3244:
737:
2889:
3988:
8995:
13162:
6599:
7771:
3624:
8269:
11692:
3239:{\displaystyle {\begin{aligned}\mathbf {r} &=r\left(\cos \theta {\hat {\mathbf {x} }}+\sin \theta {\hat {\mathbf {y} }}\right)=r{\hat {\mathbf {r} }}\\{\dot {\mathbf {r} }}&={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\mathbf {q} }}\\{\ddot {\mathbf {r} }}&=\left({\ddot {r}}-r{\dot {\theta }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right){\hat {\mathbf {q} }}\end{aligned}}}
18251:
18301:
12773:
29:
7402:
18325:
6344:
7475:
18277:
8101:
18313:
11303:
1877:) that can be computed from position and velocity, three of which have already been discussed. These elements are convenient in that of the six, five are unchanging for an unperturbed orbit (a stark contrast to two constantly changing vectors). The future location of an object within its orbit can be predicted and its new position and velocity can be easily obtained from the orbital elements.
3479:
2858:
18289:
1855:
3983:{\displaystyle H=|\mathbf {r} \times {\dot {\mathbf {r} }}|=\left|{\begin{pmatrix}r\cos(\theta )\\r\sin(\theta )\\0\end{pmatrix}}\times {\begin{pmatrix}{\dot {r}}\cos(\theta )-r\sin(\theta ){\dot {\theta }}\\{\dot {r}}\sin(\theta )+r\cos(\theta ){\dot {\theta }}\\0\end{pmatrix}}\right|=\left|{\begin{pmatrix}0\\0\\r^{2}{\dot {\theta }}\end{pmatrix}}\right|=r^{2}{\dot {\theta }}}
2390:
4955:
1838:. The orbits of the artificial satellites around the Earth are, with a fair approximation, Kepler orbits with small perturbations due to the gravitational attraction of the Sun, the Moon and the oblateness of the Earth. In high accuracy applications for which the equation of motion must be integrated numerically with all gravitational and non-gravitational forces (such as
1064:(also proven by Isaac Newton) states that the magnitude of this force is the same as if all mass was concentrated in the middle of the sphere, even if the density of the sphere varies with depth (as it does for most celestial bodies). From this immediately follows that the attraction between two homogeneous spheres is as if both had its mass concentrated to its center.
6838:
7198:
7062:
13157:{\displaystyle \tan ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{1+\cos \theta }}={\frac {1-{\frac {e-\cosh E}{e\cdot \cosh E-1}}}{1+{\frac {e-\cosh E}{e\cdot \cosh E-1}}}}={\frac {e\cdot \cosh E-e+\cosh E}{e\cdot \cosh E+e-\cosh E}}={\frac {e+1}{e-1}}\cdot {\frac {\cosh E-1}{\cosh E+1}}={\frac {e+1}{e-1}}\cdot \tanh ^{2}{\frac {E}{2}}}
3261:
2683:
6594:{\displaystyle \mathbf {H} =\mathbf {r} \times {\dot {\mathbf {r} }}=r\mathbf {u} \times {\frac {d}{dt}}(r\mathbf {u} )=r\mathbf {u} \times (r{\dot {\mathbf {u} }}+{\dot {r}}\mathbf {u} )=r^{2}(\mathbf {u} \times {\dot {\mathbf {u} }})+r{\dot {r}}(\mathbf {u} \times \mathbf {u} )=r^{2}\mathbf {u} \times {\dot {\mathbf {u} }}}
7766:{\displaystyle r={\frac {\mathbf {r} \cdot ({\dot {\mathbf {r} }}\times \mathbf {H} )}{\alpha +c\cos(\theta )}}={\frac {(\mathbf {r} \times {\dot {\mathbf {r} }})\cdot \mathbf {H} }{\alpha +c\cos(\theta )}}={\frac {|\mathbf {H} |^{2}}{\alpha +c\cos(\theta )}}={\frac {|\mathbf {H} |^{2}/\alpha }{1+(c/\alpha )\cos(\theta )}}.}
2231:
4734:
8264:{\displaystyle \mathbf {e} \triangleq {\frac {\mathbf {c} }{\alpha }}={\frac {{\dot {\mathbf {r} }}\times \mathbf {H} }{\alpha }}-\mathbf {u} ={\frac {\mathbf {v} \times \mathbf {H} }{\alpha }}-{\frac {\mathbf {r} }{r}}={\frac {\mathbf {v} \times (\mathbf {r} \times \mathbf {v} )}{\alpha }}-{\frac {\mathbf {r} }{r}}}
5127:
6606:
11687:{\displaystyle \tan ^{2}{\frac {\theta }{2}}={\frac {1-\cos \theta }{1+\cos \theta }}={\frac {1-{\frac {\cos E-e}{1-e\cos E}}}{1+{\frac {\cos E-e}{1-e\cos E}}}}={\frac {1-e\cos E-\cos E+e}{1-e\cos E+\cos E-e}}={\frac {1+e}{1-e}}\cdot {\frac {1-\cos E}{1+\cos E}}={\frac {1+e}{1-e}}\cdot \tan ^{2}{\frac {E}{2}}}
4730:
5475:
16755:
16815:
is a small "perturbing force" due to for example a faint gravitational pull from other celestial bodies. The parameters of the osculating Kepler orbit will then only slowly change and the osculating Kepler orbit is a good approximation to the real orbit for a considerable time period before and after
1079:
Planets rotate at varying rates and thus may take a slightly oblate shape because of the centrifugal force. With such an oblate shape, the gravitational attraction will deviate somewhat from that of a homogeneous sphere. At larger distances the effect of this oblateness becomes negligible. Planetary
1075:
often have a shape strongly deviating from a sphere. But the gravitational forces produced by these irregularities are generally small compared to the gravity of the central body. The difference between an irregular shape and a perfect sphere also diminishes with distances, and most orbital distances
6909:
1829:
Under these assumptions the differential equation for the two body case can be completely solved mathematically and the resulting orbit which follows Kepler's laws of planetary motion is called a "Kepler orbit". The orbits of all planets are to high accuracy Kepler orbits around the Sun. The small
1872:
Any
Keplerian trajectory can be defined by six parameters. The motion of an object moving in three-dimensional space is characterized by a position vector and a velocity vector. Each vector has three components, so the total number of values needed to define a trajectory through space is six. An
1006:
From the laws of motion and the law of universal gravitation, Newton was able to derive Kepler's laws, which are specific to orbital motion in astronomy. Since Kepler's laws were well-supported by observation data, this consistency provided strong support of the validity of Newton's generalized
2159:
are simply angular measurements defining the orientation of the trajectory in the reference frame, they are not strictly necessary when discussing the motion of the object within the orbital plane. They have been mentioned here for completeness, but are not required for the proofs below.
12303:
12158:
7397:{\displaystyle \mathbf {r} \cdot ({\dot {\mathbf {r} }}\times \mathbf {H} )=\mathbf {r} \cdot (\alpha \mathbf {u} +\mathbf {c} )=\alpha \mathbf {r} \cdot \mathbf {u} +\mathbf {r} \cdot \mathbf {c} =\alpha r(\mathbf {u} \cdot \mathbf {u} )+rc\cos(\theta )=r(\alpha +c\cos(\theta ))}
122:
In most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of similar mass can be described as Kepler orbits around their common center of mass, their
1076:
are very large when compared with the diameter of a small orbiting body. Thus for some applications, shape irregularity can be neglected without significant impact on accuracy. This effect is quite noticeable for artificial Earth satellites, especially those in low orbits.
4427:
15934:
9593:
32:
An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation
1826:'s mass is less than the Sun's by a factor of 1047, which would constitute an error of 0.096% in the value of α. Notable exceptions include the Earth-Moon system (mass ratio of 81.3), the Pluto-Charon system (mass ratio of 8.9) and binary star systems.
4576:
14280:
9253:
17000:
3474:{\displaystyle \left({\ddot {r}}-r{\dot {\theta }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}\right){\hat {\mathbf {q} }}=\left(-{\frac {\alpha }{r^{2}}}\right){\hat {\mathbf {r} }}+(0){\hat {\mathbf {q} }}}
16627:
15770:
15669:
14152:
10766:
9439:
4964:
2853:{\displaystyle {\begin{aligned}{\hat {\mathbf {r} }}&=\cos {\theta }{\hat {\mathbf {x} }}+\sin {\theta }{\hat {\mathbf {y} }}\\{\hat {\mathbf {q} }}&=-\sin {\theta }{\hat {\mathbf {x} }}+\cos {\theta }{\hat {\mathbf {y} }}\end{aligned}}}
10053:
8334:
1304:
5796:
1405:
13316:
2385:{\displaystyle {\dot {\mathbf {H} }}={\frac {d}{dt}}\left(\mathbf {r} \times {\dot {\mathbf {r} }}\right)={\dot {\mathbf {r} }}\times {\dot {\mathbf {r} }}+\mathbf {r} \times {\ddot {\mathbf {r} }}=\mathbf {0} +\mathbf {0} =\mathbf {0} }
12010:
5314:
4950:{\displaystyle {\frac {d^{2}r}{d\theta ^{2}}}\cdot \left({\frac {H}{r^{2}}}\right)^{2}+{\frac {dr}{d\theta }}\cdot \left(-{\frac {2\cdot H\cdot {\dot {r}}}{r^{3}}}\right)-r\left({\frac {H}{r^{2}}}\right)^{2}=-{\frac {\alpha }{r^{2}}}}
16311:
7183:
14430:
14765:
7123:
5291:
6833:{\displaystyle {\ddot {\mathbf {r} }}\times \mathbf {H} =-{\frac {\alpha }{r^{2}}}\mathbf {u} \times (r^{2}\mathbf {u} \times {\dot {\mathbf {u} }})=-\alpha \mathbf {u} \times (\mathbf {u} \times {\dot {\mathbf {u} }})=-\alpha }
1616:
11922:
12752:
5682:
632:
13608:
6339:
4143:
4572:
3603:
13828:
11276:
12181:
7057:{\displaystyle \mathbf {u} \cdot {\dot {\mathbf {u} }}={\frac {1}{2}}(\mathbf {u} \cdot {\dot {\mathbf {u} }}+{\dot {\mathbf {u} }}\cdot \mathbf {u} )={\frac {1}{2}}{\frac {d}{dt}}(\mathbf {u} \cdot \mathbf {u} )=0}
10948:
8973:
8718:
4283:
11856:
10325:
2226:
6905:
6176:
16811:
16383:
12036:
308:
15836:
1483:
15550:
15355:
14610:
2654:
2597:
13672:
10542:
14926:
10859:
10622:
9829:
4306:
10146:
9909:
7861:
486:
14664:
15055:
6245:
16425:
15129:
7954:
2469:
1023:
relativity in the early 20th century. For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy and is used extensively in
16866:
16047:
15473:
14553:
9151:
4041:
2894:
2688:
1060:
The shapes of large celestial bodies are close to spheres. By symmetry, the net gravitational force attracting a mass point towards a homogeneous sphere must be directed towards its centre. The
16586:
15981:
15192:
14487:
15583:
directed from the centre of the homogeneous sphere to the pericentre) defining the orientation of the conical section (ellipse, parabola or hyperbola) can then be determined with the relation
8460:
13490:
12566:
11091:
5554:
2506:
869:
9492:
17074:
16096:
15423:
14356:
12645:
11786:
1822:
This assumption is not necessary to solve the simplified two body problem, but it simplifies calculations, particularly with Earth-orbiting satellites and planets orbiting the Sun. Even
1547:
Dividing by their respective masses and subtracting the second equation from the first yields the equation of motion for the acceleration of the first object with respect to the second:
10236:
6278:
13195:
8490:
6108:
9340:
9288:
5889:
2543:
7896:
1204:
13868:
8878:
8623:
5591:
1709:
1308:
16547:
16516:
16218:
15581:
14005:
13945:
13724:
5962:
13228:
11172:
10400:
14175:
12435:
1514:
16894:
9126:
9086:
6068:
15692:
15591:
14056:
10456:
7130:
5187:
5122:{\displaystyle {\frac {H^{2}}{r^{4}}}\cdot \left({\frac {d^{2}r}{d\theta ^{2}}}-2\cdot {\frac {\left({\frac {dr}{d\theta }}\right)^{2}}{r}}-r\right)=-{\frac {\alpha }{r^{2}}}}
2051:) defines the position of the orbiting body along the trajectory, measured from periapsis. Several alternate values can be used instead of true anomaly, the most common being
1193:
1164:
13519:
11742:
9743:
9687:
8793:
16174:
10652:
9350:
7069:
17038:
9939:
8566:
8407:
8378:
8356:
8276:
8088:
8066:
8044:
7466:
7444:
6208:
2882:
2676:
2414:
1429:
727:
705:
659:
16128:
7805:
5989:
5918:
5843:
14840:
14034:
12475:
12357:
9046:
1796:
4496:
16606:
16485:
15831:
15493:
15375:
15228:
14988:
14968:
10984:
9629:
9146:
8022:
7994:
7422:
6013:
4725:{\displaystyle {\frac {d^{2}r}{d\theta ^{2}}}\cdot {\dot {\theta }}^{2}+{\frac {dr}{d\theta }}\cdot {\ddot {\theta }}-r{\dot {\theta }}^{2}=-{\frac {\alpha }{r^{2}}}}
4191:
3521:
2157:
2137:
2018:
1992:
1652:
407:
16819:
This concept can also be useful for a rocket during powered flight as it then tells which Kepler orbit the rocket would continue in case the thrust is switched off.
14807:
8821:
5708:
5470:{\displaystyle {\frac {d^{2}r}{d\theta ^{2}}}={\frac {2}{s^{3}}}\cdot \left({\frac {ds}{d\theta }}\right)^{2}-{\frac {1}{s^{2}}}\cdot {\frac {d^{2}s}{d\theta ^{2}}}}
562:
15286:
15259:
13524:
13238:
12389:
9482:
1135:
1108:
977:
948:
11932:
2049:
16750:{\displaystyle \mathbf {F} (\mathbf {r} ,{\dot {\mathbf {r} }},t)=-\alpha {\frac {\hat {\mathbf {r} }}{r^{2}}}+\mathbf {f} (\mathbf {r} ,{\dot {\mathbf {r} }},t)}
10188:
8760:
8524:
4500:
16465:
16445:
6033:
4473:
4211:
3501:
2117:
2093:
2069:
1966:
1931:
1905:
1816:
1538:
999:
916:
894:
681:
508:
381:
355:
333:
16243:
13753:
14366:
413:, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the
14687:
5213:
733:
Strictly speaking, this form of the equation only applies to an object of constant mass, which holds true based on the simplifying assumptions made below.
6849:
6115:
1555:
16760:
16332:
11861:
12671:
224:
5601:
1436:
514:
of the curve. This form of the equation is particularly useful when dealing with parabolic trajectories, for which the semi-major axis is infinite.
6283:
17217:(1952), "Book I, Chapter 4, The Movement of the Celestial Bodies Is Regular, Circular, and Everlasting-Or Else Compounded of Circular Movements",
4073:
529:
developed several concepts related to motion, gravitation and differential calculus. However, these concepts were not published until 1687 in the
12298:{\displaystyle E=2\cdot \arg \left({\sqrt {1+e}}\cdot \cos {\frac {\theta }{2}},{\sqrt {1-e}}\cdot \sin {\frac {\theta }{2}}\right)+n\cdot 2\pi }
3531:
2423:
Because the equation has symmetry around its origin, it is easier to solve in polar coordinates. However, it is important to note that equation (
11198:
530:
426:
10885:
8901:
8646:
4221:
11795:
10262:
16388:
12153:{\displaystyle \theta =2\cdot \arg \left({\sqrt {1-e}}\cdot \cos {\frac {E}{2}},{\sqrt {1+e}}\cdot \sin {\frac {E}{2}}\right)+n\cdot 2\pi }
2184:
17449:
17260:
1994:) defines the angle between the reference direction and the upward crossing of the orbit on the reference plane (the ascending node).
13347:
12495:
11020:
812:
15501:
15306:
14561:
2605:
2548:
538:
4422:{\displaystyle {\ddot {r}}={\frac {d^{2}r}{d\theta ^{2}}}\cdot {\dot {\theta }}^{2}+{\frac {dr}{d\theta }}\cdot {\ddot {\theta }}}
13630:
10482:
18128:
17221:, Great Books of the Western World, vol. 16, translated by Charles Glenn Wallis, Chicago: William Benton, pp. 497–838
14870:
10796:
10565:
9769:
2394:
Since the cross product of the position vector and its velocity stays constant, they must lie in the same plane, orthogonal to
10083:
9852:
9258:
7810:
17232:
14620:
14998:
6213:
1659:
135:
From ancient times until the 16th and 17th centuries, the motions of the planets were believed to follow perfectly circular
18188:
17288:
17095:
13966:
13699:
184:
15078:
7906:
2432:
16829:
16002:
15929:{\displaystyle V_{t}=V_{0}={\sqrt {\frac {\alpha }{p}}}={\sqrt {\frac {\alpha }{\frac {{(r\cdot V_{t})}^{2}}{\alpha }}}}}
15428:
14508:
14298:
The equation of the center relates mean anomaly to true anomaly for elliptical orbits, for small numerical eccentricity.
3997:
124:
16552:
15944:
15155:
14555:
the Kepler orbit corresponding to the solution of this initial value problem can be found with the following algorithm:
14453:
2545:
acceleration. Therefore, one must be cautious when transforming the equation. Introducing a cartesian coordinate system
9088:
the pericentre. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value
2179:
9588:{\displaystyle V_{t}=r\cdot {\dot {\theta }}={\frac {H}{r}}={\sqrt {\frac {\alpha }{p}}}\cdot (1+e\cdot \cos \theta )}
8428:
17308:
17204:
14302:
517:
Despite developing these laws from observations, Kepler was never able to develop a theory to explain these motions.
147:. Variations in the motions of the planets were explained by smaller circular paths overlaid on the larger path (see
16098:
the Kepler orbit corresponding to this state can be computed with the algorithm defined above. First the parameters
5508:
2474:
807:
to the product of the two masses and inversely proportional to the square of the distance between the point masses:
18183:
18063:
17482:
17359:
1765:
16611:
The Kepler orbit computed in this way having the same "state vector" as the solution to the "equation of motion" (
18148:
17902:
17043:
16065:
15392:
14325:
12594:
11747:
1971:
761:
which is proportional to the product of the two masses and inversely proportional to the square of the distance (
10205:
6250:
151:). As measurements of the planets became increasingly accurate, revisions to the theory were proposed. In 1543,
18221:
17860:
17851:
17588:
17318:
17187:
El'Yasberg "Theory of flight of artificial earth satellites", Israel program for
Scientific Translations (1967)
13167:
8465:
6077:
4452:
9311:
5852:
2511:
17253:
8991:
The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue)
7866:
804:
14275:{\displaystyle V_{y}=V_{r}\sin \theta +V_{t}\cos \theta ={\sqrt {\frac {\alpha }{p}}}\cdot (e+\cos \theta )}
13838:
9248:{\displaystyle -\cos ^{-1}\left(-{\frac {1}{e}}\right)<\theta <\cos ^{-1}\left(-{\frac {1}{e}}\right)}
8835:
8580:
5559:
18168:
17638:
16995:{\displaystyle \mathbf {e} ={\frac {(V_{t}-V_{0}){\hat {\mathbf {r} }}-V_{r}{\hat {\mathbf {t} }}}{V_{0}}}}
16521:
16490:
16179:
15555:
13918:
5923:
1080:
motions in the Solar System can be computed with sufficient precision if they are treated as point masses.
13200:
11124:
10355:
18113:
15765:{\displaystyle {\hat {\mathbf {y} }}=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\mathbf {t} }}}
15664:{\displaystyle {\hat {\mathbf {x} }}=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\mathbf {t} }}}
14147:{\displaystyle V_{x}=V_{r}\cos \theta -V_{t}\sin \theta =-{\sqrt {\frac {\alpha }{p}}}\cdot \sin \theta }
12398:
1490:
1196:
9091:
9051:
1830:
deviations are due to the much weaker gravitational attractions between the planets, and in the case of
163:, although he still believed that the planets traveled in perfectly circular paths centered on the Sun.
18267:
18093:
17920:
6038:
534:
112:
17158:
10761:{\displaystyle H=x\cdot {\dot {y}}-y\cdot {\dot {x}}=a\cdot b\cdot (e\cdot \cosh E-1)\cdot {\dot {E}}}
10423:
9434:{\displaystyle V_{r}={\dot {r}}={\frac {H}{p}}e\sin \theta ={\sqrt {\frac {\alpha }{p}}}e\sin \theta }
5159:
1169:
1140:
1007:
theory, and unified celestial and ordinary mechanics. These laws of motion formed the basis of modern
18231:
13495:
11703:
10048:{\displaystyle H=x\cdot {\dot {y}}-y\cdot {\dot {x}}=a\cdot b\cdot (1-e\cdot \cos E)\cdot {\dot {E}}}
9710:
9654:
8769:
8329:{\displaystyle \mathbf {H} =\mathbf {r} \times {\dot {\mathbf {r} }}=\mathbf {r} \times \mathbf {v} }
6843:
3249:
1938:
72:
1846:) being taken into account, the Kepler orbit concepts are of paramount importance and heavily used.
1056:
There are no external or internal forces acting upon the bodies other than their mutual gravitation.
18350:
18216:
17741:
17335:
17246:
16133:
1839:
1719:
When compared to the central body, the mass of the orbiting body is insignificant. Mathematically,
88:
80:
17021:
8536:
8390:
8361:
8339:
8071:
8049:
8027:
7449:
7427:
6191:
6185:
Another way to solve this equation without the use of polar differential equations is as follows:
2865:
2659:
2397:
1412:
1040:
710:
688:
642:
18226:
17534:
17235:
in an elliptic Kepler orbit around the Earth with any value for semi-major axis and eccentricity.
16101:
7778:
5967:
5896:
5821:
2600:
14812:
14010:
1299:{\displaystyle m_{1}{\ddot {\mathbf {r} }}_{1}={\frac {-Gm_{1}m_{2}}{r^{2}}}\mathbf {\hat {r}} }
219:. Mathematically, the distance between a central body and an orbiting body can be expressed as:
18088:
17690:
17610:
17598:
17429:
12442:
12324:
9025:
9022:
The point on the horizontal line going out to the right from the focal point is the point with
5791:{\displaystyle s={\frac {\alpha }{H^{2}}}\cdot \left(1+e\cdot \cos(\theta -\theta _{0})\right)}
1771:
1400:{\displaystyle m_{2}{\ddot {\mathbf {r} }}_{2}={\frac {Gm_{1}m_{2}}{r^{2}}}\mathbf {\hat {r}} }
919:
784:
553:
acting on the body, is in the direction of the net force, and is inversely proportional to the
16588:
will all vary with time as opposed to the case of a Kepler orbit for which only the parameter
14307:
13311:{\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {e+1}{e-1}}}\cdot \tanh {\frac {E}{2}}}
4478:
511:
18211:
18153:
18123:
17911:
17788:
17756:
17726:
17685:
17670:
17549:
17402:
17313:
17105:
16591:
16470:
15803:
15478:
15360:
15213:
14973:
14947:
14313:
12005:{\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {1+e}{1-e}}}\cdot \tan {\frac {E}{2}}}
10969:
9614:
9131:
8413:
of the orbit, and it has the magnitude of orbital eccentricity. This makes it very useful in
8007:
7979:
7407:
5998:
4176:
3506:
2142:
2122:
2003:
1997:
1977:
1637:
392:
148:
14786:
8800:
18236:
18058:
17842:
17731:
17700:
17628:
17603:
17578:
17539:
17520:
17475:
17443:
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17110:
15264:
15237:
12362:
10344:
9460:
1910:
1113:
1086:
955:
926:
384:
2034:
1541:
8:
18329:
18098:
17893:
17633:
17434:
17214:
16823:
16306:{\displaystyle {\ddot {\mathbf {r} }}=\mathbf {F} (\mathbf {r} ,{\dot {\mathbf {r} }},t)}
10167:
8739:
8503:
8414:
7178:{\displaystyle {\dot {\mathbf {r} }}\times \mathbf {H} =\alpha \mathbf {u} +\mathbf {c} }
1008:
152:
104:
44:
14425:{\displaystyle {\dot {\mathbf {v} }}=-\alpha \cdot {\frac {\hat {\mathbf {r} }}{r^{2}}}}
18317:
18305:
17771:
17660:
17558:
16450:
16430:
7807:
are effectively the polar coordinates of the vector function. Making the substitutions
6018:
4458:
4196:
3621:
In order to solve this equation, all time derivatives must be eliminated. This brings:
3486:
2102:
2078:
2054:
1951:
1916:
1890:
1867:
1835:
1801:
1523:
1020:
1016:
984:
901:
879:
666:
493:
366:
340:
318:
108:
14760:{\displaystyle \mathbf {v} _{0}=V_{r}{\hat {\mathbf {r} }}+V_{t}{\hat {\mathbf {t} }}}
7118:{\displaystyle {\ddot {\mathbf {r} }}\times \mathbf {H} =\alpha {\dot {\mathbf {u} }}}
5286:{\displaystyle {\frac {dr}{d\theta }}=-{\frac {1}{s^{2}}}\cdot {\frac {ds}{d\theta }}}
1798:, are widely available for Sun, major planets and Moon, which have much larger masses
179:. Kepler would spend the next five years trying to fit the observations of the planet
79:. It considers only the point-like gravitational attraction of two bodies, neglecting
18138:
18036:
17966:
17721:
17675:
17593:
17375:
17283:
17200:
17193:
17115:
11697:
9639:
207:
More generally, the path of an object undergoing
Keplerian motion may also follow a
18281:
18118:
18050:
17814:
17776:
17650:
17620:
17573:
17409:
17367:
17085:
16057:
12482:
8418:
1859:
1831:
1611:{\displaystyle {\ddot {\mathbf {r} }}=-{\frac {\alpha }{r^{2}}}\mathbf {\hat {r}} }
1046:
736:
200:
116:
96:
95:
central body, and so on. It is thus said to be a solution of a special case of the
11917:{\displaystyle \left(\cos {\tfrac {\theta }{2}},\sin {\tfrac {\theta }{2}}\right)}
18345:
18158:
17751:
17655:
17645:
17544:
17468:
17383:
17351:
17298:
17269:
13910:) that the sum of the kinetic and the potential energy for a hyperbolic orbit is
13334:
This relation is convenient for passing between "true anomaly" and the parameter
12747:{\displaystyle \cos \theta ={\frac {x}{r}}={\frac {e-\cosh E}{e\cdot \cosh E-1}}}
1012:
358:
172:
56:
15999:
which is the case that it is a circular orbit that is fitting the initial state
5677:{\displaystyle H^{2}\cdot \left({\frac {d^{2}s}{d\theta ^{2}}}+s\right)=\alpha }
1431:
is the relative position vector of mass 1 with respect to mass 2, expressed as:
18254:
18206:
18198:
18193:
18078:
18073:
18004:
17984:
17975:
17568:
17554:
17530:
17525:
17500:
17422:
17343:
17120:
17100:
17090:
2163:
1884:
1843:
627:{\displaystyle \mathbf {F} =m\mathbf {a} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}}
216:
100:
84:
13603:{\displaystyle \tanh ^{-1}x={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)}
707:
is the acceleration vector, the second time derivative of the position vector
18339:
18108:
18103:
18022:
17665:
17583:
17415:
7973:
6334:{\displaystyle {\ddot {\mathbf {r} }}=-{\tfrac {\alpha }{r^{2}}}\mathbf {u} }
1061:
1028:
136:
76:
4138:{\displaystyle {\ddot {\theta }}=-{\frac {2\cdot H\cdot {\dot {r}}}{r^{3}}}}
18293:
18173:
18083:
17957:
17940:
17798:
17695:
14308:
Determination of the Kepler orbit that corresponds to a given initial state
5964:
defining the coordinate system in the orbital plane are selected such that
4567:{\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-{\frac {\alpha }{r^{2}}}}
3598:{\displaystyle {\ddot {r}}-r{\dot {\theta }}^{2}=-{\frac {\alpha }{r^{2}}}}
2072:
2028:
546:
526:
410:
160:
156:
8994:
18178:
18013:
17783:
17763:
17680:
13823:{\displaystyle {\frac {{V_{r}}^{2}+{V_{t}}^{2}}{2}}-{\frac {\alpha }{r}}}
11271:{\displaystyle \cos \theta ={\frac {x}{r}}={\frac {\cos E-e}{1-e\cos E}}}
2417:
1945:
1517:
176:
1053:
The bodies are spherically symmetric and can be treated as point masses.
896:
is the magnitude of the gravitational force between the two point masses
183:
to various curves. In 1609, Kepler published the first two of his three
16:
Celestial orbit whose trajectory is a conic section in the orbital plane
17746:
17460:
14322:) which is a first order equation for the 6-dimensional "state vector"
10943:{\displaystyle t=a\cdot {\sqrt {\frac {a}{\alpha }}}(e\cdot \sinh E-E)}
9457:
and that the tangential component (velocity component perpendicular to
8968:{\displaystyle b={\frac {p}{\sqrt {e^{2}-1}}}=a\cdot {\sqrt {e^{2}-1}}}
8713:{\displaystyle b={\frac {p}{\sqrt {1-e^{2}}}}=a\cdot {\sqrt {1-e^{2}}}}
4278:{\displaystyle {\dot {r}}={\frac {dr}{d\theta }}\cdot {\dot {\theta }}}
1072:
796:
175:
acquired the extensive, meticulous observations of the planets made by
13690:
As the potential energy corresponding to the force field of relation (
11851:{\displaystyle \left(\cos {\tfrac {E}{2}},\sin {\tfrac {E}{2}}\right)}
10320:{\displaystyle t=a\cdot {\sqrt {\frac {a}{\alpha }}}(E-e\cdot \sin E)}
1968:) defines the angle between the orbital plane and the reference plane.
740:
The mechanisms of Newton's law of universal gravitation; a point mass
18163:
17515:
8824:
2221:{\displaystyle \mathbf {H} =\mathbf {r} \times {\dot {\mathbf {r} }}}
1024:
765:) between them. Regardless of masses or distance, the magnitudes of |
414:
212:
140:
92:
68:
8763:
6900:{\displaystyle \mathbf {u} \cdot \mathbf {u} =|\mathbf {u} |^{2}=1}
6171:{\displaystyle r={\frac {1}{s}}={\frac {p}{1+e\cdot \cos \theta }}}
3483:
This gives the ordinary differential equation in the two variables
1068:
215:, which, along with ellipses, belong to a group of curves known as
208:
64:
17238:
16806:{\displaystyle \mathbf {f} (\mathbf {r} ,{\dot {\mathbf {r} }},t)}
16378:{\displaystyle \mathbf {F} (\mathbf {r} ,{\dot {\mathbf {r} }},t)}
1713:
In many applications, a third simplifying assumption can be made:
18068:
12486:
8569:
5845:
are constants of integration depending on the initial values for
2020:) defines the angle between the ascending node and the periapsis.
1823:
303:{\displaystyle r(\theta )={\frac {a(1-e^{2})}{1+e\cos(\theta )}}}
196:
144:
60:
28:
12485:
DATAN2(y,x)) available in for example the programming language
10966:
To find what time t that corresponds to a certain true anomaly
5920:
explicitly one introduces the convention that the unit vectors
2174:
For movement under any central force, i.e. a force parallel to
803:
pointing along the line intersecting both points. The force is
1478:{\displaystyle \mathbf {r} =\mathbf {r} _{1}-\mathbf {r} _{2}}
17873:
17492:
15545:{\displaystyle ({\hat {\mathbf {x} }},{\hat {\mathbf {y} }})}
15350:{\displaystyle ({\hat {\mathbf {r} }},{\hat {\mathbf {t} }})}
14605:{\displaystyle ({\hat {\mathbf {r} }},{\hat {\mathbf {t} }})}
12478:
2649:{\displaystyle ({\hat {\mathbf {r} }},{\hat {\mathbf {q} }})}
2592:{\displaystyle ({\hat {\mathbf {x} }},{\hat {\mathbf {y} }})}
800:
550:
192:
22:
9048:
for which the distance to the focus takes the minimal value
7066:
Substituting these values into the previous equation gives:
4213:, the chain rule is used to find appropriate substitutions:
17162:
9002:
and their eccentricities. Blue is a hyperbolic trajectory (
8358:
is the velocity vector associated with the position vector
554:
549:
of a body is parallel and directly proportional to the net
180:
75:
in three-dimensional space. A Kepler orbit can also form a
18288:
16176:
and then the orthogonal unit vectors in the orbital plane
13667:{\displaystyle P=2\pi a\cdot {\sqrt {\frac {a}{\alpha }}}}
10537:{\displaystyle {\dot {x}}=-a\cdot \sinh E\cdot {\dot {E}}}
9635:
is slightly different for elliptic and hyperbolic orbits.
8336:
is the constant angular momentum vector of the orbit, and
5151:) can be solved analytically by the variable substitution
14921:{\displaystyle p={\frac {{(r\cdot V_{t})}^{2}}{\alpha }}}
10854:{\displaystyle H\cdot t=a\cdot b\cdot (e\cdot \sinh E-E)}
10617:{\displaystyle {\dot {y}}=b\cdot \cosh E\cdot {\dot {E}}}
9824:{\displaystyle {\dot {x}}=-a\cdot \sin E\cdot {\dot {E}}}
4445:
Using these four substitutions, all time derivatives in (
1854:
17040:
is a smooth differentiable function of the state vector
10190:
is selected such that the integration constant is zero.
10141:{\displaystyle H\cdot t=a\cdot b\cdot (E-e\cdot \sin E)}
9904:{\displaystyle {\dot {y}}=b\cdot \cos E\cdot {\dot {E}}}
8387:, having the same direction as the integration constant
7856:{\displaystyle p={\tfrac {|\mathbf {H} |^{2}}{\alpha }}}
481:{\displaystyle r(\theta )={\frac {p}{1+e\cos(\theta )}}}
14659:{\displaystyle \mathbf {r} _{0}=r{\hat {\mathbf {r} }}}
13750:) that the sum of the kinetic and the potential energy
13338:, the latter being connected to time through relation (
59:) is the motion of one body relative to another, as an
15050:{\displaystyle e\cos \theta ={\frac {V_{t}}{V_{0}}}-1}
13500:
11898:
11877:
11832:
11811:
9096:
9056:
8774:
7877:
7821:
6308:
6240:{\displaystyle \mathbf {u} ={\frac {\mathbf {r} }{r}}}
6082:
6049:
5857:
3904:
3745:
3679:
83:
due to gravitational interactions with other objects,
18265:
17046:
17024:
16897:
16832:
16763:
16630:
16594:
16555:
16524:
16493:
16473:
16453:
16433:
16420:{\displaystyle -\alpha {\frac {\mathbf {r} }{r^{2}}}}
16391:
16335:
16246:
16182:
16136:
16104:
16068:
16005:
15947:
15839:
15806:
15695:
15594:
15558:
15504:
15481:
15431:
15395:
15363:
15309:
15267:
15240:
15216:
15158:
15081:
15001:
14976:
14950:
14873:
14815:
14789:
14690:
14623:
14564:
14511:
14456:
14369:
14328:
14178:
14059:
14013:
13969:
13921:
13841:
13756:
13702:
13633:
13527:
13498:
13350:
13241:
13203:
13170:
12776:
12674:
12597:
12498:
12445:
12401:
12365:
12327:
12184:
12039:
11935:
11864:
11798:
11750:
11706:
11306:
11201:
11127:
11023:
10972:
10888:
10799:
10655:
10568:
10485:
10426:
10358:
10265:
10208:
10170:
10086:
9942:
9855:
9772:
9713:
9657:
9617:
9495:
9463:
9353:
9314:
9261:
9154:
9134:
9094:
9054:
9028:
8904:
8838:
8803:
8772:
8742:
8649:
8583:
8539:
8506:
8468:
8431:
8393:
8364:
8342:
8279:
8104:
8074:
8052:
8030:
8010:
7982:
7909:
7869:
7813:
7781:
7478:
7452:
7430:
7410:
7201:
7133:
7072:
6912:
6852:
6609:
6347:
6286:
6253:
6216:
6194:
6118:
6080:
6041:
6021:
6001:
5970:
5926:
5899:
5855:
5824:
5711:
5604:
5562:
5511:
5317:
5216:
5162:
4967:
4737:
4579:
4503:
4481:
4461:
4309:
4224:
4199:
4179:
4076:
4000:
3627:
3534:
3509:
3489:
3264:
2892:
2868:
2686:
2662:
2608:
2551:
2514:
2477:
2435:
2400:
2234:
2187:
2145:
2125:
2105:
2081:
2057:
2037:
2006:
1980:
1954:
1919:
1893:
1873:
orbit is generally defined by six elements (known as
1804:
1774:
1662:
1640:
1558:
1526:
1493:
1439:
1415:
1311:
1207:
1172:
1143:
1116:
1089:
987:
958:
929:
904:
882:
815:
713:
691:
669:
645:
565:
496:
429:
395:
369:
343:
321:
227:
17076:
also if this state corresponds to a circular orbit.
15124:{\displaystyle e\sin \theta ={\frac {V_{r}}{V_{0}}}}
8495:
7949:{\displaystyle r={\frac {p}{1+e\cdot \cos \theta }}}
2464:{\displaystyle \left({\ddot {\mathbf {r} }}\right),}
2164:
Mathematical solution of the differential equation (
683:
is the mass of the body on which the force is acting
107:, it also does not take into account the effects of
17191:Bate, Roger; Mueller, Donald; White, Jerry (1971).
16861:{\displaystyle \mathbf {e} =e{\hat {\mathbf {x} }}}
16042:{\displaystyle (\mathbf {r} _{0},\mathbf {v} _{0})}
15468:{\displaystyle (\mathbf {r} _{0},\mathbf {v} _{0})}
14548:{\displaystyle (\mathbf {r} _{0},\mathbf {v} _{0})}
14303:
Equation of the center § Analytical expansions
4036:{\displaystyle {\dot {\theta }}={\frac {H}{r^{2}}}}
17192:
17068:
17032:
16994:
16860:
16805:
16749:
16600:
16581:{\displaystyle \mathbf {r} ,{\dot {\mathbf {r} }}}
16580:
16541:
16510:
16479:
16459:
16439:
16419:
16377:
16305:
16212:
16168:
16122:
16090:
16041:
15976:{\displaystyle V_{t}={\sqrt {\frac {\alpha }{r}}}}
15975:
15928:
15825:
15764:
15663:
15575:
15544:
15487:
15467:
15417:
15369:
15349:
15280:
15253:
15222:
15187:{\displaystyle V_{0}={\sqrt {\frac {\alpha }{p}}}}
15186:
15123:
15049:
14982:
14962:
14920:
14834:
14801:
14759:
14658:
14604:
14547:
14482:{\displaystyle {\dot {\mathbf {r} }}=\mathbf {v} }
14481:
14424:
14350:
14274:
14146:
14028:
13999:
13939:
13862:
13822:
13718:
13666:
13602:
13513:
13484:
13344:). Note that this is a mapping between the ranges
13310:
13222:
13189:
13156:
12746:
12639:
12560:
12469:
12429:
12383:
12351:
12297:
12152:
12004:
11916:
11850:
11780:
11736:
11686:
11270:
11166:
11085:
10978:
10942:
10853:
10760:
10616:
10536:
10450:
10394:
10319:
10230:
10182:
10140:
10047:
9903:
9823:
9737:
9681:
9623:
9587:
9476:
9433:
9334:
9282:
9247:
9140:
9120:
9080:
9040:
8967:
8872:
8815:
8787:
8754:
8712:
8617:
8560:
8518:
8484:
8454:
8401:
8372:
8350:
8328:
8263:
8082:
8060:
8038:
8016:
7988:
7948:
7890:
7855:
7799:
7765:
7460:
7438:
7416:
7396:
7177:
7117:
7056:
6899:
6832:
6593:
6333:
6272:
6239:
6202:
6170:
6102:
6062:
6027:
6007:
5983:
5956:
5912:
5883:
5837:
5790:
5676:
5585:
5548:
5469:
5285:
5181:
5121:
4949:
4724:
4566:
4490:
4467:
4421:
4277:
4205:
4185:
4137:
4035:
3982:
3597:
3515:
3495:
3473:
3238:
2876:
2852:
2670:
2648:
2591:
2537:
2500:
2463:
2408:
2384:
2220:
2151:
2131:
2111:
2087:
2063:
2043:
2012:
1986:
1960:
1925:
1899:
1810:
1790:
1703:
1646:
1610:
1532:
1508:
1477:
1423:
1399:
1298:
1187:
1158:
1129:
1102:
993:
971:
942:
910:
888:
863:
721:
699:
675:
653:
626:
502:
480:
401:
375:
349:
327:
302:
139:paths as taught by the ancient Greek philosophers
11924:are in the same quadrant. One therefore has that
1880:Two define the size and shape of the trajectory:
1602:
1500:
1391:
1290:
1041:Orbit § Newtonian analysis of orbital motion
541:. His second of his three laws of motion states:
18337:
17190:
15498:The standard inertially fixed coordinate system
8455:{\displaystyle \mathbf {r} ,\mathbf {\dot {r}} }
7972:This is the equation in polar coordinates for a
4193:. In order to eliminate the time derivatives of
4173:) allow us to eliminate the time derivatives of
12492:Note that this is a mapping between the ranges
11792:-axis. From this then follows that the vectors
11696:From the geometrical construction defining the
9342:one gets that the radial velocity component is
5205:Using the chain rule for differentiation gets:
1654:is the gravitational parameter and is equal to
421:Alternately, the equation can be expressed as:
17233:JAVA applet animating the orbit of a satellite
16051:
15210:one gets a Kepler orbit that for true anomaly
14505:For any values for the initial "state vector"
13485:{\displaystyle \left\longleftrightarrow \left}
12561:{\displaystyle \left\longleftrightarrow \left}
11086:{\displaystyle \left\longleftrightarrow \left}
5549:{\displaystyle {\frac {d^{2}r}{d\theta ^{2}}}}
2501:{\displaystyle \left({\ddot {\theta }}\right)}
1034:
864:{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}}
17476:
17254:
16822:For a "close to circular" orbit the concept "
15425:of the Kepler orbit takes the desired values
5893:Instead of using the constant of integration
15303:If this Kepler orbit then also has the same
1001:is the distance between the two point masses
17069:{\displaystyle (\mathbf {r} ,\mathbf {v} )}
16624:This concept is for example useful in case
16091:{\displaystyle (\mathbf {r} ,\mathbf {v} )}
15418:{\displaystyle (\mathbf {r} ,\mathbf {v} )}
14351:{\displaystyle (\mathbf {r} ,\mathbf {v} )}
12640:{\displaystyle r=a\cdot (e\cdot \cosh E-1)}
11781:{\displaystyle (\cos \theta ,\sin \theta )}
11095:
9638:For an elliptic orbit one switches to the "
7976:with origin in a focal point. The argument
1049:, two simplifying assumptions can be made:
18250:
17483:
17469:
17261:
17247:
17219:On the Revolutions of the Heavenly Spheres
17213:
10231:{\displaystyle H={\sqrt {\alpha \cdot p}}}
9611:The connection between the polar argument
9292:Using the chain rule for differentiation (
9006:> 1). Green is a parabolic trajectory (
6273:{\displaystyle \mathbf {r} =r\mathbf {u} }
1045:To solve for the motion of an object in a
166:
16621:is said to be "osculating" at this time.
13190:{\displaystyle \tan {\frac {\theta }{2}}}
9010:= 1). Red is an elliptical orbit (0 <
8485:{\displaystyle \mathbf {r} ,\mathbf {v} }
8090:must be pointing in the direction of the
8024:is the angle between the position vector
6103:{\displaystyle {\tfrac {H^{2}}{\alpha }}}
1784:
17490:
13963:Relative the inertial coordinate system
9335:{\displaystyle {\frac {H^{2}}{\alpha }}}
9283:{\displaystyle -\pi <\theta <\pi }
8993:
7191:is a constant vector. Dotting this with
5884:{\displaystyle {\tfrac {ds}{d\theta }}.}
2538:{\displaystyle \left({\ddot {r}}\right)}
2416:. This implies the vector function is a
1853:
735:
27:
21:For broader coverage of this topic, see
7891:{\displaystyle e={\tfrac {c}{\alpha }}}
6180:
2862:We can now rewrite the vector function
18338:
18129:Transposition, docking, and extraction
17199:. Dover Publications, Inc., New York.
13863:{\displaystyle -{\frac {\alpha }{2a}}}
12359:" is the polar argument of the vector
11017:) define a mapping between the ranges
8998:A diagram of the various forms of the
8873:{\displaystyle a={\frac {p}{e^{2}-1}}}
8618:{\displaystyle a={\frac {p}{1-e^{2}}}}
8409:, also points to the direction of the
7999:
5586:{\displaystyle {\frac {dr}{d\theta }}}
1704:{\displaystyle \alpha =G(m_{1}+m_{2})}
387:, which defines the shape of the orbit
17464:
17242:
16542:{\displaystyle {\hat {\mathbf {y} }}}
16511:{\displaystyle {\hat {\mathbf {x} }}}
16213:{\displaystyle {\hat {x}},{\hat {y}}}
15576:{\displaystyle {\hat {\mathbf {x} }}}
14000:{\displaystyle {\hat {x}},{\hat {y}}}
13940:{\displaystyle {\frac {\alpha }{2a}}}
13719:{\displaystyle -{\frac {\alpha }{r}}}
10996:) for an elliptic and with relation (
10986:one computes corresponding parameter
8094:of the orbit. We can then define the
5957:{\displaystyle {\hat {x}},{\hat {y}}}
1849:
799:attracts every other point mass by a
361:, which defines the size of the orbit
16888:
16237:
15938:
15686:
15585:
15149:
15072:
14992:
14864:
14681:
14614:
14447:
14360:
14169:
14050:
13912:
13832:
13624:
13232:
13223:{\displaystyle \tanh {\frac {E}{2}}}
12665:
12588:
12175:
12030:
11926:
11192:
11167:{\displaystyle r=a\cdot (1-e\cos E)}
11118:
10879:
10790:
10646:
10559:
10476:
10417:
10395:{\displaystyle x=a\cdot (e-\cosh E)}
10349:
10343:For a hyperbolic orbit one uses the
10256:
10199:
10077:
9933:
9846:
9763:
9704:
9648:
9486:
9344:
8895:
8829:
8640:
8574:
7900:
5702:
5595:
5308:
5207:
5153:
4958:
4300:
4215:
4067:
3991:
3525:
1937:Three define the orientation of the
1549:
979:is the mass of the second point mass
791:Newton's law of gravitation states:
55:, named after the German astronomer
17268:
14558:Define the orthogonal unit vectors
14048:) that the velocity components are
13521:can be computed using the relation
13230:have the same sign it follows that
12430:{\displaystyle |E-\theta |<\pi }
9014:< 1). Grey is a circular orbit (
6070:is minimal. Defining the parameter
1509:{\displaystyle \mathbf {\hat {r}} }
1083:Two point mass objects with masses
950:is the mass of the first point mass
13:
14036:towards pericentre one gets from (
13618:) follows that the orbital period
13474:
13462:
12550:
12538:
12519:
12507:
11075:
11063:
11044:
11032:
9121:{\displaystyle {\tfrac {p}{1-e}}.}
9081:{\displaystyle {\tfrac {p}{1+e}},}
7898:, we again arrive at the equation
5995:is positive. This then means that
2180:specific relative angular momentum
2126:
1981:
14:
18362:
18189:Kepler's laws of planetary motion
17289:Kepler's laws of planetary motion
17226:
17096:Kepler's laws of planetary motion
16235:If now the equation of motion is
14316:" for the differential equation (
12439:For the numerical computation of
10990:connected to time with relation (
10784:Integrating with respect to time
10071:Integrating with respect to time
8496:Properties of trajectory equation
6063:{\displaystyle r={\tfrac {1}{s}}}
4451:) can be eliminated, yielding an
1766:standard gravitational parameters
1199:experience gravitational forces:
18323:
18311:
18299:
18287:
18275:
18249:
18184:Interplanetary Transport Network
18064:Collision avoidance (spacecraft)
17360:Epitome Astronomiae Copernicanae
17148:Bate, Mueller, White. pp 177–181
17059:
17051:
17026:
16969:
16942:
16899:
16848:
16834:
16784:
16773:
16765:
16728:
16717:
16709:
16684:
16651:
16640:
16632:
16568:
16557:
16529:
16498:
16401:
16356:
16345:
16337:
16284:
16273:
16265:
16251:
16081:
16073:
16026:
16011:
15752:
15726:
15700:
15651:
15625:
15599:
15563:
15529:
15512:
15452:
15437:
15408:
15400:
15334:
15317:
14747:
14720:
14693:
14646:
14626:
14589:
14572:
14532:
14517:
14475:
14461:
14401:
14374:
14341:
14333:
10451:{\displaystyle y=b\cdot \sinh E}
9255:and for a parabola the range is
9128:For the hyperbola the range for
8478:
8470:
8446:
8443:
8433:
8395:
8366:
8344:
8322:
8314:
8300:
8289:
8281:
8252:
8233:
8225:
8214:
8198:
8182:
8174:
8163:
8149:
8135:
8116:
8106:
8076:
8054:
8032:
7830:
7693:
7636:
7591:
7574:
7563:
7517:
7503:
7489:
7454:
7432:
7327:
7319:
7302:
7294:
7286:
7278:
7264:
7256:
7242:
7231:
7217:
7203:
7171:
7163:
7152:
7138:
7105:
7091:
7077:
7041:
7033:
6994:
6980:
6963:
6952:
6925:
6914:
6875:
6862:
6854:
6817:
6806:
6798:
6787:
6773:
6762:
6733:
6722:
6711:
6688:
6677:
6656:
6628:
6614:
6581:
6570:
6549:
6541:
6506:
6495:
6471:
6445:
6428:
6414:
6385:
6368:
6357:
6349:
6327:
6291:
6266:
6255:
6228:
6218:
6196:
5182:{\displaystyle r={\frac {1}{s}}}
3651:
3640:
3461:
3435:
3388:
3316:
3222:
3150:
3082:
3064:
3032:
2999:
2981:
2956:
2930:
2898:
2870:
2836:
2808:
2773:
2755:
2727:
2695:
2664:
2633:
2616:
2576:
2559:
2444:
2429:) refers to linear acceleration
2402:
2378:
2370:
2362:
2348:
2337:
2323:
2306:
2284:
2273:
2239:
2208:
2197:
2189:
1599:
1563:
1497:
1465:
1450:
1441:
1417:
1388:
1327:
1287:
1223:
1188:{\displaystyle \mathbf {r} _{2}}
1175:
1159:{\displaystyle \mathbf {r} _{1}}
1146:
715:
693:
647:
602:
578:
567:
71:, which forms a two-dimensional
18149:Astronomical coordinate systems
17903:Longitude of the ascending node
13514:{\displaystyle {\tfrac {E}{2}}}
11737:{\displaystyle (\cos E,\sin E)}
10164:under the assumption that time
9738:{\displaystyle y=b\cdot \sin E}
9682:{\displaystyle x=a\cdot \cos E}
8788:{\displaystyle {\tfrac {p}{2}}}
4059:Taking the time derivative of (
1972:Longitude of the ascending node
1818:than their orbiting satellites.
520:
130:
18222:Retrograde and prograde motion
17151:
17142:
17133:
17063:
17047:
16973:
16946:
16935:
16909:
16852:
16800:
16769:
16744:
16713:
16688:
16667:
16636:
16533:
16502:
16372:
16341:
16300:
16269:
16204:
16189:
16085:
16069:
16036:
16006:
15909:
15890:
15756:
15730:
15704:
15655:
15629:
15603:
15567:
15539:
15533:
15516:
15505:
15462:
15432:
15412:
15396:
15357:vectors for this true anomaly
15344:
15338:
15321:
15310:
14903:
14884:
14751:
14724:
14650:
14599:
14593:
14576:
14565:
14542:
14512:
14405:
14345:
14329:
14269:
14251:
14020:
13991:
13976:
13451:
12634:
12610:
12527:
12464:
12452:
12417:
12403:
12378:
12366:
12346:
12334:
11775:
11751:
11731:
11707:
11161:
11140:
11052:
10937:
10913:
10848:
10824:
10740:
10716:
10389:
10371:
10314:
10290:
10135:
10111:
10027:
10003:
9582:
9558:
8237:
8221:
8098:associated with the orbit as:
7836:
7825:
7794:
7782:
7754:
7748:
7739:
7725:
7699:
7688:
7675:
7669:
7642:
7631:
7618:
7612:
7584:
7559:
7547:
7541:
7521:
7496:
7391:
7388:
7382:
7364:
7355:
7349:
7331:
7315:
7268:
7249:
7235:
7210:
7195:yields an interesting result:
7045:
7029:
6998:
6948:
6881:
6870:
6827:
6810:
6794:
6783:
6758:
6755:
6743:
6718:
6698:
6663:
6553:
6537:
6516:
6491:
6475:
6435:
6418:
6407:
5948:
5933:
5780:
5761:
4453:ordinary differential equation
3857:
3851:
3836:
3830:
3793:
3787:
3772:
3766:
3719:
3713:
3697:
3691:
3662:
3635:
3465:
3454:
3448:
3439:
3392:
3320:
3226:
3154:
3068:
3036:
2985:
2960:
2934:
2840:
2812:
2777:
2759:
2731:
2699:
2643:
2637:
2620:
2609:
2586:
2580:
2563:
2552:
1698:
1672:
472:
466:
439:
433:
294:
288:
268:
249:
237:
231:
1:
17195:Fundamentals of Astrodynamics
17181:
16169:{\displaystyle r,V_{r},V_{t}}
11700:it is clear that the vectors
8526:this is a circle with radius
8046:and the integration constant
3252:"). Substituting these into (
18169:Equatorial coordinate system
17126:
17033:{\displaystyle \mathbf {e} }
17008:
16882:
16876:
16870:
16613:
16319:
16228:
16222:
15989:
15796:
15790:
15778:
15677:
15385:
15379:
15296:
15290:
15288:values as those defined by (
15200:
15137:
15063:
14934:
14858:
14852:
14846:
14773:
14672:
14495:
14438:
14288:
14160:
14044:
14038:
13953:
13906:
13900:
13894:
13888:
13876:
13746:
13740:
13734:
13728:
13680:
13614:
13340:
13324:
12760:
12653:
12582:
12576:
12311:
12166:
12018:
11788:are on the same side of the
11296:
11284:
11180:
11112:
11106:
11013:
11007:
10998:
10992:
10956:
10867:
10774:
10630:
10550:
10464:
10408:
10333:
10244:
10154:
10061:
9917:
9837:
9751:
9695:
9601:
9447:
8981:
8886:
8726:
8631:
8561:{\displaystyle 0<e<1,}
8402:{\displaystyle \mathbf {c} }
8373:{\displaystyle \mathbf {r} }
8351:{\displaystyle \mathbf {v} }
8083:{\displaystyle \mathbf {c} }
8061:{\displaystyle \mathbf {c} }
8039:{\displaystyle \mathbf {r} }
7962:
7461:{\displaystyle \mathbf {c} }
7439:{\displaystyle \mathbf {r} }
6203:{\displaystyle \mathbf {u} }
5804:
5690:
5495:
5483:
2877:{\displaystyle \mathbf {r} }
2671:{\displaystyle \mathbf {H} }
2409:{\displaystyle \mathbf {H} }
1424:{\displaystyle \mathbf {r} }
747:attracts another point mass
722:{\displaystyle \mathbf {r} }
700:{\displaystyle \mathbf {a} }
654:{\displaystyle \mathbf {F} }
539:law of universal gravitation
35:
7:
17079:
16123:{\displaystyle p,e,\theta }
16052:The osculating Kepler orbit
15552:in the orbital plane (with
14318:
13692:
9300:
9294:
7800:{\displaystyle (r,\theta )}
6015:is zero at the point where
5984:{\displaystyle \theta _{0}}
5913:{\displaystyle \theta _{0}}
5838:{\displaystyle \theta _{0}}
5501:
5299:
5195:
5147:
5145:The differential equation (
5135:
4447:
4435:
4291:
4169:
4163:
4151:
4061:
4049:
3611:
3254:
2656:in the plane orthogonal to
2425:
2166:
2095:, the time since periapsis.
1624:
1035:Simplified two body problem
1015:introduced the concepts of
533:, in which he outlined his
10:
18367:
17921:Longitude of the periapsis
16055:
14862:) follows that by setting
14835:{\displaystyle V_{t}>0}
14300:
14029:{\displaystyle {\hat {x}}}
14007:in the orbital plane with
11002:) for a hyperbolic orbit.
7996:is called "true anomaly".
5700:with the general solution
1865:
1038:
111:. Keplerian orbits can be
20:
18245:
18232:Specific angular momentum
18137:
18049:
17993:
17929:
17882:
17822:
17813:
17709:
17619:
17508:
17499:
17394:
17327:
17276:
16427:the resulting parameters
16385:is a function other than
15800:) has a singularity when
15788:Note that the relations (
13830:for an elliptic orbit is
13622:for an elliptic orbit is
12470:{\displaystyle \arg(x,y)}
12352:{\displaystyle \arg(x,y)}
11005:Note that the relations (
10640:and the angular momentum
10347:for the parameterisation
9927:and the angular momentum
9041:{\displaystyle \theta =0}
6035:is maximal and therefore
5991:takes the value zero and
1791:{\displaystyle \mu =G\,M}
17336:Mysterium Cosmographicum
16816:the time of osculation.
15377:as the ones defined by (
11096:Some additional formulae
9304:) and the definition of
8004:Notice also that, since
7127:Integrating both sides:
4491:{\displaystyle \theta .}
2884:and its derivatives as:
1840:solar radiation pressure
1197:inertial reference frame
779:| will always be equal.
187:. The first law states:
185:laws of planetary motion
89:solar radiation pressure
18227:Specific orbital energy
16601:{\displaystyle \theta }
16480:{\displaystyle \theta }
15826:{\displaystyle V_{r}=0}
15488:{\displaystyle \theta }
15370:{\displaystyle \theta }
15223:{\displaystyle \theta }
14983:{\displaystyle \theta }
14963:{\displaystyle e\geq 0}
10979:{\displaystyle \theta }
9624:{\displaystyle \theta }
9141:{\displaystyle \theta }
8017:{\displaystyle \theta }
7989:{\displaystyle \theta }
7417:{\displaystyle \theta }
6008:{\displaystyle \theta }
5493:Using the expressions (
4186:{\displaystyle \theta }
3516:{\displaystyle \theta }
2152:{\displaystyle \omega }
2132:{\displaystyle \Omega }
2013:{\displaystyle \omega }
1987:{\displaystyle \Omega }
1718:
1647:{\displaystyle \alpha }
525:Between 1665 and 1666,
402:{\displaystyle \theta }
167:Development of the laws
17639:Geostationary transfer
17430:Kepler space telescope
17139:Copernicus. pp 513–514
17070:
17034:
16996:
16862:
16807:
16751:
16602:
16582:
16543:
16512:
16481:
16461:
16441:
16421:
16379:
16307:
16214:
16170:
16124:
16092:
16043:
15977:
15930:
15827:
15766:
15665:
15577:
15546:
15489:
15469:
15419:
15371:
15351:
15282:
15255:
15224:
15188:
15125:
15051:
14984:
14964:
14922:
14836:
14803:
14802:{\displaystyle r>0}
14761:
14660:
14606:
14549:
14483:
14426:
14352:
14276:
14148:
14030:
14001:
13941:
13864:
13824:
13720:
13668:
13604:
13515:
13486:
13312:
13224:
13191:
13158:
12748:
12641:
12562:
12477:the standard function
12471:
12431:
12395:is selected such that
12385:
12353:
12299:
12154:
12006:
11918:
11852:
11782:
11738:
11688:
11272:
11168:
11087:
10980:
10944:
10855:
10762:
10618:
10538:
10452:
10396:
10321:
10232:
10184:
10142:
10049:
9905:
9825:
9739:
9683:
9625:
9589:
9478:
9435:
9336:
9284:
9249:
9142:
9122:
9082:
9042:
9019:
8969:
8874:
8817:
8816:{\displaystyle e>1}
8789:
8756:
8714:
8619:
8562:
8520:
8486:
8456:
8403:
8374:
8352:
8330:
8265:
8084:
8062:
8040:
8018:
7990:
7950:
7892:
7857:
7801:
7767:
7462:
7440:
7418:
7398:
7179:
7119:
7058:
6901:
6834:
6595:
6335:
6274:
6241:
6204:
6172:
6104:
6064:
6029:
6009:
5985:
5958:
5914:
5885:
5839:
5792:
5678:
5587:
5550:
5471:
5287:
5183:
5123:
4951:
4726:
4568:
4492:
4469:
4423:
4279:
4207:
4187:
4139:
4037:
3984:
3599:
3517:
3497:
3475:
3240:
2878:
2854:
2672:
2650:
2593:
2539:
2502:
2471:as opposed to angular
2465:
2410:
2386:
2222:
2153:
2133:
2113:
2089:
2065:
2045:
2014:
1988:
1962:
1927:
1901:
1863:
1812:
1792:
1705:
1648:
1612:
1534:
1520:in that direction and
1510:
1479:
1425:
1401:
1300:
1189:
1160:
1131:
1104:
1067:Smaller objects, like
1004:
995:
973:
944:
920:gravitational constant
912:
890:
865:
788:
785:gravitational constant
731:
723:
701:
677:
655:
628:
504:
482:
403:
377:
351:
329:
304:
205:
195:of every planet is an
40:
18212:Orbital state vectors
18154:Characteristic energy
18124:Trans-lunar injection
17912:Argument of periapsis
17589:Prograde / Retrograde
17550:Hyperbolic trajectory
17403:Die Harmonie der Welt
17106:Hyperbolic trajectory
17071:
17035:
16997:
16863:
16808:
16752:
16603:
16583:
16544:
16513:
16482:
16462:
16442:
16422:
16380:
16308:
16220:using the relations (
16215:
16171:
16125:
16093:
16062:For any state vector
16044:
15978:
15931:
15828:
15767:
15666:
15578:
15547:
15490:
15470:
15420:
15372:
15352:
15283:
15281:{\displaystyle V_{t}}
15256:
15254:{\displaystyle V_{r}}
15225:
15189:
15126:
15052:
14985:
14965:
14923:
14837:
14804:
14762:
14661:
14607:
14550:
14484:
14427:
14353:
14314:initial value problem
14301:Further information:
14277:
14149:
14031:
14002:
13942:
13865:
13825:
13721:
13669:
13605:
13516:
13487:
13313:
13225:
13192:
13159:
12749:
12642:
12563:
12472:
12432:
12386:
12384:{\displaystyle (x,y)}
12354:
12300:
12155:
12007:
11919:
11853:
11783:
11739:
11689:
11273:
11169:
11088:
10981:
10945:
10856:
10763:
10619:
10539:
10453:
10397:
10322:
10233:
10185:
10143:
10050:
9906:
9826:
9740:
9684:
9626:
9590:
9479:
9477:{\displaystyle V_{r}}
9436:
9337:
9285:
9250:
9143:
9123:
9083:
9043:
8997:
8970:
8875:
8818:
8790:
8757:
8715:
8620:
8563:
8521:
8487:
8457:
8404:
8375:
8353:
8331:
8266:
8085:
8063:
8041:
8019:
7991:
7951:
7893:
7858:
7802:
7768:
7463:
7441:
7424:is the angle between
7419:
7399:
7180:
7120:
7059:
6902:
6844:Vector triple product
6835:
6596:
6336:
6275:
6242:
6205:
6188:Define a unit vector
6173:
6105:
6065:
6030:
6010:
5986:
5959:
5915:
5886:
5840:
5793:
5679:
5588:
5551:
5472:
5288:
5184:
5124:
4952:
4727:
4569:
4493:
4470:
4424:
4280:
4208:
4188:
4140:
4038:
3985:
3600:
3518:
3498:
3476:
3241:
2879:
2855:
2673:
2651:
2594:
2540:
2503:
2466:
2411:
2387:
2223:
2154:
2134:
2114:
2090:
2066:
2046:
2015:
1998:Argument of periapsis
1989:
1963:
1928:
1902:
1857:
1813:
1793:
1706:
1649:
1613:
1535:
1511:
1480:
1426:
1402:
1301:
1190:
1161:
1137:and position vectors
1132:
1130:{\displaystyle m_{2}}
1105:
1103:{\displaystyle m_{1}}
996:
974:
972:{\displaystyle m_{2}}
945:
943:{\displaystyle m_{1}}
913:
891:
866:
793:
739:
724:
702:
678:
656:
629:
543:
505:
483:
404:
378:
352:
330:
305:
189:
31:
18059:Bi-elliptic transfer
17579:Parabolic trajectory
17444:Astronomers Monument
17215:Copernicus, Nicolaus
17111:Parabolic trajectory
17044:
17022:
16895:
16830:
16761:
16628:
16592:
16553:
16522:
16491:
16471:
16451:
16431:
16389:
16333:
16244:
16180:
16134:
16130:are determined from
16102:
16066:
16003:
15945:
15837:
15804:
15693:
15592:
15556:
15502:
15479:
15429:
15393:
15361:
15307:
15265:
15238:
15214:
15156:
15079:
14999:
14974:
14948:
14871:
14813:
14787:
14688:
14621:
14562:
14509:
14454:
14367:
14326:
14176:
14057:
14011:
13967:
13919:
13839:
13754:
13700:
13631:
13525:
13496:
13348:
13239:
13201:
13168:
12774:
12672:
12595:
12496:
12443:
12399:
12363:
12325:
12182:
12037:
11933:
11862:
11796:
11748:
11704:
11304:
11300:) then follows that
11199:
11125:
11021:
10970:
10886:
10797:
10653:
10566:
10483:
10424:
10356:
10345:hyperbolic functions
10263:
10254:this can be written
10206:
10193:As by definition of
10168:
10084:
9940:
9853:
9770:
9711:
9655:
9615:
9493:
9461:
9351:
9312:
9259:
9152:
9132:
9092:
9052:
9026:
8902:
8836:
8801:
8770:
8740:
8647:
8581:
8537:
8504:
8466:
8429:
8391:
8362:
8340:
8277:
8102:
8072:
8050:
8028:
8008:
7980:
7907:
7867:
7811:
7779:
7476:
7450:
7428:
7408:
7199:
7131:
7070:
6910:
6850:
6607:
6345:
6284:
6251:
6214:
6192:
6181:Alternate derivation
6116:
6078:
6039:
6019:
5999:
5968:
5924:
5897:
5853:
5822:
5709:
5602:
5560:
5509:
5315:
5214:
5160:
4965:
4735:
4577:
4501:
4479:
4459:
4307:
4222:
4197:
4177:
4074:
3998:
3625:
3532:
3507:
3487:
3262:
2890:
2866:
2684:
2660:
2606:
2549:
2512:
2475:
2433:
2398:
2232:
2185:
2143:
2123:
2103:
2079:
2055:
2044:{\displaystyle \nu }
2035:
2004:
1978:
1952:
1917:
1891:
1802:
1772:
1660:
1638:
1556:
1524:
1491:
1437:
1413:
1309:
1205:
1170:
1141:
1114:
1087:
985:
956:
927:
902:
880:
813:
711:
689:
667:
643:
563:
494:
427:
393:
367:
341:
319:
225:
18099:Low-energy transfer
17319:Keplerian telescope
17165:on 16 February 2011
16824:eccentricity vector
15389:) the state vector
12663:and therefore that
11190:and therefore that
10183:{\displaystyle t=0}
8755:{\displaystyle e=1}
8519:{\displaystyle e=0}
8421:of an orbit when a
8415:orbit determination
8385:eccentricity vector
8096:eccentricity vector
8000:Eccentricity Vector
1768:, often denoted as
1009:celestial mechanics
661:is the force vector
153:Nicolaus Copernicus
105:classical mechanics
45:celestial mechanics
18094:Inclination change
17742:Distant retrograde
17314:Kepler's Supernova
17066:
17030:
16992:
16858:
16803:
16747:
16598:
16578:
16539:
16508:
16477:
16457:
16437:
16417:
16375:
16303:
16210:
16166:
16120:
16088:
16039:
15973:
15926:
15823:
15762:
15661:
15573:
15542:
15485:
15465:
15415:
15367:
15347:
15278:
15251:
15220:
15184:
15121:
15047:
14980:
14960:
14918:
14832:
14799:
14757:
14656:
14602:
14545:
14479:
14422:
14348:
14272:
14144:
14026:
13997:
13937:
13860:
13820:
13716:
13664:
13600:
13511:
13509:
13482:
13308:
13220:
13187:
13154:
12744:
12637:
12558:
12467:
12427:
12381:
12349:
12295:
12150:
12002:
11914:
11907:
11886:
11848:
11841:
11820:
11778:
11734:
11684:
11268:
11164:
11083:
10976:
10940:
10851:
10758:
10614:
10534:
10474:for which one has
10448:
10392:
10317:
10228:
10180:
10138:
10045:
9901:
9821:
9735:
9679:
9621:
9585:
9474:
9431:
9332:
9280:
9245:
9138:
9118:
9113:
9078:
9073:
9038:
9020:
8965:
8870:
8813:
8785:
8783:
8766:with focal length
8752:
8710:
8615:
8558:
8516:
8482:
8452:
8399:
8370:
8348:
8326:
8261:
8080:
8058:
8036:
8014:
7986:
7946:
7888:
7886:
7853:
7851:
7797:
7763:
7458:
7436:
7414:
7394:
7175:
7115:
7054:
6897:
6830:
6591:
6341:. It follows that
6331:
6324:
6270:
6237:
6200:
6168:
6100:
6098:
6060:
6058:
6025:
6005:
5981:
5954:
5910:
5881:
5876:
5835:
5788:
5674:
5583:
5546:
5467:
5283:
5179:
5119:
4947:
4722:
4564:
4488:
4465:
4419:
4275:
4203:
4183:
4135:
4033:
3980:
3945:
3881:
3731:
3595:
3513:
3493:
3471:
3236:
3234:
2874:
2850:
2848:
2668:
2646:
2601:polar unit vectors
2589:
2535:
2498:
2461:
2406:
2382:
2218:
2149:
2129:
2109:
2085:
2061:
2041:
2010:
1984:
1958:
1923:
1897:
1875:Keplerian elements
1868:Keplerian elements
1864:
1850:Keplerian elements
1836:general relativity
1808:
1788:
1701:
1644:
1608:
1530:
1506:
1475:
1421:
1397:
1296:
1185:
1156:
1127:
1100:
991:
969:
940:
908:
886:
861:
789:
719:
697:
673:
651:
624:
500:
478:
399:
373:
347:
325:
300:
199:with the sun at a
109:general relativity
41:
18263:
18262:
18237:Two-line elements
18045:
18044:
17967:Eccentric anomaly
17809:
17808:
17676:Orbit of the Moon
17535:Highly elliptical
17458:
17457:
17450:List of namesakes
17376:Rudolphine Tables
17304:Kepler's equation
17284:Kepler conjecture
17277:Scientific career
17116:Radial trajectory
17016:
17015:
16990:
16976:
16949:
16868:is useful. From (
16855:
16791:
16735:
16703:
16691:
16658:
16575:
16536:
16505:
16460:{\displaystyle e}
16440:{\displaystyle p}
16415:
16363:
16327:
16326:
16291:
16258:
16207:
16192:
15997:
15996:
15971:
15970:
15924:
15923:
15922:
15876:
15875:
15786:
15785:
15759:
15733:
15707:
15685:
15684:
15658:
15632:
15606:
15570:
15536:
15519:
15475:for true anomaly
15341:
15324:
15208:
15207:
15182:
15181:
15145:
15144:
15119:
15071:
15070:
15039:
14942:
14941:
14916:
14781:
14780:
14754:
14727:
14680:
14679:
14653:
14596:
14579:
14503:
14502:
14468:
14446:
14445:
14420:
14408:
14381:
14296:
14295:
14246:
14245:
14168:
14167:
14130:
14129:
14023:
13994:
13979:
13961:
13960:
13935:
13884:
13883:
13858:
13818:
13805:
13726:it follows from (
13714:
13688:
13687:
13662:
13661:
13594:
13558:
13508:
13439:
13391:
13332:
13331:
13306:
13287:
13286:
13256:
13218:
13185:
13152:
13126:
13097:
13056:
13027:
12950:
12947:
12895:
12839:
12798:
12768:
12767:
12742:
12695:
12661:
12660:
12319:
12318:
12273:
12254:
12238:
12219:
12174:
12173:
12128:
12109:
12093:
12074:
12026:
12025:
12000:
11981:
11980:
11950:
11906:
11885:
11840:
11819:
11698:eccentric anomaly
11682:
11656:
11627:
11586:
11557:
11474:
11471:
11422:
11369:
11328:
11292:
11291:
11266:
11222:
11188:
11187:
10964:
10963:
10911:
10910:
10875:
10874:
10782:
10781:
10755:
10698:
10677:
10638:
10637:
10611:
10578:
10558:
10557:
10531:
10495:
10472:
10471:
10416:
10415:
10341:
10340:
10288:
10287:
10252:
10251:
10226:
10162:
10161:
10069:
10068:
10042:
9985:
9964:
9925:
9924:
9898:
9865:
9845:
9844:
9818:
9782:
9761:and consequently
9759:
9758:
9703:
9702:
9640:eccentric anomaly
9609:
9608:
9553:
9552:
9538:
9524:
9455:
9454:
9417:
9416:
9390:
9376:
9330:
9298:), the equation (
9238:
9190:
9112:
9072:
8989:
8988:
8963:
8934:
8933:
8894:
8893:
8868:
8782:
8734:
8733:
8708:
8679:
8678:
8639:
8638:
8613:
8449:
8307:
8259:
8244:
8205:
8190:
8157:
8142:
8123:
7970:
7969:
7944:
7885:
7850:
7758:
7679:
7622:
7581:
7551:
7510:
7224:
7145:
7112:
7084:
7027:
7012:
6987:
6970:
6946:
6932:
6824:
6780:
6740:
6695:
6653:
6621:
6588:
6534:
6513:
6467:
6452:
6405:
6375:
6323:
6298:
6235:
6166:
6133:
6097:
6057:
6028:{\displaystyle s}
5951:
5936:
5875:
5812:
5811:
5733:
5698:
5697:
5655:
5581:
5544:
5491:
5490:
5465:
5428:
5398:
5370:
5350:
5307:
5306:
5281:
5258:
5235:
5203:
5202:
5177:
5143:
5142:
5117:
5083:
5068:
5032:
4990:
4945:
4912:
4879:
4865:
4828:
4795:
4770:
4720:
4690:
4671:
4657:
4627:
4612:
4562:
4532:
4513:
4468:{\displaystyle r}
4443:
4442:
4416:
4402:
4372:
4357:
4319:
4299:
4298:
4272:
4258:
4234:
4206:{\displaystyle r}
4159:
4158:
4133:
4119:
4086:
4057:
4056:
4031:
4010:
3977:
3940:
3869:
3821:
3805:
3757:
3658:
3619:
3618:
3593:
3563:
3544:
3496:{\displaystyle r}
3468:
3442:
3424:
3395:
3376:
3364:
3346:
3323:
3298:
3279:
3229:
3210:
3198:
3180:
3157:
3132:
3113:
3089:
3071:
3057:
3039:
3025:
3006:
2988:
2963:
2937:
2843:
2815:
2780:
2762:
2734:
2702:
2640:
2623:
2583:
2566:
2528:
2491:
2451:
2355:
2330:
2313:
2291:
2265:
2246:
2215:
2112:{\displaystyle i}
2088:{\displaystyle T}
2064:{\displaystyle M}
1961:{\displaystyle i}
1926:{\displaystyle e}
1900:{\displaystyle a}
1811:{\displaystyle M}
1632:
1631:
1605:
1594:
1570:
1533:{\displaystyle r}
1503:
1394:
1383:
1334:
1293:
1282:
1230:
1195:relative to some
994:{\displaystyle r}
911:{\displaystyle G}
889:{\displaystyle F}
859:
676:{\displaystyle m}
622:
512:semi-latus rectum
503:{\displaystyle p}
476:
376:{\displaystyle e}
350:{\displaystyle a}
328:{\displaystyle r}
298:
119:in various ways.
103:. As a theory in
18358:
18328:
18327:
18326:
18316:
18315:
18314:
18304:
18303:
18302:
18292:
18291:
18280:
18279:
18278:
18271:
18253:
18252:
18194:Lagrangian point
18089:Hohmann transfer
18034:
18020:
18011:
18002:
17982:
17973:
17964:
17955:
17951:
17947:
17938:
17918:
17909:
17900:
17891:
17871:
17867:
17858:
17849:
17840:
17820:
17819:
17789:Heliosynchronous
17738:Lagrange points
17691:Transatmospheric
17506:
17505:
17485:
17478:
17471:
17462:
17461:
17410:Katharina Kepler
17368:Harmonices Mundi
17309:Kepler polyhedra
17263:
17256:
17249:
17240:
17239:
17222:
17210:
17198:
17175:
17174:
17172:
17170:
17161:. Archived from
17155:
17149:
17146:
17140:
17137:
17086:Two-body problem
17075:
17073:
17072:
17067:
17062:
17054:
17039:
17037:
17036:
17031:
17029:
17010:
17001:
16999:
16998:
16993:
16991:
16989:
16988:
16979:
16978:
16977:
16972:
16967:
16964:
16963:
16951:
16950:
16945:
16940:
16934:
16933:
16921:
16920:
16907:
16902:
16889:
16867:
16865:
16864:
16859:
16857:
16856:
16851:
16846:
16837:
16812:
16810:
16809:
16804:
16793:
16792:
16787:
16782:
16776:
16768:
16756:
16754:
16753:
16748:
16737:
16736:
16731:
16726:
16720:
16712:
16704:
16702:
16701:
16692:
16687:
16682:
16680:
16660:
16659:
16654:
16649:
16643:
16635:
16620:
16607:
16605:
16604:
16599:
16587:
16585:
16584:
16579:
16577:
16576:
16571:
16566:
16560:
16548:
16546:
16545:
16540:
16538:
16537:
16532:
16527:
16517:
16515:
16514:
16509:
16507:
16506:
16501:
16496:
16486:
16484:
16483:
16478:
16466:
16464:
16463:
16458:
16446:
16444:
16443:
16438:
16426:
16424:
16423:
16418:
16416:
16414:
16413:
16404:
16399:
16384:
16382:
16381:
16376:
16365:
16364:
16359:
16354:
16348:
16340:
16321:
16312:
16310:
16309:
16304:
16293:
16292:
16287:
16282:
16276:
16268:
16260:
16259:
16254:
16249:
16238:
16219:
16217:
16216:
16211:
16209:
16208:
16200:
16194:
16193:
16185:
16175:
16173:
16172:
16167:
16165:
16164:
16152:
16151:
16129:
16127:
16126:
16121:
16097:
16095:
16094:
16089:
16084:
16076:
16058:Osculating orbit
16048:
16046:
16045:
16040:
16035:
16034:
16029:
16020:
16019:
16014:
15991:
15982:
15980:
15979:
15974:
15972:
15963:
15962:
15957:
15956:
15939:
15935:
15933:
15932:
15927:
15925:
15918:
15917:
15912:
15908:
15907:
15887:
15883:
15882:
15877:
15868:
15867:
15862:
15861:
15849:
15848:
15832:
15830:
15829:
15824:
15816:
15815:
15780:
15771:
15769:
15768:
15763:
15761:
15760:
15755:
15750:
15735:
15734:
15729:
15724:
15709:
15708:
15703:
15698:
15687:
15679:
15670:
15668:
15667:
15662:
15660:
15659:
15654:
15649:
15634:
15633:
15628:
15623:
15608:
15607:
15602:
15597:
15586:
15582:
15580:
15579:
15574:
15572:
15571:
15566:
15561:
15551:
15549:
15548:
15543:
15538:
15537:
15532:
15527:
15521:
15520:
15515:
15510:
15494:
15492:
15491:
15486:
15474:
15472:
15471:
15466:
15461:
15460:
15455:
15446:
15445:
15440:
15424:
15422:
15421:
15416:
15411:
15403:
15376:
15374:
15373:
15368:
15356:
15354:
15353:
15348:
15343:
15342:
15337:
15332:
15326:
15325:
15320:
15315:
15287:
15285:
15284:
15279:
15277:
15276:
15260:
15258:
15257:
15252:
15250:
15249:
15233:
15229:
15227:
15226:
15221:
15202:
15193:
15191:
15190:
15185:
15183:
15174:
15173:
15168:
15167:
15150:
15139:
15130:
15128:
15127:
15122:
15120:
15118:
15117:
15108:
15107:
15098:
15073:
15065:
15056:
15054:
15053:
15048:
15040:
15038:
15037:
15028:
15027:
15018:
14993:
14989:
14987:
14986:
14981:
14969:
14967:
14966:
14961:
14944:and by defining
14936:
14927:
14925:
14924:
14919:
14917:
14912:
14911:
14906:
14902:
14901:
14881:
14865:
14841:
14839:
14838:
14833:
14825:
14824:
14808:
14806:
14805:
14800:
14775:
14766:
14764:
14763:
14758:
14756:
14755:
14750:
14745:
14742:
14741:
14729:
14728:
14723:
14718:
14715:
14714:
14702:
14701:
14696:
14682:
14674:
14665:
14663:
14662:
14657:
14655:
14654:
14649:
14644:
14635:
14634:
14629:
14615:
14611:
14609:
14608:
14603:
14598:
14597:
14592:
14587:
14581:
14580:
14575:
14570:
14554:
14552:
14551:
14546:
14541:
14540:
14535:
14526:
14525:
14520:
14497:
14488:
14486:
14485:
14480:
14478:
14470:
14469:
14464:
14459:
14448:
14440:
14431:
14429:
14428:
14423:
14421:
14419:
14418:
14409:
14404:
14399:
14397:
14383:
14382:
14377:
14372:
14361:
14358:when written as
14357:
14355:
14354:
14349:
14344:
14336:
14290:
14281:
14279:
14278:
14273:
14247:
14238:
14237:
14223:
14222:
14201:
14200:
14188:
14187:
14170:
14162:
14153:
14151:
14150:
14145:
14131:
14122:
14121:
14104:
14103:
14082:
14081:
14069:
14068:
14051:
14035:
14033:
14032:
14027:
14025:
14024:
14016:
14006:
14004:
14003:
13998:
13996:
13995:
13987:
13981:
13980:
13972:
13955:
13946:
13944:
13943:
13938:
13936:
13934:
13923:
13913:
13878:
13869:
13867:
13866:
13861:
13859:
13857:
13846:
13833:
13829:
13827:
13826:
13821:
13819:
13811:
13806:
13801:
13800:
13799:
13794:
13793:
13792:
13778:
13777:
13772:
13771:
13770:
13758:
13725:
13723:
13722:
13717:
13715:
13707:
13682:
13673:
13671:
13670:
13665:
13663:
13654:
13653:
13625:
13609:
13607:
13606:
13601:
13599:
13595:
13593:
13582:
13571:
13559:
13551:
13540:
13539:
13520:
13518:
13517:
13512:
13510:
13501:
13491:
13489:
13488:
13483:
13481:
13477:
13450:
13446:
13445:
13441:
13440:
13432:
13419:
13418:
13397:
13393:
13392:
13384:
13371:
13370:
13326:
13317:
13315:
13314:
13309:
13307:
13299:
13288:
13285:
13274:
13263:
13262:
13257:
13249:
13233:
13229:
13227:
13226:
13221:
13219:
13211:
13196:
13194:
13193:
13188:
13186:
13178:
13163:
13161:
13160:
13155:
13153:
13145:
13140:
13139:
13127:
13125:
13114:
13103:
13098:
13096:
13079:
13062:
13057:
13055:
13044:
13033:
13028:
13026:
12991:
12956:
12951:
12949:
12948:
12946:
12923:
12906:
12897:
12896:
12894:
12871:
12854:
12845:
12840:
12838:
12821:
12804:
12799:
12791:
12786:
12785:
12762:
12753:
12751:
12750:
12745:
12743:
12741:
12718:
12701:
12696:
12688:
12666:
12655:
12646:
12644:
12643:
12638:
12589:
12572:hyperbolic orbit
12567:
12565:
12564:
12559:
12557:
12553:
12526:
12522:
12483:double precision
12476:
12474:
12473:
12468:
12436:
12434:
12433:
12428:
12420:
12406:
12390:
12388:
12387:
12382:
12358:
12356:
12355:
12350:
12313:
12304:
12302:
12301:
12296:
12279:
12275:
12274:
12266:
12255:
12244:
12239:
12231:
12220:
12209:
12176:
12168:
12159:
12157:
12156:
12151:
12134:
12130:
12129:
12121:
12110:
12099:
12094:
12086:
12075:
12064:
12031:
12020:
12011:
12009:
12008:
12003:
12001:
11993:
11982:
11979:
11968:
11957:
11956:
11951:
11943:
11927:
11923:
11921:
11920:
11915:
11913:
11909:
11908:
11899:
11887:
11878:
11857:
11855:
11854:
11849:
11847:
11843:
11842:
11833:
11821:
11812:
11787:
11785:
11784:
11779:
11743:
11741:
11740:
11735:
11693:
11691:
11690:
11685:
11683:
11675:
11670:
11669:
11657:
11655:
11644:
11633:
11628:
11626:
11609:
11592:
11587:
11585:
11574:
11563:
11558:
11556:
11518:
11480:
11475:
11473:
11472:
11470:
11450:
11433:
11424:
11423:
11421:
11401:
11384:
11375:
11370:
11368:
11351:
11334:
11329:
11321:
11316:
11315:
11286:
11277:
11275:
11274:
11269:
11267:
11265:
11245:
11228:
11223:
11215:
11193:
11182:
11173:
11171:
11170:
11165:
11119:
11092:
11090:
11089:
11084:
11082:
11078:
11051:
11047:
10985:
10983:
10982:
10977:
10958:
10949:
10947:
10946:
10941:
10912:
10903:
10902:
10880:
10869:
10860:
10858:
10857:
10852:
10791:
10776:
10767:
10765:
10764:
10759:
10757:
10756:
10748:
10700:
10699:
10691:
10679:
10678:
10670:
10647:
10632:
10623:
10621:
10620:
10615:
10613:
10612:
10604:
10580:
10579:
10571:
10560:
10552:
10543:
10541:
10540:
10535:
10533:
10532:
10524:
10497:
10496:
10488:
10477:
10466:
10457:
10455:
10454:
10449:
10418:
10410:
10401:
10399:
10398:
10393:
10350:
10335:
10326:
10324:
10323:
10318:
10289:
10280:
10279:
10257:
10246:
10237:
10235:
10234:
10229:
10227:
10216:
10200:
10189:
10187:
10186:
10181:
10156:
10147:
10145:
10144:
10139:
10078:
10063:
10054:
10052:
10051:
10046:
10044:
10043:
10035:
9987:
9986:
9978:
9966:
9965:
9957:
9934:
9919:
9910:
9908:
9907:
9902:
9900:
9899:
9891:
9867:
9866:
9858:
9847:
9839:
9830:
9828:
9827:
9822:
9820:
9819:
9811:
9784:
9783:
9775:
9764:
9753:
9744:
9742:
9741:
9736:
9705:
9697:
9688:
9686:
9685:
9680:
9649:
9630:
9628:
9627:
9622:
9603:
9594:
9592:
9591:
9586:
9554:
9545:
9544:
9539:
9531:
9526:
9525:
9517:
9505:
9504:
9487:
9483:
9481:
9480:
9475:
9473:
9472:
9449:
9440:
9438:
9437:
9432:
9418:
9409:
9408:
9391:
9383:
9378:
9377:
9369:
9363:
9362:
9345:
9341:
9339:
9338:
9333:
9331:
9326:
9325:
9316:
9289:
9287:
9286:
9281:
9254:
9252:
9251:
9246:
9244:
9240:
9239:
9231:
9218:
9217:
9196:
9192:
9191:
9183:
9170:
9169:
9147:
9145:
9144:
9139:
9127:
9125:
9124:
9119:
9114:
9111:
9097:
9087:
9085:
9084:
9079:
9074:
9071:
9057:
9047:
9045:
9044:
9039:
8983:
8974:
8972:
8971:
8966:
8964:
8956:
8955:
8946:
8935:
8926:
8925:
8916:
8912:
8896:
8888:
8879:
8877:
8876:
8871:
8869:
8867:
8860:
8859:
8846:
8830:
8822:
8820:
8819:
8814:
8794:
8792:
8791:
8786:
8784:
8775:
8761:
8759:
8758:
8753:
8728:
8719:
8717:
8716:
8711:
8709:
8707:
8706:
8691:
8680:
8677:
8676:
8661:
8657:
8641:
8633:
8624:
8622:
8621:
8616:
8614:
8612:
8611:
8610:
8591:
8575:
8567:
8565:
8564:
8559:
8525:
8523:
8522:
8517:
8491:
8489:
8488:
8483:
8481:
8473:
8461:
8459:
8458:
8453:
8451:
8450:
8442:
8436:
8419:orbital elements
8408:
8406:
8405:
8400:
8398:
8379:
8377:
8376:
8371:
8369:
8357:
8355:
8354:
8349:
8347:
8335:
8333:
8332:
8327:
8325:
8317:
8309:
8308:
8303:
8298:
8292:
8284:
8270:
8268:
8267:
8262:
8260:
8255:
8250:
8245:
8240:
8236:
8228:
8217:
8211:
8206:
8201:
8196:
8191:
8186:
8185:
8177:
8171:
8166:
8158:
8153:
8152:
8144:
8143:
8138:
8133:
8129:
8124:
8119:
8114:
8109:
8089:
8087:
8086:
8081:
8079:
8067:
8065:
8064:
8059:
8057:
8045:
8043:
8042:
8037:
8035:
8023:
8021:
8020:
8015:
7995:
7993:
7992:
7987:
7964:
7955:
7953:
7952:
7947:
7945:
7943:
7917:
7901:
7897:
7895:
7894:
7889:
7887:
7878:
7862:
7860:
7859:
7854:
7852:
7846:
7845:
7844:
7839:
7833:
7828:
7822:
7806:
7804:
7803:
7798:
7772:
7770:
7769:
7764:
7759:
7757:
7735:
7717:
7713:
7708:
7707:
7702:
7696:
7691:
7685:
7680:
7678:
7652:
7651:
7650:
7645:
7639:
7634:
7628:
7623:
7621:
7595:
7594:
7583:
7582:
7577:
7572:
7566:
7557:
7552:
7550:
7524:
7520:
7512:
7511:
7506:
7501:
7492:
7486:
7467:
7465:
7464:
7459:
7457:
7445:
7443:
7442:
7437:
7435:
7423:
7421:
7420:
7415:
7403:
7401:
7400:
7395:
7330:
7322:
7305:
7297:
7289:
7281:
7267:
7259:
7245:
7234:
7226:
7225:
7220:
7215:
7206:
7184:
7182:
7181:
7176:
7174:
7166:
7155:
7147:
7146:
7141:
7136:
7124:
7122:
7121:
7116:
7114:
7113:
7108:
7103:
7094:
7086:
7085:
7080:
7075:
7063:
7061:
7060:
7055:
7044:
7036:
7028:
7026:
7015:
7013:
7005:
6997:
6989:
6988:
6983:
6978:
6972:
6971:
6966:
6961:
6955:
6947:
6939:
6934:
6933:
6928:
6923:
6917:
6906:
6904:
6903:
6898:
6890:
6889:
6884:
6878:
6873:
6865:
6857:
6839:
6837:
6836:
6831:
6826:
6825:
6820:
6815:
6809:
6801:
6790:
6782:
6781:
6776:
6771:
6765:
6742:
6741:
6736:
6731:
6725:
6714:
6697:
6696:
6691:
6686:
6680:
6675:
6674:
6659:
6654:
6652:
6651:
6639:
6631:
6623:
6622:
6617:
6612:
6600:
6598:
6597:
6592:
6590:
6589:
6584:
6579:
6573:
6568:
6567:
6552:
6544:
6536:
6535:
6527:
6515:
6514:
6509:
6504:
6498:
6490:
6489:
6474:
6469:
6468:
6460:
6454:
6453:
6448:
6443:
6431:
6417:
6406:
6404:
6393:
6388:
6377:
6376:
6371:
6366:
6360:
6352:
6340:
6338:
6337:
6332:
6330:
6325:
6322:
6321:
6309:
6300:
6299:
6294:
6289:
6279:
6277:
6276:
6271:
6269:
6258:
6246:
6244:
6243:
6238:
6236:
6231:
6226:
6221:
6209:
6207:
6206:
6201:
6199:
6177:
6175:
6174:
6169:
6167:
6165:
6139:
6134:
6126:
6109:
6107:
6106:
6101:
6099:
6093:
6092:
6083:
6069:
6067:
6066:
6061:
6059:
6050:
6034:
6032:
6031:
6026:
6014:
6012:
6011:
6006:
5990:
5988:
5987:
5982:
5980:
5979:
5963:
5961:
5960:
5955:
5953:
5952:
5944:
5938:
5937:
5929:
5919:
5917:
5916:
5911:
5909:
5908:
5890:
5888:
5887:
5882:
5877:
5874:
5866:
5858:
5844:
5842:
5841:
5836:
5834:
5833:
5806:
5797:
5795:
5794:
5789:
5787:
5783:
5779:
5778:
5734:
5732:
5731:
5719:
5703:
5692:
5683:
5681:
5680:
5675:
5667:
5663:
5656:
5654:
5653:
5652:
5639:
5635:
5634:
5624:
5614:
5613:
5596:
5592:
5590:
5589:
5584:
5582:
5580:
5572:
5564:
5555:
5553:
5552:
5547:
5545:
5543:
5542:
5541:
5528:
5524:
5523:
5513:
5485:
5476:
5474:
5473:
5468:
5466:
5464:
5463:
5462:
5449:
5445:
5444:
5434:
5429:
5427:
5426:
5414:
5409:
5408:
5403:
5399:
5397:
5389:
5381:
5371:
5369:
5368:
5356:
5351:
5349:
5348:
5347:
5334:
5330:
5329:
5319:
5309:
5301:
5292:
5290:
5289:
5284:
5282:
5280:
5272:
5264:
5259:
5257:
5256:
5244:
5236:
5234:
5226:
5218:
5208:
5197:
5188:
5186:
5185:
5180:
5178:
5170:
5154:
5137:
5128:
5126:
5125:
5120:
5118:
5116:
5115:
5103:
5095:
5091:
5084:
5079:
5078:
5073:
5069:
5067:
5059:
5051:
5044:
5033:
5031:
5030:
5029:
5016:
5012:
5011:
5001:
4991:
4989:
4988:
4979:
4978:
4969:
4959:
4956:
4954:
4953:
4948:
4946:
4944:
4943:
4931:
4923:
4922:
4917:
4913:
4911:
4910:
4898:
4885:
4881:
4880:
4878:
4877:
4868:
4867:
4866:
4858:
4842:
4829:
4827:
4819:
4811:
4806:
4805:
4800:
4796:
4794:
4793:
4781:
4771:
4769:
4768:
4767:
4754:
4750:
4749:
4739:
4731:
4729:
4728:
4723:
4721:
4719:
4718:
4706:
4698:
4697:
4692:
4691:
4683:
4673:
4672:
4664:
4658:
4656:
4648:
4640:
4635:
4634:
4629:
4628:
4620:
4613:
4611:
4610:
4609:
4596:
4592:
4591:
4581:
4573:
4571:
4570:
4565:
4563:
4561:
4560:
4548:
4540:
4539:
4534:
4533:
4525:
4515:
4514:
4506:
4497:
4495:
4494:
4489:
4474:
4472:
4471:
4466:
4437:
4428:
4426:
4425:
4420:
4418:
4417:
4409:
4403:
4401:
4393:
4385:
4380:
4379:
4374:
4373:
4365:
4358:
4356:
4355:
4354:
4341:
4337:
4336:
4326:
4321:
4320:
4312:
4301:
4293:
4284:
4282:
4281:
4276:
4274:
4273:
4265:
4259:
4257:
4249:
4241:
4236:
4235:
4227:
4216:
4212:
4210:
4209:
4204:
4192:
4190:
4189:
4184:
4153:
4144:
4142:
4141:
4136:
4134:
4132:
4131:
4122:
4121:
4120:
4112:
4096:
4088:
4087:
4079:
4068:
4051:
4042:
4040:
4039:
4034:
4032:
4030:
4029:
4017:
4012:
4011:
4003:
3992:
3989:
3987:
3986:
3981:
3979:
3978:
3970:
3967:
3966:
3954:
3950:
3949:
3942:
3941:
3933:
3930:
3929:
3891:
3887:
3886:
3885:
3871:
3870:
3862:
3823:
3822:
3814:
3807:
3806:
3798:
3759:
3758:
3750:
3736:
3735:
3665:
3660:
3659:
3654:
3649:
3643:
3638:
3613:
3604:
3602:
3601:
3596:
3594:
3592:
3591:
3579:
3571:
3570:
3565:
3564:
3556:
3546:
3545:
3537:
3526:
3522:
3520:
3519:
3514:
3502:
3500:
3499:
3494:
3480:
3478:
3477:
3472:
3470:
3469:
3464:
3459:
3444:
3443:
3438:
3433:
3430:
3426:
3425:
3423:
3422:
3410:
3397:
3396:
3391:
3386:
3383:
3379:
3378:
3377:
3369:
3366:
3365:
3357:
3348:
3347:
3339:
3325:
3324:
3319:
3314:
3311:
3307:
3306:
3305:
3300:
3299:
3291:
3281:
3280:
3272:
3245:
3243:
3242:
3237:
3235:
3231:
3230:
3225:
3220:
3217:
3213:
3212:
3211:
3203:
3200:
3199:
3191:
3182:
3181:
3173:
3159:
3158:
3153:
3148:
3145:
3141:
3140:
3139:
3134:
3133:
3125:
3115:
3114:
3106:
3091:
3090:
3085:
3080:
3073:
3072:
3067:
3062:
3059:
3058:
3050:
3041:
3040:
3035:
3030:
3027:
3026:
3018:
3008:
3007:
3002:
2997:
2990:
2989:
2984:
2979:
2970:
2966:
2965:
2964:
2959:
2954:
2939:
2938:
2933:
2928:
2901:
2883:
2881:
2880:
2875:
2873:
2859:
2857:
2856:
2851:
2849:
2845:
2844:
2839:
2834:
2831:
2817:
2816:
2811:
2806:
2803:
2782:
2781:
2776:
2771:
2764:
2763:
2758:
2753:
2750:
2736:
2735:
2730:
2725:
2722:
2704:
2703:
2698:
2693:
2677:
2675:
2674:
2669:
2667:
2655:
2653:
2652:
2647:
2642:
2641:
2636:
2631:
2625:
2624:
2619:
2614:
2598:
2596:
2595:
2590:
2585:
2584:
2579:
2574:
2568:
2567:
2562:
2557:
2544:
2542:
2541:
2536:
2534:
2530:
2529:
2521:
2507:
2505:
2504:
2499:
2497:
2493:
2492:
2484:
2470:
2468:
2467:
2462:
2457:
2453:
2452:
2447:
2442:
2415:
2413:
2412:
2407:
2405:
2391:
2389:
2388:
2383:
2381:
2373:
2365:
2357:
2356:
2351:
2346:
2340:
2332:
2331:
2326:
2321:
2315:
2314:
2309:
2304:
2298:
2294:
2293:
2292:
2287:
2282:
2276:
2266:
2264:
2253:
2248:
2247:
2242:
2237:
2228:stays constant:
2227:
2225:
2224:
2219:
2217:
2216:
2211:
2206:
2200:
2192:
2158:
2156:
2155:
2150:
2138:
2136:
2135:
2130:
2118:
2116:
2115:
2110:
2094:
2092:
2091:
2086:
2070:
2068:
2067:
2062:
2050:
2048:
2047:
2042:
2019:
2017:
2016:
2011:
1993:
1991:
1990:
1985:
1967:
1965:
1964:
1959:
1932:
1930:
1929:
1924:
1906:
1904:
1903:
1898:
1860:orbital elements
1844:atmospheric drag
1817:
1815:
1814:
1809:
1797:
1795:
1794:
1789:
1763:
1710:
1708:
1707:
1702:
1697:
1696:
1684:
1683:
1653:
1651:
1650:
1645:
1626:
1617:
1615:
1614:
1609:
1607:
1606:
1598:
1595:
1593:
1592:
1580:
1572:
1571:
1566:
1561:
1550:
1544:of that vector.
1539:
1537:
1536:
1531:
1515:
1513:
1512:
1507:
1505:
1504:
1496:
1484:
1482:
1481:
1476:
1474:
1473:
1468:
1459:
1458:
1453:
1444:
1430:
1428:
1427:
1422:
1420:
1406:
1404:
1403:
1398:
1396:
1395:
1387:
1384:
1382:
1381:
1372:
1371:
1370:
1361:
1360:
1347:
1342:
1341:
1336:
1335:
1330:
1325:
1321:
1320:
1305:
1303:
1302:
1297:
1295:
1294:
1286:
1283:
1281:
1280:
1271:
1270:
1269:
1260:
1259:
1243:
1238:
1237:
1232:
1231:
1226:
1221:
1217:
1216:
1194:
1192:
1191:
1186:
1184:
1183:
1178:
1165:
1163:
1162:
1157:
1155:
1154:
1149:
1136:
1134:
1133:
1128:
1126:
1125:
1109:
1107:
1106:
1101:
1099:
1098:
1000:
998:
997:
992:
978:
976:
975:
970:
968:
967:
949:
947:
946:
941:
939:
938:
917:
915:
914:
909:
895:
893:
892:
887:
870:
868:
867:
862:
860:
858:
857:
848:
847:
846:
837:
836:
826:
728:
726:
725:
720:
718:
706:
704:
703:
698:
696:
682:
680:
679:
674:
660:
658:
657:
652:
650:
633:
631:
630:
625:
623:
621:
620:
619:
606:
605:
600:
599:
589:
581:
570:
509:
507:
506:
501:
487:
485:
484:
479:
477:
475:
446:
408:
406:
405:
400:
382:
380:
379:
374:
356:
354:
353:
348:
334:
332:
331:
326:
309:
307:
306:
301:
299:
297:
271:
267:
266:
244:
117:orbital elements
97:two-body problem
85:atmospheric drag
18366:
18365:
18361:
18360:
18359:
18357:
18356:
18355:
18351:Johannes Kepler
18336:
18335:
18334:
18324:
18322:
18312:
18310:
18300:
18298:
18286:
18276:
18274:
18266:
18264:
18259:
18241:
18159:Escape velocity
18140:
18133:
18114:Rocket equation
18041:
18033:
18027:
18018:
18009:
18000:
17989:
17980:
17971:
17962:
17953:
17949:
17945:
17936:
17925:
17916:
17907:
17898:
17889:
17878:
17869:
17865:
17861:Semi-minor axis
17856:
17852:Semi-major axis
17847:
17838:
17832:
17805:
17727:Areosynchronous
17711:
17705:
17686:Sun-synchronous
17671:Near-equatorial
17615:
17495:
17489:
17459:
17454:
17436:Johannes Kepler
17390:
17363:(1618, 1620-21)
17352:Astronomia nova
17323:
17299:Kepler triangle
17272:
17270:Johannes Kepler
17267:
17229:
17207:
17184:
17179:
17178:
17168:
17166:
17157:
17156:
17152:
17147:
17143:
17138:
17134:
17129:
17082:
17058:
17050:
17045:
17042:
17041:
17025:
17023:
17020:
17019:
16984:
16980:
16968:
16966:
16965:
16959:
16955:
16941:
16939:
16938:
16929:
16925:
16916:
16912:
16908:
16906:
16898:
16896:
16893:
16892:
16886:) follows that
16847:
16845:
16844:
16833:
16831:
16828:
16827:
16783:
16781:
16780:
16772:
16764:
16762:
16759:
16758:
16727:
16725:
16724:
16716:
16708:
16697:
16693:
16683:
16681:
16679:
16650:
16648:
16647:
16639:
16631:
16629:
16626:
16625:
16618:
16593:
16590:
16589:
16567:
16565:
16564:
16556:
16554:
16551:
16550:
16528:
16526:
16525:
16523:
16520:
16519:
16497:
16495:
16494:
16492:
16489:
16488:
16472:
16469:
16468:
16452:
16449:
16448:
16432:
16429:
16428:
16409:
16405:
16400:
16398:
16390:
16387:
16386:
16355:
16353:
16352:
16344:
16336:
16334:
16331:
16330:
16283:
16281:
16280:
16272:
16264:
16250:
16248:
16247:
16245:
16242:
16241:
16199:
16198:
16184:
16183:
16181:
16178:
16177:
16160:
16156:
16147:
16143:
16135:
16132:
16131:
16103:
16100:
16099:
16080:
16072:
16067:
16064:
16063:
16060:
16054:
16030:
16025:
16024:
16015:
16010:
16009:
16004:
16001:
16000:
15961:
15952:
15948:
15946:
15943:
15942:
15913:
15903:
15899:
15889:
15888:
15881:
15866:
15857:
15853:
15844:
15840:
15838:
15835:
15834:
15811:
15807:
15805:
15802:
15801:
15751:
15749:
15748:
15725:
15723:
15722:
15699:
15697:
15696:
15694:
15691:
15690:
15650:
15648:
15647:
15624:
15622:
15621:
15598:
15596:
15595:
15593:
15590:
15589:
15562:
15560:
15559:
15557:
15554:
15553:
15528:
15526:
15525:
15511:
15509:
15508:
15503:
15500:
15499:
15480:
15477:
15476:
15456:
15451:
15450:
15441:
15436:
15435:
15430:
15427:
15426:
15407:
15399:
15394:
15391:
15390:
15362:
15359:
15358:
15333:
15331:
15330:
15316:
15314:
15313:
15308:
15305:
15304:
15272:
15268:
15266:
15263:
15262:
15245:
15241:
15239:
15236:
15235:
15231:
15215:
15212:
15211:
15172:
15163:
15159:
15157:
15154:
15153:
15113:
15109:
15103:
15099:
15097:
15080:
15077:
15076:
15033:
15029:
15023:
15019:
15017:
15000:
14997:
14996:
14975:
14972:
14971:
14949:
14946:
14945:
14907:
14897:
14893:
14883:
14882:
14880:
14872:
14869:
14868:
14820:
14816:
14814:
14811:
14810:
14788:
14785:
14784:
14746:
14744:
14743:
14737:
14733:
14719:
14717:
14716:
14710:
14706:
14697:
14692:
14691:
14689:
14686:
14685:
14645:
14643:
14642:
14630:
14625:
14624:
14622:
14619:
14618:
14588:
14586:
14585:
14571:
14569:
14568:
14563:
14560:
14559:
14536:
14531:
14530:
14521:
14516:
14515:
14510:
14507:
14506:
14474:
14460:
14458:
14457:
14455:
14452:
14451:
14414:
14410:
14400:
14398:
14396:
14373:
14371:
14370:
14368:
14365:
14364:
14340:
14332:
14327:
14324:
14323:
14310:
14305:
14236:
14218:
14214:
14196:
14192:
14183:
14179:
14177:
14174:
14173:
14120:
14099:
14095:
14077:
14073:
14064:
14060:
14058:
14055:
14054:
14015:
14014:
14012:
14009:
14008:
13986:
13985:
13971:
13970:
13968:
13965:
13964:
13927:
13922:
13920:
13917:
13916:
13850:
13845:
13840:
13837:
13836:
13810:
13795:
13788:
13784:
13783:
13782:
13773:
13766:
13762:
13761:
13760:
13759:
13757:
13755:
13752:
13751:
13706:
13701:
13698:
13697:
13652:
13632:
13629:
13628:
13612:From relation (
13583:
13572:
13570:
13566:
13550:
13532:
13528:
13526:
13523:
13522:
13499:
13497:
13494:
13493:
13458:
13454:
13431:
13427:
13423:
13411:
13407:
13383:
13379:
13375:
13363:
13359:
13355:
13351:
13349:
13346:
13345:
13298:
13275:
13264:
13261:
13248:
13240:
13237:
13236:
13210:
13202:
13199:
13198:
13177:
13169:
13166:
13165:
13144:
13135:
13131:
13115:
13104:
13102:
13080:
13063:
13061:
13045:
13034:
13032:
12992:
12957:
12955:
12924:
12907:
12905:
12898:
12872:
12855:
12853:
12846:
12844:
12822:
12805:
12803:
12790:
12781:
12777:
12775:
12772:
12771:
12719:
12702:
12700:
12687:
12673:
12670:
12669:
12596:
12593:
12592:
12574:one gets from (
12534:
12530:
12503:
12499:
12497:
12494:
12493:
12444:
12441:
12440:
12416:
12402:
12400:
12397:
12396:
12364:
12361:
12360:
12326:
12323:
12322:
12265:
12243:
12230:
12208:
12207:
12203:
12183:
12180:
12179:
12120:
12098:
12085:
12063:
12062:
12058:
12038:
12035:
12034:
11992:
11969:
11958:
11955:
11942:
11934:
11931:
11930:
11897:
11876:
11869:
11865:
11863:
11860:
11859:
11831:
11810:
11803:
11799:
11797:
11794:
11793:
11749:
11746:
11745:
11705:
11702:
11701:
11674:
11665:
11661:
11645:
11634:
11632:
11610:
11593:
11591:
11575:
11564:
11562:
11519:
11481:
11479:
11451:
11434:
11432:
11425:
11402:
11385:
11383:
11376:
11374:
11352:
11335:
11333:
11320:
11311:
11307:
11305:
11302:
11301:
11246:
11229:
11227:
11214:
11200:
11197:
11196:
11126:
11123:
11122:
11104:one gets from (
11098:
11059:
11055:
11028:
11024:
11022:
11019:
11018:
10971:
10968:
10967:
10901:
10887:
10884:
10883:
10798:
10795:
10794:
10747:
10746:
10690:
10689:
10669:
10668:
10654:
10651:
10650:
10603:
10602:
10570:
10569:
10567:
10564:
10563:
10523:
10522:
10487:
10486:
10484:
10481:
10480:
10425:
10422:
10421:
10357:
10354:
10353:
10278:
10264:
10261:
10260:
10215:
10207:
10204:
10203:
10169:
10166:
10165:
10085:
10082:
10081:
10034:
10033:
9977:
9976:
9956:
9955:
9941:
9938:
9937:
9890:
9889:
9857:
9856:
9854:
9851:
9850:
9810:
9809:
9774:
9773:
9771:
9768:
9767:
9712:
9709:
9708:
9656:
9653:
9652:
9616:
9613:
9612:
9543:
9530:
9516:
9515:
9500:
9496:
9494:
9491:
9490:
9468:
9464:
9462:
9459:
9458:
9407:
9382:
9368:
9367:
9358:
9354:
9352:
9349:
9348:
9321:
9317:
9315:
9313:
9310:
9309:
9260:
9257:
9256:
9230:
9226:
9222:
9210:
9206:
9182:
9178:
9174:
9162:
9158:
9153:
9150:
9149:
9133:
9130:
9129:
9101:
9095:
9093:
9090:
9089:
9061:
9055:
9053:
9050:
9049:
9027:
9024:
9023:
8951:
8947:
8945:
8921:
8917:
8911:
8903:
8900:
8899:
8855:
8851:
8850:
8845:
8837:
8834:
8833:
8802:
8799:
8798:
8773:
8771:
8768:
8767:
8741:
8738:
8737:
8702:
8698:
8690:
8672:
8668:
8656:
8648:
8645:
8644:
8606:
8602:
8595:
8590:
8582:
8579:
8578:
8538:
8535:
8534:
8505:
8502:
8501:
8498:
8477:
8469:
8467:
8464:
8463:
8441:
8440:
8432:
8430:
8427:
8426:
8394:
8392:
8389:
8388:
8383:Obviously, the
8365:
8363:
8360:
8359:
8343:
8341:
8338:
8337:
8321:
8313:
8299:
8297:
8296:
8288:
8280:
8278:
8275:
8274:
8251:
8249:
8232:
8224:
8213:
8212:
8210:
8197:
8195:
8181:
8173:
8172:
8170:
8162:
8148:
8134:
8132:
8131:
8130:
8128:
8115:
8113:
8105:
8103:
8100:
8099:
8075:
8073:
8070:
8069:
8053:
8051:
8048:
8047:
8031:
8029:
8026:
8025:
8009:
8006:
8005:
8002:
7981:
7978:
7977:
7921:
7916:
7908:
7905:
7904:
7876:
7868:
7865:
7864:
7840:
7835:
7834:
7829:
7824:
7823:
7820:
7812:
7809:
7808:
7780:
7777:
7776:
7731:
7718:
7709:
7703:
7698:
7697:
7692:
7687:
7686:
7684:
7653:
7646:
7641:
7640:
7635:
7630:
7629:
7627:
7596:
7590:
7573:
7571:
7570:
7562:
7558:
7556:
7525:
7516:
7502:
7500:
7499:
7488:
7487:
7485:
7477:
7474:
7473:
7453:
7451:
7448:
7447:
7431:
7429:
7426:
7425:
7409:
7406:
7405:
7326:
7318:
7301:
7293:
7285:
7277:
7263:
7255:
7241:
7230:
7216:
7214:
7213:
7202:
7200:
7197:
7196:
7170:
7162:
7151:
7137:
7135:
7134:
7132:
7129:
7128:
7104:
7102:
7101:
7090:
7076:
7074:
7073:
7071:
7068:
7067:
7040:
7032:
7019:
7014:
7004:
6993:
6979:
6977:
6976:
6962:
6960:
6959:
6951:
6938:
6924:
6922:
6921:
6913:
6911:
6908:
6907:
6885:
6880:
6879:
6874:
6869:
6861:
6853:
6851:
6848:
6847:
6846:). Notice that
6816:
6814:
6813:
6805:
6797:
6786:
6772:
6770:
6769:
6761:
6732:
6730:
6729:
6721:
6710:
6687:
6685:
6684:
6676:
6670:
6666:
6655:
6647:
6643:
6638:
6627:
6613:
6611:
6610:
6608:
6605:
6604:
6580:
6578:
6577:
6569:
6563:
6559:
6548:
6540:
6526:
6525:
6505:
6503:
6502:
6494:
6485:
6481:
6470:
6459:
6458:
6444:
6442:
6441:
6427:
6413:
6397:
6392:
6384:
6367:
6365:
6364:
6356:
6348:
6346:
6343:
6342:
6326:
6317:
6313:
6307:
6290:
6288:
6287:
6285:
6282:
6281:
6265:
6254:
6252:
6249:
6248:
6227:
6225:
6217:
6215:
6212:
6211:
6195:
6193:
6190:
6189:
6183:
6143:
6138:
6125:
6117:
6114:
6113:
6088:
6084:
6081:
6079:
6076:
6075:
6048:
6040:
6037:
6036:
6020:
6017:
6016:
6000:
5997:
5996:
5975:
5971:
5969:
5966:
5965:
5943:
5942:
5928:
5927:
5925:
5922:
5921:
5904:
5900:
5898:
5895:
5894:
5867:
5859:
5856:
5854:
5851:
5850:
5829:
5825:
5823:
5820:
5819:
5774:
5770:
5742:
5738:
5727:
5723:
5718:
5710:
5707:
5706:
5648:
5644:
5640:
5630:
5626:
5625:
5623:
5622:
5618:
5609:
5605:
5603:
5600:
5599:
5573:
5565:
5563:
5561:
5558:
5557:
5537:
5533:
5529:
5519:
5515:
5514:
5512:
5510:
5507:
5506:
5458:
5454:
5450:
5440:
5436:
5435:
5433:
5422:
5418:
5413:
5404:
5390:
5382:
5380:
5376:
5375:
5364:
5360:
5355:
5343:
5339:
5335:
5325:
5321:
5320:
5318:
5316:
5313:
5312:
5273:
5265:
5263:
5252:
5248:
5243:
5227:
5219:
5217:
5215:
5212:
5211:
5169:
5161:
5158:
5157:
5111:
5107:
5102:
5074:
5060:
5052:
5050:
5046:
5045:
5043:
5025:
5021:
5017:
5007:
5003:
5002:
5000:
4999:
4995:
4984:
4980:
4974:
4970:
4968:
4966:
4963:
4962:
4939:
4935:
4930:
4918:
4906:
4902:
4897:
4893:
4892:
4873:
4869:
4857:
4856:
4843:
4841:
4837:
4833:
4820:
4812:
4810:
4801:
4789:
4785:
4780:
4776:
4775:
4763:
4759:
4755:
4745:
4741:
4740:
4738:
4736:
4733:
4732:
4714:
4710:
4705:
4693:
4682:
4681:
4680:
4663:
4662:
4649:
4641:
4639:
4630:
4619:
4618:
4617:
4605:
4601:
4597:
4587:
4583:
4582:
4580:
4578:
4575:
4574:
4556:
4552:
4547:
4535:
4524:
4523:
4522:
4505:
4504:
4502:
4499:
4498:
4480:
4477:
4476:
4475:as function of
4460:
4457:
4456:
4408:
4407:
4394:
4386:
4384:
4375:
4364:
4363:
4362:
4350:
4346:
4342:
4332:
4328:
4327:
4325:
4311:
4310:
4308:
4305:
4304:
4264:
4263:
4250:
4242:
4240:
4226:
4225:
4223:
4220:
4219:
4198:
4195:
4194:
4178:
4175:
4174:
4127:
4123:
4111:
4110:
4097:
4095:
4078:
4077:
4075:
4072:
4071:
4025:
4021:
4016:
4002:
4001:
3999:
3996:
3995:
3969:
3968:
3962:
3958:
3944:
3943:
3932:
3931:
3925:
3921:
3918:
3917:
3911:
3910:
3900:
3899:
3895:
3880:
3879:
3873:
3872:
3861:
3860:
3813:
3812:
3809:
3808:
3797:
3796:
3749:
3748:
3741:
3740:
3730:
3729:
3723:
3722:
3701:
3700:
3675:
3674:
3673:
3669:
3661:
3650:
3648:
3647:
3639:
3634:
3626:
3623:
3622:
3587:
3583:
3578:
3566:
3555:
3554:
3553:
3536:
3535:
3533:
3530:
3529:
3508:
3505:
3504:
3488:
3485:
3484:
3460:
3458:
3457:
3434:
3432:
3431:
3418:
3414:
3409:
3405:
3401:
3387:
3385:
3384:
3368:
3367:
3356:
3355:
3338:
3337:
3333:
3329:
3315:
3313:
3312:
3301:
3290:
3289:
3288:
3271:
3270:
3269:
3265:
3263:
3260:
3259:
3250:Vector calculus
3233:
3232:
3221:
3219:
3218:
3202:
3201:
3190:
3189:
3172:
3171:
3167:
3163:
3149:
3147:
3146:
3135:
3124:
3123:
3122:
3105:
3104:
3103:
3099:
3092:
3081:
3079:
3078:
3075:
3074:
3063:
3061:
3060:
3049:
3048:
3031:
3029:
3028:
3017:
3016:
3009:
2998:
2996:
2995:
2992:
2991:
2980:
2978:
2977:
2955:
2953:
2952:
2929:
2927:
2926:
2916:
2912:
2902:
2897:
2893:
2891:
2888:
2887:
2869:
2867:
2864:
2863:
2847:
2846:
2835:
2833:
2832:
2827:
2807:
2805:
2804:
2799:
2783:
2772:
2770:
2769:
2766:
2765:
2754:
2752:
2751:
2746:
2726:
2724:
2723:
2718:
2705:
2694:
2692:
2691:
2687:
2685:
2682:
2681:
2663:
2661:
2658:
2657:
2632:
2630:
2629:
2615:
2613:
2612:
2607:
2604:
2603:
2575:
2573:
2572:
2558:
2556:
2555:
2550:
2547:
2546:
2520:
2519:
2515:
2513:
2510:
2509:
2483:
2482:
2478:
2476:
2473:
2472:
2443:
2441:
2440:
2436:
2434:
2431:
2430:
2401:
2399:
2396:
2395:
2377:
2369:
2361:
2347:
2345:
2344:
2336:
2322:
2320:
2319:
2305:
2303:
2302:
2283:
2281:
2280:
2272:
2271:
2267:
2257:
2252:
2238:
2236:
2235:
2233:
2230:
2229:
2207:
2205:
2204:
2196:
2188:
2186:
2183:
2182:
2172:
2144:
2141:
2140:
2124:
2121:
2120:
2104:
2101:
2100:
2080:
2077:
2076:
2056:
2053:
2052:
2036:
2033:
2032:
2005:
2002:
2001:
1979:
1976:
1975:
1953:
1950:
1949:
1918:
1915:
1914:
1892:
1889:
1888:
1870:
1852:
1803:
1800:
1799:
1773:
1770:
1769:
1762:
1755:
1748:
1734:
1732:
1725:
1692:
1688:
1679:
1675:
1661:
1658:
1657:
1639:
1636:
1635:
1597:
1596:
1588:
1584:
1579:
1562:
1560:
1559:
1557:
1554:
1553:
1525:
1522:
1521:
1495:
1494:
1492:
1489:
1488:
1469:
1464:
1463:
1454:
1449:
1448:
1440:
1438:
1435:
1434:
1416:
1414:
1411:
1410:
1386:
1385:
1377:
1373:
1366:
1362:
1356:
1352:
1348:
1346:
1337:
1326:
1324:
1323:
1322:
1316:
1312:
1310:
1307:
1306:
1285:
1284:
1276:
1272:
1265:
1261:
1255:
1251:
1244:
1242:
1233:
1222:
1220:
1219:
1218:
1212:
1208:
1206:
1203:
1202:
1179:
1174:
1173:
1171:
1168:
1167:
1150:
1145:
1144:
1142:
1139:
1138:
1121:
1117:
1115:
1112:
1111:
1094:
1090:
1088:
1085:
1084:
1047:two body system
1043:
1037:
1013:Albert Einstein
986:
983:
982:
963:
959:
957:
954:
953:
934:
930:
928:
925:
924:
903:
900:
899:
881:
878:
877:
853:
849:
842:
838:
832:
828:
827:
825:
814:
811:
810:
778:
771:
760:
753:
746:
714:
712:
709:
708:
692:
690:
687:
686:
668:
665:
664:
646:
644:
641:
640:
615:
611:
607:
601:
595:
591:
590:
588:
577:
566:
564:
561:
560:
523:
495:
492:
491:
450:
445:
428:
425:
424:
394:
391:
390:
368:
365:
364:
359:semi-major axis
342:
339:
338:
335:is the distance
320:
317:
316:
272:
262:
258:
245:
243:
226:
223:
222:
173:Johannes Kepler
169:
133:
99:, known as the
57:Johannes Kepler
53:Keplerian orbit
26:
17:
12:
11:
5:
18364:
18354:
18353:
18348:
18333:
18332:
18320:
18308:
18296:
18284:
18261:
18260:
18258:
18257:
18255:List of orbits
18246:
18243:
18242:
18240:
18239:
18234:
18229:
18224:
18219:
18214:
18209:
18207:Orbit equation
18204:
18196:
18191:
18186:
18181:
18176:
18171:
18166:
18161:
18156:
18151:
18145:
18143:
18135:
18134:
18132:
18131:
18126:
18121:
18116:
18111:
18106:
18101:
18096:
18091:
18086:
18081:
18079:Gravity assist
18076:
18074:Delta-v budget
18071:
18066:
18061:
18055:
18053:
18047:
18046:
18043:
18042:
18040:
18039:
18031:
18025:
18016:
18007:
18005:Orbital period
17997:
17995:
17991:
17990:
17988:
17987:
17985:True longitude
17978:
17976:Mean longitude
17969:
17960:
17943:
17933:
17931:
17927:
17926:
17924:
17923:
17914:
17905:
17896:
17886:
17884:
17880:
17879:
17877:
17876:
17863:
17854:
17845:
17835:
17833:
17831:
17830:
17827:
17823:
17817:
17811:
17810:
17807:
17806:
17804:
17803:
17802:
17801:
17793:
17792:
17791:
17786:
17781:
17780:
17779:
17766:
17761:
17760:
17759:
17754:
17749:
17744:
17736:
17735:
17734:
17732:Areostationary
17729:
17724:
17715:
17713:
17707:
17706:
17704:
17703:
17701:Very low Earth
17698:
17693:
17688:
17683:
17678:
17673:
17668:
17663:
17658:
17653:
17648:
17643:
17642:
17641:
17636:
17629:Geosynchronous
17625:
17623:
17617:
17616:
17614:
17613:
17611:Transfer orbit
17608:
17607:
17606:
17601:
17591:
17586:
17581:
17576:
17571:
17569:Lagrange point
17566:
17561:
17552:
17547:
17542:
17537:
17528:
17523:
17518:
17512:
17510:
17503:
17497:
17496:
17491:Gravitational
17488:
17487:
17480:
17473:
17465:
17456:
17455:
17453:
17452:
17447:
17440:
17432:
17427:
17419:
17413:
17407:
17398:
17396:
17392:
17391:
17389:
17388:
17380:
17372:
17364:
17356:
17348:
17344:De Stella Nova
17340:
17331:
17329:
17325:
17324:
17322:
17321:
17316:
17311:
17306:
17301:
17296:
17291:
17286:
17280:
17278:
17274:
17273:
17266:
17265:
17258:
17251:
17243:
17237:
17236:
17228:
17227:External links
17225:
17224:
17223:
17211:
17205:
17188:
17183:
17180:
17177:
17176:
17159:"NASA website"
17150:
17141:
17131:
17130:
17128:
17125:
17124:
17123:
17121:Orbit modeling
17118:
17113:
17108:
17103:
17101:Elliptic orbit
17098:
17093:
17091:Kepler problem
17088:
17081:
17078:
17065:
17061:
17057:
17053:
17049:
17028:
17014:
17013:
17004:
17002:
16987:
16983:
16975:
16971:
16962:
16958:
16954:
16948:
16944:
16937:
16932:
16928:
16924:
16919:
16915:
16911:
16905:
16901:
16854:
16850:
16843:
16840:
16836:
16802:
16799:
16796:
16790:
16786:
16779:
16775:
16771:
16767:
16746:
16743:
16740:
16734:
16730:
16723:
16719:
16715:
16711:
16707:
16700:
16696:
16690:
16686:
16678:
16675:
16672:
16669:
16666:
16663:
16657:
16653:
16646:
16642:
16638:
16634:
16597:
16574:
16570:
16563:
16559:
16535:
16531:
16504:
16500:
16476:
16456:
16436:
16412:
16408:
16403:
16397:
16394:
16374:
16371:
16368:
16362:
16358:
16351:
16347:
16343:
16339:
16325:
16324:
16315:
16313:
16302:
16299:
16296:
16290:
16286:
16279:
16275:
16271:
16267:
16263:
16257:
16253:
16206:
16203:
16197:
16191:
16188:
16163:
16159:
16155:
16150:
16146:
16142:
16139:
16119:
16116:
16113:
16110:
16107:
16087:
16083:
16079:
16075:
16071:
16056:Main article:
16053:
16050:
16038:
16033:
16028:
16023:
16018:
16013:
16008:
15995:
15994:
15985:
15983:
15969:
15966:
15960:
15955:
15951:
15921:
15916:
15911:
15906:
15902:
15898:
15895:
15892:
15886:
15880:
15874:
15871:
15865:
15860:
15856:
15852:
15847:
15843:
15822:
15819:
15814:
15810:
15784:
15783:
15774:
15772:
15758:
15754:
15747:
15744:
15741:
15738:
15732:
15728:
15721:
15718:
15715:
15712:
15706:
15702:
15683:
15682:
15673:
15671:
15657:
15653:
15646:
15643:
15640:
15637:
15631:
15627:
15620:
15617:
15614:
15611:
15605:
15601:
15569:
15565:
15541:
15535:
15531:
15524:
15518:
15514:
15507:
15484:
15464:
15459:
15454:
15449:
15444:
15439:
15434:
15414:
15410:
15406:
15402:
15398:
15366:
15346:
15340:
15336:
15329:
15323:
15319:
15312:
15275:
15271:
15248:
15244:
15219:
15206:
15205:
15196:
15194:
15180:
15177:
15171:
15166:
15162:
15143:
15142:
15133:
15131:
15116:
15112:
15106:
15102:
15096:
15093:
15090:
15087:
15084:
15069:
15068:
15059:
15057:
15046:
15043:
15036:
15032:
15026:
15022:
15016:
15013:
15010:
15007:
15004:
14979:
14959:
14956:
14953:
14940:
14939:
14930:
14928:
14915:
14910:
14905:
14900:
14896:
14892:
14889:
14886:
14879:
14876:
14831:
14828:
14823:
14819:
14798:
14795:
14792:
14779:
14778:
14769:
14767:
14753:
14749:
14740:
14736:
14732:
14726:
14722:
14713:
14709:
14705:
14700:
14695:
14678:
14677:
14668:
14666:
14652:
14648:
14641:
14638:
14633:
14628:
14601:
14595:
14591:
14584:
14578:
14574:
14567:
14544:
14539:
14534:
14529:
14524:
14519:
14514:
14501:
14500:
14491:
14489:
14477:
14473:
14467:
14463:
14444:
14443:
14434:
14432:
14417:
14413:
14407:
14403:
14395:
14392:
14389:
14386:
14380:
14376:
14347:
14343:
14339:
14335:
14331:
14309:
14306:
14294:
14293:
14284:
14282:
14271:
14268:
14265:
14262:
14259:
14256:
14253:
14250:
14244:
14241:
14235:
14232:
14229:
14226:
14221:
14217:
14213:
14210:
14207:
14204:
14199:
14195:
14191:
14186:
14182:
14166:
14165:
14156:
14154:
14143:
14140:
14137:
14134:
14128:
14125:
14119:
14116:
14113:
14110:
14107:
14102:
14098:
14094:
14091:
14088:
14085:
14080:
14076:
14072:
14067:
14063:
14022:
14019:
13993:
13990:
13984:
13978:
13975:
13959:
13958:
13949:
13947:
13933:
13930:
13926:
13882:
13881:
13872:
13870:
13856:
13853:
13849:
13844:
13817:
13814:
13809:
13804:
13798:
13791:
13787:
13781:
13776:
13769:
13765:
13713:
13710:
13705:
13686:
13685:
13676:
13674:
13660:
13657:
13651:
13648:
13645:
13642:
13639:
13636:
13598:
13592:
13589:
13586:
13581:
13578:
13575:
13569:
13565:
13562:
13557:
13554:
13549:
13546:
13543:
13538:
13535:
13531:
13507:
13504:
13480:
13476:
13473:
13470:
13467:
13464:
13461:
13457:
13453:
13449:
13444:
13438:
13435:
13430:
13426:
13422:
13417:
13414:
13410:
13406:
13403:
13400:
13396:
13390:
13387:
13382:
13378:
13374:
13369:
13366:
13362:
13358:
13354:
13330:
13329:
13320:
13318:
13305:
13302:
13297:
13294:
13291:
13284:
13281:
13278:
13273:
13270:
13267:
13260:
13255:
13252:
13247:
13244:
13217:
13214:
13209:
13206:
13184:
13181:
13176:
13173:
13151:
13148:
13143:
13138:
13134:
13130:
13124:
13121:
13118:
13113:
13110:
13107:
13101:
13095:
13092:
13089:
13086:
13083:
13078:
13075:
13072:
13069:
13066:
13060:
13054:
13051:
13048:
13043:
13040:
13037:
13031:
13025:
13022:
13019:
13016:
13013:
13010:
13007:
13004:
13001:
12998:
12995:
12990:
12987:
12984:
12981:
12978:
12975:
12972:
12969:
12966:
12963:
12960:
12954:
12945:
12942:
12939:
12936:
12933:
12930:
12927:
12922:
12919:
12916:
12913:
12910:
12904:
12901:
12893:
12890:
12887:
12884:
12881:
12878:
12875:
12870:
12867:
12864:
12861:
12858:
12852:
12849:
12843:
12837:
12834:
12831:
12828:
12825:
12820:
12817:
12814:
12811:
12808:
12802:
12797:
12794:
12789:
12784:
12780:
12766:
12765:
12756:
12754:
12740:
12737:
12734:
12731:
12728:
12725:
12722:
12717:
12714:
12711:
12708:
12705:
12699:
12694:
12691:
12686:
12683:
12680:
12677:
12659:
12658:
12649:
12647:
12636:
12633:
12630:
12627:
12624:
12621:
12618:
12615:
12612:
12609:
12606:
12603:
12600:
12556:
12552:
12549:
12546:
12543:
12540:
12537:
12533:
12529:
12525:
12521:
12518:
12515:
12512:
12509:
12506:
12502:
12466:
12463:
12460:
12457:
12454:
12451:
12448:
12426:
12423:
12419:
12415:
12412:
12409:
12405:
12380:
12377:
12374:
12371:
12368:
12348:
12345:
12342:
12339:
12336:
12333:
12330:
12317:
12316:
12307:
12305:
12294:
12291:
12288:
12285:
12282:
12278:
12272:
12269:
12264:
12261:
12258:
12253:
12250:
12247:
12242:
12237:
12234:
12229:
12226:
12223:
12218:
12215:
12212:
12206:
12202:
12199:
12196:
12193:
12190:
12187:
12172:
12171:
12162:
12160:
12149:
12146:
12143:
12140:
12137:
12133:
12127:
12124:
12119:
12116:
12113:
12108:
12105:
12102:
12097:
12092:
12089:
12084:
12081:
12078:
12073:
12070:
12067:
12061:
12057:
12054:
12051:
12048:
12045:
12042:
12024:
12023:
12014:
12012:
11999:
11996:
11991:
11988:
11985:
11978:
11975:
11972:
11967:
11964:
11961:
11954:
11949:
11946:
11941:
11938:
11912:
11905:
11902:
11896:
11893:
11890:
11884:
11881:
11875:
11872:
11868:
11846:
11839:
11836:
11830:
11827:
11824:
11818:
11815:
11809:
11806:
11802:
11777:
11774:
11771:
11768:
11765:
11762:
11759:
11756:
11753:
11733:
11730:
11727:
11724:
11721:
11718:
11715:
11712:
11709:
11681:
11678:
11673:
11668:
11664:
11660:
11654:
11651:
11648:
11643:
11640:
11637:
11631:
11625:
11622:
11619:
11616:
11613:
11608:
11605:
11602:
11599:
11596:
11590:
11584:
11581:
11578:
11573:
11570:
11567:
11561:
11555:
11552:
11549:
11546:
11543:
11540:
11537:
11534:
11531:
11528:
11525:
11522:
11517:
11514:
11511:
11508:
11505:
11502:
11499:
11496:
11493:
11490:
11487:
11484:
11478:
11469:
11466:
11463:
11460:
11457:
11454:
11449:
11446:
11443:
11440:
11437:
11431:
11428:
11420:
11417:
11414:
11411:
11408:
11405:
11400:
11397:
11394:
11391:
11388:
11382:
11379:
11373:
11367:
11364:
11361:
11358:
11355:
11350:
11347:
11344:
11341:
11338:
11332:
11327:
11324:
11319:
11314:
11310:
11290:
11289:
11280:
11278:
11264:
11261:
11258:
11255:
11252:
11249:
11244:
11241:
11238:
11235:
11232:
11226:
11221:
11218:
11213:
11210:
11207:
11204:
11186:
11185:
11176:
11174:
11163:
11160:
11157:
11154:
11151:
11148:
11145:
11142:
11139:
11136:
11133:
11130:
11102:elliptic orbit
11097:
11094:
11081:
11077:
11074:
11071:
11068:
11065:
11062:
11058:
11054:
11050:
11046:
11043:
11040:
11037:
11034:
11031:
11027:
10975:
10962:
10961:
10952:
10950:
10939:
10936:
10933:
10930:
10927:
10924:
10921:
10918:
10915:
10909:
10906:
10900:
10897:
10894:
10891:
10873:
10872:
10863:
10861:
10850:
10847:
10844:
10841:
10838:
10835:
10832:
10829:
10826:
10823:
10820:
10817:
10814:
10811:
10808:
10805:
10802:
10780:
10779:
10770:
10768:
10754:
10751:
10745:
10742:
10739:
10736:
10733:
10730:
10727:
10724:
10721:
10718:
10715:
10712:
10709:
10706:
10703:
10697:
10694:
10688:
10685:
10682:
10676:
10673:
10667:
10664:
10661:
10658:
10636:
10635:
10626:
10624:
10610:
10607:
10601:
10598:
10595:
10592:
10589:
10586:
10583:
10577:
10574:
10556:
10555:
10546:
10544:
10530:
10527:
10521:
10518:
10515:
10512:
10509:
10506:
10503:
10500:
10494:
10491:
10470:
10469:
10460:
10458:
10447:
10444:
10441:
10438:
10435:
10432:
10429:
10414:
10413:
10404:
10402:
10391:
10388:
10385:
10382:
10379:
10376:
10373:
10370:
10367:
10364:
10361:
10339:
10338:
10329:
10327:
10316:
10313:
10310:
10307:
10304:
10301:
10298:
10295:
10292:
10286:
10283:
10277:
10274:
10271:
10268:
10250:
10249:
10240:
10238:
10225:
10222:
10219:
10214:
10211:
10179:
10176:
10173:
10160:
10159:
10150:
10148:
10137:
10134:
10131:
10128:
10125:
10122:
10119:
10116:
10113:
10110:
10107:
10104:
10101:
10098:
10095:
10092:
10089:
10067:
10066:
10057:
10055:
10041:
10038:
10032:
10029:
10026:
10023:
10020:
10017:
10014:
10011:
10008:
10005:
10002:
9999:
9996:
9993:
9990:
9984:
9981:
9975:
9972:
9969:
9963:
9960:
9954:
9951:
9948:
9945:
9923:
9922:
9913:
9911:
9897:
9894:
9888:
9885:
9882:
9879:
9876:
9873:
9870:
9864:
9861:
9843:
9842:
9833:
9831:
9817:
9814:
9808:
9805:
9802:
9799:
9796:
9793:
9790:
9787:
9781:
9778:
9757:
9756:
9747:
9745:
9734:
9731:
9728:
9725:
9722:
9719:
9716:
9701:
9700:
9691:
9689:
9678:
9675:
9672:
9669:
9666:
9663:
9660:
9620:
9607:
9606:
9597:
9595:
9584:
9581:
9578:
9575:
9572:
9569:
9566:
9563:
9560:
9557:
9551:
9548:
9542:
9537:
9534:
9529:
9523:
9520:
9514:
9511:
9508:
9503:
9499:
9471:
9467:
9453:
9452:
9443:
9441:
9430:
9427:
9424:
9421:
9415:
9412:
9406:
9403:
9400:
9397:
9394:
9389:
9386:
9381:
9375:
9372:
9366:
9361:
9357:
9329:
9324:
9320:
9279:
9276:
9273:
9270:
9267:
9264:
9243:
9237:
9234:
9229:
9225:
9221:
9216:
9213:
9209:
9205:
9202:
9199:
9195:
9189:
9186:
9181:
9177:
9173:
9168:
9165:
9161:
9157:
9137:
9117:
9110:
9107:
9104:
9100:
9077:
9070:
9067:
9064:
9060:
9037:
9034:
9031:
8987:
8986:
8977:
8975:
8962:
8959:
8954:
8950:
8944:
8941:
8938:
8932:
8929:
8924:
8920:
8915:
8910:
8907:
8892:
8891:
8882:
8880:
8866:
8863:
8858:
8854:
8849:
8844:
8841:
8812:
8809:
8806:
8781:
8778:
8751:
8748:
8745:
8732:
8731:
8722:
8720:
8705:
8701:
8697:
8694:
8689:
8686:
8683:
8675:
8671:
8667:
8664:
8660:
8655:
8652:
8637:
8636:
8627:
8625:
8609:
8605:
8601:
8598:
8594:
8589:
8586:
8557:
8554:
8551:
8548:
8545:
8542:
8515:
8512:
8509:
8497:
8494:
8480:
8476:
8472:
8448:
8445:
8439:
8435:
8397:
8368:
8346:
8324:
8320:
8316:
8312:
8306:
8302:
8295:
8291:
8287:
8283:
8258:
8254:
8248:
8243:
8239:
8235:
8231:
8227:
8223:
8220:
8216:
8209:
8204:
8200:
8194:
8189:
8184:
8180:
8176:
8169:
8165:
8161:
8156:
8151:
8147:
8141:
8137:
8127:
8122:
8118:
8112:
8108:
8078:
8056:
8034:
8013:
8001:
7998:
7985:
7968:
7967:
7958:
7956:
7942:
7939:
7936:
7933:
7930:
7927:
7924:
7920:
7915:
7912:
7884:
7881:
7875:
7872:
7849:
7843:
7838:
7832:
7827:
7819:
7816:
7796:
7793:
7790:
7787:
7784:
7762:
7756:
7753:
7750:
7747:
7744:
7741:
7738:
7734:
7730:
7727:
7724:
7721:
7716:
7712:
7706:
7701:
7695:
7690:
7683:
7677:
7674:
7671:
7668:
7665:
7662:
7659:
7656:
7649:
7644:
7638:
7633:
7626:
7620:
7617:
7614:
7611:
7608:
7605:
7602:
7599:
7593:
7589:
7586:
7580:
7576:
7569:
7565:
7561:
7555:
7549:
7546:
7543:
7540:
7537:
7534:
7531:
7528:
7523:
7519:
7515:
7509:
7505:
7498:
7495:
7491:
7484:
7481:
7468:. Solving for
7456:
7434:
7413:
7393:
7390:
7387:
7384:
7381:
7378:
7375:
7372:
7369:
7366:
7363:
7360:
7357:
7354:
7351:
7348:
7345:
7342:
7339:
7336:
7333:
7329:
7325:
7321:
7317:
7314:
7311:
7308:
7304:
7300:
7296:
7292:
7288:
7284:
7280:
7276:
7273:
7270:
7266:
7262:
7258:
7254:
7251:
7248:
7244:
7240:
7237:
7233:
7229:
7223:
7219:
7212:
7209:
7205:
7173:
7169:
7165:
7161:
7158:
7154:
7150:
7144:
7140:
7111:
7107:
7100:
7097:
7093:
7089:
7083:
7079:
7053:
7050:
7047:
7043:
7039:
7035:
7031:
7025:
7022:
7018:
7011:
7008:
7003:
7000:
6996:
6992:
6986:
6982:
6975:
6969:
6965:
6958:
6954:
6950:
6945:
6942:
6937:
6931:
6927:
6920:
6916:
6896:
6893:
6888:
6883:
6877:
6872:
6868:
6864:
6860:
6856:
6829:
6823:
6819:
6812:
6808:
6804:
6800:
6796:
6793:
6789:
6785:
6779:
6775:
6768:
6764:
6760:
6757:
6754:
6751:
6748:
6745:
6739:
6735:
6728:
6724:
6720:
6717:
6713:
6709:
6706:
6703:
6700:
6694:
6690:
6683:
6679:
6673:
6669:
6665:
6662:
6658:
6650:
6646:
6642:
6637:
6634:
6630:
6626:
6620:
6616:
6587:
6583:
6576:
6572:
6566:
6562:
6558:
6555:
6551:
6547:
6543:
6539:
6533:
6530:
6524:
6521:
6518:
6512:
6508:
6501:
6497:
6493:
6488:
6484:
6480:
6477:
6473:
6466:
6463:
6457:
6451:
6447:
6440:
6437:
6434:
6430:
6426:
6423:
6420:
6416:
6412:
6409:
6403:
6400:
6396:
6391:
6387:
6383:
6380:
6374:
6370:
6363:
6359:
6355:
6351:
6329:
6320:
6316:
6312:
6306:
6303:
6297:
6293:
6268:
6264:
6261:
6257:
6234:
6230:
6224:
6220:
6198:
6182:
6179:
6164:
6161:
6158:
6155:
6152:
6149:
6146:
6142:
6137:
6132:
6129:
6124:
6121:
6096:
6091:
6087:
6056:
6053:
6047:
6044:
6024:
6004:
5978:
5974:
5950:
5947:
5941:
5935:
5932:
5907:
5903:
5880:
5873:
5870:
5865:
5862:
5832:
5828:
5810:
5809:
5800:
5798:
5786:
5782:
5777:
5773:
5769:
5766:
5763:
5760:
5757:
5754:
5751:
5748:
5745:
5741:
5737:
5730:
5726:
5722:
5717:
5714:
5696:
5695:
5686:
5684:
5673:
5670:
5666:
5662:
5659:
5651:
5647:
5643:
5638:
5633:
5629:
5621:
5617:
5612:
5608:
5579:
5576:
5571:
5568:
5540:
5536:
5532:
5527:
5522:
5518:
5489:
5488:
5479:
5477:
5461:
5457:
5453:
5448:
5443:
5439:
5432:
5425:
5421:
5417:
5412:
5407:
5402:
5396:
5393:
5388:
5385:
5379:
5374:
5367:
5363:
5359:
5354:
5346:
5342:
5338:
5333:
5328:
5324:
5305:
5304:
5295:
5293:
5279:
5276:
5271:
5268:
5262:
5255:
5251:
5247:
5242:
5239:
5233:
5230:
5225:
5222:
5201:
5200:
5191:
5189:
5176:
5173:
5168:
5165:
5141:
5140:
5131:
5129:
5114:
5110:
5106:
5101:
5098:
5094:
5090:
5087:
5082:
5077:
5072:
5066:
5063:
5058:
5055:
5049:
5042:
5039:
5036:
5028:
5024:
5020:
5015:
5010:
5006:
4998:
4994:
4987:
4983:
4977:
4973:
4942:
4938:
4934:
4929:
4926:
4921:
4916:
4909:
4905:
4901:
4896:
4891:
4888:
4884:
4876:
4872:
4864:
4861:
4855:
4852:
4849:
4846:
4840:
4836:
4832:
4826:
4823:
4818:
4815:
4809:
4804:
4799:
4792:
4788:
4784:
4779:
4774:
4766:
4762:
4758:
4753:
4748:
4744:
4717:
4713:
4709:
4704:
4701:
4696:
4689:
4686:
4679:
4676:
4670:
4667:
4661:
4655:
4652:
4647:
4644:
4638:
4633:
4626:
4623:
4616:
4608:
4604:
4600:
4595:
4590:
4586:
4559:
4555:
4551:
4546:
4543:
4538:
4531:
4528:
4521:
4518:
4512:
4509:
4487:
4484:
4464:
4441:
4440:
4431:
4429:
4415:
4412:
4406:
4400:
4397:
4392:
4389:
4383:
4378:
4371:
4368:
4361:
4353:
4349:
4345:
4340:
4335:
4331:
4324:
4318:
4315:
4297:
4296:
4287:
4285:
4271:
4268:
4262:
4256:
4253:
4248:
4245:
4239:
4233:
4230:
4202:
4182:
4157:
4156:
4147:
4145:
4130:
4126:
4118:
4115:
4109:
4106:
4103:
4100:
4094:
4091:
4085:
4082:
4055:
4054:
4045:
4043:
4028:
4024:
4020:
4015:
4009:
4006:
3976:
3973:
3965:
3961:
3957:
3953:
3948:
3939:
3936:
3928:
3924:
3920:
3919:
3916:
3913:
3912:
3909:
3906:
3905:
3903:
3898:
3894:
3890:
3884:
3878:
3875:
3874:
3868:
3865:
3859:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3820:
3817:
3811:
3810:
3804:
3801:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3756:
3753:
3747:
3746:
3744:
3739:
3734:
3728:
3725:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3680:
3678:
3672:
3668:
3664:
3657:
3653:
3646:
3642:
3637:
3633:
3630:
3617:
3616:
3607:
3605:
3590:
3586:
3582:
3577:
3574:
3569:
3562:
3559:
3552:
3549:
3543:
3540:
3512:
3492:
3467:
3463:
3456:
3453:
3450:
3447:
3441:
3437:
3429:
3421:
3417:
3413:
3408:
3404:
3400:
3394:
3390:
3382:
3375:
3372:
3363:
3360:
3354:
3351:
3345:
3342:
3336:
3332:
3328:
3322:
3318:
3310:
3304:
3297:
3294:
3287:
3284:
3278:
3275:
3268:
3228:
3224:
3216:
3209:
3206:
3197:
3194:
3188:
3185:
3179:
3176:
3170:
3166:
3162:
3156:
3152:
3144:
3138:
3131:
3128:
3121:
3118:
3112:
3109:
3102:
3098:
3095:
3093:
3088:
3084:
3077:
3076:
3070:
3066:
3056:
3053:
3047:
3044:
3038:
3034:
3024:
3021:
3015:
3012:
3010:
3005:
3001:
2994:
2993:
2987:
2983:
2976:
2973:
2969:
2962:
2958:
2951:
2948:
2945:
2942:
2936:
2932:
2925:
2922:
2919:
2915:
2911:
2908:
2905:
2903:
2900:
2896:
2895:
2872:
2842:
2838:
2830:
2826:
2823:
2820:
2814:
2810:
2802:
2798:
2795:
2792:
2789:
2786:
2784:
2779:
2775:
2768:
2767:
2761:
2757:
2749:
2745:
2742:
2739:
2733:
2729:
2721:
2717:
2714:
2711:
2708:
2706:
2701:
2697:
2690:
2689:
2666:
2645:
2639:
2635:
2628:
2622:
2618:
2611:
2588:
2582:
2578:
2571:
2565:
2561:
2554:
2533:
2527:
2524:
2518:
2496:
2490:
2487:
2481:
2460:
2456:
2450:
2446:
2439:
2404:
2380:
2376:
2372:
2368:
2364:
2360:
2354:
2350:
2343:
2339:
2335:
2329:
2325:
2318:
2312:
2308:
2301:
2297:
2290:
2286:
2279:
2275:
2270:
2263:
2260:
2256:
2251:
2245:
2241:
2214:
2210:
2203:
2199:
2195:
2191:
2171:
2162:
2148:
2128:
2108:
2097:
2096:
2084:
2060:
2040:
2022:
2021:
2009:
1995:
1983:
1969:
1957:
1935:
1934:
1922:
1908:
1896:
1885:Semimajor axis
1866:Main article:
1851:
1848:
1820:
1819:
1807:
1787:
1783:
1780:
1777:
1760:
1753:
1746:
1730:
1723:
1717:
1700:
1695:
1691:
1687:
1682:
1678:
1674:
1671:
1668:
1665:
1643:
1630:
1629:
1620:
1618:
1604:
1601:
1591:
1587:
1583:
1578:
1575:
1569:
1565:
1529:
1502:
1499:
1472:
1467:
1462:
1457:
1452:
1447:
1443:
1419:
1393:
1390:
1380:
1376:
1369:
1365:
1359:
1355:
1351:
1345:
1340:
1333:
1329:
1319:
1315:
1292:
1289:
1279:
1275:
1268:
1264:
1258:
1254:
1250:
1247:
1241:
1236:
1229:
1225:
1215:
1211:
1182:
1177:
1153:
1148:
1124:
1120:
1097:
1093:
1058:
1057:
1054:
1036:
1033:
1003:
1002:
990:
980:
966:
962:
951:
937:
933:
922:
907:
897:
885:
856:
852:
845:
841:
835:
831:
824:
821:
818:
776:
769:
758:
751:
744:
730:
729:
717:
695:
684:
672:
662:
649:
618:
614:
610:
604:
598:
594:
587:
584:
580:
576:
573:
569:
535:laws of motion
522:
519:
510:is called the
499:
474:
471:
468:
465:
462:
459:
456:
453:
449:
444:
441:
438:
435:
432:
419:
418:
398:
388:
372:
362:
346:
336:
324:
296:
293:
290:
287:
284:
281:
278:
275:
270:
265:
261:
257:
254:
251:
248:
242:
239:
236:
233:
230:
217:conic sections
168:
165:
132:
129:
101:Kepler problem
15:
9:
6:
4:
3:
2:
18363:
18352:
18349:
18347:
18344:
18343:
18341:
18331:
18321:
18319:
18309:
18307:
18297:
18295:
18290:
18285:
18283:
18273:
18272:
18269:
18256:
18248:
18247:
18244:
18238:
18235:
18233:
18230:
18228:
18225:
18223:
18220:
18218:
18215:
18213:
18210:
18208:
18205:
18203:
18202:-body problem
18201:
18197:
18195:
18192:
18190:
18187:
18185:
18182:
18180:
18177:
18175:
18172:
18170:
18167:
18165:
18162:
18160:
18157:
18155:
18152:
18150:
18147:
18146:
18144:
18142:
18136:
18130:
18127:
18125:
18122:
18120:
18117:
18115:
18112:
18110:
18107:
18105:
18104:Oberth effect
18102:
18100:
18097:
18095:
18092:
18090:
18087:
18085:
18082:
18080:
18077:
18075:
18072:
18070:
18067:
18065:
18062:
18060:
18057:
18056:
18054:
18052:
18048:
18038:
18030:
18026:
18024:
18023:Orbital speed
18017:
18015:
18008:
18006:
17999:
17998:
17996:
17992:
17986:
17979:
17977:
17970:
17968:
17961:
17959:
17944:
17942:
17935:
17934:
17932:
17928:
17922:
17915:
17913:
17906:
17904:
17897:
17895:
17888:
17887:
17885:
17881:
17875:
17864:
17862:
17855:
17853:
17846:
17844:
17837:
17836:
17834:
17828:
17825:
17824:
17821:
17818:
17816:
17812:
17800:
17797:
17796:
17794:
17790:
17787:
17785:
17782:
17778:
17777:Earth's orbit
17775:
17774:
17773:
17770:
17769:
17767:
17765:
17762:
17758:
17755:
17753:
17750:
17748:
17745:
17743:
17740:
17739:
17737:
17733:
17730:
17728:
17725:
17723:
17720:
17719:
17717:
17716:
17714:
17708:
17702:
17699:
17697:
17694:
17692:
17689:
17687:
17684:
17682:
17679:
17677:
17674:
17672:
17669:
17667:
17664:
17662:
17659:
17657:
17654:
17652:
17649:
17647:
17644:
17640:
17637:
17635:
17634:Geostationary
17632:
17631:
17630:
17627:
17626:
17624:
17622:
17618:
17612:
17609:
17605:
17602:
17600:
17597:
17596:
17595:
17592:
17590:
17587:
17585:
17582:
17580:
17577:
17575:
17572:
17570:
17567:
17565:
17562:
17560:
17556:
17553:
17551:
17548:
17546:
17543:
17541:
17538:
17536:
17532:
17529:
17527:
17524:
17522:
17519:
17517:
17514:
17513:
17511:
17507:
17504:
17502:
17498:
17494:
17486:
17481:
17479:
17474:
17472:
17467:
17466:
17463:
17451:
17448:
17446:
17445:
17441:
17439:
17437:
17433:
17431:
17428:
17425:
17424:
17420:
17417:
17416:Jakob Bartsch
17414:
17411:
17408:
17405:
17404:
17400:
17399:
17397:
17393:
17386:
17385:
17381:
17378:
17377:
17373:
17370:
17369:
17365:
17362:
17361:
17357:
17354:
17353:
17349:
17346:
17345:
17341:
17338:
17337:
17333:
17332:
17330:
17326:
17320:
17317:
17315:
17312:
17310:
17307:
17305:
17302:
17300:
17297:
17295:
17292:
17290:
17287:
17285:
17282:
17281:
17279:
17275:
17271:
17264:
17259:
17257:
17252:
17250:
17245:
17244:
17241:
17234:
17231:
17230:
17220:
17216:
17212:
17208:
17206:0-486-60061-0
17202:
17197:
17196:
17189:
17186:
17185:
17164:
17160:
17154:
17145:
17136:
17132:
17122:
17119:
17117:
17114:
17112:
17109:
17107:
17104:
17102:
17099:
17097:
17094:
17092:
17089:
17087:
17084:
17083:
17077:
17055:
17012:
17005:
17003:
16985:
16981:
16960:
16956:
16952:
16930:
16926:
16922:
16917:
16913:
16903:
16891:
16890:
16887:
16885:
16884:
16879:
16878:
16873:
16872:
16841:
16838:
16826:" defined as
16825:
16820:
16817:
16813:
16797:
16794:
16788:
16777:
16741:
16738:
16732:
16721:
16705:
16698:
16694:
16676:
16673:
16670:
16664:
16661:
16655:
16644:
16622:
16616:
16615:
16609:
16595:
16572:
16561:
16474:
16454:
16434:
16410:
16406:
16395:
16392:
16369:
16366:
16360:
16349:
16323:
16316:
16314:
16297:
16294:
16288:
16277:
16261:
16255:
16240:
16239:
16236:
16233:
16231:
16230:
16225:
16224:
16201:
16195:
16186:
16161:
16157:
16153:
16148:
16144:
16140:
16137:
16117:
16114:
16111:
16108:
16105:
16077:
16059:
16049:
16031:
16021:
16016:
15993:
15986:
15984:
15967:
15964:
15958:
15953:
15949:
15941:
15940:
15937:
15919:
15914:
15904:
15900:
15896:
15893:
15884:
15878:
15872:
15869:
15863:
15858:
15854:
15850:
15845:
15841:
15820:
15817:
15812:
15808:
15799:
15798:
15793:
15792:
15782:
15775:
15773:
15745:
15742:
15739:
15736:
15719:
15716:
15713:
15710:
15689:
15688:
15681:
15674:
15672:
15644:
15641:
15638:
15635:
15618:
15615:
15612:
15609:
15588:
15587:
15584:
15522:
15496:
15482:
15457:
15447:
15442:
15404:
15388:
15387:
15382:
15381:
15364:
15327:
15301:
15299:
15298:
15293:
15292:
15273:
15269:
15246:
15242:
15230:has the same
15217:
15204:
15197:
15195:
15178:
15175:
15169:
15164:
15160:
15152:
15151:
15148:
15141:
15134:
15132:
15114:
15110:
15104:
15100:
15094:
15091:
15088:
15085:
15082:
15075:
15074:
15067:
15060:
15058:
15044:
15041:
15034:
15030:
15024:
15020:
15014:
15011:
15008:
15005:
15002:
14995:
14994:
14991:
14977:
14957:
14954:
14951:
14938:
14931:
14929:
14913:
14908:
14898:
14894:
14890:
14887:
14877:
14874:
14867:
14866:
14863:
14861:
14860:
14855:
14854:
14849:
14848:
14842:
14829:
14826:
14821:
14817:
14796:
14793:
14790:
14777:
14770:
14768:
14738:
14734:
14730:
14711:
14707:
14703:
14698:
14684:
14683:
14676:
14669:
14667:
14639:
14636:
14631:
14617:
14616:
14613:
14582:
14556:
14537:
14527:
14522:
14499:
14492:
14490:
14471:
14465:
14450:
14449:
14442:
14435:
14433:
14415:
14411:
14393:
14390:
14387:
14384:
14378:
14363:
14362:
14359:
14337:
14321:
14320:
14315:
14312:This is the "
14304:
14299:
14292:
14285:
14283:
14266:
14263:
14260:
14257:
14254:
14248:
14242:
14239:
14233:
14230:
14227:
14224:
14219:
14215:
14211:
14208:
14205:
14202:
14197:
14193:
14189:
14184:
14180:
14172:
14171:
14164:
14157:
14155:
14141:
14138:
14135:
14132:
14126:
14123:
14117:
14114:
14111:
14108:
14105:
14100:
14096:
14092:
14089:
14086:
14083:
14078:
14074:
14070:
14065:
14061:
14053:
14052:
14049:
14047:
14046:
14041:
14040:
14017:
13988:
13982:
13973:
13957:
13950:
13948:
13931:
13928:
13924:
13915:
13914:
13911:
13909:
13908:
13903:
13902:
13897:
13896:
13891:
13890:
13880:
13873:
13871:
13854:
13851:
13847:
13842:
13835:
13834:
13831:
13815:
13812:
13807:
13802:
13796:
13789:
13785:
13779:
13774:
13767:
13763:
13749:
13748:
13743:
13742:
13737:
13736:
13731:
13730:
13711:
13708:
13703:
13695:
13694:
13684:
13677:
13675:
13658:
13655:
13649:
13646:
13643:
13640:
13637:
13634:
13627:
13626:
13623:
13621:
13617:
13616:
13610:
13596:
13590:
13587:
13584:
13579:
13576:
13573:
13567:
13563:
13560:
13555:
13552:
13547:
13544:
13541:
13536:
13533:
13529:
13505:
13502:
13478:
13471:
13468:
13465:
13459:
13455:
13447:
13442:
13436:
13433:
13428:
13424:
13420:
13415:
13412:
13408:
13404:
13401:
13398:
13394:
13388:
13385:
13380:
13376:
13372:
13367:
13364:
13360:
13356:
13352:
13343:
13342:
13337:
13328:
13321:
13319:
13303:
13300:
13295:
13292:
13289:
13282:
13279:
13276:
13271:
13268:
13265:
13258:
13253:
13250:
13245:
13242:
13235:
13234:
13231:
13215:
13212:
13207:
13204:
13182:
13179:
13174:
13171:
13149:
13146:
13141:
13136:
13132:
13128:
13122:
13119:
13116:
13111:
13108:
13105:
13099:
13093:
13090:
13087:
13084:
13081:
13076:
13073:
13070:
13067:
13064:
13058:
13052:
13049:
13046:
13041:
13038:
13035:
13029:
13023:
13020:
13017:
13014:
13011:
13008:
13005:
13002:
12999:
12996:
12993:
12988:
12985:
12982:
12979:
12976:
12973:
12970:
12967:
12964:
12961:
12958:
12952:
12943:
12940:
12937:
12934:
12931:
12928:
12925:
12920:
12917:
12914:
12911:
12908:
12902:
12899:
12891:
12888:
12885:
12882:
12879:
12876:
12873:
12868:
12865:
12862:
12859:
12856:
12850:
12847:
12841:
12835:
12832:
12829:
12826:
12823:
12818:
12815:
12812:
12809:
12806:
12800:
12795:
12792:
12787:
12782:
12778:
12764:
12757:
12755:
12738:
12735:
12732:
12729:
12726:
12723:
12720:
12715:
12712:
12709:
12706:
12703:
12697:
12692:
12689:
12684:
12681:
12678:
12675:
12668:
12667:
12664:
12657:
12650:
12648:
12631:
12628:
12625:
12622:
12619:
12616:
12613:
12607:
12604:
12601:
12598:
12591:
12590:
12587:
12585:
12584:
12579:
12578:
12573:
12568:
12554:
12547:
12544:
12541:
12535:
12531:
12523:
12516:
12513:
12510:
12504:
12500:
12490:
12489:can be used.
12488:
12484:
12480:
12461:
12458:
12455:
12449:
12446:
12437:
12424:
12421:
12413:
12410:
12407:
12394:
12375:
12372:
12369:
12343:
12340:
12337:
12331:
12328:
12315:
12308:
12306:
12292:
12289:
12286:
12283:
12280:
12276:
12270:
12267:
12262:
12259:
12256:
12251:
12248:
12245:
12240:
12235:
12232:
12227:
12224:
12221:
12216:
12213:
12210:
12204:
12200:
12197:
12194:
12191:
12188:
12185:
12178:
12177:
12170:
12163:
12161:
12147:
12144:
12141:
12138:
12135:
12131:
12125:
12122:
12117:
12114:
12111:
12106:
12103:
12100:
12095:
12090:
12087:
12082:
12079:
12076:
12071:
12068:
12065:
12059:
12055:
12052:
12049:
12046:
12043:
12040:
12033:
12032:
12029:
12022:
12015:
12013:
11997:
11994:
11989:
11986:
11983:
11976:
11973:
11970:
11965:
11962:
11959:
11952:
11947:
11944:
11939:
11936:
11929:
11928:
11925:
11910:
11903:
11900:
11894:
11891:
11888:
11882:
11879:
11873:
11870:
11866:
11844:
11837:
11834:
11828:
11825:
11822:
11816:
11813:
11807:
11804:
11800:
11791:
11772:
11769:
11766:
11763:
11760:
11757:
11754:
11728:
11725:
11722:
11719:
11716:
11713:
11710:
11699:
11694:
11679:
11676:
11671:
11666:
11662:
11658:
11652:
11649:
11646:
11641:
11638:
11635:
11629:
11623:
11620:
11617:
11614:
11611:
11606:
11603:
11600:
11597:
11594:
11588:
11582:
11579:
11576:
11571:
11568:
11565:
11559:
11553:
11550:
11547:
11544:
11541:
11538:
11535:
11532:
11529:
11526:
11523:
11520:
11515:
11512:
11509:
11506:
11503:
11500:
11497:
11494:
11491:
11488:
11485:
11482:
11476:
11467:
11464:
11461:
11458:
11455:
11452:
11447:
11444:
11441:
11438:
11435:
11429:
11426:
11418:
11415:
11412:
11409:
11406:
11403:
11398:
11395:
11392:
11389:
11386:
11380:
11377:
11371:
11365:
11362:
11359:
11356:
11353:
11348:
11345:
11342:
11339:
11336:
11330:
11325:
11322:
11317:
11312:
11308:
11299:
11298:
11288:
11281:
11279:
11262:
11259:
11256:
11253:
11250:
11247:
11242:
11239:
11236:
11233:
11230:
11224:
11219:
11216:
11211:
11208:
11205:
11202:
11195:
11194:
11191:
11184:
11177:
11175:
11158:
11155:
11152:
11149:
11146:
11143:
11137:
11134:
11131:
11128:
11121:
11120:
11117:
11115:
11114:
11109:
11108:
11103:
11093:
11079:
11072:
11069:
11066:
11060:
11056:
11048:
11041:
11038:
11035:
11029:
11025:
11016:
11015:
11010:
11009:
11003:
11001:
11000:
10995:
10994:
10989:
10973:
10960:
10953:
10951:
10934:
10931:
10928:
10925:
10922:
10919:
10916:
10907:
10904:
10898:
10895:
10892:
10889:
10882:
10881:
10878:
10871:
10864:
10862:
10845:
10842:
10839:
10836:
10833:
10830:
10827:
10821:
10818:
10815:
10812:
10809:
10806:
10803:
10800:
10793:
10792:
10789:
10787:
10778:
10771:
10769:
10752:
10749:
10743:
10737:
10734:
10731:
10728:
10725:
10722:
10719:
10713:
10710:
10707:
10704:
10701:
10695:
10692:
10686:
10683:
10680:
10674:
10671:
10665:
10662:
10659:
10656:
10649:
10648:
10645:
10643:
10634:
10627:
10625:
10608:
10605:
10599:
10596:
10593:
10590:
10587:
10584:
10581:
10575:
10572:
10562:
10561:
10554:
10547:
10545:
10528:
10525:
10519:
10516:
10513:
10510:
10507:
10504:
10501:
10498:
10492:
10489:
10479:
10478:
10475:
10468:
10461:
10459:
10445:
10442:
10439:
10436:
10433:
10430:
10427:
10420:
10419:
10412:
10405:
10403:
10386:
10383:
10380:
10377:
10374:
10368:
10365:
10362:
10359:
10352:
10351:
10348:
10346:
10337:
10330:
10328:
10311:
10308:
10305:
10302:
10299:
10296:
10293:
10284:
10281:
10275:
10272:
10269:
10266:
10259:
10258:
10255:
10248:
10241:
10239:
10223:
10220:
10217:
10212:
10209:
10202:
10201:
10198:
10196:
10191:
10177:
10174:
10171:
10158:
10151:
10149:
10132:
10129:
10126:
10123:
10120:
10117:
10114:
10108:
10105:
10102:
10099:
10096:
10093:
10090:
10087:
10080:
10079:
10076:
10074:
10065:
10058:
10056:
10039:
10036:
10030:
10024:
10021:
10018:
10015:
10012:
10009:
10006:
10000:
9997:
9994:
9991:
9988:
9982:
9979:
9973:
9970:
9967:
9961:
9958:
9952:
9949:
9946:
9943:
9936:
9935:
9932:
9930:
9921:
9914:
9912:
9895:
9892:
9886:
9883:
9880:
9877:
9874:
9871:
9868:
9862:
9859:
9849:
9848:
9841:
9834:
9832:
9815:
9812:
9806:
9803:
9800:
9797:
9794:
9791:
9788:
9785:
9779:
9776:
9766:
9765:
9762:
9755:
9748:
9746:
9732:
9729:
9726:
9723:
9720:
9717:
9714:
9707:
9706:
9699:
9692:
9690:
9676:
9673:
9670:
9667:
9664:
9661:
9658:
9651:
9650:
9647:
9645:
9641:
9636:
9634:
9618:
9605:
9598:
9596:
9579:
9576:
9573:
9570:
9567:
9564:
9561:
9555:
9549:
9546:
9540:
9535:
9532:
9527:
9521:
9518:
9512:
9509:
9506:
9501:
9497:
9489:
9488:
9485:
9469:
9465:
9451:
9444:
9442:
9428:
9425:
9422:
9419:
9413:
9410:
9404:
9401:
9398:
9395:
9392:
9387:
9384:
9379:
9373:
9370:
9364:
9359:
9355:
9347:
9346:
9343:
9327:
9322:
9318:
9307:
9303:
9302:
9297:
9296:
9290:
9277:
9274:
9271:
9268:
9265:
9262:
9241:
9235:
9232:
9227:
9223:
9219:
9214:
9211:
9207:
9203:
9200:
9197:
9193:
9187:
9184:
9179:
9175:
9171:
9166:
9163:
9159:
9155:
9135:
9115:
9108:
9105:
9102:
9098:
9075:
9068:
9065:
9062:
9058:
9035:
9032:
9029:
9017:
9013:
9009:
9005:
9001:
8996:
8992:
8985:
8978:
8976:
8960:
8957:
8952:
8948:
8942:
8939:
8936:
8930:
8927:
8922:
8918:
8913:
8908:
8905:
8898:
8897:
8890:
8883:
8881:
8864:
8861:
8856:
8852:
8847:
8842:
8839:
8832:
8831:
8828:
8826:
8810:
8807:
8804:
8795:
8779:
8776:
8765:
8749:
8746:
8743:
8730:
8723:
8721:
8703:
8699:
8695:
8692:
8687:
8684:
8681:
8673:
8669:
8665:
8662:
8658:
8653:
8650:
8643:
8642:
8635:
8628:
8626:
8607:
8603:
8599:
8596:
8592:
8587:
8584:
8577:
8576:
8573:
8571:
8555:
8552:
8549:
8546:
8543:
8540:
8531:
8529:
8513:
8510:
8507:
8493:
8474:
8437:
8424:
8420:
8417:(OD) for the
8416:
8412:
8386:
8381:
8318:
8310:
8304:
8293:
8285:
8271:
8256:
8246:
8241:
8229:
8218:
8207:
8202:
8192:
8187:
8178:
8167:
8159:
8154:
8145:
8139:
8125:
8120:
8110:
8097:
8093:
8068:, the vector
8011:
7997:
7983:
7975:
7974:conic section
7966:
7959:
7957:
7940:
7937:
7934:
7931:
7928:
7925:
7922:
7918:
7913:
7910:
7903:
7902:
7899:
7882:
7879:
7873:
7870:
7847:
7841:
7817:
7814:
7791:
7788:
7785:
7773:
7760:
7751:
7745:
7742:
7736:
7732:
7728:
7722:
7719:
7714:
7710:
7704:
7681:
7672:
7666:
7663:
7660:
7657:
7654:
7647:
7624:
7615:
7609:
7606:
7603:
7600:
7597:
7587:
7578:
7567:
7553:
7544:
7538:
7535:
7532:
7529:
7526:
7513:
7507:
7493:
7482:
7479:
7471:
7411:
7385:
7379:
7376:
7373:
7370:
7367:
7361:
7358:
7352:
7346:
7343:
7340:
7337:
7334:
7323:
7312:
7309:
7306:
7298:
7290:
7282:
7274:
7271:
7260:
7252:
7246:
7238:
7227:
7221:
7207:
7194:
7190:
7185:
7167:
7159:
7156:
7148:
7142:
7125:
7109:
7098:
7095:
7087:
7081:
7064:
7051:
7048:
7037:
7023:
7020:
7016:
7009:
7006:
7001:
6990:
6984:
6973:
6967:
6956:
6943:
6940:
6935:
6929:
6918:
6894:
6891:
6886:
6866:
6858:
6845:
6840:
6821:
6802:
6791:
6777:
6766:
6752:
6749:
6746:
6737:
6726:
6715:
6707:
6704:
6701:
6692:
6681:
6671:
6667:
6660:
6648:
6644:
6640:
6635:
6632:
6624:
6618:
6603:Now consider
6601:
6585:
6574:
6564:
6560:
6556:
6545:
6531:
6528:
6522:
6519:
6510:
6499:
6486:
6482:
6478:
6464:
6461:
6455:
6449:
6438:
6432:
6424:
6421:
6410:
6401:
6398:
6394:
6389:
6381:
6378:
6372:
6361:
6353:
6318:
6314:
6310:
6304:
6301:
6295:
6262:
6259:
6232:
6222:
6186:
6178:
6162:
6159:
6156:
6153:
6150:
6147:
6144:
6140:
6135:
6130:
6127:
6122:
6119:
6111:
6110:one has that
6094:
6089:
6085:
6073:
6054:
6051:
6045:
6042:
6022:
6002:
5994:
5976:
5972:
5945:
5939:
5930:
5905:
5901:
5891:
5878:
5871:
5868:
5863:
5860:
5848:
5830:
5826:
5817:
5808:
5801:
5799:
5784:
5775:
5771:
5767:
5764:
5758:
5755:
5752:
5749:
5746:
5743:
5739:
5735:
5728:
5724:
5720:
5715:
5712:
5705:
5704:
5701:
5694:
5687:
5685:
5671:
5668:
5664:
5660:
5657:
5649:
5645:
5641:
5636:
5631:
5627:
5619:
5615:
5610:
5606:
5598:
5597:
5594:
5577:
5574:
5569:
5566:
5538:
5534:
5530:
5525:
5520:
5516:
5504:
5503:
5498:
5497:
5487:
5480:
5478:
5459:
5455:
5451:
5446:
5441:
5437:
5430:
5423:
5419:
5415:
5410:
5405:
5400:
5394:
5391:
5386:
5383:
5377:
5372:
5365:
5361:
5357:
5352:
5344:
5340:
5336:
5331:
5326:
5322:
5311:
5310:
5303:
5296:
5294:
5277:
5274:
5269:
5266:
5260:
5253:
5249:
5245:
5240:
5237:
5231:
5228:
5223:
5220:
5210:
5209:
5206:
5199:
5192:
5190:
5174:
5171:
5166:
5163:
5156:
5155:
5152:
5150:
5149:
5139:
5132:
5130:
5112:
5108:
5104:
5099:
5096:
5092:
5088:
5085:
5080:
5075:
5070:
5064:
5061:
5056:
5053:
5047:
5040:
5037:
5034:
5026:
5022:
5018:
5013:
5008:
5004:
4996:
4992:
4985:
4981:
4975:
4971:
4961:
4960:
4957:
4940:
4936:
4932:
4927:
4924:
4919:
4914:
4907:
4903:
4899:
4894:
4889:
4886:
4882:
4874:
4870:
4862:
4859:
4853:
4850:
4847:
4844:
4838:
4834:
4830:
4824:
4821:
4816:
4813:
4807:
4802:
4797:
4790:
4786:
4782:
4777:
4772:
4764:
4760:
4756:
4751:
4746:
4742:
4715:
4711:
4707:
4702:
4699:
4694:
4687:
4684:
4677:
4674:
4668:
4665:
4659:
4653:
4650:
4645:
4642:
4636:
4631:
4624:
4621:
4614:
4606:
4602:
4598:
4593:
4588:
4584:
4557:
4553:
4549:
4544:
4541:
4536:
4529:
4526:
4519:
4516:
4510:
4507:
4485:
4482:
4462:
4454:
4450:
4449:
4439:
4432:
4430:
4413:
4410:
4404:
4398:
4395:
4390:
4387:
4381:
4376:
4369:
4366:
4359:
4351:
4347:
4343:
4338:
4333:
4329:
4322:
4316:
4313:
4303:
4302:
4295:
4288:
4286:
4269:
4266:
4260:
4254:
4251:
4246:
4243:
4237:
4231:
4228:
4218:
4217:
4214:
4200:
4180:
4172:
4171:
4166:
4165:
4155:
4148:
4146:
4128:
4124:
4116:
4113:
4107:
4104:
4101:
4098:
4092:
4089:
4083:
4080:
4070:
4069:
4066:
4064:
4063:
4053:
4046:
4044:
4026:
4022:
4018:
4013:
4007:
4004:
3994:
3993:
3990:
3974:
3971:
3963:
3959:
3955:
3951:
3946:
3937:
3934:
3926:
3922:
3914:
3907:
3901:
3896:
3892:
3888:
3882:
3876:
3866:
3863:
3854:
3848:
3845:
3842:
3839:
3833:
3827:
3824:
3818:
3815:
3802:
3799:
3790:
3784:
3781:
3778:
3775:
3769:
3763:
3760:
3754:
3751:
3742:
3737:
3732:
3726:
3716:
3710:
3707:
3704:
3694:
3688:
3685:
3682:
3676:
3670:
3666:
3655:
3644:
3631:
3628:
3615:
3608:
3606:
3588:
3584:
3580:
3575:
3572:
3567:
3560:
3557:
3550:
3547:
3541:
3538:
3528:
3527:
3524:
3510:
3490:
3481:
3451:
3445:
3427:
3419:
3415:
3411:
3406:
3402:
3398:
3380:
3373:
3370:
3361:
3358:
3352:
3349:
3343:
3340:
3334:
3330:
3326:
3308:
3302:
3295:
3292:
3285:
3282:
3276:
3273:
3266:
3257:
3256:
3251:
3246:
3214:
3207:
3204:
3195:
3192:
3186:
3183:
3177:
3174:
3168:
3164:
3160:
3142:
3136:
3129:
3126:
3119:
3116:
3110:
3107:
3100:
3096:
3094:
3086:
3054:
3051:
3045:
3042:
3022:
3019:
3013:
3011:
3003:
2974:
2971:
2967:
2949:
2946:
2943:
2940:
2923:
2920:
2917:
2913:
2909:
2906:
2904:
2885:
2860:
2828:
2824:
2821:
2818:
2800:
2796:
2793:
2790:
2787:
2785:
2747:
2743:
2740:
2737:
2719:
2715:
2712:
2709:
2707:
2679:
2626:
2602:
2569:
2531:
2525:
2522:
2516:
2494:
2488:
2485:
2479:
2458:
2454:
2448:
2437:
2428:
2427:
2421:
2419:
2392:
2374:
2366:
2358:
2352:
2341:
2333:
2327:
2316:
2310:
2299:
2295:
2288:
2277:
2268:
2261:
2258:
2254:
2249:
2243:
2212:
2201:
2193:
2181:
2177:
2169:
2168:
2161:
2146:
2106:
2082:
2074:
2058:
2038:
2030:
2027:
2026:
2025:
2024:And finally:
2007:
1999:
1996:
1973:
1970:
1955:
1947:
1944:
1943:
1942:
1940:
1939:orbital plane
1920:
1912:
1909:
1894:
1886:
1883:
1882:
1881:
1878:
1876:
1869:
1861:
1856:
1847:
1845:
1841:
1837:
1833:
1827:
1825:
1805:
1785:
1781:
1778:
1775:
1767:
1759:
1752:
1745:
1741:
1737:
1729:
1722:
1716:
1715:
1714:
1711:
1693:
1689:
1685:
1680:
1676:
1669:
1666:
1663:
1655:
1641:
1628:
1621:
1619:
1589:
1585:
1581:
1576:
1573:
1567:
1552:
1551:
1548:
1545:
1543:
1527:
1519:
1485:
1470:
1460:
1455:
1445:
1432:
1407:
1378:
1374:
1367:
1363:
1357:
1353:
1349:
1343:
1338:
1331:
1317:
1313:
1277:
1273:
1266:
1262:
1256:
1252:
1248:
1245:
1239:
1234:
1227:
1213:
1209:
1200:
1198:
1180:
1151:
1122:
1118:
1095:
1091:
1081:
1077:
1074:
1070:
1065:
1063:
1062:shell theorem
1055:
1052:
1051:
1050:
1048:
1042:
1032:
1030:
1029:astrodynamics
1026:
1022:
1018:
1014:
1010:
988:
981:
964:
960:
952:
935:
931:
923:
921:
905:
898:
883:
876:
875:
874:
871:
854:
850:
843:
839:
833:
829:
822:
819:
816:
808:
806:
802:
798:
792:
786:
782:
775:
768:
764:
757:
750:
743:
738:
734:
685:
670:
663:
639:
638:
637:
634:
616:
612:
608:
596:
592:
585:
582:
574:
571:
558:
557:of the body:
556:
552:
548:
542:
540:
536:
532:
528:
518:
515:
513:
497:
488:
469:
463:
460:
457:
454:
451:
447:
442:
436:
430:
422:
416:
412:
396:
389:
386:
370:
363:
360:
344:
337:
322:
315:
314:
313:
310:
291:
285:
282:
279:
276:
273:
263:
259:
255:
252:
246:
240:
234:
228:
220:
218:
214:
210:
204:
202:
198:
194:
188:
186:
182:
178:
174:
164:
162:
159:model of the
158:
154:
150:
146:
142:
138:
128:
126:
120:
118:
114:
110:
106:
102:
98:
94:
90:
86:
82:
81:perturbations
78:
77:straight line
74:
73:orbital plane
70:
66:
62:
58:
54:
50:
46:
38:
37:
30:
24:
19:
18330:Solar System
18217:Perturbation
18199:
18174:Ground track
18084:Gravity turn
18035:
18028:
18021:
18012:
18003:
17983:
17974:
17965:
17958:True anomaly
17956:
17941:Mean anomaly
17939:
17919:
17910:
17901:
17892:
17872:
17859:
17850:
17843:Eccentricity
17841:
17799:Lunar cycler
17772:Heliocentric
17712:other points
17661:Medium Earth
17563:
17559:Non-inclined
17442:
17435:
17421:
17418:(son-in-law)
17401:
17382:
17374:
17366:
17358:
17350:
17342:
17334:
17294:Kepler orbit
17293:
17218:
17194:
17167:. Retrieved
17163:the original
17153:
17144:
17135:
17017:
17006:
16881:
16875:
16869:
16821:
16818:
16814:
16623:
16612:
16610:
16328:
16317:
16234:
16227:
16221:
16061:
15998:
15987:
15795:
15789:
15787:
15776:
15675:
15497:
15384:
15378:
15302:
15295:
15289:
15209:
15198:
15146:
15135:
15061:
14943:
14932:
14857:
14851:
14845:
14843:
14782:
14771:
14670:
14557:
14504:
14493:
14436:
14317:
14311:
14297:
14286:
14158:
14043:
14037:
13962:
13951:
13905:
13899:
13893:
13887:
13885:
13874:
13745:
13739:
13733:
13727:
13691:
13689:
13678:
13619:
13613:
13611:
13339:
13335:
13333:
13322:
12769:
12758:
12662:
12651:
12581:
12575:
12571:
12569:
12491:
12438:
12392:
12320:
12309:
12164:
12027:
12016:
11789:
11695:
11295:
11293:
11282:
11189:
11178:
11111:
11105:
11101:
11099:
11012:
11006:
11004:
10997:
10991:
10987:
10965:
10954:
10876:
10865:
10785:
10783:
10772:
10641:
10639:
10628:
10548:
10473:
10462:
10406:
10342:
10331:
10253:
10242:
10194:
10192:
10163:
10152:
10072:
10070:
10059:
9928:
9926:
9915:
9835:
9760:
9749:
9693:
9643:
9637:
9632:
9610:
9599:
9456:
9445:
9305:
9299:
9293:
9291:
9021:
9015:
9011:
9007:
9003:
9000:Kepler Orbit
8999:
8990:
8979:
8884:
8796:
8735:
8724:
8629:
8532:
8527:
8499:
8492:] is known.
8423:state vector
8422:
8410:
8384:
8382:
8272:
8095:
8091:
8003:
7971:
7960:
7775:Notice that
7774:
7469:
7192:
7188:
7186:
7126:
7065:
6841:
6602:
6247:, such that
6187:
6184:
6112:
6071:
5992:
5892:
5846:
5815:
5813:
5802:
5699:
5688:
5500:
5494:
5492:
5481:
5297:
5204:
5193:
5146:
5144:
5133:
4446:
4444:
4433:
4289:
4168:
4162:
4160:
4149:
4060:
4058:
4047:
3620:
3609:
3482:
3258:), we find:
3253:
3247:
2886:
2861:
2680:
2424:
2422:
2393:
2175:
2173:
2165:
2098:
2073:mean anomaly
2029:True anomaly
2023:
1936:
1911:Eccentricity
1879:
1874:
1871:
1828:
1821:
1757:
1750:
1743:
1739:
1735:
1727:
1720:
1712:
1656:
1633:
1622:
1546:
1486:
1433:
1408:
1201:
1082:
1078:
1066:
1059:
1044:
1005:
872:
809:
805:proportional
794:
790:
780:
773:
766:
762:
755:
748:
741:
732:
635:
559:
547:acceleration
544:
527:Isaac Newton
524:
521:Isaac Newton
516:
489:
423:
420:
411:true anomaly
385:eccentricity
311:
221:
206:
190:
170:
161:Solar System
157:heliocentric
155:published a
134:
131:Introduction
121:
113:parametrized
52:
49:Kepler orbit
48:
42:
34:
18:
18318:Outer space
18306:Spaceflight
18179:Hill sphere
18014:Mean motion
17894:Inclination
17883:Orientation
17784:Mars cycler
17722:Areocentric
17594:Synchronous
16608:will vary.
16549:defined by
8568:this is an
4161:Equations (
2418:plane curve
1946:Inclination
1518:unit vector
754:by a force
177:Tycho Brahe
18340:Categories
18119:Rendezvous
17815:Parameters
17651:High Earth
17621:Geocentric
17574:Osculating
17531:Elliptical
17182:References
16617:) at time
14990:such that
13886:and from (
12479:ATAN2(y,x)
9646:for which
8823:this is a
8762:this is a
2508:or radial
1858:Keplerian
1756:) ≈
1073:spacecraft
1039:See also:
797:point mass
137:geocentric
125:barycenter
18282:Astronomy
18164:Ephemeris
18141:mechanics
18051:Maneuvers
17994:Variation
17757:Libration
17752:Lissajous
17656:Low Earth
17646:Graveyard
17545:Horseshoe
17169:12 August
17127:Citations
16974:^
16953:−
16947:^
16923:−
16853:^
16789:˙
16733:˙
16689:^
16677:α
16674:−
16656:˙
16596:θ
16573:˙
16534:^
16503:^
16475:θ
16396:α
16393:−
16361:˙
16289:˙
16256:¨
16205:^
16190:^
16118:θ
15965:α
15920:α
15897:⋅
15885:α
15870:α
15757:^
15746:θ
15743:
15731:^
15720:θ
15717:
15705:^
15656:^
15645:θ
15642:
15636:−
15630:^
15619:θ
15616:
15604:^
15568:^
15534:^
15517:^
15483:θ
15365:θ
15339:^
15322:^
15218:θ
15176:α
15092:θ
15089:
15042:−
15012:θ
15009:
14978:θ
14955:≥
14914:α
14891:⋅
14752:^
14725:^
14651:^
14594:^
14577:^
14466:˙
14406:^
14394:⋅
14391:α
14388:−
14379:˙
14267:θ
14264:
14249:⋅
14240:α
14231:θ
14228:
14209:θ
14206:
14142:θ
14139:
14133:⋅
14124:α
14118:−
14112:θ
14109:
14093:−
14090:θ
14087:
14021:^
13992:^
13977:^
13925:α
13848:α
13843:−
13813:α
13808:−
13709:α
13704:−
13659:α
13650:⋅
13644:π
13588:−
13564:
13542:
13534:−
13492:and that
13475:∞
13463:∞
13460:−
13452:⟷
13429:−
13421:
13413:−
13402:θ
13381:−
13373:
13365:−
13357:−
13296:
13290:⋅
13280:−
13251:θ
13246:
13208:
13180:θ
13175:
13142:
13129:⋅
13120:−
13085:
13074:−
13068:
13059:⋅
13050:−
13021:
13015:−
13003:
12997:⋅
12986:
12974:−
12968:
12962:⋅
12941:−
12935:
12929:⋅
12918:
12912:−
12889:−
12883:
12877:⋅
12866:
12860:−
12851:−
12836:θ
12833:
12819:θ
12816:
12810:−
12793:θ
12788:
12736:−
12730:
12724:⋅
12713:
12707:−
12682:θ
12679:
12629:−
12623:
12617:⋅
12608:⋅
12551:∞
12539:∞
12536:−
12528:⟷
12520:∞
12514:θ
12508:∞
12505:−
12450:
12425:π
12414:θ
12411:−
12332:
12293:π
12287:⋅
12268:θ
12263:
12257:⋅
12249:−
12233:θ
12228:
12222:⋅
12201:
12195:⋅
12148:π
12142:⋅
12118:
12112:⋅
12083:
12077:⋅
12069:−
12056:
12050:⋅
12041:θ
12028:and that
11990:
11984:⋅
11974:−
11945:θ
11940:
11901:θ
11895:
11880:θ
11874:
11829:
11808:
11773:θ
11770:
11761:θ
11758:
11726:
11714:
11672:
11659:⋅
11650:−
11621:
11604:
11598:−
11589:⋅
11580:−
11551:−
11545:
11533:
11524:−
11507:
11501:−
11495:
11486:−
11465:
11456:−
11445:−
11439:
11416:
11407:−
11396:−
11390:
11381:−
11366:θ
11363:
11349:θ
11346:
11340:−
11323:θ
11318:
11260:
11251:−
11240:−
11234:
11209:θ
11206:
11156:
11147:−
11138:⋅
11076:∞
11064:∞
11061:−
11053:⟷
11045:∞
11033:∞
11030:−
10974:θ
10932:−
10926:
10920:⋅
10908:α
10899:⋅
10843:−
10837:
10831:⋅
10822:⋅
10816:⋅
10804:⋅
10753:˙
10744:⋅
10735:−
10729:
10723:⋅
10714:⋅
10708:⋅
10696:˙
10687:⋅
10681:−
10675:˙
10666:⋅
10609:˙
10600:⋅
10594:
10588:⋅
10576:˙
10529:˙
10520:⋅
10514:
10508:⋅
10502:−
10493:˙
10443:
10437:⋅
10384:
10378:−
10369:⋅
10309:
10303:⋅
10297:−
10285:α
10276:⋅
10221:⋅
10218:α
10130:
10124:⋅
10118:−
10109:⋅
10103:⋅
10091:⋅
10040:˙
10031:⋅
10022:
10016:⋅
10010:−
10001:⋅
9995:⋅
9983:˙
9974:⋅
9968:−
9962:˙
9953:⋅
9896:˙
9887:⋅
9881:
9875:⋅
9863:˙
9816:˙
9807:⋅
9801:
9795:⋅
9789:−
9780:˙
9730:
9724:⋅
9674:
9668:⋅
9631:and time
9619:θ
9580:θ
9577:
9571:⋅
9556:⋅
9547:α
9522:˙
9519:θ
9513:⋅
9429:θ
9426:
9411:α
9402:θ
9399:
9374:˙
9328:α
9278:π
9272:θ
9266:π
9263:−
9228:−
9220:
9212:−
9201:θ
9180:−
9172:
9164:−
9156:−
9136:θ
9106:−
9030:θ
8958:−
8943:⋅
8928:−
8862:−
8825:hyperbola
8696:−
8688:⋅
8666:−
8600:−
8447:˙
8411:periapsis
8319:×
8305:˙
8294:×
8247:−
8242:α
8230:×
8219:×
8193:−
8188:α
8179:×
8160:−
8155:α
8146:×
8140:˙
8121:α
8111:≜
8092:periapsis
8012:θ
7984:θ
7941:θ
7938:
7932:⋅
7883:α
7848:α
7792:θ
7752:θ
7746:
7737:α
7715:α
7673:θ
7667:
7655:α
7616:θ
7610:
7598:α
7588:⋅
7579:˙
7568:×
7545:θ
7539:
7527:α
7514:×
7508:˙
7494:⋅
7472: :
7412:θ
7386:θ
7380:
7368:α
7353:θ
7347:
7324:⋅
7310:α
7299:⋅
7283:⋅
7275:α
7253:α
7247:⋅
7228:×
7222:˙
7208:⋅
7160:α
7149:×
7143:˙
7110:˙
7099:α
7088:×
7082:¨
7038:⋅
6991:⋅
6985:˙
6968:˙
6957:⋅
6930:˙
6919:⋅
6859:⋅
6822:˙
6803:⋅
6792:−
6778:˙
6767:⋅
6753:α
6750:−
6738:˙
6727:×
6716:×
6708:α
6705:−
6693:˙
6682:×
6661:×
6641:α
6636:−
6625:×
6619:¨
6586:˙
6575:×
6546:×
6532:˙
6511:˙
6500:×
6465:˙
6450:˙
6433:×
6390:×
6373:˙
6362:×
6311:α
6305:−
6296:¨
6163:θ
6160:
6154:⋅
6095:α
6003:θ
5973:θ
5949:^
5934:^
5902:θ
5872:θ
5827:θ
5772:θ
5768:−
5765:θ
5759:
5753:⋅
5736:⋅
5721:α
5672:α
5646:θ
5616:⋅
5578:θ
5535:θ
5456:θ
5431:⋅
5411:−
5395:θ
5373:⋅
5341:θ
5278:θ
5261:⋅
5241:−
5232:θ
5105:α
5100:−
5086:−
5065:θ
5041:⋅
5035:−
5023:θ
4993:⋅
4933:α
4928:−
4887:−
4863:˙
4854:⋅
4848:⋅
4839:−
4831:⋅
4825:θ
4773:⋅
4761:θ
4708:α
4703:−
4688:˙
4685:θ
4675:−
4669:¨
4666:θ
4660:⋅
4654:θ
4625:˙
4622:θ
4615:⋅
4603:θ
4550:α
4545:−
4530:˙
4527:θ
4517:−
4511:¨
4483:θ
4414:¨
4411:θ
4405:⋅
4399:θ
4370:˙
4367:θ
4360:⋅
4348:θ
4317:¨
4270:˙
4267:θ
4261:⋅
4255:θ
4232:˙
4181:θ
4117:˙
4108:⋅
4102:⋅
4093:−
4084:¨
4081:θ
4008:˙
4005:θ
3975:˙
3972:θ
3938:˙
3935:θ
3867:˙
3864:θ
3855:θ
3849:
3834:θ
3828:
3819:˙
3803:˙
3800:θ
3791:θ
3785:
3776:−
3770:θ
3764:
3755:˙
3738:×
3717:θ
3711:
3695:θ
3689:
3656:˙
3645:×
3581:α
3576:−
3561:˙
3558:θ
3548:−
3542:¨
3511:θ
3466:^
3440:^
3412:α
3407:−
3393:^
3374:˙
3371:θ
3362:˙
3344:¨
3341:θ
3321:^
3296:˙
3293:θ
3283:−
3277:¨
3227:^
3208:˙
3205:θ
3196:˙
3178:¨
3175:θ
3155:^
3130:˙
3127:θ
3117:−
3111:¨
3087:¨
3069:^
3055:˙
3052:θ
3037:^
3023:˙
3004:˙
2986:^
2961:^
2950:θ
2947:
2935:^
2924:θ
2921:
2841:^
2829:θ
2825:
2813:^
2801:θ
2797:
2791:−
2778:^
2760:^
2748:θ
2744:
2732:^
2720:θ
2716:
2700:^
2638:^
2621:^
2581:^
2564:^
2526:¨
2489:¨
2486:θ
2449:¨
2353:¨
2342:×
2328:˙
2317:×
2311:˙
2289:˙
2278:×
2244:˙
2213:˙
2202:×
2147:ω
2127:Ω
2039:ν
2008:ω
1982:Ω
1834:, due to
1776:μ
1726:>>
1664:α
1642:α
1603:^
1582:α
1577:−
1568:¨
1501:^
1461:−
1392:^
1332:¨
1291:^
1246:−
1228:¨
1069:asteroids
1025:astronomy
531:Principia
470:θ
464:
437:θ
415:periapsis
397:θ
292:θ
286:
256:−
235:θ
213:hyperbola
171:In 1601,
141:Aristotle
115:into six
93:spherical
69:hyperbola
17930:Position
17555:Inclined
17526:Circular
17412:(mother)
17080:See also
14612:through
10197:one has
8764:parabola
2099:Because
537:and his
209:parabola
149:epicycle
91:, a non-
65:parabola
18268:Portals
18139:Orbital
18109:Phasing
18069:Delta-v
17874:Apsides
17868:,
17666:Molniya
17584:Parking
17521:Capture
17509:General
17426:(opera)
17406:(opera)
17395:Related
17384:Somnium
16880:) and (
16226:) and (
15794:) and (
15383:) and (
15294:) and (
14856:) and (
14042:) and (
13904:) and (
13744:) and (
13164:and as
12586:) that
12580:) and (
12487:FORTRAN
12481:(or in
12321:where "
11116:) that
11110:) and (
11100:For an
11011:) and (
8570:ellipse
5499:) and (
4167:) and (
4065:) gets
2170:) above
1832:Mercury
1824:Jupiter
1764:. Such
1540:is the
1516:is the
1021:general
1017:special
918:is the
873:where:
783:is the
772:| and |
636:Where:
409:is the
383:is the
357:is the
312:where:
197:ellipse
145:Ptolemy
61:ellipse
18346:Orbits
17795:Other
17696:Tundra
17564:Kepler
17540:Escape
17493:orbits
17423:Kepler
17387:(1634)
17379:(1627)
17371:(1619)
17355:(1609)
17347:(1606)
17339:(1596)
17203:
16757:where
16329:where
15147:where
14844:From (
12570:For a
11294:From (
10075:gives
8462:] or [
8273:where
7404:where
7187:where
5814:where
5505:) for
3248:(see "
2178:, the
1634:where
1542:length
1409:where
1011:until
795:Every
490:Where
18294:Stars
18037:Epoch
17826:Shape
17764:Lunar
17718:Mars
17710:About
17681:Polar
17501:Types
17328:Works
17018:i.e.
15936:i.e.
14783:with
13696:) is
10877:i.e.
10788:gets
9484:) is
9018:= 0).
8827:with
8572:with
6842:(see
5593:gets
1733:, so
801:force
551:force
211:or a
201:focus
193:orbit
67:, or
23:Orbit
17829:Size
17768:Sun
17747:Halo
17599:semi
17201:ISBN
17171:2012
16874:), (
15833:and
15261:and
14970:and
14850:), (
14827:>
14809:and
14794:>
13898:), (
13892:), (
13738:), (
13732:), (
13530:tanh
13472:<
13466:<
13405:<
13399:<
13293:tanh
13205:tanh
13197:and
13133:tanh
13082:cosh
13065:cosh
13018:cosh
13000:cosh
12983:cosh
12965:cosh
12932:cosh
12915:cosh
12880:cosh
12863:cosh
12727:cosh
12710:cosh
12620:cosh
12548:<
12542:<
12517:<
12511:<
12422:<
12391:and
11858:and
11744:and
11073:<
11067:<
11042:<
11036:<
10923:sinh
10834:sinh
10726:cosh
10591:cosh
10511:sinh
10440:sinh
10381:cosh
9275:<
9269:<
9204:<
9198:<
8808:>
8797:For
8736:For
8550:<
8544:<
8533:For
8500:For
7863:and
7446:and
6280:and
5849:and
5818:and
5556:and
4455:for
3503:and
2599:and
2139:and
2075:and
2071:the
1842:and
1487:and
1166:and
1110:and
1027:and
1019:and
555:mass
545:The
191:The
181:Mars
143:and
51:(or
47:, a
17604:sub
17516:Box
17438:ATV
16232:).
15740:cos
15714:sin
15639:sin
15613:cos
15300:).
15086:sin
15006:cos
14261:cos
14225:cos
14203:sin
14136:sin
14106:sin
14084:cos
13409:cos
13361:cos
13243:tan
13172:tan
12830:cos
12813:cos
12779:tan
12770:As
12676:cos
12447:arg
12329:arg
12260:sin
12225:cos
12198:arg
12115:sin
12080:cos
12053:arg
11987:tan
11937:tan
11892:sin
11871:cos
11826:sin
11805:cos
11767:sin
11755:cos
11723:sin
11711:cos
11663:tan
11618:cos
11601:cos
11542:cos
11530:cos
11504:cos
11492:cos
11462:cos
11436:cos
11413:cos
11387:cos
11360:cos
11343:cos
11309:tan
11257:cos
11231:cos
11203:cos
11153:cos
10644:is
10306:sin
10127:sin
10019:cos
9931:is
9878:cos
9798:sin
9727:sin
9671:cos
9574:cos
9423:sin
9396:sin
9308:as
9208:cos
9160:cos
9148:is
7935:cos
7743:cos
7664:cos
7607:cos
7536:cos
7377:cos
7344:cos
6157:cos
6074:as
5756:cos
3846:cos
3825:sin
3782:sin
3761:cos
3708:sin
3686:cos
2944:sin
2918:cos
2822:cos
2794:sin
2741:sin
2713:cos
1071:or
461:cos
283:cos
43:In
18342::
17952:,
17948:,
17557:/
17533:/
17009:60
16883:56
16877:54
16871:53
16614:59
16518:,
16487:,
16467:,
16447:,
16320:59
16229:57
16223:56
15990:58
15797:54
15791:53
15779:57
15678:56
15495:.
15386:51
15380:50
15297:51
15291:50
15234:,
15201:55
15138:54
15064:53
14935:52
14859:19
14853:18
14847:13
14774:51
14673:50
14496:49
14439:48
14289:47
14161:46
14045:19
14039:18
13954:45
13907:19
13901:18
13895:16
13889:13
13877:44
13747:19
13741:18
13735:14
13729:13
13681:43
13615:27
13561:ln
13341:34
13325:42
12761:41
12654:40
12583:29
12577:28
12312:39
12167:38
12019:37
11297:36
11285:36
11181:35
11113:21
11107:20
11014:34
11008:27
10999:34
10993:27
10957:34
10868:33
10775:32
10631:31
10551:30
10465:29
10409:28
10334:27
10245:26
10155:25
10062:24
9918:23
9838:22
9752:21
9696:20
9642:"
9602:19
9448:18
8982:17
8887:16
8727:15
8632:14
8530:.
8380:.
7963:13
6210:,
5805:12
5691:11
5496:10
5484:10
3523::
2678::
2420:.
2119:,
1941::
1758:Gm
1749:+
1738:=
1031:.
417:).
127:.
87:,
63:,
36:13
18270::
18200:n
18032:0
18029:t
18019:v
18010:n
18001:T
17981:l
17972:L
17963:E
17954:f
17950:θ
17946:ν
17937:M
17917:ϖ
17908:ω
17899:Ω
17890:i
17870:q
17866:Q
17857:b
17848:a
17839:e
17484:e
17477:t
17470:v
17262:e
17255:t
17248:v
17209:.
17173:.
17064:)
17060:v
17056:,
17052:r
17048:(
17027:e
17011:)
17007:(
16986:0
16982:V
16970:t
16961:r
16957:V
16943:r
16936:)
16931:0
16927:V
16918:t
16914:V
16910:(
16904:=
16900:e
16849:x
16842:e
16839:=
16835:e
16801:)
16798:t
16795:,
16785:r
16778:,
16774:r
16770:(
16766:f
16745:)
16742:t
16739:,
16729:r
16722:,
16718:r
16714:(
16710:f
16706:+
16699:2
16695:r
16685:r
16671:=
16668:)
16665:t
16662:,
16652:r
16645:,
16641:r
16637:(
16633:F
16619:t
16569:r
16562:,
16558:r
16530:y
16499:x
16455:e
16435:p
16411:2
16407:r
16402:r
16373:)
16370:t
16367:,
16357:r
16350:,
16346:r
16342:(
16338:F
16322:)
16318:(
16301:)
16298:t
16295:,
16285:r
16278:,
16274:r
16270:(
16266:F
16262:=
16252:r
16202:y
16196:,
16187:x
16162:t
16158:V
16154:,
16149:r
16145:V
16141:,
16138:r
16115:,
16112:e
16109:,
16106:p
16086:)
16082:v
16078:,
16074:r
16070:(
16037:)
16032:0
16027:v
16022:,
16017:0
16012:r
16007:(
15992:)
15988:(
15968:r
15959:=
15954:t
15950:V
15915:2
15910:)
15905:t
15901:V
15894:r
15891:(
15879:=
15873:p
15864:=
15859:0
15855:V
15851:=
15846:t
15842:V
15821:0
15818:=
15813:r
15809:V
15781:)
15777:(
15753:t
15737:+
15727:r
15711:=
15701:y
15680:)
15676:(
15652:t
15626:r
15610:=
15600:x
15564:x
15540:)
15530:y
15523:,
15513:x
15506:(
15463:)
15458:0
15453:v
15448:,
15443:0
15438:r
15433:(
15413:)
15409:v
15405:,
15401:r
15397:(
15345:)
15335:t
15328:,
15318:r
15311:(
15274:t
15270:V
15247:r
15243:V
15232:r
15203:)
15199:(
15179:p
15170:=
15165:0
15161:V
15140:)
15136:(
15115:0
15111:V
15105:r
15101:V
15095:=
15083:e
15066:)
15062:(
15045:1
15035:0
15031:V
15025:t
15021:V
15015:=
15003:e
14958:0
14952:e
14937:)
14933:(
14909:2
14904:)
14899:t
14895:V
14888:r
14885:(
14878:=
14875:p
14830:0
14822:t
14818:V
14797:0
14791:r
14776:)
14772:(
14748:t
14739:t
14735:V
14731:+
14721:r
14712:r
14708:V
14704:=
14699:0
14694:v
14675:)
14671:(
14647:r
14640:r
14637:=
14632:0
14627:r
14600:)
14590:t
14583:,
14573:r
14566:(
14543:)
14538:0
14533:v
14528:,
14523:0
14518:r
14513:(
14498:)
14494:(
14476:v
14472:=
14462:r
14441:)
14437:(
14416:2
14412:r
14402:r
14385:=
14375:v
14346:)
14342:v
14338:,
14334:r
14330:(
14319:1
14291:)
14287:(
14270:)
14258:+
14255:e
14252:(
14243:p
14234:=
14220:t
14216:V
14212:+
14198:r
14194:V
14190:=
14185:y
14181:V
14163:)
14159:(
14127:p
14115:=
14101:t
14097:V
14079:r
14075:V
14071:=
14066:x
14062:V
14018:x
13989:y
13983:,
13974:x
13956:)
13952:(
13932:a
13929:2
13879:)
13875:(
13855:a
13852:2
13816:r
13803:2
13797:2
13790:t
13786:V
13780:+
13775:2
13768:r
13764:V
13712:r
13693:1
13683:)
13679:(
13656:a
13647:a
13641:2
13638:=
13635:P
13620:P
13597:)
13591:x
13585:1
13580:x
13577:+
13574:1
13568:(
13556:2
13553:1
13548:=
13545:x
13537:1
13506:2
13503:E
13479:]
13469:E
13456:[
13448:]
13443:)
13437:e
13434:1
13425:(
13416:1
13395:)
13389:e
13386:1
13377:(
13368:1
13353:[
13336:E
13327:)
13323:(
13304:2
13301:E
13283:1
13277:e
13272:1
13269:+
13266:e
13259:=
13254:2
13216:2
13213:E
13183:2
13150:2
13147:E
13137:2
13123:1
13117:e
13112:1
13109:+
13106:e
13100:=
13094:1
13091:+
13088:E
13077:1
13071:E
13053:1
13047:e
13042:1
13039:+
13036:e
13030:=
13024:E
13012:e
13009:+
13006:E
12994:e
12989:E
12980:+
12977:e
12971:E
12959:e
12953:=
12944:1
12938:E
12926:e
12921:E
12909:e
12903:+
12900:1
12892:1
12886:E
12874:e
12869:E
12857:e
12848:1
12842:=
12827:+
12824:1
12807:1
12801:=
12796:2
12783:2
12763:)
12759:(
12739:1
12733:E
12721:e
12716:E
12704:e
12698:=
12693:r
12690:x
12685:=
12656:)
12652:(
12635:)
12632:1
12626:E
12614:e
12611:(
12605:a
12602:=
12599:r
12555:]
12545:E
12532:[
12524:]
12501:[
12465:)
12462:y
12459:,
12456:x
12453:(
12418:|
12408:E
12404:|
12393:n
12379:)
12376:y
12373:,
12370:x
12367:(
12347:)
12344:y
12341:,
12338:x
12335:(
12314:)
12310:(
12290:2
12284:n
12281:+
12277:)
12271:2
12252:e
12246:1
12241:,
12236:2
12217:e
12214:+
12211:1
12205:(
12192:2
12189:=
12186:E
12169:)
12165:(
12145:2
12139:n
12136:+
12132:)
12126:2
12123:E
12107:e
12104:+
12101:1
12096:,
12091:2
12088:E
12072:e
12066:1
12060:(
12047:2
12044:=
12021:)
12017:(
11998:2
11995:E
11977:e
11971:1
11966:e
11963:+
11960:1
11953:=
11948:2
11911:)
11904:2
11889:,
11883:2
11867:(
11845:)
11838:2
11835:E
11823:,
11817:2
11814:E
11801:(
11790:x
11776:)
11764:,
11752:(
11732:)
11729:E
11720:,
11717:E
11708:(
11680:2
11677:E
11667:2
11653:e
11647:1
11642:e
11639:+
11636:1
11630:=
11624:E
11615:+
11612:1
11607:E
11595:1
11583:e
11577:1
11572:e
11569:+
11566:1
11560:=
11554:e
11548:E
11539:+
11536:E
11527:e
11521:1
11516:e
11513:+
11510:E
11498:E
11489:e
11483:1
11477:=
11468:E
11459:e
11453:1
11448:e
11442:E
11430:+
11427:1
11419:E
11410:e
11404:1
11399:e
11393:E
11378:1
11372:=
11357:+
11354:1
11337:1
11331:=
11326:2
11313:2
11287:)
11283:(
11263:E
11254:e
11248:1
11243:e
11237:E
11225:=
11220:r
11217:x
11212:=
11183:)
11179:(
11162:)
11159:E
11150:e
11144:1
11141:(
11135:a
11132:=
11129:r
11080:]
11070:E
11057:[
11049:]
11039:t
11026:[
10988:E
10959:)
10955:(
10938:)
10935:E
10929:E
10917:e
10914:(
10905:a
10896:a
10893:=
10890:t
10870:)
10866:(
10849:)
10846:E
10840:E
10828:e
10825:(
10819:b
10813:a
10810:=
10807:t
10801:H
10786:t
10777:)
10773:(
10750:E
10741:)
10738:1
10732:E
10720:e
10717:(
10711:b
10705:a
10702:=
10693:x
10684:y
10672:y
10663:x
10660:=
10657:H
10642:H
10633:)
10629:(
10606:E
10597:E
10585:b
10582:=
10573:y
10553:)
10549:(
10526:E
10517:E
10505:a
10499:=
10490:x
10467:)
10463:(
10446:E
10434:b
10431:=
10428:y
10411:)
10407:(
10390:)
10387:E
10375:e
10372:(
10366:a
10363:=
10360:x
10336:)
10332:(
10315:)
10312:E
10300:e
10294:E
10291:(
10282:a
10273:a
10270:=
10267:t
10247:)
10243:(
10224:p
10213:=
10210:H
10195:p
10178:0
10175:=
10172:t
10157:)
10153:(
10136:)
10133:E
10121:e
10115:E
10112:(
10106:b
10100:a
10097:=
10094:t
10088:H
10073:t
10064:)
10060:(
10037:E
10028:)
10025:E
10013:e
10007:1
10004:(
9998:b
9992:a
9989:=
9980:x
9971:y
9959:y
9950:x
9947:=
9944:H
9929:H
9920:)
9916:(
9893:E
9884:E
9872:b
9869:=
9860:y
9840:)
9836:(
9813:E
9804:E
9792:a
9786:=
9777:x
9754:)
9750:(
9733:E
9721:b
9718:=
9715:y
9698:)
9694:(
9677:E
9665:a
9662:=
9659:x
9644:E
9633:t
9604:)
9600:(
9583:)
9568:e
9565:+
9562:1
9559:(
9550:p
9541:=
9536:r
9533:H
9528:=
9510:r
9507:=
9502:t
9498:V
9470:r
9466:V
9450:)
9446:(
9420:e
9414:p
9405:=
9393:e
9388:p
9385:H
9380:=
9371:r
9365:=
9360:r
9356:V
9323:2
9319:H
9306:p
9301:2
9295:5
9242:)
9236:e
9233:1
9224:(
9215:1
9194:)
9188:e
9185:1
9176:(
9167:1
9116:.
9109:e
9103:1
9099:p
9076:,
9069:e
9066:+
9063:1
9059:p
9036:0
9033:=
9016:e
9012:e
9008:e
9004:e
8984:)
8980:(
8961:1
8953:2
8949:e
8940:a
8937:=
8931:1
8923:2
8919:e
8914:p
8909:=
8906:b
8889:)
8885:(
8865:1
8857:2
8853:e
8848:p
8843:=
8840:a
8811:1
8805:e
8780:2
8777:p
8750:1
8747:=
8744:e
8729:)
8725:(
8704:2
8700:e
8693:1
8685:a
8682:=
8674:2
8670:e
8663:1
8659:p
8654:=
8651:b
8634:)
8630:(
8608:2
8604:e
8597:1
8593:p
8588:=
8585:a
8556:,
8553:1
8547:e
8541:0
8528:p
8514:0
8511:=
8508:e
8479:v
8475:,
8471:r
8444:r
8438:,
8434:r
8425:[
8396:c
8367:r
8345:v
8323:v
8315:r
8311:=
8301:r
8290:r
8286:=
8282:H
8257:r
8253:r
8238:)
8234:v
8226:r
8222:(
8215:v
8208:=
8203:r
8199:r
8183:H
8175:v
8168:=
8164:u
8150:H
8136:r
8126:=
8117:c
8107:e
8077:c
8055:c
8033:r
7965:)
7961:(
7929:e
7926:+
7923:1
7919:p
7914:=
7911:r
7880:c
7874:=
7871:e
7842:2
7837:|
7831:H
7826:|
7818:=
7815:p
7795:)
7789:,
7786:r
7783:(
7761:.
7755:)
7749:(
7740:)
7733:/
7729:c
7726:(
7723:+
7720:1
7711:/
7705:2
7700:|
7694:H
7689:|
7682:=
7676:)
7670:(
7661:c
7658:+
7648:2
7643:|
7637:H
7632:|
7625:=
7619:)
7613:(
7604:c
7601:+
7592:H
7585:)
7575:r
7564:r
7560:(
7554:=
7548:)
7542:(
7533:c
7530:+
7522:)
7518:H
7504:r
7497:(
7490:r
7483:=
7480:r
7470:r
7455:c
7433:r
7392:)
7389:)
7383:(
7374:c
7371:+
7365:(
7362:r
7359:=
7356:)
7350:(
7341:c
7338:r
7335:+
7332:)
7328:u
7320:u
7316:(
7313:r
7307:=
7303:c
7295:r
7291:+
7287:u
7279:r
7272:=
7269:)
7265:c
7261:+
7257:u
7250:(
7243:r
7239:=
7236:)
7232:H
7218:r
7211:(
7204:r
7193:r
7189:c
7172:c
7168:+
7164:u
7157:=
7153:H
7139:r
7106:u
7096:=
7092:H
7078:r
7052:0
7049:=
7046:)
7042:u
7034:u
7030:(
7024:t
7021:d
7017:d
7010:2
7007:1
7002:=
6999:)
6995:u
6981:u
6974:+
6964:u
6953:u
6949:(
6944:2
6941:1
6936:=
6926:u
6915:u
6895:1
6892:=
6887:2
6882:|
6876:u
6871:|
6867:=
6863:u
6855:u
6828:]
6818:u
6811:)
6807:u
6799:u
6795:(
6788:u
6784:)
6774:u
6763:u
6759:(
6756:[
6747:=
6744:)
6734:u
6723:u
6719:(
6712:u
6702:=
6699:)
6689:u
6678:u
6672:2
6668:r
6664:(
6657:u
6649:2
6645:r
6633:=
6629:H
6615:r
6582:u
6571:u
6565:2
6561:r
6557:=
6554:)
6550:u
6542:u
6538:(
6529:r
6523:r
6520:+
6517:)
6507:u
6496:u
6492:(
6487:2
6483:r
6479:=
6476:)
6472:u
6462:r
6456:+
6446:u
6439:r
6436:(
6429:u
6425:r
6422:=
6419:)
6415:u
6411:r
6408:(
6402:t
6399:d
6395:d
6386:u
6382:r
6379:=
6369:r
6358:r
6354:=
6350:H
6328:u
6319:2
6315:r
6302:=
6292:r
6267:u
6263:r
6260:=
6256:r
6233:r
6229:r
6223:=
6219:u
6197:u
6151:e
6148:+
6145:1
6141:p
6136:=
6131:s
6128:1
6123:=
6120:r
6090:2
6086:H
6072:p
6055:s
6052:1
6046:=
6043:r
6023:s
5993:e
5977:0
5946:y
5940:,
5931:x
5906:0
5879:.
5869:d
5864:s
5861:d
5847:s
5831:0
5816:e
5807:)
5803:(
5785:)
5781:)
5776:0
5762:(
5750:e
5747:+
5744:1
5740:(
5729:2
5725:H
5716:=
5713:s
5693:)
5689:(
5669:=
5665:)
5661:s
5658:+
5650:2
5642:d
5637:s
5632:2
5628:d
5620:(
5611:2
5607:H
5575:d
5570:r
5567:d
5539:2
5531:d
5526:r
5521:2
5517:d
5502:9
5486:)
5482:(
5460:2
5452:d
5447:s
5442:2
5438:d
5424:2
5420:s
5416:1
5406:2
5401:)
5392:d
5387:s
5384:d
5378:(
5366:3
5362:s
5358:2
5353:=
5345:2
5337:d
5332:r
5327:2
5323:d
5302:)
5300:9
5298:(
5275:d
5270:s
5267:d
5254:2
5250:s
5246:1
5238:=
5229:d
5224:r
5221:d
5198:)
5196:8
5194:(
5175:s
5172:1
5167:=
5164:r
5148:7
5138:)
5136:7
5134:(
5113:2
5109:r
5097:=
5093:)
5089:r
5081:r
5076:2
5071:)
5062:d
5057:r
5054:d
5048:(
5038:2
5027:2
5019:d
5014:r
5009:2
5005:d
4997:(
4986:4
4982:r
4976:2
4972:H
4941:2
4937:r
4925:=
4920:2
4915:)
4908:2
4904:r
4900:H
4895:(
4890:r
4883:)
4875:3
4871:r
4860:r
4851:H
4845:2
4835:(
4822:d
4817:r
4814:d
4808:+
4803:2
4798:)
4791:2
4787:r
4783:H
4778:(
4765:2
4757:d
4752:r
4747:2
4743:d
4716:2
4712:r
4700:=
4695:2
4678:r
4651:d
4646:r
4643:d
4637:+
4632:2
4607:2
4599:d
4594:r
4589:2
4585:d
4558:2
4554:r
4542:=
4537:2
4520:r
4508:r
4486:.
4463:r
4448:2
4438:)
4436:6
4434:(
4396:d
4391:r
4388:d
4382:+
4377:2
4352:2
4344:d
4339:r
4334:2
4330:d
4323:=
4314:r
4294:)
4292:5
4290:(
4252:d
4247:r
4244:d
4238:=
4229:r
4201:r
4170:4
4164:3
4154:)
4152:4
4150:(
4129:3
4125:r
4114:r
4105:H
4099:2
4090:=
4062:3
4052:)
4050:3
4048:(
4027:2
4023:r
4019:H
4014:=
3964:2
3960:r
3956:=
3952:|
3947:)
3927:2
3923:r
3915:0
3908:0
3902:(
3897:|
3893:=
3889:|
3883:)
3877:0
3858:)
3852:(
3843:r
3840:+
3837:)
3831:(
3816:r
3794:)
3788:(
3779:r
3773:)
3767:(
3752:r
3743:(
3733:)
3727:0
3720:)
3714:(
3705:r
3698:)
3692:(
3683:r
3677:(
3671:|
3667:=
3663:|
3652:r
3641:r
3636:|
3632:=
3629:H
3614:)
3612:2
3610:(
3589:2
3585:r
3573:=
3568:2
3551:r
3539:r
3491:r
3462:q
3455:)
3452:0
3449:(
3446:+
3436:r
3428:)
3420:2
3416:r
3403:(
3399:=
3389:q
3381:)
3359:r
3353:2
3350:+
3335:r
3331:(
3327:+
3317:r
3309:)
3303:2
3286:r
3274:r
3267:(
3255:1
3223:q
3215:)
3193:r
3187:2
3184:+
3169:r
3165:(
3161:+
3151:r
3143:)
3137:2
3120:r
3108:r
3101:(
3097:=
3083:r
3065:q
3046:r
3043:+
3033:r
3020:r
3014:=
3000:r
2982:r
2975:r
2972:=
2968:)
2957:y
2941:+
2931:x
2914:(
2910:r
2907:=
2899:r
2871:r
2837:y
2819:+
2809:x
2788:=
2774:q
2756:y
2738:+
2728:x
2710:=
2696:r
2665:H
2644:)
2634:q
2627:,
2617:r
2610:(
2587:)
2577:y
2570:,
2560:x
2553:(
2532:)
2523:r
2517:(
2495:)
2480:(
2459:,
2455:)
2445:r
2438:(
2426:1
2403:H
2379:0
2375:=
2371:0
2367:+
2363:0
2359:=
2349:r
2338:r
2334:+
2324:r
2307:r
2300:=
2296:)
2285:r
2274:r
2269:(
2262:t
2259:d
2255:d
2250:=
2240:H
2209:r
2198:r
2194:=
2190:H
2176:r
2167:1
2107:i
2083:T
2059:M
2031:(
2000:(
1974:(
1956:i
1948:(
1933:)
1921:e
1913:(
1907:)
1895:a
1887:(
1862:.
1806:M
1786:M
1782:G
1779:=
1761:1
1754:2
1751:m
1747:1
1744:m
1742:(
1740:G
1736:α
1731:2
1728:m
1724:1
1721:m
1699:)
1694:2
1690:m
1686:+
1681:1
1677:m
1673:(
1670:G
1667:=
1627:)
1625:1
1623:(
1600:r
1590:2
1586:r
1574:=
1564:r
1528:r
1498:r
1471:2
1466:r
1456:1
1451:r
1446:=
1442:r
1418:r
1389:r
1379:2
1375:r
1368:2
1364:m
1358:1
1354:m
1350:G
1344:=
1339:2
1328:r
1318:2
1314:m
1288:r
1278:2
1274:r
1267:2
1263:m
1257:1
1253:m
1249:G
1240:=
1235:1
1224:r
1214:1
1210:m
1181:2
1176:r
1152:1
1147:r
1123:2
1119:m
1096:1
1092:m
989:r
965:2
961:m
936:1
932:m
906:G
884:F
855:2
851:r
844:2
840:m
834:1
830:m
823:G
820:=
817:F
787:.
781:G
777:2
774:F
770:1
767:F
763:r
759:2
756:F
752:2
749:m
745:1
742:m
716:r
694:a
671:m
648:F
617:2
613:t
609:d
603:r
597:2
593:d
586:m
583:=
579:a
575:m
572:=
568:F
498:p
473:)
467:(
458:e
455:+
452:1
448:p
443:=
440:)
434:(
431:r
371:e
345:a
323:r
295:)
289:(
280:e
277:+
274:1
269:)
264:2
260:e
253:1
250:(
247:a
241:=
238:)
232:(
229:r
203:.
39:)
33:(
25:.
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