1 00:00:00,000 --> 00:00:04,396 We now have a mathematical way to describe the position of a star in the 2 00:00:04,396 --> 00:00:07,768 celestial sphere. So we know where it is in the sky. What 3 00:00:07,768 --> 00:00:12,766 we need to do to finish our project is to figure out how to relate where a star is 4 00:00:12,766 --> 00:00:16,079 in the sky to where on earth it's visible at what time. 5 00:00:16,079 --> 00:00:21,137 And so let's understand exactly what this translation process is so we can make our 6 00:00:21,137 --> 00:00:24,964 next move. The challenge that we have to meet is 7 00:00:24,964 --> 00:00:29,185 explained in this simulation. Here we have the celestial sphere again 8 00:00:29,185 --> 00:00:32,795 surrounding the earth. The stars are fixed on the celestial 9 00:00:32,795 --> 00:00:35,731 sphere. We've equipped the celestial sphere with 10 00:00:35,731 --> 00:00:39,280 its equator and its prime meridian so that we can tell 11 00:00:39,280 --> 00:00:42,846 positions of a star relative to these coordinates. 12 00:00:42,846 --> 00:00:46,354 And then we have an observer here at some point on Earth. 13 00:00:46,354 --> 00:00:51,278 We want to understand what that is that the observer sees at any given instant as 14 00:00:51,278 --> 00:00:55,340 the whole thing rotates. And what the observer sees is the half of 15 00:00:55,340 --> 00:00:59,586 the sky that is above his horizon. So his horizon is this circle here. 16 00:00:59,586 --> 00:01:04,695 Which is an extension of the plane of the Earth near where their observer is to 17 00:01:04,695 --> 00:01:08,018 meet the celestial sphere. And this point over here. 18 00:01:08,018 --> 00:01:12,784 The point that he looks straight up above over his head will be his zenith. 19 00:01:12,784 --> 00:01:17,262 And so the observer's point of view is given by this diagram, extending his 20 00:01:17,262 --> 00:01:20,162 horizon. And then rotating to make the observer 21 00:01:20,162 --> 00:01:22,957 vertical. To describe where a star is, he will 22 00:01:22,957 --> 00:01:25,566 describe how high it is above his horizon. 23 00:01:25,566 --> 00:01:29,852 That's called the star's altitude. Here's a star at an altitude of 30 24 00:01:29,852 --> 00:01:33,269 degrees, and here's a star at an altitude of 43 degrees. 25 00:01:33,269 --> 00:01:38,176 And very often, we will express the same aspect of a star's position in the sky. 26 00:01:38,176 --> 00:01:41,593 By giving instead, its angular distance from the zenith. 27 00:01:41,593 --> 00:01:46,190 This is called the zenith angle, which is simply 90 degrees minus altitude. 28 00:01:46,190 --> 00:01:52,053 Once you've given either the altitude or the zenith angle, you've parameterized 29 00:01:52,053 --> 00:01:57,843 essentially a line of constant latitude. And we need an analog of longitude. 30 00:01:57,843 --> 00:02:02,445 we measure local longitude away from a particular meridian. 31 00:02:02,445 --> 00:02:07,750 The, meridian we use is the one line in the sky that. 32 00:02:07,750 --> 00:02:12,676 Starts at our northern horizon, goes through Earth's zenith, and intersects 33 00:02:12,676 --> 00:02:16,604 the horizon due south of us, so this is your locum meridian. 34 00:02:16,604 --> 00:02:21,863 It splits the sky into an eastern half and a western half and to des, designate 35 00:02:21,863 --> 00:02:26,790 the position of a star, we measure its angular distance from this meridian 36 00:02:26,790 --> 00:02:31,650 moving to the east, so that a star that is due east will be at azimuth 90 37 00:02:31,650 --> 00:02:36,776 degrees, and then various altitudes. This is how you describe to someone which 38 00:02:36,776 --> 00:02:42,838 way they need to look to find a star. Summarizing again our coordinates in our 39 00:02:42,838 --> 00:02:45,978 local view. We measure the position of a star by 40 00:02:45,978 --> 00:02:51,081 giving either its altitude above the horizon or its angular distance from the 41 00:02:51,081 --> 00:02:54,156 zenith. These two add up to 90 degrees and it's 42 00:02:54,156 --> 00:02:57,737 azimuth. The angle from north measuring east from 43 00:02:57,737 --> 00:03:01,839 zero to 360 degrees. And then we add our special points. 44 00:03:01,839 --> 00:03:07,956 The zenith the point directly overhead at altitude equal 90 degrees or zenith angle 45 00:03:07,956 --> 00:03:11,684 equals to zero degrees. And the zenith has no, azimuth in the 46 00:03:11,684 --> 00:03:15,542 same way that the North Pole has no well defined longitude. 47 00:03:15,542 --> 00:03:19,794 We have the horizon. The collection of all the points at 48 00:03:19,794 --> 00:03:22,540 altitude equals zero degrees, or zenith angle equal 90 degrees. 49 00:03:22,540 --> 00:03:27,125 And then we had the local meridian. The line that divides the sky into an 50 00:03:27,125 --> 00:03:31,172 eastern and a western half. This is the line that meets the horizon 51 00:03:31,172 --> 00:03:33,866 in both the north. And the southern points. 52 00:03:33,866 --> 00:03:39,021 And so this these are all the points with Azimuth equal zero degrees for the 53 00:03:39,021 --> 00:03:44,511 Northern half from the Northern horizon to the zenith or 180 degrees for the part 54 00:03:44,511 --> 00:03:48,997 from the Southern horizon to the zenith that is the local meridian. 55 00:03:48,997 --> 00:03:53,081 These are all the stars that are neither east nor west of us. 56 00:03:53,081 --> 00:03:58,504 Now what we need to understand is how to translate positions on the celestial 57 00:03:58,504 --> 00:04:03,391 sphere in the celestial sphere coordinates to positions in the sky in 58 00:04:03,391 --> 00:04:06,797 this altitude in this Azimuth Coordinate System. 59 00:04:06,797 --> 00:04:12,152 And so, here we have our little Earth, inside a large celestial sphere. 60 00:04:12,152 --> 00:04:15,644 Remember, the celestial sphere should be huge. 61 00:04:15,644 --> 00:04:19,292 We have the poles of the celestial sphere here. 62 00:04:19,292 --> 00:04:25,501 We have the celestial equator over here, and two diameters drawn for convenience. 63 00:04:25,501 --> 00:04:32,770 And so we will place our observer over here at some particular latitude. 64 00:04:32,770 --> 00:04:37,797 And, this observer will come with a horizon, a half of the sky that he is 65 00:04:37,797 --> 00:04:43,314 able to see and we've drawn this before. This is the line, notice that it should 66 00:04:43,314 --> 00:04:48,761 go through the center of the celestial sphere, because the celestial sphere is 67 00:04:48,761 --> 00:04:53,440 so large, and therefore, this is the visible half of the sky, this is 68 00:04:53,440 --> 00:04:58,426 2-dimensional version of what we did before in three dimensions and what this 69 00:04:58,426 --> 00:05:03,220 makes clear is if you look at the geometry for a moment, then this angle 70 00:05:03,220 --> 00:05:07,951 determining the latitude of the angle between this diameter and this line. 71 00:05:07,951 --> 00:05:11,723 Well this line is vertical so perpendicular to the horizon. 72 00:05:11,723 --> 00:05:16,453 These two diameters have a 90 degree angle between them, the net result is 73 00:05:16,453 --> 00:05:20,609 that this angle here is also the observer's latitude which means, 74 00:05:20,609 --> 00:05:25,573 remembering that this is the horizon, that your latitude is the angle from the 75 00:05:25,573 --> 00:05:28,462 horizon to whichever celestial pole is visible. 76 00:05:28,462 --> 00:05:33,504 In other words, if you can find the north star or the south celestial pole in your 77 00:05:33,504 --> 00:05:38,300 sky you now know which direction is north or south depending which hemisphere 78 00:05:38,300 --> 00:05:43,280 you're in and you also know your latitude because the higher the latitude higher 79 00:05:43,280 --> 00:05:48,382 the pole is in the sky clearly if you are at the pole then your latitude is 90 and 80 00:05:48,382 --> 00:05:53,178 the pole is directly over head, so you can use some understanding of the stars 81 00:05:53,178 --> 00:05:57,710 to figure out where you are. And in what direction you're going, stars 82 00:05:57,710 --> 00:06:02,782 are very useful for navigation. the other thing that this diagram shows 83 00:06:02,782 --> 00:06:07,718 us is that if this observer looks directly overhead, he finds his zenith, 84 00:06:07,718 --> 00:06:12,516 and his zenith is over here. And what this shows us is that his zenith 85 00:06:12,516 --> 00:06:17,314 is, at any given time, going to be at a declination, a distance from the 86 00:06:17,314 --> 00:06:20,399 celestial equator also equal to his latitude. 87 00:06:20,399 --> 00:06:25,609 And then, as the earth or the celestial sphere rotates, he will see different 88 00:06:25,609 --> 00:06:29,498 points on this line. A fixed declination directly overhead so 89 00:06:29,498 --> 00:06:33,981 the only points that are directly overhead ever for a given observer are 90 00:06:33,981 --> 00:06:37,360 the points whose declination is equal to your latitude. 91 00:06:37,360 --> 00:06:43,887 Here we have our favorite view of the sky in Athens again and in addition to the 92 00:06:43,887 --> 00:06:50,092 green celestial grid that I drew before centered on the north celestial pole, 93 00:06:50,092 --> 00:06:56,619 I've drawn in red, our local altitude in Azimuth grid, and so, we see our zenith 94 00:06:56,619 --> 00:07:02,552 directly overhead, we see lines of constant altitude in concentric circles 95 00:07:02,552 --> 00:07:06,273 around it and we see lines of constant azimuth. 96 00:07:06,273 --> 00:07:12,766 And as you see we are looking due south at the moment and so this is our prime 97 00:07:12,766 --> 00:07:16,488 meridian. The line that splits the sky in to an 98 00:07:16,488 --> 00:07:20,685 eastern half over here and a western half over there. 99 00:07:20,685 --> 00:07:26,307 And because the prime meridian goes through both the zenith and whichever 100 00:07:26,307 --> 00:07:32,569 celestial pole is visible because it in this case reaches the northern horizon it 101 00:07:32,569 --> 00:07:37,431 coincides with some specific celestial meridian for its long part, 102 00:07:37,431 --> 00:07:42,736 for the part including the zenith. At this point I think that I can tell 103 00:07:42,736 --> 00:07:49,145 that because this is zero hour celestial meridian this is the two hour celestial 104 00:07:49,145 --> 00:07:52,902 meridian. Our local meridian correspond with the 105 00:07:52,902 --> 00:07:59,093 celestial meridian for one hour. And when I allow time to elapse, then as 106 00:07:59,093 --> 00:08:04,637 we saw, the green grid on which the stars are fixed, will move from east to west, 107 00:08:04,637 --> 00:08:09,759 rotating about the celestial pole. The red grid, which is fixed to us, will 108 00:08:09,759 --> 00:08:14,600 appear to us to be stationary. So as time goes by, the celestial lines 109 00:08:14,600 --> 00:08:17,758 of, celestial meridians will move to the east. 110 00:08:17,758 --> 00:08:21,056 And, whereas the lines of fixed Asimoth will not. 111 00:08:21,056 --> 00:08:26,178 So that, if our local meridian corresponded previously to the meridian 112 00:08:26,178 --> 00:08:28,531 for one. hour of right ascension. 113 00:08:28,531 --> 00:08:31,848 Now we have two hours of right ascension overhead. 114 00:08:31,848 --> 00:08:33,307 Three hours. And so on. 115 00:08:33,307 --> 00:08:36,590 And so forth. This animation is truly wonderful. 116 00:08:36,590 --> 00:08:41,146 We have here two, the two views, simultaneously and synchronously. 117 00:08:41,146 --> 00:08:45,987 So, here we have our observer. I've put him in the vicinity of Athens, 118 00:08:45,987 --> 00:08:49,404 Greece. we see the celestial sphere inside it, 119 00:08:49,404 --> 00:08:53,320 the Earth, with an observer, position located upon it. 120 00:08:53,320 --> 00:08:58,303 And, as time goes by, the Earth will rotate inside the celestial sphere. 121 00:08:58,303 --> 00:09:02,147 Here, simultaneously, is the local view of the observer. 122 00:09:02,147 --> 00:09:07,060 We see, his horizon, his north, northern horizon over here, and, 123 00:09:07,060 --> 00:09:12,319 This, this is the half of the sky that the observer will see and as we begin our 124 00:09:12,319 --> 00:09:16,922 animation I've also drawn a few star patterns on the celestial sphere. 125 00:09:16,922 --> 00:09:22,444 We see here the big dipper near the North Pole Orion near the equator and the other 126 00:09:22,444 --> 00:09:27,441 southern celestial pole the southern cross and as our animation begins note 127 00:09:27,441 --> 00:09:32,372 that the observer looks to the east, Orion is just rising above his eastern 128 00:09:32,372 --> 00:09:35,200 horizon as it was exactly in our animation. 129 00:09:35,200 --> 00:09:41,981 in our simulation and our Orion will rise high in the sky, crosses meridian and set 130 00:09:41,981 --> 00:09:45,239 in the west. And this will happen as the celestial 131 00:09:45,239 --> 00:09:50,192 sphere rotates about the observer or as the Earth rotates, and you see Orion 132 00:09:50,192 --> 00:09:53,776 rising and setting. On the other hand, if we look at the 133 00:09:53,776 --> 00:09:58,989 motion of the Big Dipper we notice that as the Big Dipper moves, or as the Earth 134 00:09:58,989 --> 00:10:04,202 rotates, the Big Dipper's circle around the pole essentially never carries it 135 00:10:04,202 --> 00:10:07,917 below the horizon. It performs circular motion around the 136 00:10:07,917 --> 00:10:09,968 pole, and it never, ever since. 137 00:10:09,968 --> 00:10:15,931 And so, that makes the Big Dipper part of what we call As said the circumpolar star 138 00:10:15,931 --> 00:10:20,773 there is the whole region around the pole for each observer of stars that as they 139 00:10:20,773 --> 00:10:25,159 celestial sphere rotates or as the earth rotates never set below the horizon. 140 00:10:25,159 --> 00:10:29,773 And correspondingly if you look at the southern cross you see there are observer 141 00:10:29,773 --> 00:10:34,386 in Athens who will never be able to see it, it will never actually rise above his 142 00:10:34,386 --> 00:10:38,542 horizon so there is a, a similar region around the opposite pole 143 00:10:38,542 --> 00:10:42,275 that never rises. And so these stars are never visible. 144 00:10:42,275 --> 00:10:47,528 These stars are always visible. And then the stars in between rise in the 145 00:10:47,528 --> 00:10:51,959 east and set in the west. And it is this relation of the two 146 00:10:51,959 --> 00:10:55,540 systems of coordinates that we need to understand. 147 00:10:57,300 --> 00:11:02,975 Repeating as the sky looped, it's about the loop, first visible celestial pole. 148 00:11:02,975 --> 00:11:06,237 Stars close to whatever pole you can see never set. 149 00:11:06,237 --> 00:11:11,175 Stars near the opposite pole never rise. And stars in between, or near the equator 150 00:11:11,175 --> 00:11:14,712 rise in the east, move west across the sky, set in the west. 151 00:11:14,712 --> 00:11:19,589 Of course, for example if you're at the North Pole then you have half of the sky 152 00:11:19,589 --> 00:11:23,125 that is circumpolar and never sets. Half of it never rises. 153 00:11:23,125 --> 00:11:27,454 And stars can't rise in the east. And therefore at the North Pole their 154 00:11:27,454 --> 00:11:31,416 trajectories are just horizontal circles parallel to the horizon. 155 00:11:31,416 --> 00:11:35,013 And we see this in these collection of beautiful images. 156 00:11:35,013 --> 00:11:39,264 The two images on the left. Are star trails, in other words, somebody 157 00:11:39,264 --> 00:11:44,472 left the shutter open, around the South celestial pole on the left, and the North 158 00:11:44,472 --> 00:11:48,964 celestial pole on the right. So the stars in this image would have 159 00:11:48,964 --> 00:11:53,782 been moving counter-clockwise, and the stars in this image would have been 160 00:11:53,782 --> 00:11:57,493 moving clockwise. And then, on the left, we see a beautiful 161 00:11:57,493 --> 00:12:02,571 image of star trails near the horizon, this is Orion rising, and you can see 162 00:12:02,571 --> 00:12:07,470 from the fact that Orion is rising. And moving to the left you should be able 163 00:12:07,470 --> 00:12:12,244 to figure out that this means this image was taken in the Southern Hemisphere, and 164 00:12:12,244 --> 00:12:16,902 indeed, if you're a Northern Hemisphere denizen like me and you look at Orion, he 165 00:12:16,902 --> 00:12:20,686 seems to be doing a headstand. this is because in the Southern 166 00:12:20,686 --> 00:12:24,646 Hemisphere, one's head is pointing in a different direction in space. 167 00:12:24,646 --> 00:12:29,071 And so we have the beginnings of an understanding of what it is that we see 168 00:12:29,071 --> 00:12:33,238 in any given place and we're almost ready to make the calculation. 169 00:12:33,238 --> 00:12:37,766 And so the thing that we need is this description of the zenith. 170 00:12:37,766 --> 00:12:43,284 So I already described to you that our zenith at any point is a point on a 171 00:12:43,284 --> 00:12:47,528 celestial sphere whose declination is equal to our latitude. 172 00:12:47,528 --> 00:12:52,410 Now over the course of a 24 hour rotation of the celestial sphere 173 00:12:52,410 --> 00:12:56,610 which point along that celestial parallel we see changes. 174 00:12:56,610 --> 00:13:02,504 You see all of the stars over a 24-hour period which have that declination pass 175 00:13:02,504 --> 00:13:05,894 overhead. And so what we're going to is at any 176 00:13:05,894 --> 00:13:10,301 given instant we can look. Which celestial meridian contains our 177 00:13:10,301 --> 00:13:12,030 zenith? And 178 00:13:12,030 --> 00:13:17,049 We can look at the right extension of the celestial meridian and call that our 179 00:13:17,049 --> 00:13:22,006 local sidereal time and this will of course change with time as the celestial 180 00:13:22,006 --> 00:13:25,120 sphere rotates to the east or, to the west or the. 181 00:13:25,120 --> 00:13:28,936 Earth rotates to the east. Either way, your sidereal time will 182 00:13:28,936 --> 00:13:33,692 increase by about an hour each hour. And a complete rotation of the Earth or 183 00:13:33,692 --> 00:13:38,635 equivalent of the celestial sphere will advance your sidereal time by 24 hours. 184 00:13:38,635 --> 00:13:43,391 So, sidereal time is the name of the celestial meridian which coincides with 185 00:13:43,391 --> 00:13:46,644 the local meridian, the one that includes our zenith. 186 00:13:46,644 --> 00:13:51,589 And of course, over 24 sidereal hours. When this right ascension advances by 24 187 00:13:51,589 --> 00:13:54,474 hours, this is one full rotation of the Earth. 188 00:13:54,474 --> 00:13:59,733 So we can use the stars to measure time. The motion of the stars across the sky is 189 00:13:59,733 --> 00:14:04,607 a way to measure time, and indeed, in one sidereal hour the celestial spheres 190 00:14:04,607 --> 00:14:09,609 shifts by one hour of right ascension. One parallel giving way to the parallel 191 00:14:09,609 --> 00:14:12,110 moved by another hour. And until the 192 00:14:12,110 --> 00:14:16,309 20th century, this was, in fact, the definition of our units of time. 193 00:14:16,309 --> 00:14:20,391 A second was defined in terms of the rate at which the Earth rotates. 194 00:14:20,391 --> 00:14:25,418 this is a pretty stately and fixed rate and it's no coincidence, that, until the 195 00:14:25,418 --> 00:14:30,387 19th century, this was the most precise way to measure time that we had at our 196 00:14:30,387 --> 00:14:34,531 disposal. thing to note, of course, is that 197 00:14:34,531 --> 00:14:40,018 different locations on earth will measure different sidereal times, because 198 00:14:40,018 --> 00:14:46,090 depending on your longitude you will see overhead different celestial meridians. 199 00:14:46,090 --> 00:14:51,870 And as you move to the east you will see later and later sidereal time, because 200 00:14:51,870 --> 00:14:56,570 moving to the east, you see stars overhead that are farther and farther to 201 00:14:56,570 --> 00:15:01,461 the east along the celestial sphere. And so, fifteen degrees of longitude east 202 00:15:01,461 --> 00:15:05,780 advances your sidereal time by an hour but this varies continuously. 203 00:15:05,780 --> 00:15:10,735 So that moving east by twenty meters changes your sidereal time by some small 204 00:15:10,735 --> 00:15:13,931 amount. Returning to our view of Athens, we can 205 00:15:13,931 --> 00:15:18,730 now read off the sidereal time. In our picture, remember, the sidereal 206 00:15:18,730 --> 00:15:23,881 time is the time indicated by the celestial meridian that coincides with 207 00:15:23,881 --> 00:15:29,244 our local meridian, and we've seen that before, this is the one hour meridian. 208 00:15:29,244 --> 00:15:32,278 So the sidereal time in Athens at 900 p.m. 209 00:15:32,278 --> 00:15:36,230 On November 27th was one a.m. and of course, as. 210 00:15:36,230 --> 00:15:42,027 We allow time to progress by an hour sidereal time will progress. 211 00:15:42,027 --> 00:15:46,120 We can see that by, allowing time to progress. 212 00:15:46,120 --> 00:15:50,980 And an hour later, it is 10:01.01. The sidereal time is changed. 213 00:15:50,980 --> 00:15:54,220 It is now two hours sidereal time. Now, 214 00:15:54,220 --> 00:15:59,765 we can use this, now that we understand the position of our zenith. 215 00:15:59,765 --> 00:16:04,014 And the location of our, your local meridian, we 216 00:16:04,014 --> 00:16:09,421 can use this to find a particular star. So let's suppose that at 900 p.m., we 217 00:16:09,421 --> 00:16:14,418 wanted to find a particular star. Let's imagine that we wanted to find, in 218 00:16:14,418 --> 00:16:19,003 the sky, this particular star, Alpheratz in the constellation Pegasus. 219 00:16:19,003 --> 00:16:23,931 So, if you look, Alpheratz you can look it up, has celestial coordinates. 220 00:16:23,931 --> 00:16:29,202 It has a declination of about 30 degrees, and a right ascension of about zero 221 00:16:29,202 --> 00:16:32,419 hours. It lies at the intersection of this line 222 00:16:32,419 --> 00:16:35,910 of, declination and the prime celestial meridian. 223 00:16:35,910 --> 00:16:40,020 That means that Alpheratz will rise in our East, 224 00:16:40,020 --> 00:16:44,050 become as high overhead as it's going to get. 225 00:16:44,050 --> 00:16:48,263 When it's on our local meridian and then start setting in the west. 226 00:16:48,263 --> 00:16:53,608 So the best time to view Alpheratz in fact, is when it is directly overhead. 227 00:16:53,608 --> 00:16:57,884 Because it is on the prime celestial meridian, best time to view the Alpheratz 228 00:16:58,890 --> 00:17:02,537 would have been at zero hours, at select, sidereal midnight. 229 00:17:02,537 --> 00:17:05,430 Which means at 9 p.m. we're an hour too late. 230 00:17:05,430 --> 00:17:09,766 But that's okay. We can fix this we can just though the 231 00:17:09,766 --> 00:17:13,817 wonders of technology go and observe Alpheratz at 8 p.m. 232 00:17:13,817 --> 00:17:19,078 Here we can move time backwards and indeed, as advertised, as a, at eight pm, 233 00:17:19,078 --> 00:17:24,196 Alpheratz is right on our celestial meridian, it has risen as high as it's 234 00:17:24,196 --> 00:17:27,252 going to rise and has not yet begun to set. 235 00:17:27,252 --> 00:17:32,460 So at that time, 8 p.m. or more importantly. 236 00:17:32,460 --> 00:17:36,929 Midnight sidereal time, Alpheratz will be on our local meridian. 237 00:17:36,929 --> 00:17:40,058 Where will it be along the local meridian? 238 00:17:40,058 --> 00:17:44,229 Well, this is where Alpheratz declination comes into play. 239 00:17:44,229 --> 00:17:47,283 Alpheratz is at a declination of 30 degrees. 240 00:17:47,283 --> 00:17:52,945 Our zenith, we said, because we are in Athens, is always at its declination of, 241 00:17:52,945 --> 00:17:56,967 about, 37 degrees. This means that the angle, the zenith 242 00:17:56,967 --> 00:18:01,810 angle of Alpheratz the distance between Alpheratz and our zenith at 243 00:18:01,810 --> 00:18:09,342 it's meridian crossing at the time when Alpheratz is as high in the sky as it 244 00:18:09,342 --> 00:18:12,981 gets. Well, its zenith angle is given by our 245 00:18:12,981 --> 00:18:19,074 latitude minus its declination. in general, the absolute value of this 246 00:18:19,074 --> 00:18:23,560 which in our case, is about, 30 minus 30 or eight degrees. 247 00:18:23,560 --> 00:18:30,161 So, Alpheratz will attain a minimal zenith angle of eight degrees or a 248 00:18:30,161 --> 00:18:37,586 maximal altitude of 82 degrees above the horizon, quite high. 249 00:18:37,586 --> 00:18:40,496 Now, this tells us almost where it is. 250 00:18:40,496 --> 00:18:46,065 This defines it to within a circle but because it's on our local meridian, it 251 00:18:46,065 --> 00:18:49,536 can only be either north or south of the zenith. 252 00:18:49,536 --> 00:18:55,539 And because the declination of Alpheratz is less than our latitude, our zenith is 253 00:18:55,539 --> 00:18:59,010 at 38 degrees. Alpheratz only at, 30 degrees of 254 00:18:59,010 --> 00:19:01,976 declination. It's Azimuth will be 180. 255 00:19:01,976 --> 00:19:07,834 It'll be due south of our zenith at the time that it is as high as it can get. 256 00:19:07,834 --> 00:19:12,859 And so if we observe our Alpheratz at. sidereal midnight we know exactly where 257 00:19:12,859 --> 00:19:15,923 to find it. And now if we need to observe Alfrax 258 00:19:15,923 --> 00:19:21,382 later or earlier, we simply use the fact that we understand that over time the sky 259 00:19:21,382 --> 00:19:26,707 rotates about the celestial pole at the rate of fifteen degrees an hour so that 260 00:19:26,707 --> 00:19:31,900 if we wait an hour, and we indeed are observing at, 9 p.m. Alpheratz will have 261 00:19:31,900 --> 00:19:36,745 moved fifteen degrees to the next celestial meridian and indeed, that is 262 00:19:36,745 --> 00:19:40,315 what we will observe. So, we have a way of finding the 263 00:19:40,315 --> 00:19:44,356 approximate position of a star over the course of the night. 264 00:19:44,356 --> 00:19:49,879 Notice that there is an explicit formula that gives you altitude and azimuth as a 265 00:19:49,879 --> 00:19:54,055 function of sidereal time and declination and right ascension. 266 00:19:54,055 --> 00:19:57,760 The formula involves trigonometry, so we won't apply it. 267 00:19:57,760 --> 00:20:02,000 What I want us to understand is the principle from which it's derived. 268 00:20:02,000 --> 00:20:06,891 So, we've achieved what we wanted, we did not complete the mathematics because it 269 00:20:06,891 --> 00:20:10,803 would involve trigonometry. But I think we have the idea, we know, 270 00:20:10,803 --> 00:20:15,694 given the sidereal time and our latitude, how to figure out where in the sky, the 271 00:20:15,694 --> 00:20:20,646 pole is, which line of right ascension is directly over head, and then moving east 272 00:20:20,646 --> 00:20:25,476 or west we count an hour per fifteen degrees and we can find where the rest of 273 00:20:25,476 --> 00:20:30,367 the sky is, so we have an understanding which part of the sky will be visible to 274 00:20:30,367 --> 00:20:33,057 us at any given time at a position on earth. 275 00:20:33,057 --> 00:20:38,050 We've a sensually solved the problem. the one missing ingredient is that we 276 00:20:38,050 --> 00:20:42,712 still need to figure out what this sidereal time thing is because our 277 00:20:42,712 --> 00:20:47,908 watches do not measure sidereal time. It was one am sidereal in Athens when it 278 00:20:47,908 --> 00:20:52,237 was nine pm on November 27. If we could get the translation right 279 00:20:52,237 --> 00:20:55,767 we'd be set. So that's the next thing we have to take 280 00:20:55,767 --> 00:20:55,967 on.