1561:
3213:
158:
4085:
43:
133:
3090:
3227:
Dashed ellipses are orbits relative to Earth. Solid curves are perturbations relative to the satellite: in one orbit, (1) and (2) return to the satellite having made a clockwise loop on either side of the satellite. Unintuitively, (3) spirals farther and farther behind whereas (4) spirals
2503:
2323:
2695:
2922:
2216:
1469:
1542:
2903:
2584:
974:
2362:
1568:
of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy remains constant throughout the constant speed circular orbit.
717:
2019:
1892:
2079:
1389:
2761:
3207:
1832:
1685:
1643:
3146:
2819:
1178:
1307:
2354:
2101:
1934:
2786:
2044:
2227:
1075:
838:
1595:
1257:
903:
800:
1106:
1336:
1560:
1213:
3085:{\displaystyle \gamma =g_{\mu \nu }u^{\mu }v^{\nu }=\left(1-{\frac {2M}{r}}\right){\sqrt {\frac {r}{r-3M}}}{\sqrt {\frac {r}{r-2M}}}={\sqrt {\frac {r-2M}{r-3M}}}}
880:
859:
771:
744:
2595:
1774:
1025:
1000:
2116:
1404:
53:
1484:
1077:, although in common practice, if the greater mass is significantly larger, the lesser mass is often neglected, with minimal change in the result.
553:
3962:
2827:
4022:
325:
2514:
919:
394:
3282:
2498:{\displaystyle {\frac {M}{r^{2}}}\left(1-{\frac {2M}{r}}\right){\dot {t}}^{2}-r\left(1-{\frac {2M}{r}}\right){\dot {\phi }}^{2}=0}
4017:
3897:
3316:
1109:
3982:
3736:
660:
1949:
4055:
3694:
3685:
3422:
1840:
215:
111:
68:
2049:
1716:
times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.
83:
1344:
4002:
3472:
647:
to velocity) causes a change in direction. If it is constant in magnitude and changing in direction with the velocity,
3216:
At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass:
3947:
2706:
546:
479:
90:
3157:
2916:
of the observer and the orbiting body equals the gamma factor for the orbiting body relative to the observer, hence:
1782:
3927:
3754:
1649:
1603:
4065:
3105:
1216:
2795:
818:, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value
97:
4050:
3575:
652:
474:
389:
1139:
4060:
3368:
1573:
345:
1269:
79:
3922:
3524:
3444:
3432:
1741:
1476:
1003:
815:
539:
262:
136:
31:
2331:
4045:
3987:
3957:
3745:
3622:
3590:
3560:
3519:
3504:
3383:
2084:
1905:
447:
282:
190:
3212:
2318:{\displaystyle {\ddot {x}}^{\mu }+\Gamma _{\nu \sigma }^{\mu }{\dot {x}}^{\nu }{\dot {x}}^{\sigma }=0}
1736:, although the latter implies getting arbitrarily far away and having more energy than needed for the
4070:
3892:
3676:
3565:
3534:
3462:
3437:
3412:
3373:
3354:
3309:
1753:
1549:
320:
277:
267:
195:
2769:
2027:
3932:
3727:
3467:
1725:
1034:
618:
610:
362:
200:
821:
3605:
3494:
3392:
1579:
1226:
435:
310:
885:
782:
3972:
3870:
3800:
3555:
3509:
3427:
3278:
614:
350:
287:
166:
1082:
3952:
3884:
3648:
3610:
3484:
3454:
3407:
3248:
587:
520:
469:
234:
178:
104:
2690:{\displaystyle \left(1-{\frac {2M}{r}}\right){\dot {t}}^{2}-{\frac {M}{r}}{\dot {t}}^{2}=1}
1319:
4109:
3992:
3585:
3489:
3479:
3378:
3302:
1706:
1189:
648:
525:
430:
340:
315:
864:
843:
755:
728:
4088:
4040:
4032:
4027:
3912:
3907:
3838:
3818:
3809:
3402:
3388:
3364:
3334:
3243:
3238:
3096:
1759:
1691:
1395:
1313:
1122:
1010:
985:
747:
651:
ensues. Taking two derivatives of the particle's coordinates concerning time gives the
622:
591:
497:
413:
407:
330:
255:
249:
2211:{\displaystyle \left(1-{\frac {2M}{r}}\right){\dot {t}}^{2}-r^{2}{\dot {\phi }}^{2}=1}
1464:{\displaystyle T_{\text{ff}}={\frac {\pi }{2{\sqrt {2}}}}{\sqrt {r^{3} \over {\mu }}}}
4103:
3942:
3937:
3856:
3499:
3417:
2913:
2789:
2107:
1940:
1737:
913:
The speed (or the magnitude of velocity) relative to the central object is constant:
803:
644:
629:
625:
606:
583:
502:
335:
292:
157:
4007:
3917:
3791:
3774:
3632:
3529:
3397:
1740:
of the circular orbit. It is also a matter of maneuvering into the orbit. See also
1733:
1698:
the kinetic energy of the system is equal to the absolute value of the total energy
1565:
1548:
The fact that the formulas only differ by a constant factor is a priori clear from
599:
220:
210:
205:
1564:
A circular orbit is depicted in the top-left quadrant of this diagram, where the
1537:{\displaystyle T_{\text{par}}={\frac {\sqrt {2}}{3}}{\sqrt {r^{3} \over {\mu }}}}
4012:
3847:
3617:
3597:
3514:
2909:
640:
384:
42:
17:
3580:
1902:
For the sake of convenience, the derivation will be written in units in which
575:
464:
420:
379:
3997:
3349:
595:
2081:). The dot above a variable denotes derivation with respect to proper time
2046:
is constant on a circular orbit, and the coordinates can be chosen so that
132:
3275:
Fundamental
Planetary Sciences : physics, chemistry, and habitability
3902:
3294:
1729:
60:
2898:{\displaystyle v^{\mu }=\left({\sqrt {\frac {r}{r-2M}}},0,0,0\right)}
2788:, who is not moving with respect to the central body, that is, their
1701:
the potential energy of the system is equal to twice the total energy
807:
774:
579:
840:
is measured in meters per second squared, then the numerical values
2579:{\displaystyle {\dot {\phi }}^{2}={\frac {M}{r^{3}}}{\dot {t}}^{2}}
969:{\displaystyle v={\sqrt {GM\! \over {r}}}={\sqrt {\mu \over {r}}}}
3707:
3326:
2589:
Substituting this into the equation for a massive particle gives:
571:
185:
131:
1028:
3298:
3277:. New York, NY, USA: Cambridge University Press. p. 604.
36:
2821:. The normalization condition implies that it is equal to:
582:. In this case, not only the distance, but also the speed,
712:{\displaystyle a\,={\frac {v^{2}}{r}}\,={\omega ^{2}}{r}}
2014:{\displaystyle u^{\mu }=({\dot {t}},0,0,{\dot {\phi }})}
1756:, the orbital velocity for a circular orbit with radius
1887:{\displaystyle \scriptstyle r_{S}={\frac {2GM}{c^{2}}}}
621:, and the axis mentioned above is the line through the
64:
2799:
2773:
2335:
2088:
2074:{\displaystyle \scriptstyle \theta ={\frac {\pi }{2}}}
2053:
2031:
1909:
1844:
3160:
3108:
2925:
2830:
2798:
2772:
2709:
2598:
2517:
2365:
2334:
2230:
2119:
2087:
2052:
2030:
1952:
1908:
1843:
1785:
1762:
1652:
1606:
1582:
1487:
1407:
1347:
1322:
1272:
1229:
1192:
1142:
1085:
1037:
1013:
988:
922:
888:
867:
846:
824:
785:
758:
731:
663:
3971:
3883:
3827:
3763:
3716:
3656:
3647:
3543:
3453:
3342:
3333:
1384:{\displaystyle T=2\pi {\sqrt {r^{3} \over {\mu }}}}
3201:
3140:
3084:
2897:
2813:
2780:
2755:
2689:
2578:
2497:
2348:
2317:
2210:
2095:
2073:
2038:
2013:
1928:
1886:
1826:
1768:
1679:
1637:
1589:
1536:
1463:
1383:
1330:
1301:
1251:
1207:
1172:
1100:
1069:
1019:
994:
968:
897:
874:
853:
832:
794:
765:
738:
711:
3224:(4 - black) backwards in the direction of travel.
1894:is the Schwarzschild radius of the central body.
1471:(17.7% of the orbital period in a circular orbit)
1327:
939:
2756:{\displaystyle {\dot {t}}^{2}={\frac {r}{r-3M}}}
1724:Maneuvering into a large circular orbit, e.g. a
1544:(7.5% of the orbital period in a circular orbit)
3202:{\displaystyle v={\sqrt {\frac {GM}{r-r_{S}}}}}
1827:{\displaystyle v={\sqrt {\frac {GM}{r-r_{S}}}}}
27:Orbit with a fixed distance from the barycenter
3268:
3266:
3264:
2106:For a massive particle, the components of the
3310:
1125:in polar coordinates, which in general gives
547:
8:
2328:The only nontrivial equation is the one for
1694:applies even without taking a time-average:
1680:{\displaystyle \epsilon =-{\mu \over {2r}}}
1638:{\displaystyle \epsilon =-{v^{2} \over {2}}}
69:introducing citations to additional sources
3141:{\displaystyle v={\sqrt {\frac {M}{r-2M}}}}
1943:of a body on a circular orbit is given by:
4084:
3653:
3339:
3317:
3303:
3295:
3273:Lissauer, Jack J.; de Pater, Imke (2019).
3222:(3 - grey) in the direction of travel, and
2814:{\displaystyle \scriptstyle \partial _{t}}
1475:and the time to fall to a point mass in a
554:
540:
142:
3189:
3167:
3159:
3115:
3107:
3048:
3022:
2999:
2979:
2959:
2949:
2936:
2924:
2849:
2835:
2829:
2804:
2797:
2771:
2732:
2723:
2712:
2711:
2708:
2675:
2664:
2663:
2652:
2643:
2632:
2631:
2610:
2597:
2570:
2559:
2558:
2549:
2540:
2531:
2520:
2519:
2516:
2483:
2472:
2471:
2450:
2427:
2416:
2415:
2394:
2375:
2366:
2364:
2333:
2303:
2292:
2291:
2284:
2273:
2272:
2265:
2257:
2244:
2233:
2232:
2229:
2196:
2185:
2184:
2177:
2164:
2153:
2152:
2131:
2118:
2086:
2060:
2051:
2029:
1997:
1996:
1970:
1969:
1957:
1951:
1907:
1875:
1858:
1849:
1842:
1814:
1792:
1784:
1761:
1667:
1662:
1651:
1628:
1622:
1616:
1605:
1586:
1581:
1526:
1520:
1513:
1501:
1492:
1486:
1453:
1447:
1440:
1430:
1421:
1412:
1406:
1398:(time to fall to a point mass from rest)
1394:Compare two proportional quantities, the
1373:
1367:
1360:
1346:
1326:
1321:
1287:
1277:
1271:
1243:
1228:
1191:
1163:
1156:
1151:
1149:
1141:
1084:
1058:
1045:
1036:
1012:
987:
958:
952:
941:
929:
921:
887:
871:
866:
850:
845:
825:
823:
784:
762:
757:
735:
730:
704:
697:
692:
688:
677:
671:
667:
662:
3211:
1559:
1173:{\displaystyle r={{h^{2}} \over {\mu }}}
59:Relevant discussion may be found on the
3260:
445:
360:
164:
150:
3963:Transposition, docking, and extraction
1748:Orbital velocity in general relativity
2766:Assume we have an observer at radius
1302:{\displaystyle \omega ^{2}r^{3}=\mu }
613:under standard assumptions. Here the
7:
605:Listed below is a circular orbit in
2349:{\displaystyle \scriptstyle \mu =r}
1776:is given by the following formula:
2801:
2254:
2096:{\displaystyle \scriptstyle \tau }
1929:{\displaystyle \scriptstyle c=G=1}
139:. Path C depicts a circular orbit.
25:
4023:Kepler's laws of planetary motion
1720:Delta-v to reach a circular orbit
574:with a fixed distance around the
326:Kepler's laws of planetary motion
4083:
4018:Interplanetary Transport Network
3898:Collision avoidance (spacecraft)
2110:satisfy the following equation:
1262:Angular speed and orbital period
1110:standard gravitational parameter
826:
156:
52:relies largely or entirely on a
41:
3983:Astronomical coordinate systems
3737:Longitude of the ascending node
598:or apoapsis. This orbit has no
30:For other uses of "orbit", see
4056:Retrograde and prograde motion
2792:is proportional to the vector
2781:{\displaystyle \scriptstyle r}
2221:We use the geodesic equation:
2039:{\displaystyle \scriptstyle r}
2008:
1966:
1064:
1038:
861:will be in meters per second,
1:
1070:{\displaystyle (M_{1}+M_{2})}
578:; that is, in the shape of a
4003:Equatorial coordinate system
1566:gravitational potential well
833:{\displaystyle \mathbf {a} }
1590:{\displaystyle \epsilon \,}
1252:{\displaystyle \mu =rv^{2}}
480:Tsiolkovsky rocket equation
4126:
3755:Longitude of the periapsis
3220:(2 - red) away from Earth,
623:center of the central mass
594:are constant. There is no
449:Engineering and efficiency
268:Bi-elliptic transfer orbit
29:
4079:
4066:Specific angular momentum
3218:(1 - blue) towards Earth,
1217:specific angular momentum
898:{\displaystyle \omega \ }
795:{\displaystyle \omega \ }
137:Isaac Newton's Cannonball
1031:of both orbiting bodies
653:centripetal acceleration
475:Propellant mass fraction
374:Gravitational influences
4061:Specific orbital energy
1574:specific orbital energy
1101:{\displaystyle \mu =GM}
905:in radians per second.
346:Specific orbital energy
3473:Geostationary transfer
3229:
3203:
3142:
3086:
2899:
2815:
2782:
2757:
2691:
2580:
2499:
2350:
2319:
2212:
2097:
2075:
2040:
2015:
1930:
1888:
1828:
1770:
1742:Hohmann transfer orbit
1681:
1639:
1591:
1569:
1538:
1477:radial parabolic orbit
1465:
1385:
1338:) can be computed as:
1332:
1303:
1253:
1209:
1174:
1102:
1071:
1021:
1004:gravitational constant
996:
970:
899:
876:
855:
834:
796:
767:
740:
713:
263:Hohmann transfer orbit
140:
32:Orbit (disambiguation)
4046:Orbital state vectors
3988:Characteristic energy
3958:Trans-lunar injection
3746:Argument of periapsis
3423:Prograde / Retrograde
3384:Hyperbolic trajectory
3215:
3204:
3143:
3087:
2900:
2816:
2783:
2758:
2692:
2581:
2500:
2351:
2320:
2213:
2098:
2076:
2041:
2016:
1931:
1889:
1829:
1771:
1709:from any distance is
1682:
1640:
1592:
1563:
1539:
1466:
1386:
1333:
1331:{\displaystyle T\,\!}
1304:
1254:
1219:of the orbiting body.
1210:
1175:
1103:
1072:
1022:
997:
971:
900:
877:
856:
835:
797:
768:
750:of the orbiting body,
741:
714:
636:Circular acceleration
459:Preflight engineering
191:Argument of periapsis
135:
3893:Bi-elliptic transfer
3413:Parabolic trajectory
3158:
3106:
2923:
2828:
2796:
2770:
2707:
2596:
2515:
2363:
2332:
2228:
2117:
2085:
2050:
2028:
1950:
1906:
1841:
1783:
1760:
1754:Schwarzschild metric
1728:, requires a larger
1650:
1604:
1580:
1550:dimensional analysis
1485:
1405:
1345:
1320:
1270:
1227:
1208:{\displaystyle h=rv}
1190:
1140:
1083:
1035:
1011:
986:
920:
886:
865:
844:
822:
783:
756:
748:the orbital velocity
729:
661:
515:Propulsive maneuvers
65:improve this article
3933:Low-energy transfer
2508:From this, we get:
2270:
1726:geostationary orbit
1597:) is negative, and
875:{\displaystyle r\,}
854:{\displaystyle v\,}
766:{\displaystyle r\,}
739:{\displaystyle v\,}
619:gravitational force
611:celestial mechanics
492:Efficiency measures
395:Sphere of influence
364:Celestial mechanics
146:Part of a series on
3928:Inclination change
3576:Distant retrograde
3230:
3199:
3138:
3082:
2895:
2811:
2810:
2778:
2777:
2753:
2687:
2576:
2495:
2346:
2345:
2315:
2253:
2208:
2093:
2092:
2071:
2070:
2036:
2035:
2011:
1926:
1925:
1884:
1883:
1824:
1766:
1677:
1635:
1587:
1570:
1534:
1461:
1381:
1328:
1299:
1249:
1205:
1170:
1117:Equation of motion
1098:
1067:
1017:
992:
966:
895:
872:
851:
830:
792:
763:
736:
709:
311:Dynamical friction
141:
4097:
4096:
4071:Two-line elements
3879:
3878:
3801:Eccentric anomaly
3643:
3642:
3510:Orbit of the Moon
3369:Highly elliptical
3197:
3196:
3151:Or, in SI units:
3136:
3135:
3080:
3079:
3043:
3042:
3020:
3019:
2992:
2870:
2869:
2751:
2720:
2672:
2660:
2640:
2623:
2567:
2555:
2528:
2480:
2463:
2424:
2407:
2381:
2300:
2281:
2241:
2193:
2161:
2144:
2068:
2005:
1978:
1881:
1822:
1821:
1769:{\displaystyle r}
1675:
1633:
1532:
1531:
1511:
1507:
1495:
1459:
1458:
1438:
1435:
1415:
1379:
1378:
1168:
1020:{\displaystyle M}
995:{\displaystyle G}
964:
963:
947:
946:
894:
791:
686:
615:centripetal force
564:
563:
414:Lagrangian points
351:Vis-viva equation
321:Kepler's equation
168:Orbital mechanics
130:
129:
115:
16:(Redirected from
4117:
4087:
4086:
4028:Lagrangian point
3923:Hohmann transfer
3868:
3854:
3845:
3836:
3816:
3807:
3798:
3789:
3785:
3781:
3772:
3752:
3743:
3734:
3725:
3705:
3701:
3692:
3683:
3674:
3654:
3623:Heliosynchronous
3572:Lagrange points
3525:Transatmospheric
3340:
3319:
3312:
3305:
3296:
3289:
3288:
3270:
3249:Two-body problem
3208:
3206:
3205:
3200:
3198:
3195:
3194:
3193:
3177:
3169:
3168:
3147:
3145:
3144:
3139:
3137:
3134:
3117:
3116:
3091:
3089:
3088:
3083:
3081:
3078:
3064:
3050:
3049:
3044:
3041:
3024:
3023:
3021:
3018:
3001:
3000:
2998:
2994:
2993:
2988:
2980:
2964:
2963:
2954:
2953:
2944:
2943:
2904:
2902:
2901:
2896:
2894:
2890:
2871:
2868:
2851:
2850:
2840:
2839:
2820:
2818:
2817:
2812:
2809:
2808:
2787:
2785:
2784:
2779:
2762:
2760:
2759:
2754:
2752:
2750:
2733:
2728:
2727:
2722:
2721:
2713:
2696:
2694:
2693:
2688:
2680:
2679:
2674:
2673:
2665:
2661:
2653:
2648:
2647:
2642:
2641:
2633:
2629:
2625:
2624:
2619:
2611:
2585:
2583:
2582:
2577:
2575:
2574:
2569:
2568:
2560:
2556:
2554:
2553:
2541:
2536:
2535:
2530:
2529:
2521:
2504:
2502:
2501:
2496:
2488:
2487:
2482:
2481:
2473:
2469:
2465:
2464:
2459:
2451:
2432:
2431:
2426:
2425:
2417:
2413:
2409:
2408:
2403:
2395:
2382:
2380:
2379:
2367:
2355:
2353:
2352:
2347:
2324:
2322:
2321:
2316:
2308:
2307:
2302:
2301:
2293:
2289:
2288:
2283:
2282:
2274:
2269:
2264:
2249:
2248:
2243:
2242:
2234:
2217:
2215:
2214:
2209:
2201:
2200:
2195:
2194:
2186:
2182:
2181:
2169:
2168:
2163:
2162:
2154:
2150:
2146:
2145:
2140:
2132:
2102:
2100:
2099:
2094:
2080:
2078:
2077:
2072:
2069:
2061:
2045:
2043:
2042:
2037:
2020:
2018:
2017:
2012:
2007:
2006:
1998:
1980:
1979:
1971:
1962:
1961:
1935:
1933:
1932:
1927:
1893:
1891:
1890:
1885:
1882:
1880:
1879:
1870:
1859:
1854:
1853:
1833:
1831:
1830:
1825:
1823:
1820:
1819:
1818:
1802:
1794:
1793:
1775:
1773:
1772:
1767:
1715:
1714:
1686:
1684:
1683:
1678:
1676:
1674:
1663:
1644:
1642:
1641:
1636:
1634:
1632:
1627:
1626:
1617:
1596:
1594:
1593:
1588:
1543:
1541:
1540:
1535:
1533:
1530:
1525:
1524:
1515:
1514:
1512:
1503:
1502:
1497:
1496:
1493:
1470:
1468:
1467:
1462:
1460:
1457:
1452:
1451:
1442:
1441:
1439:
1437:
1436:
1431:
1422:
1417:
1416:
1413:
1390:
1388:
1387:
1382:
1380:
1377:
1372:
1371:
1362:
1361:
1337:
1335:
1334:
1329:
1308:
1306:
1305:
1300:
1292:
1291:
1282:
1281:
1258:
1256:
1255:
1250:
1248:
1247:
1223:This is because
1214:
1212:
1211:
1206:
1179:
1177:
1176:
1171:
1169:
1167:
1162:
1161:
1160:
1150:
1107:
1105:
1104:
1099:
1076:
1074:
1073:
1068:
1063:
1062:
1050:
1049:
1026:
1024:
1023:
1018:
1001:
999:
998:
993:
975:
973:
972:
967:
965:
962:
954:
953:
948:
945:
940:
931:
930:
904:
902:
901:
896:
892:
881:
879:
878:
873:
860:
858:
857:
852:
839:
837:
836:
831:
829:
801:
799:
798:
793:
789:
772:
770:
769:
764:
745:
743:
742:
737:
718:
716:
715:
710:
708:
703:
702:
701:
687:
682:
681:
672:
556:
549:
542:
521:Orbital maneuver
470:Payload fraction
450:
431:Lissajous orbits
365:
336:Orbital velocity
283:Hyperbolic orbit
179:Orbital elements
169:
160:
143:
125:
122:
116:
114:
80:"Circular orbit"
73:
45:
37:
21:
4125:
4124:
4120:
4119:
4118:
4116:
4115:
4114:
4100:
4099:
4098:
4093:
4075:
3993:Escape velocity
3974:
3967:
3948:Rocket equation
3875:
3867:
3861:
3852:
3843:
3834:
3823:
3814:
3805:
3796:
3787:
3783:
3779:
3770:
3759:
3750:
3741:
3732:
3723:
3712:
3703:
3699:
3695:Semi-minor axis
3690:
3686:Semi-major axis
3681:
3672:
3666:
3639:
3561:Areosynchronous
3545:
3539:
3520:Sun-synchronous
3505:Near-equatorial
3449:
3329:
3323:
3293:
3292:
3285:
3272:
3271:
3262:
3257:
3235:
3226:
3225:
3223:
3221:
3219:
3217:
3185:
3178:
3170:
3156:
3155:
3121:
3104:
3103:
3095:This gives the
3065:
3051:
3028:
3005:
2981:
2972:
2968:
2955:
2945:
2932:
2921:
2920:
2914:four-velocities
2855:
2848:
2844:
2831:
2826:
2825:
2800:
2794:
2793:
2768:
2767:
2737:
2710:
2705:
2704:
2662:
2630:
2612:
2603:
2599:
2594:
2593:
2557:
2545:
2518:
2513:
2512:
2470:
2452:
2443:
2439:
2414:
2396:
2387:
2383:
2371:
2361:
2360:
2330:
2329:
2290:
2271:
2231:
2226:
2225:
2183:
2173:
2151:
2133:
2124:
2120:
2115:
2114:
2083:
2082:
2048:
2047:
2026:
2025:
1953:
1948:
1947:
1904:
1903:
1900:
1871:
1860:
1845:
1839:
1838:
1810:
1803:
1795:
1781:
1780:
1758:
1757:
1750:
1722:
1712:
1710:
1707:escape velocity
1648:
1647:
1618:
1602:
1601:
1578:
1577:
1558:
1516:
1488:
1483:
1482:
1443:
1426:
1408:
1403:
1402:
1363:
1343:
1342:
1318:
1317:
1283:
1273:
1268:
1267:
1264:
1239:
1225:
1224:
1188:
1187:
1152:
1138:
1137:
1119:
1081:
1080:
1054:
1041:
1033:
1032:
1009:
1008:
984:
983:
932:
918:
917:
911:
884:
883:
882:in meters, and
863:
862:
842:
841:
820:
819:
814:The formula is
781:
780:
754:
753:
727:
726:
693:
673:
659:
658:
649:circular motion
638:
560:
531:
530:
526:Orbit insertion
516:
508:
507:
493:
485:
484:
460:
452:
448:
441:
440:
436:Lyapunov orbits
427:
426:
410:
400:
399:
375:
367:
363:
356:
355:
341:Surface gravity
316:Escape velocity
306:
298:
297:
278:Parabolic orbit
274:
273:
240:
238:
235:two-body orbits
226:
225:
216:Semi-major axis
181:
171:
167:
126:
120:
117:
74:
72:
58:
46:
35:
28:
23:
22:
18:Circular orbits
15:
12:
11:
5:
4123:
4121:
4113:
4112:
4102:
4101:
4095:
4094:
4092:
4091:
4089:List of orbits
4080:
4077:
4076:
4074:
4073:
4068:
4063:
4058:
4053:
4048:
4043:
4041:Orbit equation
4038:
4030:
4025:
4020:
4015:
4010:
4005:
4000:
3995:
3990:
3985:
3979:
3977:
3969:
3968:
3966:
3965:
3960:
3955:
3950:
3945:
3940:
3935:
3930:
3925:
3920:
3915:
3913:Gravity assist
3910:
3908:Delta-v budget
3905:
3900:
3895:
3889:
3887:
3881:
3880:
3877:
3876:
3874:
3873:
3865:
3859:
3850:
3841:
3839:Orbital period
3831:
3829:
3825:
3824:
3822:
3821:
3819:True longitude
3812:
3810:Mean longitude
3803:
3794:
3777:
3767:
3765:
3761:
3760:
3758:
3757:
3748:
3739:
3730:
3720:
3718:
3714:
3713:
3711:
3710:
3697:
3688:
3679:
3669:
3667:
3665:
3664:
3661:
3657:
3651:
3645:
3644:
3641:
3640:
3638:
3637:
3636:
3635:
3627:
3626:
3625:
3620:
3615:
3614:
3613:
3600:
3595:
3594:
3593:
3588:
3583:
3578:
3570:
3569:
3568:
3566:Areostationary
3563:
3558:
3549:
3547:
3541:
3540:
3538:
3537:
3535:Very low Earth
3532:
3527:
3522:
3517:
3512:
3507:
3502:
3497:
3492:
3487:
3482:
3477:
3476:
3475:
3470:
3463:Geosynchronous
3459:
3457:
3451:
3450:
3448:
3447:
3445:Transfer orbit
3442:
3441:
3440:
3435:
3425:
3420:
3415:
3410:
3405:
3403:Lagrange point
3400:
3395:
3386:
3381:
3376:
3371:
3362:
3357:
3352:
3346:
3344:
3337:
3331:
3330:
3325:Gravitational
3324:
3322:
3321:
3314:
3307:
3299:
3291:
3290:
3283:
3259:
3258:
3256:
3253:
3252:
3251:
3246:
3244:List of orbits
3241:
3239:Elliptic orbit
3234:
3231:
3210:
3209:
3192:
3188:
3184:
3181:
3176:
3173:
3166:
3163:
3149:
3148:
3133:
3130:
3127:
3124:
3120:
3114:
3111:
3093:
3092:
3077:
3074:
3071:
3068:
3063:
3060:
3057:
3054:
3047:
3040:
3037:
3034:
3031:
3027:
3017:
3014:
3011:
3008:
3004:
2997:
2991:
2987:
2984:
2978:
2975:
2971:
2967:
2962:
2958:
2952:
2948:
2942:
2939:
2935:
2931:
2928:
2906:
2905:
2893:
2889:
2886:
2883:
2880:
2877:
2874:
2867:
2864:
2861:
2858:
2854:
2847:
2843:
2838:
2834:
2807:
2803:
2776:
2764:
2763:
2749:
2746:
2743:
2740:
2736:
2731:
2726:
2719:
2716:
2698:
2697:
2686:
2683:
2678:
2671:
2668:
2659:
2656:
2651:
2646:
2639:
2636:
2628:
2622:
2618:
2615:
2609:
2606:
2602:
2587:
2586:
2573:
2566:
2563:
2552:
2548:
2544:
2539:
2534:
2527:
2524:
2506:
2505:
2494:
2491:
2486:
2479:
2476:
2468:
2462:
2458:
2455:
2449:
2446:
2442:
2438:
2435:
2430:
2423:
2420:
2412:
2406:
2402:
2399:
2393:
2390:
2386:
2378:
2374:
2370:
2344:
2341:
2338:
2326:
2325:
2314:
2311:
2306:
2299:
2296:
2287:
2280:
2277:
2268:
2263:
2260:
2256:
2252:
2247:
2240:
2237:
2219:
2218:
2207:
2204:
2199:
2192:
2189:
2180:
2176:
2172:
2167:
2160:
2157:
2149:
2143:
2139:
2136:
2130:
2127:
2123:
2091:
2067:
2064:
2059:
2056:
2034:
2022:
2021:
2010:
2004:
2001:
1995:
1992:
1989:
1986:
1983:
1977:
1974:
1968:
1965:
1960:
1956:
1924:
1921:
1918:
1915:
1912:
1899:
1896:
1878:
1874:
1869:
1866:
1863:
1857:
1852:
1848:
1835:
1834:
1817:
1813:
1809:
1806:
1801:
1798:
1791:
1788:
1765:
1749:
1746:
1721:
1718:
1703:
1702:
1699:
1692:virial theorem
1688:
1687:
1673:
1670:
1666:
1661:
1658:
1655:
1645:
1631:
1625:
1621:
1615:
1612:
1609:
1585:
1557:
1554:
1546:
1545:
1529:
1523:
1519:
1510:
1506:
1500:
1491:
1473:
1472:
1456:
1450:
1446:
1434:
1429:
1425:
1420:
1411:
1396:free-fall time
1392:
1391:
1376:
1370:
1366:
1359:
1356:
1353:
1350:
1325:
1314:orbital period
1310:
1309:
1298:
1295:
1290:
1286:
1280:
1276:
1263:
1260:
1246:
1242:
1238:
1235:
1232:
1221:
1220:
1204:
1201:
1198:
1195:
1181:
1180:
1166:
1159:
1155:
1148:
1145:
1133:, reduces to:
1123:orbit equation
1118:
1115:
1114:
1113:
1097:
1094:
1091:
1088:
1078:
1066:
1061:
1057:
1053:
1048:
1044:
1040:
1016:
1006:
991:
977:
976:
961:
957:
951:
944:
938:
935:
928:
925:
910:
907:
891:
870:
849:
828:
812:
811:
810:per unit time.
806:, measured in
788:
778:
761:
751:
734:
720:
719:
707:
700:
696:
691:
685:
680:
676:
670:
666:
643:acceleration (
637:
634:
600:radial version
592:kinetic energy
568:circular orbit
562:
561:
559:
558:
551:
544:
536:
533:
532:
529:
528:
523:
517:
514:
513:
510:
509:
506:
505:
500:
498:Gravity assist
494:
491:
490:
487:
486:
483:
482:
477:
472:
467:
461:
458:
457:
454:
453:
446:
443:
442:
439:
438:
433:
425:
424:
416:
412:
411:
406:
405:
402:
401:
398:
397:
392:
387:
382:
376:
373:
372:
369:
368:
361:
358:
357:
354:
353:
348:
343:
338:
333:
331:Orbital period
328:
323:
318:
313:
307:
304:
303:
300:
299:
296:
295:
293:Decaying orbit
290:
285:
280:
272:
271:
265:
258:
256:Transfer orbit
254:
253:
252:
250:Elliptic orbit
247:
245:Circular orbit
241:
232:
231:
228:
227:
224:
223:
218:
213:
208:
203:
198:
193:
188:
182:
177:
176:
173:
172:
165:
162:
161:
153:
152:
148:
147:
128:
127:
63:. Please help
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4122:
4111:
4108:
4107:
4105:
4090:
4082:
4081:
4078:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4036:-body problem
4035:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3980:
3978:
3976:
3970:
3964:
3961:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3938:Oberth effect
3936:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3914:
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3890:
3888:
3886:
3882:
3872:
3864:
3860:
3858:
3857:Orbital speed
3851:
3849:
3842:
3840:
3833:
3832:
3830:
3826:
3820:
3813:
3811:
3804:
3802:
3795:
3793:
3778:
3776:
3769:
3768:
3766:
3762:
3756:
3749:
3747:
3740:
3738:
3731:
3729:
3722:
3721:
3719:
3715:
3709:
3698:
3696:
3689:
3687:
3680:
3678:
3671:
3670:
3668:
3662:
3659:
3658:
3655:
3652:
3650:
3646:
3634:
3631:
3630:
3628:
3624:
3621:
3619:
3616:
3612:
3611:Earth's orbit
3609:
3608:
3607:
3604:
3603:
3601:
3599:
3596:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3573:
3571:
3567:
3564:
3562:
3559:
3557:
3554:
3553:
3551:
3550:
3548:
3542:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3474:
3471:
3469:
3468:Geostationary
3466:
3465:
3464:
3461:
3460:
3458:
3456:
3452:
3446:
3443:
3439:
3436:
3434:
3431:
3430:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3347:
3345:
3341:
3338:
3336:
3332:
3328:
3320:
3315:
3313:
3308:
3306:
3301:
3300:
3297:
3286:
3284:9781108411981
3280:
3276:
3269:
3267:
3265:
3261:
3254:
3250:
3247:
3245:
3242:
3240:
3237:
3236:
3232:
3214:
3190:
3186:
3182:
3179:
3174:
3171:
3164:
3161:
3154:
3153:
3152:
3131:
3128:
3125:
3122:
3118:
3112:
3109:
3102:
3101:
3100:
3098:
3075:
3072:
3069:
3066:
3061:
3058:
3055:
3052:
3045:
3038:
3035:
3032:
3029:
3025:
3015:
3012:
3009:
3006:
3002:
2995:
2989:
2985:
2982:
2976:
2973:
2969:
2965:
2960:
2956:
2950:
2946:
2940:
2937:
2933:
2929:
2926:
2919:
2918:
2917:
2915:
2911:
2891:
2887:
2884:
2881:
2878:
2875:
2872:
2865:
2862:
2859:
2856:
2852:
2845:
2841:
2836:
2832:
2824:
2823:
2822:
2805:
2791:
2790:four-velocity
2774:
2747:
2744:
2741:
2738:
2734:
2729:
2724:
2717:
2714:
2703:
2702:
2701:
2684:
2681:
2676:
2669:
2666:
2657:
2654:
2649:
2644:
2637:
2634:
2626:
2620:
2616:
2613:
2607:
2604:
2600:
2592:
2591:
2590:
2571:
2564:
2561:
2550:
2546:
2542:
2537:
2532:
2525:
2522:
2511:
2510:
2509:
2492:
2489:
2484:
2477:
2474:
2466:
2460:
2456:
2453:
2447:
2444:
2440:
2436:
2433:
2428:
2421:
2418:
2410:
2404:
2400:
2397:
2391:
2388:
2384:
2376:
2372:
2368:
2359:
2358:
2357:
2342:
2339:
2336:
2312:
2309:
2304:
2297:
2294:
2285:
2278:
2275:
2266:
2261:
2258:
2250:
2245:
2238:
2235:
2224:
2223:
2222:
2205:
2202:
2197:
2190:
2187:
2178:
2174:
2170:
2165:
2158:
2155:
2147:
2141:
2137:
2134:
2128:
2125:
2121:
2113:
2112:
2111:
2109:
2108:four-velocity
2104:
2089:
2065:
2062:
2057:
2054:
2032:
2002:
1999:
1993:
1990:
1987:
1984:
1981:
1975:
1972:
1963:
1958:
1954:
1946:
1945:
1944:
1942:
1941:four-velocity
1937:
1922:
1919:
1916:
1913:
1910:
1897:
1895:
1876:
1872:
1867:
1864:
1861:
1855:
1850:
1846:
1815:
1811:
1807:
1804:
1799:
1796:
1789:
1786:
1779:
1778:
1777:
1763:
1755:
1747:
1745:
1743:
1739:
1738:orbital speed
1735:
1731:
1727:
1719:
1717:
1708:
1700:
1697:
1696:
1695:
1693:
1671:
1668:
1664:
1659:
1656:
1653:
1646:
1629:
1623:
1619:
1613:
1610:
1607:
1600:
1599:
1598:
1583:
1575:
1567:
1562:
1555:
1553:
1551:
1527:
1521:
1517:
1508:
1504:
1498:
1489:
1481:
1480:
1479:
1478:
1454:
1448:
1444:
1432:
1427:
1423:
1418:
1409:
1401:
1400:
1399:
1397:
1374:
1368:
1364:
1357:
1354:
1351:
1348:
1341:
1340:
1339:
1323:
1315:
1296:
1293:
1288:
1284:
1278:
1274:
1266:
1265:
1261:
1259:
1244:
1240:
1236:
1233:
1230:
1218:
1202:
1199:
1196:
1193:
1186:
1185:
1184:
1164:
1157:
1153:
1146:
1143:
1136:
1135:
1134:
1132:
1128:
1124:
1116:
1111:
1095:
1092:
1089:
1086:
1079:
1059:
1055:
1051:
1046:
1042:
1030:
1014:
1007:
1005:
989:
982:
981:
980:
959:
955:
949:
942:
936:
933:
926:
923:
916:
915:
914:
908:
906:
889:
868:
847:
817:
816:dimensionless
809:
805:
804:angular speed
786:
779:
777:of the circle
776:
759:
752:
749:
732:
725:
724:
723:
705:
698:
694:
689:
683:
678:
674:
668:
664:
657:
656:
655:
654:
650:
646:
645:perpendicular
642:
635:
633:
631:
630:orbital plane
627:
626:perpendicular
624:
620:
616:
612:
608:
607:astrodynamics
603:
601:
597:
593:
589:
585:
584:angular speed
581:
577:
573:
569:
557:
552:
550:
545:
543:
538:
537:
535:
534:
527:
524:
522:
519:
518:
512:
511:
504:
503:Oberth effect
501:
499:
496:
495:
489:
488:
481:
478:
476:
473:
471:
468:
466:
463:
462:
456:
455:
451:
444:
437:
434:
432:
429:
428:
422:
418:
417:
415:
409:
408:N-body orbits
404:
403:
396:
393:
391:
390:Perturbations
388:
386:
383:
381:
378:
377:
371:
370:
366:
359:
352:
349:
347:
344:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
317:
314:
312:
309:
308:
302:
301:
294:
291:
289:
286:
284:
281:
279:
276:
275:
269:
266:
264:
260:
259:
257:
251:
248:
246:
243:
242:
236:
230:
229:
222:
219:
217:
214:
212:
211:Orbital nodes
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
183:
180:
175:
174:
170:
163:
159:
155:
154:
151:Astrodynamics
149:
145:
144:
138:
134:
124:
113:
110:
106:
103:
99:
96:
92:
89:
85:
82: –
81:
77:
76:Find sources:
70:
66:
62:
56:
55:
54:single source
50:This article
48:
44:
39:
38:
33:
19:
4051:Perturbation
4033:
4008:Ground track
3918:Gravity turn
3869:
3862:
3855:
3846:
3837:
3817:
3808:
3799:
3792:True anomaly
3790:
3775:Mean anomaly
3773:
3753:
3744:
3735:
3726:
3706:
3693:
3684:
3677:Eccentricity
3675:
3633:Lunar cycler
3606:Heliocentric
3546:other points
3495:Medium Earth
3393:Non-inclined
3359:
3274:
3150:
3094:
2907:
2765:
2699:
2588:
2507:
2356:. It gives:
2327:
2220:
2105:
2023:
1938:
1901:
1836:
1751:
1734:escape orbit
1723:
1704:
1689:
1571:
1547:
1474:
1393:
1311:
1222:
1182:
1130:
1129:in terms of
1126:
1120:
978:
912:
813:
721:
639:
604:
567:
565:
288:Radial orbit
244:
239:eccentricity
221:True anomaly
206:Mean anomaly
196:Eccentricity
118:
108:
101:
94:
87:
75:
51:
4013:Hill sphere
3848:Mean motion
3728:Inclination
3717:Orientation
3618:Mars cycler
3556:Areocentric
3428:Synchronous
2910:dot product
421:Halo orbits
385:Hill sphere
201:Inclination
3953:Rendezvous
3649:Parameters
3485:High Earth
3455:Geocentric
3408:Osculating
3365:Elliptical
3255:References
1898:Derivation
1312:Hence the
641:Transverse
576:barycenter
465:Mass ratio
380:Barycenter
121:April 2020
91:newspapers
3998:Ephemeris
3975:mechanics
3885:Maneuvers
3828:Variation
3591:Libration
3586:Lissajous
3490:Low Earth
3480:Graveyard
3379:Horseshoe
3183:−
3126:−
3070:−
3056:−
3033:−
3010:−
2977:−
2961:ν
2951:μ
2941:ν
2938:μ
2927:γ
2860:−
2837:μ
2802:∂
2742:−
2718:˙
2670:˙
2650:−
2638:˙
2608:−
2565:˙
2526:˙
2523:ϕ
2478:˙
2475:ϕ
2448:−
2434:−
2422:˙
2392:−
2337:μ
2305:σ
2298:˙
2286:ν
2279:˙
2267:μ
2262:σ
2259:ν
2255:Γ
2246:μ
2239:¨
2191:˙
2188:ϕ
2171:−
2159:˙
2129:−
2090:τ
2063:π
2055:θ
2003:˙
2000:ϕ
1976:˙
1959:μ
1808:−
1690:Thus the
1665:μ
1660:−
1654:ϵ
1614:−
1608:ϵ
1584:ϵ
1528:μ
1455:μ
1424:π
1375:μ
1358:π
1297:μ
1275:ω
1231:μ
1165:μ
1108:, is the
1087:μ
1027:, is the
1002:, is the
956:μ
890:ω
787:ω
695:ω
596:periapsis
588:potential
305:Equations
233:Types of
61:talk page
4104:Category
3764:Position
3389:Inclined
3360:Circular
3233:See also
3097:velocity
1732:than an
909:Velocity
3973:Orbital
3943:Phasing
3903:Delta-v
3708:Apsides
3702:,
3500:Molniya
3418:Parking
3355:Capture
3343:General
2912:of the
2700:Hence:
1730:delta-v
1711:√
1183:where:
979:where:
808:radians
722:where:
628:to the
617:is the
105:scholar
4110:Orbits
3629:Other
3530:Tundra
3398:Kepler
3374:Escape
3327:orbits
3281:
3228:ahead.
1837:where
1556:Energy
893:
790:
775:radius
580:circle
570:is an
107:
100:
93:
86:
78:
3871:Epoch
3660:Shape
3598:Lunar
3552:Mars
3544:About
3515:Polar
3335:Types
572:orbit
186:Apsis
112:JSTOR
98:books
3663:Size
3602:Sun
3581:Halo
3433:semi
3279:ISBN
2908:The
1939:The
1705:The
1572:The
1121:The
1029:mass
590:and
84:news
3438:sub
3350:Box
1752:In
1494:par
1215:is
802:is
773:is
746:is
609:or
237:by
67:by
4106::
3786:,
3782:,
3391:/
3367:/
3263:^
3099::
2103:.
1936:.
1744:.
1552:.
1414:ff
632:.
602:.
586:,
566:A
4034:n
3866:0
3863:t
3853:v
3844:n
3835:T
3815:l
3806:L
3797:E
3788:f
3784:θ
3780:ν
3771:M
3751:ϖ
3742:ω
3733:Ω
3724:i
3704:q
3700:Q
3691:b
3682:a
3673:e
3318:e
3311:t
3304:v
3287:.
3191:S
3187:r
3180:r
3175:M
3172:G
3165:=
3162:v
3132:M
3129:2
3123:r
3119:M
3113:=
3110:v
3076:M
3073:3
3067:r
3062:M
3059:2
3053:r
3046:=
3039:M
3036:2
3030:r
3026:r
3016:M
3013:3
3007:r
3003:r
2996:)
2990:r
2986:M
2983:2
2974:1
2970:(
2966:=
2957:v
2947:u
2934:g
2930:=
2892:)
2888:0
2885:,
2882:0
2879:,
2876:0
2873:,
2866:M
2863:2
2857:r
2853:r
2846:(
2842:=
2833:v
2806:t
2775:r
2748:M
2745:3
2739:r
2735:r
2730:=
2725:2
2715:t
2685:1
2682:=
2677:2
2667:t
2658:r
2655:M
2645:2
2635:t
2627:)
2621:r
2617:M
2614:2
2605:1
2601:(
2572:2
2562:t
2551:3
2547:r
2543:M
2538:=
2533:2
2493:0
2490:=
2485:2
2467:)
2461:r
2457:M
2454:2
2445:1
2441:(
2437:r
2429:2
2419:t
2411:)
2405:r
2401:M
2398:2
2389:1
2385:(
2377:2
2373:r
2369:M
2343:r
2340:=
2313:0
2310:=
2295:x
2276:x
2251:+
2236:x
2206:1
2203:=
2198:2
2179:2
2175:r
2166:2
2156:t
2148:)
2142:r
2138:M
2135:2
2126:1
2122:(
2066:2
2058:=
2033:r
2024:(
2009:)
1994:,
1991:0
1988:,
1985:0
1982:,
1973:t
1967:(
1964:=
1955:u
1923:1
1920:=
1917:G
1914:=
1911:c
1877:2
1873:c
1868:M
1865:G
1862:2
1856:=
1851:S
1847:r
1816:S
1812:r
1805:r
1800:M
1797:G
1790:=
1787:v
1764:r
1713:2
1672:r
1669:2
1657:=
1630:2
1624:2
1620:v
1611:=
1576:(
1522:3
1518:r
1509:3
1505:2
1499:=
1490:T
1449:3
1445:r
1433:2
1428:2
1419:=
1410:T
1369:3
1365:r
1355:2
1352:=
1349:T
1324:T
1316:(
1294:=
1289:3
1285:r
1279:2
1245:2
1241:v
1237:r
1234:=
1203:v
1200:r
1197:=
1194:h
1158:2
1154:h
1147:=
1144:r
1131:θ
1127:r
1112:.
1096:M
1093:G
1090:=
1065:)
1060:2
1056:M
1052:+
1047:1
1043:M
1039:(
1015:M
990:G
960:r
950:=
943:r
937:M
934:G
927:=
924:v
869:r
848:v
827:a
760:r
733:v
706:r
699:2
690:=
684:r
679:2
675:v
669:=
665:a
555:e
548:t
541:v
423:)
419:(
270:)
261:(
123:)
119:(
109:·
102:·
95:·
88:·
71:.
57:.
34:.
20:)
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