Any flower you like

Satellite constellations are becoming very popular nowaday. SpaceX, Amazon, and other commercial entities are currently breaking any record in terms of number of satellites launched in orbit. Constellations are meant to provide continuous coverage on Earth, normally for remote sensing or communication. The classic design foresees circular orbits of the same size whose orbital planes are equally spaced along the RAAN direction (also called Walker constellations). Looking at other possible implementations, I was fascinated by another approach that uses elliptical orbits instead to provide a continuous, closed loop patterns in a rotating reference system called Flower Constellation.

In an inertial reference frame the orbit is an ellipse, but if you look at it from a reference frame rotating with the Earth you'll see how the resulting trajectory starts looking different. Using specific parameters values for the orbit size and orientation (that I won't discuss here because it would be too long for this article), it is possible to make such trajectories closed in the rotating reference system. Flower constellations are called this way because the patterns are reminiscent of flower petals.?The video below shows how two phased satellites in two distinct Molniya orbits (in white) actually follow a closed space track in the rotating reference frame (in blue), that looks like a two-petals flower. The same track is run by both satellites:

The main parameters to set in a Flower Constellations are the number of days to repeat the pattern and the number of petals to get. The latter parameters specifies how many apogees the satellite will cross before the pattern is repeated. The figure below shows typical patterns obtained setting the days to repeat to one and changing the number of petals. The higher this number, the lower the apogee altitude since each satellite has more orbits to travel in just one day:

The figure below shows the patterns created by just changing the number of days to repeat instead. The higher this number, the higher the apogee altitude:

Once you establish the shape of the orbit in the rotating reference frame (also called compatibility analysis), you can start putting additional satellites following the same space track. A specific spacing in RAAN and mean anomaly have to be designed in order to distribute satellites among a bunch of admissible locations (that are upper bounded and define the phasing analysis).

In the next two videos I'm showing two of the most iconic constellations made by using the Flower Constellation concept. The first one is called Lone Star, that forms the shape of a five pointed star spinning about the axis of symmetry, which in this case is the spin axis of the Earth. The constellation itself is composed by 77 satellites, with 23-days repeating pattern and 38 petals. The specific design of the parameters allows the creation of five independent petals synchronized with the constellation's motion:

The second one is called SCP (Secondary Closed Paths), where the satellite distribution highlights the existence of secondary paths whose dynamics is intricate and beautiful to see. The constellation itself is composed by 90 satellites, with 1-day repeating pattern and 8 petals. The secondary paths rotate together with the constellation and follow a closed pattern defined on top of the original relative path:

I used ANSYS Systems ToolKit (STK) to generate the orbits and generate the videos and pictures for this article.

The Flower Constellation theory has been developed by professor Daniele Mortari from Texas university. Here below the link to download his paper.