In physics, an orbit is the gravitationally curved trajectory of an object,[1] such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse,[2] as described by Kepler's laws of planetary motion. Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics. For ease of calculation, in most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse square law.[3] Contents 1 History 2 Planetary orbits 2.1 Understanding orbits 3
3.1 Newton's law of gravitation and laws of motion for two-body problems 3.2 Defining gravitational potential energy 3.3 Orbital energies and orbit shapes 3.4 Kepler's laws 3.5 Limitations of Newton's law of gravitation 3.6 Approaches to many-body problems 4 Newtonian analysis of orbital motion 5 Relativistic orbital motion 6 Orbital planes 7 Orbital period 8 Specifying orbits 9 Orbital perturbations 9.1 Radial, prograde and transverse perturbations 9.2 Orbital decay 9.3 Oblateness 9.4 Multiple gravitating bodies 9.5 Light radiation and stellar wind 10 Strange orbits
11 Astrodynamics
12
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v t e Historically, the apparent motions of the planets were described by
European and Arabic philosophers using the idea of celestial spheres.
This model posited the existence of perfect moving spheres or rings to
which the stars and planets were attached. It assumed the heavens were
fixed apart from the motion of the spheres, and was developed without
any understanding of gravity. After the planets' motions were more
accurately measured, theoretical mechanisms such as deferent and
epicycles were added. Although the model was capable of reasonably
accurately predicting the planets' positions in the sky, more and more
epicycles were required as the measurements became more accurate,
hence the model became increasingly unwieldy. Originally geocentric it
was modified by
The lines traced out by orbits dominated by the gravity of a central source are conic sections: the shapes of the curves of intersection between a plane and a cone. Parabolic (1) and hyperbolic (3) orbits are escape orbits, whereas elliptical and circular orbits (2) are captive. This image shows the four trajectory categories with the gravitational potential well of the central mass's field of potential energy shown in black and the height of the kinetic energy of the moving body shown in red extending above that, correlating to changes in speed as distance changes according to Kepler's laws.
A force, such as gravity, pulls an object into a curved path as it attempts to fly off in a straight line. As the object is pulled toward the massive body, it falls toward that body. However, if it has enough tangential velocity it will not fall into the body but will instead continue to follow the curved trajectory caused by that body indefinitely. The object is then said to be orbiting the body. As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). This is a 'thought experiment', in which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle speed. The effects of air friction on the cannonball are ignored (or perhaps the mountain is high enough that the cannon will be above the Earth's atmosphere, which comes to the same thing).[7] Newton's cannonball, an illustration of how objects can "fall" in a curve Conic sections describe the possible orbits (yellow) of small objects
around the Earth. A projection of these orbits onto the gravitational
potential (blue) of the
If the cannon fires its ball with a low initial speed, the trajectory
of the ball curves downward and hits the ground (A). As the firing
speed is increased, the cannonball hits the ground farther (B) away
from the cannon, because while the ball is still falling towards the
ground, the ground is increasingly curving away from it (see first
point, above). All these motions are actually "orbits" in a technical
sense – they are describing a portion of an elliptical path around
the center of gravity – but the orbits are interrupted by striking
the Earth.
If the cannonball is fired with sufficient speed, the ground curves
away from the ball at least as much as the ball falls – so the ball
never strikes the ground. It is now in what could be called a
non-interrupted, or circumnavigating, orbit. For any specific
combination of height above the center of gravity and mass of the
planet, there is one specific firing speed (unaffected by the mass of
the ball, which is assumed to be very small relative to the Earth's
mass) that produces a circular orbit, as shown in (C).
As the firing speed is increased beyond this, non-interrupted elliptic
orbits are produced; one is shown in (D). If the initial firing is
above the surface of the
No orbit Suborbital trajectories Range of interrupted elliptical paths Orbital trajectories (or simply "orbits") Range of elliptical paths with closest point opposite firing point Circular path Range of elliptical paths with closest point at firing point Open (or escape) trajectories Parabolic paths Hyperbolic paths It is worth noting that orbital rockets are launched vertically at first to lift the rocket above the atmosphere (which causes frictional drag), and then slowly pitch over and finish firing the rocket engine parallel to the atmosphere to achieve orbit speed. Once in orbit, their speed keeps them in orbit above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall). Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an aerobraking maneuver
The orbit of a planet around the
Limitations of Newton's law of gravitation[edit]
Note that while bound orbits of a point mass or a spherical body with
a
One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. Still, there are secular phenomena that have to be dealt with by post-Newtonian methods. The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces acting on a body will equal the mass of the body times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach. Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large assemblages of objects have been simulated.[citation needed] Newtonian analysis of orbital motion[edit] (See also Kepler orbit, orbit equation and Kepler's first law.) The
eq 1. F 2 = − G m 1 m 2 r 2 displaystyle F_ 2 =- frac Gm_ 1 m_ 2 r^ 2 Where F2 is the force acting on the mass m2 caused by the gravitational attraction mass m1 has for m2, G is the universal gravitational constant, and r is the distance between the two masses centers. From Newton's Second Law, the summation of the forces acting on m2 related to that bodies acceleration: eq 2. F 2 = m 2 A 2 displaystyle F_ 2 =m_ 2 A_ 2 Where A2 is the acceleration of m2 caused by the force of gravitational attraction F2 of m1 acting on m2. Combining Eq 1 and 2: − G m 1 m 2 r 2 = m 2 A 2 displaystyle - frac Gm_ 1 m_ 2 r^ 2 =m_ 2 A_ 2 Solving for the acceleration, A2: A 2 = F 2 m 2 = − 1 m 2 G m 1 m 2 r 2 = − μ r 2 displaystyle A_ 2 = frac F_ 2 m_ 2 =- frac 1 m_ 2 frac Gm_ 1 m_ 2 r^ 2 =- frac mu r^ 2 where μ displaystyle mu , is the standard gravitational parameter, in this case G m 1 displaystyle Gm_ 1 . It is understood that the system being described is m2, hence the subscripts can be dropped. We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of general relativity. When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance A = F / m = − k r . displaystyle A=F/m=-kr. Due to the way vectors add, the component of the force in the x ^ displaystyle hat mathbf x or in the y ^ displaystyle hat mathbf y directions are also proportionate to the respective components of the distances, r x ″ = A x = − k r x displaystyle r''_ x =A_ x =-kr_ x . Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations x = A cos ( t ) displaystyle x=Acos(t) and y = B sin ( t ) displaystyle y=Bsin(t) of the ellipse. In contrast, with the decreasing relationship A = μ / r 2 displaystyle A=mu /r^ 2 , the dimensions cannot be separated.[citation needed] The location of the orbiting object at the current time t displaystyle t is located in the plane using
r displaystyle r be the distance between the object and the center and θ displaystyle theta be the angle it has rotated. Let x ^ displaystyle hat mathbf x and y ^ displaystyle hat mathbf y be the standard Euclidean bases and let r ^ = cos ( θ ) x ^ + sin ( θ ) y ^ displaystyle hat mathbf r =cos(theta ) hat mathbf x +sin(theta ) hat mathbf y and θ ^ = − sin ( θ ) x ^ + cos ( θ ) y ^ displaystyle hat boldsymbol theta =-sin(theta ) hat mathbf x +cos(theta ) hat mathbf y be the radial and transverse polar basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is O ^ = r cos ( θ ) x ^ + r sin ( θ ) y ^ = r r ^ displaystyle hat mathbf O =rcos(theta ) hat mathbf x +rsin(theta ) hat mathbf y =r hat mathbf r We use r ˙ displaystyle dot r and θ ˙ displaystyle dot theta to denote the standard derivatives of how this distance and angle change over time. We take the derivative of a vector to see how it changes over time by subtracting its location at time t displaystyle t from that at time t + δ t displaystyle t+delta t and dividing by δ t displaystyle delta t . The result is also a vector. Because our basis vector r ^ displaystyle hat mathbf r moves as the object orbits, we start by differentiating it. From time t displaystyle t to t + δ t displaystyle t+delta t , the vector r ^ displaystyle hat mathbf r keeps its beginning at the origin and rotates from angle θ displaystyle theta to θ + θ ˙ δ t displaystyle theta + dot theta delta t which moves its head a distance θ ˙ δ t displaystyle dot theta delta t in the perpendicular direction θ ^ displaystyle hat boldsymbol theta giving a derivative of θ ˙ θ ^ displaystyle dot theta hat boldsymbol theta . r ^ = cos ( θ ) x ^ + sin ( θ ) y ^ displaystyle hat mathbf r =cos(theta ) hat mathbf x +sin(theta ) hat mathbf y δ r ^ δ t = r ˙ = − sin ( θ ) θ ˙ x ^ + cos ( θ ) θ ˙ y ^ = θ ˙ θ ^ displaystyle frac delta hat mathbf r delta t = dot mathbf r =-sin(theta ) dot theta hat mathbf x +cos(theta ) dot theta hat mathbf y = dot theta hat boldsymbol theta θ ^ = − sin ( θ ) x ^ + cos ( θ ) y ^ displaystyle hat boldsymbol theta =-sin(theta ) hat mathbf x +cos(theta ) hat mathbf y δ θ ^ δ t = θ ˙ = − cos ( θ ) θ ˙ x ^ − sin ( θ ) θ ˙ y ^ = − θ ˙ r ^ displaystyle frac delta hat boldsymbol theta delta t = dot boldsymbol theta =-cos(theta ) dot theta hat mathbf x -sin(theta ) dot theta hat mathbf y =- dot theta hat mathbf r We can now find the velocity and acceleration of our orbiting object. O ^ = r r ^ displaystyle hat mathbf O =r hat mathbf r O ˙ = δ r δ t r ^ + r δ r ^ δ t = r ˙ r ^ + r [ θ ˙ θ ^ ] displaystyle dot mathbf O = frac delta r delta t hat mathbf r +r frac delta hat mathbf r delta t = dot r hat mathbf r +r[ dot theta hat boldsymbol theta ] O ¨ = [ r ¨ r ^ + r ˙ θ ˙ θ ^ ] + [ r ˙ θ ˙ θ ^ + r θ ¨ θ ^ − r θ ˙ 2 r ^ ] displaystyle ddot mathbf O =[ ddot r hat mathbf r + dot r dot theta hat boldsymbol theta ]+[ dot r dot theta hat boldsymbol theta +r ddot theta hat boldsymbol theta -r dot theta ^ 2 hat mathbf r ] = [ r ¨ − r θ ˙ 2 ] r ^ + [ r θ ¨ + 2 r ˙ θ ˙ ] θ ^ displaystyle =[ ddot r -r dot theta ^ 2 ] hat mathbf r +[r ddot theta +2 dot r dot theta ] hat boldsymbol theta The coefficients of r ^ displaystyle hat mathbf r and θ ^ displaystyle hat boldsymbol theta give the accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity is − μ / r 2 displaystyle -mu /r^ 2 and the second is zero. r ¨ − r θ ˙ 2 = − μ r 2 displaystyle ddot r -r dot theta ^ 2 =- frac mu r^ 2
(1) r θ ¨ + 2 r ˙ θ ˙ = 0 displaystyle r ddot theta +2 dot r dot theta =0
(2) Equation (2) can be rearranged using integration by parts. r θ ¨ + 2 r ˙ θ ˙ = 1 r d d t ( r 2 θ ˙ ) = 0 displaystyle r ddot theta +2 dot r dot theta = frac 1 r frac d dt left(r^ 2 dot theta right)=0 We can multiply through by r displaystyle r because it is not zero unless the orbiting object crashes. Then having the derivative be zero gives that the function is a constant. r 2 θ ˙ = h displaystyle r^ 2 dot theta =h
(3) which is actually the theoretical proof of
r displaystyle r of the orbiting object from the center as a function of its angle θ displaystyle theta . However, it is easier to introduce the auxiliary variable u = 1 / r displaystyle u=1/r and to express u displaystyle u as a function of θ displaystyle theta . Derivatives of r displaystyle r with respect to time may be rewritten as derivatives of u displaystyle u with respect to angle. u = 1 r displaystyle u= 1 over r θ ˙ = h r 2 = h u 2 displaystyle dot theta = frac h r^ 2 =hu^ 2 (reworking (3)) δ u δ θ = δ δ t ( 1 r ) δ t δ θ = − r ˙ r 2 θ ˙ = − r ˙ h δ 2 u δ θ 2 = − 1 h δ r ˙ δ t δ t δ θ = − r ¨ h θ ˙ = − r ¨ h 2 u 2 o r r ¨ = − h 2 u 2 δ 2 u δ θ 2 displaystyle begin aligned & frac delta u delta theta = frac delta delta t left( frac 1 r right) frac delta t delta theta =- frac dot r r^ 2 dot theta =- frac dot r h \& frac delta ^ 2 u delta theta ^ 2 =- frac 1 h frac delta dot r delta t frac delta t delta theta =- frac ddot r h dot theta =- frac ddot r h^ 2 u^ 2 or ddot r =-h^ 2 u^ 2 frac delta ^ 2 u delta theta ^ 2 \end aligned Plugging these into (1) gives r ¨ − r θ ˙ 2 = − μ r 2 displaystyle ddot r -r dot theta ^ 2 =- frac mu r^ 2 − h 2 u 2 δ 2 u δ θ 2 − 1 u ( h u 2 ) 2 = − μ u 2 displaystyle -h^ 2 u^ 2 frac delta ^ 2 u delta theta ^ 2 - frac 1 u (hu^ 2 )^ 2 =-mu u^ 2 δ 2 u δ θ 2 + u = μ h 2 displaystyle frac delta ^ 2 u delta theta ^ 2 +u= frac mu h^ 2 So for the gravitational force – or, more generally, for any inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is: u ( θ ) = μ h 2 − A cos ( θ − θ 0 ) displaystyle u(theta )= frac mu h^ 2 -Acos(theta -theta _ 0 ) where A and θ0 are arbitrary constants. This resulting equation of the orbit of the object is that of an ellipse in Polar form relative to one of the focal points. This is put into a more standard form by letting e ≡ h 2 A / μ displaystyle eequiv h^ 2 A/mu be the eccentricity, letting a ≡ h 2 / ( μ ( 1 − e 2 ) ) displaystyle aequiv h^ 2 /(mu (1-e^ 2 )) be the semi-major axis. Finally, letting θ 0 ≡ 0 displaystyle theta _ 0 equiv 0 so the long axis of the ellipse is along the positive x coordinate. r ( θ ) = a ( 1 − e 2 ) 1 − e cos θ displaystyle r(theta )= frac a(1-e^ 2 ) 1-ecos theta Relativistic orbital motion[edit]
The above classical (Newtonian) analysis of orbital mechanics assumes
that the more subtle effects of general relativity, such as frame
dragging and gravitational time dilation are negligible. Relativistic
effects cease to be negligible when near very massive bodies (as with
the precession of Mercury's orbit about the Sun), or when extreme
precision is needed (as with calculations of the orbital elements and
time signal references for
In principle once the orbital elements are known for a body, its
position can be calculated forward and backwards indefinitely in time.
However, in practice, orbits are affected or perturbed, by other
forces than simple gravity from an assumed point source (see the next
section), and thus the orbital elements change over time.
Orbital perturbations[edit]
An orbital perturbation is when a force or impulse which is much
smaller than the overall force or average impulse of the main
gravitating body and which is external to the two orbiting bodies
causes an acceleration, which changes the parameters of the orbit over
time.
Radial, prograde and transverse perturbations[edit]
A small radial impulse given to a body in orbit changes the
eccentricity, but not the orbital period (to first order). A prograde
or retrograde impulse (i.e. an impulse applied along the orbital
motion) changes both the eccentricity and the orbital period. Notably,
a prograde impulse at periapsis raises the altitude at apoapsis, and
vice versa, and a retrograde impulse does the opposite. A transverse
impulse (out of the orbital plane) causes rotation of the orbital
plane without changing the period or eccentricity. In all instances, a
closed orbit will still intersect the perturbation point.
Orbital decay[edit]
Main article: Orbital decay
If an orbit is about a planetary body with significant atmosphere, its
orbit can decay because of drag. Particularly at each periapsis, the
object experiences atmospheric drag, losing energy. Each time, the
orbit grows less eccentric (more circular) because the object loses
kinetic energy precisely when that energy is at its maximum. This is
similar to the effect of slowing a pendulum at its lowest point; the
highest point of the pendulum's swing becomes lower. With each
successive slowing more of the orbit's path is affected by the
atmosphere and the effect becomes more pronounced. Eventually, the
effect becomes so great that the maximum kinetic energy is not enough
to return the orbit above the limits of the atmospheric drag effect.
When this happens the body will rapidly spiral down and intersect the
central body.
The bounds of an atmosphere vary wildly. During a solar maximum, the
Earth's atmosphere causes drag up to a hundred kilometres higher than
during a solar minimum.
Some satellites with long conductive tethers can also experience
orbital decay because of electromagnetic drag from the Earth's
magnetic field. As the wire cuts the magnetic field it acts as a
generator, moving electrons from one end to the other. The orbital
energy is converted to heat in the wire.
Orbits can be artificially influenced through the use of rocket
engines which change the kinetic energy of the body at some point in
its path. This is the conversion of chemical or electrical energy to
kinetic energy. In this way changes in the orbit shape or orientation
can be facilitated.
Another method of artificially influencing an orbit is through the use
of solar sails or magnetic sails. These forms of propulsion require no
propellant or energy input other than that of the Sun, and so can be
used indefinitely. See statite for one such proposed use.
Comparison of geostationary
Main article: List of orbits Low
Scaling in gravity[edit] The gravitational constant G has been calculated as: (6.6742 ± 0.001) × 10−11 (kg/m3)−1s−2. Thus the constant has dimension density−1 time−2. This corresponds to the following properties. Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the Earth. Scaling of distances while keeping the masses the same (in the case of point masses, or by reducing the densities) gives similar orbits; if distances are multiplied by 4, gravitational forces and accelerations are divided by 16, velocities are halved and orbital periods are multiplied by 8. When all densities are multiplied by 4, orbits are the same; gravitational forces are multiplied by 16 and accelerations by 4, velocities are doubled and orbital periods are halved. When all densities are multiplied by 4, and all sizes are halved, orbits are similar; masses are divided by 2, gravitational forces are the same, gravitational accelerations are doubled. Hence velocities are the same and orbital periods are halved. In all these cases of scaling. if densities are multiplied by 4, times are halved; if velocities are doubled, forces are multiplied by 16. These properties are illustrated in the formula (derived from the formula for the orbital period) G T 2 σ = 3 π ( a r ) 3 , displaystyle GT^ 2 sigma =3pi left( frac a r right)^ 3 , for an elliptical orbit with semi-major axis a, of a small body around
a spherical body with radius r and average density σ, where T is the
orbital period. See also Kepler's Third Law.
Patents[edit]
The application of certain orbits or orbital maneuvers to specific
useful purposes have been the subject of patents.[17]
Tidal locking[edit]
Main article: Tidal locking
Some bodies are tidally locked with other bodies, meaning that one
side of the celestial body is permanently facing its host object. This
is the case for Sun-Mercury, Earth-
Astronomy portal
Klemperer rosette List of orbits Molniya orbit Orbital spaceflight Perifocal coordinate system Polar Orbits Radial trajectory Rosetta (orbit) VSOP (planets) Notes[edit] ^ Orbital periods and speeds are calculated using the relations
4π²R³ = T²GM and V²R = GM, where R = radius
of orbit in metres, T = orbital period in seconds, V = orbital speed
in m/s, G = gravitational constant ≈ 6.673×10−11 Nm²/kg²,
M = mass of
References[edit] ^ orbit (astronomy) – Britannica Online Encyclopedia
^ The Space Place :: What's a Barycenter
^ Kuhn, The Copernican Revolution, pp. 238, 246–252
^ Encyclopædia Britannica, 1968, vol. 2, p. 645
^ M Caspar, Kepler (1959, Abelard-Schuman), at pp.131–140; A Koyré,
The Astronomical Revolution: Copernicus, Kepler, Borelli (1973,
Methuen), pp. 277–279
^ Jones, Andrew. "Kepler's Laws of Planetary Motion". about.com.
Retrieved 2008-06-01.
^ See pages 6 to 8 in Newton's "Treatise of the System of the World"
(written 1685, translated into English 1728, see Newton's 'Principia'
– A preliminary version), for the original version of this
'cannonball' thought-experiment.
^ Fitzpatrick, Richard (2006-02-02). "Planetary orbits". Classical
Mechanics – an introductory course. The University of Texas at
Austin. Archived from the original on 3 March 2001. Retrieved
2009-01-14.
^ Pogge, Richard W.; "Real-World Relativity: The
Further reading[edit] Abell; Morrison & Wolff (1987). Exploration of the Universe (fifth
ed.). Saunders College Publishing.
Linton, Christopher (2004). From Eudoxus to Einstein. Cambridge:
University Press. ISBN 0-521-82750-7
Swetz, Frank; et al. (1997). Learn from the Masters!. Mathematical
Association of America. ISBN 0-88385-703-0
Andrea Milani and Giovanni F. Gronchi. Theory of
External links[edit] Look up orbit in Wiktionary, the free dictionary. Wikimedia Commons has media related to Orbits. CalcTool:
v t e Gravitational orbits Types General Box
Capture
Circular
semi sub Transfer orbit Geocentric Geosynchronous
Geostationary
Sun-synchronous
Low Earth
Medium Earth
High Earth
Molniya
Near-equatorial
About other points Areosynchronous Areostationary Halo Lissajous Lunar Heliocentric Heliosynchronous Parameters Shape Size e Eccentricity a Semi-major axis b Semi-minor axis Q, q Apsides Orientation i Inclination Ω Longitude of the ascending node ω Argument of periapsis ϖ Longitude of the periapsis Position M Mean anomaly ν, θ, f True anomaly E Eccentric anomaly L Mean longitude l True longitude Variation T Orbital period n Mean motion v Orbital speed t0 Epoch Maneuvers Collision avoidance (spacecraft)
Delta-v
Orbital mechanics Celestial coordinate system
Characteristic energy
Escape velocity
Ephemeris
Equatorial coordinate system
Ground track
Hill sphere
Interplanetary Transport Network
Kepler's laws of planetary motion
Lagrangian point
n-body problem
Li |