Angular distance in globular clusters

[June_cosmo]

Homework Statement

Go to the following web site: http://www.astro.utoronto.ca/~cclement/cat/listngc.html
This is the portal to a large on-line catalog of variable stars in globular clusters. Each link is a cluster name and will take you to a simple text page that lists near the top the coordinates of the cluster. Below that is an explanation of the data in the table, followed by a list of variable stars in the cluster. Each variable star entry lists the ID, RA and Declination of the star along with other information not relevant to this problem.
Choose a few of the clusters until you find one that has at least ten variable stars listed in its catalog page.

For these ten (or more) stars, use the spherical sine and cosine laws to calculate i) the angular distance of each star from the cluster center in arcmin and ii) the ‘position angle’ (or PA) of each star relative to the cluster center in deg. The PA is defined so that if a star is located directly north of the cluster center is at =0deg=360 , while a star directly east is at =90

Homework Equations

The Attempt at a Solution

How do I get the central point of the cluster? And I don't know how to interpret those position coordinates?

 

[CWatters]

How do I get the central point of the cluster?

Not my subject at all but it would seem reasonable to me to assume that the "coordinates of the cluster" are the same as "the coordinates of the centre of the cluster".

The coordinate system appears to be explained here.

http://spiff.rit.edu/classes/phys373/lectures/radec/radec.html#radec

 

[June_cosmo]

Not my subject at all but it would seem reasonable to me to assume that the "coordinates of the cluster" are the same as "the coordinates of the centre of the cluster".

The coordinate system appears to be explained here.

http://spiff.rit.edu/classes/phys373/lectures/radec/radec.html#radec

Thank you! I think you're right. But how do I calculate the angular distance from the coordinates?

 

[haruspex]

Thank you! I think you're right. But how do I calculate the angular distance from the coordinates?

As far as I understand it, angular distance is based on the position of an observer. In this case, the observer must be someone on Earth. So it would be the angle that the pair of points (star, cluster centre) subtends at Earth. That can be calculated from the ascensions and declinations. See http://www.gyes.eu/calculator/calculator_page1.htm.

 

[June_cosmo]

As far as I understand it, angular distance is based on the position of an observer. In this case, the observer must be someone on Earth. So it would be the angle that the pair of points (star, cluster centre) subtends at Earth. That can be calculated from the ascensions and declinations. See http://www.gyes.eu/calculator/calculator_page1.htm.

Actually the "observing point" should be the north or south celestial pole. And I was looking at the webpage too but had no idea how they derived that formula : (

 

[haruspex]

Actually the "observing point" should be the north or south celestial pole.

Are you sure? It needs to be a specific point in space. Celestial poles are just directions, aren't they?
It would help to know what the data in the table means. Where it gives the ascension and declination of a star in the cluster, is that from the point of view of an observer at the cluster centre or an observer on Earth? I would have thought that the 3D location within the cluster was unknowable. Even though we may know the distance to the cluster, don't we only get a 2D map of the stars within it? That would be consistent with the last part of the question, where it only asks for one angle to specify the position of the star within the cluster.

 

[June_cosmo]

Are you sure? It needs to be a specific point in space. Celestial poles are just directions, aren't they?
It would help to know what the data in the table means. Where it gives the ascension and declination of a star in the cluster, is that from the point of view of an observer at the cluster centre or an observer on Earth? I would have thought that the 3D location within the cluster was unknowable. Even though we may know the distance to the cluster, don't we only get a 2D map of the stars within it? That would be consistent with the last part of the question, where it only asks for one angle to specify the position of the star within the cluster.

If you look at http://spiff.rit.edu/classes/phys373/lectures/radec/radec.html#radec that @CWatters kindly provided, that would explain a lot.

 

[haruspex]

If you look at http://spiff.rit.edu/classes/phys373/lectures/radec/radec.html#radec that @CWatters kindly provided, that would explain a lot.

That supports what I'm saying. The observation point is from Earth. The celestial pole just determines the orientation of the reference frame. So the angle asked for is the angle that the pair (star, cluster centre) subtends at Earth.

 

[haruspex]

Actually the "observing point" should be the north or south celestial pole. And I was looking at the webpage too but had no idea how they derived that formula : (

Do you need to derive it? From what starting point? In CWatter's link, there's an equation based on the law of cosine, just after:

In this case, the law of cosines becomes
​

I can walk you through from there to the approximation $\gamma=...$.
It's based on $\cos(\theta)=1-\frac 12\theta^2+$ smaller terms, for small theta. The differences $\delta_1-\delta_2$ and $\alpha_1-\alpha_2$ are small, as is $\gamma$.

 

[June_cosmo]

That supports what I'm saying. The observation point is from Earth. The celestial pole just determines the orientation of the reference frame. So the angle asked for is the angle that the pair (star, cluster centre) subtends at Earth.

Yes, after a second thought I think you're right. I now know how to derive that formula from cosine law. But what about the position angle?

 

[haruspex]

Yes, after a second thought I think you're right. I now know how to derive that formula from cosine law. But what about the position angle?

Are you asking what it is or how to find it? Think of the cluster as seen from Earth. If the star is due north of the centre the PA is 0, if due east then it is 90, etc.
To find a formula for it, consider two points near each other on the surface of the earth, at about latitude $\phi$, latitude difference $\delta\phi$, and longitude difference $\delta\theta$. Draw the (almost) rectangle with those points at diagonally opposite corners. What are the dimensions of the rectangle, approximately?