Technical note | Changing Coordinates in the Context of Orbital Mechanics
This note works through an example of switching between many coordinate systems using a modern matrix language that lends itself to describing arenas with multiple entities such as found in many Defence scenarios. To this end, it describes an example in planetary orbital theory, whose various Sun- and Earth-centred coordinate systems makes that theory a good test-bed for such an exposition of changing coordinates. In particular, we predict the look direction to Jupiter from a given place on Earth at a given time, highlighting the careful bookkeeping that is required along the way. To avoid much of the rather antiquated jargon and notation that pervades orbital theory, we explain the first principles of 2-body orbital motion (Kepler's theory), beginning with Newton's laws and proving all the necessary expressions. The systematic and modern approach to changing coordinates described here can also be applied just as readily in contexts such as a Defence aerospace engagement, which follows the interaction of multiple entities that each carry their own coordinate system.
Real-world defence scenarios might be described or managed by any of their participants, and a core element of this description is the ability to transform between the many coordinate systems that typically quantify the entities involved. Switching between coordinates is often seen as a classical yet difficult problem. This report attempts to show that the task can be made easier and more transparent by using unambiguous notation that carefully describes all relationships of the relevant entities.
A worked example in planetary orbital theory is a useful test-bed for such an exposition. The various Sun- and Earth-centred coordinate systems involved in predicting, say, the look direction to Jupiter from a given place on Earth at a given time require careful book-keeping of the plethora of numbers involved in the calculation.
With that worked example in our sights, and to avoid much of the rather antiquated jargon and notation that pervades orbital theory, we cover the first principles of 2-body orbital motion by beginning with Newton's laws and proving all the necessary expressions. The main focus here is to show how to interrelate the various coordinate systems that are necessary to the worked example.
We are content to consider the 2-body problem (Kepler's theory) — which can be solved analytically, unlike the many-body problem — because the relevant concepts of changing coordinates are sufficiently illustrated in a 2-body scenario. We thus decouple the Solar System into two 2-body systems that are gravitationally independent: Sun-Earth, and Sun-Jupiter. The resulting high accuracy in the prediction of Jupiter's look direction from Earth supports the validity of this decoupling.
The exposition begins with the relevant classical mechanics and time concepts, proves Kepler's three laws, then establishes and describes how to relate the different coordinate systems involved with the Earth-centred and Sun-centred inertial frames, the Earth-centred Earth-fixed frame, and the observer's local \at Earth" frame. It describes the necessary celestial geometry and orbital elements, and finishes with the worked example of locating Jupiter at a given time.
The explanations in the pages that follow are written in an expansive style that describes related concepts, such as what equinoxes and solstices are and when they occur, how julian days are defined, and how orbital calculations can be extended to more complex motion, such as that of the Moon. In particular, the theory of how to change coordinates more generally in an elegant but also powerful way is explained in detail.
Although predicting where Jupiter can be found is not of any great utility in a Defence context, anyone who understands the procedure described in this note will have no problem attempting the simpler task of working, for example, in the area of research into satellite positioning systems such as GPS.