3717:
80:
3579:
1415:
4319:
4486:, 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
961:, 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
32:
3700:(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
3593:
126:
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40:
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982:
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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
2241:
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
1054:
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
5816:
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.
4490:, 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,
2603:
1660:
3303:
5825:
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
3574:
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.
2211:
3686:
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
2916:
956:
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'
1173:
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:
3736:
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
3695:
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,
4429:, 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
1442:, 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
2426:
1483:
2745:
1844:
4602:
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
4341:
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
2387:
5225:"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"
2092:
3592:
3263:
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
902:. 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
2109:
965:
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.
5174:"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"
2812:
2019:
4651:
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.
1906:
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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
3222:
5786:
5552:
3284:
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
4509:
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
7449:
1946:
866:
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
6990:
3514:{\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}}}}
4413:
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
1116:, 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 '
3115:{\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}}}}
1090:
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
4510:
2612:
1711:
604:
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
870:
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
55:
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 (
5873:
2287:
4395:
2026:
3280:
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
5980:
5510:
5767:
4807:
4520:
would provide very important observations not only for Earth forecasts, but also for deep space support (Mars predictions and for crewed missions to
3653:
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
1424:
7427:
7017:
5797:
521:
4540:
allows comparatively easy access to Lunar and Earth orbits with minimal change in velocity and this has as an advantage to position a habitable
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7007:
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5655:
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670:
points are stable points, meaning that objects can orbit them and that they have a tendency to pull objects into them. Several planets have
5984:
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1159:
927:
90:
6843:
6050:
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showing a negative direction. The percentage columns show the distance from the orbit compared to the semimajor axis. E.g. for the Moon, L
2598:{\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}}}}
1655:{\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}}}}
1418:
Visualisation of the relationship between the
Lagrange points (red) of a planet (blue) orbiting a star (yellow) counterclockwise, and the
293:
79:
4835:
2098:
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
3265:
2774:
1981:
958:
362:
1849:
7027:
4735:
4379:
4293:
1103:
805:
3183:
7047:
5019:
3634:
does not contain these periodic orbits, but does contain quasi-periodic (i.e. bounded but not precisely repeating) orbits following
788:". In the first chapter he considered the general three-body problem. From that, in the second chapter, he demonstrated two special
5695:
1004:
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
6838:
6718:
6137:
5750:
4706:
3642:
are what most of
Lagrangian-point space missions have used until now. Although they are not perfectly stable, a modest effort of
709:
to study solar wind coming toward Earth from the Sun and to monitor Earth's climate, by taking images and sending them back. The
5642:
3781:
from Earth's center, which is 99.3% of the EarthâMoon distance or 0.7084% inside (Earthward) of the Moon's 'negative' position.
717:. This allows the satellite's large sunshield to protect the telescope from the light and heat of the Sun, Earth and Moon. The L
7202:
6803:
6557:
4513:
4343:
3662:
35:
Lagrange points in the SunâEarth system (not to scale). This view is from the north, so that Earth's orbit is counterclockwise.
7577:
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7022:
6876:
6515:
6506:
6243:
4347:
1394:
are sometimes erroneously described as trojans, but do not occupy
Lagrange points. Known objects on horseshoe orbits include
702:
183:
1055:(either gravity or angular momentum-induced speed) will also increase or decrease, bending the object's path into a stable,
1340:
about the Lagrange points, with Polydeuces describing the largest deviations, moving up to 32° away from the SaturnâDione L
7773:
5910:
4696:
4391:
4308:
6092:
5928:
1688:
are the masses of the large and small object, respectively. The quantity in parentheses on the right is the distance of L
7768:
7143:
6823:
6293:
6016:
5993:
5001:
4414:
952:
point lies on the line defined by the two large masses, beyond the larger of the two. Within the SunâEarth system, the L
5684:
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˛
7555:
7391:
7138:
6985:
6768:
5404:(May 2007). "Equipotential Surfaces and Lagrangian Points in Nonsynchronous, Eccentric Binary and Planetary Systems".
4725:
4434:
1918:
689:
Some Lagrange points are being used for space exploration. Two important Lagrange points in the Sun-Earth system are L
627:, all in the orbital plane of the two large bodies. There are five Lagrange points for the SunâEarth system, and five
514:
447:
20:
6026:
1120:' or 'trojan asteroids'. The name derives from the names that were given to asteroids discovered orbiting at the Sunâ
4464:
of roughly 23 days. Satellites at these points will wander off in a few months unless course corrections are made.
7627:
7396:
7069:
6748:
6575:
4628:
4438:
4327:
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from Earth's center, which is 84.9% of the EarthâMoon distance or 15.1% "in front of" (Earthwards from) the Moon; L
923:
710:
7550:
7432:
7170:
6935:
6886:
2206:{\displaystyle \rho _{2}\left({\frac {d_{2}}{r}}\right)^{3}\approx 3\rho _{1}\left({\frac {d_{1}}{R}}\right)^{3}}
1220:
5069:
4346:(ISEE-3) mission used as an interplanetary early warning storm monitor for solar disturbances. Since June 2015,
3716:
3578:
7665:
7165:
7153:
7125:
6871:
6396:
4691:
4564:
1450:
442:
357:
7384:
5874:"TYCHO: Supporting Permanently Crewed Lunar Exploration with High-Speed Optical Communication from Everywhere"
5593:"Proposals for the masses of the three largest asteroids, the Moon-Earth mass ratio and the Astronomical Unit"
4318:
5694:
Tantardini, Marco; Fantino, Elena; Ren, Yuan; Pergola, Pierpaolo; GĂłmez, Gerard; Masdemont, Josep J. (2010).
7669:
7605:
7286:
7197:
7101:
7074:
6975:
6881:
6189:
4568:
3643:
3250:
1348:
1091:
697:, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Currently, an
613:
586:
313:
6040:
7717:
7679:
7113:
7032:
6970:
6743:
6345:
6265:
6253:
5954:
4367:
1075:
507:
230:
6106:
See the Lagrange Points and Halo Orbits subsection under the section on Geosynchronous Transfer Orbit in
7567:
7374:
7012:
6866:
6808:
6778:
6566:
6443:
6411:
6381:
6340:
6325:
6204:
6044:
5838:
4997:
3697:
778:
415:
250:
158:
5087:
3669:
keeps a probe out of Earth's shadow and therefore ensures continuous illumination of its solar panels.
1465:, allowing the smaller third body to remain stationary (in this frame) with respect to the first two.
7736:
7637:
7620:
7257:
7192:
7185:
7042:
6928:
6891:
6713:
6497:
6386:
6355:
6283:
6258:
6233:
6194:
6175:
6130:
5854:
5714:
5604:
5528:
5476:
5423:
5366:
5289:
5246:
5195:
5136:
4740:
4730:
4604:
1414:
1001:
698:
656:
288:
245:
235:
163:
6075:
An online calculator to compute the precise positions of the 5 Lagrange points for any 2-body system
3230:
to the larger object is less than the separation of the two objects (although the distance between L
1422:
in the plane containing the orbit (grey rubber-sheet model with purple contours of equal potential).
7632:
7610:
7545:
7264:
7108:
6753:
6548:
6288:
6096:
3281:
1458:
1419:
1383:
932:
918:
from Earth (away from the sun). An example of a spacecraft designed to operate near the EarthâSun L
733:
535:
330:
168:
48:
7743:
7747:
7530:
7456:
7311:
7207:
7059:
6426:
6315:
6213:
5768:"Chang'e-4 relay satellite enters halo orbit around Earth-Moon L2, microsatellite in lunar orbit"
5730:
5663:
5620:
5492:
5466:
5439:
5413:
5382:
5356:
5315:
5236:
5185:
5154:
5126:
4942:
4682:
4430:
3285:
1700:
1431:
1117:
789:
785:
671:
598:
403:
278:
6101:
4952:
4425:, avoiding partial eclipses of the Sun to maintain a constant temperature. From locations near L
3234:
and the barycentre is greater than the distance between the smaller object and the barycentre).
4978:
4870:
3769:
from Earth's center, which is 116.8% of the EarthâMoon distance or 16.8% beyond the Moon; and L
7702:
7342:
7237:
7212:
7158:
7148:
6965:
6793:
6691:
6621:
6376:
6330:
6248:
6080:
6074:
5588:
4867:
4842:
4711:
4632:
4587:
4521:
4495:
4383:
4323:
3708:
are the projections of the orbits on a plane (e.g. the ecliptic) and not the full 3-D orbits.
3597:
3269:
2883:
1462:
1399:
1329:
1224:
915:
879:
729:
652:
609:
318:
255:
134:
72:
52:
7367:
5050:
1227:. Stability at these specific points is greatly complicated by solar gravitational influence.
7615:
7444:
7401:
7332:
7269:
7222:
7064:
6773:
6705:
6469:
6431:
6305:
6275:
6228:
5722:
5612:
5584:
5484:
5431:
5374:
5297:
5280:
5254:
5203:
5144:
4668:
4572:
3627:
3600:
of the Roche potential of two orbiting bodies, rendered half as a surface and half as a mesh
1704:
1454:
867:
616:, and hence fuel requirements, needed to maintain the desired orbit are kept at a minimum.
545:
488:
437:
202:
146:
612:
balance each other. This can make Lagrange points an excellent location for satellites, as
7660:
7560:
7513:
7411:
7352:
7337:
7291:
7274:
7180:
7175:
6813:
6406:
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6300:
6199:
6123:
6107:
6020:
5997:
5754:
5344:
5340:
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4644:
4616:
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4355:
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2887:
2879:
1443:
1391:
741:
493:
398:
308:
283:
4888:
3665:
on Earthâspacecraft communications. Similarly, a large-amplitude Lissajous orbit around L
3300:
of an object in orbit at a point along the line passing through both bodies is given by:
6056:
5858:
5718:
5608:
5480:
5427:
5370:
5293:
5250:
5199:
5140:
2913:
is the solution to the following equation, gravitation providing the centripetal force:
2423:
is the solution to the following equation, gravitation providing the centripetal force:
1480:
is the solution to the following equation, gravitation providing the centripetal force:
7707:
7694:
7587:
7518:
7508:
7493:
7471:
7461:
7357:
7347:
7296:
7217:
6909:
6861:
6853:
6733:
6728:
6659:
6639:
6630:
6209:
6185:
6180:
6155:
6009:
5401:
5276:"A Search for Natural or Artificial Objects Located at the EarthâMoon Libration Points"
4974:
3277:
2246:
1379:
1352:
1313:
1309:
1243:
793:
774:
465:
375:
298:
223:
217:
212:
31:
5842:
5747:
5511:"Widnall, Lecture L18 - Exploring the Neighborhood: the Restricted Three-Body Problem"
4552:
Lagrangian Point on 11 November 2004 and passed into the area dominated by the Moon's
4374:
up to an hour before Earth. Solar and heliospheric missions currently located around L
7762:
7655:
7572:
7503:
7466:
7306:
7227:
7091:
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6758:
6677:
6320:
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5301:
5275:
5158:
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4476:
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1395:
1321:
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874:
point, the object's orbital period becomes exactly equal to Earth's orbital period. L
470:
303:
260:
5496:
5443:
5386:
1438:, there are five positions in space where a third body, of comparatively negligible
125:
7712:
7647:
7316:
7301:
7002:
6997:
6828:
6738:
6612:
6595:
6453:
6350:
6218:
5989:"Essay on the Three-Body Problem" by J.-L. Lagrange, translated from the above, in
5106:
4457:
4399:
3631:
1403:
1378:; if one of the stars expands past its Roche lobe, then it will lose matter to its
1317:
188:
178:
173:
44:
5748:
SMART-1: On Course for Lunar Capture | Moon Today â Your Daily Source of Moon News
5023:
725:
Lagrange points are located about 1,500,000 km (930,000 mi) from earth.
4913:
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7406:
7096:
7079:
6951:
6833:
6668:
6438:
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6335:
5821:
is not just a great gatewayâit is a great place to store propellants. ... L
4701:
2768:
2095:
1975:
1697:
1367:
1257:
1253:
1056:
605:
352:
5932:
5149:
5114:
1059:-shaped orbit around the point (as seen in the corotating frame of reference).
39:
7674:
7582:
7525:
7485:
7279:
6980:
6401:
6062:
6005:
5892:
5726:
5616:
5088:"NASA - NASA's Wise Mission Finds First Trojan Asteroid Sharing Earth's Orbit"
4716:
4658:
4591:
4483:
4363:
3738:
3617:
2891:
2740:{\displaystyle x^{5}+x^{4}(3-\mu )+x^{3}(3-2\mu )-x^{2}(\mu )-x(2\mu )-\mu =0}
1839:{\displaystyle x^{5}+(\mu -3)x^{4}+(3-2\mu )x^{3}-(\mu )x^{2}+(2\mu )x-\mu =0}
1435:
1371:
1147:
749:
432:
388:
347:
6013:
5990:
3616:
points are nominally unstable, there are quasi-stable periodic orbits called
7684:
7535:
7252:
7133:
7054:
7037:
6818:
6170:
6068:
5259:
5224:
5208:
5173:
4938:
4631:, which aimed to look back towards Earth's orbit and compile a catalogue of
4387:
3268:
of the system. Additionally, the geometry of the triangle ensures that the
1363:
point and crashed into Earth after its orbit destabilized, forming the Moon.
1273:
1143:
619:
For any combination of two orbital bodies, there are five Lagrange points, L
2816:
The same remarks about tidal influence and apparent size apply as for the L
2399:
5471:
5418:
5361:
4544:
intended to help transport cargo and personnel to the Moon and back. The
1284:
1262:
1199:
1188:
4409:
is a good spot for space-based observatories. Because an object around L
2249:, corresponding to a circular orbit with this distance as radius around
6723:
6115:
5457:
Seidov, Zakir F. (March 1, 2004). "The Roche Problem: Some Analytics".
4575:
as part of the proposed depot-based space transportation architecture.
4553:
4545:
2382:{\displaystyle T_{s,M_{2}}(r)={\frac {T_{M_{2},M_{1}}(R)}{\sqrt {3}}}.}
1337:
1231:
1121:
981:
683:
590:
4303:
1434:. For example, given two massive bodies in orbits around their common
7498:
7362:
3582:
Net radial acceleration of a point orbiting along the EarthâMoon line
2087:{\displaystyle {\frac {M_{2}}{r^{3}}}\approx 3{\frac {M_{1}}{R^{3}}}}
1430:
Lagrange points are the constant-pattern solutions of the restricted
75:
of the potential. At the points themselves these forces are balanced.
5592:
1012:) the smaller mass with regard to its orbit around the larger mass.
643:
are on the line through the centers of the two large bodies, while L
5488:
5435:
5378:
5241:
5190:
5131:
4948:
Dynamical Systems, the Three-Body Problem, and Space Mission Design
1457:
of the two co-orbiting bodies, at the Lagrange points the combined
1132:
points, which were taken from mythological characters appearing in
6528:
6147:
5911:"B612 studying smallsat missions to search for near Earth objects"
4620:
4491:
4317:
3715:
3590:
3577:
3273:
2820:
point. For example, the angular radius of the sun as viewed from L
2412:
2398:
1413:
1138:
1133:
980:
863:
855:
594:
153:
104:
78:
38:
30:
3646:
keeps a spacecraft in a desired Lissajous orbit for a long time.
1154:
point, ahead of Jupiter, are named after Greek characters in the
3548:. The terms in this function represent respectively: force from
3272:
acceleration is to the distance from the barycenter in the same
1439:
1249:
1166:
point are named after Trojan characters and referred to as the "
792:, the collinear and the equilateral, for any three masses, with
6924:
6119:
4358:, because it provides an uninterrupted view of the Sun and any
3720:
Sunâplanet Lagrange points to scale (Click for clearer points.)
1948:
is the normalised distance. If the mass of the smaller object (
1186:
points contain interplanetary dust and at least two asteroids,
7379:
6920:
5979:, Tome 6, ÂŤ Essai sur le Problème des Trois Corps Âťâ
5893:"TYCHO mission to Earth-Moon libration point EML-4 @ IAC 2013"
2408:
859:
810:
The five Lagrange points are labelled and defined as follows:
5955:"NASA proposes a magnetic shield to protect Mars' atmosphere"
5787:"Evolving to a Depot-Based Space Transportation Architecture"
3661:, because the line between Sun and Earth has increased solar
3276:
as for the two massive bodies. The barycenter being both the
563:
5002:"Tome 6, Chapitre II: Essai sur le problème des trois corps"
2106:
point is about three times of that body. We may also write:
826:
point lies on the line defined between the two large masses
662:
When the mass ratio of the two bodies is large enough, the L
4802:
4764:
4762:
4571:, launched in 2018, and would be "an ideal location" for a
4524:). In 2010, spacecraft transfer trajectories to SunâEarth L
1170:". Both camps are considered to be types of trojan bodies.
906:
point, that orbital period becomes equal to Earth's. Like L
551:
5223:
SlĂz-Balogh, Judit; Barta, AndrĂĄs; HorvĂĄth, GĂĄbor (2019).
5172:
SlĂz-Balogh, Judit; Barta, AndrĂĄs; HorvĂĄth, GĂĄbor (2018).
2852:) â 0.264°, whereas that of the earth is arcsin(
597:
bodies. Mathematically, this involves the solution of the
569:
5643:
Stability of Lagrange Points: James Webb Space Telescope"
557:
840:. It is the point where the gravitational attraction of
713:, a powerful infrared space observatory, is located at L
5843:"Photographische Untersuchungen des Librationspunktes L
4980:
De motu rectilineo trium corporum se mutuo attrahentium
5070:"First Asteroid Companion of Earth Discovered at Last"
3536:
is the distance between the two main objects, and sgn(
3197:
3171:) is much smaller than the mass of the larger object (
2756:) is much smaller than the mass of the larger object (
2245:
This distance can be described as being such that the
1955:) is much smaller than the mass of the larger object (
4354:
point. Conversely, it is also useful for space-based
3306:
3186:
2919:
2777:
2615:
2429:
2290:
2112:
2029:
1984:
1921:
1852:
1714:
1486:
566:
1974:
from the smaller object, equal to the radius of the
1028:) are stable equilibria, provided that the ratio of
560:
548:
7693:
7646:
7598:
7484:
7420:
7325:
7245:
7236:
7124:
6958:
6792:
6704:
6648:
6584:
6537:
6477:
6468:
6364:
6274:
6163:
6154:
6032:J R Stockton - Includes translations of Lagrange's
5022:. Space Telescope Science Institute. Archived from
4417:, so solar radiation is not completely blocked at L
3622:around these points in a three-body system. A full
3255:
The reason these points are in balance is that at L
2874:) â 0.242°. Looking toward the sun from L
554:
4829:
4827:
4528:were studied and several designs were considered.
3741:especially in the case of Moon and Jupiter) with L
3513:
3216:
3114:
2807:{\displaystyle r\approx R{\sqrt{\frac {\mu }{3}}}}
2806:
2739:
2597:
2381:
2205:
2086:
2014:{\displaystyle r\approx R{\sqrt{\frac {\mu }{3}}}}
2013:
1940:
1900:
1838:
1674:is the distance between the two main objects, and
1654:
854:combine to produce an equilibrium. An object that
6069:Locations of Lagrange points, with approximations
5645:, University of Arizona. Retrieved 17 Sept. 2018.
5229:Monthly Notices of the Royal Astronomical Society
5178:Monthly Notices of the Royal Astronomical Society
5058:, Neil J. Cornish, with input from Jeremy Goodman
1901:{\displaystyle \mu ={\frac {M_{2}}{M_{1}+M_{2}}}}
659:formed with the centers of the two large bodies.
5008:(in French). Gauthier-Villars. pp. 229â334.
2227:are the average densities of the two bodies and
1242:points contain several dozen known objects, the
3217:{\displaystyle r\approx R{\tfrac {7}{12}}\mu .}
2898:in order for its solar panels to get full sun.
878:is about 1.5 million kilometers, or 0.01
777:around 1750, a decade before the Italian-born
6936:
6131:
5785:Zegler, Frank; Kutter, Bernard (2010-09-02).
5553:"Stability of the Lagrange Points, L4 and L5"
4841:. WMAP Education and Outreach. Archived from
3493:
3438:
914:is about 1.5 million kilometers or 0.01
784:In 1772, Lagrange published an "Essay on the
773:) were discovered by the Swiss mathematician
686:has more than one million of these trojans.
515:
8:
5546:
5544:
5542:
2609:case. The corresponding quintic equation is
1000:points lie at the third vertices of the two
631:Lagrange points for the EarthâMoon system. L
6006:Considerationes de motu corporum coelestium
5794:AIAA SPACE 2010 Conference & Exposition
5707:Celestial Mechanics and Dynamical Astronomy
5597:Celestial Mechanics and Dynamical Astronomy
5274:Freitas, Robert; Valdes, Francisco (1980).
4567:covering the Moon's far side, for example,
4396:Interstellar Mapping and Acceleration Probe
16:Equilibrium points near two orbiting bodies
7242:
6943:
6929:
6921:
6905:
6474:
6160:
6138:
6124:
6116:
5700:point of the SunâEarth three-body problem"
5045:
5043:
5041:
2749:Again, if the mass of the smaller object (
1692:from the center of mass. The solution for
882:, from Earth in the direction of the Sun.
522:
508:
110:
83:An example of a spacecraft at Sun-Earth L2
5470:
5417:
5360:
5347:(2005). "Where Did The Moon Come From?".
5258:
5240:
5207:
5189:
5148:
5130:
3503:
3492:
3491:
3482:
3463:
3450:
3437:
3436:
3430:
3397:
3373:
3363:
3337:
3326:
3316:
3305:
3196:
3185:
3152:being defined such that the distance of L
3104:
3093:
3080:
3073:
3044:
3031:
3020:
3014:
2998:
2969:
2963:
2952:
2926:
2920:
2918:
2882:. It is necessary for a spacecraft, like
2797:
2787:
2776:
2692:
2661:
2633:
2620:
2614:
2587:
2576:
2563:
2556:
2533:
2520:
2509:
2503:
2487:
2477:
2471:
2459:
2436:
2430:
2428:
2351:
2338:
2333:
2326:
2306:
2295:
2289:
2197:
2182:
2176:
2165:
2149:
2134:
2128:
2117:
2111:
2076:
2066:
2060:
2046:
2036:
2030:
2028:
2004:
1994:
1983:
1928:
1920:
1889:
1876:
1865:
1859:
1851:
1800:
1778:
1747:
1719:
1713:
1644:
1633:
1620:
1613:
1590:
1577:
1566:
1560:
1544:
1534:
1528:
1516:
1493:
1487:
1485:
4516:. Moreover, a satellite near SunâEarth L
4302:
3783:
3156:from the centre of the larger object is
1300:has two smaller moons of Saturn in its L
7428:Effect of spaceflight on the human body
4823:
4758:
3793:Semimajor axis, SMA (×10 m)
608:forces of the two large bodies and the
413:
328:
132:
118:
7450:Psychological and sociological effects
6784:Transposition, docking, and extraction
4505:A spacecraft orbiting near SunâEarth L
4441:was positioned in a halo orbit about L
2767:is at approximately the radius of the
761:The three collinear Lagrange points (L
5551:Greenspan, Thomas (January 7, 2014).
4421:. Spacecraft generally orbit around L
3562:; and centripetal force. The points L
3164:. If the mass of the smaller object (
1970:are at approximately equal distances
1094:in order to maintain their position.
7:
5400:Sepinsky, Jeremy F.; Willems, Bart;
3525:is the distance from the large body
2605:with parameters defined as for the L
928:Wilkinson Microwave Anisotropy Probe
728:The European Space Agency's earlier
6059:âalso attributed to Neil J. Cornish
6036:and of two related papers by Euler
5975:Joseph-Louis, Comte Lagrange, from
4643:In 2017, the idea of positioning a
3724:This table lists sample values of L
3638:trajectories. These quasi-periodic
1446:that matches their orbital motion.
1320:also has two Lagrange co-orbitals,
7159:Weather and environment monitoring
6012:âtranscription and translation at
5115:"The Second Earth Trojan 2020 XL5"
4736:List of objects at Lagrange points
4380:Solar and Heliospheric Observatory
4344:International Sun Earth Explorer 3
4294:List of objects at Lagrange points
3785:Lagrangian points in Solar System
1461:of two massive bodies balance the
1104:List of objects at Lagrange points
1098:Natural objects at Lagrange points
806:List of objects at Lagrange points
732:telescope, and its newly launched
693:, between the Sun and Earth, and L
14:
6844:Kepler's laws of planetary motion
5931:. B612 Foundation. Archived from
5796:. AIAA. p. 4. Archived from
5696:"Spacecraft trajectories to the L
5119:The Astrophysical Journal Letters
5113:; FĂśhring, Dora (November 2021).
5068:Choi, Charles Q. (27 July 2011).
4875:Eric Weisstein's World of Physics
1410:Physical and mathematical details
1219:points contain concentrations of
1108:Due to the natural stability of L
589:for small-mass objects under the
294:Kepler's laws of planetary motion
7742:
7732:
7731:
6904:
6839:Interplanetary Transport Network
6719:Collision avoidance (spacecraft)
5851:Acta Astronomica, Vol. 11, p.165
4707:Interplanetary Transport Network
4675:
4661:
4627:point to position their planned
4460:and exponentially unstable with
4330:(blue) orbits around SunâEarth L
1941:{\displaystyle x={\frac {r}{R}}}
1351:postulates that an object named
985:Gravitational accelerations at L
682:points with respect to the Sun;
544:
124:
7203:Space launch market competition
6804:Astronomical coordinate systems
6558:Longitude of the ascending node
6102:Earth, a lone Trojan discovered
5322:from the original on 2011-07-25
4514:Space Weather Prediction Center
4394:. Planned missions include the
1670:point from the smaller object,
1398:with Earth, and Saturn's moons
926:. Earlier examples include the
7440:Health threat from cosmic rays
6877:Retrograde and prograde motion
6063:Explanation of Lagrange points
6051:Explanation of Lagrange points
5891:Hornig, Andreas (2013-10-06).
5872:Hornig, Andreas (2022-05-01).
5529:"Stability of Lagrange Points"
4471:was a popular place to put a "
4311:in an orbit around SunâEarth L
3680:
3673:
3469:
3443:
3424:
3412:
3394:
3381:
3357:
3351:
2722:
2713:
2704:
2698:
2682:
2667:
2651:
2639:
2456:
2443:
2365:
2359:
2320:
2314:
1818:
1809:
1793:
1787:
1771:
1756:
1740:
1728:
1513:
1500:
1449:Alternatively, when seen in a
781:discovered the remaining two.
703:Deep Space Climate Observatory
1:
5853:. Vol. 11. p. 165.
4893:NASA Solar System Exploration
4548:Mission passed through the L
4392:Advanced Composition Explorer
736:, also occupy orbits around L
599:restricted three-body problem
6824:Equatorial coordinate system
6109:NASA: Basics of Space Flight
5766:Jones, Andrew (2018-06-14).
5302:10.1016/0019-1035(80)90106-2
63:). Counterintuitively, the L
7556:Self-replicating spacecraft
7392:International Space Station
6093:The Five Points of Lagrange
4951:. p. 9. Archived from
4726:Lagrange point colonization
4435:cosmic microwave background
2023:We may also write this as:
448:Tsiolkovsky rocket equation
7790:
7070:Space Liability Convention
6576:Longitude of the periapsis
6084:âEp. 76: "Lagrange Points"
6047:page, with good animations
4697:Euler's three-body problem
4586:are the locations for the
4439:James Webb Space Telescope
4328:James Webb Space Telescope
4291:
3248:
1101:
1078:. Any object orbiting at L
924:James Webb Space Telescope
803:
790:constant-pattern solutions
711:James Webb Space Telescope
417:Engineering and efficiency
236:Bi-elliptic transfer orbit
18:
7726:
7433:Space adaptation syndrome
6900:
6887:Specific angular momentum
6041:What are Lagrange points?
5727:10.1007/s10569-010-9299-x
5637:Cacolici, Gianna Nicole,
5617:10.1007/s10569-009-9203-8
5459:The Astrophysical Journal
5406:The Astrophysical Journal
5316:"List Of Neptune Trojans"
4945:; Ross, Shane D. (2006).
4834:Cornish, Neil J. (1998).
4224:
4165:
4106:
4063:
4017:
3974:
3931:
3885:
3839:
3830:
3823:
3816:
3809:
3802:
3795:
3792:
3789:
1374:has its apex located at L
1355:formed at the SunâEarth L
753:Earth for data transfer.
748:, while Euclid follows a
593:influence of two massive
7166:Communications satellite
5660:Solar System Exploration
5349:The Astronomical Journal
5150:10.3847/2041-8213/ac37bf
4692:Co-orbital configuration
4647:shield at the SunâMars L
4594:'s name comes from the L
4565:communications satellite
4433:and observations of the
4286:Spaceflight applications
3296:The radial acceleration
3226:Thus the distance from L
1908:is the mass fraction of
1666:is the distance of the L
1463:centrifugal pseudo-force
1451:rotating reference frame
1158:and referred to as the "
1020:The triangular points (L
705:(DSCOVR) is located at L
443:Propellant mass fraction
342:Gravitational influences
19:Not to be confused with
7670:reusable launch systems
7287:Extravehicular activity
7198:Commercial use of space
7102:Militarisation of space
7075:Registration Convention
6991:Accidents and incidents
6882:Specific orbital energy
5753:2 November 2005 at the
5318:. Minor Planet Center.
4588:Kordylewski dust clouds
4502:every 20 months).
3821:/SMA â 1 (%)
3251:Trojan (celestial body)
2242:the earth and the sun.
1349:giant impact hypothesis
1316:. Another Saturn moon,
740:. Gaia keeps a tighter
314:Specific orbital energy
51:due to gravity and the
7718:Mission control center
7680:Non-rocket spacelaunch
7114:Billionaire space race
6294:Geostationary transfer
5929:"The Sentinel Mission"
5839:Kordylewski, Kazimierz
5662:. NASA. Archived from
5535:. University of Texas.
5527:Fitzpatrick, Richard.
4998:Lagrange, Joseph-Louis
4494:comes within 0.3
4368:coronal mass ejections
4334:
4315:
3721:
3601:
3583:
3515:
3218:
3116:
2808:
2741:
2599:
2416:
2383:
2207:
2088:
2015:
1942:
1902:
1840:
1656:
1427:
989:
651:each act as the third
231:Hohmann transfer orbit
108:
76:
59:) and away from them (
36:
7568:Spacecraft propulsion
7018:European Space Agency
6867:Orbital state vectors
6809:Characteristic energy
6779:Trans-lunar injection
6567:Argument of periapsis
6244:Prograde / Retrograde
6205:Hyperbolic trajectory
6045:European Space Agency
5260:10.1093/mnras/sty2630
5209:10.1093/mnras/sty2049
5051:"The Lagrange Points"
4836:"The Lagrange Points"
4619:were planning to use
4445:on January 24, 2022.
4321:
4306:
3719:
3698:Coriolis acceleration
3596:
3581:
3516:
3249:Further information:
3219:
3117:
2809:
2742:
2600:
2402:
2384:
2208:
2089:
2016:
1943:
1903:
1841:
1657:
1417:
1390:Objects which are on
1002:equilateral triangles
984:
779:Joseph-Louis Lagrange
427:Preflight engineering
159:Argument of periapsis
82:
42:
34:
7774:Lagrangian mechanics
7193:Satellite navigation
6714:Bi-elliptic transfer
6234:Parabolic trajectory
5847:im System Erde-Mond"
5345:Gott III, J. Richard
4848:on September 7, 2015
4741:Lunar space elevator
4731:Lagrangian mechanics
4633:near-Earth asteroids
4605:geosynchronous orbit
4563:has been used for a
4522:near-Earth asteroids
3304:
3184:
3140:defined as for the L
2917:
2775:
2613:
2427:
2288:
2110:
2027:
1982:
1919:
1850:
1712:
1484:
1459:gravitational fields
1425:Click for animation.
1150:. Asteroids at the L
1076:unstable equilibrium
699:artificial satellite
657:equilateral triangle
483:Propulsive maneuvers
7769:Trojans (astronomy)
7578:Electric propulsion
7265:Life-support system
7149:Imagery and mapping
7109:Private spaceflight
6754:Low-energy transfer
6097:Neil deGrasse Tyson
6086:by Fraser Cain and
5859:1961AcA....11..165K
5719:2010CeMDA.108..215T
5609:2009CeMDA.103..365P
5481:2004ApJ...603..283S
5428:2007ApJ...660.1624S
5371:2005AJ....129.1724B
5294:1980Icar...42..442F
5251:2019MNRAS.482..762S
5200:2018MNRAS.480.5550S
5141:2021ApJ...922L..25H
4943:Marsden, Jerrold E.
3786:
3712:Solar System values
3292:Radial acceleration
1420:effective potential
1384:Roche lobe overflow
1347:One version of the
1336:. The moons wander
1221:interplanetary dust
930:and its successor,
536:celestial mechanics
460:Efficiency measures
363:Sphere of influence
332:Celestial mechanics
114:Part of a series on
49:effective potential
7531:Robotic spacecraft
7457:Space and survival
7312:Space colonization
7208:Space architecture
7060:Outer Space Treaty
6749:Inclination change
6397:Distant retrograde
6071:âDavid Peter Stern
6057:A NASA explanation
6019:2020-08-03 at the
6014:merlyn.demon.co.uk
5996:2019-06-23 at the
5991:merlyn.demon.co.uk
5589:Standish, E. Myles
5533:Newtonian Dynamics
5026:on 3 February 2014
5006:Ĺuvres de Lagrange
4889:"DSCOVR: In-Depth"
4868:Weisstein, Eric W.
4683:Spaceflight portal
4629:Sentinel telescope
4615:Scientists at the
4431:infrared astronomy
4335:
4316:
3828:(×10 m)
3814:(×10 m)
3800:(×10 m)
3784:
3722:
3602:
3584:
3511:
3286:three-body problem
3214:
3206:
3112:
2804:
2737:
2595:
2417:
2379:
2256:in the absence of
2203:
2084:
2011:
1938:
1898:
1836:
1652:
1432:three-body problem
1428:
1252:has four accepted
1225:Kordylewski clouds
990:
862:more closely than
786:three-body problem
279:Dynamical friction
109:
77:
37:
7756:
7755:
7703:Flight controller
7480:
7479:
7238:Human spaceflight
7213:Space exploration
7139:Earth observation
6918:
6917:
6892:Two-line elements
6700:
6699:
6622:Eccentric anomaly
6464:
6463:
6331:Orbit of the Moon
6190:Highly elliptical
5585:Pitjeva, Elena V.
5565:on April 18, 2018
4937:Koon, Wang Sang;
4871:"Lagrange Points"
4753:Explanatory notes
4712:Klemperer rosette
4607:cover the Earth.
4388:Aditya-L1 Mission
4350:has orbited the L
4283:
4282:
3657:than to stay at L
3509:
3404:
3343:
3205:
3110:
3051:
3004:
2958:
2909:The location of L
2802:
2796:
2593:
2540:
2493:
2466:
2419:The location of L
2374:
2373:
2191:
2143:
2082:
2052:
2009:
2003:
1936:
1896:
1703:of the following
1650:
1597:
1550:
1523:
1476:The location of L
1453:that matches the
1162:". Those at the L
1074:are positions of
614:orbit corrections
610:centrifugal force
579:Lagrangian points
532:
531:
382:Lagrangian points
319:Vis-viva equation
289:Kepler's equation
136:Orbital mechanics
53:centrifugal force
7781:
7746:
7735:
7734:
7445:Space psychology
7270:Animals in space
7243:
7223:Space technology
7065:Rescue Agreement
6945:
6938:
6931:
6922:
6908:
6907:
6849:Lagrangian point
6744:Hohmann transfer
6689:
6675:
6666:
6657:
6637:
6628:
6619:
6610:
6606:
6602:
6593:
6573:
6564:
6555:
6546:
6526:
6522:
6513:
6504:
6495:
6475:
6444:Heliosynchronous
6393:Lagrange points
6346:Transatmospheric
6161:
6140:
6133:
6126:
6117:
6053:âNeil J. Cornish
5963:
5962:
5951:
5945:
5944:
5942:
5940:
5925:
5919:
5918:
5917:. June 20, 2017.
5907:
5901:
5900:
5888:
5882:
5881:
5869:
5863:
5862:
5835:
5829:
5828:
5809:
5808:
5802:
5791:
5782:
5776:
5775:
5763:
5757:
5745:
5739:
5738:
5704:
5691:
5685:
5682:
5676:
5675:
5673:
5671:
5666:on July 20, 2015
5652:
5646:
5635:
5629:
5628:
5581:
5575:
5574:
5572:
5570:
5564:
5558:. Archived from
5557:
5548:
5537:
5536:
5524:
5518:
5517:
5515:
5507:
5501:
5500:
5474:
5472:astro-ph/0311272
5454:
5448:
5447:
5421:
5419:astro-ph/0612508
5412:(2): 1624â1635.
5397:
5391:
5390:
5364:
5362:astro-ph/0405372
5355:(3): 1724â1745.
5341:Belbruno, Edward
5337:
5331:
5330:
5328:
5327:
5312:
5306:
5305:
5271:
5265:
5264:
5262:
5244:
5220:
5214:
5213:
5211:
5193:
5184:(4): 5550â5559.
5169:
5163:
5162:
5152:
5134:
5111:Tholen, David J.
5107:Wiegert, Paul A.
5102:
5096:
5095:
5084:
5078:
5077:
5065:
5059:
5057:
5055:
5047:
5036:
5035:
5033:
5031:
5016:
5010:
5009:
4994:
4988:
4987:
4985:
4971:
4965:
4963:
4961:
4960:
4934:
4928:
4927:
4925:
4924:
4910:
4904:
4903:
4901:
4900:
4885:
4879:
4878:
4864:
4858:
4857:
4855:
4853:
4847:
4840:
4831:
4811:
4805:
4799:
4798:
4795:
4789:
4787:
4786:
4783:
4780:
4779:
4778:
4766:
4685:
4680:
4679:
4678:
4671:
4669:Astronomy portal
4666:
4665:
4664:
4573:propellant depot
4356:solar telescopes
4279:
4278:
4271:
4269:
4262:
4257:
4255:
4248:
4243:
4241:
4234:
4232:
4220:
4219:
4212:
4210:
4203:
4198:
4196:
4189:
4184:
4182:
4175:
4173:
4161:
4160:
4153:
4151:
4144:
4139:
4137:
4130:
4125:
4123:
4116:
4114:
4102:
4101:
4094:
4089:
4084:
4079:
4074:
4069:
4059:
4058:
4055:
4048:
4043:
4038:
4033:
4028:
4023:
4013:
4012:
4005:
4000:
3995:
3990:
3985:
3980:
3970:
3969:
3962:
3957:
3952:
3947:
3942:
3937:
3927:
3926:
3923:
3916:
3911:
3906:
3901:
3896:
3891:
3881:
3876:
3875:
3868:
3863:
3858:
3853:
3852:
3845:
3787:
3780:
3778:
3768:
3766:
3756:
3754:
3640:Lissajous orbits
3628:dynamical system
3595:
3520:
3518:
3517:
3512:
3510:
3508:
3507:
3498:
3497:
3496:
3487:
3486:
3468:
3467:
3455:
3454:
3442:
3441:
3431:
3405:
3403:
3402:
3401:
3379:
3378:
3377:
3364:
3344:
3342:
3341:
3332:
3331:
3330:
3317:
3223:
3221:
3220:
3215:
3207:
3198:
3122:with parameters
3121:
3119:
3118:
3113:
3111:
3109:
3108:
3099:
3098:
3097:
3085:
3084:
3074:
3072:
3068:
3052:
3050:
3049:
3048:
3036:
3035:
3025:
3024:
3015:
3005:
3003:
3002:
2997:
2993:
2974:
2973:
2964:
2959:
2957:
2956:
2951:
2947:
2931:
2930:
2921:
2873:
2871:
2870:
2869:
2867:
2861:
2858:
2851:
2849:
2848:
2847:
2845:
2839:
2836:
2835:
2833:
2813:
2811:
2810:
2805:
2803:
2801:
2789:
2788:
2746:
2744:
2743:
2738:
2697:
2696:
2666:
2665:
2638:
2637:
2625:
2624:
2604:
2602:
2601:
2596:
2594:
2592:
2591:
2582:
2581:
2580:
2568:
2567:
2557:
2555:
2551:
2541:
2539:
2538:
2537:
2525:
2524:
2514:
2513:
2504:
2494:
2492:
2491:
2482:
2481:
2472:
2467:
2465:
2464:
2463:
2441:
2440:
2431:
2403:The Lagrangian L
2388:
2386:
2385:
2380:
2375:
2369:
2368:
2358:
2357:
2356:
2355:
2343:
2342:
2327:
2313:
2312:
2311:
2310:
2283:
2282:
2212:
2210:
2209:
2204:
2202:
2201:
2196:
2192:
2187:
2186:
2177:
2170:
2169:
2154:
2153:
2148:
2144:
2139:
2138:
2129:
2122:
2121:
2093:
2091:
2090:
2085:
2083:
2081:
2080:
2071:
2070:
2061:
2053:
2051:
2050:
2041:
2040:
2031:
2020:
2018:
2017:
2012:
2010:
2008:
1996:
1995:
1947:
1945:
1944:
1939:
1937:
1929:
1907:
1905:
1904:
1899:
1897:
1895:
1894:
1893:
1881:
1880:
1870:
1869:
1860:
1845:
1843:
1842:
1837:
1805:
1804:
1783:
1782:
1752:
1751:
1724:
1723:
1705:quintic function
1661:
1659:
1658:
1653:
1651:
1649:
1648:
1639:
1638:
1637:
1625:
1624:
1614:
1612:
1608:
1598:
1596:
1595:
1594:
1582:
1581:
1571:
1570:
1561:
1551:
1549:
1548:
1539:
1538:
1529:
1524:
1522:
1521:
1520:
1498:
1497:
1488:
1455:angular velocity
1392:horseshoe orbits
1292:
1290:
1289:
1281:
1279:
1278:
1270:
1268:
1267:
1211:The EarthâMoon L
1207:
1205:
1204:
1196:
1194:
1193:
1053:
1051:
1050:
1042:
1039:
672:trojan asteroids
585:) are points of
583:libration points
576:
575:
572:
571:
568:
565:
562:
559:
556:
553:
550:
524:
517:
510:
489:Orbital maneuver
438:Payload fraction
418:
399:Lissajous orbits
333:
304:Orbital velocity
251:Hyperbolic orbit
147:Orbital elements
137:
128:
111:
107:
102:
93:
88:
62:
58:
7789:
7788:
7784:
7783:
7782:
7780:
7779:
7778:
7759:
7758:
7757:
7752:
7722:
7689:
7661:Escape velocity
7642:
7594:
7561:Space telescope
7514:Reentry capsule
7476:
7416:
7321:
7292:Overview effect
7275:Bioastronautics
7232:
7120:
6954:
6949:
6919:
6914:
6896:
6814:Escape velocity
6795:
6788:
6769:Rocket equation
6696:
6688:
6682:
6673:
6664:
6655:
6644:
6635:
6626:
6617:
6608:
6604:
6600:
6591:
6580:
6571:
6562:
6553:
6544:
6533:
6524:
6520:
6516:Semi-minor axis
6511:
6507:Semi-major axis
6502:
6493:
6487:
6460:
6382:Areosynchronous
6366:
6360:
6341:Sun-synchronous
6326:Near-equatorial
6270:
6150:
6144:
6021:Wayback Machine
5998:Wayback Machine
5985:Tome 6 (Viewer)
5972:
5967:
5966:
5953:
5952:
5948:
5938:
5936:
5935:on 30 June 2012
5927:
5926:
5922:
5909:
5908:
5904:
5890:
5889:
5885:
5871:
5870:
5866:
5846:
5837:
5836:
5832:
5824:
5820:
5815:
5806:
5804:
5800:
5789:
5784:
5783:
5779:
5765:
5764:
5760:
5755:Wayback Machine
5746:
5742:
5702:
5699:
5693:
5692:
5688:
5683:
5679:
5669:
5667:
5654:
5653:
5649:
5636:
5632:
5583:
5582:
5578:
5568:
5566:
5562:
5555:
5550:
5549:
5540:
5526:
5525:
5521:
5513:
5509:
5508:
5504:
5456:
5455:
5451:
5402:Kalogera, Vicky
5399:
5398:
5394:
5339:
5338:
5334:
5325:
5323:
5314:
5313:
5309:
5273:
5272:
5268:
5222:
5221:
5217:
5171:
5170:
5166:
5104:
5103:
5099:
5086:
5085:
5081:
5067:
5066:
5062:
5053:
5049:
5048:
5039:
5029:
5027:
5018:
5017:
5013:
4996:
4995:
4991:
4983:
4975:Euler, Leonhard
4973:
4972:
4968:
4958:
4956:
4936:
4935:
4931:
4922:
4920:
4912:
4911:
4907:
4898:
4896:
4887:
4886:
4882:
4866:
4865:
4861:
4851:
4849:
4845:
4838:
4833:
4832:
4825:
4820:
4815:
4814:
4801:
4796:
4793:
4791:
4784:
4781:
4776:
4774:
4772:
4771:
4769:
4767:
4760:
4755:
4750:
4720:
4681:
4676:
4674:
4667:
4662:
4660:
4657:
4650:
4645:magnetic dipole
4641:
4626:
4617:B612 Foundation
4613:
4601:
4597:
4585:
4581:
4562:
4551:
4539:
4534:
4527:
4519:
4508:
4501:
4489:
4480:science fiction
4470:
4455:
4451:
4444:
4428:
4424:
4420:
4412:
4408:
4398:(IMAP) and the
4377:
4373:
4362:(including the
4353:
4340:
4333:
4314:
4301:
4296:
4288:
4276:
4274:
4267:
4265:
4260:
4253:
4251:
4246:
4239:
4237:
4230:
4228:
4217:
4215:
4208:
4206:
4201:
4194:
4192:
4187:
4180:
4178:
4171:
4169:
4158:
4156:
4149:
4147:
4142:
4135:
4133:
4128:
4121:
4119:
4112:
4110:
4099:
4097:
4092:
4087:
4082:
4077:
4072:
4067:
4056:
4053:
4051:
4046:
4041:
4036:
4031:
4026:
4021:
4010:
4008:
4003:
3998:
3993:
3988:
3983:
3978:
3967:
3965:
3960:
3955:
3950:
3945:
3940:
3935:
3924:
3921:
3919:
3914:
3909:
3904:
3899:
3894:
3889:
3879:
3873:
3871:
3866:
3861:
3856:
3850:
3848:
3843:
3834:
3827:
3820:
3813:
3806:
3803:1 â L
3799:
3776:
3774:
3772:
3764:
3762:
3760:
3752:
3750:
3748:
3744:
3735:
3731:
3727:
3714:
3707:
3703:
3694:
3690:
3684:
3677:
3668:
3660:
3656:
3652:
3649:For SunâEarth-L
3644:station keeping
3636:Lissajous-curve
3615:
3611:
3607:
3591:
3589:
3573:
3569:
3565:
3561:
3554:
3531:
3499:
3478:
3459:
3446:
3432:
3393:
3380:
3369:
3365:
3333:
3322:
3318:
3302:
3301:
3294:
3262:
3258:
3253:
3247:
3245:
3241:
3233:
3229:
3182:
3181:
3177:
3170:
3155:
3147:
3143:
3135:
3128:
3100:
3089:
3076:
3075:
3040:
3027:
3026:
3016:
3013:
3009:
2980:
2976:
2975:
2965:
2937:
2933:
2932:
2922:
2915:
2914:
2912:
2907:
2905:
2897:
2888:Lissajous orbit
2880:annular eclipse
2877:
2865:
2863:
2862:
2859:
2856:
2855:
2853:
2843:
2841:
2840:
2837:
2831:
2829:
2828:
2827:
2825:
2823:
2819:
2773:
2772:
2766:
2762:
2755:
2688:
2657:
2629:
2616:
2611:
2610:
2608:
2583:
2572:
2559:
2558:
2529:
2516:
2515:
2505:
2502:
2498:
2483:
2473:
2455:
2442:
2432:
2425:
2424:
2422:
2406:
2397:
2395:
2347:
2334:
2329:
2328:
2302:
2291:
2286:
2285:
2280:
2278:
2276:
2269:
2262:
2255:
2240:
2233:
2226:
2219:
2178:
2172:
2171:
2161:
2130:
2124:
2123:
2113:
2108:
2107:
2105:
2101:
2072:
2062:
2042:
2032:
2025:
2024:
1980:
1979:
1969:
1965:
1961:
1954:
1917:
1916:
1913:
1885:
1872:
1871:
1861:
1848:
1847:
1796:
1774:
1743:
1715:
1710:
1709:
1691:
1687:
1680:
1669:
1640:
1629:
1616:
1615:
1586:
1573:
1572:
1562:
1559:
1555:
1540:
1530:
1512:
1499:
1489:
1482:
1481:
1479:
1474:
1472:
1444:circular motion
1423:
1412:
1377:
1362:
1358:
1343:
1335:
1327:
1307:
1303:
1287:
1286:
1283:
1276:
1275:
1272:
1265:
1264:
1261:
1244:Neptune trojans
1241:
1237:
1218:
1214:
1202:
1201:
1198:
1191:
1190:
1187:
1185:
1181:
1178:The SunâEarth L
1165:
1153:
1146:set during the
1131:
1127:
1115:
1111:
1106:
1100:
1092:station keeping
1089:
1085:
1081:
1073:
1069:
1065:
1049:
1043:
1040:
1038:
1032:
1031:
1029:
1027:
1023:
1018:
1011:
1007:
999:
995:
988:
979:
976:
972:
964:
955:
951:
946:
943:
921:
913:
909:
905:
901:
897:
892:
889:
877:
873:
868:Earth's gravity
853:
846:
839:
832:
825:
820:
817:
808:
802:
800:Lagrange points
794:circular orbits
772:
768:
764:
759:
747:
742:Lissajous orbit
739:
724:
720:
716:
708:
696:
692:
681:
677:
669:
665:
650:
646:
642:
638:
634:
626:
622:
547:
543:
540:Lagrange points
528:
499:
498:
494:Orbit insertion
484:
476:
475:
461:
453:
452:
428:
420:
416:
409:
408:
404:Lyapunov orbits
395:
394:
378:
368:
367:
343:
335:
331:
324:
323:
309:Surface gravity
284:Escape velocity
274:
266:
265:
246:Parabolic orbit
242:
241:
208:
206:
203:two-body orbits
194:
193:
184:Semi-major axis
149:
139:
135:
100:
99:
86:
85:
84:
71:points are the
70:
66:
60:
56:
27:
17:
12:
11:
5:
7787:
7785:
7777:
7776:
7771:
7761:
7760:
7754:
7753:
7751:
7750:
7739:
7727:
7724:
7723:
7721:
7720:
7715:
7710:
7708:Ground station
7705:
7699:
7697:
7695:Ground segment
7691:
7690:
7688:
7687:
7682:
7677:
7672:
7663:
7658:
7652:
7650:
7644:
7643:
7641:
7640:
7635:
7630:
7628:Interplanetary
7625:
7624:
7623:
7621:Geosynchronous
7618:
7608:
7602:
7600:
7596:
7595:
7593:
7592:
7591:
7590:
7588:Gravity assist
7585:
7580:
7575:
7565:
7564:
7563:
7558:
7553:
7548:
7543:
7538:
7528:
7523:
7522:
7521:
7519:Service module
7516:
7511:
7509:Orbital module
7501:
7496:
7494:Launch vehicle
7490:
7488:
7482:
7481:
7478:
7477:
7475:
7474:
7472:Space sexology
7469:
7464:
7462:Space medicine
7459:
7454:
7453:
7452:
7442:
7437:
7436:
7435:
7424:
7422:
7418:
7417:
7415:
7414:
7409:
7404:
7399:
7394:
7389:
7388:
7387:
7377:
7372:
7371:
7370:
7365:
7355:
7350:
7345:
7340:
7335:
7329:
7327:
7323:
7322:
7320:
7319:
7314:
7309:
7304:
7299:
7297:Weightlessness
7294:
7289:
7284:
7283:
7282:
7277:
7272:
7262:
7261:
7260:
7249:
7247:
7240:
7234:
7233:
7231:
7230:
7225:
7220:
7218:Space research
7215:
7210:
7205:
7200:
7195:
7190:
7189:
7188:
7183:
7178:
7173:
7163:
7162:
7161:
7156:
7154:Reconnaissance
7151:
7146:
7136:
7130:
7128:
7122:
7121:
7119:
7118:
7117:
7116:
7106:
7105:
7104:
7099:
7094:
7084:
7083:
7082:
7077:
7072:
7067:
7062:
7052:
7051:
7050:
7045:
7040:
7035:
7030:
7025:
7023:European Union
7020:
7015:
7010:
7000:
6995:
6994:
6993:
6988:
6983:
6978:
6968:
6962:
6960:
6956:
6955:
6950:
6948:
6947:
6940:
6933:
6925:
6916:
6915:
6913:
6912:
6910:List of orbits
6901:
6898:
6897:
6895:
6894:
6889:
6884:
6879:
6874:
6869:
6864:
6862:Orbit equation
6859:
6851:
6846:
6841:
6836:
6831:
6826:
6821:
6816:
6811:
6806:
6800:
6798:
6790:
6789:
6787:
6786:
6781:
6776:
6771:
6766:
6761:
6756:
6751:
6746:
6741:
6736:
6734:Gravity assist
6731:
6729:Delta-v budget
6726:
6721:
6716:
6710:
6708:
6702:
6701:
6698:
6697:
6695:
6694:
6686:
6680:
6671:
6662:
6660:Orbital period
6652:
6650:
6646:
6645:
6643:
6642:
6640:True longitude
6633:
6631:Mean longitude
6624:
6615:
6598:
6588:
6586:
6582:
6581:
6579:
6578:
6569:
6560:
6551:
6541:
6539:
6535:
6534:
6532:
6531:
6518:
6509:
6500:
6490:
6488:
6486:
6485:
6482:
6478:
6472:
6466:
6465:
6462:
6461:
6459:
6458:
6457:
6456:
6448:
6447:
6446:
6441:
6436:
6435:
6434:
6421:
6416:
6415:
6414:
6409:
6404:
6399:
6391:
6390:
6389:
6387:Areostationary
6384:
6379:
6370:
6368:
6362:
6361:
6359:
6358:
6356:Very low Earth
6353:
6348:
6343:
6338:
6333:
6328:
6323:
6318:
6313:
6308:
6303:
6298:
6297:
6296:
6291:
6284:Geosynchronous
6280:
6278:
6272:
6271:
6269:
6268:
6266:Transfer orbit
6263:
6262:
6261:
6256:
6246:
6241:
6236:
6231:
6226:
6224:Lagrange point
6221:
6216:
6207:
6202:
6197:
6192:
6183:
6178:
6173:
6167:
6165:
6158:
6152:
6151:
6146:Gravitational
6145:
6143:
6142:
6135:
6128:
6120:
6114:
6113:
6104:
6099:
6090:
6082:Astronomy Cast
6078:
6072:
6066:
6060:
6054:
6048:
6030:
6029:
6024:
6010:Leonhard Euler
6003:
6002:
6001:
5971:
5970:External links
5968:
5965:
5964:
5946:
5920:
5902:
5883:
5864:
5844:
5830:
5822:
5818:
5813:
5777:
5758:
5740:
5713:(3): 215â232.
5697:
5686:
5677:
5647:
5630:
5603:(4): 365â372.
5591:(2009-04-01).
5576:
5538:
5519:
5502:
5489:10.1086/381315
5465:(1): 283â284.
5449:
5436:10.1086/513736
5392:
5379:10.1086/427539
5332:
5307:
5288:(3): 442â447.
5266:
5235:(1): 762â770.
5215:
5164:
5097:
5079:
5060:
5037:
5011:
4989:
4966:
4929:
4905:
4880:
4859:
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4426:
4422:
4418:
4410:
4406:
4375:
4371:
4351:
4338:
4331:
4312:
4307:The satellite
4300:
4297:
4292:Main article:
4287:
4284:
4281:
4280:
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4263:
4258:
4249:
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3692:
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3675:
3666:
3658:
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3613:
3609:
3605:
3604:Although the L
3588:
3585:
3571:
3567:
3563:
3559:
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3529:
3506:
3502:
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3429:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3400:
3396:
3392:
3389:
3386:
3383:
3376:
3372:
3368:
3362:
3359:
3356:
3353:
3350:
3347:
3340:
3336:
3329:
3325:
3321:
3315:
3312:
3309:
3293:
3290:
3278:center of mass
3260:
3256:
3246:
3243:
3239:
3236:
3231:
3227:
3213:
3210:
3204:
3201:
3195:
3192:
3189:
3175:
3168:
3153:
3145:
3141:
3133:
3126:
3107:
3103:
3096:
3092:
3088:
3083:
3079:
3071:
3067:
3064:
3061:
3058:
3055:
3047:
3043:
3039:
3034:
3030:
3023:
3019:
3012:
3008:
3001:
2996:
2992:
2989:
2986:
2983:
2979:
2972:
2968:
2962:
2955:
2950:
2946:
2943:
2940:
2936:
2929:
2925:
2910:
2906:
2903:
2900:
2895:
2886:, to follow a
2875:
2821:
2817:
2800:
2795:
2792:
2786:
2783:
2780:
2764:
2760:
2753:
2736:
2733:
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2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2695:
2691:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2664:
2660:
2656:
2653:
2650:
2647:
2644:
2641:
2636:
2632:
2628:
2623:
2619:
2606:
2590:
2586:
2579:
2575:
2571:
2566:
2562:
2554:
2550:
2547:
2544:
2536:
2532:
2528:
2523:
2519:
2512:
2508:
2501:
2497:
2490:
2486:
2480:
2476:
2470:
2462:
2458:
2454:
2451:
2448:
2445:
2439:
2435:
2420:
2407:point for the
2404:
2396:
2393:
2390:
2378:
2372:
2367:
2364:
2361:
2354:
2350:
2346:
2341:
2337:
2332:
2325:
2322:
2319:
2316:
2309:
2305:
2301:
2298:
2294:
2274:
2267:
2260:
2253:
2247:orbital period
2238:
2231:
2224:
2217:
2200:
2195:
2190:
2185:
2181:
2175:
2168:
2164:
2160:
2157:
2152:
2147:
2142:
2137:
2133:
2127:
2120:
2116:
2103:
2099:
2079:
2075:
2069:
2065:
2059:
2056:
2049:
2045:
2039:
2035:
2007:
2002:
1999:
1993:
1990:
1987:
1967:
1963:
1959:
1952:
1935:
1932:
1927:
1924:
1911:
1892:
1888:
1884:
1879:
1875:
1868:
1864:
1858:
1855:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1803:
1799:
1795:
1792:
1789:
1786:
1781:
1777:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1750:
1746:
1742:
1739:
1736:
1733:
1730:
1727:
1722:
1718:
1689:
1685:
1678:
1667:
1647:
1643:
1636:
1632:
1628:
1623:
1619:
1611:
1607:
1604:
1601:
1593:
1589:
1585:
1580:
1576:
1569:
1565:
1558:
1554:
1547:
1543:
1537:
1533:
1527:
1519:
1515:
1511:
1508:
1505:
1502:
1496:
1492:
1477:
1473:
1470:
1467:
1411:
1408:
1388:
1387:
1380:companion star
1375:
1364:
1360:
1356:
1345:
1341:
1333:
1325:
1305:
1301:
1296:Saturn's moon
1294:
1247:
1239:
1235:
1228:
1216:
1212:
1209:
1183:
1179:
1163:
1151:
1129:
1125:
1113:
1109:
1102:Main article:
1099:
1096:
1087:
1083:
1079:
1071:
1067:
1063:
1047:
1036:
1025:
1021:
1017:
1014:
1009:
1008:) or behind (L
1005:
997:
993:
986:
978:
974:
970:
967:
962:
953:
949:
945:
941:
938:
919:
911:
907:
903:
899:
895:
891:
887:
884:
875:
871:
851:
844:
837:
830:
823:
819:
815:
812:
801:
798:
775:Leonhard Euler
770:
766:
762:
758:
755:
745:
737:
722:
718:
714:
706:
694:
690:
679:
675:
667:
663:
648:
644:
640:
636:
632:
624:
620:
530:
529:
527:
526:
519:
512:
504:
501:
500:
497:
496:
491:
485:
482:
481:
478:
477:
474:
473:
468:
466:Gravity assist
462:
459:
458:
455:
454:
451:
450:
445:
440:
435:
429:
426:
425:
422:
421:
414:
411:
410:
407:
406:
401:
393:
392:
384:
380:
379:
374:
373:
370:
369:
366:
365:
360:
355:
350:
344:
341:
340:
337:
336:
329:
326:
325:
322:
321:
316:
311:
306:
301:
299:Orbital period
296:
291:
286:
281:
275:
272:
271:
268:
267:
264:
263:
261:Decaying orbit
258:
253:
248:
240:
239:
233:
226:
224:Transfer orbit
222:
221:
220:
218:Elliptic orbit
215:
213:Circular orbit
209:
200:
199:
196:
195:
192:
191:
186:
181:
176:
171:
166:
161:
156:
150:
145:
144:
141:
140:
133:
130:
129:
121:
120:
116:
115:
68:
64:
22:Lagrange Point
15:
13:
10:
9:
6:
4:
3:
2:
7786:
7775:
7772:
7770:
7767:
7766:
7764:
7749:
7745:
7740:
7738:
7729:
7728:
7725:
7719:
7716:
7714:
7711:
7709:
7706:
7704:
7701:
7700:
7698:
7696:
7692:
7686:
7683:
7681:
7678:
7676:
7673:
7671:
7667:
7664:
7662:
7659:
7657:
7656:Direct ascent
7654:
7653:
7651:
7649:
7645:
7639:
7638:Intergalactic
7636:
7634:
7631:
7629:
7626:
7622:
7619:
7617:
7614:
7613:
7612:
7609:
7607:
7604:
7603:
7601:
7597:
7589:
7586:
7584:
7581:
7579:
7576:
7574:
7573:Rocket engine
7571:
7570:
7569:
7566:
7562:
7559:
7557:
7554:
7552:
7549:
7547:
7544:
7542:
7539:
7537:
7534:
7533:
7532:
7529:
7527:
7524:
7520:
7517:
7515:
7512:
7510:
7507:
7506:
7505:
7504:Space capsule
7502:
7500:
7497:
7495:
7492:
7491:
7489:
7487:
7483:
7473:
7470:
7468:
7467:Space nursing
7465:
7463:
7460:
7458:
7455:
7451:
7448:
7447:
7446:
7443:
7441:
7438:
7434:
7431:
7430:
7429:
7426:
7425:
7423:
7421:Health issues
7419:
7413:
7410:
7408:
7405:
7403:
7400:
7398:
7395:
7393:
7390:
7386:
7383:
7382:
7381:
7378:
7376:
7375:Space Shuttle
7373:
7369:
7366:
7364:
7361:
7360:
7359:
7356:
7354:
7351:
7349:
7346:
7344:
7341:
7339:
7336:
7334:
7331:
7330:
7328:
7324:
7318:
7315:
7313:
7310:
7308:
7307:Space tourism
7305:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7281:
7278:
7276:
7273:
7271:
7268:
7267:
7266:
7263:
7259:
7256:
7255:
7254:
7251:
7250:
7248:
7244:
7241:
7239:
7235:
7229:
7228:Space weather
7226:
7224:
7221:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
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7187:
7184:
7182:
7179:
7177:
7174:
7172:
7169:
7168:
7167:
7164:
7160:
7157:
7155:
7152:
7150:
7147:
7145:
7142:
7141:
7140:
7137:
7135:
7132:
7131:
7129:
7127:
7123:
7115:
7112:
7111:
7110:
7107:
7103:
7100:
7098:
7095:
7093:
7092:Space command
7090:
7089:
7088:
7087:Space warfare
7085:
7081:
7078:
7076:
7073:
7071:
7068:
7066:
7063:
7061:
7058:
7057:
7056:
7053:
7049:
7048:United States
7046:
7044:
7041:
7039:
7036:
7034:
7031:
7029:
7026:
7024:
7021:
7019:
7016:
7014:
7011:
7009:
7006:
7005:
7004:
7001:
6999:
6996:
6992:
6989:
6987:
6984:
6982:
6979:
6977:
6974:
6973:
6972:
6969:
6967:
6966:Astrodynamics
6964:
6963:
6961:
6957:
6953:
6946:
6941:
6939:
6934:
6932:
6927:
6926:
6923:
6911:
6903:
6902:
6899:
6893:
6890:
6888:
6885:
6883:
6880:
6878:
6875:
6873:
6870:
6868:
6865:
6863:
6860:
6858:
6857:-body problem
6856:
6852:
6850:
6847:
6845:
6842:
6840:
6837:
6835:
6832:
6830:
6827:
6825:
6822:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6801:
6799:
6797:
6791:
6785:
6782:
6780:
6777:
6775:
6772:
6770:
6767:
6765:
6762:
6760:
6759:Oberth effect
6757:
6755:
6752:
6750:
6747:
6745:
6742:
6740:
6737:
6735:
6732:
6730:
6727:
6725:
6722:
6720:
6717:
6715:
6712:
6711:
6709:
6707:
6703:
6693:
6685:
6681:
6679:
6678:Orbital speed
6672:
6670:
6663:
6661:
6654:
6653:
6651:
6647:
6641:
6634:
6632:
6625:
6623:
6616:
6614:
6599:
6597:
6590:
6589:
6587:
6583:
6577:
6570:
6568:
6561:
6559:
6552:
6550:
6543:
6542:
6540:
6536:
6530:
6519:
6517:
6510:
6508:
6501:
6499:
6492:
6491:
6489:
6483:
6480:
6479:
6476:
6473:
6471:
6467:
6455:
6452:
6451:
6449:
6445:
6442:
6440:
6437:
6433:
6432:Earth's orbit
6430:
6429:
6428:
6425:
6424:
6422:
6420:
6417:
6413:
6410:
6408:
6405:
6403:
6400:
6398:
6395:
6394:
6392:
6388:
6385:
6383:
6380:
6378:
6375:
6374:
6372:
6371:
6369:
6363:
6357:
6354:
6352:
6349:
6347:
6344:
6342:
6339:
6337:
6334:
6332:
6329:
6327:
6324:
6322:
6319:
6317:
6314:
6312:
6309:
6307:
6304:
6302:
6299:
6295:
6292:
6290:
6289:Geostationary
6287:
6286:
6285:
6282:
6281:
6279:
6277:
6273:
6267:
6264:
6260:
6257:
6255:
6252:
6251:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6227:
6225:
6222:
6220:
6217:
6215:
6211:
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6187:
6184:
6182:
6179:
6177:
6174:
6172:
6169:
6168:
6166:
6162:
6159:
6157:
6153:
6149:
6141:
6136:
6134:
6129:
6127:
6122:
6121:
6118:
6112:
6110:
6105:
6103:
6100:
6098:
6094:
6091:
6089:
6088:Pamela L. Gay
6085:
6083:
6079:
6076:
6073:
6070:
6067:
6064:
6061:
6058:
6055:
6052:
6049:
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5956:
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5916:
5915:SpaceNews.com
5912:
5906:
5903:
5898:
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5884:
5879:
5875:
5868:
5865:
5860:
5856:
5852:
5848:
5840:
5834:
5831:
5827:
5803:on 2014-06-24
5799:
5795:
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5781:
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5690:
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5554:
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5506:
5503:
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5486:
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5473:
5468:
5464:
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5453:
5450:
5445:
5441:
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5407:
5403:
5396:
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5388:
5384:
5380:
5376:
5372:
5368:
5363:
5358:
5354:
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5346:
5342:
5336:
5333:
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5308:
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5230:
5226:
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5205:
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5192:
5187:
5183:
5179:
5175:
5168:
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5151:
5146:
5142:
5138:
5133:
5128:
5124:
5120:
5116:
5112:
5108:
5105:Hui, Man-To;
5101:
5098:
5093:
5089:
5083:
5080:
5075:
5071:
5064:
5061:
5056:. NASA. 1998.
5052:
5046:
5044:
5042:
5038:
5025:
5021:
5015:
5012:
5007:
5003:
4999:
4993:
4990:
4982:
4981:
4976:
4970:
4967:
4955:on 2008-05-27
4954:
4950:
4949:
4944:
4940:
4939:Lo, Martin W.
4933:
4930:
4919:
4915:
4914:"About Orbit"
4909:
4906:
4894:
4890:
4884:
4881:
4876:
4872:
4869:
4863:
4860:
4844:
4837:
4830:
4828:
4824:
4817:
4809:
4804:
4765:
4763:
4759:
4752:
4747:
4746:Oberth effect
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4715:
4713:
4710:
4708:
4705:
4703:
4700:
4698:
4695:
4693:
4690:
4689:
4684:
4673:
4670:
4659:
4654:
4652:
4646:
4638:
4636:
4634:
4630:
4622:
4618:
4610:
4608:
4606:
4593:
4589:
4576:
4574:
4570:
4566:
4557:
4555:
4554:gravitational
4547:
4543:
4542:space station
4531:
4529:
4523:
4515:
4512:
4503:
4497:
4493:
4485:
4481:
4478:
4474:
4473:Counter-Earth
4465:
4463:
4462:time constant
4459:
4458:saddle points
4446:
4440:
4436:
4432:
4416:
4403:
4401:
4397:
4393:
4389:
4385:
4381:
4369:
4365:
4361:
4360:space weather
4357:
4349:
4345:
4329:
4326:(yellow) and
4325:
4320:
4310:
4305:
4298:
4295:
4290:
4285:
4273:
4264:
4259:
4250:
4245:
4236:
4227:
4223:
4214:
4205:
4200:
4191:
4186:
4177:
4168:
4164:
4155:
4146:
4141:
4132:
4127:
4118:
4109:
4105:
4096:
4091:
4086:
4081:
4076:
4071:
4066:
4062:
4050:
4045:
4040:
4035:
4030:
4025:
4020:
4016:
4007:
4002:
3997:
3992:
3987:
3982:
3977:
3973:
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3954:
3949:
3944:
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3934:
3930:
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3908:
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3888:
3884:
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3870:
3865:
3860:
3855:
3847:
3842:
3838:
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3740:
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3711:
3709:
3699:
3685:
3678:
3670:
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3645:
3641:
3637:
3633:
3629:
3625:
3621:
3620:
3599:
3594:
3586:
3580:
3576:
3558:
3555:; force from
3551:
3547:
3543:
3542:sign function
3539:
3535:
3528:
3524:
3504:
3500:
3488:
3483:
3479:
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3460:
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3415:
3409:
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3354:
3348:
3345:
3338:
3334:
3327:
3323:
3319:
3313:
3310:
3307:
3299:
3291:
3289:
3287:
3283:
3279:
3275:
3271:
3267:
3252:
3237:
3235:
3224:
3211:
3208:
3202:
3199:
3193:
3190:
3187:
3179:
3174:
3167:
3163:
3160: â
3159:
3151:
3139:
3132:
3125:
3105:
3101:
3094:
3090:
3086:
3081:
3077:
3069:
3065:
3062:
3059:
3056:
3053:
3045:
3041:
3037:
3032:
3028:
3021:
3017:
3010:
3006:
2999:
2994:
2990:
2987:
2984:
2981:
2977:
2970:
2966:
2960:
2953:
2948:
2944:
2941:
2938:
2934:
2927:
2923:
2901:
2899:
2893:
2889:
2885:
2881:
2814:
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2793:
2790:
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2781:
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2770:
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2679:
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2673:
2670:
2662:
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2654:
2648:
2645:
2642:
2634:
2630:
2626:
2621:
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2588:
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2577:
2573:
2569:
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2560:
2552:
2548:
2545:
2542:
2534:
2530:
2526:
2521:
2517:
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2499:
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2478:
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2468:
2460:
2452:
2449:
2446:
2437:
2433:
2414:
2410:
2401:
2391:
2389:
2376:
2370:
2362:
2352:
2348:
2344:
2339:
2335:
2330:
2323:
2317:
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2303:
2299:
2296:
2292:
2277:, divided by
2273:
2266:
2263:, is that of
2259:
2252:
2248:
2243:
2237:
2230:
2223:
2216:
2198:
2193:
2188:
2183:
2179:
2173:
2166:
2162:
2158:
2155:
2150:
2145:
2140:
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2114:
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2077:
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2063:
2057:
2054:
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2043:
2037:
2033:
2021:
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2000:
1997:
1991:
1988:
1985:
1977:
1973:
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1951:
1933:
1930:
1925:
1922:
1914:
1890:
1886:
1882:
1877:
1873:
1866:
1862:
1856:
1853:
1833:
1830:
1827:
1824:
1821:
1815:
1812:
1806:
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1784:
1779:
1775:
1768:
1765:
1762:
1759:
1753:
1748:
1744:
1737:
1734:
1731:
1725:
1720:
1716:
1707:
1706:
1702:
1699:
1695:
1684:
1677:
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1665:
1645:
1641:
1634:
1630:
1626:
1621:
1617:
1609:
1605:
1602:
1599:
1591:
1587:
1583:
1578:
1574:
1567:
1563:
1556:
1552:
1545:
1541:
1535:
1531:
1525:
1517:
1509:
1506:
1503:
1494:
1490:
1468:
1466:
1464:
1460:
1456:
1452:
1447:
1445:
1441:
1437:
1433:
1426:
1421:
1416:
1409:
1407:
1405:
1401:
1397:
1396:3753 Cruithne
1393:
1385:
1381:
1373:
1369:
1365:
1354:
1350:
1346:
1339:
1331:
1323:
1319:
1315:
1311:
1299:
1295:
1291:
1280:
1269:
1259:
1255:
1251:
1248:
1245:
1233:
1229:
1226:
1222:
1210:
1206:
1195:
1177:
1176:
1175:
1171:
1169:
1161:
1157:
1149:
1145:
1141:
1140:
1135:
1123:
1119:
1105:
1097:
1095:
1093:
1077:
1060:
1058:
1046:
1035:
1015:
1013:
1003:
983:
968:
966:
960:
939:
937:
935:
934:
929:
925:
917:
885:
883:
881:
869:
865:
861:
857:
850:
843:
836:
829:
813:
811:
807:
799:
797:
795:
791:
787:
782:
780:
776:
756:
754:
751:
743:
735:
731:
726:
712:
704:
700:
687:
685:
673:
660:
658:
654:
630:
617:
615:
611:
607:
606:gravitational
602:
600:
596:
592:
591:gravitational
588:
584:
580:
574:
541:
537:
525:
520:
518:
513:
511:
506:
505:
503:
502:
495:
492:
490:
487:
486:
480:
479:
472:
471:Oberth effect
469:
467:
464:
463:
457:
456:
449:
446:
444:
441:
439:
436:
434:
431:
430:
424:
423:
419:
412:
405:
402:
400:
397:
396:
390:
386:
385:
383:
377:
376:N-body orbits
372:
371:
364:
361:
359:
358:Perturbations
356:
354:
351:
349:
346:
345:
339:
338:
334:
327:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
276:
270:
269:
262:
259:
257:
254:
252:
249:
247:
244:
243:
237:
234:
232:
228:
227:
225:
219:
216:
214:
211:
210:
204:
198:
197:
190:
187:
185:
182:
180:
179:Orbital nodes
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
151:
148:
143:
142:
138:
131:
127:
123:
122:
119:Astrodynamics
117:
113:
112:
106:
97:
92:
81:
74:
54:
50:
46:
41:
33:
29:
25:
23:
7648:Space launch
7633:Interstellar
7599:Destinations
7368:ApolloâSoyuz
7317:Space diving
7302:Space toilet
7126:Applications
7043:Soviet Union
7003:Space policy
6998:Space launch
6872:Perturbation
6854:
6848:
6829:Ground track
6739:Gravity turn
6690:
6683:
6676:
6667:
6658:
6638:
6629:
6620:
6613:True anomaly
6611:
6596:Mean anomaly
6594:
6574:
6565:
6556:
6547:
6527:
6514:
6505:
6498:Eccentricity
6496:
6454:Lunar cycler
6427:Heliocentric
6367:other points
6316:Medium Earth
6223:
6214:Non-inclined
6108:
6081:
6033:
6031:
5976:
5958:
5949:
5937:. Retrieved
5933:the original
5923:
5914:
5905:
5896:
5886:
5877:
5867:
5850:
5833:
5811:
5805:. Retrieved
5798:the original
5793:
5780:
5771:
5761:
5743:
5710:
5706:
5689:
5680:
5668:. Retrieved
5664:the original
5659:
5656:"ISEE-3/ICE"
5650:
5638:
5633:
5600:
5596:
5579:
5569:February 28,
5567:. Retrieved
5560:the original
5532:
5522:
5505:
5462:
5458:
5452:
5409:
5405:
5395:
5352:
5348:
5335:
5324:. Retrieved
5310:
5285:
5279:
5269:
5232:
5228:
5218:
5181:
5177:
5167:
5122:
5118:
5100:
5092:www.nasa.gov
5091:
5082:
5073:
5063:
5028:. Retrieved
5024:the original
5014:
5005:
4992:
4979:
4969:
4957:. Retrieved
4953:the original
4947:
4932:
4921:. Retrieved
4917:
4908:
4897:. Retrieved
4892:
4883:
4874:
4862:
4850:. Retrieved
4843:the original
4642:
4614:
4578:EarthâMoon L
4577:
4559:EarthâMoon L
4558:
4536:EarthâMoon L
4535:
4504:
4466:
4447:
4404:
4400:NEO Surveyor
4378:include the
4336:
4289:
4225:SunâNeptune
4064:SunâJupiter
3886:SunâMercury
3723:
3671:
3663:interference
3648:
3632:Solar System
3630:such as the
3623:
3618:
3603:
3598:STL 3D model
3556:
3549:
3545:
3537:
3533:
3526:
3522:
3297:
3295:
3254:
3225:
3180:
3172:
3165:
3161:
3157:
3149:
3137:
3130:
3123:
2908:
2878:one sees an
2815:
2771:, given by:
2757:
2750:
2748:
2418:
2271:
2264:
2257:
2250:
2244:
2235:
2228:
2221:
2214:
2022:
1978:, given by:
1971:
1956:
1949:
1909:
1708:
1696:is the only
1693:
1682:
1675:
1671:
1663:
1475:
1448:
1429:
1389:
1368:binary stars
1254:Mars trojans
1172:
1155:
1137:
1107:
1062:The points L
1061:
1044:
1033:
1019:
991:
947:
931:
893:
848:
847:and that of
841:
834:
827:
821:
809:
783:
760:
727:
688:
674:near their L
661:
628:
618:
603:
582:
578:
539:
533:
381:
256:Radial orbit
207:eccentricity
189:True anomaly
174:Mean anomaly
164:Eccentricity
95:
45:contour plot
28:
24:(video game)
21:
7606:Sub-orbital
7541:Space probe
7407:New Shepard
7385:ShuttleâMir
7144:Archaeology
7097:Space force
7080:Moon Treaty
6952:Spaceflight
6834:Hill sphere
6669:Mean motion
6549:Inclination
6538:Orientation
6439:Mars cycler
6377:Areocentric
6249:Synchronous
6111:, Chapter 5
5981:Essai (PDF)
5000:(1867â92).
4702:Gegenschein
4556:influence.
4484:comic books
4467:SunâEarth L
4448:SunâEarth L
4405:SunâEarth L
4370:) reaches L
4337:SunâEarth L
4166:SunâUranus
4107:SunâSaturn
3840:EarthâMoon
3773:is located
3761:is located
3619:halo orbits
3282:equilibrium
3148:cases, and
2769:Hill sphere
2102:or at the L
1976:Hill sphere
1382:, known as
1338:azimuthally
1258:5261 Eureka
1223:, known as
1168:Trojan camp
1057:kidney bean
701:called the
587:equilibrium
389:Halo orbits
353:Hill sphere
169:Inclination
73:high points
7763:Categories
7675:Launch pad
7666:Expendable
7616:Geocentric
7583:Solar sail
7526:Spaceplane
7486:Spacecraft
7280:Space suit
7258:commercial
7186:Television
6981:Space Race
6774:Rendezvous
6470:Parameters
6306:High Earth
6276:Geocentric
6229:Osculating
6186:Elliptical
6077:âTony Dunn
6065:âJohn Baez
5939:1 February
5826:penalties.
5807:2011-01-25
5326:2010-10-27
5242:1910.07471
5191:1910.07466
5132:2111.05058
5125:(2): L25.
5020:"L2 Orbit"
4959:2008-06-09
4923:2022-01-01
4899:2021-10-27
4818:References
4800:(sequence
4592:L5 Society
4532:EarthâMoon
4364:solar wind
3975:SunâEarth
3932:SunâVenus
3790:Body pair
3739:barycenter
3266:barycenter
2892:halo orbit
2824:is arcsin(
2094:Since the
1436:barycenter
1400:Epimetheus
1372:Roche lobe
1330:Polydeuces
1328:point and
1160:Greek camp
1148:Trojan War
959:barycenter
804:See also:
750:halo orbit
433:Mass ratio
348:Barycenter
7685:Spaceport
7536:Satellite
7253:Astronaut
7181:Telephone
7134:Astronomy
7055:Space law
7008:Australia
6819:Ephemeris
6796:mechanics
6706:Maneuvers
6649:Variation
6412:Libration
6407:Lissajous
6311:Low Earth
6301:Graveyard
6200:Horseshoe
5983:; source
5772:SpaceNews
5735:121179935
5670:August 8,
5625:121374703
5159:243860678
5074:Space.com
5030:28 August
4768:Actually
4611:SunâVenus
4498:of this L
4299:SunâEarth
4018:SunâMars
3835:/SMA (%)
3807:/SMA (%)
3587:Stability
3540:) is the
3476:−
3419:−
3410:
3388:−
3349:
3314:−
3270:resultant
3209:μ
3191:≈
3178:), then:
3063:−
2988:−
2942:−
2791:μ
2782:≈
2729:μ
2726:−
2720:μ
2708:−
2702:μ
2686:−
2680:μ
2674:−
2649:μ
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7171:Internet
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6176:Capture
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