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Identification of a Circumpolar Star
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Introduction :

 

 

                                     Circumpolar means to circle around the pole. Circumpolar star is a star that never sets (that is, it never disappears below the horizon at all times, irrespective of day and night or year), as seen from a given latitude, due to its proximity to one of the celestial poles. They move in a counterclockwise direction. However it is not visible during the day because of sunshine. Designation of a star as circumpolar depends on the observer's latitude. At either of Earth's poles all stars of that hemisphere are circumpolar, whereas at the equator none are, since only one half of the celestial sphere can ever be seen. For an observer (with a latitude ϕ), a star whose declination is greater than 90ϕ will be circumpolar, appearing to circle the celestial pole and remaining always above the horizon. However, there will be some stars which can never be seen beyond a certain latitude.

 

 

Demonstration:

                                                    

                         Fig 1: The nautical triangle for deriving transformations between the horizontal and equatorial frames.
 

Using spherical trigonometry and transformations between the horizontal and equatorial frames, the relation between various relevant positions can be found to be:

                                                                        «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathvariant=¨normal¨»sin«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mo»§nbsp;«/mo»«mi»h«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mo»§nbsp;«/mo»«mi»§#948;«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mo»§nbsp;«/mo»«mi»§#934;«/mi»«mo»+«/mo»«mi mathvariant=¨normal¨»sin«/mi»«mo»§nbsp;«/mo»«mi»§#948;«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»sin«/mi»«mo»§nbsp;«/mo»«mi»§#934;«/mi»«mo»,«/mo»«/math»

where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«/math»=altitude,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«/math»=hour angle of the star,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#948;«/mi»«/math»=declination of the star,«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#934;«/mi»«/math»=observer's latitude. These are shown in Fig. 1. Any object's altitude is greatest when it is on the south meridian (the great circle arc between the celestial poles containing the zenith). The hour angle isat that instance and it is called upper culmination, or transit. When «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»12«/mn»«mi»h«/mi»«mi»r«/mi»«mo»,«/mo»«/math» it is called lower culmination. Using previous relation, for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»0«/mn»«mi»h«/mi»«mi»r«/mi»«/math», we find,

                                                                        «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi mathvariant=¨normal¨»sin«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mi»§#948;«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mi»§#934;«/mi»«mo»+«/mo»«mi mathvariant=¨normal¨»sin«/mi»«mi»§#948;«/mi»«mi mathvariant=¨normal¨»sin«/mi»«mi»§#934;«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mfenced»«mrow»«mi»§#934;«/mi»«mo»-«/mo»«mi»§#948;«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»sin«/mi»«mfenced»«mrow»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«mo»-«/mo»«mi»§#934;«/mi»«mo»+«/mo»«mi»§#948;«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/math»

So the altitude at the upper culmination is,

                                      «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mrow»«mi»m«/mi»«mi»a«/mi»«mi»x«/mi»«/mrow»«/msub»«mo»=«/mo»«mfenced close=¨¨ open=¨{¨»«mtable»«mtr»«mtd»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«mo»-«/mo»«mi»§#934;«/mi»«mo»+«/mo»«mi»§#948;«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»i«/mi»«mi»f«/mi»«mo»§nbsp;«/mo»«mi»t«/mi»«mi»h«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»b«/mi»«mi»j«/mi»«mi»e«/mi»«mi»c«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»u«/mi»«mi»l«/mi»«mi»m«/mi»«mi»i«/mi»«mi»n«/mi»«mi»a«/mi»«mi»t«/mi»«mi»e«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mi»S«/mi»«mi»o«/mi»«mi»u«/mi»«mi»t«/mi»«mi»h«/mi»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»f«/mi»«mo»§nbsp;«/mo»«mi»z«/mi»«mi»e«/mi»«mi»n«/mi»«mi»i«/mi»«mi»t«/mi»«mi»h«/mi»«/mtd»«/mtr»«mtr»«mtd»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«mo»+«/mo»«mi»§#934;«/mi»«mo»-«/mo»«mi»§#948;«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mi»i«/mi»«mi»f«/mi»«mo»§nbsp;«/mo»«mi»t«/mi»«mi»h«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»b«/mi»«mi»j«/mi»«mi»e«/mi»«mi»c«/mi»«mi»t«/mi»«mo»§nbsp;«/mo»«mi»c«/mi»«mi»u«/mi»«mi»l«/mi»«mi»m«/mi»«mi»i«/mi»«mi»n«/mi»«mi»a«/mi»«mi»t«/mi»«mi»e«/mi»«mi»s«/mi»«mo»§nbsp;«/mo»«mi»N«/mi»«mi»o«/mi»«mi»r«/mi»«mi»t«/mi»«mi»h«/mi»«mo»§nbsp;«/mo»«mi»o«/mi»«mi»f«/mi»«mo»§nbsp;«/mo»«mi»z«/mi»«mi»e«/mi»«mi»n«/mi»«mi»i«/mi»«mi»t«/mi»«mi»h«/mi»«/mtd»«/mtr»«/mtable»«/mfenced»«/math»

The altitude is positive for objects with «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#948;«/mi»«mo»§gt;«/mo»«mi»§#934;«/mi»«mo»-«/mo»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«/math».  Objects with declinations less than«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#934;«/mi»«mo»-«/mo»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«/math»can never be seen from the latitude «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#934;«/mi»«/math» Using the relation between various relevant positions, for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»h«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»§nbsp;«/mo»«mn»12«/mn»«mi»h«/mi»«mi»r«/mi»«/math», we get

                                                                        «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mi mathvariant=¨normal¨»sin«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»-«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mi»§#948;«/mi»«mo»§nbsp;«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mi»§#934;«/mi»«mo»+«/mo»«mi mathvariant=¨normal¨»sin«/mi»«mi»§#948;«/mi»«mi mathvariant=¨normal¨»sin«/mi»«mi»§#934;«/mi»«/mtd»«/mtr»«mtr»«mtd»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»=«/mo»«mo»-«/mo»«mi mathvariant=¨normal¨»cos«/mi»«mfenced»«mrow»«mi»§#948;«/mi»«mo»+«/mo»«mi»§#934;«/mi»«/mrow»«/mfenced»«mo»=«/mo»«mi mathvariant=¨normal¨»sin«/mi»«mfenced»«mrow»«mi»§#948;«/mi»«mo»+«/mo»«mi»§#934;«/mi»«mo»-«/mo»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«/mrow»«/mfenced»«mo».«/mo»«/mtd»«/mtr»«/mtable»«/math»

Likewise, at the lower culmination, the altitude is«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mrow»«mi»m«/mi»«mi»i«/mi»«mi»n«/mi»«/mrow»«/msub»«mo»=«/mo»«mi»§#948;«/mi»«mo»+«/mo»«mi»§#934;«/mi»«mo»-«/mo»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«/math». Stars with a declination «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#948;«/mi»«mo»§gt;«/mo»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«mo»-«/mo»«mi»§#934;«/mi»«/math» will never set.

                                                

                                        Fig 2: The altitude of a circumpolar star at upper and lower culmination.

Suppose we are observing a circumpolar star at its upper and lower culmination, as shown in Fig. 2. Eliminating«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#948;«/mi»«/math»from«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mrow»«mi»m«/mi»«mi»a«/mi»«mi»x«/mi»«/mrow»«/msub»«mo»=«/mo»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«mo»-«/mo»«mi»§#934;«/mi»«mo»+«/mo»«mi»§#948;«/mi»«mo»,«/mo»«/math» at the upper transit and«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mrow»«mi»m«/mi»«mi»i«/mi»«mi»n«/mi»«/mrow»«/msub»«mo»=«/mo»«mi»§#948;«/mi»«mo»+«/mo»«mi»§#934;«/mi»«mo»-«/mo»«msup»«mn»90«/mn»«mo»§#8728;«/mo»«/msup»«/math» at the lower transit, we get«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#948;«/mi»«mo»=«/mo»«mfenced»«mrow»«msub»«mi»a«/mi»«mrow»«mi»m«/mi»«mi»i«/mi»«mi»n«/mi»«/mrow»«/msub»«mo»+«/mo»«msub»«mi»a«/mi»«mrow»«mi»m«/mi»«mi»a«/mi»«mi»x«/mi»«/mrow»«/msub»«/mrow»«/mfenced»«mo».«/mo»«/math» Hence, the declination will be same and independent of the observer's location (latitude).
 

 


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