What does a 6-month GTO to GEO transfer look like in practice?

Question

Based on How much time does it take to circularize a GTO orbit using ion propulsion? and the launch and entry into service dates, it takes about 6 months of continuous ion propulsion to transfer a GTO to a GEO.

According to a spacenews.com article, the journey was 7 months for the all-electric Eutelsat 117 West B. They have an illustration, which suggests the GTO's apogee was higher than GEO:

Source: spacenews.com

The post Thrust and rotation strategy to circularize a standard GTO orbit using ion propulsion? offers a theoretical approach. To a mere Kerbonaut, who only knows how to circularize an orbit by firing at apogee: when a continuous very, very, very small thrust is used in practice such as that payload (Eutelsat 117 West B) aboard the June 2016 Falcon 9 launch, how does the orbit look like as it circularizes, since the thrust is not always at apogee?

Of note, I think, is the movable arms with the thrusters:

(...) these twin arms can be moved freely about its body so their thrusts can always be aligned precisely with the satellite's center of gravity for orbit raising and stationkeeping – saving propellant to elongate mission life (eoportal.org).

Answer 1

I don't have an exact GTO-to-GEO transfer on hand, but I have this figure from a class assignment from a few years back. While definitely not a practical trajectory, it shows the characteristics of how to transfer from an elliptical to a circular orbit.

This solution was computed using indirect optimization. This problem assumed constant thrust magnitude (so the thruster is always firing), with the control variable being the thrust angle. The objective function was min(t_f).

As the figure shows, the thrust vectors are approximately in the velocity direction near periapsis and apoapsis, and they are approximately perpendicular between the apses. In general, when the thrust vectors are in line with the velocity vector (parallel or anti-parallel), they are adjusting the orbital energy (and therefore increasing/decreasing semi-major axis). When the thrust vectors are perpendicular to velocity, they are adjusting the eccentricity (shape) and argument of periapsis (orientation) of the orbit.

For a 6-month transfer from GTO to GEO, you would have a LOT of spirals. A GTO has a period of around 10.7 hr, and GEO has a 24 hr period. If you take their average, you're looking at around 250 spirals.