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Astrodynamics is the study and creation of orbits, especially those of artificial satellites. It uses the principles of Celestial Mechanics, which is the application of Newton's Laws of Motion and Law of Universal Gravitation to the determination of the motion of objects in space. Orbits of astronomical bodies, such as planets, asteroids, and comets are calculated using the principles of celestial mechanics. Astrodynamics is principally concerned with spacecraft trajectories, from launch to atmospheric re-entry, including all orbital manoeuvres. See spacecraft propulsion, Tsiolkovsky rocket equation.
Laws of Astrodynamics
The fundamental physical laws are Newton's law of universal gravitation, and Newton's laws of motion, while the fundamental mathematical tool is his differential calculus. Kepler's laws of planetary motion may be derived from these laws. It is also possible to derive a formula for escape velocity as follows: The specific potential energy associated with a planet of mass M is - GM / r, while the specific kinetic energy of an object is given by v2 / 2. Since energy is conserved, the total specific orbital energy v2 / 2 - GM / r does not depend on r. Therefore the object can reach infinite r only if this quantity is nonnegative, which implies .
Formulae for ellipse
Orbits are ellipses, so, naturally, the formula for the distance of a body for a given angle corresponds to the formula for an ellipse in polar coordinates. The parameters of the ellipse are given by the orbital elements.
Until the rise of space travel in the twentieth century, there was little distinction between astrodynamics and celestial mechanics. The fundamental techniques, such as those used to solve the Kepler problem, are therefore the same in both fields. Furthermore, the history of the fields is essentially identical.
Kepler was the first to successfully model planetary orbits to a high degree of accuracy.
To 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.
The problem is to find the time T at which the satellite reaches point S, given that it is at periapsis P at time t = 0. We are given that the semimajor axis of the orbit is a, and the semiminor axis is b; the eccentricity is e, and the planet is at Q, at a distance of ae from the center 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 a / b in the direction of the minor axis, so all area measures on the circle are magnified by a factor of 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 a / b further from the ellipse's major axis. If we do this mapping for the position S of the satellite at time T, we arrive at a point R on the circumscribed circle. Kepler defines the angle PCR to be the eccentric anomaly angle 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 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 PQS swept out by the satellite.
First, the area PQR is a magnified version of the area PQS:
Furthermore, area PQS is the area swept out by the satellite in time T. We know that, in one orbital period τ, the satellite sweeps out the whole area πab of the orbital ellipse. PQS is the T / τ fraction of this area, and substituting, we arrive at this expression for PQR:
Second, the area PQR is also formed by removing area QCR from PCR:
Area PCR is a fraction of the circumscribed circle, whose total area is πa2. The fraction is E / 2π, thus:
Meanwhile, area QCR is a triangle whose base is the line segment QC of length ae, and whose height is asinE:
Combining all of the above:
Dividing through by a2 / 2:
To understand the significance of this formula, consider an analogous formula giving an angle θ during circular motion with constant angular velocity M:
Setting M = 2π / τ and θ = E - esinE gives us Kepler's equation. Kepler referred to M as the mean motion, and E - esinE 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 2π per orbital period τ, so the mean angular velocity is always 2π / τ.
Substituting M into the formula we derived above gives this:
This formula is commonly referred to as Kepler's equation.
With Kepler's formula, finding the time-of-flight 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 time-of-flight T from the eccentric anomaly E
Finding the angle at a given time is harder. Kepler's equation is transcendental in E, meaning it cannot be solved for E analytically, and so numerical approaches must be used. In effect, one must guess a value of E and solve for time-of-flight; then adjust 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 e is nearly 1, and plugging e = 1 into the formula for mean anomaly, E - sinE, 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 doesn't hold for parabolic or hyperbolic orbits at all. These difficulties are what led to the development of the universal variable formulation, described below.
You can deal with perturbations just by summing the forces and integrating, but that's not always best. Historically, people (who?) did variation of parameters, which works better in some ways.
Today, we don't use the same techniques that Kepler used, in general.
For simple things like computing the delta-v for coplanar transfer ellipses, traditional approaches work pretty well. But time-of-flight is harder, especially for nearly-circular orbits, or for hyperbolic orbits.
Transfer orbits get you from one orbit to another. Usually they require a burn at the start, a burn at the end, and sometimes a burn in the middle. The Hohmann transfer orbit requires the least delta-v, but any orbit that intersects both your origin and destination will work.
The patched conic approximation
The 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 uselessly 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' gravity is considered during the final portion of the trajectory where Mars' 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 things like 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 formulation
To 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.
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 and v0 at a given epoch 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 x0(t) and the velocity element as v0(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 x0(t) and v0(t).
The following are some effects which make real orbits differ from the mathematical models. 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.
- Equatorial bulges render inaccurate the approximation of each body as a point source of gravity
- Tidal forces cause precession, and can also alter the other orbital elements over time
- Relativistic effects make the basic Newtonian laws of gravitation and motion inaccurate
However other things are not so small, and necessitate a different approach. For instance, a spacecraft's thrust while its engines are active can dominate its trajectory.
Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behaviour can become chaotic.
- Celestial mechanics
- Chaos theory
- Lagrangian point
- N-body problem
- Roche limit
- Bate, Roger R., Mueller, Donald D., and White, Jerry E. (1971). Fundamentals of Astrodynamics. Dover Publications. ISBN 0-48-660061-0
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