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I have stumbled upon two variations of Hill’s equations across numerous scientific journals:

  1. enter image description here

  2. enter image description here

Why are the mean motion’s positive and negative signs of these two Hill equations reversed? And are these expression equivalent, if so why?

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There is no standard definition of the local vertical, local horizontal frame. There are four key choices:

  1. Is local vertical (aka radial) toward or away from the center of the Earth?
  2. Is another axis along or against the angular velocity vector ($\vec r \times \vec v$)?
  3. Should we use a right-handed or left-handed coordinate system?
  4. Which axis is which regarding the xyz order of the axes?

Item (3) has been pretty much standardized in favor of a right-handed coordinate system. The remaining three have not been standardized. When reading about Hill's equations, also known as the Clohessy-Wiltshire equations, it is very important to understand the coordinate system used in that reference and to be able to transform the equations to your chosen form.

The first set of equations in the question appears to be using $\hat x$ as pointing away from the center of the Earth, $\hat z$ as pointing along the angular velocity vector, and $\hat y$ as completing a right-handed coordinate system. The second set of equations appears to be using $\hat z$ as pointing toward the center of the Earth as opposed to radially away.

Every writeup I have seen have uniformly made choices for items 1, 2, and 4 such that the third axis more or less points along rather than against the velocity vector. (The CW equations can be used for orbits that are very slightly elliptical, which is why I wrote "more or less".) The choices of items 1, 2, and 4 are however not standardized.

Caveat reader!

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  • $\begingroup$ Do not use the wikipedia page on the CW equations. It does not specify what $x$, $y$, and $z$ mean. $\endgroup$
    – David Hammen
    Commented Nov 18, 2022 at 12:54
  • $\begingroup$ Regarding item (3), never, never use a left-handed coordinate system (tl;dr). The change from or to left-handed coordinate system affects the sign of all vectors defined by cross products, as they are not normal vectors but the result of antisymmetric operators. It is in theory possible to use left-handed reference frames, but the extra work is so large that people decided to use once and for all, only right-handed reference frames and avoid the problem completely. $\endgroup$
    – Paulo Gil
    Commented Nov 18, 2022 at 15:26
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    $\begingroup$ @PauloGil That move to exclusively using right-handed coordinate systems is surprisingly recent. For example, airports until recently used north-east-up, which is left-handed. The cross product can be consistently defined in terms of left-handed coordinate systems such that $\hat x \times \hat y$ is still equal to $\hat z$ (but you need to use your left hand so that your thumb points along $\hat x$, your index finger along $\hat y$, and your middle finger along $\hat z$). That said, I do agree: Do not use left-handed coordinate systems. $\endgroup$
    – David Hammen
    Commented Nov 18, 2022 at 15:44
  • $\begingroup$ I only talk about reference frame, not "real life". Yes, the choice of using right-handed reference frames is arbitrary, we could have selected left-handed ones. Yes, we would have to use the left hand, meaning that for the usual determinant rules a minus sign must be added. We could even not forbid left r right-handed reference frames and deal with the sign at vector level: for each axial (aka pseudovector) we include $(-1)^\alpha$ where $\alpha$ is the sign of the determinant of the transformation matrix. But that is really annoying so people prefer to abolish one "hand" $\endgroup$
    – Paulo Gil
    Commented Nov 19, 2022 at 15:08
  • $\begingroup$ @PauloGil The determinant of a transformation matrix from one left-handed coordinate system to another left-handed coordinate system that only involves a rotation is still one. The determinant of a transformation matrix from a left-handed coordinate system to its reflection is still minus one. Using only left-handed coordinate systems is self-consistent, just as is using only right-handed coordinate systems. Problems arise only with mixing and matching. $\endgroup$
    – David Hammen
    Commented Nov 19, 2022 at 15:25

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