Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics 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, orbital 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 it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).
Until the rise of space travel in the twentieth century, there was little distinction between orbital and celestial mechanics. At the time of Sputnik, the field was termed 'space dynamics'.[1] 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 the first edition of Philosophiæ Naturalis Principia Mathematica (1687), which gave a method for finding the orbit of a body following a parabolic path from three observations.[2] This was used by Edmund Halley to establish the orbits of various comets, including that which bears his name. Newton's method of successive approximation was formalised into an analytic method by Leonhard Euler in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits by Johann Lambert in 1761–1777.
Another milestone in orbit determination was Carl Friedrich Gauss's assistance in the "recovery" of the dwarf planet Ceres in 1801. Gauss's method was able to use just three observations (in the form of pairs of right ascension and declination), to find the six orbital 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 observed minor 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 astronomer Samuel Herrick beginning in the 1930s. He consulted the rocket scientist Robert 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 by classical 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 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 actually slow down relative to the leading craft, missing the target. The space 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, simple atmospheric drag is another complicating factor for objects in low 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 n-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 between classical mechanics and general relativity also become important.
See also: Laplace–Runge–Lenz vector. The fundamental laws of astrodynamics are Newton's law of universal gravitation and Newton's laws of motion, while the fundamental mathematical tool is differential calculus.
In a Newtonian framework, the laws governing orbits and trajectories are in principle time-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:
See main article: Escape velocity. The formula for an escape velocity is 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
\epsilonp=-
| GM | |
| r |
where G is the gravitational constant and r is the distance between the two bodies;
while the specific kinetic energy of an object is given by
\epsilonk=
| v2 | |
| 2 |
where v is its Velocity;
and so the total specific orbital energy is
\epsilon=\epsilonk+\epsilonp=
| v2 | |
| 2 |
-
| GM | |
| r |
Since energy is conserved,
\epsilon
r
r
| v\geq\sqrt{ | 2GM |
| r |
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 are conic sections, so the formula for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is:
r=
| p | |
| 1+e\cos\theta |
\mu=G(m1+m2)
p=h2/\mu
\mu
m1
m2
h
\theta
p
e
See main article: Circular orbit. 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 can be derived as follows:
Centrifugal acceleration matches the acceleration due to gravity.
So,
Therefore,
v=\sqrt{
| GM | |
| r |
}
where
G
6.6743 × 10-11 m3/(kg·s2)
To properly use this formula, the units must be consistent; for example,
M
r
The quantity
GM
Once the circular orbital velocity is known, the escape velocity is easily found by multiplying by
\sqrt{2}
v=\sqrt2\sqrt{
| GM | |
| r |
}=\sqrt{
| 2GM | |
| r |
}.
To escape from gravity, the kinetic energy must at least match the negative potential energy. Therefore,
v=\sqrt{
| 2GM | |
| r |
}.
If
0<e<1
\theta
rp
| r | ||||
|
The maximum value
r
\theta=180\circ
ra
| r | ||||
|
Let
2a
P
A
2a=rp+ra
Substituting the equations above, we get:
| a= | p |
| 1-e2 |
a is the semimajor axis of the ellipse. Solving for
p
| r= | a(1-e2) |
| 1+e\cos\theta |
Under standard assumptions the orbital period (
T
T=2\pi\sqrt{a3\over{\mu}}
\mu
a
a
Under standard assumptions the orbital speed (
v
v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}
\mu
r
a
The velocity equation for a hyperbolic trajectory is
v=\sqrt{\mu\left({2\over{r}}+\left\vert{1\over{a}}\right\vert\right)}
Under standard assumptions, specific orbital energy (
\epsilon
{v2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0
v
r
a
\mu
Using the virial theorem we find:
2\epsilon
r-1
a-1
-\epsilon
If the eccentricity equals 1, then the orbit equation becomes:
r={{h2}\over{\mu}}{{1}\over{1+\cos\theta}}
r
h
\theta
\mu
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:
\epsilon={v2\over2}-{\mu\over{r}}=0
v
In other words, the speed anywhere on a parabolic path is:
v=\sqrt{2\mu\over{r}}
If
e>1
r={{h2}\over{\mu}}{{1}\over{1+e\cos\theta}}
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
\cos\theta=-1/e
\thetainfty=\cos-1\left(-
| 1e | |
| \right) |
\thetainfty
\thetainfty
\sin2\theta+\cos2\theta=1
\sin\thetainfty=
| 1e | |
| \sqrt{e |
2-1}
Under standard assumptions, specific orbital energy (
\epsilon
\epsilon={v2\over2}-{\mu\over{r}}={\mu\over{-2a}}
v
r
a
\mu
See also: Characteristic energy.
Under standard assumptions the body traveling along a hyperbolic trajectory will attain at
r=
vinfty
vinfty=\sqrt{\mu\over{-a}}
\mu
a
The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by
2\epsilon=C3=v
| 2 | |
| infty |
One approach to calculating orbits (mainly used historically) is to use Kepler's equation:
M=E-\epsilon ⋅ \sinE
where M is the mean anomaly, E is the eccentric anomaly, and
\epsilon
With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of
\theta
E
\theta
t
E
Finding the eccentric anomaly at a given time (the inverse problem) is more difficult. Kepler's equation is transcendental in
E
E
E
A solution of Kepler's equation, valid for all real values of
style\epsilon
\displaystyle \sum_^ \lim_ \left(\frac \left[\left(\frac{\theta}{ \sqrt[3] } \right) ^n \right]\right), & \epsilon = 1 \\
\displaystyle \sum_^\lim_ \left(\frac \left[\left(\frac{ \theta }{ \theta - \epsilon \cdot \sin(\theta)} \right) ^n \right]\right), & \epsilon \ne 1
\end
Evaluating this yields:
, & \epsilon \ne 1
\end
Alternatively, Kepler's Equation can be solved numerically. First one must guess a value of
E
E
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
\epsilon
e=1
E-\sinE
For simple procedures, such as computing the delta-v for coplanar transfer ellipses, traditional approaches are fairly effective. Others, such as time-of-flight are far more complicated, especially for near-circular and hyperbolic orbits.
See main article: Patched conic approximation. The Hohmann 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 a first-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 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 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 a gravity assist.[3]
The size of the "neighborhoods" (or spheres of influence) vary with radius
rSOI
rSOI=
| a | ||||
|
\right)2/5
ap
mp
ms
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, 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.
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
x0
v0
t=0
However, perturbations cause the orbital elements to change over time. Hence, the position element is written as
x0(t)
v0(t)
x0(t)
v0(t)
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 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 main article: Orbital maneuver. 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. 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.
See main article: Interplanetary Transport Network.
See also: Low energy transfers. 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's Trojan points. Collectively referred to as the Interplanetary 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 as Genesis. This spacecraft visited the Earth-Sun point and returned using very little propellant.
Many of the options, procedures, and supporting theory are covered in standard works such as: