Simultaneously view periapsis to body and inclination/nodes relative to specified satellite

[simon-dima]

First, a description of what I originally wanted to achieve, to avoid XY problem: given an interplanetary trajectory passing close to Eve, I want to adjust the approach in order to prepare capture and subsequent rendezvous with Gilly. More specifically, the tweaked trajectory should

• have an Eve periapsis which is low (for maximum use of the Oberth effect during capture) but remains above the atmosphere
• be suitable for a transfer to Gilly, either by
  • having a small inclination relative to Gilly's orbital plane (if lucky), or
  • having the capture burn at Eve periapsis coincide with the ascending/descending node of the trajectory relative to Gilly's orbital plane (in order to first capture into a highly elliptical orbit, then match inclinations at apoapsis).

To tune the manœuvre, it is desirable to be able to visualize both the periapsis relative to Eve and the inclination and ascending/descending nodes relative to Gilly's orbital plane, simultaneously. (Although Gilly's orbit is significantly nonplanar, for my use case it does not matter much whether inclination and nodes are measured relative to a moving osculating plane or relative to an inertially fixed plane which is somewhat close to Gilly's average orbital plane.)

This situation is not specific to Eve and Gilly: seeing both periapsis and inclination simultaneously is useful when approaching any body while planning an approach to a satellite (celestial body or vessel) in an inclined orbit. This information is readily available in stock KSP when Gilly is set as target.

The reference frames currently available in Principia do not show this information:

• GEO does not show the Eve periapsis (Eve moves quite a lot in this frame, since Gilly's orbit is very eccentric).
• GEL has a fixed Eve but distances pulsate and the apsides are not shown.
• ECI does not show the inclination relative to Gilly's orbital plane.

One possible way I see to display this information simultaneously would be to enable displaying markers computed in one reference frame on the trajectory plotted in another reference frame, e.g. show the GEO ascending/descending node markers when plotting in ECI (or the ECI Eve periapsis marker when plotting in GEO). From a user interface perspective I think keeping pinned markers would work quite well: switch to GEO, pin ascending node, switch to ECI.

Another approach would involve a new reference frame in which this information is available. I see several possibilities:

• An "Eve-Gilly-Orbit" frame, exactly analogous to Gilly-Eve-Orbit: Eve is fixed, Gilly is in a fixed direction but oscillating radially, the horizontal plane (against which ascending/descending nodes are measured) corresponds to the plane osculating to Eve's orbit in the Gilly-centered nonrotating frame (this is the same plane as the plane osculating to Gilly's orbit in the ECI frame). I don't know how the satellite-body-orbit frames are implemented in Principia, but if the choice of central body and satellite is free, then this frame would presumably be easy to add with just an tweak to the UI.
• An "Eve-Gilly-aligned" frame, in which Eve is fixed and Gilly orbits in the horizontal plane. Similarly to GEO, the plane which is horizontal in the proposed frame is osculating to Gilly's orbit in the nonrotating Eve-centered plane. However, the movement of this plane relative to inertial space is different: where the horizontal plane of GEO moves in order to ensure that Gilly stays on a specific ray, the horizontal plane of this proposed frame would move without twisting around its normal axis, only guaranteeing that Gilly stays on the plane. I visualize this as similar to parallel transport of a tangent plane around a sphere. In the case where Gilly's (or more generally the satellite's) orbit were perfectly planar, the "Gilly-Eve-aligned" frame would be a nonrotating reference frame centered on Eve, with the difference from ECI being the position of the horizontal plane which makes ascending/descending nodes relative to Gilly's orbit around Eve.
  Implementing this frame would probably be tricky, as the transformation matrix between coordinates in ECI and coordinates in the proposed frame cannot be calculated simply from knowing the instantaneous position and velocity of Gilly relative to Eve (as is the case with GEO); rather, it can be obtained by solving some differential equations involving the whole time history of Gilly's trajectory.
• An inclined Eve-centered nonrotating frame: like ECI, but with the horizontal plane being tilted by two fixed configurable angles relative to the horizontal plane of ECI. The user would be responsible for entering angles which position the fixed plane approximately in the same orientation as Gilly's (or any target's) orbit.