Time of Flight — Lesson 4

To determine how long a transfer takes or how long it takes a satellite to move to a new point in an orbit, the concepts of mean anomaly and eccentric anomaly are introduced.

Mean anomaly comes from the concept of mean motion. This is the average angular velocity a satellite would be traveling in a circular orbit.

n = √(µ/a³)

The mean anomaly is the location within a circular orbit found by multiplying the time it takes for a satellite to pass from a reference location to a location of interest by the mean motion.

M = n (t₁- t₀)

To relate mean anomaly to true anomaly the concept of eccentric anomaly is introduced.

To find eccentric anomaly draw a line perpendicular to the major axis of the ellipse which intersects the satellite and the circle. Then draw a line from this point on the circle to the circle’s center. Eccentric anomaly is the angle between the satellite’s periapsis and this line.

Eccentric anomaly can be geometrically related to true anomaly (ν) and eccentricity.

E= cos^-1 ((e+cos⁡(ν))/(1+e cos⁡(ν)))

With eccentric and mean anomaly defined, a relationship between time of flight and position within an orbit can be defined by utilizing Kepler’s 2nd Law and geometry relations.

(t₁- t₀) = √(a³/µ) * (E - e sin⁡(E))

The above equation can be put in terms of mean anomaly and eccentric anomaly. Thus, with a, e, ν₀, and ν₁ known, the time of flight can be determined.

M = n (t₁ - t₀) = E - e sin⁡(E)

A more general equation can be used if the satellite passes through periapsis where k is the number of times the satellite has passed through periapsis. Eccentric anomaly and true anomaly will be in the same half plane.

(t₁ - t₀)=√(a³/µ)  [2kπ+(E - e sin(E))-(E₀ - e sin(E₀))]