How Do You Calculate Angular Separation of Stars Using Different Coordinates?

[TheSource007]

Hi all.
I have been looking for a formula that gives me the angular separation of two stars.
So far I just found this one :http://www.skythisweek.info/angsep.pdf
but it uses the RA and DEC of the 2 stars, and I need to use the hour angle and DEC. Is it the same thing? Do I just replace HA by RA?

Also I need the equation that gives me the angular separation given the altitude and azimuth of two stars. Can I use the same equation by replacing RA by azimuth and so on?

My goals is to see if the angular distance between two stars given the RA, DEC is different from the angular distance given the ALT, AZ. I think it should be the same but I want to prove it. Also I want to know how that distance (given the ALT, AZ) changes over time. Does it stay the same or does it change?
I would appreciate answers but I really need the equations for a computer code.
Thank you

 

[Philosophaie]

HA is Hour Angle. RA is Right Ascension. LMST is the Local Mean Sidereal Time. d is the time in days since Jan 1,2000.

HA = LMST - RA

where LMST = 24*( 18.697374558 + 24.0657098244191 * d - longitude / 15)
and d = (Julian Date - 2451545)

 

[Filip Larsen]

Yes, you can use any pair of spherical coordinates in place of right ascension and declination, as long as the alpha coordinate in the equation you cite is the angle measured in the reference plane (equator) of the coordinate system and the delta coordinate is the angle measured from the reference plane and up, that is, delta is an elevation angle. As you can see, the alpha coordinates do only appear as an difference in angle and in addition in a way that makes the sign of the difference irrelevant, so the two alphas can be replaced by the angle subtended in the reference plane by the two objects.

You should very much end up with the same angular separation when you apply right ascension and declination as when you apply azimuth and elevation for the same two stars.

You can see a bit more about formulas for angular separation on [1]. Depending on what language you are using, it should be fairly straight forward to turn this into code. Note the comment on the wiki-page regarding the benefit of using the atan2 function to calculate inverse tangent [2].


[1] http://en.wikipedia.org/wiki/Great-circle_distance
[2] http://en.wikipedia.org/wiki/Atan2

 

[TheSource007]

Thanks a lot. I used the same formula for both coordinate systems and I got approximately the same result.

 

[pmacias]

for your question and interest in understanding the angular separation of two stars. I am happy to provide you with some information and equations to help you with your research.

First, let's clarify the difference between RA and hour angle (HA). RA, or right ascension, is the angular distance measured eastward along the celestial equator from the vernal equinox to the hour circle passing through a celestial object. HA, on the other hand, is the angular distance measured westward along the celestial equator from the observer's meridian to the hour circle passing through a celestial object. So, while they are related, they are not the same thing and cannot be simply replaced in an equation.

To calculate the angular separation of two stars using HA and DEC, you can use the following formula:

Angular separation = arccos(sin(DEC1) * sin(DEC2) + cos(DEC1) * cos(DEC2) * cos(HA1-HA2))

Where DEC1 and DEC2 are the declinations of the two stars, and HA1 and HA2 are their respective hour angles.

To calculate the angular separation using altitude and azimuth, you can use the following formula:

Angular separation = arccos(sin(ALT1) * sin(ALT2) + cos(ALT1) * cos(ALT2) * cos(AZ1-AZ2))

Where ALT1 and ALT2 are the altitudes of the two stars, and AZ1 and AZ2 are their respective azimuths.

It is important to note that the angular separation will vary depending on the time of observation, as the positions of the stars will change in the sky. So, the distance given by ALT and AZ will change over time. However, the distance given by RA and DEC should remain constant.

I hope this helps you with your research and calculations. If you require any further assistance, I suggest consulting with a professional astronomer or using specialized software for accurate and precise results. Happy observing!