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Wikipedia , on several asteroid articles, claims that right ascension and declination of the North Pole can give you the axial tilt of the object. What formula is used to convert these 2 values into the axial tilt? (Ceres, for example, has an R.A. Of 291 degrees and a declination of 59 degrees; its axial tilt is approximately 3 degrees. How is this figured?

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  • $\begingroup$ You measure the distance from the body's north pole to the ecliptic north pole: en.wikipedia.org/wiki/Ecliptic_pole $\endgroup$ – barrycarter Sep 23 '15 at 1:16
  • $\begingroup$ Could you show how this is done? $\endgroup$ – Damon Blevins Sep 23 '15 at 13:23
  • $\begingroup$ Being too lazy to answer that <G>, I will point out that the Wikipedia Ceres article gives an RA of 29.41 degrees, a declination of 66.79 degrees, and an inclination of 10.593 degrees to the ecliptic. Could you source your data or perhaps you're thinking of another dwarf planet? $\endgroup$ – barrycarter Sep 23 '15 at 16:28
  • $\begingroup$ No I saw it before $\endgroup$ – Damon Blevins Sep 23 '15 at 16:35
  • $\begingroup$ OK, it's on the wikipedia page, but it's followed by "Dawn would later determine that the axis points in a different direction." $\endgroup$ – barrycarter Sep 23 '15 at 16:37
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This is wrong, but may help someone find the right answer:

You can convert right ascension and declination to a 3 dimensional unit vector in the J2000.0 ICRF reference frame using the standard formula for converting spherical coordinates to rectangular coordinates and using a radius of 1:

{Cos[dec] Cos[ra], Cos[dec] Sin[ra], Sin[dec]}

The ra and dec of the north ecliptic pole is ra,dec of {270, 66.5607083333} per Wikipedia, so the unit vector representing the north ecliptic pole is:

{0, -0.3977771648286046, 0.9174820582119942}

As it turns out, there is conflicting data for Ceres, perhaps because of the semi-recent DAWN flyby.

Instead, I'll use Vesta, where wikipedia explicitly states (https://en.wikipedia.org/wiki/4_Vesta#Rotation):

north pole pointing in the direction of right ascension 20 h 32 min, declination +48 [... which] gives an axial tilt of 29 [degrees]

Converting to degrees, Vesta's north pole's ra,dec is {308,48}. Using the formula above to find the unit vector, we get:

{0.411957936296447, -0.527282113378167, 0.743144825477394}

If we take the dot product of two vectors and divide by the product of their lengths, we get the cosine of the angle between them. In this case, both vectors have length 1, and the dot product is 0.891563 whose arc-cosine is right around 27 degrees.

So, if Vesta's orbit were the same plane as Earth's orbit, the axial tilt would be 27 degrees.

However, since Vesta's orbit is inclined 7.14043 degrees to the ecliptic, this answer is incorrect.

I don't think you can find the correct answer without using Vesta's inclination and the longitude of its ascending node.

Both of these values are known, but I can't figure out how to use them to get the correct asnwer.

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