1.

How to Calculate the Angular Distance Between Two Stars

 
Calculating angular distances on the celestial sphere involves no more than intermediate level geometry. Below, the proper formula is presented "without proof" together with an angular distance calculator that does all the tedious work for you.

Let RA.1 and Decl.1 be the right ascension and declinations of star 1 and Let RA.2 and Decl.2 be the right ascension and declination of star 2 in degrees. I.e.: Right Ascencion takes values between 0 and 360 degrees and Declination has a value between -90 and +90 degrees).

Then the angular distance A, in degrees, between the two stars, 1 and 2 is determined by the following relation:

 
  • cos(A) = sin(Decl.1)sin(Decl.2) + cos(Decl.1)cos(Decl.2)cos(RA.1 - RA.2) and thus, A = arccos(A)

Note: The above formula is analytically exact. However, for pairs of stars with small distances the above formula becomes numerically unstable to solve for ordinary electronic calculators. This is because cos(A) is  very close to 1 for small A-values and varying very litte with changes in A when A is very small. Thus, a calculator having about seven significant digits for its internal calculations cannot distinguish the cosines of any distances smaller than about one minute of arc. However, most modern computers) use 64-bit floating-point numbers, which provide 15 significant figures of precision. With this precision, the formula gives well-behaved results down to much smaller distances and with the use of a contemporary computer the formula should be applicable for most practical purposes

Normally, declination is expressed in degrees, arcminutes and arcseconds (+ or - DD:MM:SS) which must the be converted to decimal degrees:

  • Decl.(decimal degrees) = + or - (DD + MM/60 + SS/3600)

Likewise, Right Ascension is normally expressed in hours, minutes and seconds (HH:MM:SS) which must be converted to decimal degrees as follows:

First convert HH:MM:SS to decimal hours: RA(decimal hours) = HH + MM/60 + SS/3600. Then convert to degrees using that 1 hour in RA represents 15 degrees on the celestial sphere:

  • RA (degrees) = 15 * (HH + MM/60 + SS/3600)

In the angular distance calculator below, you may either enter right ascensions and declinations as decimal hours and decimal degrees or, you may enter RA values as read from most tables in hours followed by a punctuation and then minutes and seconds of time as the following digits; i.e.: The format to be used below is HH.MMSS. When you have entered this value, clik on the corresponding  "convert to decimal hours" button to the right. Likewise, you may enter declination values as read from most tables in the similar format as + or - DD.MMSS and then click the "convert to decimal degrees" button to the right.

When all co-ordinates have been entered as or converted to decimal hours and decimal degrees, you may clik the "Calculate" button to get your result.

 

Angular Distance Calculator
 
RA.1(in decimal hours) or in Hours.MinutesSeconds (HH.MMSS) --->
Decl.1(in decimal degrees) or in Degrees.ArcminutesArcseconds (+DD.MMSS) --->
RA.2(in decimal hours) or in Hours.MinutesSeconds (HH.MMSS) --->
Decl.2(in decimal degrees)   or in Degrees.ArcminutesArcseconds (+DD.MM.SS) --->
Angular Distance = decimal degrees
Angular Distance = degrees(d) arcminutes(m) arcseconds(s)

 

 

 
 
2.

How to Calculate the Angular Field of View for a Camera

 
The focal length, F of a lens (an optical system) is the distance to the focal point (sensor or film plane) when the lens/optical system is focused on an object at infinity. This is a fundamental characteristics of the lens and thus, does not depend upon what camera the lens is used on.

The field of view of a camera lens, FOV  is defined as the angle in space over which objects are recorded on the film or sensor in the camera. The FOV depends on both the focal length of the lens, F and the physical size of the film or sensor. Thus, it can only be calculated, measured or stated if the size of the film or sensor is known. For camera lenses, three fields of view are often stated: the horizontal FOV, the vertical FOV and the diagonal FOV.

Since current digital cameras come with a number of different CCD/CMOS-sizes, you will often find references to a "conversion factor" named Digital Multiplier  or Focal Length Multiplier used for lens/camera systems with sensors smaller than the frame size of a 35mm camera which is 36 mm *24 mm.. This "multiplier" is used to state the factor by which a lens's focal length would have to be increased to give the same angle of view on 35 mm film as what this lens provides on the specific digital sensor it has been designed for. (I.e.: a lens, that is considered "short" or "wide" on a 35 mm camera would be considered as "long" or "tele" on a compact digital camera). One should note that while the aspect ration (width : height) of 35 mm film is 3:2 many compact digital cameras have sensors with an aspect ratio of 4:3, while most digital SLR cameras (but not all) exhibit the same aspect ratio as that of 35 mm film. Thus, you cannot sensibly characterize differences between different designs by merely one "conversion factor". One has to state whether that applies to image width, height, diagonal or all!

Now then, for a normal (not macro or fish-eye) lens projecting a rectilinear image, the angle of view (v) can be calculated from the chosen dimension (d), and effective focal length (F) as follows:

 
  • v = 2 * arctan( d/(2*F) )

Please note that this formula assumes that horizontal and vertical FOVs are measured along the axis passing through the centre of film/sensor - i.e.: the centre of the the (uncropped) image. Focusing at infinity means that you are effectively projecting great circles (meridians) and parallel circles (declination circles) of the celestial spere onto a flat sensor/film plane. Thus, only the central meridian and the declination circle passing through the center of your image will be projected as straight lines. Meridians and declination circles towards the edges will be projected as curves and the field of view will be larger along the edges than the center horizonthal and vertical FOVs. For wide-angle lenses, this effect may be quite significant.

The calculator below computes the angular field of view for a lens of a specified focal length and specified dimension of the camera's film / sensor. You may use it in two ways:

Either input the "Lens focal length" and the "Digital multiplier" as stated by the lens/camera manufacturer. This assumes that your camera provides pictures with a 3:2 aspect ratio and "Sensor Width" and "Sensor Height" should be left at 36 mm and 24 mm accordingly.

Or you may input "Lens focal length" and known dimensions of your film/camera sensor wile leaving "Digital multiplier" input as 1.0. This will apply to sensors of all aspect ratios.

Note: Typical sensor sizes may be found at the sensor size page at this site.

 

 

Angular Field of View Calculator

Photography Calculators
Lens focal length (mm):
Digital multiplier:
Sensor Width (mm):
Sensor Height (mm):
Horizontal FOV (degrees):
Vertical FOV (degrees):
Diagonal FOV (degrees):
     
 

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Copyright 2010 - Steen G. Bruun