Astrodynamics

Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newton's laws of motion and his law of universal gravitation. It is a specific and distinct branch of celestial mechanics, which focuses more broadly on Newtonian gravitation and includes the orbital motions of artificial and natural astronomical bodies such as planets, moons, and comets. Astrodynamics is principally concerned with spacecraft trajectories, from launch to atmospheric re-entry, including all orbital maneuvers, orbit plane changes, and interplanetary transfers.

Laws of astrodynamics

The fundamental laws of astrodynamics 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, 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 second law of motion applies, and Kepler's laws are temporarily invalidated.

The formula for escape velocity is easily 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

while the specific kinetic energy of an object is given by

Since energy is conserved, the total specific orbital energy

does not depend on the distance, <math>r<math>, from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite <math>r<math> 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 to infinite distance, because of the Sun. To escape the solar system from the vicinity of the Earth requires around 42 km/s velocity, but there will be "part 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.

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.

Historical approaches

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 Keplerian problem, are therefore the same in both fields. Furthermore, the history of the fields is essentially identical.

Kepler's equation

Kepler was the first to successfully 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 to find the time <math>T<math> at which the satellite reaches point <math>S<math>, given that it is at periapsis <math>P<math> at time <math>t = 0<math>. 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 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 <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 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>

Second, the area <math>PQR<math> is also formed by removing area <math>QCR<math> from <math>PCR<math>:

<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

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

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 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> analytically, 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 does not 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 is not always best. Historically, people (who?) did variation of parameters, which works better in some ways.

Modern techniques

Today, we do not 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

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.

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 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 <math>x_0(t)<math> and the velocity element 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

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.

• Equatorial bulges cause precession of the node and the perigee
• Tesseral harmonics [1] (http://mathworld.wolfram.com/TesseralHarmonic.html) of the gravity field introduce additional perturbations
• lunar and solar gravity perturbations alter the orbits
• Atmospheric drag reduces the semi-major axis unless make-up thrust is used

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

Many of the options, procedures, and supporting theory are covered in standard works such as:

1. Bate, R.R., Mueller, D.D., White, J.E., Fundamentals of Astrodynamics, Dover Publications, New York, 1971.

2. Vallado, D. A., Fundamentals of Astrodynamics and Applications, 2nd Edition, McGraw-Hill, 2001

3. Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, New York, 1987.

4. Chobotov, V.A. (ed.), Orbital Mechanics, 3rd Edition, AIAA, Washington, DC, 2002.

5. Herrick, S. Astrodynamics, Van Nostrand Reinhold, London, 1971 (two volumes).

6. Kaplan, M.H., Modern Spacecraft Dynamics and Controls , Wiley, New York, 1976.

7. Logsdon, T., Orbital Mechanics, Wiley-Interscience, New York, 1997.

8. Prussing, J.E., and B.A. Conway, Orbital Mechanics, Oxford University Press, New York, 1993.

9. Sidi, M.J., Spacecraft Dynamics and Control, Cambridge University Press, New York, 1997.

10. Wiesel, W.E., Spaceflight Dynamics, McGraw-Hill, New York, 1996, 2nd edition.

11. Vinti, J.P., Orbital and Celestial Mechanics , AIAA, Reston, VA, 1998.

or, on line:

[2] (http://www.psatellite.com/products/manuals/SCTUsersGuideBook1.pdf) and [3] (http://www.psatellite.com/products/manuals/SCTUsersGuideBook2.pdf)

The most elementary but very widely used reference is Bate, Mueller and White. It has several useful graphs off which one can read the rates of change of perigee and node due to earth oblateness, but there are typographical errors in a few equations. For example, in Eq. (9.7.5) the term in (3/2) J2 needs (r_e/r) squared and the term in J3 needs it cubed. The coefficient 315 in the J6 term, Eq.(9.7.6.) should be 245 (but the 315 in the J5 term is just fine). Battin's book may be too mathematical for many users.

Interplanetary superhighway and fuzzy orbits

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 Superhighway, these highly perturbative, even chaotic, orbital trajectories in principle need no fuel (in practice keeping to the trajectory requires some course corrections). The biggest problem with them is they are usually exceedingly slow taking many years to arrive.

They have, however, been employed on projects such as Genesis. This spacecraft visited Earth's lagrange L1 point and returned using very little propellant.

See also

• Spacecraft propulsion
• Tsiolkovsky rocket equation
• Aerodynamics
• Astrophysics
• Celestial mechanics
• Chaos theory
• Lagrangian point
• N-body problem
• Orbit
• Roche limit

Reference

• Bate, Roger R., Mueller, Donald D., and White, Jerry E. (1971). Fundamentals of Astrodynamics. Dover Publications. ISBN 0-48-660061-0fr:Mécanique spatiale

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