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Orbital mechanics orastrodynamics is the application ofballistics andcelestial mechanics torockets,satellites, and otherspacecraft. The motion of these objects is usually calculated fromNewton's laws of motion and thelaw of universal gravitation. Astrodynamics is a core discipline withinspace-mission design and control.
Celestial mechanics treats more broadly the orbital dynamics of systems under the influence ofgravity, including both spacecraft and naturalastronomical bodies such asstar systems,planets,moons, andcomets. Orbital mechanics focuses on spacecrafttrajectories, includingorbital maneuvers,orbital plane changes, and interplanetary transfers, and is used by mission planners to predict the results ofpropulsive maneuvers.
General relativity is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).
Until the rise ofspace travel in the twentieth century, there was little distinction between orbital and celestial mechanics. At the time ofSputnik, the field was termed 'space dynamics'.[1] The fundamental techniques, such as those used to solve theKeplerian 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, publishinghis laws in 1609.Isaac Newton published more general laws of celestial motion in the first edition ofPhilosophiæ Naturalis Principia Mathematica (1687), which gave a method for finding the orbit of a body following aparabolic path from three observations.[2] This was used byEdmund Halley to establish the orbits of various comets, includingthat which bears his name. Newton's method of successive approximation was formalised into an analytic method byLeonhard Euler in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits byJohann Lambert in 1761–1777.
Another milestone in orbit determination wasCarl Friedrich Gauss's assistance in the "recovery" of the dwarf planetCeres in 1801.Gauss's method was able to use just three observations (in the form of pairs ofright ascension anddeclination), to find the sixorbital elements that completely describe an orbit. The theory of orbit determination has subsequently been developed to the point where today it is applied in GPS receivers as well as the tracking and cataloguing of newly observedminor planets. Modern orbit determination and prediction are used to operate all types of satellites and space probes, as it is necessary to know their future positions to a high degree of accuracy.
Astrodynamics was developed by astronomerSamuel Herrick beginning in the 1930s. He consulted the rocket scientistRobert Goddard and was encouraged to continue his work on space navigation techniques, as Goddard believed they would be needed in the future. Numerical techniques of astrodynamics were coupled with new powerful computers in the 1960s, and humans were ready to travel to the Moon and return.
The following rules of thumb are useful for situations approximated byclassical mechanics under the standard assumptions of astrodynamics outlined below. 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 spacecrafts are in the same circular orbit and wish to dock, the trailing craft cannot simply fire its engines to accelerate towards the leading craft. This will change the shape of its orbit, causing it to gain altitude and slow down relative to the leading craft, thus moving away from the target. Thespace rendezvous before docking normally takes multiple precisely calculated engine firings in multiple orbital periods, requiring hours or even days to complete.
To the extent that the standard assumptions of astrodynamics do not hold, actual trajectories will vary from those calculated. For example, simpleatmospheric drag is another complicating factor for objects inlow Earth orbit.
These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as abinary star system (seen-body problem).Celestial mechanics uses more general rules applicable to a wider variety of situations. 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 non-gravitational forces; they also describe parabolic and hyperbolic trajectories. In the close proximity of large objects like stars the differences betweenclassical mechanics andgeneral relativity also become important.
The fundamental laws of astrodynamics areNewton's law of universal gravitation andNewton's laws of motion, while the fundamental mathematical tool isdifferential calculus.
In a Newtonian framework, the laws governing orbits and trajectories are in principletime-symmetric.
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 which have been set out above. The three laws are:
The formula for anescape velocity is derived as follows. Thespecific energy (energy per unitmass) of any space vehicle is composed of two components, thespecific potential energy and thespecific kinetic energy. The specific potential energy associated with a planet ofmassM is given by
whereG is thegravitational constant andr is the distance between the two bodies;
while thespecific kinetic energy of an object is given by
wherev is its Velocity;
and so the totalspecific orbital energy is
Sinceenergy is conserved, cannot depend on the distance,, from the center of the central body to the space vehicle in question, i.e.v must vary withr to keep the specific orbital energy constant. Therefore, the object can reach infinite 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 Sun–Earth, but not close to the Earth, requires around 42 km/s velocity, but there will be "partial 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.
Orbits areconic sections, so the formula for the distance of a body for a given angle corresponds to the formula for that curve inpolar coordinates, which is:
is called thegravitational parameter. and are the masses of objects 1 and 2, and is thespecific angular momentum of object 2 with respect to object 1. The parameter is known as thetrue anomaly, is thesemi-latus rectum, while is theorbital eccentricity, all obtainable from the various forms of the six independentorbital elements.
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 distancer from the center of gravity of massM can be derived as follows:
Centrifugal acceleration matches the acceleration due to gravity.
So,
Therefore,
where is thegravitational constant, equal to
To properly use this formula, the units must be consistent; for example, must be in kilograms, and must be in meters. The answer will be in meters per second.
The quantity is often termed thestandard gravitational parameter, which has a different value for every planet or moon in theSolar System.
Once the circular orbital velocity is known, theescape velocity is easily found by multiplying by:
To escape from gravity, the kinetic energy must at least match the negative potential energy. Therefore,
If, then the denominator of the equation of free orbits varies with the true anomaly, but remains positive, never becoming zero. Therefore, the relative position vector remains bounded, having its smallest magnitude at periapsis, which is given by:
The maximum value is reached when. This point is called the apoapsis, and its radial coordinate, denoted, is
Let be the distance measured along the apse line from periapsis to apoapsis, as illustrated in the equation below:
Substituting the equations above, we get:
a is the semimajor axis of the ellipse. Solving for, and substituting the result in the conic section curve formula above, we get:
Under standard assumptions theorbital period () of a body traveling along an elliptic orbit can be computed as:
where:
Conclusions:
Under standard assumptions theorbital speed () of a body traveling along anelliptic orbit can be computed from theVis-viva equation as:
where:
The velocity equation for ahyperbolic trajectory is.
Under standard assumptions,specific orbital energy () of elliptic orbit is negative and the orbital energy conservation equation (theVis-viva equation) for this orbit can take the form:
where:
Conclusions:
Using thevirial theorem we find:
If the eccentricity equals 1, then the orbit equation becomes:
where:
As the true anomaly θ approaches 180°, the denominator approaches zero, so thatr tends towards infinity. Hence, the energy of the trajectory for whiche=1 is zero, and is given by:
where:
In other words, the speed anywhere on a parabolic path is:
If, 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. we denote this value of true anomaly
since the radial distance approaches infinity as the true anomaly approaches, known as thetrue anomaly of the asymptote. Observe that lies between 90° and 180°. From the trigonometric identity it follows that:
Under standard assumptions,specific orbital energy () of ahyperbolic trajectory is greater than zero and theorbital energy conservation equation for this kind of trajectory takes form:
where:
Under standard assumptions the body traveling along a hyperbolic trajectory will attain at infinity anorbital velocity called hyperbolic excess velocity () that can be computed as:
where:
The hyperbolic excess velocity is related to thespecific orbital energy or characteristic energy by
One approach to calculating orbits (mainly used historically) is to useKepler's equation:
whereM is themean anomaly,E is theeccentric anomaly, and is theeccentricity.
With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of fromperiapsis is broken into two steps:
Finding the eccentric anomaly at a given time (the inverse problem) is more difficult. Kepler's equation istranscendental in, meaning it cannot be solved foralgebraically. Kepler's equation can be solved foranalytically 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 and solve for time-of-flight; then adjust 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 is nearly 1, and substituting into the formula for mean anomaly,, we find ourselves subtracting two nearly-equal values, and 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. These difficulties are what led to the development of theuniversal variable formulation, described below.
For simple procedures, such as computing thedelta-v for coplanar transfer ellipses, traditional approaches[clarification needed] are fairly effective. Others, such as time-of-flight are far more complicated, especially for near-circular and hyperbolic orbits.
TheHohmann transfer orbit alone is a poor approximation for interplanetary trajectories because it neglects the planets' own gravity. Planetary gravity dominates the behavior of the spacecraft in the vicinity of a planet and in most cases Hohmann severely overestimates delta-v, and produces highly inaccurate prescriptions for burn timings. A relatively simple way to get afirst-order approximation of delta-v 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 givenescape velocity to send it on its way to interplanetary space. Next, one would consider only the Sun's gravity until the trajectory reaches the neighborhood 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 behavior. The spacecraft would approach Mars on a hyperbolic orbit, and a final retrograde burn would slow the spacecraft enough to be captured by Mars.Friedrich Zander was one of the first to apply the patched-conics approach for astrodynamics purposes, when proposing the use of intermediary bodies' gravity for interplanetary travels, in what is known today as agravity assist.[3]
The size of the "neighborhoods" (orspheres of influence) vary with radius:
where is thesemimajor axis of the planet's orbit relative to theSun; and are themasses 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.
To address computational shortcomings of traditional approaches for solving the 2-body problem, theuniversal 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.
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 and at a given epoch. 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 velocitywould 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, the position element is written as and the velocity element as, 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 and.
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 two-body effects.
Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behavior can becomechaotic. On the other hand, the various perturbations can be orchestrated by clever astrodynamicists to assist with orbit maintenance tasks, such asstation-keeping,ground track maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude.
Inspaceflight, anorbital maneuver is the use ofpropulsion systems to change theorbit of aspacecraft. For spacecraft far from Earth—for example those in orbits around the Sun—an orbital maneuver is called adeep-space maneuver (DSM).[not verified in body]
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.
For the case of orbital transfer between non-coplanar orbits, thechange-of-plane thrust must be made at the point where the orbital planes intersect (the "node"). As the objective is to change the direction of the velocity vector by an angle equal to the angle between the planes, almost all of this thrust should be made when the spacecraft is at the node near the apoapse, when the magnitude of the velocity vector is at its lowest. However, a small fraction of the orbital inclination change can be made at the node near the periapse, by slightly angling the transfer orbit injection thrust in the direction of the desired inclination change. This works because the cosine of a small angle is very nearly one, resulting in the small plane change being effectively "free" despite the high velocity of the spacecraft near periapse, as the Oberth Effect due to the increased, slightly angled thrust exceeds the cost of the thrust in the orbit-normal axis.
In agravity 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 anelastic 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.
TheOberth 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 effectivedelta-v.
It 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'strojan points.[citation needed] Collectively referred to as theInterplanetary Transport Network, these highly perturbative, even chaotic, orbital trajectories in principle need no fuel beyond that needed to reach the Lagrange point (in practice keeping to the trajectory requires some course corrections). The biggest problem with them is they can be exceedingly slow, taking many years. In addition launch windows can be very far apart.
They have, however, been employed on projects such asGenesis. This spacecraft visited the Earth-SunL1 point and returned using very little propellant.
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