{"id":50,"date":"2018-11-16T06:06:01","date_gmt":"2018-11-16T06:06:01","guid":{"rendered":"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=50"},"modified":"2018-11-16T07:16:40","modified_gmt":"2018-11-16T07:16:40","slug":"diurnal-motion-of-stars","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/chapter\/diurnal-motion-of-stars\/","title":{"rendered":"Diurnal Motion of Stars"},"content":{"raw":"<div>\r\n\r\n&nbsp;\r\n\r\n<strong>1.\u00a0\u00a0Learning Outcomes<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">After studying this module, you should be able to<\/p>\r\n\r\n<ul style=\"text-align: justify\">\r\n \t<li>understand why stars and other celestial objects describe circular orbits from east to west<\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">grasp the meaning of rising and setting of stars<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">explain why some stars are circumpolar for an observer and why some are never visible to this observer<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">derive the condition for a star to be circumpolar<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">appreciate the phenomenon of \u2018midnight Sun\u2019<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">infer that from north pole (south pole) the stars of only the northern (southern) hemisphere are visible<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">conclude that for an observer at the equator, no star is circumpolar; the observer can see all stars in both the hemispheres but only for half of their orbits<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>2.\u00a0 <\/strong><strong>Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In the last three modules we have dealt with the various coordinate systems for celestial objects. <strong>This subject is collectively called positional astronomy<\/strong>.\u00a0 In particular, we discussed the <strong>horizon system<\/strong>, the <strong>equatorial system <\/strong>of both kinds, that is, the hour angle \u2013 declination system and right ascension \u2013 declination system, the <strong>ecliptic system <\/strong>and the <strong>galactic system<\/strong>. We also learnt the situations where each of these systems is important. Depending on the need of the observer, it is necessary to transform coordinates from one system to another. We saw how this can be done using spherical trigonometry.\u00a0 This module is the last in positional astronomy where we discuss the motion of stars as seen by observers at various locations.<\/p>\r\n&nbsp;\r\n\r\n<strong>3.\u00a0 Diurnal Motion of Stars\u00a0<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As we already know, the earth rotates on its axis completing one rotation in 24 hours (Fig. 4.1). Since we share the motion of the earth, we see the celestial objects rotate in the direction opposite to that of earth\u2019s rotation, completing one orbit in 24 hours. These orbits are in planes<\/p>\r\n<img class=\"aligncenter size-full wp-image-63\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.1.png\" alt=\"\" width=\"688\" height=\"447\" \/>\r\n<p style=\"text-align: justify\">Fig. 3.1. Earth rotates from west to east on its axis. Seen from above the north pole the rotation is counterclockwise.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">perpendicular to the axis of rotation of the earth and, therefore, parallel to the plane of the celestial equator.\u00a0 This daily motion of objects is called <\/span><strong style=\"text-align: initial;font-size: 1em\">diurnal (occurring daily) motion<\/strong><span style=\"text-align: initial;font-size: 1em\">.\u00a0\u00a0 The circular orbits in which objects execute their daily motion are called <\/span><strong style=\"text-align: initial;font-size: 1em\">diurnal circles <\/strong><span style=\"text-align: initial;font-size: 1em\">(Fig. 4.2). Since the earth rotates on its axis from west to east (anticlockwise as seen from above the north pole) the celestial objects appear to rotate from east to west, or clockwise as seen from above, <\/span><strong style=\"text-align: initial;font-size: 1em\"><em>even in the southern hemisphere<\/em><\/strong><span style=\"text-align: initial;font-size: 1em\">. \u00a0So, if you look up north, you see the objects rotating counterclockwise.<\/span><\/p>\r\n<img class=\"aligncenter size-full wp-image-62\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.2.png\" alt=\"\" width=\"695\" height=\"477\" \/>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Fig. 3.2. \u00a0While the earth rotates on its axis from west to east, the stars appear to rotate from east to west on their diurnal circles in planes parallel to the celestial equator. \u00a0P and Q are the North and South celestial poles, respectively.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>3.1.\u00a0 Rising and Setting of Stars \u2013 Circumpolar Stars\u00a0<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">As a star crosses the horizon so that it is visible, we say that <strong>the star has risen<\/strong>. When the star goes below the horizon, we say that <strong>the star has set<\/strong>. On the portion of the diurnal circle which\u00a0<span style=\"text-align: initial;font-size: 1em\">is above the horizon of an observer the star is visible to the observer; below the horizon, the star is not visible (Fig. 4.3).\u00a0 In Fig. 4.3 the portions of the orbit above the horizon are shown in red; those below the horizon are shown in black. Interestingly, there are some stars whose diurnal circles are completely above the horizon; these stars are always visible to the observer, rotating round the Pole star. These stars never set.\u00a0 These are called <\/span><strong style=\"text-align: initial;font-size: 1em\">circumpolar stars<\/strong><span style=\"text-align: initial;font-size: 1em\">, because of their proximity to the poles.\u00a0<\/span>There are also stars which never cross the horizon for the observer; they are never visible to the observer.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-61\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.3.png\" alt=\"\" width=\"702\" height=\"507\" \/>\r\n<p style=\"text-align: justify\">Fig. 3.3. The stars whose diurnal circles cross the horizon are visible for the duration their diurnal circles are above the horizon.\u00a0 For the rest of the duration, they are not visible.\u00a0 These stars are said to rise and set.\u00a0 The stars whose diurnal circles do not cross the horizon are either visible all the time or are never visible. The former are called circumpolar stars.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>3.2.\u00a0 Condition for a Star to become Circumpolar\u00a0<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">To find the condition under which a star is circumpolar for an observer, notice that the declination of the star whose diurnal circle just touches the horizon is ?(Fig. 4.4). Also, the altitude of the Pole is ?, the latitude of the observer in the northern hemisphere. Since the distance of the Pole from the equator is 90\u00b0, we have ?+ ?= 90\u00b0. This means that a star<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-60\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.4.png\" alt=\"\" width=\"741\" height=\"566\" \/>\r\n<p style=\"text-align: justify\">Fig. 3.4. The arc length RN is the declination of the star whose diurnal circle just touches the horizon.\u00a0 The arc length from N to the Pole is equal to the latitude of the observer. The sum of the two must be 90\u00b0.\u00a0 This gives the condition for a star to be circumpolar.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">which has ?= 90\u00b0 \u2212 ?just touches the horizon for the observer at latitude ?. All those stars whose declination exceeds 90\u00b0 \u2212 ?are circumpolar at latitude ?. Thus, the condition for a star to be circumpolar at latitude ?is:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For observers in the Northern Hemisphere (Fig. 3.4)<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">? + ? \u2265 ??\u00b0,\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(3.1)<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">or,<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">? \u2265 (??\u00b0 \u2212 ? ).\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (3.2)<\/span><\/p>\r\nFor observers in the Southern Hemisphere (Fig. 3.5)\r\n\r\n? + ? \u2264 \u2212 ??\u00b0 . <span style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>(3.3)\r\n\r\n<img class=\"aligncenter size-full wp-image-59\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.5.png\" alt=\"\" width=\"724\" height=\"511\" \/>\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Fig. 4.5. Circumpolar stars and stars that never rise for an observer in the southern hemisphere at latitude <\/span><em style=\"font-size: 1em\">-<\/em><span style=\"font-size: 1em\">?.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">By the same argument, stars which always remain below the horizon must have:<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For observers in the Northern Hemisphere<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">?\u2212 ?\u2264 \u2212 ?0\u00b0,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3.4)<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">or,<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">?\u2264 (?\u2212 ?0\u00b0).\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3.5)<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For observers in the Southern Hemisphere<\/span><\/p>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">?\u2212 ?\u2265 ?0\u00b0.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3.6)<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Stars which never come up the horizon are never visible to the observer at latitude ?. Consider, for example, the constellation Big Bear (Saptarishi). The declination of the star of this constellation which is farthest from the Pole Star is 49\u00b0. Therefore, for all those living at latitudes 41\u00b0 or higher, this constellation will be circumpolar. It will be visible all the time, if the sunlight did not mask it during the day.<\/p>\r\n<img class=\"aligncenter size-full wp-image-58\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.6.png\" alt=\"\" width=\"682\" height=\"303\" \/>\r\n<p style=\"text-align: center\">Fig. 3.6. For people living at latitude 41\u00b0, or higher, Big Bear is circumpolar.<\/p>\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Fig. 4.7 shows the circumpolar stars in the northern hemisphere. For photograph and animation, visit: https:\/\/en.wikipedia.org\/wiki\/Circumpolar_star<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"size-full wp-image-57 aligncenter\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.7.png\" alt=\"\" width=\"191\" height=\"191\" \/>\r\n<p style=\"text-align: justify\">Fig. 4.7. The North Star and circumpolar stars in a photograph with a long shutter speed of several hours. (Source: Wikipedia, photo by LCGS Russ)<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">As another example, consider that you live at a place on the tropic of cancer, ?= -23.5\u00b0. \u00a0Then all stars satisfying the condition ?+ ?\u2264 \u2212 ?0\u00b0, or having ?\u2264 \u221266.5\u00b0 are circumpolar to you. At the same time, stars satisfying ?\u2212 ?\u2265 ?0\u00b0, or having ?\u2265 66.5\u00b0 will never be visible to you.<\/p>\r\n&nbsp;\r\n\r\n<strong>3.3.\u00a0 Circumpolar Stars for an Observer on the North Pole\u00a0<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Suppose an observer is stationed at the north pole. \u00a0As we have seen earlier, the horizon of this observer coincides with the celestial equator. All stars of the northern hemisphere are circumpolar. No star of the southern hemisphere is visible to this observer. Since the latitude of this observer is 90\u00b0, the stars of all positive declinations are circumpolar, as is clear from Equation (3.1). \u00a0These stars never set, as shown in Fig. 4.8. By the same logic, all stars south of the equator are invisible.\u00a0 For this observer, they never rise.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><img class=\"aligncenter size-full wp-image-56\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.8.png\" alt=\"\" width=\"678\" height=\"477\" \/><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Fig. 3.8.\u00a0 Circumpolar stars for the observer at the north pole. The situation is exactly the reverse for the observer at the south pole.<\/p>\r\n&nbsp;\r\n\r\n<strong>3.4.\u00a0 The Sun as Circumpolar Star\u00a0<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Note that the declination of the Sun varies between 0\u00b0 and 23.5\u00b0 during the six months from 21 March to 23 September. For the observer at the North Pole (indeed, for latitudes &gt; 66.5\u00b0), the Sun is circumpolar during these six months; it is always up, even at midnight (called <strong>midnight Sun, Fig. 4.9<\/strong>), and does not set.\u00a0 During the six months from 23 September to 21 June when the declination of the Sun varies from 0\u00b0 and -23.5\u00b0, the Sun is never visible to these observers in the northern polar region; it never rises. For the south polar region, though, the Sun is circumpolar\u00a0during 23 September to 21 March and visible continuously, while during 21 March to 23 September it is not visible at all.\u00a0 In popular parlance, it is said that in the polar regions the day and night are of six month\u2019s duration.<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-55\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.9.png\" alt=\"\" width=\"271\" height=\"203\" \/>\r\n<p style=\"text-align: center\">Fig. 3.9. The Altafjord in Alta, Norway bathed in the Midnight Sun. Source: Wikipedia, photo by Vberger.<\/p>\r\n&nbsp;\r\n\r\n<strong>3.5.\u00a0 Observer at the Equator\u00a0<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">What happens if the observer is located at the equator? This situation is shown in Fig. 4.10. Since the horizon is now perpendicular to the equator, it divides the diurnal circles in two equal<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"aligncenter size-full wp-image-54\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.10.png\" alt=\"\" width=\"680\" height=\"480\" \/>\r\n<p style=\"text-align: center\">Fig. 3.10. An observer at the equator is able to see all stars but only for half the time.<\/p>\r\n&nbsp;\r\n\r\n<span style=\"font-size: 1em;text-align: initial\">halves.\u00a0 All the stars, of both the hemispheres, are visible to this observer for half the time; for the remaining half they are below the horizon.<\/span>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For any observer between the equator and the pole, the situation is similar to that shown in Fig.\u00a0<\/span><span style=\"text-align: justify;font-size: 1em\">3.4. Depending on the latitude of the observer, some stars are circumpolar and some stars are not accessible to this observer at all. \u00a0Other stars rise and set for different durations.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>4. Summary\u00a0<\/strong>\r\n<ul>\r\n \t<li>The earth rotates on its axis from West to East.<\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Seen from above the North Pole this motion is anticlockwise.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">We on the earth see all celestial objects moving from East to West in circular orbits.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">The planes of these orbits are parallel to the Celestial Equator.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">If the orbit of a star crosses the horizon of the observer, the star is visible when above the horizon and it is invisible as goes below the horizon.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">When a star just comes up the horizon, it is said to have risen.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">The moment it goes below the horizon; it is said to have set.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">If the orbit of a star is always above the horizon, the star is always visible and is called a circumpolar star.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">If its orbit is completely below the horizon, the star is never visible to the observer.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Whether a star is circumpolar or not depends on the latitude of the observer.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">For an observer on the North Pole all stars in the northern hemisphere are circumpolar, and all those in the southern hemisphere are not visible.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">For an observer on the South Pole, all stars of the southern hemisphere are circumpolar and no star of the northern hemisphere is visible.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">In the polar regions, the Sun rises and sets for six months at a stretch.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Sun can be seen even at midnight at these places.<\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">An observer at the equator can see all stars from both the hemispheres, but only for half the time, because the horizon divides the orbits of stars in two equal halves.<\/span><\/li>\r\n<\/ul>\r\n<div class=\"toolbar\"><\/div>\r\n<div id=\"viewerContainer\">\r\n<div id=\"viewer\" class=\"pdfViewer\">\r\n<div id=\"pageContainer1\" class=\"page\">\r\n<div class=\"textLayer\">\r\n\r\n&nbsp;\r\n\r\n<strong>Know More<\/strong>\r\n\r\n&nbsp;\r\n\r\n<strong>Helpful Websites:<\/strong>\r\n<ul>\r\n \t<li>https:\/\/en.wikipedia.org\/wiki\/Circumpolar_star<\/li>\r\n \t<li>http:\/\/www.backyardastronomy.com\/Backyard_Astronomy\/Sky_Motion_Movies\/Pages\/Circumpolar_Stars-Looking_South_%28Southern_Hemisphere%29.html<\/li>\r\n \t<li>http:\/\/www.astro.rug.nl\/~sctrager\/teaching\/OA\/PositionalAstronomy.pdf<\/li>\r\n \t<li>https:\/\/www.youtube.com\/watch?v=soy04EG3VcE (animation)<\/li>\r\n \t<li>https:\/\/www.youtube.com\/watch?v=soy04EG3VcE (animation)<\/li>\r\n<\/ul>\r\n<span style=\"font-size: 1em\">You can\u00a0<\/span><span style=\"font-size: 1em\">have a look at the book at this site:\u00a0<\/span><span style=\"font-size: 1em\">https:\/\/www.willbell.com\/MATH\/mc8.htm<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div>\n<p>&nbsp;<\/p>\n<p><strong>1.\u00a0\u00a0Learning Outcomes<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">After studying this module, you should be able to<\/p>\n<ul style=\"text-align: justify\">\n<li>understand why stars and other celestial objects describe circular orbits from east to west<\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">grasp the meaning of rising and setting of stars<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">explain why some stars are circumpolar for an observer and why some are never visible to this observer<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">derive the condition for a star to be circumpolar<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">appreciate the phenomenon of \u2018midnight Sun\u2019<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">infer that from north pole (south pole) the stars of only the northern (southern) hemisphere are visible<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">conclude that for an observer at the equator, no star is circumpolar; the observer can see all stars in both the hemispheres but only for half of their orbits<\/span><\/li>\n<\/ul>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>2.\u00a0 <\/strong><strong>Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In the last three modules we have dealt with the various coordinate systems for celestial objects. <strong>This subject is collectively called positional astronomy<\/strong>.\u00a0 In particular, we discussed the <strong>horizon system<\/strong>, the <strong>equatorial system <\/strong>of both kinds, that is, the hour angle \u2013 declination system and right ascension \u2013 declination system, the <strong>ecliptic system <\/strong>and the <strong>galactic system<\/strong>. We also learnt the situations where each of these systems is important. Depending on the need of the observer, it is necessary to transform coordinates from one system to another. We saw how this can be done using spherical trigonometry.\u00a0 This module is the last in positional astronomy where we discuss the motion of stars as seen by observers at various locations.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.\u00a0 Diurnal Motion of Stars\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As we already know, the earth rotates on its axis completing one rotation in 24 hours (Fig. 4.1). Since we share the motion of the earth, we see the celestial objects rotate in the direction opposite to that of earth\u2019s rotation, completing one orbit in 24 hours. These orbits are in planes<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-63\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.1.png\" alt=\"\" width=\"688\" height=\"447\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.1.png 688w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.1-300x195.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.1-65x42.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.1-225x146.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.1-350x227.png 350w\" sizes=\"auto, (max-width: 688px) 100vw, 688px\" \/><\/p>\n<p style=\"text-align: justify\">Fig. 3.1. Earth rotates from west to east on its axis. Seen from above the north pole the rotation is counterclockwise.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">perpendicular to the axis of rotation of the earth and, therefore, parallel to the plane of the celestial equator.\u00a0 This daily motion of objects is called <\/span><strong style=\"text-align: initial;font-size: 1em\">diurnal (occurring daily) motion<\/strong><span style=\"text-align: initial;font-size: 1em\">.\u00a0\u00a0 The circular orbits in which objects execute their daily motion are called <\/span><strong style=\"text-align: initial;font-size: 1em\">diurnal circles <\/strong><span style=\"text-align: initial;font-size: 1em\">(Fig. 4.2). Since the earth rotates on its axis from west to east (anticlockwise as seen from above the north pole) the celestial objects appear to rotate from east to west, or clockwise as seen from above, <\/span><strong style=\"text-align: initial;font-size: 1em\"><em>even in the southern hemisphere<\/em><\/strong><span style=\"text-align: initial;font-size: 1em\">. \u00a0So, if you look up north, you see the objects rotating counterclockwise.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-62\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.2.png\" alt=\"\" width=\"695\" height=\"477\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.2.png 695w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.2-300x206.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.2-65x45.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.2-225x154.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.2-350x240.png 350w\" sizes=\"auto, (max-width: 695px) 100vw, 695px\" \/><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Fig. 3.2. \u00a0While the earth rotates on its axis from west to east, the stars appear to rotate from east to west on their diurnal circles in planes parallel to the celestial equator. \u00a0P and Q are the North and South celestial poles, respectively.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>3.1.\u00a0 Rising and Setting of Stars \u2013 Circumpolar Stars\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As a star crosses the horizon so that it is visible, we say that <strong>the star has risen<\/strong>. When the star goes below the horizon, we say that <strong>the star has set<\/strong>. On the portion of the diurnal circle which\u00a0<span style=\"text-align: initial;font-size: 1em\">is above the horizon of an observer the star is visible to the observer; below the horizon, the star is not visible (Fig. 4.3).\u00a0 In Fig. 4.3 the portions of the orbit above the horizon are shown in red; those below the horizon are shown in black. Interestingly, there are some stars whose diurnal circles are completely above the horizon; these stars are always visible to the observer, rotating round the Pole star. These stars never set.\u00a0 These are called <\/span><strong style=\"text-align: initial;font-size: 1em\">circumpolar stars<\/strong><span style=\"text-align: initial;font-size: 1em\">, because of their proximity to the poles.\u00a0<\/span>There are also stars which never cross the horizon for the observer; they are never visible to the observer.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-61\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.3.png\" alt=\"\" width=\"702\" height=\"507\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.3.png 702w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.3-300x217.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.3-65x47.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.3-225x163.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.3-350x253.png 350w\" sizes=\"auto, (max-width: 702px) 100vw, 702px\" \/><\/p>\n<p style=\"text-align: justify\">Fig. 3.3. The stars whose diurnal circles cross the horizon are visible for the duration their diurnal circles are above the horizon.\u00a0 For the rest of the duration, they are not visible.\u00a0 These stars are said to rise and set.\u00a0 The stars whose diurnal circles do not cross the horizon are either visible all the time or are never visible. The former are called circumpolar stars.<\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>3.2.\u00a0 Condition for a Star to become Circumpolar\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">To find the condition under which a star is circumpolar for an observer, notice that the declination of the star whose diurnal circle just touches the horizon is ?(Fig. 4.4). Also, the altitude of the Pole is ?, the latitude of the observer in the northern hemisphere. Since the distance of the Pole from the equator is 90\u00b0, we have ?+ ?= 90\u00b0. This means that a star<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-60\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.4.png\" alt=\"\" width=\"741\" height=\"566\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.4.png 741w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.4-300x229.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.4-65x50.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.4-225x172.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.4-350x267.png 350w\" sizes=\"auto, (max-width: 741px) 100vw, 741px\" \/><\/p>\n<p style=\"text-align: justify\">Fig. 3.4. The arc length RN is the declination of the star whose diurnal circle just touches the horizon.\u00a0 The arc length from N to the Pole is equal to the latitude of the observer. The sum of the two must be 90\u00b0.\u00a0 This gives the condition for a star to be circumpolar.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">which has ?= 90\u00b0 \u2212 ?just touches the horizon for the observer at latitude ?. All those stars whose declination exceeds 90\u00b0 \u2212 ?are circumpolar at latitude ?. Thus, the condition for a star to be circumpolar at latitude ?is:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For observers in the Northern Hemisphere (Fig. 3.4)<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">? + ? \u2265 ??\u00b0,\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(3.1)<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">or,<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">? \u2265 (??\u00b0 \u2212 ? ).\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (3.2)<\/span><\/p>\n<p>For observers in the Southern Hemisphere (Fig. 3.5)<\/p>\n<p>? + ? \u2264 \u2212 ??\u00b0 . <span style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>(3.3)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-59\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.5.png\" alt=\"\" width=\"724\" height=\"511\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.5.png 724w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.5-300x212.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.5-65x46.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.5-225x159.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.5-350x247.png 350w\" sizes=\"auto, (max-width: 724px) 100vw, 724px\" \/><\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em\">Fig. 4.5. Circumpolar stars and stars that never rise for an observer in the southern hemisphere at latitude <\/span><em style=\"font-size: 1em\">&#8211;<\/em><span style=\"font-size: 1em\">?.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"font-size: 1em;text-align: initial\">By the same argument, stars which always remain below the horizon must have:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For observers in the Northern Hemisphere<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">?\u2212 ?\u2264 \u2212 ?0\u00b0,\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3.4)<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">or,<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">?\u2264 (?\u2212 ?0\u00b0).\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3.5)<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For observers in the Southern Hemisphere<\/span><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">?\u2212 ?\u2265 ?0\u00b0.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (3.6)<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Stars which never come up the horizon are never visible to the observer at latitude ?. Consider, for example, the constellation Big Bear (Saptarishi). The declination of the star of this constellation which is farthest from the Pole Star is 49\u00b0. Therefore, for all those living at latitudes 41\u00b0 or higher, this constellation will be circumpolar. It will be visible all the time, if the sunlight did not mask it during the day.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-58\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.6.png\" alt=\"\" width=\"682\" height=\"303\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.6.png 682w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.6-300x133.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.6-65x29.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.6-225x100.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.6-350x155.png 350w\" sizes=\"auto, (max-width: 682px) 100vw, 682px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 3.6. For people living at latitude 41\u00b0, or higher, Big Bear is circumpolar.<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Fig. 4.7 shows the circumpolar stars in the northern hemisphere. For photograph and animation, visit: https:\/\/en.wikipedia.org\/wiki\/Circumpolar_star<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-57 aligncenter\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.7.png\" alt=\"\" width=\"191\" height=\"191\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.7.png 191w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.7-150x150.png 150w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.7-65x65.png 65w\" sizes=\"auto, (max-width: 191px) 100vw, 191px\" \/><\/p>\n<p style=\"text-align: justify\">Fig. 4.7. The North Star and circumpolar stars in a photograph with a long shutter speed of several hours. (Source: Wikipedia, photo by LCGS Russ)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">As another example, consider that you live at a place on the tropic of cancer, ?= -23.5\u00b0. \u00a0Then all stars satisfying the condition ?+ ?\u2264 \u2212 ?0\u00b0, or having ?\u2264 \u221266.5\u00b0 are circumpolar to you. At the same time, stars satisfying ?\u2212 ?\u2265 ?0\u00b0, or having ?\u2265 66.5\u00b0 will never be visible to you.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.3.\u00a0 Circumpolar Stars for an Observer on the North Pole\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Suppose an observer is stationed at the north pole. \u00a0As we have seen earlier, the horizon of this observer coincides with the celestial equator. All stars of the northern hemisphere are circumpolar. No star of the southern hemisphere is visible to this observer. Since the latitude of this observer is 90\u00b0, the stars of all positive declinations are circumpolar, as is clear from Equation (3.1). \u00a0These stars never set, as shown in Fig. 4.8. By the same logic, all stars south of the equator are invisible.\u00a0 For this observer, they never rise.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-56\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.8.png\" alt=\"\" width=\"678\" height=\"477\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.8.png 678w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.8-300x211.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.8-65x46.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.8-225x158.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.8-350x246.png 350w\" sizes=\"auto, (max-width: 678px) 100vw, 678px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Fig. 3.8.\u00a0 Circumpolar stars for the observer at the north pole. The situation is exactly the reverse for the observer at the south pole.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.4.\u00a0 The Sun as Circumpolar Star\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Note that the declination of the Sun varies between 0\u00b0 and 23.5\u00b0 during the six months from 21 March to 23 September. For the observer at the North Pole (indeed, for latitudes &gt; 66.5\u00b0), the Sun is circumpolar during these six months; it is always up, even at midnight (called <strong>midnight Sun, Fig. 4.9<\/strong>), and does not set.\u00a0 During the six months from 23 September to 21 June when the declination of the Sun varies from 0\u00b0 and -23.5\u00b0, the Sun is never visible to these observers in the northern polar region; it never rises. For the south polar region, though, the Sun is circumpolar\u00a0during 23 September to 21 March and visible continuously, while during 21 March to 23 September it is not visible at all.\u00a0 In popular parlance, it is said that in the polar regions the day and night are of six month\u2019s duration.<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-55\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.9.png\" alt=\"\" width=\"271\" height=\"203\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.9.png 271w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.9-65x49.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.9-225x169.png 225w\" sizes=\"auto, (max-width: 271px) 100vw, 271px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 3.9. The Altafjord in Alta, Norway bathed in the Midnight Sun. Source: Wikipedia, photo by Vberger.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.5.\u00a0 Observer at the Equator\u00a0<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">What happens if the observer is located at the equator? This situation is shown in Fig. 4.10. Since the horizon is now perpendicular to the equator, it divides the diurnal circles in two equal<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-54\" src=\"http:\/\/phyp06.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.10.png\" alt=\"\" width=\"680\" height=\"480\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.10.png 680w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.10-300x212.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.10-65x46.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.10-225x159.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-content\/uploads\/sites\/97\/2018\/11\/6.3.10-350x247.png 350w\" sizes=\"auto, (max-width: 680px) 100vw, 680px\" \/><\/p>\n<p style=\"text-align: center\">Fig. 3.10. An observer at the equator is able to see all stars but only for half the time.<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 1em;text-align: initial\">halves.\u00a0 All the stars, of both the hemispheres, are visible to this observer for half the time; for the remaining half they are below the horizon.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">For any observer between the equator and the pole, the situation is similar to that shown in Fig.\u00a0<\/span><span style=\"text-align: justify;font-size: 1em\">3.4. Depending on the latitude of the observer, some stars are circumpolar and some stars are not accessible to this observer at all. \u00a0Other stars rise and set for different durations.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>4. Summary\u00a0<\/strong><\/p>\n<ul>\n<li>The earth rotates on its axis from West to East.<\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Seen from above the North Pole this motion is anticlockwise.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">We on the earth see all celestial objects moving from East to West in circular orbits.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">The planes of these orbits are parallel to the Celestial Equator.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">If the orbit of a star crosses the horizon of the observer, the star is visible when above the horizon and it is invisible as goes below the horizon.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">When a star just comes up the horizon, it is said to have risen.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">The moment it goes below the horizon; it is said to have set.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">If the orbit of a star is always above the horizon, the star is always visible and is called a circumpolar star.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">If its orbit is completely below the horizon, the star is never visible to the observer.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Whether a star is circumpolar or not depends on the latitude of the observer.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">For an observer on the North Pole all stars in the northern hemisphere are circumpolar, and all those in the southern hemisphere are not visible.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">For an observer on the South Pole, all stars of the southern hemisphere are circumpolar and no star of the northern hemisphere is visible.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">In the polar regions, the Sun rises and sets for six months at a stretch.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Sun can be seen even at midnight at these places.<\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">An observer at the equator can see all stars from both the hemispheres, but only for half the time, because the horizon divides the orbits of stars in two equal halves.<\/span><\/li>\n<\/ul>\n<div class=\"toolbar\"><\/div>\n<div id=\"viewerContainer\">\n<div id=\"viewer\" class=\"pdfViewer\">\n<div id=\"pageContainer1\" class=\"page\">\n<div class=\"textLayer\">\n<p>&nbsp;<\/p>\n<p><strong>Know More<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Helpful Websites:<\/strong><\/p>\n<ul>\n<li>https:\/\/en.wikipedia.org\/wiki\/Circumpolar_star<\/li>\n<li>http:\/\/www.backyardastronomy.com\/Backyard_Astronomy\/Sky_Motion_Movies\/Pages\/Circumpolar_Stars-Looking_South_%28Southern_Hemisphere%29.html<\/li>\n<li>http:\/\/www.astro.rug.nl\/~sctrager\/teaching\/OA\/PositionalAstronomy.pdf<\/li>\n<li>https:\/\/www.youtube.com\/watch?v=soy04EG3VcE (animation)<\/li>\n<li>https:\/\/www.youtube.com\/watch?v=soy04EG3VcE (animation)<\/li>\n<\/ul>\n<p><span style=\"font-size: 1em\">You can\u00a0<\/span><span style=\"font-size: 1em\">have a look at the book at this site:\u00a0<\/span><span style=\"font-size: 1em\">https:\/\/www.willbell.com\/MATH\/mc8.htm<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":4,"menu_order":3,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-v-b-bhatia"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-50","chapter","type-chapter","status-publish","hentry","contributor-prof-v-b-bhatia"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/50","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/users\/4"}],"version-history":[{"count":5,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions"}],"predecessor-version":[{"id":65,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions\/65"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapters\/50\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/media?parent=50"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/pressbooks\/v2\/chapter-type?post=50"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/contributor?post=50"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp06\/wp-json\/wp\/v2\/license?post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}