# Transform asteroid rotation to heliocentric ecliptic coordinates

I'm working with data from the DAMIT database of asteroid shape models. I'm adding them to a visualization in which the sun is at [0, 0, 0] and the X, Y axes constitute the ecliptic plane of the solar system.

Each asteroid model comes with some attributes that define its orientation and spin:

• λ (ecliptic longitude),
• β (ecliptic latitude)
• P (sidereal rotation period)
• φ0 (initial rotation angle)
• JD0 (initial date).

I've applied the matrix formulas suggested by the folks at DAMIT:

in which

This is where I get confused. My understanding is that my visualization uses a heliocentric ecliptic coordinate system.

I set r_ast to the XYZ location of a vertex in my visualization. I'm not sure this is correct.

Then I compute r_ecl from the equation above. But I think variables λ and β are angles in a geocentric ecliptic coordinate system and I need to do some transformation into heliocentric ecliptic coordinates.

Is my approach correct and if so, how can I convert the asteroid's rotation to my visualization's heliocentric ecliptic coordinate system?

• $\lambda$ and $\beta$ are the co-ordinates of the asteroid's pole in a heliocentric ecliptic frame so I think you are OK if this is also what your visualization is using Feb 7, 2019 at 15:23
• Thanks @astrosnapper. How can you tell λ and β are heliocentric rather than geocentric in this case? From this table of ecliptic coordinate notation on Wikipedia I guessed they were geocentric: en.wikipedia.org/wiki/…
– ty.
Feb 7, 2019 at 16:23
• Technically it doesn't specify the origin (and it should) but since the asteroid is in orbit around the Sun and not the Earth, heliocentric vs geocentric seems most likely Feb 7, 2019 at 16:46
• Thank you @astrosnapper. One more point of clarification. Is it correct to set r_ast to the asteroid's heliocentric ecliptic rectangular coordinates? The website describes r_ast as a vector in the "asteroid co-rotating coordinate frame" but I'm not sure what that means here.
– ty.
Feb 8, 2019 at 18:29

r_ast is a vector that goes from the center of the asteroid to each vertex of the shape model as if you had stuck a set of rods in the center of a potato and pointing along the long axis, to the pole on top and 90 degrees to these to give X,Y,Z directions.
The matrices above that you give then transform these r_ast vectors for the co-ordinates of the surface of the asteroid into the ecliptic plane as a function of time as the asteroid rotates, giving you r_ecl. It says in the Kaasalainen paper (I think) that r_ecl has an origin shifted to the center of the asteroid. If you are wanting a visualization of what the asteroid looks like relative to the Earth or Sun etc, then you will need to calculate the Sun->asteroid vector and the Sun->Earth vector using e.g. JPL HORIZONS and using the Vector Table ephemeris type or a programmatic wrapper to this (e.g. astroquery) or the NAIF SPICE toolkit and getting or generating a SPK kernel file for the asteroid (the normal e.g. de430.bsp JPL planetary ephemeris will handle the Sun->Earth part)