Keplerian problem

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To compute the position of a satellite at a given time using Kepler's laws of planetary motion (the Keplerian problem, or Kepler 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, also called Kepler's equation here.

Kepler's construction for deriving the time-of-flight equation. The bold ellipse is the satellite's orbit, with the star or planet at one focus Q. The goal is to compute the time required for a satellite to travel from periapsis P to a given point S. Kepler circumscribed the blue circle around the ellipse, and used it to derive his time-of-flight equation in terms of eccentric anomaly.

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 $a e$ 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 $P C R$ 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 $P Q S$ .

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 $P Q S$ swept out by the satellite. First, the area $P Q R$ is a magnified version of the area $P Q S$ :

$PQR = \frac{a}{b} PQS$

Furthermore, area $P Q S$ 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 $π a b$ of the orbital ellipse. $P Q S$ is the $T / τ$ fraction of this area, and substituting, we arrive at this expression for $P Q R$ :

$PQR = \frac{T}{\tau} \pi a^2$

Another expression for $P Q R$ is found by a simple conglomeration of adjacent areas:

$PQR = PCR - QCR \;$

Area $P C R$ is a fraction of the circumscribed circle, whose total area is $π a 2$ . The fraction is $E / 2π$ , thus:

$PCR = \frac{a^2}{2}E$

Meanwhile, area $Q C R$ is a triangle whose base is the line segment $Q C$ of length $a e$ , and whose height is $a sin E$ :

$QCR = \frac{a^2}{2} e \sin E$

Combining all of the above:

$PQR = \frac{T}{\tau} \pi a^2 = \frac{a^2}{2}E - \frac{a^2}{2} e \sin E$

Dividing through by $a 2 / 2$ :

$\frac{2 \pi}{\tau}T = E - e \sin E$

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

$MT = \theta \;$

Setting $M = 2π / τ$ and $θ = E - e sin E$ gives us Kepler's equation. Kepler referred to $M$ as the mean motion, and $E - e sin E$ 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 $2π$ per orbital period $τ$ , so the mean angular velocity is always $2π / τ$ .

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

$MT = E - e \sin E \;$

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

[edit] Application

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 cannot be solved for $E$ analytically, and so a numerical root-finding algorithm must be used.

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, $E - sin E$ , 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.