# Diagram of Hayabusa 2 in "Hill Coordinate System"; what is that exactly? How to convert it to inertial?

These JAXA's Hayabusa-2's two Tweets: 1 2 show a diagram of the spacecraft's motion relative to Ruygu in what they call a Hill Coordinate System,

This is a diagram of the trajectory during solar conjunction. It is drawn in the Hill coordinate system; think of this as the coordinate system where the Sun is in the negative x-axis direction. (1/2)

Hayabusa2 departed from the home position (altitude 20km) on November 23 and will pass the furthest point from Ryugu tomorrow (December 11). Red dots on the previous image show where a trajectory control manoeuvre was performed. (2/2)

What is this coordinate system exactly? How would it be converted back to inertial coordinates relative to Ryugu? Hill coordinates represent a local reference frame widely used in relative motion when only the main body gravity (in this case the Sun) is considered. The Hill coordinates origin and axes are defined as follows

$$Ox_hy_hz_h$$: centred in the target center of gravity (in this case Ryugu gravity center)

$$x_H$$: the line joining the target and the Sun, positive in the direction of the Sun to the target

$$y_H$$: is defined using the cross product of $$x$$ and $$z$$ to form a right-handed system. If the target orbit is circular is aligned with its velocity.

$$z_H$$: parallel to the angular momentum vector of the target orbit

If you want to transform the system to an inertial one (understanding the Sun reference frame), you just have to add the position of the target to the relative position

$$\vec{r}_{hayabusa/I}$$=$$\vec{r}_{ryugu/I}$$+$$\vec{r}_{hayabusa/ryugu}$$, being the last term the one given by the Hill frame (maybe you have to rotate something depending on your definition of the inertial axes). Note that Ryugu movement around the Sun is keplerian and its position is considered to be known Figure representing the Hill frame for relative motion in geocentric orbits. Source: "Controlled orbital dynamics of low altitude formations by means of electrical propulsion" (downloadable PDF in ResearchGate)

As a curiosity the Space Shuttle does not use the Hill frame for relative motion operations, it uses the local-vertical/local-horizontal frame

Bonus: Space Shuttle LVLH definition illustration (Source:https://slideplayer.com/slide/8865753/) • It seems they are using the direction from the Sun to Ruygu as $+x$, so Hayabusa-2 is at negative $x$ to have the Sun behind it's back. Also, to get an inertial frame, I would also have to undo the rotation of Ryugu about the Sun. This $x$ axis is not inertial, it's in a rotating, or synodic frame.
– uhoh
Dec 16, 2018 at 1:34
• Yep, you are right Dec 22, 2018 at 23:00
• I found a copy of your cited source and added a link. The caption to your figure is "Figure 1 LVLH (also known as Hill’s) reference frame". Is the caption wrong, or are they the same? Your last sentence seems to draw a distinction. I'd like to accept your answer but I just need to clear this up. Thanks!
– uhoh
Jul 12, 2019 at 1:08
• Yep, interesting remark. In a strict way, Hill frame and "NASA" frame for rendezvous operations can be considered local-vertical local-horizontal. The "NASA" LVLH frame comes as an heritage from the aircraft community. The correspondance between them is this: $x_{Hill}$ <-> $-z_{NASA}$ / $y_{Hill}$ <-> $x_{NASA}$ / $z_{Hill}$ <-> $-y_{NASA}$ Jul 12, 2019 at 10:26
• Thanks! If it's possible, can you add a little bit of that to the end of the answer?
– uhoh
Jul 12, 2019 at 10:29