Orbital mechanics
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Orbital mechanics or astrodynamics is the study of 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, collectively known as classical mechanics. Celestial mechanics focuses more broadly on the orbital motions of artificial and natural astronomical bodies such as planets, moons, and comets. Orbital mechanics is a subfield[citation needed] which focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsion. General relativity provides more exact equations for calculating orbits, sometimes necessary for greater accuracy or high-gravity situations (such as orbits close to the Sun).
Rules of thumbThe 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.
The consequences of the rules of orbital mechanics are sometimes counter-intuitive. 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 re-circularize 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 assumptions do not hold, actual trajectories will vary from those calculated. Atmospheric drag is one major complicating factor for objects in Earth orbit. The differences between classical mechanics and general relativity can become important for large objects like planets. These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system. Laws of astrodynamicsThe 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. Standard assumptions in astrodynamics include non-interference 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. Escape velocityThe 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, Failed to parse (Missing texvc executable; please see math/README to configure.): r , from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite Failed to parse (Missing texvc executable; please see math/README to configure.): 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 the vicinity of 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 orbitsOrbits are conic sections, so, naturally, the formula for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is:
. The parameters are given by the orbital elements. Circular orbitsAlthough most orbits are elliptical in nature, a special case 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
is the gravitational constant, equal to
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:
Historical approachesUntil 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, are therefore the same in both fields. Furthermore, the history of the fields is almost entirely shared. Kepler's equation
Kepler was the first to successfully model planetary orbits to a high degree of accuracy. DerivationTo compute the position of a satellite at a given time (the Keplerian problem) is a difficult problem. The opposite problem—to compute the time-of-flight given the starting and ending positions—is simpler. We present a derivation for the time-of-flight equation here. 
Kepler's construction for deriving the time-of-flight equation. The bold ellipse is the satellite's orbit, with the star or planet at one focus Q. The goal is to compute the time required for a satellite to travel from periapsis P to a given point S. Kepler circumscribed the blue auxiliary circle around the ellipse, and used it to derive his time-of-flight equation in terms of eccentric anomaly.
The problem is to find the time Failed to parse (Missing texvc executable; please see math/README to configure.): t at which the satellite reaches point Failed to parse (Missing texvc executable; please see math/README to configure.): S , given that it is at periapsis Failed to parse (Missing texvc executable; please see math/README to configure.): P at time Failed to parse (Missing texvc executable; please see math/README to configure.): t = 0 . We are given that the semimajor axis of the orbit is Failed to parse (Missing texvc executable; please see math/README to configure.): a , and the semiminor axis is Failed to parse (Missing texvc executable; please see math/README to configure.): b
, and the planet is at Failed to parse (Missing texvc executable; please see math/README to configure.): Q , at a distance of Failed to parse (Missing texvc executable; please see math/README to configure.): ae from the center Failed to parse (Missing texvc executable; please see math/README to configure.): C of the ellipse. The key construction that will allow us to analyse this situation is the auxiliary circle (shown in blue) circumscribed on the orbital ellipse. This circle is taller than the ellipse by a factor of Failed to parse (Missing texvc executable; please see math/README to configure.): a/b in the direction of the minor axis, so all area measures on the circle are magnified by a factor of Failed to parse (Missing texvc executable; please see math/README to configure.): a/b with respect to the analogous area measures on the ellipse. Any given point on the ellipse can be mapped to the corresponding point on the circle that is Failed to parse (Missing texvc executable; please see math/README to configure.): a/b further from the ellipse's major axis. If we do this mapping for the position Failed to parse (Missing texvc executable; please see math/README to configure.): S of the satellite at time Failed to parse (Missing texvc executable; please see math/README to configure.): t , we arrive at a point Failed to parse (Missing texvc executable; please see math/README to configure.): R on the circumscribed circle. Kepler defines the angle Failed to parse (Missing texvc executable; please see math/README to configure.): PCR to be the eccentric anomaly angle Failed to parse (Missing texvc executable; please see math/README to configure.): E . (Kepler's terminology often refers to angles as "anomalies".) This definition makes the time-of-flight equation easier to derive than it would be using the true anomaly angle Failed to parse (Missing texvc executable; please see math/README to configure.): PQS . To compute the time-of-flight from this construction, we note that Kepler's second law allows us to compute time-of-flight from the area swept out by the satellite, and so we will set about computing the area Failed to parse (Missing texvc executable; please see math/README to configure.): PQS swept out by the satellite. First, the area Failed to parse (Missing texvc executable; please see math/README to configure.): PQR is a magnified version of the area Failed to parse (Missing texvc executable; please see math/README to configure.): PQS
is the area swept out by the satellite in time Failed to parse (Missing texvc executable; please see math/README to configure.): t . We know that, in one orbital period Failed to parse (Missing texvc executable; please see math/README to configure.): T , the satellite sweeps out the whole area Failed to parse (Missing texvc executable; please see math/README to configure.): \pi a b of the orbital ellipse. Failed to parse (Missing texvc executable; please see math/README to configure.): PQS is the Failed to parse (Missing texvc executable; please see math/README to configure.): t / T fraction of this area, and substituting, we arrive at this expression for Failed to parse (Missing texvc executable; please see math/README to configure.): PQR
is also formed by removing area Failed to parse (Missing texvc executable; please see math/README to configure.): QCR from Failed to parse (Missing texvc executable; please see math/README to configure.): PCR
is a fraction of the circumscribed circle, whose total area is Failed to parse (Missing texvc executable; please see math/README to configure.): \pi a^2 . The fraction is Failed to parse (Missing texvc executable; please see math/README to configure.): E / 2 \pi , thus:
is a triangle whose base is the line segment Failed to parse (Missing texvc executable; please see math/README to configure.): QC of length Failed to parse (Missing texvc executable; please see math/README to configure.): ae , and whose height is Failed to parse (Missing texvc executable; please see math/README to configure.): a \sin E
during circular motion with constant angular velocity Failed to parse (Missing texvc executable; please see math/README to configure.): n
and Failed to parse (Missing texvc executable; please see math/README to configure.): M = E - e \sin E gives us Kepler's equation. Kepler referred to Failed to parse (Missing texvc executable; please see math/README to configure.): n as the mean motion, and Failed to parse (Missing texvc executable; please see math/README to configure.): E - e \sin E as the mean anomaly. The term "mean" in this case refers to the fact that we have "averaged" the satellite's non-constant angular velocity over an entire period to make the satellite's motion amenable to analysis. All satellites traverse an angle of Failed to parse (Missing texvc executable; please see math/README to configure.): 2 \pi per orbital period Failed to parse (Missing texvc executable; please see math/README to configure.): T , so the mean angular velocity is always Failed to parse (Missing texvc executable; please see math/README to configure.): 2 \pi / T . Substituting Failed to parse (Missing texvc executable; please see math/README to configure.): n into the formula we derived above gives this:
ApplicationWith Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of Failed to parse (Missing texvc executable; please see math/README to configure.): \theta from periapsis is broken into two steps:
from true anomaly Failed to parse (Missing texvc executable; please see math/README to configure.): \theta
from the eccentric anomaly Failed to parse (Missing texvc executable; please see math/README to configure.): E
analytically, and so numerical approaches must be used. In effect, one must guess a value of Failed to parse (Missing texvc executable; please see math/README to configure.): E and solve for time-of-flight; then adjust Failed to parse (Missing texvc executable; please see math/README to configure.): E as necessary to bring the computed time-of-flight 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 near-parabolic orbits, eccentricity Failed to parse (Missing texvc executable; please see math/README to configure.): e is nearly 1, and plugging Failed to parse (Missing texvc executable; please see math/README to configure.): e = 1 into the formula for mean anomaly, Failed to parse (Missing texvc executable; please see math/README to configure.): E - \sin E , we find ourselves subtracting two nearly-equal values, and so accuracy suffers. For near-circular 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 at all. These difficulties are what led to the development of the universal variable formulation, described below. Perturbation theoryOne can deal with perturbations just by summing the forces and integrating, but that is not always best. Historically, variation of parameters has been used which is easier to mathematically apply with when perturbations are small. Practical techniques
Transfer orbitsTransfer orbits allow spacecraft to move from one 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. The Hohmann transfer orbit typically requires the least delta-v, but any orbit that intersects both the origin orbit and destination orbit may be used. Gravity assist and the Oberth effectIn a gravity assist, a spacecraft swings by a planet and leaves in a different direction, at a different velocity. 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 delta-v. Interplanetary Transport Network and fuzzy orbitsIt 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 are usually 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 L1 point and returned using very little propellant. Modern mathematical techniques
Conic orbitsFor simple things like computing the delta-v for coplanar transfer ellipses, traditional approaches work pretty well. But time-of-flight is harder, especially for near-circular and hyperbolic orbits. The patched conic approximationThe transfer orbit alone is not a good 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, so it severely underestimates delta-v, and produces highly inaccurate prescriptions for burn timings. One relatively simple way to get a first-order approximation of delta-v is based on the patched conic approximation technique. The idea is to 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 Failed to parse (Missing texvc executable; please see math/README to configure.): r_{SOI}
where Failed to parse (Missing texvc executable; please see math/README to configure.): a_p is the semimajor axis of the planet's orbit relative to the Sun; Failed to parse (Missing texvc executable; please see math/README to configure.): m_p and Failed to parse (Missing texvc executable; please see math/README to configure.): m_s are the masses of the planet and Sun, respectively. This simplification is sufficient to compute rough estimates of fuel requirements, and rough time-of-flight estimates, but it is not generally accurate enough to guide a spacecraft to its destination. For that, numerical methods are required. The universal variable formulationTo address the shortcomings of the traditional approaches, the universal variable approach was developed. It works equally well on circular, elliptical, parabolic, and hyperbolic orbits; and also works well with perturbation theory. The differential equations converge nicely when integrated for any orbit. PerturbationsThe 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 Failed to parse (Missing texvc executable; please see math/README to configure.): x_0 and Failed to parse (Missing texvc executable; please see math/README to configure.): v_0 at a given epoch Failed to parse (Missing texvc executable; please see math/README to configure.): t = 0 . In a two-body 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 two-body 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 Failed to parse (Missing texvc executable; please see math/README to configure.): x_0(t) and the velocity element as Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): x_0(t) and Failed to parse (Missing texvc executable; please see math/README to configure.): v_0(t) . Non-ideal orbitsThe 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 two-body effects.
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 station-keeping, ground track maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude. See also Wikibooks has a book on the topic of
References
External links
Further readingMany of the options, procedures, and supporting theory are covered in standard works such as:
or, on line: [2] and [3] (link doesn't work at the moment) The most elementary but very widely used reference is Bate, Mueller and White. It has several useful graphs off which one can read the rates of change of perigee and node due to earth oblateness, but there are typographical errors in a few equations. For example, in Eq. (9.7.5) the term in (3/2) J2 needs (re/r) squared and the term in J3 needs it cubed. The coefficient 315 in the J6 term, Eq.(9.7.6.) should be 245 (but the 315 in the J5 term is just fine). Battin's book may be too mathematical for many users.ca:Astrodinàmica es:Astrodinámica fr:Mécanique spatiale it:Astrodinamica nl:Astrodynamica pl:Astrodynamika pt:Astrodinâmica sk:Astrodynamika |
