I am stuck and a little embarrassed.

I'm trying to implement this answer. I have a unit vector $w$ pointing in the direction of a radio source on the celestial sphere.

$u$ and $v$ are the two other, mutually orthogonal unit vectors that point east and north from that point, tangent to the sphere.

Without back-converting to R.A. and Dec or using trigonometry, is there a simple way to generate $u$ and $v$ vectorially?

Slide 38 here may or may not be helpful, I think my description is sufficient by itself.

  • $\begingroup$ How did you determine w? $\endgroup$
    – Mike G
    Apr 11 '19 at 11:42
  • $\begingroup$ @MikeG In my question, $w$ is a given. It can come from any source. If it comes from orbital mechanics, it's the normalized relative position vector $\mathbf{r}/|r|$, or it can come from an initial RA/Dec value that's been spun around the Earth's axis for 24 hours like I did here when I built SgrA_star (which is what would be $w$), or it can be from somewhere else entirely. From any given $w$ I'm asking how to get $u$ and $v$ without going back to trigonometry. Can it be done using just vectors somehow? $\endgroup$
    – uhoh
    Apr 11 '19 at 11:49

If $\mathbf{\hat{n}}$ points to the north celestial pole, then the eastward tangent vector is

$$\mathbf{\hat{u} = \frac{\hat{n} \times \hat{w}}{\|\hat{n} \times \hat{w}\|}}$$

and the northward tangent vector is

$$\mathbf{\hat{v} = \frac{\hat{w} \times \hat{u}}{\|\hat{w} \times \hat{u}\|}}$$

$\mathbf{\|\hat{w} \times \hat{u}\|}$ analytically should be 1 but can be slightly different numerically.

  • $\begingroup$ Bingo! renormalize, facepalm, repeat... $\endgroup$
    – uhoh
    Apr 11 '19 at 21:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.