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Astrodynamics

Astrodynamics 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 astrodynamics, as are spacecraft trajectories, from launch to atmospheric re-entry, including all orbital manoeuvres.

Laws of Astrodynamics

The fundamental physical laws are Newton's law of universal gravitation, and Newton's laws of motion.

Throwing Newton's differential calculus into the mix, it is possible to derive Kepler's laws of planetary motion {show how?}, and a formula for escape velocity {show how}.

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. People also use orbital elements.

Historical approaches

First, you've got your Aristotle and Ptolemy.

Next, you have your Johannes Kepler and Tycho Brahe, followed by your Isaac Newton.

The Keplerian problem was addressed by lots of very able mathematicians, such as (examples here ... Bernoulli ??? Alan Turing ??? Henri Poincaré, Joseph Louis Lagrange, Pierre-Simon Laplace ).

This stuff is useful for calculating the behaviour of planets and comets and such. More recently, it has also become useful to calculate spacecraft trajectories. See spacecraft propulsion, Tsiolkovsky rocket equation.

Kepler's equation

Kepler was the first to succesfully model planetary orbits to a high degree of accuracy.

Derivation

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 as follows: We are given that the semimajor axis of the orbit is <math>a<math>, and the semiminor axis is <math>b<math>. The eccentricity is <math>e<math>, and the planet is at <math>Q<math>, at a distance of <math>ae<math> from the center <math>C<math> of the ellipse. The satellite is at periapsis <math>P<math> at time <math>t = 0<math>. The goal is to find the time <math>T<math> at which the satellite reaches point <math>S<math>.

The key construction that will allow us to analyze this situation is the circle (shown in blue) circumscribed on the orbital ellipse. This circle is taller than the ellipse by a factor of <math>a/b<math> in the direction of the minor axis, so all area measures on the circle are magnified by a factor of <math>a/b<math> 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 <math>a/b<math> further from the ellipse's major axis. If we do this mapping for the position <math>S<math> of the satellite at time <math>T<math>, we arrive at a point <math>R<math> on the circumscribed circle. Kepler defines the angle <math>PCR<math> to be the eccentric anomaly angle <math>E<math>. (Kepler's terminology often refers to angles as "anomalies.") This definition makes the the time-of-flight equation easier to derive than it would be using the true anomaly angle <math>PQS<math>.

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 <math>PQS<math> swept out by the satellite. First, the area <math>PQR<math> is a magnified version of the area <math>PQS<math>:

<math>PQR = \frac{a}{b} PQS<math>

Furthermore, area <math>PQS<math> is the area swept out by the satellite in time <math>T<math>. We know that, in one orbital period <math>\tau<math>, the satellite sweeps out the whole area <math>\pi a b<math> of the orbital ellipse. <math>PQS<math> is the <math>T / \tau<math> fraction of this area, and substituting, we arrive at this expression for <math>PQR<math>:

<math>PQR = \frac{T}{\tau} \pi a^2<math>

Another expression for <math>PQR<math> is found by a simple conglomeration of adjacent areas:

<math>PQR = PCR - QCR \;<math>

Area <math>PCR<math> is a fraction of the circumscribed circle, whose total area is <math>\pi a^2<math>. The fraction is <math>E / 2 \pi<math>, thus:

<math>PCR = \frac{a^2}{2}E <math>

Meanwhile, area <math>QCR<math> is a triangle whose base is the line segment <math>QC<math> of length <math>ae<math>, and whose height is <math>a \sin E<math>:

<math>QCR = \frac{a^2}{2} e \sin E <math>

Combining all of the above:

<math>PQR = \frac{T}{\tau} \pi a^2

Dividing through by <math>a^2 / 2<math>:

<math>\frac{2 \pi}{\tau}T = E - e \sin E<math>

To understand the significance of this formula, consider an analogous formula giving an angle <math>\theta<math> during circular motion with constant angular velocity <math>M<math>:

<math>MT = \theta \;<math>

Setting <math>M = 2 \pi / \tau<math> and <math>\theta = E - e \sin E<math> gives us Kepler's equation. Kepler referred to <math>M<math> as the mean motion, and <math>E - e \sin E<math> 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 <math>2 \pi<math> per orbital period <math>\tau<math>, so the mean angular velocity is always <math>2 \pi / \tau<math>.

Substituting <math>M<math> into the formula we derived above gives this:

<math>MT = E - e \sin E \;<math>

This formula is commonly referred to as Kepler's equation.

Application

With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of <math>\theta<math> from periapsis is broken into two steps:

1. Compute the the eccentric anomaly <math>E<math> from true anomaly <math>\theta<math>
2. Compute the time-of-flight <math>T<math> from the eccentric anomaly <math>E<math>

Finding the angle at a given time is harder. Kepler's equation is transcendental in <math>E<math>, meaning it cannot be solved for <math>E<math> analitically, and so numerical approaches must be used. In effect, one must guess a value of <math>E<math> and solve for time-of-flight; then adjust <math>E<math> 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 <math>e<math> is nearly 1, and plugging <math>e = 1<math> into the formula for mean anomaly, <math>E - \sin E<math>, 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.

Perturbation theory

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.

Modern techniques

Today, we don't use the same techniques that Kepler used, in general.

Conic orbits

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

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

Transfer orbits alone don't cut it when you are going from one planet to another. For instance, the Hohmann transfer orbit from Earth to Mars neglects these two bodies' own gravity; an effect which dominates whenever the spacecraft is orbiting a planet. The Patched conic approximation deals with just one gravitating body at a time, thereby allowing us to use conic sections for each phase of the trip, and simplifying the calculations. In this case, we first must escape Earth via a hyperbola; then follow a transfer orbit around the sun, until finally we approach Mars via another hyperbola.

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, several smart people invented the universal variable approach. 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.

Perturbations

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 <math>x_0<math> and <math>v_0<math> at a given epoch <math>t = 0<math>. In a two-body simulation, these elements are sufficient to compute the satellite's position and veloticy 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 position as <math>x_0(t)<math> and velocity as <math>v_0(t)<math>, 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 <math>x_0(t)<math> and <math>v_0(t)<math>.

Non-ideal orbits

You've got your equatorial bulges, which mean you can't treat each body as a point source of gravity. You've got your tidal forces which can cause precession, and can also alter the other orbital elements over time. You've also got relativistic effects. These sorts of effects can be treated using the usual perturbation theory, because they are usually small relative to the two-body effects.

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, this weirdness dominates, and some interesting things happen. For one thing, chaos can take over.

See also

• N-body problem
• orbit
• Lagrangian point
• Chaos theory
• Aerodynamics
• Roche limit
• Celestial mechanics

References

Fundamentals of Astrodynamics. ISBN 0-486-60061-0