39:
2660:
31:
66:
5344:
4369:
4129:
2845:
If the mass of the central body is not known, its standard gravitational parameter, and hence its mass, can be determined by the deflection of the smaller body together with the impact parameter and approach speed. Because typically all these variables can be determined accurately, a spacecraft flyby
540:
Like an elliptical orbit, a hyperbolic trajectory for a given system can be defined (ignoring orientation) by its semi major axis and the eccentricity. However, with a hyperbolic orbit other parameters may be more useful in understanding a body's motion. The following table lists the main parameters
46:
of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases as the speed decreases and distance increases according to Kepler's laws. The part of the kinetic energy that remains above zero total energy is
2663:
Hyperbolic trajectories followed by objects approaching central object (small dot) with same hyperbolic excess velocity (and semi-major axis (=1)) and from same direction but with different impact parameters and eccentricities. The yellow line indeed passes around the central dot, approaching it
2679:
In the situation of a spacecraft or comet approaching a planet, the impact parameter and excess velocity will be known accurately. If the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planet's radius an
2825:
4364:{\displaystyle \delta =2\arcsin \left({\frac {{\sqrt {1-{\frac {p_{e}}{R_{\text{SOI}}}}}}{\sqrt {1+{\frac {p_{e}}{R_{\text{SOI}}}}-{\frac {2\mu p_{e}}{v_{\infty }^{2}R_{\text{SOI}}^{2}}}}}}{1+{\frac {v_{\infty }^{2}p_{e}}{\mu }}-{\frac {2p_{e}}{R_{\text{SOI}}}}}}\right)}
2085:) is not immediately visible with an hyperbolic trajectory but can be constructed as it is the distance from periapsis to the point where the two asymptotes cross. Usually, by convention, it is negative, to keep various equations consistent with elliptical orbits.
3964:
above that needed to accelerate to the escape speed results in a relatively large speed at infinity. For example, at a place where escape speed is 11.2 km/s, the addition of 0.4 km/s yields a hyperbolic excess speed of 3.02 km/s.
3331:
3248:
4025:. The converse is also true - a body does not need to be slowed by much compared to its hyperbolic excess speed (e.g. by atmospheric drag near periapsis) for velocity to fall below escape velocity and so for the body to be captured.
1429:
4383:, trajectories of objects with enough energy to escape the gravitational pull of the other no longer are shaped like a hyperbola. Nonetheless, the term "hyperbolic trajectory" is still used to describe orbits of this type.
3574:
3165:
4041:. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1 this is not a parabolic orbit.
3688:
2686:
3467:
3065:
2244:
2594:
770:
3952:
2905:
4016:
2842:(radius 70000 km) from the outer Solar System with a speed of 5.5 km/s, will need the impact parameter to be at least 770,000 km or 11 times Jupiter radius to avoid collision.
2947:
2649:
2015:
2297:
718:
3494:
The flight path angle (φ) is the angle between the direction of velocity and the perpendicular to the radial direction, so it is zero at periapsis and tends to 90 degrees at infinity.
840:
1779:
1324:
893:
1256:
1032:
2520:
2489:
2456:
the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line.
1939:
1654:
1506:
1139:
988:
1883:
1196:
1101:
509:
Under simplistic assumptions a body traveling along this trajectory will coast towards infinity, settling to a final excess velocity relative to the central body. Similarly to
3872:
2170:
4094:
2043:
2428:
3393:
1970:
1807:
3843:
2331:
2111:
1065:
584:
1594:
1842:
4067:
2984:
1728:
3715:
2454:
653:
4121:
3770:
3253:
3170:
2400:) is greater than 1. The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At
2362:
2141:
2083:
1689:
1551:
1287:
924:
3803:
3742:
3607:
3005:
2398:
1911:
1462:
799:
624:
604:
1523:
1330:
2838:) is to avoid a collision with Earth, the impact parameter will need to be at least 8600 km, or 34% more than the Earth's radius. A body approaching
3500:
3091:
3626:
2834:(effective radius ~6400 km) with a velocity of 12.5 km/s (the approximate minimum approach speed of a body coming from the outer
2676:. With bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola.
2820:{\displaystyle r_{p}=-a(e-1)={\frac {\mu }{v_{\infty }^{2}}}\left({\sqrt {1+\left(b{\frac {v_{\infty }^{2}}{\mu }}\right)^{2}}}-1\right)}
541:
describing the path of body following a hyperbolic trajectory around another under standard assumptions and the formula connecting them.
495:
4543:
38:
4548:
4538:
4480:
3404:
461:
2367:
Note that the total energy is positive in the case of a hyperbolic trajectory (whereas it is negative for an elliptical orbit).
494:
with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to
5221:
4380:
3017:
2178:
4445:
2528:
724:
5281:
4469:. Porto Alegre: Department of Astronomy - Institute of Physics of Federal University of Rio Grande do Sul. pp. 97–106.
3880:
1976:
233:
2852:
2143:
of the orbit, and to the velocity the body attains at as the distance tends to infinity, the hyperbolic excess velocity (
529:
302:
3971:
2910:
2599:
5276:
5156:
4575:
3718:
3474:
2334:
630:
17:
4523:
1983:
5241:
4995:
2249:
659:
5314:
4953:
4944:
4681:
123:
805:
1734:
1293:
846:
2672:
is the distance by which a body, if it continued on an unperturbed path, would miss the central body at its
5261:
4731:
2659:
1202:
994:
2494:
5206:
2466:
1917:
1600:
1468:
454:
387:
1114:
930:
5186:
5013:
1848:
1145:
1071:
5324:
4422:
3848:
2146:
1889:
4072:
2021:
5309:
4834:
2403:
382:
297:
3357:
1945:
1792:
5319:
4627:
3812:
2680:
impact should be expected. The distance of closest approach, or periapsis distance, is given by:
2305:
2095:
2089:
1785:
1043:
562:
518:
253:
1557:
5181:
4783:
4703:
4691:
447:
170:
3326:{\displaystyle \tanh {\frac {E}{2}}={\sqrt {\frac {e-1}{e+1}}}\cdot \tan {\frac {\theta }{2}}}
3243:{\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}}
1813:
5304:
5246:
5216:
5004:
4881:
4849:
4819:
4778:
4763:
4052:
2969:
2114:
2058:
1695:
355:
98:
3699:
2433:
637:
42:
A hyperbolic trajectory is depicted in the bottom-right quadrant of this diagram, where the
5329:
5151:
4935:
4824:
4793:
4721:
4696:
4671:
4632:
4613:
4568:
4099:
3753:
3348:
2522:
is the external angle between approach and departure directions (between asymptotes). Then
2459:
The angle between the direction of periapsis and an asymptote from the central body is the
2376:
2340:
2119:
2066:
1667:
1536:
1269:
907:
776:
510:
503:
228:
185:
175:
103:
3780:
Under standard assumptions, at any position in the orbit the following relation holds for
8:
5191:
4986:
4726:
4506:
4417:
3787:
3726:
3591:
2989:
2382:
479:
270:
108:
4864:
4753:
4651:
1896:
1447:
1424:{\displaystyle 2\sin ^{-1}{\bigg (}{\frac {1}{(1+r_{p}v_{\infty }^{2}/\mu )}}{\bigg )}}
784:
609:
589:
343:
218:
4484:
1512:
5231:
5129:
5059:
4814:
4768:
4686:
4441:
4034:
3614:
3075:
1529:
514:
258:
195:
74:
5211:
5143:
4907:
4869:
4743:
4713:
4666:
2673:
2669:
1435:
428:
377:
142:
86:
5368:
5251:
4844:
4748:
4738:
4637:
4561:
4412:
4038:
3806:
3773:
3482:
1439:
899:
433:
338:
248:
223:
3569:{\displaystyle \tan(\phi )={\frac {e\cdot \sin \theta }{1+e\cdot \cos \theta }}}
5347:
5299:
5291:
5286:
5171:
5166:
5097:
5077:
5068:
4661:
4647:
4623:
4618:
4593:
4524:
https://orbital-mechanics.space/the-orbit-equation/hyperbolic-trajectories.html
4407:
4397:
3781:
3008:
525:
405:
321:
315:
238:
163:
157:
152:
3160:{\displaystyle \cosh {E}={{\cos {\theta }+e} \over {1+e\cdot \cos {\theta }}}}
2949:
is the angle the smaller body is deflected from a straight line in its course.
5362:
5201:
5196:
5115:
4758:
4676:
4022:
3585:
475:
410:
243:
200:
2364:
is characteristic energy, commonly used in planning interplanetary missions
5266:
5176:
5050:
5033:
4891:
4788:
4656:
4402:
3745:
3341:
2964:
2835:
2460:
491:
128:
118:
113:
43:
502:. In more technical terms this can be expressed by the condition that the
30:
5271:
5106:
4876:
4856:
4773:
2052:
292:
3683:{\displaystyle v={\sqrt {\mu \left({2 \over {r}}+{1 \over {a}}\right)}}}
2654:
2370:
4839:
4553:
372:
328:
287:
65:
5256:
4608:
499:
34:
The blue path in this image is an example of a hyperbolic trajectory
5161:
3958:
2839:
535:
4037:
where the relative speed of the two objects always exceeds the
4966:
4585:
4438:
4392:
4044:
2831:
1660:
93:
3462:{\displaystyle M={\sqrt {\frac {\mu }{-a^{3}}}}.(t-\tau ),}
3060:{\displaystyle r={\frac {\ell }{1+e\cdot \cos \theta }}}
2239:{\displaystyle v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a}
2986:
is linked to the distance between the orbiting bodies (
1515:
4522:
Orbital
Mechanics & Astrodynamics by Bryan Weber:
2655:
Impact parameter and the distance of closest approach
2589:{\displaystyle \theta {_{\infty }}=\cos ^{-1}(-1/e)\,}
2053:
Semi-major axis, energy and hyperbolic excess velocity
765:{\displaystyle bv_{\infty }^{2}\cot \theta _{\infty }}
4132:
4102:
4075:
4055:
3974:
3947:{\displaystyle v^{2}={v_{esc}}^{2}+{v_{\infty }}^{2}}
3883:
3851:
3815:
3790:
3756:
3729:
3702:
3629:
3594:
3503:
3407:
3360:
3256:
3173:
3094:
3020:
2992:
2972:
2913:
2855:
2689:
2602:
2531:
2497:
2469:
2436:
2406:
2385:
2371:
Eccentricity and angle between approach and departure
2343:
2308:
2252:
2181:
2149:
2122:
2098:
2069:
2024:
1986:
1948:
1920:
1899:
1851:
1816:
1795:
1737:
1698:
1670:
1603:
1560:
1539:
1471:
1450:
1333:
1296:
1272:
1205:
1148:
1117:
1074:
1046:
997:
933:
910:
849:
808:
787:
727:
662:
640:
612:
592:
565:
47:
that associated with the hyperbolic excess velocity.
3957:Note that this means that a relatively small extra
2900:{\displaystyle \mu =bv_{\infty }^{2}\tan \delta /2}
4363:
4115:
4088:
4061:
4010:
3946:
3866:
3837:
3797:
3764:
3736:
3709:
3682:
3601:
3568:
3461:
3387:
3325:
3242:
3159:
3059:
2999:
2978:
2941:
2899:
2819:
2643:
2588:
2514:
2483:
2448:
2422:
2392:
2356:
2325:
2291:
2238:
2164:
2135:
2105:
2077:
2037:
2009:
1964:
1933:
1905:
1877:
1836:
1801:
1773:
1722:
1683:
1648:
1588:
1545:
1517:
1500:
1456:
1423:
1318:
1281:
1250:
1190:
1133:
1095:
1059:
1026:
982:
918:
887:
834:
793:
764:
712:
647:
618:
598:
578:
4049:A more accurate formula for the deflection angle
4033:A radial hyperbolic trajectory is a non-periodic
3863:
3761:
2322:
2161:
2074:
1416:
1355:
915:
5360:
4374:
4011:{\displaystyle {\sqrt {11.6^{2}-11.2^{2}}}=3.02}
1262:Angle between asymptotes and the conjugate axis
4465:S.O., Kepler; Saraiva, Maria de Fátima (2014).
4028:
2942:{\displaystyle \delta =2\theta _{\infty }-\pi }
2846:will provide a good estimate of a body's mass.
2644:{\displaystyle e=-1/\cos \theta {_{\infty }}\,}
2088:The semi major axis is directly linked to the
521:of a hyperbolic trajectory orbit is positive.
4569:
4096:of the deflecting body, assuming a periapsis
536:Parameters describing a hyperbolic trajectory
455:
2010:{\displaystyle {\frac {\Delta A}{\Delta t}}}
4481:"Basics of Space Flight: Orbital Mechanics"
4464:
4069:considering the sphere of influence radius
2292:{\displaystyle a=-{\mu /{v_{\infty }^{2}}}}
5343:
4576:
4562:
4045:Deflection with finite sphere of influence
2430:the asymptotes are at right angles. With
713:{\displaystyle {\frac {v^{2}}{(2/r-1/a)}}}
462:
448:
3862:
3834:
3794:
3760:
3744:is radial distance of orbiting body from
3733:
3706:
3598:
3398:The mean anomaly is proportional to time
2996:
2640:
2585:
2511:
2480:
2389:
2321:
2160:
2102:
2073:
914:
644:
490:is the trajectory of any object around a
4583:
2658:
835:{\displaystyle {\frac {\ell }{r_{p}}}-1}
37:
29:
4435:
1774:{\displaystyle {\sqrt {a^{2}+b^{2}}}+a}
1319:{\displaystyle 2\theta _{\infty }-\pi }
888:{\displaystyle {\sqrt {1+b^{2}/a^{2}}}}
528:, can be described within the planet's
513:, all hyperbolic trajectories are also
14:
5361:
5222:Transposition, docking, and extraction
4381:two-body problem in general relativity
3081:(alternatively the hyperbolic anomaly
3070:The relation between the true anomaly
2953:
4557:
1251:{\displaystyle \pi +2\tan ^{-1}(b/a)}
1027:{\displaystyle -\mu /v_{\infty }^{2}}
4504:
3489:
2515:{\displaystyle 2\theta _{\infty }\,}
4440:. Hawthorne, CA.: Hawthorne Press.
2484:{\displaystyle \theta _{\infty }\,}
1934:{\displaystyle {\sqrt {\mu \ell }}}
1649:{\displaystyle -b^{2}/a=h^{2}/\mu }
1501:{\displaystyle -a{\sqrt {e^{2}-1}}}
1109:(External) Angle between asymptotes
24:
4296:
4250:
3932:
3857:
3845:) and hyperbolic excess velocity (
2928:
2870:
2778:
2737:
2634:
2540:
2506:
2475:
2277:
2187:
2155:
1998:
1990:
1957:
1857:
1390:
1305:
1264:of the hyperbolic path of approach
1134:{\displaystyle 2\theta _{\infty }}
1126:
1052:
1014:
983:{\displaystyle 1/(2/r-v^{2}/\mu )}
757:
736:
571:
25:
5380:
5282:Kepler's laws of planetary motion
4532:
4507:"Spacecraft Dynamics and Control"
4505:Peet, Matthew M. (13 June 2019).
2375:With a hyperbolic trajectory the
1878:{\displaystyle v_{\infty }^{2}/2}
1191:{\displaystyle 2\cos ^{-1}(-1/e)}
1096:{\displaystyle {\sqrt {-\mu /a}}}
545:Hyperbolic trajectory equations
498:such an orbit has the shape of a
234:Kepler's laws of planetary motion
5342:
5277:Interplanetary Transport Network
5157:Collision avoidance (spacecraft)
3719:standard gravitational parameter
2335:standard gravitational parameter
631:Standard gravitational parameter
64:
5242:Astronomical coordinate systems
4996:Longitude of the ascending node
3867:{\displaystyle v_{\infty }\,\!}
3584:Under standard assumptions the
3167: or
2963:In a hyperbolic trajectory the
2463:as distance tends to infinity (
2165:{\displaystyle v_{\infty }\,\!}
532:using hyperbolic trajectories.
5315:Retrograde and prograde motion
4516:
4498:
4473:
4458:
4089:{\displaystyle R_{\text{SOI}}}
3609:) of a body traveling along a
3516:
3510:
3453:
3441:
2721:
2709:
2582:
2565:
2038:{\displaystyle {\frac {h}{2}}}
1717:
1705:
1583:
1564:
1408:
1366:
1245:
1231:
1185:
1168:
977:
942:
704:
676:
13:
1:
4428:
4375:Relativistic two-body problem
4035:trajectory on a straight line
2423:{\displaystyle e={\sqrt {2}}}
5262:Equatorial coordinate system
4029:Radial hyperbolic trajectory
3388:{\displaystyle M=e\sinh E-E}
1965:{\displaystyle bv_{\infty }}
1802:{\displaystyle \varepsilon }
44:gravitational potential well
7:
4386:
3838:{\displaystyle {v_{esc}}\,}
3579:
2958:
2326:{\displaystyle \mu =Gm\,\!}
2106:{\displaystyle \epsilon \,}
1060:{\displaystyle v_{\infty }}
579:{\displaystyle v_{\infty }}
524:Planetary flybys, used for
388:Tsiolkovsky rocket equation
10:
5385:
5014:Longitude of the periapsis
4436:Vallado, David A. (2007).
4021:This is an example of the
2830:So if a comet approaching
2056:
1589:{\displaystyle a(1-e^{2})}
1038:Hyperbolic excess velocity
357:Engineering and efficiency
176:Bi-elliptic transfer orbit
5338:
5325:Specific angular momentum
5230:
5142:
5086:
5022:
4975:
4915:
4906:
4802:
4712:
4601:
4592:
4423:List of hyperbolic comets
3613:can be computed from the
1890:Specific angular momentum
4467:Astronomia e Astrofísica
3250: or
1837:{\displaystyle -\mu /2a}
526:gravitational slingshots
383:Propellant mass fraction
282:Gravitational influences
27:Concept in astrodynamics
5320:Specific orbital energy
4062:{\displaystyle \delta }
3475:gravitational parameter
2979:{\displaystyle \theta }
2090:specific orbital energy
1786:Specific orbital energy
1723:{\displaystyle -a(e-1)}
254:Specific orbital energy
4732:Geostationary transfer
4365:
4117:
4090:
4063:
4012:
3948:
3868:
3839:
3799:
3766:
3738:
3711:
3710:{\displaystyle \mu \,}
3684:
3603:
3570:
3463:
3389:
3336:The eccentric anomaly
3327:
3244:
3161:
3061:
3001:
2980:
2943:
2901:
2821:
2665:
2645:
2590:
2516:
2485:
2450:
2449:{\displaystyle e>2}
2424:
2394:
2358:
2327:
2293:
2240:
2166:
2137:
2107:
2079:
2039:
2011:
1977:Area swept up per time
1966:
1935:
1907:
1879:
1838:
1803:
1775:
1724:
1685:
1650:
1590:
1547:
1519:
1502:
1458:
1425:
1320:
1283:
1252:
1192:
1135:
1097:
1061:
1028:
984:
920:
889:
836:
795:
766:
714:
649:
648:{\displaystyle \mu \,}
620:
600:
580:
511:parabolic trajectories
171:Hohmann transfer orbit
48:
35:
5305:Orbital state vectors
5247:Characteristic energy
5217:Trans-lunar injection
5005:Argument of periapsis
4682:Prograde / Retrograde
4643:Hyperbolic trajectory
4366:
4118:
4116:{\displaystyle p_{e}}
4091:
4064:
4013:
3949:
3869:
3840:
3800:
3767:
3765:{\displaystyle a\,\!}
3739:
3712:
3685:
3611:hyperbolic trajectory
3604:
3571:
3464:
3390:
3328:
3245:
3162:
3062:
3002:
2981:
2944:
2902:
2822:
2662:
2646:
2591:
2517:
2486:
2451:
2425:
2395:
2359:
2357:{\displaystyle C_{3}}
2328:
2294:
2241:
2167:
2138:
2136:{\displaystyle C_{3}}
2115:characteristic energy
2108:
2080:
2078:{\displaystyle a\,\!}
2063:The semi major axis (
2059:Characteristic energy
2040:
2012:
1967:
1936:
1908:
1880:
1839:
1804:
1776:
1725:
1686:
1684:{\displaystyle r_{p}}
1651:
1591:
1548:
1546:{\displaystyle \ell }
1520:
1503:
1459:
1426:
1321:
1284:
1282:{\displaystyle 2\nu }
1253:
1193:
1136:
1098:
1062:
1029:
985:
921:
919:{\displaystyle a\,\!}
890:
837:
796:
767:
715:
650:
621:
601:
581:
506:is greater than one.
484:hyperbolic trajectory
367:Preflight engineering
99:Argument of periapsis
41:
33:
5152:Bi-elliptic transfer
4672:Parabolic trajectory
4130:
4100:
4073:
4053:
3972:
3881:
3849:
3813:
3788:
3754:
3727:
3700:
3627:
3592:
3501:
3405:
3358:
3254:
3171:
3092:
3018:
2990:
2970:
2911:
2853:
2687:
2600:
2529:
2495:
2467:
2434:
2404:
2383:
2377:orbital eccentricity
2341:
2306:
2250:
2179:
2147:
2120:
2096:
2067:
2022:
1984:
1946:
1918:
1897:
1849:
1814:
1793:
1735:
1696:
1668:
1601:
1558:
1537:
1513:
1469:
1448:
1331:
1294:
1270:
1203:
1146:
1115:
1072:
1044:
995:
931:
908:
847:
806:
785:
725:
660:
638:
610:
590:
563:
504:orbital eccentricity
423:Propulsive maneuvers
5192:Low-energy transfer
4418:Hyperbolic asteroid
4305:
4274:
4259:
3798:{\displaystyle v\,}
3737:{\displaystyle r\,}
3602:{\displaystyle v\,}
3000:{\displaystyle r\,}
2954:Equations of motion
2879:
2787:
2746:
2393:{\displaystyle e\,}
2286:
2196:
1866:
1399:
1023:
745:
546:
530:sphere of influence
515:escape trajectories
480:celestial mechanics
400:Efficiency measures
303:Sphere of influence
272:Celestial mechanics
54:Part of a series on
5187:Inclination change
4835:Distant retrograde
4379:In context of the
4361:
4291:
4260:
4245:
4113:
4086:
4059:
4008:
3944:
3864:
3835:
3795:
3772:is the (negative)
3762:
3734:
3707:
3680:
3599:
3566:
3459:
3385:
3340:is related to the
3323:
3240:
3157:
3057:
2997:
2976:
2939:
2897:
2865:
2817:
2773:
2732:
2666:
2641:
2586:
2512:
2481:
2446:
2420:
2390:
2354:
2323:
2289:
2272:
2236:
2182:
2162:
2133:
2103:
2075:
2035:
2007:
1962:
1931:
1903:
1875:
1852:
1834:
1799:
1771:
1720:
1681:
1661:Periapsis distance
1646:
1586:
1543:
1498:
1454:
1421:
1385:
1316:
1279:
1248:
1188:
1131:
1093:
1057:
1024:
1009:
980:
916:
885:
832:
791:
762:
731:
710:
645:
616:
596:
576:
544:
219:Dynamical friction
49:
36:
5356:
5355:
5330:Two-line elements
5138:
5137:
5060:Eccentric anomaly
4902:
4901:
4769:Orbit of the Moon
4628:Highly elliptical
4447:978-1-881883-14-2
4355:
4352:
4349:
4320:
4278:
4276:
4267:
4219:
4216:
4187:
4185:
4182:
4083:
4000:
3678:
3671:
3656:
3564:
3490:Flight path angle
3436:
3435:
3349:Kepler's equation
3321:
3302:
3301:
3271:
3238:
3219:
3218:
3188:
3155:
3076:eccentric anomaly
3055:
2804:
2791:
2747:
2418:
2050:
2049:
2033:
2005:
1929:
1906:{\displaystyle h}
1763:
1530:Semi-latus rectum
1496:
1457:{\displaystyle b}
1412:
1091:
883:
824:
794:{\displaystyle e}
708:
619:{\displaystyle b}
599:{\displaystyle a}
472:
471:
322:Lagrangian points
259:Vis-viva equation
229:Kepler's equation
76:Orbital mechanics
16:(Redirected from
5376:
5346:
5345:
5287:Lagrangian point
5182:Hohmann transfer
5127:
5113:
5104:
5095:
5075:
5066:
5057:
5048:
5044:
5040:
5031:
5011:
5002:
4993:
4984:
4964:
4960:
4951:
4942:
4933:
4913:
4912:
4882:Heliosynchronous
4831:Lagrange points
4784:Transatmospheric
4599:
4598:
4578:
4571:
4564:
4555:
4554:
4526:
4520:
4514:
4513:
4511:
4502:
4496:
4495:
4493:
4492:
4483:. Archived from
4477:
4471:
4470:
4462:
4451:
4370:
4368:
4367:
4362:
4360:
4356:
4354:
4353:
4351:
4350:
4347:
4341:
4340:
4339:
4326:
4321:
4316:
4315:
4314:
4304:
4299:
4289:
4280:
4279:
4277:
4275:
4273:
4268:
4265:
4258:
4253:
4243:
4242:
4241:
4225:
4220:
4218:
4217:
4214:
4208:
4207:
4198:
4190:
4188:
4186:
4184:
4183:
4180:
4174:
4173:
4164:
4156:
4153:
4122:
4120:
4119:
4114:
4112:
4111:
4095:
4093:
4092:
4087:
4085:
4084:
4081:
4068:
4066:
4065:
4060:
4017:
4015:
4014:
4009:
4001:
3999:
3998:
3986:
3985:
3976:
3953:
3951:
3950:
3945:
3943:
3942:
3937:
3936:
3935:
3921:
3920:
3915:
3914:
3913:
3893:
3892:
3873:
3871:
3870:
3865:
3861:
3860:
3844:
3842:
3841:
3836:
3833:
3832:
3831:
3804:
3802:
3801:
3796:
3782:orbital velocity
3771:
3769:
3768:
3763:
3743:
3741:
3740:
3735:
3716:
3714:
3713:
3708:
3689:
3687:
3686:
3681:
3679:
3677:
3673:
3672:
3670:
3662:
3657:
3655:
3647:
3637:
3608:
3606:
3605:
3600:
3575:
3573:
3572:
3567:
3565:
3563:
3540:
3523:
3468:
3466:
3465:
3460:
3437:
3434:
3433:
3432:
3416:
3415:
3394:
3392:
3391:
3386:
3332:
3330:
3329:
3324:
3322:
3314:
3303:
3300:
3289:
3278:
3277:
3272:
3264:
3249:
3247:
3246:
3241:
3239:
3231:
3220:
3217:
3206:
3195:
3194:
3189:
3181:
3166:
3164:
3163:
3158:
3156:
3154:
3153:
3129:
3122:
3110:
3105:
3073:
3066:
3064:
3063:
3058:
3056:
3054:
3028:
3006:
3004:
3003:
2998:
2985:
2983:
2982:
2977:
2948:
2946:
2945:
2940:
2932:
2931:
2906:
2904:
2903:
2898:
2893:
2878:
2873:
2826:
2824:
2823:
2818:
2816:
2812:
2805:
2803:
2802:
2797:
2793:
2792:
2786:
2781:
2772:
2755:
2748:
2745:
2740:
2728:
2699:
2698:
2674:closest approach
2670:impact parameter
2650:
2648:
2647:
2642:
2639:
2638:
2637:
2619:
2595:
2593:
2592:
2587:
2578:
2561:
2560:
2545:
2544:
2543:
2521:
2519:
2518:
2513:
2510:
2509:
2490:
2488:
2487:
2482:
2479:
2478:
2455:
2453:
2452:
2447:
2429:
2427:
2426:
2421:
2419:
2414:
2399:
2397:
2396:
2391:
2363:
2361:
2360:
2355:
2353:
2352:
2332:
2330:
2329:
2324:
2298:
2296:
2295:
2290:
2288:
2287:
2285:
2280:
2270:
2245:
2243:
2242:
2237:
2232:
2218:
2217:
2195:
2190:
2171:
2169:
2168:
2163:
2159:
2158:
2142:
2140:
2139:
2134:
2132:
2131:
2112:
2110:
2109:
2104:
2084:
2082:
2081:
2076:
2044:
2042:
2041:
2036:
2034:
2026:
2016:
2014:
2013:
2008:
2006:
2004:
1996:
1988:
1971:
1969:
1968:
1963:
1961:
1960:
1940:
1938:
1937:
1932:
1930:
1922:
1912:
1910:
1909:
1904:
1884:
1882:
1881:
1876:
1871:
1865:
1860:
1843:
1841:
1840:
1835:
1827:
1808:
1806:
1805:
1800:
1780:
1778:
1777:
1772:
1764:
1762:
1761:
1749:
1748:
1739:
1729:
1727:
1726:
1721:
1690:
1688:
1687:
1682:
1680:
1679:
1655:
1653:
1652:
1647:
1642:
1637:
1636:
1621:
1616:
1615:
1595:
1593:
1592:
1587:
1582:
1581:
1552:
1550:
1549:
1544:
1524:
1522:
1521:
1518:{\displaystyle }
1516:
1507:
1505:
1504:
1499:
1497:
1489:
1488:
1479:
1463:
1461:
1460:
1455:
1436:Impact parameter
1430:
1428:
1427:
1422:
1420:
1419:
1413:
1411:
1404:
1398:
1393:
1384:
1383:
1361:
1359:
1358:
1349:
1348:
1325:
1323:
1322:
1317:
1309:
1308:
1288:
1286:
1285:
1280:
1257:
1255:
1254:
1249:
1241:
1227:
1226:
1197:
1195:
1194:
1189:
1181:
1164:
1163:
1140:
1138:
1137:
1132:
1130:
1129:
1102:
1100:
1099:
1094:
1092:
1087:
1076:
1066:
1064:
1063:
1058:
1056:
1055:
1033:
1031:
1030:
1025:
1022:
1017:
1008:
989:
987:
986:
981:
973:
968:
967:
952:
941:
925:
923:
922:
917:
894:
892:
891:
886:
884:
882:
881:
872:
867:
866:
851:
841:
839:
838:
833:
825:
823:
822:
810:
800:
798:
797:
792:
771:
769:
768:
763:
761:
760:
744:
739:
719:
717:
716:
711:
709:
707:
700:
686:
674:
673:
664:
654:
652:
651:
646:
625:
623:
622:
617:
605:
603:
602:
597:
585:
583:
582:
577:
575:
574:
547:
543:
496:Newtonian theory
488:hyperbolic orbit
464:
457:
450:
429:Orbital maneuver
378:Payload fraction
358:
339:Lissajous orbits
273:
244:Orbital velocity
191:Hyperbolic orbit
87:Orbital elements
77:
68:
51:
50:
21:
18:Hyperbolic orbit
5384:
5383:
5379:
5378:
5377:
5375:
5374:
5373:
5359:
5358:
5357:
5352:
5334:
5252:Escape velocity
5233:
5226:
5207:Rocket equation
5134:
5126:
5120:
5111:
5102:
5093:
5082:
5073:
5064:
5055:
5046:
5042:
5038:
5029:
5018:
5009:
5000:
4991:
4982:
4971:
4962:
4958:
4954:Semi-minor axis
4949:
4945:Semi-major axis
4940:
4931:
4925:
4898:
4820:Areosynchronous
4804:
4798:
4779:Sun-synchronous
4764:Near-equatorial
4708:
4588:
4582:
4535:
4530:
4529:
4521:
4517:
4509:
4503:
4499:
4490:
4488:
4479:
4478:
4474:
4463:
4459:
4454:
4448:
4431:
4413:Planetary flyby
4389:
4377:
4346:
4342:
4335:
4331:
4327:
4325:
4310:
4306:
4300:
4295:
4290:
4288:
4281:
4269:
4264:
4254:
4249:
4244:
4237:
4233:
4226:
4224:
4213:
4209:
4203:
4199:
4197:
4189:
4179:
4175:
4169:
4165:
4163:
4155:
4154:
4152:
4148:
4131:
4128:
4127:
4107:
4103:
4101:
4098:
4097:
4080:
4076:
4074:
4071:
4070:
4054:
4051:
4050:
4047:
4039:escape velocity
4031:
3994:
3990:
3981:
3977:
3975:
3973:
3970:
3969:
3938:
3931:
3927:
3926:
3925:
3916:
3903:
3899:
3898:
3897:
3888:
3884:
3882:
3879:
3878:
3856:
3852:
3850:
3847:
3846:
3821:
3817:
3816:
3814:
3811:
3810:
3807:escape velocity
3789:
3786:
3785:
3774:semi-major axis
3755:
3752:
3751:
3728:
3725:
3724:
3701:
3698:
3697:
3666:
3661:
3651:
3646:
3645:
3641:
3636:
3628:
3625:
3624:
3593:
3590:
3589:
3582:
3541:
3524:
3522:
3502:
3499:
3498:
3492:
3483:semi-major axis
3428:
3424:
3420:
3414:
3406:
3403:
3402:
3359:
3356:
3355:
3313:
3290:
3279:
3276:
3263:
3255:
3252:
3251:
3230:
3207:
3196:
3193:
3180:
3172:
3169:
3168:
3149:
3130:
3118:
3111:
3109:
3101:
3093:
3090:
3089:
3071:
3032:
3027:
3019:
3016:
3015:
2991:
2988:
2987:
2971:
2968:
2967:
2961:
2956:
2927:
2923:
2912:
2909:
2908:
2889:
2874:
2869:
2854:
2851:
2850:
2798:
2782:
2777:
2771:
2767:
2763:
2762:
2754:
2753:
2749:
2741:
2736:
2727:
2694:
2690:
2688:
2685:
2684:
2657:
2633:
2630:
2629:
2615:
2601:
2598:
2597:
2574:
2553:
2549:
2539:
2536:
2535:
2530:
2527:
2526:
2505:
2501:
2496:
2493:
2492:
2474:
2470:
2468:
2465:
2464:
2435:
2432:
2431:
2413:
2405:
2402:
2401:
2384:
2381:
2380:
2373:
2348:
2344:
2342:
2339:
2338:
2307:
2304:
2303:
2281:
2276:
2271:
2266:
2262:
2251:
2248:
2247:
2228:
2213:
2209:
2191:
2186:
2180:
2177:
2176:
2154:
2150:
2148:
2145:
2144:
2127:
2123:
2121:
2118:
2117:
2097:
2094:
2093:
2068:
2065:
2064:
2061:
2055:
2025:
2023:
2020:
2019:
1997:
1989:
1987:
1985:
1982:
1981:
1956:
1952:
1947:
1944:
1943:
1921:
1919:
1916:
1915:
1898:
1895:
1894:
1867:
1861:
1856:
1850:
1847:
1846:
1823:
1815:
1812:
1811:
1794:
1791:
1790:
1757:
1753:
1744:
1740:
1738:
1736:
1733:
1732:
1697:
1694:
1693:
1675:
1671:
1669:
1666:
1665:
1638:
1632:
1628:
1617:
1611:
1607:
1602:
1599:
1598:
1577:
1573:
1559:
1556:
1555:
1538:
1535:
1534:
1514:
1511:
1510:
1484:
1480:
1478:
1470:
1467:
1466:
1449:
1446:
1445:
1440:semi-minor axis
1415:
1414:
1400:
1394:
1389:
1379:
1375:
1365:
1360:
1354:
1353:
1341:
1337:
1332:
1329:
1328:
1304:
1300:
1295:
1292:
1291:
1271:
1268:
1267:
1263:
1237:
1219:
1215:
1204:
1201:
1200:
1177:
1156:
1152:
1147:
1144:
1143:
1125:
1121:
1116:
1113:
1112:
1083:
1075:
1073:
1070:
1069:
1051:
1047:
1045:
1042:
1041:
1018:
1013:
1004:
996:
993:
992:
969:
963:
959:
948:
937:
932:
929:
928:
909:
906:
905:
900:Semi-major axis
877:
873:
868:
862:
858:
850:
848:
845:
844:
818:
814:
809:
807:
804:
803:
786:
783:
782:
756:
752:
740:
735:
726:
723:
722:
696:
682:
675:
669:
665:
663:
661:
658:
657:
639:
636:
635:
611:
608:
607:
591:
588:
587:
570:
566:
564:
561:
560:
538:
519:specific energy
468:
439:
438:
434:Orbit insertion
424:
416:
415:
401:
393:
392:
368:
360:
356:
349:
348:
344:Lyapunov orbits
335:
334:
318:
308:
307:
283:
275:
271:
264:
263:
249:Surface gravity
224:Escape velocity
214:
206:
205:
186:Parabolic orbit
182:
181:
148:
146:
143:two-body orbits
134:
133:
124:Semi-major axis
89:
79:
75:
28:
23:
22:
15:
12:
11:
5:
5382:
5372:
5371:
5354:
5353:
5351:
5350:
5348:List of orbits
5339:
5336:
5335:
5333:
5332:
5327:
5322:
5317:
5312:
5307:
5302:
5300:Orbit equation
5297:
5289:
5284:
5279:
5274:
5269:
5264:
5259:
5254:
5249:
5244:
5238:
5236:
5228:
5227:
5225:
5224:
5219:
5214:
5209:
5204:
5199:
5194:
5189:
5184:
5179:
5174:
5172:Gravity assist
5169:
5167:Delta-v budget
5164:
5159:
5154:
5148:
5146:
5140:
5139:
5136:
5135:
5133:
5132:
5124:
5118:
5109:
5100:
5098:Orbital period
5090:
5088:
5084:
5083:
5081:
5080:
5078:True longitude
5071:
5069:Mean longitude
5062:
5053:
5036:
5026:
5024:
5020:
5019:
5017:
5016:
5007:
4998:
4989:
4979:
4977:
4973:
4972:
4970:
4969:
4956:
4947:
4938:
4928:
4926:
4924:
4923:
4920:
4916:
4910:
4904:
4903:
4900:
4899:
4897:
4896:
4895:
4894:
4886:
4885:
4884:
4879:
4874:
4873:
4872:
4859:
4854:
4853:
4852:
4847:
4842:
4837:
4829:
4828:
4827:
4825:Areostationary
4822:
4817:
4808:
4806:
4800:
4799:
4797:
4796:
4794:Very low Earth
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4746:
4741:
4736:
4735:
4734:
4729:
4722:Geosynchronous
4718:
4716:
4710:
4709:
4707:
4706:
4704:Transfer orbit
4701:
4700:
4699:
4694:
4684:
4679:
4674:
4669:
4664:
4662:Lagrange point
4659:
4654:
4645:
4640:
4635:
4630:
4621:
4616:
4611:
4605:
4603:
4596:
4590:
4589:
4584:Gravitational
4581:
4580:
4573:
4566:
4558:
4552:
4551:
4546:
4541:
4534:
4533:External links
4531:
4528:
4527:
4515:
4497:
4472:
4456:
4455:
4453:
4452:
4446:
4432:
4430:
4427:
4426:
4425:
4420:
4415:
4410:
4408:List of orbits
4405:
4400:
4398:Orbit equation
4395:
4388:
4385:
4376:
4373:
4372:
4371:
4359:
4345:
4338:
4334:
4330:
4324:
4319:
4313:
4309:
4303:
4298:
4294:
4287:
4284:
4272:
4263:
4257:
4252:
4248:
4240:
4236:
4232:
4229:
4223:
4212:
4206:
4202:
4196:
4193:
4178:
4172:
4168:
4162:
4159:
4151:
4147:
4144:
4141:
4138:
4135:
4110:
4106:
4079:
4058:
4046:
4043:
4030:
4027:
4019:
4018:
4007:
4004:
3997:
3993:
3989:
3984:
3980:
3955:
3954:
3941:
3934:
3930:
3924:
3919:
3912:
3909:
3906:
3902:
3896:
3891:
3887:
3859:
3855:
3830:
3827:
3824:
3820:
3793:
3778:
3777:
3759:
3749:
3732:
3722:
3705:
3691:
3690:
3676:
3669:
3665:
3660:
3654:
3650:
3644:
3640:
3635:
3632:
3597:
3581:
3578:
3577:
3576:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3539:
3536:
3533:
3530:
3527:
3521:
3518:
3515:
3512:
3509:
3506:
3491:
3488:
3487:
3486:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3431:
3427:
3423:
3419:
3413:
3410:
3396:
3395:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3334:
3333:
3320:
3317:
3312:
3309:
3306:
3299:
3296:
3293:
3288:
3285:
3282:
3275:
3270:
3267:
3262:
3259:
3237:
3234:
3229:
3226:
3223:
3216:
3213:
3210:
3205:
3202:
3199:
3192:
3187:
3184:
3179:
3176:
3152:
3148:
3145:
3142:
3139:
3136:
3133:
3128:
3125:
3121:
3117:
3114:
3108:
3104:
3100:
3097:
3068:
3067:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3031:
3026:
3023:
3009:orbit equation
2995:
2975:
2960:
2957:
2955:
2952:
2951:
2950:
2938:
2935:
2930:
2926:
2922:
2919:
2916:
2896:
2892:
2888:
2885:
2882:
2877:
2872:
2868:
2864:
2861:
2858:
2828:
2827:
2815:
2811:
2808:
2801:
2796:
2790:
2785:
2780:
2776:
2770:
2766:
2761:
2758:
2752:
2744:
2739:
2735:
2731:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2697:
2693:
2656:
2653:
2652:
2651:
2636:
2632:
2628:
2625:
2622:
2618:
2614:
2611:
2608:
2605:
2584:
2581:
2577:
2573:
2570:
2567:
2564:
2559:
2556:
2552:
2548:
2542:
2538:
2534:
2508:
2504:
2500:
2477:
2473:
2445:
2442:
2439:
2417:
2412:
2409:
2388:
2372:
2369:
2351:
2347:
2320:
2317:
2314:
2311:
2300:
2299:
2284:
2279:
2275:
2269:
2265:
2261:
2258:
2255:
2235:
2231:
2227:
2224:
2221:
2216:
2212:
2208:
2205:
2202:
2199:
2194:
2189:
2185:
2157:
2153:
2130:
2126:
2101:
2072:
2054:
2051:
2048:
2047:
2045:
2032:
2029:
2017:
2003:
2000:
1995:
1992:
1979:
1973:
1972:
1959:
1955:
1951:
1941:
1928:
1925:
1913:
1902:
1892:
1886:
1885:
1874:
1870:
1864:
1859:
1855:
1844:
1833:
1830:
1826:
1822:
1819:
1809:
1798:
1788:
1782:
1781:
1770:
1767:
1760:
1756:
1752:
1747:
1743:
1730:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1691:
1678:
1674:
1663:
1657:
1656:
1645:
1641:
1635:
1631:
1627:
1624:
1620:
1614:
1610:
1606:
1596:
1585:
1580:
1576:
1572:
1569:
1566:
1563:
1553:
1542:
1532:
1526:
1525:
1508:
1495:
1492:
1487:
1483:
1477:
1474:
1464:
1453:
1443:
1432:
1431:
1418:
1410:
1407:
1403:
1397:
1392:
1388:
1382:
1378:
1374:
1371:
1368:
1364:
1357:
1352:
1347:
1344:
1340:
1336:
1326:
1315:
1312:
1307:
1303:
1299:
1289:
1278:
1275:
1265:
1259:
1258:
1247:
1244:
1240:
1236:
1233:
1230:
1225:
1222:
1218:
1214:
1211:
1208:
1198:
1187:
1184:
1180:
1176:
1173:
1170:
1167:
1162:
1159:
1155:
1151:
1141:
1128:
1124:
1120:
1110:
1106:
1105:
1103:
1090:
1086:
1082:
1079:
1067:
1054:
1050:
1039:
1035:
1034:
1021:
1016:
1012:
1007:
1003:
1000:
990:
979:
976:
972:
966:
962:
958:
955:
951:
947:
944:
940:
936:
926:
913:
903:
896:
895:
880:
876:
871:
865:
861:
857:
854:
842:
831:
828:
821:
817:
813:
801:
790:
780:
773:
772:
759:
755:
751:
748:
743:
738:
734:
730:
720:
706:
703:
699:
695:
692:
689:
685:
681:
678:
672:
668:
655:
643:
633:
627:
626:
615:
595:
573:
569:
557:
554:
551:
537:
534:
470:
469:
467:
466:
459:
452:
444:
441:
440:
437:
436:
431:
425:
422:
421:
418:
417:
414:
413:
408:
406:Gravity assist
402:
399:
398:
395:
394:
391:
390:
385:
380:
375:
369:
366:
365:
362:
361:
354:
351:
350:
347:
346:
341:
333:
332:
324:
320:
319:
314:
313:
310:
309:
306:
305:
300:
295:
290:
284:
281:
280:
277:
276:
269:
266:
265:
262:
261:
256:
251:
246:
241:
239:Orbital period
236:
231:
226:
221:
215:
212:
211:
208:
207:
204:
203:
201:Decaying orbit
198:
193:
188:
180:
179:
173:
166:
164:Transfer orbit
162:
161:
160:
158:Elliptic orbit
155:
153:Circular orbit
149:
140:
139:
136:
135:
132:
131:
126:
121:
116:
111:
106:
101:
96:
90:
85:
84:
81:
80:
73:
70:
69:
61:
60:
56:
55:
26:
9:
6:
4:
3:
2:
5381:
5370:
5367:
5366:
5364:
5349:
5341:
5340:
5337:
5331:
5328:
5326:
5323:
5321:
5318:
5316:
5313:
5311:
5308:
5306:
5303:
5301:
5298:
5296:
5295:-body problem
5294:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5239:
5237:
5235:
5229:
5223:
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5198:
5197:Oberth effect
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5149:
5147:
5145:
5141:
5131:
5123:
5119:
5117:
5116:Orbital speed
5110:
5108:
5101:
5099:
5092:
5091:
5089:
5085:
5079:
5072:
5070:
5063:
5061:
5054:
5052:
5037:
5035:
5028:
5027:
5025:
5021:
5015:
5008:
5006:
4999:
4997:
4990:
4988:
4981:
4980:
4978:
4974:
4968:
4957:
4955:
4948:
4946:
4939:
4937:
4930:
4929:
4927:
4921:
4918:
4917:
4914:
4911:
4909:
4905:
4893:
4890:
4889:
4887:
4883:
4880:
4878:
4875:
4871:
4870:Earth's orbit
4868:
4867:
4866:
4863:
4862:
4860:
4858:
4855:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4833:
4832:
4830:
4826:
4823:
4821:
4818:
4816:
4813:
4812:
4810:
4809:
4807:
4801:
4795:
4792:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4740:
4737:
4733:
4730:
4728:
4727:Geostationary
4725:
4724:
4723:
4720:
4719:
4717:
4715:
4711:
4705:
4702:
4698:
4695:
4693:
4690:
4689:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4653:
4649:
4646:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4606:
4604:
4600:
4597:
4595:
4591:
4587:
4579:
4574:
4572:
4567:
4565:
4560:
4559:
4556:
4550:
4547:
4545:
4542:
4540:
4537:
4536:
4525:
4519:
4508:
4501:
4487:on 2012-02-04
4486:
4482:
4476:
4468:
4461:
4457:
4449:
4443:
4439:
4434:
4433:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4390:
4384:
4382:
4357:
4343:
4336:
4332:
4328:
4322:
4317:
4311:
4307:
4301:
4292:
4285:
4282:
4270:
4261:
4255:
4246:
4238:
4234:
4230:
4227:
4221:
4210:
4204:
4200:
4194:
4191:
4176:
4170:
4166:
4160:
4157:
4149:
4145:
4142:
4139:
4136:
4133:
4126:
4125:
4124:
4108:
4104:
4077:
4056:
4042:
4040:
4036:
4026:
4024:
4023:Oberth effect
4005:
4002:
3995:
3991:
3987:
3982:
3978:
3968:
3967:
3966:
3963:
3962:
3939:
3928:
3922:
3917:
3910:
3907:
3904:
3900:
3894:
3889:
3885:
3877:
3876:
3875:
3853:
3828:
3825:
3822:
3818:
3808:
3791:
3783:
3775:
3757:
3750:
3747:
3730:
3723:
3720:
3703:
3696:
3695:
3694:
3674:
3667:
3663:
3658:
3652:
3648:
3642:
3638:
3633:
3630:
3623:
3622:
3621:
3619:
3617:
3612:
3595:
3587:
3586:orbital speed
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3537:
3534:
3531:
3528:
3525:
3519:
3513:
3507:
3504:
3497:
3496:
3495:
3485:of the orbit.
3484:
3480:
3476:
3472:
3456:
3450:
3447:
3444:
3438:
3429:
3425:
3421:
3417:
3411:
3408:
3401:
3400:
3399:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3354:
3353:
3352:
3350:
3346:
3343:
3339:
3318:
3315:
3310:
3307:
3304:
3297:
3294:
3291:
3286:
3283:
3280:
3273:
3268:
3265:
3260:
3257:
3235:
3232:
3227:
3224:
3221:
3214:
3211:
3208:
3203:
3200:
3197:
3190:
3185:
3182:
3177:
3174:
3150:
3146:
3143:
3140:
3137:
3134:
3131:
3126:
3123:
3119:
3115:
3112:
3106:
3102:
3098:
3095:
3088:
3087:
3086:
3084:
3080:
3077:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3029:
3024:
3021:
3014:
3013:
3012:
3010:
2993:
2973:
2966:
2936:
2933:
2924:
2920:
2917:
2914:
2894:
2890:
2886:
2883:
2880:
2875:
2866:
2862:
2859:
2856:
2849:
2848:
2847:
2843:
2841:
2837:
2833:
2813:
2809:
2806:
2799:
2794:
2788:
2783:
2774:
2768:
2764:
2759:
2756:
2750:
2742:
2733:
2729:
2724:
2718:
2715:
2712:
2706:
2703:
2700:
2695:
2691:
2683:
2682:
2681:
2677:
2675:
2671:
2661:
2631:
2626:
2623:
2620:
2616:
2612:
2609:
2606:
2603:
2579:
2575:
2571:
2568:
2562:
2557:
2554:
2550:
2546:
2537:
2532:
2525:
2524:
2523:
2502:
2498:
2471:
2462:
2457:
2443:
2440:
2437:
2415:
2410:
2407:
2386:
2378:
2368:
2365:
2349:
2345:
2336:
2318:
2315:
2312:
2309:
2282:
2273:
2267:
2263:
2259:
2256:
2253:
2233:
2229:
2225:
2222:
2219:
2214:
2210:
2206:
2203:
2200:
2197:
2192:
2183:
2175:
2174:
2173:
2151:
2128:
2124:
2116:
2099:
2091:
2086:
2070:
2060:
2046:
2030:
2027:
2018:
2001:
1993:
1980:
1978:
1975:
1974:
1953:
1949:
1942:
1926:
1923:
1914:
1900:
1893:
1891:
1888:
1887:
1872:
1868:
1862:
1853:
1845:
1831:
1828:
1824:
1820:
1817:
1810:
1796:
1789:
1787:
1784:
1783:
1768:
1765:
1758:
1754:
1750:
1745:
1741:
1731:
1714:
1711:
1708:
1702:
1699:
1692:
1676:
1672:
1664:
1662:
1659:
1658:
1643:
1639:
1633:
1629:
1625:
1622:
1618:
1612:
1608:
1604:
1597:
1578:
1574:
1570:
1567:
1561:
1554:
1540:
1533:
1531:
1528:
1527:
1509:
1493:
1490:
1485:
1481:
1475:
1472:
1465:
1451:
1444:
1441:
1437:
1434:
1433:
1405:
1401:
1395:
1386:
1380:
1376:
1372:
1369:
1362:
1350:
1345:
1342:
1338:
1334:
1327:
1313:
1310:
1301:
1297:
1290:
1276:
1273:
1266:
1261:
1260:
1242:
1238:
1234:
1228:
1223:
1220:
1216:
1212:
1209:
1206:
1199:
1182:
1178:
1174:
1171:
1165:
1160:
1157:
1153:
1149:
1142:
1122:
1118:
1111:
1108:
1107:
1104:
1088:
1084:
1080:
1077:
1068:
1048:
1040:
1037:
1036:
1019:
1010:
1005:
1001:
998:
991:
974:
970:
964:
960:
956:
953:
949:
945:
938:
934:
927:
911:
904:
901:
898:
897:
878:
874:
869:
863:
859:
855:
852:
843:
829:
826:
819:
815:
811:
802:
788:
781:
778:
775:
774:
753:
749:
746:
741:
732:
728:
721:
701:
697:
693:
690:
687:
683:
679:
670:
666:
656:
641:
634:
632:
629:
628:
613:
593:
567:
558:
555:
552:
549:
548:
542:
533:
531:
527:
522:
520:
516:
512:
507:
505:
501:
497:
493:
489:
485:
481:
477:
476:astrodynamics
465:
460:
458:
453:
451:
446:
445:
443:
442:
435:
432:
430:
427:
426:
420:
419:
412:
411:Oberth effect
409:
407:
404:
403:
397:
396:
389:
386:
384:
381:
379:
376:
374:
371:
370:
364:
363:
359:
353:
352:
345:
342:
340:
337:
336:
330:
326:
325:
323:
317:
316:N-body orbits
312:
311:
304:
301:
299:
298:Perturbations
296:
294:
291:
289:
286:
285:
279:
278:
274:
268:
267:
260:
257:
255:
252:
250:
247:
245:
242:
240:
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225:
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220:
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216:
210:
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199:
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189:
187:
184:
183:
177:
174:
172:
168:
167:
165:
159:
156:
154:
151:
150:
144:
138:
137:
130:
127:
125:
122:
120:
119:Orbital nodes
117:
115:
112:
110:
107:
105:
102:
100:
97:
95:
92:
91:
88:
83:
82:
78:
72:
71:
67:
63:
62:
59:Astrodynamics
58:
57:
53:
52:
45:
40:
32:
19:
5310:Perturbation
5292:
5267:Ground track
5177:Gravity turn
5128:
5121:
5114:
5105:
5096:
5076:
5067:
5058:
5051:True anomaly
5049:
5034:Mean anomaly
5032:
5012:
5003:
4994:
4985:
4965:
4952:
4943:
4936:Eccentricity
4934:
4892:Lunar cycler
4865:Heliocentric
4805:other points
4754:Medium Earth
4652:Non-inclined
4642:
4539:Trajectories
4518:
4500:
4489:. Retrieved
4485:the original
4475:
4466:
4460:
4437:
4403:Kepler orbit
4378:
4048:
4032:
4020:
3960:
3956:
3779:
3746:central body
3692:
3615:
3610:
3583:
3493:
3478:
3470:
3397:
3344:
3342:mean anomaly
3337:
3335:
3082:
3078:
3069:
2965:true anomaly
2962:
2844:
2836:Solar System
2829:
2678:
2667:
2461:true anomaly
2458:
2374:
2366:
2301:
2087:
2062:
777:Eccentricity
539:
523:
508:
492:central body
487:
483:
473:
196:Radial orbit
190:
147:eccentricity
129:True anomaly
114:Mean anomaly
104:Eccentricity
5272:Hill sphere
5107:Mean motion
4987:Inclination
4976:Orientation
4877:Mars cycler
4815:Areocentric
4687:Synchronous
329:Halo orbits
293:Hill sphere
109:Inclination
5212:Rendezvous
4908:Parameters
4744:High Earth
4714:Geocentric
4667:Osculating
4624:Elliptical
4549:Hyperbolic
4491:2012-02-28
4429:References
2057:See also:
373:Mass ratio
288:Barycenter
5257:Ephemeris
5234:mechanics
5144:Maneuvers
5087:Variation
4850:Libration
4845:Lissajous
4749:Low Earth
4739:Graveyard
4638:Horseshoe
4323:−
4318:μ
4297:∞
4251:∞
4231:μ
4222:−
4161:−
4146:
4134:δ
4057:δ
3988:−
3933:∞
3858:∞
3805:), local
3704:μ
3639:μ
3561:θ
3558:
3552:⋅
3538:θ
3535:
3529:⋅
3514:ϕ
3508:
3451:τ
3448:−
3422:−
3418:μ
3380:−
3374:
3316:θ
3311:
3305:⋅
3284:−
3261:
3228:
3222:⋅
3212:−
3183:θ
3178:
3151:θ
3147:
3141:⋅
3120:θ
3116:
3099:
3052:θ
3049:
3043:⋅
3030:ℓ
3007:) by the
2974:θ
2937:π
2934:−
2929:∞
2925:θ
2915:δ
2887:δ
2884:
2871:∞
2857:μ
2807:−
2789:μ
2779:∞
2738:∞
2730:μ
2716:−
2704:−
2635:∞
2627:θ
2624:
2610:−
2569:−
2563:
2555:−
2541:∞
2533:θ
2507:∞
2503:θ
2476:∞
2472:θ
2310:μ
2278:∞
2264:μ
2260:−
2226:μ
2223:−
2204:ϵ
2188:∞
2156:∞
2100:ϵ
1999:Δ
1991:Δ
1958:∞
1927:ℓ
1924:μ
1858:∞
1821:μ
1818:−
1797:ε
1712:−
1700:−
1644:μ
1605:−
1571:−
1541:ℓ
1491:−
1473:−
1406:μ
1391:∞
1351:
1343:−
1314:π
1311:−
1306:∞
1302:θ
1277:ν
1229:
1221:−
1207:π
1172:−
1166:
1158:−
1127:∞
1123:θ
1081:μ
1078:−
1053:∞
1015:∞
1002:μ
999:−
975:μ
957:−
827:−
812:ℓ
758:∞
754:θ
750:
737:∞
691:−
642:μ
572:∞
500:hyperbola
213:Equations
141:Types of
5363:Category
5023:Position
4648:Inclined
4619:Circular
4387:See also
3618:equation
3616:vis-viva
3580:Velocity
3074:and the
2959:Position
2664:closely.
5232:Orbital
5202:Phasing
5162:Delta-v
4967:Apsides
4961:,
4759:Molniya
4677:Parking
4614:Capture
4602:General
3693:where:
3481:is the
2840:Jupiter
2333:is the
2302:where:
902:(<0)
779:(>1)
606:), and
556:Formula
550:Element
5369:Orbits
4888:Other
4789:Tundra
4657:Kepler
4633:Escape
4586:orbits
4544:Orbits
4444:
4143:arcsin
3959:delta-
3469:where
3085:) is:
2907:where
2491:), so
559:using
553:Symbol
517:. The
5130:Epoch
4919:Shape
4857:Lunar
4811:Mars
4803:About
4774:Polar
4594:Types
4510:(PDF)
4393:Orbit
3473:is a
2832:Earth
2113:) or
94:Apsis
4922:Size
4861:Sun
4840:Halo
4692:semi
4442:ISBN
4123:is:
4006:3.02
3992:11.2
3979:11.6
3620:as:
3477:and
3371:sinh
3258:tanh
3225:tanh
3096:cosh
2668:The
2441:>
2337:and
2172:).
586:(or
482:, a
4697:sub
4609:Box
4348:SOI
4266:SOI
4215:SOI
4181:SOI
4082:SOI
3874:):
3717:is
3555:cos
3532:sin
3505:tan
3347:by
3308:tan
3175:tan
3144:cos
3113:cos
3046:cos
2881:tan
2621:cos
2596:or
2551:cos
2246:or
1339:sin
1217:tan
1154:cos
747:cot
486:or
478:or
474:In
145:by
5365::
5045:,
5041:,
4650:/
4626:/
3351::
3011::
5293:n
5125:0
5122:t
5112:v
5103:n
5094:T
5074:l
5065:L
5056:E
5047:f
5043:θ
5039:ν
5030:M
5010:ϖ
5001:ω
4992:Ω
4983:i
4963:q
4959:Q
4950:b
4941:a
4932:e
4577:e
4570:t
4563:v
4512:.
4494:.
4450:.
4358:)
4344:R
4337:e
4333:p
4329:2
4312:e
4308:p
4302:2
4293:v
4286:+
4283:1
4271:2
4262:R
4256:2
4247:v
4239:e
4235:p
4228:2
4211:R
4205:e
4201:p
4195:+
4192:1
4177:R
4171:e
4167:p
4158:1
4150:(
4140:2
4137:=
4109:e
4105:p
4078:R
4003:=
3996:2
3983:2
3961:v
3940:2
3929:v
3923:+
3918:2
3911:c
3908:s
3905:e
3901:v
3895:=
3890:2
3886:v
3854:v
3829:c
3826:s
3823:e
3819:v
3809:(
3792:v
3784:(
3776:.
3758:a
3748:,
3731:r
3721:,
3675:)
3668:a
3664:1
3659:+
3653:r
3649:2
3643:(
3634:=
3631:v
3596:v
3588:(
3549:e
3546:+
3543:1
3526:e
3520:=
3517:)
3511:(
3479:a
3471:μ
3457:,
3454:)
3445:t
3442:(
3439:.
3430:3
3426:a
3412:=
3409:M
3383:E
3377:E
3368:e
3365:=
3362:M
3345:M
3338:E
3319:2
3298:1
3295:+
3292:e
3287:1
3281:e
3274:=
3269:2
3266:E
3236:2
3233:E
3215:1
3209:e
3204:1
3201:+
3198:e
3191:=
3186:2
3138:e
3135:+
3132:1
3127:e
3124:+
3107:=
3103:E
3083:H
3079:E
3072:θ
3040:e
3037:+
3034:1
3025:=
3022:r
2994:r
2921:2
2918:=
2895:2
2891:/
2876:2
2867:v
2863:b
2860:=
2814:)
2810:1
2800:2
2795:)
2784:2
2775:v
2769:b
2765:(
2760:+
2757:1
2751:(
2743:2
2734:v
2725:=
2722:)
2719:1
2713:e
2710:(
2707:a
2701:=
2696:p
2692:r
2617:/
2613:1
2607:=
2604:e
2583:)
2580:e
2576:/
2572:1
2566:(
2558:1
2547:=
2499:2
2444:2
2438:e
2416:2
2411:=
2408:e
2387:e
2379:(
2350:3
2346:C
2319:m
2316:G
2313:=
2283:2
2274:v
2268:/
2257:=
2254:a
2234:a
2230:/
2220:=
2215:3
2211:C
2207:=
2201:2
2198:=
2193:2
2184:v
2152:v
2129:3
2125:C
2092:(
2071:a
2031:2
2028:h
2002:t
1994:A
1954:v
1950:b
1901:h
1873:2
1869:/
1863:2
1854:v
1832:a
1829:2
1825:/
1769:a
1766:+
1759:2
1755:b
1751:+
1746:2
1742:a
1718:)
1715:1
1709:e
1706:(
1703:a
1677:p
1673:r
1640:/
1634:2
1630:h
1626:=
1623:a
1619:/
1613:2
1609:b
1584:)
1579:2
1575:e
1568:1
1565:(
1562:a
1494:1
1486:2
1482:e
1476:a
1452:b
1442:)
1438:(
1417:)
1409:)
1402:/
1396:2
1387:v
1381:p
1377:r
1373:+
1370:1
1367:(
1363:1
1356:(
1346:1
1335:2
1298:2
1274:2
1246:)
1243:a
1239:/
1235:b
1232:(
1224:1
1213:2
1210:+
1186:)
1183:e
1179:/
1175:1
1169:(
1161:1
1150:2
1119:2
1089:a
1085:/
1049:v
1020:2
1011:v
1006:/
978:)
971:/
965:2
961:v
954:r
950:/
946:2
943:(
939:/
935:1
912:a
879:2
875:a
870:/
864:2
860:b
856:+
853:1
830:1
820:p
816:r
789:e
742:2
733:v
729:b
705:)
702:a
698:/
694:1
688:r
684:/
680:2
677:(
671:2
667:v
614:b
594:a
568:v
463:e
456:t
449:v
331:)
327:(
178:)
169:(
20:)
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