# Object in or out of a circular field of view? (celestial coordinate system)

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In summary, to check if an object with position RA and dec is within a given circular field of view of radius R (in arcminutes) and centred at (0,0), you need to compute the distance d of the object from the center using the formula d = (RA^2+dec^2)^0.5 and check that d is less than R. However, this may not be accurate for small angular separations, as a change in RA gets smaller as you get nearer to the poles. A more accurate formula is sqrt(δdec^2 + cos^2(dec)*δRA^2), which can be found in the provided reference.
TL;DR Summary
Check that object with position RA and dec is within circle of radius R (in arcminute) ?
In celestial coordinate system (right ascension/declination), how to check if an object with position RA and dec is within a given circular field of view of radius R (in arcminutes) and centred at (0,0)?
R is small in this case so I assumed that I could compute the distance d of the object from the center as in cartesian coordinates: d = (RA^2+dec^2)^0.5 and check that d is less than R. Is that correct (at least for small angular separation) ?

Thanks !

Summary:: Check that object with position RA and dec is within circle of radius R (in arcminute) ?

In celestial coordinate system (right ascension/declination), how to check if an object with position RA and dec is within a given circular field of view of radius R (in arcminutes) and centred at (0,0)?
R is small in this case so I assumed that I could compute the distance d of the object from the center as in cartesian coordinates: d = (RA^2+dec^2)^0.5 and check that d is less than R. Is that correct (at least for small angular separation) ?

Thanks !
Not really. A change δdec is the same size everywhere, but a change δRA gets smaller as you get nearer the poles. So for small changes, the angular separation between two objects is give by sqrt(δdec^2 + cos^2(dec)*δRA^2). Of course this won't make much difference if you are at (0,0), which is on the equator.

Thanks a lot and sorry for the late reply. Do you have any reference for the formula that you give? I'd like to see how it is derived.
Thanks again.

Thanks a lot and sorry for the late reply. Do you have any reference for the formula that you give? I'd like to see how it is derived.
Thanks again.
Here's a good link. The formula I gave is down near the bottom of the page. It also has more general formulae for when the differences of the coordinates are not small.

Great, thanks !

## 1. What is a circular field of view in the celestial coordinate system?

A circular field of view in the celestial coordinate system is a representation of the sky as seen from a specific location on Earth. It is a circular area on the celestial sphere, which is an imaginary sphere surrounding Earth, used to map and locate celestial objects.

## 2. How is an object's position determined in the celestial coordinate system?

An object's position in the celestial coordinate system is determined by its celestial coordinates, which are its right ascension and declination. Right ascension is measured in hours, minutes, and seconds eastward from the vernal equinox, while declination is measured in degrees north or south of the celestial equator.

## 3. Can an object be in or out of a circular field of view?

Yes, an object can be either in or out of a circular field of view in the celestial coordinate system. If an object's celestial coordinates fall within the circular field of view, it is considered to be in the field of view. If its coordinates fall outside of the circular field of view, it is considered to be out of the field of view.

## 4. How does the size of a circular field of view affect the visibility of objects?

The size of a circular field of view can affect the visibility of objects in the celestial coordinate system. A larger field of view can show more objects, while a smaller field of view may only show a select few. Additionally, the size of the field of view can also impact the level of detail and resolution of the objects that are visible.

## 5. How can I determine the circular field of view for a specific location and time?

The circular field of view for a specific location and time can be determined by using a star chart or a planetarium software. These tools allow you to input your location and time, and then display the circular field of view for that moment. You can also calculate the field of view using mathematical formulas if you know your location and the time of observation.

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