1 00:00:00,135 --> 00:00:02,490 The following content is provided under a Creative 2 00:00:02,490 --> 00:00:04,030 Commons license. 3 00:00:04,030 --> 00:00:06,330 Your support will help MIT OpenCourseWare 4 00:00:06,330 --> 00:00:10,720 continue to offer high-quality educational resources for free. 5 00:00:10,720 --> 00:00:13,320 To make a donation or view additional materials 6 00:00:13,320 --> 00:00:15,780 from hundreds of MIT courses, visit 7 00:00:15,780 --> 00:00:19,849 mitopencourseware@ocw.mit.edu. 8 00:00:19,849 --> 00:00:22,140 MARK HARTMAN: So we've been taking a look at this X-ray 9 00:00:22,140 --> 00:00:25,170 binary star system model, and we're 10 00:00:25,170 --> 00:00:27,390 seeing how with different parameters 11 00:00:27,390 --> 00:00:28,890 we can make different predictions, 12 00:00:28,890 --> 00:00:31,950 and see if those match up with what we're actually seeing. 13 00:00:31,950 --> 00:00:35,400 That helps us take observations and then think about, 14 00:00:35,400 --> 00:00:38,010 what kind of model do we want apply to it. 15 00:00:38,010 --> 00:00:40,350 But we've discovered that if we're 16 00:00:40,350 --> 00:00:43,950 looking for things like linear sizes of these objects, 17 00:00:43,950 --> 00:00:46,650 if we just look at the image, and if it's just a point 18 00:00:46,650 --> 00:00:50,340 source because those objects are so far away, it's hard for us 19 00:00:50,340 --> 00:00:52,500 to figure out what's the actual linear size. 20 00:00:52,500 --> 00:00:54,540 Now we have a prediction of about 10 21 00:00:54,540 --> 00:00:57,330 to the 15th meters would be the size 22 00:00:57,330 --> 00:01:00,370 of whatever this object is. 23 00:01:00,370 --> 00:01:06,630 10 to the 15th meters could be the diameter of the orbit. 24 00:01:06,630 --> 00:01:08,860 Or maybe it's a little bit bigger, 25 00:01:08,860 --> 00:01:10,980 but it's going to be representative 26 00:01:10,980 --> 00:01:13,320 of the whole thing. 27 00:01:13,320 --> 00:01:16,065 Right now, what we want to try and find out is-- 28 00:01:16,065 --> 00:01:18,300 and this is where want to take a couple of notes on. 29 00:01:24,700 --> 00:01:32,950 How can we find linear size using a light curve? 30 00:01:37,510 --> 00:01:38,260 That's a question. 31 00:01:43,250 --> 00:01:44,730 So I just want to do a simple-- 32 00:01:47,240 --> 00:01:49,230 oops, thank you. 33 00:01:49,230 --> 00:01:51,150 I just want to do a simple estimation here. 34 00:01:51,150 --> 00:01:54,420 If we have a source of light that's small 35 00:01:54,420 --> 00:01:56,790 and we have an object that's large, 36 00:01:56,790 --> 00:01:59,440 obviously we can measure the size of this object. 37 00:01:59,440 --> 00:02:05,820 But if this object is moving in front of this, 38 00:02:05,820 --> 00:02:09,286 that light gets blocked for a certain amount of time. 39 00:02:09,286 --> 00:02:15,270 If you can measure how long the object is blocked 40 00:02:15,270 --> 00:02:19,320 and you know how fast this object is moving, 41 00:02:19,320 --> 00:02:22,230 you can figure out how big this object is. 42 00:02:22,230 --> 00:02:24,150 For instance, if I move faster, what 43 00:02:24,150 --> 00:02:27,714 happens to the amount of time that that light is blocked? 44 00:02:27,714 --> 00:02:28,970 AUDIENCE: Decrease. 45 00:02:28,970 --> 00:02:30,810 MARK HARTMAN: And if I move slower. 46 00:02:30,810 --> 00:02:33,059 What happens, the amount of time that this is blocked? 47 00:02:33,059 --> 00:02:34,364 AUDIENCE: It increases. 48 00:02:34,364 --> 00:02:36,140 MARK HARTMAN: It increases. 49 00:02:36,140 --> 00:02:45,710 In this simple situation, here's what's 50 00:02:45,710 --> 00:02:49,190 happening on the light curve. 51 00:02:49,190 --> 00:02:52,310 We're starting out at a certain luminosity or certain flux 52 00:02:52,310 --> 00:02:53,450 that we're collecting. 53 00:02:53,450 --> 00:02:55,460 It drops down. 54 00:02:55,460 --> 00:02:57,530 And then it stays at zero for a while. 55 00:02:57,530 --> 00:02:59,300 And then it comes back up. 56 00:02:59,300 --> 00:03:04,200 Because my object is much bigger in angular size than my source. 57 00:03:04,200 --> 00:03:05,980 So right when it gets to the edge-- 58 00:03:05,980 --> 00:03:08,700 bloop-- it covers up that source. 59 00:03:08,700 --> 00:03:12,960 And then-- bloop-- it comes back really quickly. 60 00:03:12,960 --> 00:03:16,040 So this is going to be time. 61 00:03:16,040 --> 00:03:17,900 This is going to be the light curve, which 62 00:03:17,900 --> 00:03:20,610 is flux as a function of time. 63 00:03:26,160 --> 00:03:27,690 Right here what's happening, what 64 00:03:27,690 --> 00:03:33,140 does this system look like from the from the front. 65 00:03:37,660 --> 00:03:47,790 At this point right here, here's our orange poster. 66 00:03:47,790 --> 00:03:49,600 And there's a light-- 67 00:03:49,600 --> 00:03:51,670 it's putting out light-- 68 00:03:51,670 --> 00:03:53,622 right at the edge of it. 69 00:03:53,622 --> 00:03:55,330 And right there there, this orange poster 70 00:03:55,330 --> 00:03:59,230 is moving that way with a certain velocity. 71 00:03:59,230 --> 00:04:00,370 So we're moving that way. 72 00:04:00,370 --> 00:04:03,520 Right here at this time it blocks it out. 73 00:04:03,520 --> 00:04:05,140 And then at a later time-- this is 74 00:04:05,140 --> 00:04:07,184 going to be a later time-- we're going to see, 75 00:04:07,184 --> 00:04:08,225 here's the orange poster. 76 00:04:13,170 --> 00:04:15,960 And at this point, where is the light going to be seen? 77 00:04:19,480 --> 00:04:20,060 At the end. 78 00:04:20,060 --> 00:04:23,659 We're going to see the light over here on this side. 79 00:04:23,659 --> 00:04:26,200 And the orange poster is still moving with a certain velocity 80 00:04:26,200 --> 00:04:27,284 that way. 81 00:04:27,284 --> 00:04:29,200 Let's take a look and see if that makes sense. 82 00:04:29,200 --> 00:04:31,660 Here we've got-- if you guys were recording the amount 83 00:04:31,660 --> 00:04:34,250 of flux that you get from this object, it's constant, 84 00:04:34,250 --> 00:04:35,010 and then-- 85 00:04:35,010 --> 00:04:37,231 bloop-- it drops. 86 00:04:37,231 --> 00:04:38,605 And then-- bloop-- it comes back. 87 00:04:42,910 --> 00:04:48,010 So this is a front view of what happens at each of these times. 88 00:04:48,010 --> 00:04:50,412 This is going to be t start. 89 00:04:50,412 --> 00:04:53,410 This is going to be t stop. 90 00:04:53,410 --> 00:04:56,290 This is going to be the start of the eclipse 91 00:04:56,290 --> 00:04:57,685 or the stop of the eclipse. 92 00:05:05,290 --> 00:05:06,750 AUDIENCE: t start. 93 00:05:06,750 --> 00:05:09,870 MARK HARTMAN: t start, t stop. 94 00:05:09,870 --> 00:05:15,120 If I know that this poster is moving in a certain velocity, 95 00:05:15,120 --> 00:05:23,970 I can say, if I wanted to find let's just say 96 00:05:23,970 --> 00:05:35,286 the width of the poster, I know that in general the distance 97 00:05:35,286 --> 00:05:36,285 that something travels-- 98 00:05:39,859 --> 00:05:41,490 and this is just a general formula 99 00:05:41,490 --> 00:05:43,130 that I'm hoping most of you have seen-- 100 00:05:43,130 --> 00:05:44,580 the distances something travels is 101 00:05:44,580 --> 00:05:51,390 the speed of the object times-- 102 00:05:51,390 --> 00:05:53,201 oops, let's put these in parentheses-- 103 00:05:56,360 --> 00:06:00,420 the time it takes to travel. 104 00:06:06,090 --> 00:06:10,190 Distance equals velocity times time. 105 00:06:10,190 --> 00:06:11,432 So let's check the units. 106 00:06:11,432 --> 00:06:13,140 Whenever we're looking at a new equation, 107 00:06:13,140 --> 00:06:14,556 we always want to check the units. 108 00:06:14,556 --> 00:06:16,910 The units of distance traveled would be-- 109 00:06:16,910 --> 00:06:21,840 so this is going to be units, that's going to be meters. 110 00:06:21,840 --> 00:06:28,950 Speed of the object is going to be in meters per second. 111 00:06:28,950 --> 00:06:33,380 And then the time it takes could be in seconds. 112 00:06:33,380 --> 00:06:34,670 Does that make sense? 113 00:06:34,670 --> 00:06:38,170 Do meters equal meters per second times seconds? 114 00:06:38,170 --> 00:06:39,530 Yeah. 115 00:06:39,530 --> 00:06:41,930 Seconds cancel out, and we're just 116 00:06:41,930 --> 00:06:44,200 left with meters equals meters. 117 00:06:44,200 --> 00:06:46,010 That makes sense. 118 00:06:46,010 --> 00:06:47,960 Pretty amazing. 119 00:06:47,960 --> 00:06:54,980 Now in our case, let me go ahead and move this. 120 00:06:54,980 --> 00:06:59,579 I'm going to just get rid of this. 121 00:06:59,579 --> 00:07:01,370 Hopefully everybody still has this picture. 122 00:07:04,620 --> 00:07:07,820 We're going to redraw it several times, though. 123 00:07:07,820 --> 00:07:14,630 In our case, we can rewrite this as the width of the poster 124 00:07:14,630 --> 00:07:18,440 because how far did the orange poster travel. 125 00:07:18,440 --> 00:07:24,350 Well, the front of the orange poster traveled from this place 126 00:07:24,350 --> 00:07:30,140 all the way over to this place here in the amount of time 127 00:07:30,140 --> 00:07:32,870 that it took for this poster to move in front. 128 00:07:32,870 --> 00:07:37,160 So the length that the front of the object had traveled 129 00:07:37,160 --> 00:07:39,670 was from there to here. 130 00:07:39,670 --> 00:07:42,200 That's the width of my poster. 131 00:07:42,200 --> 00:07:43,370 Here's where it starts-- 132 00:07:43,370 --> 00:07:45,030 blink-- stops. 133 00:07:45,030 --> 00:07:46,980 It starts back up. 134 00:07:46,980 --> 00:07:48,710 So I configure the width of my poster. 135 00:07:52,530 --> 00:07:56,100 That's going to be equal to the velocity of the poster. 136 00:07:56,100 --> 00:08:02,660 Well, let's keep it at speed-- 137 00:08:02,660 --> 00:08:08,680 speed of the poster times the difference in time, 138 00:08:08,680 --> 00:08:12,230 the time that we stopped to the time that we started. 139 00:08:12,230 --> 00:08:16,330 So we're going to say t stop minus t start. 140 00:08:22,500 --> 00:08:24,420 So if I was really, really far away 141 00:08:24,420 --> 00:08:28,020 and you couldn't come up and measure this with a ruler, 142 00:08:28,020 --> 00:08:31,140 say, I was standing really, really far back there, you guys 143 00:08:31,140 --> 00:08:33,809 could measure the amount of time that you 144 00:08:33,809 --> 00:08:36,090 would see this blocked. 145 00:08:36,090 --> 00:08:39,539 And I can tell you how fast I was moving this object. 146 00:08:39,539 --> 00:08:43,630 Then you could figure out how wide this poster is. 147 00:08:43,630 --> 00:08:45,220 Any questions on that? 148 00:08:45,220 --> 00:08:47,160 So these all look really good. 149 00:08:47,160 --> 00:08:49,650 This is what you should have. 150 00:08:49,650 --> 00:08:51,540 From the front view of the telescope, 151 00:08:51,540 --> 00:08:55,140 first, we see the compact object on this side. 152 00:08:55,140 --> 00:08:57,420 The companion is moving across. 153 00:08:57,420 --> 00:09:00,510 In the middle, where is the compact object? 154 00:09:00,510 --> 00:09:01,252 AUDIENCE: Behind. 155 00:09:01,252 --> 00:09:02,460 MARK HARTMAN: It's behind it. 156 00:09:02,460 --> 00:09:04,668 And on this case, the compact object is on that side. 157 00:09:04,668 --> 00:09:07,174 The companion has moved further to that side. 158 00:09:07,174 --> 00:09:09,340 We can see the same thing over here in the top view. 159 00:09:09,340 --> 00:09:11,800 Here, the object is moving this way, 160 00:09:11,800 --> 00:09:14,530 it hasn't quite intersected our line of sight. 161 00:09:14,530 --> 00:09:16,740 Here, we've got the compact object 162 00:09:16,740 --> 00:09:18,740 behind the companion star. 163 00:09:18,740 --> 00:09:20,490 And here, we've got that the companion has 164 00:09:20,490 --> 00:09:23,640 moved a little bit further so that we can actually 165 00:09:23,640 --> 00:09:25,920 see the compact object again. 166 00:09:25,920 --> 00:09:30,990 So how would we estimate the size of our companion star? 167 00:09:33,635 --> 00:09:35,980 AUDIENCE: [INAUDIBLE] 168 00:09:35,980 --> 00:09:39,570 MARK HARTMAN: So if we looked at how long it took, 169 00:09:39,570 --> 00:09:50,950 we could estimate the diameter of companion star 170 00:09:50,950 --> 00:09:56,350 is equal to the speed of the companion 171 00:09:56,350 --> 00:10:03,370 star times the amount of time that it 172 00:10:03,370 --> 00:10:10,270 took to move in front of the compact object, 173 00:10:10,270 --> 00:10:17,100 because this distance here is the diameter of our star. 174 00:10:17,100 --> 00:10:21,000 So linear diameter of a companion star 175 00:10:21,000 --> 00:10:30,200 is equal to the speed of the companion star in orbit times 176 00:10:30,200 --> 00:10:36,950 time to eclipse the compact object. 177 00:10:42,140 --> 00:10:53,240 So we could say, linear diameter equals speed times t 178 00:10:53,240 --> 00:10:59,270 stop minus t start, because that difference-- t 179 00:10:59,270 --> 00:11:04,910 stop minus t start-- is how long it took for our object 180 00:11:04,910 --> 00:11:07,150 to move all the way in front.