To compute the position of a satellite at a given time using Kepler's laws of planetary motion (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. A satellite is an object that orbits another object (known as its primary). ... Johannes Keplers primary contributions to astronomy/astrophysics were the three laws of planetary motion. ...

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 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. Diagram for derivation of Keplers equation for time-of-flight. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipses circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. ... In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ... (This page refers to eccitricity in astrodynamics. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

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. The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipses circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. ... In astronomy, the true anomaly (, also written ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). ...

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: Johannes Keplers primary contributions to astronomy/astrophysics were the three laws of planetary motion. ...

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:

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 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: In mathematics, a line segment is a part of a line that is bounded by two end points. ...

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.

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:

1. Compute the eccentric anomaly E from true anomaly θ
2. 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. Transcendental in philosophical contexts In philosophy, transcendental experiences are experiences of an exclusively human nature that are other-worldly or beyond the human realm of understanding. ... In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...

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.