1 00:00:01,640 --> 00:00:06,001 We now understand, pretty well, how to tell which part of the sky is going to be 2 00:00:06,001 --> 00:00:09,147 visible, at any given time, at any given location on Earth. 3 00:00:09,147 --> 00:00:13,067 we have a missing ingredient, which is this mysterious sidereal time. 4 00:00:13,067 --> 00:00:17,152 We need to understand sidereal time. Associated with this is another small 5 00:00:17,152 --> 00:00:19,801 omission. We've discussed the position of all of 6 00:00:19,801 --> 00:00:23,279 the stars on the celestial sphere but we've forgotten one star. 7 00:00:23,279 --> 00:00:25,763 One that might have some impact on our lives. 8 00:00:25,763 --> 00:00:29,722 Namely our local star, the sun. The sun, like everything else that is not 9 00:00:29,722 --> 00:00:32,306 on earth is somewhere on the celestial sphere. 10 00:00:32,306 --> 00:00:36,744 In the sense, that it rises in the east and sets in the west as the earth spins, 11 00:00:36,744 --> 00:00:40,957 or as the celestial sphere spins. And so we can locate the sun somewhere on 12 00:00:40,957 --> 00:00:43,765 the celestial sphere. But I haven't told you where. 13 00:00:43,765 --> 00:00:48,090 And where that is, this is somewhat interesting because if you put the sun 14 00:00:48,090 --> 00:00:52,303 somewhere on the celestial sphere there's a whole section of sidereal time. 15 00:00:52,303 --> 00:00:55,392 A whole period of the 24 hour rotation of the earth. 16 00:00:55,392 --> 00:00:59,100 When the sidereal time is close to the right extension of the sun. 17 00:00:59,100 --> 00:01:03,673 When the stars overhead are, include the Sun, and we do not do astronomical 18 00:01:03,673 --> 00:01:06,701 observations. At least not in visible wavelengths. 19 00:01:06,701 --> 00:01:10,532 And so it's somewhat important to figure out where the Sun is. 20 00:01:10,532 --> 00:01:14,982 And this will turn out to be at the root of this, sidereal time issue. 21 00:01:14,982 --> 00:01:20,530 And so let's see how that works. This simulation might help us to explain 22 00:01:20,530 --> 00:01:24,706 what's going on. We know that in addition to spinning 23 00:01:24,706 --> 00:01:29,750 around its axis daily, the Earth also orbits the sun once a year. 24 00:01:29,750 --> 00:01:34,219 Equivalently, if you prefer a stationary earth, the sun orbits the earth once a 25 00:01:34,219 --> 00:01:37,715 year from, again, from the point of view of who sees what when. 26 00:01:37,715 --> 00:01:42,127 the two are completely equivalent. the earth orbits the sun in the same 27 00:01:42,127 --> 00:01:46,540 sense in which it spins about its axis. And what that means is, that as we see. 28 00:01:46,540 --> 00:01:51,510 It, the sun also orbits us in the same sense in which the Earth spins about its 29 00:01:51,510 --> 00:01:56,543 axis, in other words, the sun appears to orbit Earth once a year moving from west 30 00:01:56,543 --> 00:02:01,388 to east along the celestial sphere. What that tells us is that the sun cannot 31 00:02:01,388 --> 00:02:05,981 be assigned affixed right ascension. Its right ascension changes over the 32 00:02:05,981 --> 00:02:08,812 year. As the Earth orbits, if we start at some 33 00:02:08,812 --> 00:02:14,097 point on the Earth's orbit with the sun in this direction in the sky, imagine the 34 00:02:14,097 --> 00:02:17,810 celestial sphere outside this figure at celestial midnight. 35 00:02:17,810 --> 00:02:21,810 Then three months later when the earth has moved. 36 00:02:21,810 --> 00:02:26,898 To this point in its orbit, the Sun has changed orientation relative to the 37 00:02:26,898 --> 00:02:29,816 Earth. The Sun's right ascension is now six 38 00:02:29,816 --> 00:02:32,869 hours. And furthermore, three months later, the 39 00:02:32,869 --> 00:02:36,940 Sun's right ascension is twelve. And three months further on. 40 00:02:36,940 --> 00:02:41,415 It is eighteen hours and after a complete rotation, the sun will have made a 41 00:02:41,415 --> 00:02:46,126 complete circuit of the celestial sphere moving to the east in the direction of 42 00:02:46,126 --> 00:02:50,896 increasing right ascension that the sun's right ascension goes through a full 24 43 00:02:50,896 --> 00:02:55,254 hours in the course of the year. You can't assign the sun affixed position 44 00:02:55,254 --> 00:03:00,342 on the celestial sphere, rather, it moves along the celestial sphere to the east at 45 00:03:00,342 --> 00:03:05,979 the rate of one rotation per year. Now note the celestial sphere is moving 46 00:03:05,979 --> 00:03:09,711 from east to west as it rotates around the Earth. 47 00:03:09,711 --> 00:03:14,510 The Sun is moving along the celestial sphere from west to east. 48 00:03:14,510 --> 00:03:17,780 This is important so let's think about it again. 49 00:03:17,780 --> 00:03:22,230 As the earth spins about its axis once a day, it also orbits the sun once a year 50 00:03:22,230 --> 00:03:25,568 moving in the same direction. Which means, as seen from Earth, 51 00:03:25,568 --> 00:03:29,352 the sun orbits us once a year. So that means the sun moves along the 52 00:03:29,352 --> 00:03:33,747 celestial sphere moving from west to east in the direction of increasing right 53 00:03:33,747 --> 00:03:37,252 ascension to the east. And it completes one revolution per year 54 00:03:37,252 --> 00:03:41,146 around the celestial sphere. This means that which stars are invisible 55 00:03:41,146 --> 00:03:44,040 because they're only up at the same time as the sun. 56 00:03:44,040 --> 00:03:47,010 Well, that changes over the course of a year. 57 00:03:47,010 --> 00:03:50,453 So all stars get their chance to shine, so to speak. 58 00:03:50,453 --> 00:03:53,288 That is good, we get to see the entire sky. 59 00:03:53,288 --> 00:03:58,621 It also means, that since the sun is moving across the sky from west to east 60 00:03:58,621 --> 00:04:02,469 while the celestial sphere is rotating from east to west. 61 00:04:02,469 --> 00:04:07,576 The sun is carried by the celestial sphere, so it rises in the east and sets 62 00:04:07,576 --> 00:04:11,203 in the west. But its motion across the sky is a little 63 00:04:11,203 --> 00:04:15,031 bit slower than the motion of the stars. How much slower? 64 00:04:15,031 --> 00:04:20,736 Over the course of a year the Sun moves backwards along the celestial sphere by 65 00:04:20,736 --> 00:04:25,230 one complete revolution. Let's see that extra day one more time. 66 00:04:25,230 --> 00:04:29,305 This simulation will help us to understand the consequences of the fact 67 00:04:29,305 --> 00:04:32,192 that as the Earth spins it's also orbiting the Sun. 68 00:04:32,192 --> 00:04:35,532 And therefore the Sun is moving along the celestial sphere. 69 00:04:35,532 --> 00:04:40,909 to make things a little more clear, we have pretended that the sidereal day, or 70 00:04:40,909 --> 00:04:46,385 a day is not 24 but 240 hours long. This will make the effect much clearer. 71 00:04:46,385 --> 00:04:52,075 So imagine that we begin our simulation with the sun directly overhead for this 72 00:04:52,075 --> 00:04:56,983 tallest observer on the Earth. So this observer will presumably think 73 00:04:56,983 --> 00:05:00,468 that it is noon. And now, let a sidereal day go by. 74 00:05:00,468 --> 00:05:04,380 The Earth will have completed a 360 degree rotation. 75 00:05:04,380 --> 00:05:08,790 Note it's, it's oriented in exactly the same way it was before. 76 00:05:08,790 --> 00:05:15,914 However, it is not yet noon. The time it takes from noon back to noon 77 00:05:15,914 --> 00:05:20,858 again is longer than the sidereal day because in the course of this rotation, 78 00:05:20,858 --> 00:05:25,160 exaggerated by a factor of ten the earth has moved along its orbit. 79 00:05:25,160 --> 00:05:29,113 There's this extra bit of rotation required to get to noon. 80 00:05:29,113 --> 00:05:34,474 From the point of view of earth this is a consequence of the fact that over the 81 00:05:34,474 --> 00:05:40,104 course of the day as earth spun about its axis but also moved the sun moved in this 82 00:05:40,104 --> 00:05:43,387 direction to the east along the celestial sphere. 83 00:05:43,387 --> 00:05:47,341 We can do that again. 360 degrees rotation and a little bit 84 00:05:47,341 --> 00:05:51,790 required to realign us to the sun. Indeed, 85 00:05:51,790 --> 00:05:58,162 the time from noon to noon is longer than the time it takes earth to rotate by 30, 86 00:05:58,162 --> 00:06:04,380 360 degrees, by about one over 365 of a day because the sun moves back along the 87 00:06:04,380 --> 00:06:10,598 celestial sphere by one over 365 of it's complete rot, annual rotation of the 88 00:06:10,598 --> 00:06:13,940 celestial sphere. That's about four minutes. 89 00:06:13,940 --> 00:06:17,041 So, which of these is 24 hours do you think? 90 00:06:17,041 --> 00:06:22,706 Is an hour going to be a twenty-fourth of the time it takes the earth to rotate 360 91 00:06:22,706 --> 00:06:28,100 degrees, or would you define an hour to be a twenty-fourth of the time from noon 92 00:06:28,100 --> 00:06:31,624 to noon? Clearly, you want it to be the time from 93 00:06:31,624 --> 00:06:34,609 noon to noon. Our clocks keep solar time. 94 00:06:34,609 --> 00:06:40,280 The reason for this is, because as we saw, while you, the difference between a 95 00:06:40,280 --> 00:06:45,631 sidereal day, 24 hours, and the time from noon to noon is only four minutes, these 96 00:06:45,631 --> 00:06:49,927 four minutes accumulate over the course of a year, to a complete day. 97 00:06:49,927 --> 00:06:54,666 If you tried to work with a sidereal clock, then if you started it out such 98 00:06:54,666 --> 00:06:59,404 that noon fell at twelve hours sidereal, six months later, noon, the sun over 99 00:06:59,404 --> 00:07:04,080 head, would fall at zero hours sidereal. And if you tried to operate by the 100 00:07:04,080 --> 00:07:06,860 sidereal clock, you would want to have lunch. 101 00:07:06,860 --> 00:07:10,425 When it was darkest. This does not work very well for 102 00:07:10,425 --> 00:07:15,470 agriculture, though it is very well for astronomy, cause sidereal time keeps 103 00:07:15,470 --> 00:07:19,169 track of the stars. And the clocks that run our life, of 104 00:07:19,169 --> 00:07:21,860 course, keep not sidereal, but solar time. 105 00:07:21,860 --> 00:07:26,165 We call that local time. And so what that means is that these 106 00:07:26,165 --> 00:07:29,891 solar clocks. Our local clocks that keep solar time run 107 00:07:29,891 --> 00:07:34,560 slower than a sidereal clock. 24 sidereal hours, one 360 degree 108 00:07:34,560 --> 00:07:40,942 rotation of the earth is less than 24 solar hours, the time kept by our clocks 109 00:07:40,942 --> 00:07:46,097 by about four minutes. A little less than four minutes, one over 110 00:07:46,097 --> 00:07:51,743 365 of a day to be precise. So our sidereal clocks run faster than 111 00:07:51,743 --> 00:07:54,785 the clocks that are. Used to measure time. 112 00:07:54,785 --> 00:07:59,268 So how do you relate the two? Well, we have this issue of two, clocks 113 00:07:59,268 --> 00:08:04,487 one of which runs faster than the other. They're both 24 hour clocks and so at 114 00:08:04,487 --> 00:08:08,260 some point they agree. And then, the faster the clock runs 115 00:08:08,260 --> 00:08:11,152 ahead. And it will remain ahead until it has 116 00:08:11,152 --> 00:08:15,227 completed one full 24 hour rotation more that the other clock. 117 00:08:15,227 --> 00:08:18,250 And this takes, by definition, precisely a year. 118 00:08:18,250 --> 00:08:23,377 Once a year there is a day when sidereal and solar clocks agree and by convention 119 00:08:23,377 --> 00:08:27,846 that day is set to happen. We'll see why in our next lesson on or 120 00:08:27,846 --> 00:08:32,447 about September 21st. So on or about September twenty first 121 00:08:32,447 --> 00:08:37,968 wherever you are on Earth you're sidereal time is approximately equal to your local 122 00:08:37,968 --> 00:08:38,560 time. Now. 123 00:08:38,560 --> 00:08:43,003 I should qualify this. Remember local time varies from position 124 00:08:43,003 --> 00:08:47,819 to position continuously. Local time also varies from position to 125 00:08:47,819 --> 00:08:50,752 position. If you move about fifteen degrees east in 126 00:08:50,752 --> 00:08:53,224 longitude then local time is an hour later. 127 00:08:53,224 --> 00:08:57,593 That's what time zones are about but for the convenience of arranging train 128 00:08:57,593 --> 00:09:02,078 schedules, we don't let local time vary continuously so that each town has its 129 00:09:02,078 --> 00:09:04,953 own local time. Local time is fixed over an entire 130 00:09:04,953 --> 00:09:08,403 fifteen degree slice of the Earth, and then jumps by an hour. 131 00:09:08,403 --> 00:09:12,082 Whereas sidereal time is defined locally so varies continuously. 132 00:09:12,082 --> 00:09:15,590 So, even when I say sidereal time equals to local time I mean, 133 00:09:15,590 --> 00:09:19,985 to within half an hour which if the precision that we are working is good 134 00:09:19,985 --> 00:09:23,370 enough. So on September 21st to then this half an 135 00:09:23,370 --> 00:09:27,884 hour in precession, sidereal time is equal to local time and then if we know 136 00:09:27,884 --> 00:09:33,606 that we can compare that to add any time before or after September 21st because we 137 00:09:33,606 --> 00:09:39,061 know that the sidereal clock runs faster. A solar day is four minutes longer than a 138 00:09:39,061 --> 00:09:42,479 sidereal day. So, a day later, the sidereal clock will 139 00:09:42,479 --> 00:09:45,568 have become, run ahead and be four minutes fast. 140 00:09:45,568 --> 00:09:50,891 And D days after September 21st, the sidereal clock is ahead of the local 141 00:09:50,891 --> 00:09:54,966 clock by D * four minutes. D days before September 21st, 142 00:09:54,966 --> 00:09:57,990 it was behind by 4 minutes, and catching up. 143 00:09:57,990 --> 00:10:01,555 Now of course this is approximate. 4 minutes is not precise. 144 00:10:01,555 --> 00:10:05,695 We are in any event ignoring time zones. Beware of Daylight Savings Time. 145 00:10:05,695 --> 00:10:10,009 This expression of course ignores the jump by an hour that we artificially 146 00:10:10,009 --> 00:10:12,884 introduce into our clock. So this is Standard Time. 147 00:10:12,884 --> 00:10:17,427 But if you want to, this is good for dates that are near September 21st. 148 00:10:17,427 --> 00:10:22,877 We can rescale this at the four points of the compass if you will of Earth's orbit. 149 00:10:22,877 --> 00:10:27,924 So on December respectively March respectively June 21st, you can reset 150 00:10:27,924 --> 00:10:33,099 things so sidereal time is local time plus 6, 12, 18 hours and then on 151 00:10:33,099 --> 00:10:38,530 September 21st the difference will become 24 hours, which is the same as zero. 152 00:10:38,530 --> 00:10:41,150 So now, we know how to find sidereal time. 153 00:10:41,150 --> 00:10:44,884 And. Now that we know how to find sidereal 154 00:10:44,884 --> 00:10:49,809 time, let's use this to, solve an actual problem that might interest us. 155 00:10:49,809 --> 00:10:55,005 So suppose that we are interested in looking at the star, bright star Vega in 156 00:10:55,005 --> 00:10:59,391 the constellation Lyra. And we want to know when, Vega might be 157 00:10:59,391 --> 00:11:03,170 visible in the sky as high as it can around midnight. 158 00:11:03,170 --> 00:11:08,527 Now, Vega, if we look it up, has a right ascension of 18 hours and 36 minutes. 159 00:11:08,527 --> 00:11:12,296 According to what we said, Vega is the highest in the sky. 160 00:11:12,296 --> 00:11:17,720 It is crossing our meridian when our local sidereal time is 18 hours and 36 161 00:11:17,720 --> 00:11:20,960 minutes. When our local meridian corresponds with 162 00:11:20,960 --> 00:11:25,193 the meridian on which Vega lies. I want to know, when is this going to 163 00:11:25,193 --> 00:11:29,695 happen at midnight? Well, this will happen when local time of 164 00:11:29,695 --> 00:11:35,394 24 hours, which is the same as zero hours, corresponds with sidereal time 165 00:11:35,394 --> 00:11:40,856 equals 86 hours and 36 minutes. Well, if you look back at the previous 166 00:11:40,856 --> 00:11:48,025 slide, you'll realize that on a. June twenty-first we had sidereal time, 167 00:11:48,025 --> 00:11:57,820 was approximately local time +18 hours and so if we wait nine days later. 168 00:11:57,820 --> 00:12:05,153 Sidereal time will be local time plus 18 hours plus 4 minutes times D, if we want 169 00:12:05,153 --> 00:12:10,450 4 minutes times D to be 36 minutes, D is going to be nine days. 170 00:12:10,450 --> 00:12:16,356 And Vega will be high overhead at midnight, 9 days after June 21st until 171 00:12:16,356 --> 00:12:20,697 about June 30th. This might explain to you why the group 172 00:12:20,697 --> 00:12:25,892 of stars, that, the bright triangle of bright stars that included Vega, was 173 00:12:25,892 --> 00:12:28,881 something they called the summer triangle. 174 00:12:28,881 --> 00:12:34,290 And so now, we finally have a way of understanding what the sidereal time is. 175 00:12:34,290 --> 00:12:37,512 Predicting which stars will be overhead in which season. 176 00:12:37,512 --> 00:12:42,058 And at any given time, we can map what the picture of the sky is that we'll see. 177 00:12:42,058 --> 00:12:46,547 So we have a complete solution of the mathematics problem we set out to solve, 178 00:12:46,547 --> 00:12:50,230 describing which part of the sky will be visible where and when. 179 00:12:50,230 --> 00:12:51,656 Congratulations.