119:
1582:
2537:
3572:
3204:
1126:
22:
4231:
2260:
3291:
2937:
3964:
1577:{\displaystyle {\begin{aligned}{\vec {e}}&={{\vec {v}}\times {\vec {h}} \over {\mu }}-{{\vec {r}} \over {\left|{\vec {r}}\right|}}=e{\vec {i}}\\&=\left({{\left|{\vec {v}}\right|}^{2} \over {\mu }}-{1 \over {\left|{\vec {r}}\right|}}\right){\vec {r}}-{{\vec {r}}\cdot {\vec {v}} \over {\mu }}{\vec {v}}\\&={\frac {1}{\mu }}\left\end{aligned}}}
2173:
2532:{\displaystyle {\begin{aligned}\cos(\Omega )&={\frac {{\vec {I}}\cdot {\vec {n}}}{n}}={\frac {n_{I}}{n}}=\cos(360-\Omega )\\\Rightarrow \Omega &=\arccos \left({\frac {{\vec {I}}\cdot {\vec {n}}}{n}}\right)=\Omega _{0},{\text{ or }}\\\Rightarrow \Omega &=360^{\circ }-\Omega _{0},{\text{ if }}n_{J}<0,\\\end{aligned}}}
3567:{\displaystyle {\begin{aligned}\cos(\nu )&={\frac {{\vec {e}}\cdot {\vec {r}}}{er}}=\cos(360-\nu )\\\Rightarrow \nu &=\arccos \left({\frac {{\vec {e}}\cdot {\vec {r}}}{er}}\right)=\nu _{0},{\text{ or }}\\\Rightarrow \nu &=360^{\circ }-\nu _{0},{\text{ if }}{\vec {r}}\cdot {\vec {v}}<0.\\\end{aligned}}}
3199:{\displaystyle {\begin{aligned}\cos(\omega )&={\frac {{\vec {n}}\cdot {\vec {e}}}{ne}}=\cos(360-\omega )\\\Rightarrow \omega &=\arccos \left({\frac {{\vec {n}}\cdot {\vec {e}}}{ne}}\right)=\omega _{0},{\text{ or }}\\\Rightarrow \omega &=360^{\circ }-\omega _{0},{\text{ if }}e_{K}<0,\\\end{aligned}}}
446:
If the position and velocity relative to the observer are available (as is the case with radar observations), these observational data can be adjusted by the known position and velocity of the observer relative to the attracting body at the times of observation. This yields the position and velocity
1972:
210:
calculations may use this data to calculate the probability that one orbiting object will collide with another. A satellite's operator may decide to adjust the orbit, if the risk of collision in the present orbit is unacceptable. (It is not possible to adjust the orbit for events of very low
171:
After orbits are determined, mathematical propagation techniques can be used to predict the future positions of orbiting objects. As time goes by, the actual path of an orbiting object tends to diverge from the predicted path (especially if the object is subject to difficult-to-predict
591:
of its central body. The central bodies are the sources of the gravitational forces, like the Sun, Earth, Moon and other planets. The orbiting bodies, on the other hand, include planets around the Sun, artificial satellites around the Earth, and spacecraft around planets. Newton's
4226:{\displaystyle {\begin{aligned}\cos(u)&={\frac {{\vec {n}}\cdot {\vec {r}}}{nr}}=\cos(360-u)\\\Rightarrow u&=\arccos \left({\frac {{\vec {n}}\cdot {\vec {r}}}{nr}}\right)=u_{0},{\text{ or }}\\\Rightarrow u&=360^{\circ }-u_{0},{\text{ if }}r_{K}<0,\\\end{aligned}}}
419:
Orbit determination must take into account that the apparent celestial motion of the body is influenced by the observer's own motion. For instance, an observer on Earth tracking an asteroid must take into account the motion of the Earth around the
752:
353:. This corresponds to knowing the object's relative direction in space, measured from the observer, but without knowledge of the distance of the object, i.e. the resultant measurement contains only direction information, like a
943:
of the orbital plane of the orbiting body. Since the line of ascending node is the line of intersection between the orbital plane and the reference plane, it is perpendicular to both the normal vectors of the reference plane
473:. In this procedure Gauss's method is used twice, both before and after a close interaction between two asteroids. After both orbits have been determined the mass of one or both of the asteroids can be worked out.
3921:
2168:{\displaystyle {\begin{aligned}\cos(i)&={\frac {{\vec {K}}\cdot {\vec {h}}}{h}}={\frac {h_{K}}{h}}\\\Rightarrow i&=\arccos \left({\frac {{\vec {K}}\cdot {\vec {h}}}{h}}\right),&i\in ,\end{aligned}}}
447:
with respect to the attracting body. If two such observations are available, along with the time difference between them, the orbit can be determined using
Lambert's method, invented in the 18th century. See
163:
apparatus are used, because naked-eye observations are inadequate for precise orbit determination. With more or better observations, the accuracy of the orbit determination process also improves, and fewer
783:
is the unit vector of the z-axis of the orbital plane. The specific angular momentum is a constant vector for an orbiting body, with its direction perpendicular to the orbital plane of the orbiting body.
937:
3969:
3296:
2942:
2265:
1977:
1131:
1865:
2703:
642:
3616:
537:
1624:
191:
1915:
585:
3826:
3959:
454:
Even if no distance information is available, an orbit can still be determined if three or more observations of the body's right ascension and declination have been made.
4287:
3260:
2767:
2735:
2620:
2229:
1121:
1064:
1029:
1000:
971:
875:
846:
817:
781:
637:
3776:
3724:
3656:
2932:
2255:
1941:
1653:
4258:
3679:
3231:
2906:
2879:
2833:
2813:
2591:
2564:
2200:
3636:
3286:
1777:
202:
gathers observations of all objects in Earth orbit. The observations are used in new orbit determination calculations that maintain the overall accuracy of the
2853:
2787:
1967:
1797:
1747:
1727:
1697:
1677:
1088:
2881:. Therefore, we have to make the judgment based on the sign of the Y-coordinate of the vector in the plane where the angle is measured. In this case,
137:
and verify that they have not been previously discovered. The basic methods were discovered in the 17th century and have been continuously refined.
3659:
313:
that completely describe an orbit. The theory of orbit determination has subsequently been developed to the point where today it is applied in
3831:
263:
880:
143:
are the raw data fed into orbit determination algorithms. Observations made by a ground-based observer typically consist of time-tagged
1802:
4330:
122:
Diagram showing how orbit-determination data was handled in a 1962 NASA mission. (Of historical interest only.)
877:
representing the unit vector of the Z-axis of the reference plane, which is perpendicular to the reference plane of the central body:
333:
of its motion with time are required. In early modern astronomy, the only available observational data for celestial objects were the
86:
424:, the rotation of the Earth, and the observer's local latitude and longitude, as these affect the apparent position of the body.
187:
58:
1586:
1869:
279:, including that which bears his name. Newton's method of successive approximation was formalised into an analytic method by
3288:
at epoch, which is the angle between the position vector and the periapsis at the particular time ('epoch') of observation:
252:
65:
39:
4404:
195:
4399:
4384:
4369:
4354:
4339:
105:
72:
2625:
1700:
1656:
384:
of the range to the object. This generally limits radar observations to objects relatively near the Earth, such as
207:
4425:
392:. Larger apertures permit tracking of transponders on interplanetary spacecraft throughout the solar system, and
258:
The mathematical methods for orbit determination originated with the publication in 1687 of the first edition of
3577:
492:
54:
404:
43:
399:
Various space agencies and commercial providers operate tracking networks to provide these observations. See
199:
133:
of objects such as moons, planets, and spacecraft. One major application is to allow tracking newly observed
118:
546:
400:
2235:
3961:
at epoch, which is the angle between the position vector and the ascending line at the particular time:
3781:
1094:
of the orbit. This direction is often defined as the x-axis of the orbital plane and has a unit vector
180:), and a new orbit determination using new observations serves to re-calibrate knowledge of the orbit.
747:{\displaystyle {\vec {h}}={\vec {r}}\times {\vec {v}}=\left|{\vec {h}}\right|{\vec {k}}=h{\vec {k}},}
608:
440:
3932:
173:
1031:). Therefore, the ascending node vector can be defined by the cross product of these two vectors.
284:
216:
79:
32:
4263:
3236:
2740:
2708:
2596:
2205:
1097:
1040:
1005:
976:
947:
851:
822:
793:
757:
613:
4435:
3729:
3684:
3641:
2917:
2912:
2240:
1920:
1629:
540:
403:
for a partial listing. Space-based tracking of satellites is also regularly performed. See
306:
251:
to deduce the elliptical shape of its orbit and its orientation in space, deriving his three
4309:
4236:
3927:
3664:
3209:
2884:
2858:
2818:
2792:
2569:
2542:
2178:
1067:
463:
448:
439:
drawn from the attracting body to the body at different points in time will all lie in the
385:
298:
291:
148:
3621:
3271:
8:
4420:
1756:
1035:
427:
A key observation is that (to a close approximation) all objects move in orbits that are
235:
Orbit determination has a long history, beginning with the prehistoric discovery of the
2838:
2772:
2257:, which is the angle between the ascending line and the X-axis of the reference frame:
1952:
1782:
1732:
1712:
1682:
1662:
1073:
486:
183:
4395:
4380:
4365:
4350:
4335:
1707:
459:
455:
389:
380:. However, the returned signal strength from radar decreases rapidly, as the inverse
350:
302:
203:
482:
436:
310:
283:
in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits by
177:
603:
The steps of orbit determination from one state vector are summarized as follows:
4430:
1750:
588:
393:
377:
342:
334:
240:
940:
787:
593:
470:
373:
165:
4414:
428:
408:
272:
939:
The ascending node vector is a vector pointing from the central body to the
3266:
597:
381:
318:
314:
295:
259:
1947:
432:
354:
346:
338:
330:
244:
152:
3916:{\displaystyle {\vec {r}}\cdot {\vec {v}}=rv\cos(90-\phi )=h\tan(\phi )}
212:
156:
1091:
21:
2934:, which is the angle between the periapsis and the ascending line:
369:
365:
268:
134:
267:, which gave a method for finding the orbit of a body following a
481:
The basic orbit determination task is to determine the classical
431:, with the attracting body (such as the Sun or the Earth) in the
144:
1066:
of the orbit. The eccentricity vector has the magnitude of the
276:
236:
220:
3681:. And, the sign of the fly-path angle is always positive when
361:
280:
224:
160:
130:
469:
One use is in the determination of asteroid masses via the
248:
1969:
of the orbital plane with respect to the reference plane:
1123:. According to the law of motion, it can be expressed as:
596:
will explain the trajectory of an orbiting body, known as
476:
317:
as well as the tracking and cataloguing of newly observed
421:
368:
measurements (by timing of the radar echo) and relative
329:
In order to determine the unknown orbit of a body, some
305:
was able to use just three observations (in the form of
932:{\displaystyle {\vec {n}}={\vec {K}}\times {\vec {h}}.}
4266:
4239:
3967:
3935:
3834:
3784:
3732:
3687:
3667:
3644:
3624:
3580:
3294:
3274:
3239:
3212:
2940:
2920:
2887:
2861:
2841:
2821:
2795:
2775:
2743:
2711:
2628:
2599:
2572:
2545:
2263:
2243:
2208:
2181:
1975:
1955:
1923:
1872:
1805:
1785:
1759:
1735:
1715:
1685:
1665:
1632:
1589:
1129:
1100:
1076:
1043:
1008:
979:
950:
883:
854:
825:
796:
760:
645:
616:
549:
495:
341:, obtained by observing the body as it moved in its
1090:, of the orbit, and points to the direction of the
46:. Unsourced material may be challenged and removed.
4305:
4303:
4281:
4252:
4225:
3953:
3915:
3820:
3770:
3718:
3673:
3650:
3630:
3610:
3566:
3280:
3254:
3225:
3198:
2926:
2900:
2873:
2847:
2827:
2807:
2781:
2761:
2729:
2697:
2614:
2585:
2558:
2531:
2249:
2223:
2194:
2167:
1961:
1935:
1909:
1860:{\displaystyle p={\frac {h^{2}}{\mu }}=a(1-e^{2})}
1859:
1791:
1771:
1741:
1721:
1691:
1671:
1647:
1618:
1576:
1115:
1082:
1058:
1023:
994:
965:
931:
869:
840:
811:
775:
746:
631:
579:
531:
458:, made famous in his 1801 "recovery" of the first
239:and subsequent attempts to predict their motions.
4412:
2593:are the X- and Y- coordinates, respectively, of
4300:
271:path from three observations. This was used by
4314:. Courier Corporation; 1971. Ch 2 p 51 et seq.
2698:{\displaystyle \cos(A)=\cos(-A)=\cos(360-A)=C}
290:Another milestone in orbit determination was
2231:when it is projected to the reference frame.
639:of the orbiting body from its state vector:
435:, and that the orbit lies in a fixed plane.
4349:, Chapters 7, 8; Wiley-Interscience (1985)
3658:angle, because it has the same sign as the
2769:is ambiguous in that there are two angles,
587:], of an orbiting body with respect to the
4331:Orbital Mechanics for Engineering Students
3611:{\displaystyle {\vec {r}}\cdot {\vec {v}}}
2835:value. It could actually return the angle
532:{\displaystyle a,e,i,\Omega ,\omega ,\nu }
190:and partner countries, to the extent that
1619:{\displaystyle e=\left|{\vec {e}}\right|}
106:Learn how and when to remove this message
117:
2622:, in the reference frame. Notice that
1910:{\displaystyle a={\frac {p}{1-e^{2}}},}
477:Orbit determination from a state vector
294:'s assistance in the "recovery" of the
4413:
1799:is undefined or defined as infinity):
1729:of the orbit, and its semi-major axis
376:of the radar echo) are possible using
211:probability; it would soon use up the
186:is another major application. For the
3618:can be used to check the quadrant of
580:{\displaystyle {\vec {r}},{\vec {v}}}
324:
405:List of radio telescopes#Space-based
44:adding citations to reliable sources
15:
466:, has been subsequently polished.
275:to establish the orbits of various
13:
4322:
3821:{\displaystyle h=rv\sin(90-\phi )}
2489:
2465:
2441:
2377:
2364:
2277:
2244:
514:
14:
4447:
219:.) Other countries, including
2737:is defined only in degrees. So
1701:universal gravitational constant
1657:standard gravitational parameter
227:, have similar tracking assets.
20:
4392:Statistical Orbit Determination
4310:Bate RR, Mueller DD, White JE.
3926:Optionally, we may compute the
372:measurements (by measuring the
31:needs additional citations for
4364:, Chapters 2, 5; Dover (1971)
4273:
4156:
4110:
4095:
4063:
4056:
4044:
4018:
4003:
3984:
3978:
3954:{\displaystyle u=\omega +\nu }
3910:
3904:
3889:
3877:
3856:
3841:
3815:
3803:
3765:
3739:
3713:
3694:
3602:
3587:
3548:
3533:
3483:
3437:
3422:
3390:
3383:
3371:
3345:
3330:
3311:
3305:
3246:
3129:
3083:
3068:
3036:
3029:
3017:
2991:
2976:
2957:
2951:
2908:can be used for such judgment.
2756:
2750:
2724:
2718:
2686:
2674:
2662:
2653:
2641:
2635:
2606:
2462:
2421:
2406:
2374:
2367:
2355:
2314:
2299:
2280:
2274:
2215:
2155:
2136:
2109:
2094:
2062:
2026:
2011:
1992:
1986:
1854:
1835:
1606:
1559:
1549:
1543:
1528:
1519:
1509:
1485:
1448:
1399:
1379:
1364:
1346:
1322:
1279:
1244:
1219:
1202:
1177:
1162:
1140:
1107:
1050:
1015:
986:
957:
920:
905:
890:
861:
832:
803:
767:
735:
717:
701:
682:
667:
652:
623:
571:
556:
1:
4407:, Coastal Bend College, Texas
4405:Satellite Orbit Determination
4362:Fundamentals of Astrodynamics
4334:, Chapter 5; Elsevier (2005)
4312:Fundamentals of astrodynamics
4293:
1659:for the central body of mass
200:Joint Space Operations Center
4379:, Chapter 3; Krieger (1997)
401:Category:Deep space networks
7:
2236:longitude of ascending node
247:'s careful observations of
10:
4452:
4282:{\displaystyle {\vec {r}}}
3255:{\displaystyle {\vec {e}}}
2762:{\displaystyle \arccos(C)}
2730:{\displaystyle \arccos(C)}
2615:{\displaystyle {\vec {n}}}
2224:{\displaystyle {\vec {h}}}
1116:{\displaystyle {\vec {i}}}
1059:{\displaystyle {\vec {e}}}
1024:{\displaystyle {\vec {h}}}
995:{\displaystyle {\vec {k}}}
966:{\displaystyle {\vec {K}}}
870:{\displaystyle {\vec {K}}}
841:{\displaystyle {\vec {h}}}
812:{\displaystyle {\vec {n}}}
776:{\displaystyle {\vec {k}}}
632:{\displaystyle {\vec {h}}}
414:
230:
215:the satellite carries for
973:) and the orbital plane (
609:specific angular momentum
3771:{\displaystyle \nu \in }
3719:{\displaystyle \nu \in }
3651:{\displaystyle \arccos }
253:laws of planetary motion
4289:in the reference frame.
4260:is the Z-coordinate of
3262:in the reference frame.
3233:is the Z-coordinate of
2927:{\displaystyle \omega }
2815:in , who have the same
2250:{\displaystyle \Omega }
2202:is the Z-coordinate of
1936:{\displaystyle e\neq 1}
1648:{\displaystyle \mu =GM}
217:orbital station-keeping
159:values. Telescopes or
4426:Spaceflight technology
4390:Schutz, Tapley, Born;
4360:Bate, Mueller, White;
4283:
4254:
4227:
3955:
3917:
3822:
3778:. Both are related by
3772:
3720:
3675:
3652:
3632:
3612:
3568:
3282:
3256:
3227:
3200:
2928:
2902:
2875:
2849:
2829:
2809:
2783:
2763:
2731:
2699:
2616:
2587:
2560:
2533:
2251:
2225:
2196:
2169:
1963:
1937:
1911:
1861:
1793:
1773:
1743:
1723:
1693:
1673:
1649:
1620:
1578:
1117:
1084:
1060:
1025:
996:
967:
933:
871:
842:
813:
777:
748:
633:
581:
533:
123:
4284:
4255:
4253:{\displaystyle r_{K}}
4228:
3956:
3918:
3823:
3773:
3721:
3676:
3674:{\displaystyle \phi }
3653:
3633:
3613:
3569:
3283:
3257:
3228:
3226:{\displaystyle e_{K}}
3201:
2929:
2913:argument of periapsis
2903:
2901:{\displaystyle n_{J}}
2876:
2874:{\displaystyle 360-A}
2850:
2830:
2828:{\displaystyle \cos }
2810:
2808:{\displaystyle 360-A}
2784:
2764:
2732:
2700:
2617:
2588:
2586:{\displaystyle n_{J}}
2561:
2559:{\displaystyle n_{I}}
2534:
2252:
2226:
2197:
2195:{\displaystyle h_{K}}
2170:
1964:
1938:
1912:
1862:
1794:
1774:
1744:
1724:
1694:
1674:
1650:
1621:
1579:
1118:
1085:
1061:
1026:
997:
968:
934:
872:
843:
814:
778:
749:
634:
582:
541:orbital state vectors
534:
386:artificial satellites
307:celestial coordinates
198:resources allow, the
129:is the estimation of
121:
55:"Orbit determination"
4264:
4237:
3965:
3933:
3928:argument of latitude
3832:
3782:
3730:
3726:, and negative when
3685:
3665:
3642:
3631:{\displaystyle \nu }
3622:
3578:
3292:
3281:{\displaystyle \nu }
3272:
3237:
3210:
2938:
2918:
2885:
2859:
2839:
2819:
2793:
2773:
2741:
2709:
2626:
2597:
2570:
2543:
2261:
2241:
2206:
2179:
1973:
1953:
1921:
1870:
1803:
1783:
1757:
1733:
1713:
1683:
1663:
1630:
1587:
1127:
1098:
1074:
1041:
1006:
977:
948:
881:
852:
823:
794:
758:
643:
614:
547:
493:
351:an optical telescope
292:Carl Friedrich Gauss
40:improve this article
4347:Celestial Mechanics
1772:{\displaystyle e=1}
1036:eccentricity vector
396:of natural bodies.
208:Collision avoidance
127:Orbit determination
4394:, Academic Press.
4279:
4250:
4223:
4221:
3951:
3913:
3818:
3768:
3716:
3671:
3648:
3628:
3608:
3564:
3562:
3278:
3252:
3223:
3196:
3194:
2924:
2898:
2871:
2845:
2825:
2805:
2779:
2759:
2727:
2695:
2612:
2583:
2556:
2529:
2527:
2247:
2221:
2192:
2165:
2163:
1959:
1933:
1907:
1857:
1789:
1769:
1739:
1719:
1689:
1669:
1645:
1616:
1574:
1572:
1113:
1080:
1056:
1021:
992:
963:
929:
867:
838:
809:
773:
744:
629:
577:
529:
487:Keplerian elements
390:Near-Earth objects
345:, relative to the
325:Observational data
309:) to find the six
184:Satellite tracking
124:
4377:Orbital Mechanics
4276:
4198:
4150:
4125:
4113:
4098:
4033:
4021:
4006:
3859:
3844:
3605:
3590:
3551:
3536:
3525:
3477:
3452:
3440:
3425:
3360:
3348:
3333:
3249:
3171:
3123:
3098:
3086:
3071:
3006:
2994:
2979:
2848:{\displaystyle A}
2782:{\displaystyle A}
2609:
2504:
2456:
2431:
2424:
2409:
2344:
2324:
2317:
2302:
2218:
2119:
2112:
2097:
2056:
2036:
2029:
2014:
1962:{\displaystyle i}
1902:
1827:
1792:{\displaystyle a}
1742:{\displaystyle a}
1722:{\displaystyle p}
1708:semi-latus rectum
1692:{\displaystyle G}
1672:{\displaystyle M}
1609:
1562:
1546:
1531:
1512:
1496:
1488:
1451:
1423:
1402:
1391:
1382:
1367:
1349:
1333:
1325:
1301:
1282:
1247:
1230:
1222:
1205:
1189:
1180:
1165:
1143:
1110:
1083:{\displaystyle e}
1053:
1018:
989:
960:
923:
908:
893:
864:
835:
806:
770:
738:
720:
704:
685:
670:
655:
626:
574:
559:
460:lost minor planet
449:Lambert's problem
204:satellite catalog
116:
115:
108:
90:
4443:
4316:
4307:
4288:
4286:
4285:
4280:
4278:
4277:
4269:
4259:
4257:
4256:
4251:
4249:
4248:
4232:
4230:
4229:
4224:
4222:
4209:
4208:
4199:
4196:
4191:
4190:
4178:
4177:
4151:
4148:
4143:
4142:
4130:
4126:
4124:
4116:
4115:
4114:
4106:
4100:
4099:
4091:
4087:
4034:
4032:
4024:
4023:
4022:
4014:
4008:
4007:
3999:
3995:
3960:
3958:
3957:
3952:
3922:
3920:
3919:
3914:
3861:
3860:
3852:
3846:
3845:
3837:
3827:
3825:
3824:
3819:
3777:
3775:
3774:
3769:
3764:
3763:
3751:
3750:
3725:
3723:
3722:
3717:
3712:
3711:
3680:
3678:
3677:
3672:
3657:
3655:
3654:
3649:
3638:and correct the
3637:
3635:
3634:
3629:
3617:
3615:
3614:
3609:
3607:
3606:
3598:
3592:
3591:
3583:
3573:
3571:
3570:
3565:
3563:
3553:
3552:
3544:
3538:
3537:
3529:
3526:
3523:
3518:
3517:
3505:
3504:
3478:
3475:
3470:
3469:
3457:
3453:
3451:
3443:
3442:
3441:
3433:
3427:
3426:
3418:
3414:
3361:
3359:
3351:
3350:
3349:
3341:
3335:
3334:
3326:
3322:
3287:
3285:
3284:
3279:
3261:
3259:
3258:
3253:
3251:
3250:
3242:
3232:
3230:
3229:
3224:
3222:
3221:
3205:
3203:
3202:
3197:
3195:
3182:
3181:
3172:
3169:
3164:
3163:
3151:
3150:
3124:
3121:
3116:
3115:
3103:
3099:
3097:
3089:
3088:
3087:
3079:
3073:
3072:
3064:
3060:
3007:
3005:
2997:
2996:
2995:
2987:
2981:
2980:
2972:
2968:
2933:
2931:
2930:
2925:
2907:
2905:
2904:
2899:
2897:
2896:
2880:
2878:
2877:
2872:
2854:
2852:
2851:
2846:
2834:
2832:
2831:
2826:
2814:
2812:
2811:
2806:
2788:
2786:
2785:
2780:
2768:
2766:
2765:
2760:
2736:
2734:
2733:
2728:
2704:
2702:
2701:
2696:
2621:
2619:
2618:
2613:
2611:
2610:
2602:
2592:
2590:
2589:
2584:
2582:
2581:
2565:
2563:
2562:
2557:
2555:
2554:
2538:
2536:
2535:
2530:
2528:
2515:
2514:
2505:
2502:
2497:
2496:
2484:
2483:
2457:
2454:
2449:
2448:
2436:
2432:
2427:
2426:
2425:
2417:
2411:
2410:
2402:
2398:
2345:
2340:
2339:
2330:
2325:
2320:
2319:
2318:
2310:
2304:
2303:
2295:
2291:
2256:
2254:
2253:
2248:
2230:
2228:
2227:
2222:
2220:
2219:
2211:
2201:
2199:
2198:
2193:
2191:
2190:
2174:
2172:
2171:
2166:
2164:
2154:
2153:
2124:
2120:
2115:
2114:
2113:
2105:
2099:
2098:
2090:
2086:
2057:
2052:
2051:
2042:
2037:
2032:
2031:
2030:
2022:
2016:
2015:
2007:
2003:
1968:
1966:
1965:
1960:
1942:
1940:
1939:
1934:
1916:
1914:
1913:
1908:
1903:
1901:
1900:
1899:
1880:
1866:
1864:
1863:
1858:
1853:
1852:
1828:
1823:
1822:
1813:
1798:
1796:
1795:
1790:
1778:
1776:
1775:
1770:
1749:(if it is not a
1748:
1746:
1745:
1740:
1728:
1726:
1725:
1720:
1698:
1696:
1695:
1690:
1678:
1676:
1675:
1670:
1654:
1652:
1651:
1646:
1625:
1623:
1622:
1617:
1615:
1611:
1610:
1602:
1583:
1581:
1580:
1575:
1573:
1569:
1565:
1564:
1563:
1555:
1552:
1548:
1547:
1539:
1533:
1532:
1524:
1514:
1513:
1505:
1502:
1498:
1497:
1495:
1494:
1490:
1489:
1481:
1470:
1465:
1464:
1463:
1458:
1457:
1453:
1452:
1444:
1424:
1416:
1408:
1404:
1403:
1395:
1392:
1390:
1385:
1384:
1383:
1375:
1369:
1368:
1360:
1356:
1351:
1350:
1342:
1339:
1335:
1334:
1332:
1331:
1327:
1326:
1318:
1307:
1302:
1300:
1295:
1294:
1289:
1288:
1284:
1283:
1275:
1266:
1253:
1249:
1248:
1240:
1231:
1229:
1228:
1224:
1223:
1215:
1207:
1206:
1198:
1195:
1190:
1188:
1183:
1182:
1181:
1173:
1167:
1166:
1158:
1154:
1145:
1144:
1136:
1122:
1120:
1119:
1114:
1112:
1111:
1103:
1089:
1087:
1086:
1081:
1065:
1063:
1062:
1057:
1055:
1054:
1046:
1030:
1028:
1027:
1022:
1020:
1019:
1011:
1001:
999:
998:
993:
991:
990:
982:
972:
970:
969:
964:
962:
961:
953:
938:
936:
935:
930:
925:
924:
916:
910:
909:
901:
895:
894:
886:
876:
874:
873:
868:
866:
865:
857:
847:
845:
844:
839:
837:
836:
828:
818:
816:
815:
810:
808:
807:
799:
782:
780:
779:
774:
772:
771:
763:
753:
751:
750:
745:
740:
739:
731:
722:
721:
713:
710:
706:
705:
697:
687:
686:
678:
672:
671:
663:
657:
656:
648:
638:
636:
635:
630:
628:
627:
619:
586:
584:
583:
578:
576:
575:
567:
561:
560:
552:
538:
536:
535:
530:
483:orbital elements
378:radio telescopes
311:orbital elements
255:in the process.
178:atmospheric drag
111:
104:
100:
97:
91:
89:
48:
24:
16:
4451:
4450:
4446:
4445:
4444:
4442:
4441:
4440:
4411:
4410:
4325:
4323:Further reading
4320:
4319:
4308:
4301:
4296:
4268:
4267:
4265:
4262:
4261:
4244:
4240:
4238:
4235:
4234:
4220:
4219:
4204:
4200:
4195:
4186:
4182:
4173:
4169:
4162:
4153:
4152:
4147:
4138:
4134:
4117:
4105:
4104:
4090:
4089:
4088:
4086:
4082:
4069:
4060:
4059:
4025:
4013:
4012:
3998:
3997:
3996:
3994:
3987:
3968:
3966:
3963:
3962:
3934:
3931:
3930:
3851:
3850:
3836:
3835:
3833:
3830:
3829:
3783:
3780:
3779:
3759:
3755:
3746:
3742:
3731:
3728:
3727:
3707:
3703:
3686:
3683:
3682:
3666:
3663:
3662:
3643:
3640:
3639:
3623:
3620:
3619:
3597:
3596:
3582:
3581:
3579:
3576:
3575:
3561:
3560:
3543:
3542:
3528:
3527:
3522:
3513:
3509:
3500:
3496:
3489:
3480:
3479:
3474:
3465:
3461:
3444:
3432:
3431:
3417:
3416:
3415:
3413:
3409:
3396:
3387:
3386:
3352:
3340:
3339:
3325:
3324:
3323:
3321:
3314:
3295:
3293:
3290:
3289:
3273:
3270:
3269:
3241:
3240:
3238:
3235:
3234:
3217:
3213:
3211:
3208:
3207:
3193:
3192:
3177:
3173:
3168:
3159:
3155:
3146:
3142:
3135:
3126:
3125:
3120:
3111:
3107:
3090:
3078:
3077:
3063:
3062:
3061:
3059:
3055:
3042:
3033:
3032:
2998:
2986:
2985:
2971:
2970:
2969:
2967:
2960:
2941:
2939:
2936:
2935:
2919:
2916:
2915:
2892:
2888:
2886:
2883:
2882:
2860:
2857:
2856:
2840:
2837:
2836:
2820:
2817:
2816:
2794:
2791:
2790:
2774:
2771:
2770:
2742:
2739:
2738:
2710:
2707:
2706:
2627:
2624:
2623:
2601:
2600:
2598:
2595:
2594:
2577:
2573:
2571:
2568:
2567:
2550:
2546:
2544:
2541:
2540:
2526:
2525:
2510:
2506:
2501:
2492:
2488:
2479:
2475:
2468:
2459:
2458:
2453:
2444:
2440:
2416:
2415:
2401:
2400:
2399:
2397:
2393:
2380:
2371:
2370:
2335:
2331:
2329:
2309:
2308:
2294:
2293:
2292:
2290:
2283:
2264:
2262:
2259:
2258:
2242:
2239:
2238:
2210:
2209:
2207:
2204:
2203:
2186:
2182:
2180:
2177:
2176:
2162:
2161:
2149:
2145:
2128:
2104:
2103:
2089:
2088:
2087:
2085:
2081:
2068:
2059:
2058:
2047:
2043:
2041:
2021:
2020:
2006:
2005:
2004:
2002:
1995:
1976:
1974:
1971:
1970:
1954:
1951:
1950:
1922:
1919:
1918:
1895:
1891:
1884:
1879:
1871:
1868:
1867:
1848:
1844:
1818:
1814:
1812:
1804:
1801:
1800:
1784:
1781:
1780:
1758:
1755:
1754:
1751:parabolic orbit
1734:
1731:
1730:
1714:
1711:
1710:
1684:
1681:
1680:
1664:
1661:
1660:
1631:
1628:
1627:
1601:
1600:
1596:
1588:
1585:
1584:
1571:
1570:
1554:
1553:
1538:
1537:
1523:
1522:
1518:
1504:
1503:
1480:
1479:
1475:
1474:
1469:
1459:
1443:
1442:
1438:
1437:
1436:
1435:
1434:
1430:
1429:
1425:
1415:
1406:
1405:
1394:
1393:
1386:
1374:
1373:
1359:
1358:
1357:
1355:
1341:
1340:
1317:
1316:
1312:
1311:
1306:
1296:
1290:
1274:
1273:
1269:
1268:
1267:
1265:
1264:
1260:
1251:
1250:
1239:
1238:
1214:
1213:
1209:
1208:
1197:
1196:
1194:
1184:
1172:
1171:
1157:
1156:
1155:
1153:
1146:
1135:
1134:
1130:
1128:
1125:
1124:
1102:
1101:
1099:
1096:
1095:
1075:
1072:
1071:
1045:
1044:
1042:
1039:
1038:
1010:
1009:
1007:
1004:
1003:
981:
980:
978:
975:
974:
952:
951:
949:
946:
945:
915:
914:
900:
899:
885:
884:
882:
879:
878:
856:
855:
853:
850:
849:
827:
826:
824:
821:
820:
798:
797:
795:
792:
791:
762:
761:
759:
756:
755:
730:
729:
712:
711:
696:
695:
691:
677:
676:
662:
661:
647:
646:
644:
641:
640:
618:
617:
615:
612:
611:
598:Keplerian orbit
589:reference frame
566:
565:
551:
550:
548:
545:
544:
494:
491:
490:
479:
417:
394:radar astronomy
343:observation arc
335:right ascension
327:
241:Johannes Kepler
233:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
4449:
4439:
4438:
4433:
4428:
4423:
4409:
4408:
4402:
4400:978-0126836301
4388:
4373:
4358:
4343:
4324:
4321:
4318:
4317:
4298:
4297:
4295:
4292:
4291:
4290:
4275:
4272:
4247:
4243:
4218:
4215:
4212:
4207:
4203:
4197: if
4194:
4189:
4185:
4181:
4176:
4172:
4168:
4165:
4163:
4161:
4158:
4155:
4154:
4149: or
4146:
4141:
4137:
4133:
4129:
4123:
4120:
4112:
4109:
4103:
4097:
4094:
4085:
4081:
4078:
4075:
4072:
4070:
4068:
4065:
4062:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4031:
4028:
4020:
4017:
4011:
4005:
4002:
3993:
3990:
3988:
3986:
3983:
3980:
3977:
3974:
3971:
3970:
3950:
3947:
3944:
3941:
3938:
3924:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3858:
3855:
3849:
3843:
3840:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3767:
3762:
3758:
3754:
3749:
3745:
3741:
3738:
3735:
3715:
3710:
3706:
3702:
3699:
3696:
3693:
3690:
3670:
3660:fly-path angle
3647:
3627:
3604:
3601:
3595:
3589:
3586:
3559:
3556:
3550:
3547:
3541:
3535:
3532:
3524: if
3521:
3516:
3512:
3508:
3503:
3499:
3495:
3492:
3490:
3488:
3485:
3482:
3481:
3476: or
3473:
3468:
3464:
3460:
3456:
3450:
3447:
3439:
3436:
3430:
3424:
3421:
3412:
3408:
3405:
3402:
3399:
3397:
3395:
3392:
3389:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3358:
3355:
3347:
3344:
3338:
3332:
3329:
3320:
3317:
3315:
3313:
3310:
3307:
3304:
3301:
3298:
3297:
3277:
3263:
3248:
3245:
3220:
3216:
3191:
3188:
3185:
3180:
3176:
3170: if
3167:
3162:
3158:
3154:
3149:
3145:
3141:
3138:
3136:
3134:
3131:
3128:
3127:
3122: or
3119:
3114:
3110:
3106:
3102:
3096:
3093:
3085:
3082:
3076:
3070:
3067:
3058:
3054:
3051:
3048:
3045:
3043:
3041:
3038:
3035:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3004:
3001:
2993:
2990:
2984:
2978:
2975:
2966:
2963:
2961:
2959:
2956:
2953:
2950:
2947:
2944:
2943:
2923:
2909:
2895:
2891:
2870:
2867:
2864:
2844:
2824:
2804:
2801:
2798:
2778:
2758:
2755:
2752:
2749:
2746:
2726:
2723:
2720:
2717:
2714:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2608:
2605:
2580:
2576:
2553:
2549:
2524:
2521:
2518:
2513:
2509:
2503: if
2500:
2495:
2491:
2487:
2482:
2478:
2474:
2471:
2469:
2467:
2464:
2461:
2460:
2455: or
2452:
2447:
2443:
2439:
2435:
2430:
2423:
2420:
2414:
2408:
2405:
2396:
2392:
2389:
2386:
2383:
2381:
2379:
2376:
2373:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2343:
2338:
2334:
2328:
2323:
2316:
2313:
2307:
2301:
2298:
2289:
2286:
2284:
2282:
2279:
2276:
2273:
2270:
2267:
2266:
2246:
2232:
2217:
2214:
2189:
2185:
2160:
2157:
2152:
2148:
2144:
2141:
2138:
2135:
2132:
2129:
2127:
2123:
2118:
2111:
2108:
2102:
2096:
2093:
2084:
2080:
2077:
2074:
2071:
2069:
2067:
2064:
2061:
2060:
2055:
2050:
2046:
2040:
2035:
2028:
2025:
2019:
2013:
2010:
2001:
1998:
1996:
1994:
1991:
1988:
1985:
1982:
1979:
1978:
1958:
1944:
1932:
1929:
1926:
1906:
1898:
1894:
1890:
1887:
1883:
1878:
1875:
1856:
1851:
1847:
1843:
1840:
1837:
1834:
1831:
1826:
1821:
1817:
1811:
1808:
1788:
1768:
1765:
1762:
1738:
1718:
1704:
1688:
1668:
1644:
1641:
1638:
1635:
1614:
1608:
1605:
1599:
1595:
1592:
1568:
1561:
1558:
1551:
1545:
1542:
1536:
1530:
1527:
1521:
1517:
1511:
1508:
1501:
1493:
1487:
1484:
1478:
1473:
1468:
1462:
1456:
1450:
1447:
1441:
1433:
1428:
1422:
1419:
1414:
1411:
1409:
1407:
1401:
1398:
1389:
1381:
1378:
1372:
1366:
1363:
1354:
1348:
1345:
1338:
1330:
1324:
1321:
1315:
1310:
1305:
1299:
1293:
1287:
1281:
1278:
1272:
1263:
1259:
1256:
1254:
1252:
1246:
1243:
1237:
1234:
1227:
1221:
1218:
1212:
1204:
1201:
1193:
1187:
1179:
1176:
1170:
1164:
1161:
1152:
1149:
1147:
1142:
1139:
1133:
1132:
1109:
1106:
1079:
1052:
1049:
1032:
1017:
1014:
988:
985:
959:
956:
941:ascending node
928:
922:
919:
913:
907:
904:
898:
892:
889:
863:
860:
834:
831:
805:
802:
788:ascending node
784:
769:
766:
743:
737:
734:
728:
725:
719:
716:
709:
703:
700:
694:
690:
684:
681:
675:
669:
666:
660:
654:
651:
625:
622:
594:laws of motion
573:
570:
564:
558:
555:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
498:
478:
475:
471:dynamic method
456:Gauss's method
429:conic sections
416:
413:
374:Doppler effect
326:
323:
303:Gauss's method
287:in 1761–1777.
232:
229:
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
4448:
4437:
4436:Astrodynamics
4434:
4432:
4429:
4427:
4424:
4422:
4419:
4418:
4416:
4406:
4403:
4401:
4397:
4393:
4389:
4386:
4385:0-89464-010-0
4382:
4378:
4375:Madonna, R.;
4374:
4371:
4370:0-486-60061-0
4367:
4363:
4359:
4356:
4355:0-471-89316-1
4352:
4348:
4344:
4341:
4340:0-7506-6169-0
4337:
4333:
4332:
4327:
4326:
4315:
4313:
4306:
4304:
4299:
4270:
4245:
4241:
4216:
4213:
4210:
4205:
4201:
4192:
4187:
4183:
4179:
4174:
4170:
4166:
4164:
4159:
4144:
4139:
4135:
4131:
4127:
4121:
4118:
4107:
4101:
4092:
4083:
4079:
4076:
4073:
4071:
4066:
4053:
4050:
4047:
4041:
4038:
4035:
4029:
4026:
4015:
4009:
4000:
3991:
3989:
3981:
3975:
3972:
3948:
3945:
3942:
3939:
3936:
3929:
3925:
3907:
3901:
3898:
3895:
3892:
3886:
3883:
3880:
3874:
3871:
3868:
3865:
3862:
3853:
3847:
3838:
3812:
3809:
3806:
3800:
3797:
3794:
3791:
3788:
3785:
3760:
3756:
3752:
3747:
3743:
3736:
3733:
3708:
3704:
3700:
3697:
3691:
3688:
3668:
3661:
3645:
3625:
3599:
3593:
3584:
3557:
3554:
3545:
3539:
3530:
3519:
3514:
3510:
3506:
3501:
3497:
3493:
3491:
3486:
3471:
3466:
3462:
3458:
3454:
3448:
3445:
3434:
3428:
3419:
3410:
3406:
3403:
3400:
3398:
3393:
3380:
3377:
3374:
3368:
3365:
3362:
3356:
3353:
3342:
3336:
3327:
3318:
3316:
3308:
3302:
3299:
3275:
3268:
3264:
3243:
3218:
3214:
3189:
3186:
3183:
3178:
3174:
3165:
3160:
3156:
3152:
3147:
3143:
3139:
3137:
3132:
3117:
3112:
3108:
3104:
3100:
3094:
3091:
3080:
3074:
3065:
3056:
3052:
3049:
3046:
3044:
3039:
3026:
3023:
3020:
3014:
3011:
3008:
3002:
2999:
2988:
2982:
2973:
2964:
2962:
2954:
2948:
2945:
2921:
2914:
2910:
2893:
2889:
2868:
2865:
2862:
2842:
2822:
2802:
2799:
2796:
2776:
2753:
2747:
2744:
2721:
2715:
2712:
2692:
2689:
2683:
2680:
2677:
2671:
2668:
2665:
2659:
2656:
2650:
2647:
2644:
2638:
2632:
2629:
2603:
2578:
2574:
2551:
2547:
2522:
2519:
2516:
2511:
2507:
2498:
2493:
2485:
2480:
2476:
2472:
2470:
2450:
2445:
2437:
2433:
2428:
2418:
2412:
2403:
2394:
2390:
2387:
2384:
2382:
2361:
2358:
2352:
2349:
2346:
2341:
2336:
2332:
2326:
2321:
2311:
2305:
2296:
2287:
2285:
2271:
2268:
2237:
2233:
2212:
2187:
2183:
2158:
2150:
2146:
2142:
2139:
2133:
2130:
2125:
2121:
2116:
2106:
2100:
2091:
2082:
2078:
2075:
2072:
2070:
2065:
2053:
2048:
2044:
2038:
2033:
2023:
2017:
2008:
1999:
1997:
1989:
1983:
1980:
1956:
1949:
1945:
1930:
1927:
1924:
1904:
1896:
1892:
1888:
1885:
1881:
1876:
1873:
1849:
1845:
1841:
1838:
1832:
1829:
1824:
1819:
1815:
1809:
1806:
1786:
1766:
1763:
1760:
1752:
1736:
1716:
1709:
1705:
1702:
1686:
1666:
1658:
1642:
1639:
1636:
1633:
1612:
1603:
1597:
1593:
1590:
1566:
1556:
1540:
1534:
1525:
1515:
1506:
1499:
1491:
1482:
1476:
1471:
1466:
1460:
1454:
1445:
1439:
1431:
1426:
1420:
1417:
1412:
1410:
1396:
1387:
1376:
1370:
1361:
1352:
1343:
1336:
1328:
1319:
1313:
1308:
1303:
1297:
1291:
1285:
1276:
1270:
1261:
1257:
1255:
1241:
1235:
1232:
1225:
1216:
1210:
1199:
1191:
1185:
1174:
1168:
1159:
1150:
1148:
1137:
1104:
1093:
1077:
1069:
1047:
1037:
1033:
1012:
983:
954:
942:
926:
917:
911:
902:
896:
887:
858:
829:
800:
789:
785:
764:
741:
732:
726:
723:
714:
707:
698:
692:
688:
679:
673:
664:
658:
649:
620:
610:
606:
605:
604:
601:
599:
595:
590:
568:
562:
553:
542:
526:
523:
520:
517:
511:
508:
505:
502:
499:
496:
488:
484:
474:
472:
467:
465:
461:
457:
452:
451:for details.
450:
444:
442:
441:orbital plane
438:
434:
430:
425:
423:
412:
410:
409:Space Network
406:
402:
397:
395:
391:
387:
383:
379:
375:
371:
367:
363:
358:
356:
352:
348:
344:
340:
336:
332:
322:
320:
319:minor planets
316:
315:GPS receivers
312:
308:
304:
300:
297:
293:
288:
286:
282:
278:
274:
273:Edmund Halley
270:
266:
265:
261:
256:
254:
250:
246:
242:
238:
228:
226:
222:
218:
214:
209:
205:
201:
197:
193:
189:
188:United States
185:
181:
179:
175:
174:perturbations
169:
167:
162:
158:
154:
150:
146:
142:
138:
136:
132:
128:
120:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
4391:
4376:
4361:
4346:
4329:
4328:Curtis, H.;
4311:
3574:The sign of
3267:true anomaly
3265:Compute the
2911:Compute the
2234:Compute the
1946:Compute the
1706:Compute the
1068:eccentricity
1034:Compute the
786:Compute the
607:Compute the
602:
480:
468:
453:
445:
426:
418:
398:
382:fourth power
359:
331:observations
328:
296:dwarf planet
289:
262:
257:
234:
182:
170:
166:false alarms
141:Observations
140:
139:
126:
125:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
1948:inclination
539:, from the
433:prime focus
364:, relative
355:unit vector
347:fixed stars
339:declination
245:Tycho Brahe
96:August 2009
4421:Astrometry
4415:Categories
4345:Taff, L.;
4294:References
301:in 1801.
213:propellant
168:" result.
157:range rate
66:newspapers
4274:→
4180:−
4175:∘
4157:⇒
4111:→
4102:⋅
4096:→
4080:
4064:⇒
4051:−
4042:
4019:→
4010:⋅
4004:→
3976:
3949:ν
3943:ω
3908:ϕ
3902:
3887:ϕ
3884:−
3875:
3857:→
3848:⋅
3842:→
3813:ϕ
3810:−
3801:
3761:∘
3748:∘
3737:∈
3734:ν
3709:∘
3692:∈
3689:ν
3669:ϕ
3626:ν
3603:→
3594:⋅
3588:→
3549:→
3540:⋅
3534:→
3511:ν
3507:−
3502:∘
3487:ν
3484:⇒
3463:ν
3438:→
3429:⋅
3423:→
3407:
3394:ν
3391:⇒
3381:ν
3378:−
3369:
3346:→
3337:⋅
3331:→
3309:ν
3303:
3276:ν
3247:→
3157:ω
3153:−
3148:∘
3133:ω
3130:⇒
3109:ω
3084:→
3075:⋅
3069:→
3053:
3040:ω
3037:⇒
3027:ω
3024:−
3015:
2992:→
2983:⋅
2977:→
2955:ω
2949:
2922:ω
2866:−
2800:−
2748:
2716:
2681:−
2672:
2657:−
2651:
2633:
2607:→
2490:Ω
2486:−
2481:∘
2466:Ω
2463:⇒
2442:Ω
2422:→
2413:⋅
2407:→
2391:
2378:Ω
2375:⇒
2365:Ω
2362:−
2353:
2315:→
2306:⋅
2300:→
2278:Ω
2272:
2245:Ω
2216:→
2151:∘
2134:∈
2110:→
2101:⋅
2095:→
2079:
2063:⇒
2027:→
2018:⋅
2012:→
1984:
1928:≠
1889:−
1842:−
1825:μ
1634:μ
1607:→
1560:→
1544:→
1535:⋅
1529:→
1516:−
1510:→
1486:→
1472:μ
1467:−
1449:→
1421:μ
1400:→
1388:μ
1380:→
1371:⋅
1365:→
1353:−
1347:→
1323:→
1304:−
1298:μ
1280:→
1245:→
1220:→
1203:→
1192:−
1186:μ
1178:→
1169:×
1163:→
1141:→
1108:→
1092:periapsis
1051:→
1016:→
987:→
958:→
921:→
912:×
906:→
891:→
862:→
833:→
804:→
768:→
736:→
718:→
702:→
683:→
674:×
668:→
653:→
624:→
572:→
557:→
527:ν
521:ω
515:Ω
269:parabolic
264:Principia
155:, and/or
149:elevation
135:asteroids
1753:, where
370:velocity
366:distance
349:, using
260:Newton's
176:such as
1699:is the
1655:is the
848:, with
790:vector
437:Vectors
415:Methods
285:Lambert
237:planets
231:History
192:optical
145:azimuth
80:scholar
4431:Orbits
4398:
4383:
4368:
4353:
4338:
4233:where
4077:arccos
3646:arccos
3404:arccos
3206:where
3050:arccos
2745:arccos
2713:arccos
2705:, but
2539:where
2388:arccos
2175:where
2076:arccos
1679:, and
1626:where
754:where
277:comets
221:Russia
131:orbits
82:
75:
68:
61:
53:
819:from
464:Ceres
362:radar
360:With
299:Ceres
281:Euler
243:used
225:China
196:radar
161:radar
153:range
87:JSTOR
73:books
4396:ISBN
4381:ISBN
4366:ISBN
4351:ISBN
4336:ISBN
4211:<
3828:and
3555:<
3184:<
2789:and
2566:and
2517:<
1917:(if
1779:and
407:and
388:and
337:and
249:Mars
223:and
194:and
59:news
4171:360
4048:360
4039:cos
3973:cos
3899:tan
3872:cos
3798:sin
3757:360
3744:180
3705:180
3498:360
3375:360
3366:cos
3300:cos
3144:360
3021:360
3012:cos
2946:cos
2863:360
2855:or
2823:cos
2797:360
2678:360
2669:cos
2648:cos
2630:cos
2477:360
2359:360
2350:cos
2269:cos
2147:180
1981:cos
1002:or
485:or
422:Sun
206:.
42:by
4417::
4302:^
3881:90
3807:90
3558:0.
1943:).
1070:,
600:.
489:,
462:,
443:.
411:.
357:.
321:.
151:,
147:,
4387:.
4372:.
4357:.
4342:.
4271:r
4246:K
4242:r
4217:,
4214:0
4206:K
4202:r
4193:,
4188:0
4184:u
4167:=
4160:u
4145:,
4140:0
4136:u
4132:=
4128:)
4122:r
4119:n
4108:r
4093:n
4084:(
4074:=
4067:u
4057:)
4054:u
4045:(
4036:=
4030:r
4027:n
4016:r
4001:n
3992:=
3985:)
3982:u
3979:(
3946:+
3940:=
3937:u
3923:.
3911:)
3905:(
3896:h
3893:=
3890:)
3878:(
3869:v
3866:r
3863:=
3854:v
3839:r
3816:)
3804:(
3795:v
3792:r
3789:=
3786:h
3766:]
3753:,
3740:[
3714:]
3701:,
3698:0
3695:[
3600:v
3585:r
3546:v
3531:r
3520:,
3515:0
3494:=
3472:,
3467:0
3459:=
3455:)
3449:r
3446:e
3435:r
3420:e
3411:(
3401:=
3384:)
3372:(
3363:=
3357:r
3354:e
3343:r
3328:e
3319:=
3312:)
3306:(
3244:e
3219:K
3215:e
3190:,
3187:0
3179:K
3175:e
3166:,
3161:0
3140:=
3118:,
3113:0
3105:=
3101:)
3095:e
3092:n
3081:e
3066:n
3057:(
3047:=
3030:)
3018:(
3009:=
3003:e
3000:n
2989:e
2974:n
2965:=
2958:)
2952:(
2894:J
2890:n
2869:A
2843:A
2803:A
2777:A
2757:)
2754:C
2751:(
2725:)
2722:C
2719:(
2693:C
2690:=
2687:)
2684:A
2675:(
2666:=
2663:)
2660:A
2654:(
2645:=
2642:)
2639:A
2636:(
2604:n
2579:J
2575:n
2552:I
2548:n
2523:,
2520:0
2512:J
2508:n
2499:,
2494:0
2473:=
2451:,
2446:0
2438:=
2434:)
2429:n
2419:n
2404:I
2395:(
2385:=
2368:)
2356:(
2347:=
2342:n
2337:I
2333:n
2327:=
2322:n
2312:n
2297:I
2288:=
2281:)
2275:(
2213:h
2188:K
2184:h
2159:,
2156:]
2143:,
2140:0
2137:[
2131:i
2126:,
2122:)
2117:h
2107:h
2092:K
2083:(
2073:=
2066:i
2054:h
2049:K
2045:h
2039:=
2034:h
2024:h
2009:K
2000:=
1993:)
1990:i
1987:(
1957:i
1931:1
1925:e
1905:,
1897:2
1893:e
1886:1
1882:p
1877:=
1874:a
1855:)
1850:2
1846:e
1839:1
1836:(
1833:a
1830:=
1820:2
1816:h
1810:=
1807:p
1787:a
1767:1
1764:=
1761:e
1737:a
1717:p
1703:.
1687:G
1667:M
1643:M
1640:G
1637:=
1613:|
1604:e
1598:|
1594:=
1591:e
1567:]
1557:v
1550:)
1541:v
1526:r
1520:(
1507:r
1500:)
1492:|
1483:r
1477:|
1461:2
1455:|
1446:v
1440:|
1432:(
1427:[
1418:1
1413:=
1397:v
1377:v
1362:r
1344:r
1337:)
1329:|
1320:r
1314:|
1309:1
1292:2
1286:|
1277:v
1271:|
1262:(
1258:=
1242:i
1236:e
1233:=
1226:|
1217:r
1211:|
1200:r
1175:h
1160:v
1151:=
1138:e
1105:i
1078:e
1048:e
1013:h
984:k
955:K
944:(
927:.
918:h
903:K
897:=
888:n
859:K
830:h
801:n
765:k
742:,
733:k
727:h
724:=
715:k
708:|
699:h
693:|
689:=
680:v
665:r
659:=
650:h
621:h
569:v
563:,
554:r
543:[
524:,
518:,
512:,
509:i
506:,
503:e
500:,
497:a
164:"
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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