# Why is proper motion always along a great circle?

I came across this question about proper motion and intersections of 2 stars proper motions on a celestial sphere, and in the solution they make use of the fact the proper motions are always along a great circle. I am clearly misunderstanding the situation, as the diagram i drew clearly has the motion not as a great circle. Could anyone clarify? I successfully answered the part about the position angles of the proper motions.

• Great circle in the sky corresponds to a plane in space. Thé plane is defined by a line of sight and velocity vector. Commented Jul 30 at 18:31
• I thought a plane of a great circle must intersect the centre of the sphere?
– n014
Commented Jul 30 at 18:34
• Yes, the plane of a great circle must past through the center: en.wikipedia.org/wiki/Great_circle Commented Jul 30 at 19:58
• Yes, the center of the sphere is you (observer). Commented Jul 30 at 20:42
• Who says that proper motion against the celestial sphere is always along a great circle? That doesn't make sense. Proper motion in the sky can be any number of things in any direction. I think there's something that your question is leaving out. Whoever it was was probably talking about ecliptic motion. Commented Jul 30 at 21:06

It's an approximation, but a pretty good one.

The actual motion of a star over many thousands of years will depend on the local gravitational potential of this region of the galaxy. Stars won't be moving in straight lines

But over a few years, even over a few hundred years, the motion of a star can be reasonably approximated by a constant velocity vector. And this vector, when projected back through the centre of the Earth means that the motion projects to an arc of a great circle, to a high degree of accuracy.

In your diagram, the motion of the star could be achieved only by a star moving in a curved path in space. To visualise this, project the star back from the sphere to the point in space. To describe a small circle, the actual motion of the star must be curved. But an arc of a great circle on the sky can be produced by a star moving in a straight line in space.

• Thank you, this makes sense. Could you explain what is wrong with my diagram? And how to visualise it? because your point makes sense but i cant see whats wrong with mine
– n014
Commented Jul 31 at 10:01
• @n014 Your standing on a small moon. You decide to throw a rock into orbit around the moon. You choose a random direction and throw. If you succeed at getting it into orbit, it will follow the geodesic (says Einstein) and that means it follows a great circle. You can not put an object into orbit at latitude 30 degrees, but you can at longitude 30 degrees because that is a geodesic. Bottom line, you drew the wrong circle for the chosen proper motion vector in your diagram. Commented Jul 31 at 21:32
• Thank you! This is very helpful
– n014
Commented Aug 1 at 9:05