Circle about a point on the Celestial Sphere

Question

Given a star at some celestial (spherical) coordinates (RA,Dec) or (Az,Alt) and an angle that represents the radius of the circle surrounding the star- what is the general form of the equation of this circle in terms of Az,Alt (or RA,Dec) and the angle?

This answer seems so close... but I cannot put it together in my head: Circle On Sphere

I am guessing that

x = r cos θ

y = r sin θ

r = x^2 + y^2

BUT... I doubt this works on a sphere.

The article reference above has what appears to be polar like coordinates on a sphere... which is great- but it references x,y, and z and I can not figure out how these relate to Alt, AZ, and the angle.

Any help would be greatly appreciated!

Welcome to MSE. Please read this text about how to ask a good question. — José Carlos Santos, Apr 10 '20 at 05:37
Thank you Jose Carlos Santos. I do not yet know how to write mathematical equations. I will add information to flesh out the question- but I am pretty certain I need to somehow substitute the RA, Dec, and angle into the equations(s) of the referenced article. — ngc1535, Apr 10 '20 at 06:26

score 1 · Answer 1 · answered Apr 10 '20 at 07:11

The spherical-to-Cartesian coordinate transformation is, on a unit sphere,

$$\begin{cases}x=\cos\theta\sin\phi,\\ y=\sin\theta\sin\phi,\\ z=\cos\phi.\end{cases}$$

Now it suffices to express by a dot product that the angle formed by the origin, the pole of the circle $(u,v,w)=(\cos\theta_0\sin\phi_0,\sin\theta_0\sin\phi_0,\cos\phi_0)$ and any point on the circle is $\rho$:

$$u\cos\theta\sin\phi+v\sin\theta\sin\phi+w\cos\phi=\cos\rho.$$

This can be rewritten

$$\cos\left(\theta-\theta_0\right)\sin\phi_0\sin\phi+\cos\phi_0\cos\phi=\cos\rho$$

and you can draw $\theta$ in terms of $\phi$, or

$$\sqrt{\cos^2(\theta-\theta_0)\sin^2\phi_0+\cos^2\phi_0}\cos\left(\phi-\arctan(\cos(\theta-\theta_0)\tan\theta_0)\right)=\cos\rho,$$

giving $\phi$ in terms of $\theta$.

Circle about a point on the Celestial Sphere

1 Answers1