Orbital mechanics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers. General relativity is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in highgravity situations (such as orbits close to the Sun).
Contents
History
Until the rise of space travel in the twentieth century, there was little distinction between orbital and celestial mechanics. The fundamental techniques, such as those used to solve the Keplerian problem (determining position as a function of time), are therefore the same in both fields. Furthermore, the history of the fields is almost entirely shared.
Johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws in 1605. Isaac Newton published more general laws of celestial motion in his 1687 book, Philosophiæ Naturalis Principia Mathematica.
Practical techniques
Further information: List of orbitsRules of thumb
The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics. The specific example discussed is of a satellite orbiting a planet, but the rules of thumb could also apply to other situations, such as orbits of small bodies around a star such as the Sun.
 Kepler's laws of planetary motion, which can be mathematically derived from Newton's laws, hold strictly only in describing the motion of two gravitating bodies, in the absence of nongravitational forces, or approximately when the gravity of a single massive body like the Sun dominates other effects:
 Orbits are elliptical, with the planet at one focus of the ellipse. Special cases of this are circular orbits (a circle being simply an ellipse of zero eccentricity) with the planet at the center, and parabolic orbits (which are ellipses with eccentricity of exactly 1, which is simply an infinitely long ellipse) with the planet at the focus.
 A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured.
 The square of a satellite's orbital period is proportional to the cube of its average distance from the planet.
 Without applying thrust (such as firing a rocket engine), the height and shape of the satellite's orbit won't change, and it will maintain the same orientation with respect to the fixed stars.
 A satellite in a low orbit (or low part of an elliptical orbit) moves more quickly with respect to the surface of the planet than a satellite in a higher orbit (or a high part of an elliptical orbit), due to the stronger gravitational attraction closer to the planet.
 If thrust is applied at only one point in the satellite's orbit, it will return to that same point on each subsequent orbit, though the rest of its path will change. Thus to move from one circular orbit to another, at least two brief applications of thrust are needed.
 From a circular orbit, thrust in a direction which slows the satellite down will create an elliptical orbit with a lower periapse (lowest orbital point) at 180 degrees away from the firing point. If thrust is applied to speed the satellite, it will create an elliptical orbit with a higher apoapse 180 degrees away from the firing point.
The consequences of the rules of orbital mechanics are sometimes counterintuitive. For example, if two spacecraft are in the same circular orbit and wish to dock, unless they are very close, the trailing craft cannot simply fire its engines to go faster. This will change the shape of its orbit, causing it to gain altitude and miss its target. One approach is to actually fire a reverse thrust to slow down, and then fire again to recircularize the orbit at a lower altitude. Because lower orbits are faster than higher orbits, the trailing craft will begin to catch up. A third firing at the right time will put the trailing craft in an elliptical orbit which will intersect the path of the leading craft, approaching from below.
To the degree that the standard assumptions of astrodynamics do not hold, actual trajectories will vary from those calculated. For example, simple atmospheric drag is another complicating factor for objects in Earth orbit. These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system (see nbody problem). (Celestial mechanics uses more general rules applicable to a wider variety of situations.) The differences between classical mechanics and general relativity can also become important for large objects like planets.
Orbital transfer
Transfer orbits are usually elliptical orbits that allow spacecraft to move from one (usually substantially circular) orbit to another. Usually they require a burn at the start, a burn at the end, and sometimes one or more burns in the middle.
 Hohmann transfer orbit requires a minimal deltav.
 Bielliptic transfers can sometimes require less energy than a Hohmann transfer.
 For faster transfers, any orbit that intersects both the original orbit and destination orbit may be used.
Gravity assist and the Oberth effect
In a gravity assist, a spacecraft swings by a planet and leaves in a different direction, at a different speed. This is useful to speed or slow a spacecraft instead of carrying more fuel.
This maneuver can be approximated by an elastic collision at large distances, though the flyby does not involve any physical contact. Due to Newton's Third Law (equal and opposite reaction), any momentum gained by a spacecraft must be lost by the planet, or vice versa. However, because the planet is much, much more massive than the spacecraft, the effect on the planet's orbit is negligible.
The Oberth effect can be employed, particularly during a gravity assist operation. This effect is that use of a propulsion system works better at high speeds, and hence course changes are best done when close to a gravitating body; this can multiply the effective deltav.
Interplanetary Transport Network and fuzzy orbits
Main article: Interplanetary Transport NetworkSee also: Low energy transfersIt is now possible to use computers to search for routes using the nonlinearities in the gravity of the planets and moons of the solar system. For example, it is possible to plot an orbit from high earth orbit to Mars, passing close to one of the Earth's Trojan points. Collectively referred to as the Interplanetary Transport Network, these highly perturbative, even chaotic, orbital trajectories in principle need no fuel (in practice keeping to the trajectory requires some course corrections). The biggest problem with them is they can be exceedingly slow, taking many years to arrive. In addition launch windows can be very far apart.
They have, however, been employed on projects such as Genesis. This spacecraft visited Earth's Lagrange L_{1} point and returned using very little propellant.
Laws of astrodynamics
See also: Laplace–Runge–Lenz vectorThe fundamental laws of astrodynamics are Newton's law of universal gravitation and Newton's laws of motion, while the fundamental mathematical tool is his differential calculus.
Every orbit and trajectory outside atmospheres is in principle reversible, i.e., in the spacetime function the time is reversed. The velocities are reversed and the accelerations are the same, including those due to rocket bursts. Thus if a rocket burst is in the direction of the velocity, in the reversed case it is opposite to the velocity. Of course in the case of rocket bursts there is no full reversal of events, both ways the same deltav is used and the same mass ratio applies.
Standard assumptions in astrodynamics include noninterference from outside bodies, negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc.). More accurate calculations can be made without these simplifying assumptions, but they are more complicated. The increased accuracy often does not make enough of a difference in the calculation to be worthwhile.
Kepler's laws of planetary motion may be derived from Newton's laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor. When an engine thrust or propulsive force is present, Newton's laws still apply, but Kepler's laws are invalidated. When the thrust stops, the resulting orbit will be different but will once again be described by Kepler's laws. The three laws are:
 The orbit of every planet is an ellipse with the sun at one of the foci.
 A line joining a planet and the sun sweeps out equal areas during equal intervals of time.
 The squares of the orbital periods of planets are directly proportional to the cubes of the semimajor axis of the orbits.
Escape velocity
The formula for escape velocity is easily derived as follows. The specific energy (energy per unit mass) of any space vehicle is composed of two components, the specific potential energy and the specific kinetic energy. The specific potential energy associated with a planet of mass M is given by
while the specific kinetic energy of an object is given by
Since energy is conserved, the total specific orbital energy
does not depend on the distance, r, from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite r only if this quantity is nonnegative, which implies
The escape velocity from the Earth's surface is about 11 km/s, but that is insufficient to send the body an infinite distance because of the gravitational pull of the Sun. To escape the solar system from a location at a distance from the Sun equal to the distance SunEarth, but not close to the Earth, requires around 42 km/s velocity, but there will be "part credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration (due to the propulsion system) carries them in the same direction as Earth travels in its orbit.
Formulae for free orbits
Orbits are conic sections, so, naturally, the formulas for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is:
where μ is called the gravitational parameter, G is the gravitational constant, m_{1} and m_{2} are the masses of objects 1 and 2, and h is the angular momentum of object 2 with respect to object 1. The parameter θ is known as the true anomaly, p is the semilatus rectum, while e is the orbital eccentricity, all obtainable from the various forms of the six independent orbital elements.
Circular orbits
All bounded orbits where the gravity of a central body dominates are elliptical in nature. A special case of this is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit at distance r from the center of gravity of mass M is
where G is the gravitational constant, equal to
 6.672 598 × 10^{−11} m^{3}/(kg·s^{2})
To properly use this formula, the units must be consistent; for example, M must be in kilograms, and r must be in meters. The answer will be in meters per second.
The quantity GM is often termed the standard gravitational parameter, which has a different value for every planet or moon in the solar system.
Once the circular orbital velocity is known, the escape velocity is easily found by multiplying by the square root of 2:
Elliptical Orbits
If 0<e<1, then the denominator of the equation of free orbits varies with the true anomaly θ, but remains positive, never becoming zero. Therefore, the relative position vector remains bounded, having its smallest magnitude at periapsis r_{p} which is given by:
The maximum value r is reached when θ = 180. This point is called the apoapsis, and its radial coordinate, denoted r_{a}, is
Let 2a be the distance measured along the apse line from periapsis P to apoapsis A, as illustrated in the equation below:
 2a = r_{p} + r_{a}
Substituting the equations above, we get:
a is the semimajor axis of the ellipse. Solving for r we get:
Orbital period
Under standard assumptions the orbital period () of a body traveling along an elliptic orbit can be computed as:
where:
 is standard gravitational parameter,
 is length of semimajor axis.
Conclusions:
 The orbital period is equal to that for a circular orbit with the orbit radius equal to the semimajor axis (),
 For a given semimajor axis the orbital period does not depend on the eccentricity (See also: Kepler's third law).
Velocity
Under standard assumptions the orbital speed () of a body traveling along elliptic orbit can be computed from the Visviva equation as:
where:
 is the standard gravitational parameter,
 is the distance between the orbiting bodies.
 is the length of the semimajor axis.
The velocity equation for a hyperbolic trajectory has either + , or it is the same with the convention that in that case a is negative.
Energy
Under standard assumptions, specific orbital energy () of elliptic orbit is negative and the orbital energy conservation equation (the Visviva equation) for this orbit can take the form:
where:
 is orbital speed of orbiting body,
 is distance of orbiting body from central body,
 is length of semimajor axis,
 is standard gravitational parameter.
Conclusions:
 For a given semimajor axis the specific orbital energy is independent of the eccentricity.
Using the virial theorem we find:
 the timeaverage of the specific potential energy is equal to 2ε
 the timeaverage of r^{−1} is a^{−1}
 the timeaverage of the specific kinetic energy is equal to ε
Parabolic Orbits
If the eccentricity equals 1, then the orbit equation becomes:
where:
 is radial distance of orbiting body from central body,
 is specific angular momentum of the orbiting body,
 is a true anomaly of the orbiting body,
 is the standard gravitational parameter.
As the true anomaly θ approaches 180°,the denominator approaches zero, so that r tends towards infinity. Hence, the energy of the trajectory for which e=1 is zero, and is given by:
where:
 is the orbital velocity of the orbiting body.
In other words, the speed anywhere on a parabolic path is:
Hyperbolic Orbits
If e>1, the orbit formula,
describes the geometry of the hyperbolic orbit. The system consists of two symmetric curves. the orbiting body occupies one of them. The other one is its empty mathematical image. Clearly, the denominator of the equation above goes to zero when cosθ = 1/e. we denote this value of true anomaly

θ_{∞} = cos^{1}(1/e)
since the radial distance approaches infinity as the true anomaly approaches θ_{∞}. θ_{∞} is known as the true anomaly of the asymptote. Observe that θ_{∞} lies between 90° and 180°. From the trig identity sin^{2}θ+cos^{2}θ=1 it follows that:

sinθ_{∞} = (e^{2}1)^{1/2}/e
Energy
Under standard assumptions, specific orbital energy () of a hyperbolic trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form:
where:
 is orbital velocity of orbiting body,
 is radial distance of orbiting body from central body,
 is the negative semimajor axis,
 is standard gravitational parameter.
Hyperbolic excess velocity
See also: Characteristic energyUnder standard assumptions the body traveling along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity () that can be computed as:
where:
 is standard gravitational parameter,
 is the negative semimajor axis of orbit's hyperbola.
The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by
Mathematical techniques
Kepler's equation
One approach to calculating orbits (mainly used historically) is to use Kepler's equation:
 .
where M is the mean anomaly, E is the eccentric anomaly, and is the eccentricity.
With Kepler's formula, finding the timeofflight to reach an angle (true anomaly) of θ from periapsis is broken into two steps:
 Compute the eccentric anomaly E from true anomaly θ
 Compute the timeofflight t from the eccentric anomaly E
Finding the eccentric anomaly at a given time (the inverse problem) is more difficult. Kepler's equation is transcendental in E, meaning it cannot be solved for E algebraically. Kepler's equation can be solved for E analytically by inversion.
A solution of Kepler's equation, valid for all real values of is:
Evaluating this yields:
Alternatively, Kepler's Equation can be solved numerically. First one must guess a value of E and solve for timeofflight; then adjust E as necessary to bring the computed timeofflight closer to the desired value until the required precision is achieved. Usually, Newton's method is used to achieve relatively fast convergence.The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits. For nearparabolic orbits, eccentricity is nearly 1, and plugging e = 1 into the formula for mean anomaly, E − sin E, we find ourselves subtracting two nearlyequal values, and accuracy suffers. For nearcircular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all). Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it does not hold for parabolic or hyperbolic orbits. These difficulties are what led to the development of the universal variable formulation, described below.
Conic orbits
For simple procedures, such as computing the deltav for coplanar transfer ellipses, traditional approaches^{[clarification needed]} are fairly effective. Others, such as timeofflight are far more complicated, especially for nearcircular and hyperbolic orbits.
The patched conic approximation
Main article: Patched Conic ApproximationThe Hohmann transfer orbit alone is a poor approximation for interplanetary trajectories because it neglects the planets' own gravity. Planetary gravity dominates the behaviour of the spacecraft in the vicinity of a planet and in most cases Hohmann severely overestimates deltav, and produces highly inaccurate prescriptions for burn timings.
A relatively simple way to get a firstorder approximation of deltav is based on the 'Patched Conic Approximation' technique. One must choose the one dominant gravitating body in each region of space through which the trajectory will pass, and to model only that body's effects in that region. For instance, on a trajectory from the Earth to Mars, one would begin by considering only the Earth's gravity until the trajectory reaches a distance where the Earth's gravity no longer dominates that of the Sun. The spacecraft would be given escape velocity to send it on its way to interplanetary space. Next, one would consider only the Sun's gravity until the trajectory reaches the neighbourhood of Mars. During this stage, the transfer orbit model is appropriate. Finally, only Mars's gravity is considered during the final portion of the trajectory where Mars's gravity dominates the spacecraft's behaviour. The spacecraft would approach Mars on a hyperbolic orbit, and a final retrograde burn would slow the spacecraft enough to be captured by Mars.
The size of the "neighborhoods" (or spheres of influence) vary with radius r_{SOI}:
where a_{p} is the semimajor axis of the planet's orbit relative to the Sun; m_{p} and m_{s} are the masses of the planet and Sun, respectively.
This simplification is sufficient to compute rough estimates of fuel requirements, and rough timeofflight estimates, but it is not generally accurate enough to guide a spacecraft to its destination. For that, numerical methods are required.
The universal variable formulation
To address computational shortcomings of traditional approaches for solving the 2body problem, the universal variable formulation was developed. It works equally well for the circular, elliptical, parabolic, and hyperbolic cases, the differential equations converging well when integrated for any orbit. It also generalizes well to problems incorporating perturbation theory.
Perturbations
The universal variable formulation works well with the variation of parameters technique, except now, instead of the six Keplerian orbital elements, we use a different set of orbital elements: namely, the satellite's initial position and velocity vectors x_{0} and v_{0} at a given epoch t = 0. In a twobody simulation, these elements are sufficient to compute the satellite's position and velocity at any time in the future, using the universal variable formulation. Conversely, at any moment in the satellite's orbit, we can measure its position and velocity, and then use the universal variable approach to determine what its initial position and velocity would have been at the epoch. In perfect twobody motion, these orbital elements would be invariant (just like the Keplerian elements would be).
However, perturbations cause the orbital elements to change over time. Hence, we write the position element as x_{0}(t) and the velocity element as v_{0}(t), indicating that they vary with time. The technique to compute the effect of perturbations becomes one of finding expressions, either exact or approximate, for the functions x_{0}(t) and v_{0}(t).
Nonideal orbits
The following are some effects which make real orbits differ from the simple models based on a spherical earth. Most of them can be handled on short timescales (perhaps less than a few thousand orbits) by perturbation theory because they are small relative to the corresponding twobody effects.
 Equatorial bulges cause precession of the node and the perigee
 Tesseral harmonics^{[1]} of the gravity field introduce additional perturbations
 Lunar and solar gravity perturbations alter the orbits
 Atmospheric drag reduces the semimajor axis unless makeup thrust is used
Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behaviour can become chaotic. On the other hand, the various perturbations can be orchestrated by clever astrodynamicists to assist with orbit maintenance tasks, such as stationkeeping, ground track maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude.
See also
 Kepler orbit
 Spacecraft propulsion
 Tsiolkovsky rocket equation
 Aerodynamics
 Astrophysics
 Celestial mechanics
 Chaos theory
 Lagrangian point
 Nbody problem
 Orbit
 Roche limit
 Canonical units
 Aerospace Engineering
 Mechanical Engineering
References
 Curtis, Howard D., (2009). Orbital Mechanics for Engineering Students, 2e. New York: Elsevier. ISBN 9780123747785.
 Bate, Roger R.; Mueller, Donald D., and White, Jerry E. (1971). Fundamentals of Astrodynamics. New York: Dover Publications. ISBN 0486600610.
 Sellers, Jerry J.; Astore, William J., Giffen, Robert B., Larson, Wiley J. (2004). Kirkpatrick, Douglas H.. ed. Understanding Space: An Introduction to Astronautics (2 ed.). McGraw Hill. pp. 228. ISBN 0072424680.
 "Air University Space Primer, Chapter 8  Orbital Mechanics". USAF. http://space.au.af.mil/primer/orbital_mechanics.pdf.
External links
 ORBITAL MECHANICS (Rocket and Space Technology)
 Java Astrodynamics Toolkit
Further reading
Many of the options, procedures, and supporting theory are covered in standard works such as:
 Bate, R.R., Mueller, D.D., White, J.E., (1971). Fundamentals of Astrodynamics. Dover Publications, New York. ISBN 9780486600611.
 Vallado, D. A. (2001). Fundamentals of Astrodynamics and Applications, 2nd Edition. Springer. ISBN 9780792369035.
 Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics & Ast, Washington, D.C.. ISBN 9781563473425.
 Chobotov, V.A. (ed.) (2002). Orbital Mechanics, 3rd Edition. American Institute of Aeronautics & Ast, Washington, D.C.. ISBN 9781563475375.
 Herrick, S. (1971). Astrodynamics: Orbit Determination, Space Navigation, Celestial Mechanics, Volume 1. Van Nostrand Reinhold, London. ISBN 9780442033705.
 Herrick, S. (1972). Astrodynamics: Orbit Correction, Perturbation Theory, Integration, Volume 2. Van Nostrand Reinhold, London. ISBN 9780442033712.
 Kaplan, M.H. (1976). Modern Spacecraft Dynamics and Controls. Wiley, New York. ISBN 9780471457039.
 Tom Logsdon (1997). Orbital Mechanics. WileyInterscience, New York. ISBN 9780471146360.
 John E. Prussing and Bruce A. Conway (1993). Orbital Mechanics. Oxford University Press, New York. ISBN 9780195078343.
 M.J. Sidi (2000). Spacecraft Dynamics and Control. Cambridge University Press, New York. ISBN 9780521787802.
 W.E. Wiesel (1996). Spaceflight Dynamics, 2nd edition. McGrawHill, New York. ISBN 9780070701106.
 J.P. Vinti (1998). Orbital and Celestial Mechanics. American Institute of Aeronautics & Ast, Reston, Virginia. ISBN 9781563472565.
 P. Gurfil (2006). Modern Astrodynamics. ButterworthHeinemann. ISBN 9780123735621.
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 Kepler's laws of planetary motion, which can be mathematically derived from Newton's laws, hold strictly only in describing the motion of two gravitating bodies, in the absence of nongravitational forces, or approximately when the gravity of a single massive body like the Sun dominates other effects:
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