stationary (in this frame) with respect to the first two.
7747:
7648:
7631:
7268:
7203:
7196:
7053:
6939:
6902:
6724:
6508:
6397:
6366:
6294:
6269:
6244:
6205:
6186:
6141:
5865:
5725:
5615:
5539:
5487:
5434:
5377:
5300:
5257:
5206:
5147:
4751:
4741:
4615:
1425:
1012:
709:
667:
299:
256:
246:
174:
6086:
An online calculator to compute the precise positions of the 5 Lagrange points for any 2-body system
3241:
to the larger object is less than the separation of the two objects (although the distance between L
1433:
in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).
7643:
7621:
7556:
7275:
7119:
6764:
6559:
6299:
6107:
3292:
1469:
1430:
1394:
943:
929:
from Earth (away from the sun). An example of a spacecraft designed to operate near the EarthâSun L
744:
546:
341:
179:
59:
7754:
7758:
7541:
7467:
7322:
7218:
7070:
6437:
6326:
6224:
5779:"Chang'e-4 relay satellite enters halo orbit around Earth-Moon L2, microsatellite in lunar orbit"
5741:
5674:
5631:
5503:
5477:
5450:
5424:
5393:
5367:
5326:
5247:
5196:
5165:
5137:
4953:
4693:
4441:
3296:
1711:
1442:
1128:
800:
796:
682:
609:
414:
289:
6112:
4963:
4436:, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L
3245:
and the barycentre is greater than the distance between the smaller object and the barycentre).
4989:
4881:
3780:
from Earth's center, which is 116.8% of the EarthâMoon distance or 16.8% beyond the Moon; and L
7713:
7353:
7248:
7223:
7169:
7159:
6976:
6804:
6702:
6632:
6387:
6341:
6259:
6091:
6085:
5599:
4878:
4853:
4722:
4643:
4598:
4532:
4506:
4394:
4334:
3719:
are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.
3608:
3280:
2894:
1473:
1410:
1340:
1235:
926:
890:
740:
663:
620:
329:
266:
145:
83:
63:
7378:
5061:
1238:. Stability at these specific points is greatly complicated by solar gravitational influence.
7626:
7455:
7412:
7343:
7280:
7233:
7075:
6784:
6716:
6480:
6442:
6316:
6286:
6239:
5733:
5623:
5595:
5495:
5442:
5385:
5308:
5291:
5265:
5214:
5155:
4679:
4583:
3638:
3611:
of the Roche potential of two orbiting bodies, rendered half as a surface and half as a mesh
1715:
1465:
878:
627:, and hence fuel requirements, needed to maintain the desired orbit are kept at a minimum.
556:
499:
448:
213:
157:
623:
balance each other. This can make Lagrange points an excellent location for satellites, as
7671:
7571:
7524:
7422:
7363:
7348:
7302:
7285:
7191:
7186:
6824:
6417:
6321:
6311:
6210:
6134:
6118:
6031:
6008:
5765:
5355:
5351:
5121:
4655:
4627:
4490:
4366:
3650:
3646:
2898:
2890:
1454:
1402:
752:
504:
409:
319:
294:
4899:
3676:
on Earthâspacecraft communications. Similarly, a large-amplitude Lissajous orbit around L
3311:
of an object in orbit at a point along the line passing through both bodies is given by:
6067:
5869:
5729:
5619:
5491:
5438:
5381:
5304:
5261:
5210:
5151:
2924:
is the solution to the following equation, gravitation providing the centripetal force:
2434:
is the solution to the following equation, gravitation providing the centripetal force:
1491:
is the solution to the following equation, gravitation providing the centripetal force:
7718:
7705:
7598:
7529:
7519:
7504:
7482:
7472:
7368:
7358:
7307:
7228:
6920:
6872:
6864:
6744:
6739:
6670:
6650:
6641:
6220:
6196:
6191:
6166:
6020:
5412:
5287:"A Search for Natural or Artificial Objects Located at the EarthâMoon Libration Points"
4985:
3288:
2257:
1390:
1363:
1324:
1320:
1254:
804:
785:
476:
386:
309:
234:
228:
223:
42:
5853:
5758:
5522:"Widnall, Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem"
4563:
Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon's
4385:
up to an hour before Earth. Solar and heliospheric missions currently located around L
7773:
7666:
7583:
7514:
7477:
7317:
7238:
7102:
7097:
6774:
6769:
6688:
6331:
6249:
6098:
5745:
5635:
5312:
5286:
5169:
4756:
4552:
4487:
4483:
4472:
4370:
3552:
1406:
1332:
1308:
885:
point, the object's orbital period becomes exactly equal to Earth's orbital period. L
481:
314:
271:
5507:
5454:
5397:
1449:, there are five positions in space where a third body, of comparatively negligible
136:
7723:
7658:
7327:
7312:
7013:
7008:
6839:
6749:
6623:
6606:
6464:
6361:
6229:
6000:"Essay on the Three-Body Problem" by J.-L. Lagrange, translated from the above, in
5117:
4468:
4410:
3642:
1414:
1389:; if one of the stars expands past its Roche lobe, then it will lose matter to its
1328:
199:
189:
184:
55:
5759:
SMART-1: On Course for Lunar Capture | Moon Today â Your Daily Source of Moon News
5034:
736:
Lagrange points are located about 1,500,000 km (930,000 mi) from earth.
4924:
7551:
7417:
7107:
7090:
6962:
6844:
6679:
6449:
6429:
6346:
5832:
is not just a great gatewayâit is a great place to store propellants. ... L
4712:
2779:
2106:
1986:
1708:
1378:
1268:
1264:
1067:
616:
363:
5943:
5160:
5125:
1070:-shaped orbit around the point (as seen in the corotating frame of reference).
50:
7685:
7593:
7536:
7496:
7290:
6991:
6412:
6073:
6016:
5903:
5737:
5627:
5099:"NASA - NASA's Wise Mission Finds First Trojan Asteroid Sharing Earth's Orbit"
4727:
4669:
4602:
4494:
4374:
3749:
3628:
2902:
2751:{\displaystyle x^{5}+x^{4}(3-\mu )+x^{3}(3-2\mu )-x^{2}(\mu )-x(2\mu )-\mu =0}
1850:{\displaystyle x^{5}+(\mu -3)x^{4}+(3-2\mu )x^{3}-(\mu )x^{2}+(2\mu )x-\mu =0}
1446:
1382:
1158:
760:
443:
399:
358:
6024:
6001:
3627:
points are nominally unstable, there are quasi-stable periodic orbits called
17:
7695:
7546:
7263:
7144:
7065:
7048:
6829:
6181:
6079:
5270:
5235:
5219:
5184:
4949:
4642:, which aimed to look back towards Earth's orbit and compile a catalogue of
4398:
3279:
of the system. Additionally, the geometry of the triangle ensures that the
1374:
point and crashed into Earth after its orbit destabilized, forming the Moon.
1284:
1154:
630:
For any combination of two orbital bodies, there are five Lagrange points, L
2827:
The same remarks about tidal influence and apparent size apply as for the L
2410:
5482:
5429:
5372:
4555:
intended to help transport cargo and personnel to the Moon and back. The
1295:
1273:
1210:
1199:
4420:
is a good spot for space-based observatories. Because an object around L
2260:, corresponding to a circular orbit with this distance as radius around
6734:
6126:
5468:
Seidov, Zakir F. (March 1, 2004). "The Roche Problem: Some Analytics".
4586:
as part of the proposed depot-based space transportation architecture.
4564:
4556:
2393:{\displaystyle T_{s,M_{2}}(r)={\frac {T_{M_{2},M_{1}}(R)}{\sqrt {3}}}.}
1348:
1242:
1132:
992:
694:
601:
4314:
1445:. For example, given two massive bodies in orbits around their common
7509:
7373:
3593:
Net radial acceleration of a point orbiting along the EarthâMoon line
2098:{\displaystyle {\frac {M_{2}}{r^{3}}}\approx 3{\frac {M_{1}}{R^{3}}}}
1441:
Lagrange points are the constant-pattern solutions of the restricted
86:
of the potential. At the points themselves these forces are balanced.
5603:
1023:) the smaller mass with regard to its orbit around the larger mass.
654:
are on the line through the centers of the two large bodies, while L
5499:
5446:
5389:
5252:
5201:
5142:
4959:
Dynamical Systems, the Three-Body Problem, and Space Mission Design
1468:
of the two co-orbiting bodies, at the Lagrange points the combined
1143:
points, which were taken from mythological characters appearing in
6539:
6158:
5922:"B612 studying smallsat missions to search for near Earth objects"
4631:
4502:
4328:
3726:
3601:
3588:
3284:
2831:
point. For example, the angular radius of the sun as viewed from L
2423:
2409:
1424:
1149:
1144:
991:
874:
866:
605:
164:
115:
89:
49:
41:
3657:
keeps a spacecraft in a desired Lissajous orbit for a long time.
1165:
point, ahead of Jupiter, are named after Greek characters in the
3559:. The terms in this function represent respectively: force from
3283:
acceleration is to the distance from the barycenter in the same
1450:
1260:
1177:
point are named after Trojan characters and referred to as the "
803:, the collinear and the equilateral, for any three masses, with
6935:
6130:
4369:, because it provides an uninterrupted view of the Sun and any
3731:
Sunâplanet Lagrange points to scale (Click for clearer points.)
1959:
is the normalised distance. If the mass of the smaller object (
1197:
points contain interplanetary dust and at least two asteroids,
7390:
6931:
5990:, Tome 6, ÂŤ Essai sur le Problème des Trois Corps Âťâ
5904:"TYCHO mission to Earth-Moon libration point EML-4 @ IAC 2013"
2419:
870:
821:
The five Lagrange points are labelled and defined as follows:
5966:"NASA proposes a magnetic shield to protect Mars' atmosphere"
5798:"Evolving to a Depot-Based Space Transportation Architecture"
3672:, because the line between Sun and Earth has increased solar
3287:
as for the two massive bodies. The barycenter being both the
574:
5013:"Tome 6, Chapitre II: Essai sur le problème des trois corps"
2117:
point is about three times of that body. We may also write:
837:
point lies on the line defined between the two large masses
673:
When the mass ratio of the two bodies is large enough, the L
4813:
4775:
4773:
4582:, launched in 2018, and would be "an ideal location" for a
4535:). In 2010, spacecraft transfer trajectories to SunâEarth L
1181:". Both camps are considered to be types of trojan bodies.
917:
point, that orbital period becomes equal to Earth's. Like L
562:
5234:
SlĂz-Balogh, Judit; Barta, AndrĂĄs; HorvĂĄth, GĂĄbor (2019).
5183:
SlĂz-Balogh, Judit; Barta, AndrĂĄs; HorvĂĄth, GĂĄbor (2018).
2863:) â 0.264°, whereas that of the earth is arcsin(
608:
bodies. Mathematically, this involves the solution of the
580:
5654:
Stability of Lagrange Points: James Webb Space Telescope"
568:
851:. It is the point where the gravitational attraction of
724:, a powerful infrared space observatory, is located at L
5854:"Photographische Untersuchungen des Librationspunktes L
4991:
De motu rectilineo trium corporum se mutuo attrahentium
5081:"First Asteroid Companion of Earth Discovered at Last"
3547:
is the distance between the two main objects, and sgn(
3208:
3182:) is much smaller than the mass of the larger object (
2767:) is much smaller than the mass of the larger object (
2256:
This distance can be described as being such that the
1966:) is much smaller than the mass of the larger object (
4365:
point. Conversely, it is also useful for space-based
3317:
3197:
2930:
2788:
2626:
2440:
2301:
2123:
2040:
1995:
1932:
1863:
1725:
1497:
577:
1985:
from the smaller object, equal to the radius of the
1039:) are stable equilibria, provided that the ratio of
571:
559:
7704:
7657:
7609:
7495:
7431:
7336:
7256:
7247:
7135:
6969:
6803:
6715:
6659:
6595:
6548:
6488:
6479:
6375:
6285:
6174:
6165:
6043:J R Stockton - Includes translations of Lagrange's
5033:. Space Telescope Science Institute. Archived from
4428:, so solar radiation is not completely blocked at L
3633:around these points in a three-body system. A full
3266:
The reason these points are in balance is that at L
2885:) â 0.242°. Looking toward the sun from L
565:
4840:
4838:
4539:were studied and several designs were considered.
3752:especially in the case of Moon and Jupiter) with L
3524:
3227:
3125:
2818:{\displaystyle r\approx R{\sqrt{\frac {\mu }{3}}}}
2817:
2750:
2608:
2392:
2216:
2097:
2025:{\displaystyle r\approx R{\sqrt{\frac {\mu }{3}}}}
2024:
1951:
1911:
1849:
1685:is the distance between the two main objects, and
1665:
865:combine to produce an equilibrium. An object that
6080:Locations of Lagrange points, with approximations
5656:, University of Arizona. Retrieved 17 Sept. 2018.
5240:Monthly Notices of the Royal Astronomical Society
5189:Monthly Notices of the Royal Astronomical Society
5069:, Neil J. Cornish, with input from Jeremy Goodman
1912:{\displaystyle \mu ={\frac {M_{2}}{M_{1}+M_{2}}}}
670:formed with the centers of the two large bodies.
5019:(in French). Gauthier-Villars. pp. 229â334.
2238:are the average densities of the two bodies and
1253:points contain several dozen known objects, the
3228:{\displaystyle r\approx R{\tfrac {7}{12}}\mu .}
2909:in order for its solar panels to get full sun.
889:is about 1.5 million kilometers, or 0.01
788:around 1750, a decade before the Italian-born
6947:
6142:
5796:Zegler, Frank; Kutter, Bernard (2010-09-02).
5564:"Stability of the Lagrange Points, L4 and L5"
4852:. WMAP Education and Outreach. Archived from
3504:
3449:
925:is about 1.5 million kilometers or 0.01
795:In 1772, Lagrange published an "Essay on the
784:) were discovered by the Swiss mathematician
697:has more than one million of these trojans.
526:
8:
5557:
5555:
5553:
2620:case. The corresponding quintic equation is
1011:points lie at the third vertices of the two
642:Lagrange points for the EarthâMoon system. L
6017:Considerationes de motu corporum coelestium
5805:AIAA SPACE 2010 Conference & Exposition
5718:Celestial Mechanics and Dynamical Astronomy
5608:Celestial Mechanics and Dynamical Astronomy
5285:Freitas, Robert; Valdes, Francisco (1980).
4578:covering the Moon's far side, for example,
4407:Interstellar Mapping and Acceleration Probe
27:Equilibrium points near two orbiting bodies
7253:
6954:
6940:
6932:
6916:
6485:
6171:
6149:
6135:
6127:
5711:point of the SunâEarth three-body problem"
5056:
5054:
5052:
2760:Again, if the mass of the smaller object (
1703:from the center of mass. The solution for
893:, from Earth in the direction of the Sun.
533:
519:
121:
94:An example of a spacecraft at Sun-Earth L2
5481:
5428:
5371:
5358:(2005). "Where Did The Moon Come From?".
5269:
5251:
5218:
5200:
5159:
5141:
3514:
3503:
3502:
3493:
3474:
3461:
3448:
3447:
3441:
3408:
3384:
3374:
3348:
3337:
3327:
3316:
3207:
3196:
3163:being defined such that the distance of L
3115:
3104:
3091:
3084:
3055:
3042:
3031:
3025:
3009:
2980:
2974:
2963:
2937:
2931:
2929:
2893:. It is necessary for a spacecraft, like
2808:
2798:
2787:
2703:
2672:
2644:
2631:
2625:
2598:
2587:
2574:
2567:
2544:
2531:
2520:
2514:
2498:
2488:
2482:
2470:
2447:
2441:
2439:
2362:
2349:
2344:
2337:
2317:
2306:
2300:
2208:
2193:
2187:
2176:
2160:
2145:
2139:
2128:
2122:
2087:
2077:
2071:
2057:
2047:
2041:
2039:
2015:
2005:
1994:
1939:
1931:
1900:
1887:
1876:
1870:
1862:
1811:
1789:
1758:
1730:
1724:
1655:
1644:
1631:
1624:
1601:
1588:
1577:
1571:
1555:
1545:
1539:
1527:
1504:
1498:
1496:
4527:. Moreover, a satellite near SunâEarth L
4313:
3794:
3167:from the centre of the larger object is
1311:has two smaller moons of Saturn in its L
7439:Effect of spaceflight on the human body
4834:
4769:
3804:Semimajor axis, SMA (×10 m)
619:forces of the two large bodies and the
424:
339:
143:
129:
7461:Psychological and sociological effects
6795:Transposition, docking, and extraction
4516:A spacecraft orbiting near SunâEarth L
4452:was positioned in a halo orbit about L
2778:is at approximately the radius of the
772:The three collinear Lagrange points (L
5562:Greenspan, Thomas (January 7, 2014).
4432:. Spacecraft generally orbit around L
3573:; and centripetal force. The points L
3175:. If the mass of the smaller object (
1981:are at approximately equal distances
1105:in order to maintain their position.
7:
5411:Sepinsky, Jeremy F.; Willems, Bart;
3536:is the distance from the large body
2616:with parameters defined as for the L
939:Wilkinson Microwave Anisotropy Probe
739:The European Space Agency's earlier
6070:âalso attributed to Neil J. Cornish
6047:and of two related papers by Euler
5986:Joseph-Louis, Comte Lagrange, from
4654:In 2017, the idea of positioning a
3735:This table lists sample values of L
3649:trajectories. These quasi-periodic
1457:that matches their orbital motion.
1331:also has two Lagrange co-orbitals,
7170:Weather and environment monitoring
6023:âtranscription and translation at
5126:"The Second Earth Trojan 2020 XL5"
4747:List of objects at Lagrange points
4391:Solar and Heliospheric Observatory
4355:International Sun Earth Explorer 3
4305:List of objects at Lagrange points
3796:Lagrangian points in Solar System
1472:of two massive bodies balance the
1115:List of objects at Lagrange points
1109:Natural objects at Lagrange points
817:List of objects at Lagrange points
743:telescope, and its newly launched
704:, between the Sun and Earth, and L
25:
6855:Kepler's laws of planetary motion
5942:. B612 Foundation. Archived from
5807:. AIAA. p. 4. Archived from
5707:"Spacecraft trajectories to the L
5130:The Astrophysical Journal Letters
5124:; FĂśhring, Dora (November 2021).
5079:Choi, Charles Q. (27 July 2011).
4886:Eric Weisstein's World of Physics
1421:Physical and mathematical details
1230:points contain concentrations of
1119:Due to the natural stability of L
600:for small-mass objects under the
305:Kepler's laws of planetary motion
7753:
7743:
7742:
6915:
6850:Interplanetary Transport Network
6730:Collision avoidance (spacecraft)
5862:Acta Astronomica, Vol. 11, p.165
4718:Interplanetary Transport Network
4686:
4672:
4638:point to position their planned
4471:and exponentially unstable with
4341:(blue) orbits around SunâEarth L
1952:{\displaystyle x={\frac {r}{R}}}
1362:postulates that an object named
996:Gravitational accelerations at L
693:points with respect to the Sun;
555:
135:
7214:Space launch market competition
6815:Astronomical coordinate systems
6569:Longitude of the ascending node
6113:Earth, a lone Trojan discovered
5333:from the original on 2011-07-25
4525:Space Weather Prediction Center
4405:. Planned missions include the
1681:point from the smaller object,
1409:with Earth, and Saturn's moons
937:. Earlier examples include the
7451:Health threat from cosmic rays
6888:Retrograde and prograde motion
6074:Explanation of Lagrange points
6062:Explanation of Lagrange points
5902:Hornig, Andreas (2013-10-06).
5883:Hornig, Andreas (2022-05-01).
5540:"Stability of Lagrange Points"
4482:was a popular place to put a "
4322:in an orbit around SunâEarth L
3691:
3684:
3480:
3454:
3435:
3423:
3405:
3392:
3368:
3362:
2733:
2724:
2715:
2709:
2693:
2678:
2662:
2650:
2467:
2454:
2376:
2370:
2331:
2325:
1829:
1820:
1804:
1798:
1782:
1767:
1751:
1739:
1524:
1511:
1460:Alternatively, when seen in a
792:discovered the remaining two.
714:Deep Space Climate Observatory
1:
5864:. Vol. 11. p. 165.
4904:NASA Solar System Exploration
4559:Mission passed through the L
4403:Advanced Composition Explorer
747:, also occupy orbits around L
610:restricted three-body problem
6835:Equatorial coordinate system
6120:NASA: Basics of Space Flight
5777:Jones, Andrew (2018-06-14).
5313:10.1016/0019-1035(80)90106-2
74:). Counterintuitively, the L
7567:Self-replicating spacecraft
7403:International Space Station
6104:The Five Points of Lagrange
4962:. p. 9. Archived from
4737:Lagrange point colonization
4446:cosmic microwave background
2034:We may also write this as:
459:Tsiolkovsky rocket equation
7801:
7081:Space Liability Convention
6587:Longitude of the periapsis
6095:âEp. 76: "Lagrange Points"
6058:page, with good animations
4708:Euler's three-body problem
4597:are the locations for the
4450:James Webb Space Telescope
4339:James Webb Space Telescope
4302:
3259:
1112:
1089:. Any object orbiting at L
935:James Webb Space Telescope
814:
801:constant-pattern solutions
722:James Webb Space Telescope
428:Engineering and efficiency
247:Bi-elliptic transfer orbit
29:
7737:
7444:Space adaptation syndrome
6911:
6898:Specific angular momentum
6052:What are Lagrange points?
5738:10.1007/s10569-010-9299-x
5648:Cacolici, Gianna Nicole,
5628:10.1007/s10569-009-9203-8
5470:The Astrophysical Journal
5417:The Astrophysical Journal
5327:"List Of Neptune Trojans"
4956:; Ross, Shane D. (2006).
4845:Cornish, Neil J. (1998).
4235:
4176:
4117:
4074:
4028:
3985:
3942:
3896:
3850:
3841:
3834:
3827:
3820:
3813:
3806:
3803:
3800:
1385:has its apex located at L
1366:formed at the SunâEarth L
764:Earth for data transfer.
759:, while Euclid follows a
604:influence of two massive
7177:Communications satellite
5671:Solar System Exploration
5360:The Astronomical Journal
5161:10.3847/2041-8213/ac37bf
4703:Co-orbital configuration
4658:shield at the SunâMars L
4605:'s name comes from the L
4576:communications satellite
4444:and observations of the
4297:Spaceflight applications
3307:The radial acceleration
3237:Thus the distance from L
1919:is the mass fraction of
1677:is the distance of the L
1474:centrifugal pseudo-force
1462:rotating reference frame
1169:and referred to as the "
1031:The triangular points (L
716:(DSCOVR) is located at L
454:Propellant mass fraction
353:Gravitational influences
30:Not to be confused with
7681:reusable launch systems
7298:Extravehicular activity
7209:Commercial use of space
7113:Militarisation of space
7086:Registration Convention
7002:Accidents and incidents
6893:Specific orbital energy
5764:2 November 2005 at the
5329:. Minor Planet Center.
4599:Kordylewski dust clouds
4513:every 20 months).
3832:/SMA â 1 (%)
3262:Trojan (celestial body)
2253:the earth and the sun.
1360:giant impact hypothesis
1327:. Another Saturn moon,
751:. Gaia keeps a tighter
325:Specific orbital energy
62:due to gravity and the
7729:Mission control center
7691:Non-rocket spacelaunch
7125:Billionaire space race
6305:Geostationary transfer
5940:"The Sentinel Mission"
5850:Kordylewski, Kazimierz
5673:. NASA. Archived from
5546:. University of Texas.
5538:Fitzpatrick, Richard.
5009:Lagrange, Joseph-Louis
4505:comes within 0.3
4379:coronal mass ejections
4345:
4326:
3732:
3612:
3594:
3526:
3229:
3127:
2819:
2752:
2610:
2427:
2394:
2218:
2099:
2026:
1953:
1913:
1851:
1667:
1438:
1000:
662:each act as the third
242:Hohmann transfer orbit
119:
87:
70:) and away from them (
47:
7579:Spacecraft propulsion
7029:European Space Agency
6878:Orbital state vectors
6820:Characteristic energy
6790:Trans-lunar injection
6578:Argument of periapsis
6255:Prograde / Retrograde
6216:Hyperbolic trajectory
6056:European Space Agency
5271:10.1093/mnras/sty2630
5220:10.1093/mnras/sty2049
5062:"The Lagrange Points"
4847:"The Lagrange Points"
4630:were planning to use
4456:on January 24, 2022.
4332:
4317:
3730:
3709:Coriolis acceleration
3607:
3592:
3527:
3260:Further information:
3230:
3128:
2820:
2753:
2611:
2413:
2395:
2219:
2100:
2027:
1954:
1914:
1852:
1668:
1428:
1401:Objects which are on
1013:equilateral triangles
995:
790:Joseph-Louis Lagrange
438:Preflight engineering
170:Argument of periapsis
93:
53:
45:
7785:Lagrangian mechanics
7204:Satellite navigation
6725:Bi-elliptic transfer
6245:Parabolic trajectory
5858:im System Erde-Mond"
5356:Gott III, J. Richard
4859:on September 7, 2015
4752:Lunar space elevator
4742:Lagrangian mechanics
4644:near-Earth asteroids
4616:geosynchronous orbit
4574:has been used for a
4533:near-Earth asteroids
3315:
3195:
3151:defined as for the L
2928:
2786:
2624:
2438:
2299:
2121:
2038:
1993:
1930:
1861:
1723:
1495:
1470:gravitational fields
1436:Click for animation.
1161:. Asteroids at the L
1087:unstable equilibrium
710:artificial satellite
668:equilateral triangle
494:Propulsive maneuvers
7780:Trojans (astronomy)
7589:Electric propulsion
7276:Life-support system
7160:Imagery and mapping
7120:Private spaceflight
6765:Low-energy transfer
6108:Neil deGrasse Tyson
6097:by Fraser Cain and
5870:1961AcA....11..165K
5730:2010CeMDA.108..215T
5620:2009CeMDA.103..365P
5492:2004ApJ...603..283S
5439:2007ApJ...660.1624S
5382:2005AJ....129.1724B
5305:1980Icar...42..442F
5262:2019MNRAS.482..762S
5211:2018MNRAS.480.5550S
5152:2021ApJ...922L..25H
4954:Marsden, Jerrold E.
3797:
3723:Solar System values
3303:Radial acceleration
1431:effective potential
1395:Roche lobe overflow
1358:One version of the
1347:. The moons wander
1232:interplanetary dust
941:and its successor,
547:celestial mechanics
471:Efficiency measures
374:Sphere of influence
343:Celestial mechanics
125:Part of a series on
60:effective potential
7542:Robotic spacecraft
7468:Space and survival
7323:Space colonization
7219:Space architecture
7071:Outer Space Treaty
6760:Inclination change
6408:Distant retrograde
6082:âDavid Peter Stern
6068:A NASA explanation
6030:2020-08-03 at the
6025:merlyn.demon.co.uk
6007:2019-06-23 at the
6002:merlyn.demon.co.uk
5600:Standish, E. Myles
5544:Newtonian Dynamics
5037:on 3 February 2014
5017:Ĺuvres de Lagrange
4900:"DSCOVR: In-Depth"
4879:Weisstein, Eric W.
4694:Spaceflight portal
4640:Sentinel telescope
4626:Scientists at the
4442:infrared astronomy
4346:
4327:
3839:(×10 m)
3825:(×10 m)
3811:(×10 m)
3795:
3733:
3613:
3595:
3522:
3297:three-body problem
3225:
3217:
3123:
2815:
2748:
2606:
2428:
2390:
2267:in the absence of
2214:
2095:
2022:
1949:
1909:
1847:
1663:
1443:three-body problem
1439:
1263:has four accepted
1236:Kordylewski clouds
1001:
873:more closely than
797:three-body problem
290:Dynamical friction
120:
88:
48:
7767:
7766:
7714:Flight controller
7491:
7490:
7249:Human spaceflight
7224:Space exploration
7150:Earth observation
6929:
6928:
6903:Two-line elements
6711:
6710:
6633:Eccentric anomaly
6475:
6474:
6342:Orbit of the Moon
6201:Highly elliptical
5596:Pitjeva, Elena V.
5576:on April 18, 2018
4948:Koon, Wang Sang;
4882:"Lagrange Points"
4764:Explanatory notes
4723:Klemperer rosette
4618:cover the Earth.
4399:Aditya-L1 Mission
4361:has orbited the L
4294:
4293:
3668:than to stay at L
3520:
3415:
3354:
3216:
3121:
3062:
3015:
2969:
2920:The location of L
2813:
2807:
2604:
2551:
2504:
2477:
2430:The location of L
2385:
2384:
2202:
2154:
2093:
2063:
2020:
2014:
1947:
1907:
1714:of the following
1661:
1608:
1561:
1534:
1487:The location of L
1464:that matches the
1173:". Those at the L
1085:are positions of
625:orbit corrections
621:centrifugal force
590:Lagrangian points
543:
542:
393:Lagrangian points
330:Vis-viva equation
300:Kepler's equation
147:Orbital mechanics
64:centrifugal force
16:(Redirected from
7792:
7757:
7746:
7745:
7456:Space psychology
7281:Animals in space
7254:
7234:Space technology
7076:Rescue Agreement
6956:
6949:
6942:
6933:
6919:
6918:
6860:Lagrangian point
6755:Hohmann transfer
6700:
6686:
6677:
6668:
6648:
6639:
6630:
6621:
6617:
6613:
6604:
6584:
6575:
6566:
6557:
6537:
6533:
6524:
6515:
6506:
6486:
6455:Heliosynchronous
6404:Lagrange points
6357:Transatmospheric
6172:
6151:
6144:
6137:
6128:
6064:âNeil J. Cornish
5974:
5973:
5962:
5956:
5955:
5953:
5951:
5936:
5930:
5929:
5928:. June 20, 2017.
5918:
5912:
5911:
5899:
5893:
5892:
5880:
5874:
5873:
5846:
5840:
5839:
5820:
5819:
5813:
5802:
5793:
5787:
5786:
5774:
5768:
5756:
5750:
5749:
5715:
5702:
5696:
5693:
5687:
5686:
5684:
5682:
5677:on July 20, 2015
5663:
5657:
5646:
5640:
5639:
5592:
5586:
5585:
5583:
5581:
5575:
5569:. Archived from
5568:
5559:
5548:
5547:
5535:
5529:
5528:
5526:
5518:
5512:
5511:
5485:
5483:astro-ph/0311272
5465:
5459:
5458:
5432:
5430:astro-ph/0612508
5423:(2): 1624â1635.
5408:
5402:
5401:
5375:
5373:astro-ph/0405372
5366:(3): 1724â1745.
5352:Belbruno, Edward
5348:
5342:
5341:
5339:
5338:
5323:
5317:
5316:
5282:
5276:
5275:
5273:
5255:
5231:
5225:
5224:
5222:
5204:
5195:(4): 5550â5559.
5180:
5174:
5173:
5163:
5145:
5122:Tholen, David J.
5118:Wiegert, Paul A.
5113:
5107:
5106:
5095:
5089:
5088:
5076:
5070:
5068:
5066:
5058:
5047:
5046:
5044:
5042:
5027:
5021:
5020:
5005:
4999:
4998:
4996:
4982:
4976:
4974:
4972:
4971:
4945:
4939:
4938:
4936:
4935:
4921:
4915:
4914:
4912:
4911:
4896:
4890:
4889:
4875:
4869:
4868:
4866:
4864:
4858:
4851:
4842:
4822:
4816:
4810:
4809:
4806:
4800:
4798:
4797:
4794:
4791:
4790:
4789:
4777:
4696:
4691:
4690:
4689:
4682:
4680:Astronomy portal
4677:
4676:
4675:
4584:propellant depot
4367:solar telescopes
4290:
4289:
4282:
4280:
4273:
4268:
4266:
4259:
4254:
4252:
4245:
4243:
4231:
4230:
4223:
4221:
4214:
4209:
4207:
4200:
4195:
4193:
4186:
4184:
4172:
4171:
4164:
4162:
4155:
4150:
4148:
4141:
4136:
4134:
4127:
4125:
4113:
4112:
4105:
4100:
4095:
4090:
4085:
4080:
4070:
4069:
4066:
4059:
4054:
4049:
4044:
4039:
4034:
4024:
4023:
4016:
4011:
4006:
4001:
3996:
3991:
3981:
3980:
3973:
3968:
3963:
3958:
3953:
3948:
3938:
3937:
3934:
3927:
3922:
3917:
3912:
3907:
3902:
3892:
3887:
3886:
3879:
3874:
3869:
3864:
3863:
3856:
3798:
3791:
3789:
3779:
3777:
3767:
3765:
3651:Lissajous orbits
3639:dynamical system
3606:
3531:
3529:
3528:
3523:
3521:
3519:
3518:
3509:
3508:
3507:
3498:
3497:
3479:
3478:
3466:
3465:
3453:
3452:
3442:
3416:
3414:
3413:
3412:
3390:
3389:
3388:
3375:
3355:
3353:
3352:
3343:
3342:
3341:
3328:
3234:
3232:
3231:
3226:
3218:
3209:
3133:with parameters
3132:
3130:
3129:
3124:
3122:
3120:
3119:
3110:
3109:
3108:
3096:
3095:
3085:
3083:
3079:
3063:
3061:
3060:
3059:
3047:
3046:
3036:
3035:
3026:
3016:
3014:
3013:
3008:
3004:
2985:
2984:
2975:
2970:
2968:
2967:
2962:
2958:
2942:
2941:
2932:
2884:
2882:
2881:
2880:
2878:
2872:
2869:
2862:
2860:
2859:
2858:
2856:
2850:
2847:
2846:
2844:
2824:
2822:
2821:
2816:
2814:
2812:
2800:
2799:
2757:
2755:
2754:
2749:
2708:
2707:
2677:
2676:
2649:
2648:
2636:
2635:
2615:
2613:
2612:
2607:
2605:
2603:
2602:
2593:
2592:
2591:
2579:
2578:
2568:
2566:
2562:
2552:
2550:
2549:
2548:
2536:
2535:
2525:
2524:
2515:
2505:
2503:
2502:
2493:
2492:
2483:
2478:
2476:
2475:
2474:
2452:
2451:
2442:
2414:The Lagrangian L
2399:
2397:
2396:
2391:
2386:
2380:
2379:
2369:
2368:
2367:
2366:
2354:
2353:
2338:
2324:
2323:
2322:
2321:
2294:
2293:
2223:
2221:
2220:
2215:
2213:
2212:
2207:
2203:
2198:
2197:
2188:
2181:
2180:
2165:
2164:
2159:
2155:
2150:
2149:
2140:
2133:
2132:
2104:
2102:
2101:
2096:
2094:
2092:
2091:
2082:
2081:
2072:
2064:
2062:
2061:
2052:
2051:
2042:
2031:
2029:
2028:
2023:
2021:
2019:
2007:
2006:
1958:
1956:
1955:
1950:
1948:
1940:
1918:
1916:
1915:
1910:
1908:
1906:
1905:
1904:
1892:
1891:
1881:
1880:
1871:
1856:
1854:
1853:
1848:
1816:
1815:
1794:
1793:
1763:
1762:
1735:
1734:
1716:quintic function
1672:
1670:
1669:
1664:
1662:
1660:
1659:
1650:
1649:
1648:
1636:
1635:
1625:
1623:
1619:
1609:
1607:
1606:
1605:
1593:
1592:
1582:
1581:
1572:
1562:
1560:
1559:
1550:
1549:
1540:
1535:
1533:
1532:
1531:
1509:
1508:
1499:
1466:angular velocity
1403:horseshoe orbits
1303:
1301:
1300:
1292:
1290:
1289:
1281:
1279:
1278:
1222:The EarthâMoon L
1218:
1216:
1215:
1207:
1205:
1204:
1064:
1062:
1061:
1053:
1050:
683:trojan asteroids
596:) are points of
594:libration points
587:
586:
583:
582:
579:
576:
573:
570:
567:
564:
561:
535:
528:
521:
500:Orbital maneuver
449:Payload fraction
429:
410:Lissajous orbits
344:
315:Orbital velocity
262:Hyperbolic orbit
158:Orbital elements
148:
139:
122:
118:
113:
104:
99:
73:
69:
21:
7800:
7799:
7795:
7794:
7793:
7791:
7790:
7789:
7770:
7769:
7768:
7763:
7733:
7700:
7672:Escape velocity
7653:
7605:
7572:Space telescope
7525:Reentry capsule
7487:
7427:
7332:
7303:Overview effect
7286:Bioastronautics
7243:
7131:
6965:
6960:
6930:
6925:
6907:
6825:Escape velocity
6806:
6799:
6780:Rocket equation
6707:
6699:
6693:
6684:
6675:
6666:
6655:
6646:
6637:
6628:
6619:
6615:
6611:
6602:
6591:
6582:
6573:
6564:
6555:
6544:
6535:
6531:
6527:Semi-minor axis
6522:
6518:Semi-major axis
6513:
6504:
6498:
6471:
6393:Areosynchronous
6377:
6371:
6352:Sun-synchronous
6337:Near-equatorial
6281:
6161:
6155:
6032:Wayback Machine
6009:Wayback Machine
5996:Tome 6 (Viewer)
5983:
5978:
5977:
5964:
5963:
5959:
5949:
5947:
5946:on 30 June 2012
5938:
5937:
5933:
5920:
5919:
5915:
5901:
5900:
5896:
5882:
5881:
5877:
5857:
5848:
5847:
5843:
5835:
5831:
5826:
5817:
5815:
5811:
5800:
5795:
5794:
5790:
5776:
5775:
5771:
5766:Wayback Machine
5757:
5753:
5713:
5710:
5704:
5703:
5699:
5694:
5690:
5680:
5678:
5665:
5664:
5660:
5647:
5643:
5594:
5593:
5589:
5579:
5577:
5573:
5566:
5561:
5560:
5551:
5537:
5536:
5532:
5524:
5520:
5519:
5515:
5467:
5466:
5462:
5413:Kalogera, Vicky
5410:
5409:
5405:
5350:
5349:
5345:
5336:
5334:
5325:
5324:
5320:
5284:
5283:
5279:
5233:
5232:
5228:
5182:
5181:
5177:
5115:
5114:
5110:
5097:
5096:
5092:
5078:
5077:
5073:
5064:
5060:
5059:
5050:
5040:
5038:
5029:
5028:
5024:
5007:
5006:
5002:
4994:
4986:Euler, Leonhard
4984:
4983:
4979:
4969:
4967:
4947:
4946:
4942:
4933:
4931:
4923:
4922:
4918:
4909:
4907:
4898:
4897:
4893:
4877:
4876:
4872:
4862:
4860:
4856:
4849:
4844:
4843:
4836:
4831:
4826:
4825:
4812:
4807:
4804:
4802:
4795:
4792:
4787:
4785:
4783:
4782:
4780:
4778:
4771:
4766:
4761:
4731:
4692:
4687:
4685:
4678:
4673:
4671:
4668:
4661:
4656:magnetic dipole
4652:
4637:
4628:B612 Foundation
4624:
4612:
4608:
4596:
4592:
4573:
4562:
4550:
4545:
4538:
4530:
4519:
4512:
4500:
4491:science fiction
4481:
4466:
4462:
4455:
4439:
4435:
4431:
4423:
4419:
4409:(IMAP) and the
4388:
4384:
4373:(including the
4364:
4351:
4344:
4325:
4312:
4307:
4299:
4287:
4285:
4278:
4276:
4271:
4264:
4262:
4257:
4250:
4248:
4241:
4239:
4228:
4226:
4219:
4217:
4212:
4205:
4203:
4198:
4191:
4189:
4182:
4180:
4169:
4167:
4160:
4158:
4153:
4146:
4144:
4139:
4132:
4130:
4123:
4121:
4110:
4108:
4103:
4098:
4093:
4088:
4083:
4078:
4067:
4064:
4062:
4057:
4052:
4047:
4042:
4037:
4032:
4021:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3978:
3976:
3971:
3966:
3961:
3956:
3951:
3946:
3935:
3932:
3930:
3925:
3920:
3915:
3910:
3905:
3900:
3890:
3884:
3882:
3877:
3872:
3867:
3861:
3859:
3854:
3845:
3838:
3831:
3824:
3817:
3814:1 â L
3810:
3787:
3785:
3783:
3775:
3773:
3771:
3763:
3761:
3759:
3755:
3746:
3742:
3738:
3725:
3718:
3714:
3705:
3701:
3695:
3688:
3679:
3671:
3667:
3663:
3660:For SunâEarth-L
3655:station keeping
3647:Lissajous-curve
3626:
3622:
3618:
3602:
3600:
3584:
3580:
3576:
3572:
3565:
3542:
3510:
3489:
3470:
3457:
3443:
3404:
3391:
3380:
3376:
3344:
3333:
3329:
3313:
3312:
3305:
3273:
3269:
3264:
3258:
3256:
3252:
3244:
3240:
3193:
3192:
3188:
3181:
3166:
3158:
3154:
3146:
3139:
3111:
3100:
3087:
3086:
3051:
3038:
3037:
3027:
3024:
3020:
2991:
2987:
2986:
2976:
2948:
2944:
2943:
2933:
2926:
2925:
2923:
2918:
2916:
2908:
2899:Lissajous orbit
2891:annular eclipse
2888:
2876:
2874:
2873:
2870:
2867:
2866:
2864:
2854:
2852:
2851:
2848:
2842:
2840:
2839:
2838:
2836:
2834:
2830:
2784:
2783:
2777:
2773:
2766:
2699:
2668:
2640:
2627:
2622:
2621:
2619:
2594:
2583:
2570:
2569:
2540:
2527:
2526:
2516:
2513:
2509:
2494:
2484:
2466:
2453:
2443:
2436:
2435:
2433:
2417:
2408:
2406:
2358:
2345:
2340:
2339:
2313:
2302:
2297:
2296:
2291:
2289:
2287:
2280:
2273:
2266:
2251:
2244:
2237:
2230:
2189:
2183:
2182:
2172:
2141:
2135:
2134:
2124:
2119:
2118:
2116:
2112:
2083:
2073:
2053:
2043:
2036:
2035:
1991:
1990:
1980:
1976:
1972:
1965:
1928:
1927:
1924:
1896:
1883:
1882:
1872:
1859:
1858:
1807:
1785:
1754:
1726:
1721:
1720:
1702:
1698:
1691:
1680:
1651:
1640:
1627:
1626:
1597:
1584:
1583:
1573:
1570:
1566:
1551:
1541:
1523:
1510:
1500:
1493:
1492:
1490:
1485:
1483:
1455:circular motion
1434:
1423:
1388:
1373:
1369:
1354:
1346:
1338:
1318:
1314:
1298:
1297:
1294:
1287:
1286:
1283:
1276:
1275:
1272:
1255:Neptune trojans
1252:
1248:
1229:
1225:
1213:
1212:
1209:
1202:
1201:
1198:
1196:
1192:
1189:The SunâEarth L
1176:
1164:
1157:set during the
1142:
1138:
1126:
1122:
1117:
1111:
1103:station keeping
1100:
1096:
1092:
1084:
1080:
1076:
1060:
1054:
1051:
1049:
1043:
1042:
1040:
1038:
1034:
1029:
1022:
1018:
1010:
1006:
999:
990:
987:
983:
975:
966:
962:
957:
954:
932:
924:
920:
916:
912:
908:
903:
900:
888:
884:
879:Earth's gravity
864:
857:
850:
843:
836:
831:
828:
819:
813:
811:Lagrange points
805:circular orbits
783:
779:
775:
770:
758:
753:Lissajous orbit
750:
735:
731:
727:
719:
707:
703:
692:
688:
680:
676:
661:
657:
653:
649:
645:
637:
633:
558:
554:
551:Lagrange points
539:
510:
509:
505:Orbit insertion
495:
487:
486:
472:
464:
463:
439:
431:
427:
420:
419:
415:Lyapunov orbits
406:
405:
389:
379:
378:
354:
346:
342:
335:
334:
320:Surface gravity
295:Escape velocity
285:
277:
276:
257:Parabolic orbit
253:
252:
219:
217:
214:two-body orbits
205:
204:
195:Semi-major axis
160:
150:
146:
111:
110:
97:
96:
95:
82:points are the
81:
77:
71:
67:
38:
28:
23:
22:
15:
12:
11:
5:
7798:
7796:
7788:
7787:
7782:
7772:
7771:
7765:
7764:
7762:
7761:
7750:
7738:
7735:
7734:
7732:
7731:
7726:
7721:
7719:Ground station
7716:
7710:
7708:
7706:Ground segment
7702:
7701:
7699:
7698:
7693:
7688:
7683:
7674:
7669:
7663:
7661:
7655:
7654:
7652:
7651:
7646:
7641:
7639:Interplanetary
7636:
7635:
7634:
7632:Geosynchronous
7629:
7619:
7613:
7611:
7607:
7606:
7604:
7603:
7602:
7601:
7599:Gravity assist
7596:
7591:
7586:
7576:
7575:
7574:
7569:
7564:
7559:
7554:
7549:
7539:
7534:
7533:
7532:
7530:Service module
7527:
7522:
7520:Orbital module
7512:
7507:
7505:Launch vehicle
7501:
7499:
7493:
7492:
7489:
7488:
7486:
7485:
7483:Space sexology
7480:
7475:
7473:Space medicine
7470:
7465:
7464:
7463:
7453:
7448:
7447:
7446:
7435:
7433:
7429:
7428:
7426:
7425:
7420:
7415:
7410:
7405:
7400:
7399:
7398:
7388:
7383:
7382:
7381:
7376:
7366:
7361:
7356:
7351:
7346:
7340:
7338:
7334:
7333:
7331:
7330:
7325:
7320:
7315:
7310:
7308:Weightlessness
7305:
7300:
7295:
7294:
7293:
7288:
7283:
7273:
7272:
7271:
7260:
7258:
7251:
7245:
7244:
7242:
7241:
7236:
7231:
7229:Space research
7226:
7221:
7216:
7211:
7206:
7201:
7200:
7199:
7194:
7189:
7184:
7174:
7173:
7172:
7167:
7165:Reconnaissance
7162:
7157:
7147:
7141:
7139:
7133:
7132:
7130:
7129:
7128:
7127:
7117:
7116:
7115:
7110:
7105:
7095:
7094:
7093:
7088:
7083:
7078:
7073:
7063:
7062:
7061:
7056:
7051:
7046:
7041:
7036:
7034:European Union
7031:
7026:
7021:
7011:
7006:
7005:
7004:
6999:
6994:
6989:
6979:
6973:
6971:
6967:
6966:
6961:
6959:
6958:
6951:
6944:
6936:
6927:
6926:
6924:
6923:
6921:List of orbits
6912:
6909:
6908:
6906:
6905:
6900:
6895:
6890:
6885:
6880:
6875:
6873:Orbit equation
6870:
6862:
6857:
6852:
6847:
6842:
6837:
6832:
6827:
6822:
6817:
6811:
6809:
6801:
6800:
6798:
6797:
6792:
6787:
6782:
6777:
6772:
6767:
6762:
6757:
6752:
6747:
6745:Gravity assist
6742:
6740:Delta-v budget
6737:
6732:
6727:
6721:
6719:
6713:
6712:
6709:
6708:
6706:
6705:
6697:
6691:
6682:
6673:
6671:Orbital period
6663:
6661:
6657:
6656:
6654:
6653:
6651:True longitude
6644:
6642:Mean longitude
6635:
6626:
6609:
6599:
6597:
6593:
6592:
6590:
6589:
6580:
6571:
6562:
6552:
6550:
6546:
6545:
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6542:
6529:
6520:
6511:
6501:
6499:
6497:
6496:
6493:
6489:
6483:
6477:
6476:
6473:
6472:
6470:
6469:
6468:
6467:
6459:
6458:
6457:
6452:
6447:
6446:
6445:
6432:
6427:
6426:
6425:
6420:
6415:
6410:
6402:
6401:
6400:
6398:Areostationary
6395:
6390:
6381:
6379:
6373:
6372:
6370:
6369:
6367:Very low Earth
6364:
6359:
6354:
6349:
6344:
6339:
6334:
6329:
6324:
6319:
6314:
6309:
6308:
6307:
6302:
6295:Geosynchronous
6291:
6289:
6283:
6282:
6280:
6279:
6277:Transfer orbit
6274:
6273:
6272:
6267:
6257:
6252:
6247:
6242:
6237:
6235:Lagrange point
6232:
6227:
6218:
6213:
6208:
6203:
6194:
6189:
6184:
6178:
6176:
6169:
6163:
6162:
6157:Gravitational
6156:
6154:
6153:
6146:
6139:
6131:
6125:
6124:
6115:
6110:
6101:
6093:Astronomy Cast
6089:
6083:
6077:
6071:
6065:
6059:
6041:
6040:
6035:
6021:Leonhard Euler
6014:
6013:
6012:
5982:
5981:External links
5979:
5976:
5975:
5957:
5931:
5913:
5894:
5875:
5855:
5841:
5833:
5829:
5824:
5788:
5769:
5751:
5724:(3): 215â232.
5708:
5697:
5688:
5658:
5641:
5614:(4): 365â372.
5602:(2009-04-01).
5587:
5549:
5530:
5513:
5500:10.1086/381315
5476:(1): 283â284.
5460:
5447:10.1086/513736
5403:
5390:10.1086/427539
5343:
5318:
5299:(3): 442â447.
5277:
5246:(1): 762â770.
5226:
5175:
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5090:
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5048:
5022:
5000:
4977:
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4382:
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4349:
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4318:The satellite
4311:
4308:
4303:Main article:
4298:
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4260:
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3620:
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3615:Although the L
3599:
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3407:
3403:
3400:
3397:
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3387:
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3379:
3373:
3370:
3367:
3364:
3361:
3358:
3351:
3347:
3340:
3336:
3332:
3326:
3323:
3320:
3304:
3301:
3289:center of mass
3271:
3267:
3257:
3254:
3250:
3247:
3242:
3238:
3224:
3221:
3215:
3212:
3206:
3203:
3200:
3186:
3179:
3164:
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3078:
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3023:
3019:
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3007:
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2997:
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2957:
2954:
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2940:
2936:
2921:
2917:
2914:
2911:
2906:
2897:, to follow a
2886:
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2811:
2806:
2803:
2797:
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2775:
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2647:
2643:
2639:
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2630:
2617:
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2597:
2590:
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2577:
2573:
2565:
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2539:
2534:
2530:
2523:
2519:
2512:
2508:
2501:
2497:
2491:
2487:
2481:
2473:
2469:
2465:
2462:
2459:
2456:
2450:
2446:
2431:
2418:point for the
2415:
2407:
2404:
2401:
2389:
2383:
2378:
2375:
2372:
2365:
2361:
2357:
2352:
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2333:
2330:
2327:
2320:
2316:
2312:
2309:
2305:
2285:
2278:
2271:
2264:
2258:orbital period
2249:
2242:
2235:
2228:
2211:
2206:
2201:
2196:
2192:
2186:
2179:
2175:
2171:
2168:
2163:
2158:
2153:
2148:
2144:
2138:
2131:
2127:
2114:
2110:
2090:
2086:
2080:
2076:
2070:
2067:
2060:
2056:
2050:
2046:
2018:
2013:
2010:
2004:
2001:
1998:
1978:
1974:
1970:
1963:
1946:
1943:
1938:
1935:
1922:
1903:
1899:
1895:
1890:
1886:
1879:
1875:
1869:
1866:
1846:
1843:
1840:
1837:
1834:
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1828:
1825:
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1800:
1797:
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1788:
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1778:
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1772:
1769:
1766:
1761:
1757:
1753:
1750:
1747:
1744:
1741:
1738:
1733:
1729:
1700:
1696:
1689:
1678:
1658:
1654:
1647:
1643:
1639:
1634:
1630:
1622:
1618:
1615:
1612:
1604:
1600:
1596:
1591:
1587:
1580:
1576:
1569:
1565:
1558:
1554:
1548:
1544:
1538:
1530:
1526:
1522:
1519:
1516:
1513:
1507:
1503:
1488:
1484:
1481:
1478:
1422:
1419:
1399:
1398:
1391:companion star
1386:
1375:
1371:
1367:
1356:
1352:
1344:
1336:
1316:
1312:
1307:Saturn's moon
1305:
1258:
1250:
1246:
1239:
1227:
1223:
1220:
1194:
1190:
1174:
1162:
1140:
1136:
1124:
1120:
1113:Main article:
1110:
1107:
1098:
1094:
1090:
1082:
1078:
1074:
1058:
1047:
1036:
1032:
1028:
1025:
1020:
1019:) or behind (L
1016:
1008:
1004:
997:
989:
985:
981:
978:
973:
964:
960:
956:
952:
949:
930:
922:
918:
914:
910:
906:
902:
898:
895:
886:
882:
862:
855:
848:
841:
834:
830:
826:
823:
812:
809:
786:Leonhard Euler
781:
777:
773:
769:
766:
756:
748:
733:
729:
725:
717:
705:
701:
690:
686:
678:
674:
659:
655:
651:
647:
643:
635:
631:
541:
540:
538:
537:
530:
523:
515:
512:
511:
508:
507:
502:
496:
493:
492:
489:
488:
485:
484:
479:
477:Gravity assist
473:
470:
469:
466:
465:
462:
461:
456:
451:
446:
440:
437:
436:
433:
432:
425:
422:
421:
418:
417:
412:
404:
403:
395:
391:
390:
385:
384:
381:
380:
377:
376:
371:
366:
361:
355:
352:
351:
348:
347:
340:
337:
336:
333:
332:
327:
322:
317:
312:
310:Orbital period
307:
302:
297:
292:
286:
283:
282:
279:
278:
275:
274:
272:Decaying orbit
269:
264:
259:
251:
250:
244:
237:
235:Transfer orbit
233:
232:
231:
229:Elliptic orbit
226:
224:Circular orbit
220:
211:
210:
207:
206:
203:
202:
197:
192:
187:
182:
177:
172:
167:
161:
156:
155:
152:
151:
144:
141:
140:
132:
131:
127:
126:
79:
75:
33:Lagrange Point
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7797:
7786:
7783:
7781:
7778:
7777:
7775:
7760:
7756:
7751:
7749:
7740:
7739:
7736:
7730:
7727:
7725:
7722:
7720:
7717:
7715:
7712:
7711:
7709:
7707:
7703:
7697:
7694:
7692:
7689:
7687:
7684:
7682:
7678:
7675:
7673:
7670:
7668:
7667:Direct ascent
7665:
7664:
7662:
7660:
7656:
7650:
7649:Intergalactic
7647:
7645:
7642:
7640:
7637:
7633:
7630:
7628:
7625:
7624:
7623:
7620:
7618:
7615:
7614:
7612:
7608:
7600:
7597:
7595:
7592:
7590:
7587:
7585:
7584:Rocket engine
7582:
7581:
7580:
7577:
7573:
7570:
7568:
7565:
7563:
7560:
7558:
7555:
7553:
7550:
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7545:
7544:
7543:
7540:
7538:
7535:
7531:
7528:
7526:
7523:
7521:
7518:
7517:
7516:
7515:Space capsule
7513:
7511:
7508:
7506:
7503:
7502:
7500:
7498:
7494:
7484:
7481:
7479:
7478:Space nursing
7476:
7474:
7471:
7469:
7466:
7462:
7459:
7458:
7457:
7454:
7452:
7449:
7445:
7442:
7441:
7440:
7437:
7436:
7434:
7432:Health issues
7430:
7424:
7421:
7419:
7416:
7414:
7411:
7409:
7406:
7404:
7401:
7397:
7394:
7393:
7392:
7389:
7387:
7386:Space Shuttle
7384:
7380:
7377:
7375:
7372:
7371:
7370:
7367:
7365:
7362:
7360:
7357:
7355:
7352:
7350:
7347:
7345:
7342:
7341:
7339:
7335:
7329:
7326:
7324:
7321:
7319:
7318:Space tourism
7316:
7314:
7311:
7309:
7306:
7304:
7301:
7299:
7296:
7292:
7289:
7287:
7284:
7282:
7279:
7278:
7277:
7274:
7270:
7267:
7266:
7265:
7262:
7261:
7259:
7255:
7252:
7250:
7246:
7240:
7239:Space weather
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7215:
7212:
7210:
7207:
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7140:
7138:
7134:
7126:
7123:
7122:
7121:
7118:
7114:
7111:
7109:
7106:
7104:
7103:Space command
7101:
7100:
7099:
7098:Space warfare
7096:
7092:
7089:
7087:
7084:
7082:
7079:
7077:
7074:
7072:
7069:
7068:
7067:
7064:
7060:
7059:United States
7057:
7055:
7052:
7050:
7047:
7045:
7042:
7040:
7037:
7035:
7032:
7030:
7027:
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7020:
7017:
7016:
7015:
7012:
7010:
7007:
7003:
7000:
6998:
6995:
6993:
6990:
6988:
6985:
6984:
6983:
6980:
6978:
6977:Astrodynamics
6975:
6974:
6972:
6968:
6964:
6957:
6952:
6950:
6945:
6943:
6938:
6937:
6934:
6922:
6914:
6913:
6910:
6904:
6901:
6899:
6896:
6894:
6891:
6889:
6886:
6884:
6881:
6879:
6876:
6874:
6871:
6869:
6868:-body problem
6867:
6863:
6861:
6858:
6856:
6853:
6851:
6848:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6826:
6823:
6821:
6818:
6816:
6813:
6812:
6810:
6808:
6802:
6796:
6793:
6791:
6788:
6786:
6783:
6781:
6778:
6776:
6773:
6771:
6770:Oberth effect
6768:
6766:
6763:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6722:
6720:
6718:
6714:
6704:
6696:
6692:
6690:
6689:Orbital speed
6683:
6681:
6674:
6672:
6665:
6664:
6662:
6658:
6652:
6645:
6643:
6636:
6634:
6627:
6625:
6610:
6608:
6601:
6600:
6598:
6594:
6588:
6581:
6579:
6572:
6570:
6563:
6561:
6554:
6553:
6551:
6547:
6541:
6530:
6528:
6521:
6519:
6512:
6510:
6503:
6502:
6500:
6494:
6491:
6490:
6487:
6484:
6482:
6478:
6466:
6463:
6462:
6460:
6456:
6453:
6451:
6448:
6444:
6443:Earth's orbit
6441:
6440:
6439:
6436:
6435:
6433:
6431:
6428:
6424:
6421:
6419:
6416:
6414:
6411:
6409:
6406:
6405:
6403:
6399:
6396:
6394:
6391:
6389:
6386:
6385:
6383:
6382:
6380:
6374:
6368:
6365:
6363:
6360:
6358:
6355:
6353:
6350:
6348:
6345:
6343:
6340:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6318:
6315:
6313:
6310:
6306:
6303:
6301:
6300:Geostationary
6298:
6297:
6296:
6293:
6292:
6290:
6288:
6284:
6278:
6275:
6271:
6268:
6266:
6263:
6262:
6261:
6258:
6256:
6253:
6251:
6248:
6246:
6243:
6241:
6238:
6236:
6233:
6231:
6228:
6226:
6222:
6219:
6217:
6214:
6212:
6209:
6207:
6204:
6202:
6198:
6195:
6193:
6190:
6188:
6185:
6183:
6180:
6179:
6177:
6173:
6170:
6168:
6164:
6160:
6152:
6147:
6145:
6140:
6138:
6133:
6132:
6129:
6123:
6121:
6116:
6114:
6111:
6109:
6105:
6102:
6100:
6099:Pamela L. Gay
6096:
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5999:
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5993:
5989:
5985:
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5971:
5967:
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5958:
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5935:
5932:
5927:
5926:SpaceNews.com
5923:
5917:
5914:
5909:
5905:
5898:
5895:
5890:
5886:
5879:
5876:
5871:
5867:
5863:
5859:
5851:
5845:
5842:
5838:
5814:on 2014-06-24
5810:
5806:
5799:
5792:
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5760:
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5609:
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5597:
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5523:
5517:
5514:
5509:
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5497:
5493:
5489:
5484:
5479:
5475:
5471:
5464:
5461:
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5452:
5448:
5444:
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5426:
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5418:
5414:
5407:
5404:
5399:
5395:
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5387:
5383:
5379:
5374:
5369:
5365:
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5357:
5353:
5347:
5344:
5332:
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5306:
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5294:
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5288:
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5272:
5267:
5263:
5259:
5254:
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5241:
5237:
5230:
5227:
5221:
5216:
5212:
5208:
5203:
5198:
5194:
5190:
5186:
5179:
5176:
5171:
5167:
5162:
5157:
5153:
5149:
5144:
5139:
5135:
5131:
5127:
5123:
5119:
5116:Hui, Man-To;
5112:
5109:
5104:
5100:
5094:
5091:
5086:
5082:
5075:
5072:
5067:. NASA. 1998.
5063:
5057:
5055:
5053:
5049:
5036:
5032:
5026:
5023:
5018:
5014:
5010:
5004:
5001:
4993:
4992:
4987:
4981:
4978:
4966:on 2008-05-27
4965:
4961:
4960:
4955:
4951:
4950:Lo, Martin W.
4944:
4941:
4930:
4926:
4925:"About Orbit"
4920:
4917:
4905:
4901:
4895:
4892:
4887:
4883:
4880:
4874:
4871:
4855:
4848:
4841:
4839:
4835:
4828:
4820:
4815:
4776:
4774:
4770:
4763:
4758:
4757:Oberth effect
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4738:
4735:
4733:
4726:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4706:
4704:
4701:
4700:
4695:
4684:
4681:
4670:
4665:
4663:
4657:
4649:
4647:
4645:
4641:
4633:
4629:
4621:
4619:
4617:
4604:
4600:
4587:
4585:
4581:
4577:
4568:
4566:
4565:gravitational
4558:
4554:
4553:space station
4542:
4540:
4534:
4526:
4523:
4514:
4508:
4504:
4496:
4492:
4489:
4485:
4484:Counter-Earth
4476:
4474:
4473:time constant
4470:
4469:saddle points
4457:
4451:
4447:
4443:
4427:
4414:
4412:
4408:
4404:
4400:
4396:
4392:
4380:
4376:
4372:
4371:space weather
4368:
4360:
4356:
4340:
4337:(yellow) and
4336:
4331:
4321:
4316:
4309:
4306:
4301:
4296:
4284:
4275:
4270:
4261:
4256:
4247:
4238:
4234:
4225:
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4211:
4202:
4197:
4188:
4179:
4175:
4166:
4157:
4152:
4143:
4138:
4129:
4120:
4116:
4107:
4102:
4097:
4092:
4087:
4082:
4077:
4073:
4061:
4056:
4051:
4046:
4041:
4036:
4031:
4027:
4018:
4013:
4008:
4003:
3998:
3993:
3988:
3984:
3975:
3970:
3965:
3960:
3955:
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3945:
3941:
3929:
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3899:
3895:
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3881:
3876:
3871:
3866:
3858:
3853:
3849:
3799:
3793:
3751:
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3722:
3720:
3710:
3696:
3689:
3681:
3675:
3658:
3656:
3652:
3648:
3644:
3640:
3636:
3632:
3631:
3610:
3605:
3597:
3591:
3587:
3569:
3566:; force from
3562:
3558:
3554:
3553:sign function
3550:
3546:
3539:
3535:
3515:
3511:
3499:
3494:
3490:
3486:
3483:
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3359:
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3349:
3345:
3338:
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3324:
3321:
3318:
3310:
3302:
3300:
3298:
3294:
3290:
3286:
3282:
3278:
3263:
3248:
3246:
3235:
3222:
3219:
3213:
3210:
3204:
3201:
3198:
3190:
3185:
3178:
3174:
3171: â
3170:
3162:
3150:
3143:
3136:
3116:
3112:
3105:
3101:
3097:
3092:
3088:
3080:
3076:
3073:
3070:
3067:
3064:
3056:
3052:
3048:
3043:
3039:
3032:
3028:
3021:
3017:
3010:
3005:
3001:
2998:
2995:
2992:
2988:
2981:
2977:
2971:
2964:
2959:
2955:
2952:
2949:
2945:
2938:
2934:
2912:
2910:
2904:
2900:
2896:
2892:
2825:
2809:
2804:
2801:
2795:
2792:
2789:
2781:
2770:
2763:
2758:
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2742:
2739:
2736:
2730:
2727:
2721:
2718:
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2704:
2700:
2696:
2690:
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2684:
2681:
2673:
2669:
2665:
2659:
2656:
2653:
2645:
2641:
2637:
2632:
2628:
2599:
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2588:
2584:
2580:
2575:
2571:
2563:
2559:
2556:
2553:
2545:
2541:
2537:
2532:
2528:
2521:
2517:
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2499:
2495:
2489:
2485:
2479:
2471:
2463:
2460:
2457:
2448:
2444:
2425:
2421:
2412:
2402:
2400:
2387:
2381:
2373:
2363:
2359:
2355:
2350:
2346:
2341:
2334:
2328:
2318:
2314:
2310:
2307:
2303:
2288:, divided by
2284:
2277:
2274:, is that of
2270:
2263:
2259:
2254:
2248:
2241:
2234:
2227:
2209:
2204:
2199:
2194:
2190:
2184:
2177:
2173:
2169:
2166:
2161:
2156:
2151:
2146:
2142:
2136:
2129:
2125:
2108:
2088:
2084:
2078:
2074:
2068:
2065:
2058:
2054:
2048:
2044:
2032:
2016:
2011:
2008:
2002:
1999:
1996:
1988:
1984:
1969:
1962:
1944:
1941:
1936:
1933:
1925:
1901:
1897:
1893:
1888:
1884:
1877:
1873:
1867:
1864:
1844:
1841:
1838:
1835:
1832:
1826:
1823:
1817:
1812:
1808:
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1795:
1790:
1786:
1779:
1776:
1773:
1770:
1764:
1759:
1755:
1748:
1745:
1742:
1736:
1731:
1727:
1718:
1717:
1713:
1710:
1706:
1695:
1688:
1684:
1676:
1656:
1652:
1645:
1641:
1637:
1632:
1628:
1620:
1616:
1613:
1610:
1602:
1598:
1594:
1589:
1585:
1578:
1574:
1567:
1563:
1556:
1552:
1546:
1542:
1536:
1528:
1520:
1517:
1514:
1505:
1501:
1479:
1477:
1475:
1471:
1467:
1463:
1458:
1456:
1452:
1448:
1444:
1437:
1432:
1427:
1420:
1418:
1416:
1412:
1408:
1407:3753 Cruithne
1404:
1396:
1392:
1384:
1380:
1376:
1365:
1361:
1357:
1350:
1342:
1334:
1330:
1326:
1322:
1310:
1306:
1302:
1291:
1280:
1270:
1266:
1262:
1259:
1256:
1244:
1240:
1237:
1233:
1221:
1217:
1206:
1188:
1187:
1186:
1182:
1180:
1172:
1168:
1160:
1156:
1152:
1151:
1146:
1134:
1130:
1116:
1108:
1106:
1104:
1088:
1071:
1069:
1057:
1046:
1026:
1024:
1014:
994:
979:
977:
971:
950:
948:
946:
945:
940:
936:
928:
896:
894:
892:
880:
876:
872:
868:
861:
854:
847:
840:
824:
822:
818:
810:
808:
806:
802:
798:
793:
791:
787:
767:
765:
762:
754:
746:
742:
737:
723:
715:
711:
698:
696:
684:
671:
669:
665:
641:
628:
626:
622:
618:
617:gravitational
613:
611:
607:
603:
602:gravitational
599:
595:
591:
585:
552:
548:
536:
531:
529:
524:
522:
517:
516:
514:
513:
506:
503:
501:
498:
497:
491:
490:
483:
482:Oberth effect
480:
478:
475:
474:
468:
467:
460:
457:
455:
452:
450:
447:
445:
442:
441:
435:
434:
430:
423:
416:
413:
411:
408:
407:
401:
397:
396:
394:
388:
387:N-body orbits
383:
382:
375:
372:
370:
369:Perturbations
367:
365:
362:
360:
357:
356:
350:
349:
345:
338:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
296:
293:
291:
288:
287:
281:
280:
273:
270:
268:
265:
263:
260:
258:
255:
254:
248:
245:
243:
239:
238:
236:
230:
227:
225:
222:
221:
215:
209:
208:
201:
198:
196:
193:
191:
190:Orbital nodes
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
162:
159:
154:
153:
149:
142:
138:
134:
133:
130:Astrodynamics
128:
124:
123:
117:
108:
103:
92:
85:
65:
61:
57:
52:
44:
40:
36:
34:
19:
18:EarthâMoon L2
7659:Space launch
7644:Interstellar
7610:Destinations
7379:ApolloâSoyuz
7328:Space diving
7313:Space toilet
7137:Applications
7054:Soviet Union
7014:Space policy
7009:Space launch
6883:Perturbation
6865:
6859:
6840:Ground track
6750:Gravity turn
6701:
6694:
6687:
6678:
6669:
6649:
6640:
6631:
6624:True anomaly
6622:
6607:Mean anomaly
6605:
6585:
6576:
6567:
6558:
6538:
6525:
6516:
6509:Eccentricity
6507:
6465:Lunar cycler
6438:Heliocentric
6378:other points
6327:Medium Earth
6234:
6225:Non-inclined
6119:
6092:
6044:
6042:
5987:
5969:
5960:
5948:. Retrieved
5944:the original
5934:
5925:
5916:
5907:
5897:
5888:
5878:
5861:
5844:
5822:
5816:. Retrieved
5809:the original
5804:
5791:
5782:
5772:
5754:
5721:
5717:
5700:
5691:
5679:. Retrieved
5675:the original
5670:
5667:"ISEE-3/ICE"
5661:
5649:
5644:
5611:
5607:
5590:
5580:February 28,
5578:. Retrieved
5571:the original
5543:
5533:
5516:
5473:
5469:
5463:
5420:
5416:
5406:
5363:
5359:
5346:
5335:. Retrieved
5321:
5296:
5290:
5280:
5243:
5239:
5229:
5192:
5188:
5178:
5133:
5129:
5111:
5103:www.nasa.gov
5102:
5093:
5084:
5074:
5039:. Retrieved
5035:the original
5025:
5016:
5003:
4990:
4980:
4968:. Retrieved
4964:the original
4958:
4943:
4932:. Retrieved
4928:
4919:
4908:. Retrieved
4903:
4894:
4885:
4873:
4861:. Retrieved
4854:the original
4653:
4625:
4589:EarthâMoon L
4588:
4570:EarthâMoon L
4569:
4547:EarthâMoon L
4546:
4515:
4477:
4458:
4415:
4411:NEO Surveyor
4389:include the
4347:
4300:
4236:SunâNeptune
4075:SunâJupiter
3897:SunâMercury
3734:
3682:
3674:interference
3659:
3643:Solar System
3641:such as the
3634:
3629:
3614:
3609:STL 3D model
3567:
3560:
3556:
3548:
3544:
3537:
3533:
3308:
3306:
3265:
3236:
3191:
3183:
3176:
3172:
3168:
3160:
3148:
3141:
3134:
2919:
2889:one sees an
2826:
2782:, given by:
2768:
2761:
2759:
2429:
2282:
2275:
2268:
2261:
2255:
2246:
2239:
2232:
2225:
2033:
1989:, given by:
1982:
1967:
1960:
1920:
1719:
1707:is the only
1704:
1693:
1686:
1682:
1674:
1486:
1459:
1440:
1400:
1379:binary stars
1265:Mars trojans
1183:
1166:
1148:
1118:
1073:The points L
1072:
1055:
1044:
1030:
1002:
958:
942:
904:
859:
858:and that of
852:
845:
838:
832:
820:
794:
771:
738:
699:
685:near their L
672:
639:
629:
614:
593:
589:
550:
544:
392:
267:Radial orbit
218:eccentricity
200:True anomaly
185:Mean anomaly
175:Eccentricity
106:
56:contour plot
39:
35:(video game)
32:
7617:Sub-orbital
7552:Space probe
7418:New Shepard
7396:ShuttleâMir
7155:Archaeology
7108:Space force
7091:Moon Treaty
6963:Spaceflight
6845:Hill sphere
6680:Mean motion
6560:Inclination
6549:Orientation
6450:Mars cycler
6388:Areocentric
6260:Synchronous
6122:, Chapter 5
5992:Essai (PDF)
5011:(1867â92).
4713:Gegenschein
4567:influence.
4495:comic books
4478:SunâEarth L
4459:SunâEarth L
4416:SunâEarth L
4381:) reaches L
4348:SunâEarth L
4177:SunâUranus
4118:SunâSaturn
3851:EarthâMoon
3784:is located
3772:is located
3630:halo orbits
3293:equilibrium
3159:cases, and
2780:Hill sphere
2113:or at the L
1987:Hill sphere
1393:, known as
1349:azimuthally
1269:5261 Eureka
1234:, known as
1179:Trojan camp
1068:kidney bean
712:called the
598:equilibrium
400:Halo orbits
364:Hill sphere
180:Inclination
84:high points
7774:Categories
7686:Launch pad
7677:Expendable
7627:Geocentric
7594:Solar sail
7537:Spaceplane
7497:Spacecraft
7291:Space suit
7269:commercial
7197:Television
6992:Space Race
6785:Rendezvous
6481:Parameters
6317:High Earth
6287:Geocentric
6240:Osculating
6197:Elliptical
6088:âTony Dunn
6076:âJohn Baez
5950:1 February
5837:penalties.
5818:2011-01-25
5337:2010-10-27
5253:1910.07471
5202:1910.07466
5143:2111.05058
5136:(2): L25.
5031:"L2 Orbit"
4970:2008-06-09
4934:2022-01-01
4910:2021-10-27
4829:References
4811:(sequence
4603:L5 Society
4543:EarthâMoon
4375:solar wind
3986:SunâEarth
3943:SunâVenus
3801:Body pair
3750:barycenter
3277:barycenter
2903:halo orbit
2835:is arcsin(
2105:Since the
1447:barycenter
1411:Epimetheus
1383:Roche lobe
1341:Polydeuces
1339:point and
1171:Greek camp
1159:Trojan War
970:barycenter
815:See also:
761:halo orbit
444:Mass ratio
359:Barycenter
7696:Spaceport
7547:Satellite
7264:Astronaut
7192:Telephone
7145:Astronomy
7066:Space law
7019:Australia
6830:Ephemeris
6807:mechanics
6717:Maneuvers
6660:Variation
6423:Libration
6418:Lissajous
6322:Low Earth
6312:Graveyard
6211:Horseshoe
5994:; source
5783:SpaceNews
5746:121179935
5681:August 8,
5636:121374703
5170:243860678
5085:Space.com
5041:28 August
4779:Actually
4622:SunâVenus
4509:of this L
4310:SunâEarth
4029:SunâMars
3846:/SMA (%)
3818:/SMA (%)
3598:Stability
3551:) is the
3487:−
3430:−
3421:
3399:−
3360:
3325:−
3281:resultant
3220:μ
3202:≈
3189:), then:
3074:−
2999:−
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1155:epic poem
1027:Stability
640:different
284:Equations
212:Types of
7748:Category
7413:Tiangong
7408:Shenzhou
7337:Programs
7182:Internet
6987:Timeline
6596:Position
6221:Inclined
6192:Circular
6038:ZIP file
6028:Archived
6005:Archived
5970:phys.org
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5762:Archived
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5455:15519581
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7349:Mercury
7257:General
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6982:History
6970:General
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6775:Phasing
6735:Delta-v
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6534:,
6332:Molniya
6250:Parking
6187:Capture
6175:General
5908:IAC2013
5866:Bibcode
5726:Bibcode
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5616:Bibcode
5488:Bibcode
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5301:Bibcode
5258:Bibcode
5207:Bibcode
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4817:in the
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1243:Neptune
1211:2020 XL
1200:2010 TK
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1081:, and L
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768:History
695:Jupiter
650:, and L
588:; also
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7049:Russia
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6230:Kepler
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745:Euclid
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5812:(PDF)
5801:(PDF)
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5714:(PDF)
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5574:(PDF)
5567:(PDF)
5525:(PDF)
5504:S2CID
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5425:arXiv
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