# Time of flight between two anomalies in a known orbital trajectory

Suppose I know all of the orbital elements of a particular orbital trajectory (not necessarily an elliptical one). Suppose I know the position and true anomalies at two different points in this trajectory, but not the time it takes to get from one to the other. Assuming a prograde trajectory from point 1 to point 2, how can I determine the time of flight between it takes to get from the first position to the second position? Is there a single algorithm I can use to do this independently of the shape of the trajectory?

• couldn't you just calculate time at position 1 and time at position 2 and find the difference between them? – Iamsodarncool Sep 21 '17 at 2:08
• @Iamsodarncool: How? – Paul Sep 21 '17 at 3:37
• @Paul does this answer help? – uhoh Apr 17 '18 at 1:43

Try Kepler's method, which works for orbits with a (1/r$^2$) central force. See Nathaniel Grossman, The Sheer Joy Of Celestial Mechanics, http://spiff.rit.edu/classes/phys440/lectures/ellipse/ellipse.html, or http://www.bogan.ca/orbits/kepler/orbteqtn.html.

Example. For an ellipse around Sol, suppose you know perihelion r$_P$ and aphelion r$_A$. You are given the true anomaly f$_1$, the angle between perihelion and the object's position as measured from the focus occupied by Sol.

Then eccentricity e = (r$_A$-r$_P$)/(r$_A$+r$_P$),

semi-major axis a = (r$_A$+r$_P$)/2, and

period T = 2$\pi$ sqrt(a$^3$/(MG)),

where M is the mass of Sol and G is Newton's constant.

Compute the eccentric anomaly E$_1$, the angle between perihelion and the object's position as measured from the center of the ellipse.

cos(E$_1$) = (e+cos(f$_1$))/(1+ecos(f$_1$))

The travel time deltaT$_1$ from perihelion to f$_1$ is

deltaT$_1$ = (E$_1$-esin(E$_1$))T/(2$\pi$)

Find deltaT$_2$ for the second position. Then the travel time from f$_1$ to f$_2$ = deltaT$_2$ - deltaT$_1$.

The third reference above tabulates the variations needed for circles, parabolae, and hyperbolae.

-MBMelcon

• This is a great start on an answer! It drifts away from the question when you start talking about ion engines. Are you sure it can't be extended to parabolic or hyperbolic orbits? (I don't think the OP is asking about trajectories with propulsion, just not necessarily elliptical) It also needs some clean-up in the MathJax, and several typos fixed. – uhoh Sep 20 '17 at 18:26
• Indeed, i'm looking for an approach that is agnostic to the eccentricity of the orbit (i.e. it should work the same way for ellipses, parabola, and hyperbola). – Paul Sep 20 '17 at 20:36
• I removed the extraneous information on powered orbits and added a reference with explicit information for the other conic sections. - MBM – MBM Sep 20 '17 at 23:08
• It turns out that I've transcribed equations for all conics from your link in this answer to a different question. – uhoh Apr 17 '18 at 1:46