Determining Phase Angle

The calculations below assume that you are already familiar with performing Hohmann Transfer Orbits.

Thanks to Kepler’s Third Law, we don’t need to know the mass of the central body or the actual periods of the orbiting bodies to figure out what angle the destination body will move through during your transit time.

All depicted bodies are moving counterclockwise in the image, and are assumed to be in circular orbits in the same plane.

The starting orbit is in red, the transfer orbit in yellow, and the destination orbit in green.

The solid red circle is  the entry point of the transfer orbit, the hollow green circle is the exit point, and where your destination object should be at arrival.

The solid green circle is a representation of where the destination object might be  at the  beginning of the transfer orbit.

h1 = Altitude of initial orbit.

h2 = Altitude of final orbit.

r = radius of central body.

ϴ = The Sweep angle, defined as the angle destination object will travel along a circular orbit during transfer time, in degrees, measured counterclockwise.

ϕ = Phase Angle, defined as the angle between the starting point and the position of the destination object at the optimal transfer time, measured counterclockwise.

pt is the number of orbits your destination object will complete in the time it takes your spacecraft to complete the transfer orbit.

Define ft as the fractional part of  pt. For example if pt = 1.8392, ft = 0.8932  If pt<1, ft = pt.

When transferring from a lower circular orbit to a higher  circular orbit (h2>h1),  ϴ will never be less than approximately 63.6°

Working out how to set up your escape trajectory from Kerbin to leave directly into the transfer orbit is beyond the scope of this document.  Players looking to do so should see Kosmo-Not’s Interplanetary How-To Guide thread.

Measuring Phase angle with the Navball

(Probably no longer all that useful, was written before targeting and closest-approach information was available in the UI.)

In Kerbal Space Program, the Navball has a pink mark on it that always points towards one of the Kerbal Space Centers. As a result, in interplanetary space, it always points towards Kerbin.  The 90° dot in the center of the brown hemisphere on the navball always points towards the center of the object your spacecraft is orbiting.  

As a result, if your spacecraft is in a circular orbit in interplanetary space, it’s possible to determine the angle  between the Kerbol Dot and the Kerbin Marker that corresponds to the proper phase angle for your return burn, effectively allowing you to use the navball as a protractor.

We define rk as the semimajor axis of Kerbin’s Orbit, and rs as the semimajor axis of your spacecraft’s orbit

Using the Law of Cosines, we can then determine d, the distance between your spacecraft and Kerbin at the desired phase angle, φ.

Knowing d, we can then determine  the angle between Kerbol and Kerbin on the navball using the Law of Sines.

If you don’t wish to go through these calculations yourself, there are a number of  online triangle calculators that can solve for the angles of a triangle given two sides and the angle between them.

And of course, to find the desired pitch on the navball for the Kerbin marker, subtract λ from 90°.

Keep in mind, you’re going to have to make sure that Kerbin is on the proper side of Kerbol yourself, and depending on altitude, eyeballing the angle or using a physical protractor or screen-measuring software may be a more attractive solution.

To-Do: Making course corrections for returning to Kerbin using the navball.