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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 the universal variable formulation , described below. For simple procedures, such as computing the delta-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. The 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 delta-v, and produces highly inaccurate prescriptions for burn timings.

A relatively simple way to get a first-order approximation of delta-v is based on the 'Patched Conic Approximation' technique.

US6725012B1 - Method for deploying an orbiting sparse array antenna - Google Patents

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.

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.

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For that, numerical methods are required. To address computational shortcomings of traditional approaches for solving the 2-body 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. 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. 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 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.

In spaceflight , an orbital maneuver is the use of propulsion systems to change the orbit of a spacecraft. For spacecraft far from Earth—for example those in orbits around the Sun—an orbital maneuver is called a deep-space maneuver DSM. 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, the change-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 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.

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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.

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.

Lecture 24: Perturbation Theory I (International Winter School on Gravity and Light 2015)

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. 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 as Genesis. This spacecraft visited the Earth-Sun L 1 point and returned using very little propellant.

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Orbital mechanics. Orbital parameters. Types of two-body orbits , by eccentricity.

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