Celestial Mechanics ::: Project ETERNITY

Celestial Mechanics is the study of the effects of gravity on celestial bodies.

It is distinguished from Astrodynamics, which is the study of the creation of

artifical satellite orbits.

Historical approaches

Ancient Greeks developed theories regarding celestial mechanics, most of which were geocentric in nature. (Help in elaborating)

Johannes Kepler was the first to develop laws of orbits, which he did by examining and cataloging the motion of the planets. Years before Isaac Newton had even developed his law of gravitation, Kepler had developed his three laws of planetary motion from empirical observation.

Using Newton's law of gravitation, proving Kepler's Laws for the case of a circular orbit is simple. Elliptical orbits involve more complex calculations.

Using Lagrangian Mechanics it is possible to develop a single polar coordinate equation that can be used to describe any orbit, even those that are parabolic and hyperbolic.

This 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 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 as follows: 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 satellite is at periapsis $P$ at time $t\; =\; 0$. The goal is to find the time $T$ at which the satellite reaches point $S$.

The key construction that will allow us to analyse this situation is the 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 $\backslash tau$, the satellite sweeps out the whole area $\backslash pi\; a\; b$ of the orbital ellipse. $PQS$ is the $T\; /\; \backslash tau$ fraction of this area, and substituting, we arrive at this expression for $PQR$:

Another expression for $PQR$ is found by a simple conglomeration of adjacent areas:

Area $PCR$ is a fraction of the circumscribed circle, whose total area is $\backslash pi\; a^2$. The fraction is $E\; /\; 2\; \backslash pi$, thus:

Meanwhile, area $QCR$ is a triangle whose base is the line segment $QC$ of length $ae$, and whose height is $a\; \backslash sin\; E$:

Combining all of the above:

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

Dividing through by $a^2\; /\; 2$:

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

Setting $M\; =\; 2\; \backslash pi\; /\; \backslash tau$ and

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

Substituting $M$ into the formula we derived above gives this:

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 $\backslash theta$ from periapsis is broken into two steps:

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, $E\; -\; \backslash sin\; E$, 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.

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