<p>It's an approximation, but a pretty good one.</p>
<p>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</p>
<p>But over a few years, even over a few hundred years, the motion of a star can be reasonably approximated by a <em>constant velocity vector</em>.  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.</p>
<p>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.</p>

CC BY-SA 4.0
2024-07-31T10:01:33.320Z
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
CC BY-SA 4.0
2024-07-31T21:32:24.353Z
@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.
eshaya
CC BY-SA 4.0
2024-08-01T09:05:18.013Z
Thank you! This is very helpful
n014
CC BY-SA 4.0
2024-07-30T22:45:56.993Z
2024-07-31T10:57:32.233Z
2024-07-31T10:57:32.233Z
James K
<p>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.<a href="https://i.sstatic.net/ZHeI0hmS.jpg" rel="noreferrer"><img src="https://i.sstatic.net/ZHeI0hmS.jpg" alt="all necessary information" /></a></p>

CC BY-SA 4.0
2024-07-30T18:31:59.320Z
Great circle in the sky corresponds to a plane in space. Thé plane is defined by a line of sight and velocity vector.
Leos Ondra
CC BY-SA 4.0
2024-07-30T18:34:40.893Z
I thought a plane of a great circle must intersect the centre of the sphere?
n014
CC BY-SA 4.0
2024-07-30T19:58:53.450Z
Yes, the plane of a great circle must past through the center: https://en.wikipedia.org/wiki/Great_circle
Greg Miller
CC BY-SA 4.0
2024-07-30T20:42:07.693Z
Yes, the center of the sphere is you (observer).
Leos Ondra
CC BY-SA 4.0
2024-07-30T21:06:46.277Z
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.
Miss Understands
CC BY-SA 4.0
2024-07-30T21:49:10.373Z
YEah, this is what confused me. I dont really see how they can say that; would you say it is worth it to just move on and assume it’s a flawed solution? Or is there a reason in this particular context (ie finding the point where the paths intersect) we can say this just for this problem?
n014
CC BY-SA 4.0
2024-07-31T20:42:27.477Z
Draw the sphere again.  Next, draw any straight line path.  Then sketch the projection of the line onto the sphere from the perspective of the center of the sphere.  That is the path on the celestial sphere of an object moving in a straight line in 3-d space.  It is a segment of a great circle.  The proper motion varies as distance and perspective change.
eshaya
<proper-motion>
Why is proper motion always along a great circle?
n014
# Why is proper motion always along a great circle?

(Asked by: username_0 on 2024-07-30)

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.[![all necessary information](https://i.sstatic.net/ZHeI0hmS.jpg)](https://i.sstatic.net/ZHeI0hmS.jpg)

**Tags:** proper-motion

> **Comments:**
> **username_1**: Great circle in the sky corresponds to a plane in space. Thé plane is defined by a line of sight and velocity vector.  
> **username_0**: I thought a plane of a great circle must intersect the centre of the sphere?  
> **username_2**: Yes, the plane of a great circle must past through the center: https://en.wikipedia.org/wiki/Great_circle  
> **username_1**: Yes, the center of the sphere is you (observer).  
> **username_3**: 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.  
> **username_0**: YEah, this is what confused me. I dont really see how they can say that; would you say it is worth it to just move on and assume it’s a flawed solution? Or is there a reason in this particular context (ie finding the point where the paths intersect) we can say this just for this problem?  
> **username_4**: Draw the sphere again.  Next, draw any straight line path.  Then sketch the projection of the line onto the sphere from the perspective of the center of the sphere.  That is the path on the celestial sphere of an object moving in a straight line in 3-d space.  It is a segment of a great circle.  The proper motion varies as distance and perspective change.  

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## 1 answers

### Answer by username_5 on 2024-07-30. Score: 3

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.

> **Comments:**
> **username_0**: 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  
> **username_4**: @username_0 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.  
> **username_0**: Thank you! This is very helpful  

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