# Experiment 9: Circumpolar Star

### Objective :

To Identify a Circumpolar Star.

### 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 ${\small \phi}$), a star whose declination is greater than ${\small 90^\circ-\phi}$ 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 \begin{eqnarray} \sin a &=& \cos h \cos \delta \cos \phi + \sin \delta \sin \phi \label{eq1}\,, \end{eqnarray} where ${\small a}$ = altitude, ${\small h}$ = hour angle of the star, ${\small \delta}$ = declination of the star, ${\small \phi}$ = 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 is ${\small h = 0\,hr}$ at that instance and it is called upper culmination, or transit. When ${\small h = 12\,hr}$, it is called lower culmination. Using previous relation, for ${\small h = 0\,hr}$, we find, \begin{eqnarray} \sin a &=& \cos \delta \cos \phi + \sin \delta \sin \phi\\ &=&\cos(\phi-\delta)\\ &=&\sin (90^\circ-\phi+\delta) \end{eqnarray} So the altitude at the upper culmination is \begin{eqnarray} a_{max} = \left\{ \begin{array}{ll} 90^\circ-\phi+\delta, & \mbox{if the object culminates South of zenith}\\ 90^\circ+\phi-\delta, & \mbox{if the object culminates North of zenith.} \end{array} \right. \end{eqnarray} The altitude is positive for objects with ${\small \delta > \phi - 90^\circ}$ . Objects with declinations less than ${\small \phi - 90^\circ}$ can never be seen from the latitude ${\small \phi}$. Using the relation between various relevant positions, for ${\small h = 12\,hr}$, we get \begin{eqnarray} \sin a &=& - \cos \delta \cos \phi + \sin \delta \sin \phi\\ &=& - \cos(\delta + \phi) = \sin(\delta + \phi - 90^\circ )\,. \end{eqnarray} Likewise, at the lower culmination, the altitude is ${\small a_{min} = \delta + \phi-90^\circ}$ . Stars with a declination ${\small \delta > 90^\circ - \phi}$ 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 ${\small \delta}$ from ${\small a_{max} = 90^\circ -\phi+\delta}$, at the upper transit and ${\small a_{min} = \delta + \phi -90^\circ}$ at the lower transit, we get ${\small \delta = (a_{min} + a_{max})}$. Hence, the declination will be same and independent of the observer's location (latitude).

### Example : An observer at IIT Kanpur

IIT Kanpur is approximately located on ${\small 26^\circ 28' 12''}$ North latitude. From here, all stars with a declination higher than ${\small +63^\circ 31' 48''}$ (towards North pole) are circumpolar. Stars with a declination less than ${\small -63^\circ 31' 48''}$ (towards South pole) can never be seen by an observed located here .

• Ever visible star from IIT Kanpur : ${\small 30-}$Cepheus has a declination of ${\small +63^\circ35'3.9''}$, which has closest angular separation ${\small \sim7'}$ with IIT Kanpur, over a period of year. ${\small \epsilon-}$Cassiopeia is at a declination of ${\small +63^\circ40'12.27''}$, which has closest angular separation ${\small \sim12'}$ with IIT Kanpur, over a period of year.
• Never visible star from IIT Kanpur : ${\small \epsilon-}$Circinus has a declination of ${\small -63^\circ36'37.83''}$, which has closest angular separation ${\small \sim5'}$ with IIT Kanpur, over a period of year. ${\small \pi-}$Pavo has a declination of ${\small -63^\circ40'6.06''}$, , which has closest angular separation ${\small \sim6'}$ with IIT Kanpur, over a period of year.
• The stars with a declination between ${\small +63^\circ 31' 48''}$ to ${\small -63^\circ 31' 48''}$ are sometimes visible from IIT Kanpur. It is very instructive to do this astronomy experiment "virtually" using the freely available planetarium simulator Stellarium.

The following figures are useful to understand the idea of a circumpolar star.