1 00:00:00,060 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:19,797 at ocw.mit.edu. 8 00:00:19,797 --> 00:00:21,228 MARK HARTMAN: Are we ready? 9 00:00:21,228 --> 00:00:22,790 AUDIENCE: Yes. 10 00:00:22,790 --> 00:00:24,290 MARK HARTMAN: So I think in general, 11 00:00:24,290 --> 00:00:31,330 we saw that the black-body model fits our observations 12 00:00:31,330 --> 00:00:32,390 pretty well. 13 00:00:32,390 --> 00:00:37,504 And I think what we'd like to do now is to just-- 14 00:00:37,504 --> 00:00:53,330 let's think about how can we tell a good fit. 15 00:00:58,690 --> 00:01:01,390 Because we're saying we're taking-- 16 00:01:01,390 --> 00:01:03,220 again, write down today's date, the time, 17 00:01:03,220 --> 00:01:04,810 [INAUDIBLE] for a few minutes. 18 00:01:10,910 --> 00:01:13,130 What is our observation in this case? 19 00:01:16,030 --> 00:01:17,660 Chris, what is our observation? 20 00:01:17,660 --> 00:01:21,140 AUDIENCE: That the black body fits better than the power law. 21 00:01:21,140 --> 00:01:23,960 MARK HARTMAN: Well, don't go even that far yet. 22 00:01:23,960 --> 00:01:24,650 Let's back up. 23 00:01:24,650 --> 00:01:26,840 What is the data that we collected? 24 00:01:26,840 --> 00:01:28,929 What is our observation about this object? 25 00:01:28,929 --> 00:01:29,720 AUDIENCE: The plot. 26 00:01:29,720 --> 00:01:30,678 MARK HARTMAN: The plot. 27 00:01:30,678 --> 00:01:31,800 Which plot? 28 00:01:31,800 --> 00:01:34,550 AUDIENCE: The black body and the power [? log. ?] 29 00:01:34,550 --> 00:01:38,048 MARK HARTMAN: Are the black body and power law an observation? 30 00:01:38,048 --> 00:01:39,511 AUDIENCE: No, no. 31 00:01:39,511 --> 00:01:40,760 MARK HARTMAN: No, they're not. 32 00:01:40,760 --> 00:01:41,210 AUDIENCE: [INAUDIBLE] 33 00:01:41,210 --> 00:01:43,340 MARK HARTMAN: What did we actually collect, Chris? 34 00:01:46,190 --> 00:01:49,990 AUDIENCE: We collected the [? counts ?] and the energy. 35 00:01:49,990 --> 00:01:52,180 MARK HARTMAN: OK, we looked at the number of counts 36 00:01:52,180 --> 00:01:54,700 as a function of energy, or the intensity 37 00:01:54,700 --> 00:01:55,740 as a function of energy. 38 00:01:55,740 --> 00:01:56,670 What do we call that? 39 00:02:03,020 --> 00:02:04,550 AUDIENCE: Power? 40 00:02:04,550 --> 00:02:05,770 MARK HARTMAN: No. 41 00:02:05,770 --> 00:02:06,565 Chris? 42 00:02:06,565 --> 00:02:07,900 AUDIENCE: The flux? 43 00:02:07,900 --> 00:02:10,810 MARK HARTMAN: OK, we're collecting the flux 44 00:02:10,810 --> 00:02:12,610 at each energy. 45 00:02:12,610 --> 00:02:18,580 If I had a graph that had intensity 46 00:02:18,580 --> 00:02:24,320 as a function of energy, what do I call this-- 47 00:02:24,320 --> 00:02:28,310 let's say there's a stairstep thing. 48 00:02:28,310 --> 00:02:29,200 What do we call this? 49 00:02:34,030 --> 00:02:35,000 AUDIENCE: Spectrum. 50 00:02:35,000 --> 00:02:35,708 MARK HARTMAN: OK. 51 00:02:35,708 --> 00:02:36,840 Nice and loud, Steve. 52 00:02:36,840 --> 00:02:37,720 AUDIENCE: Spectrum. 53 00:02:37,720 --> 00:02:39,340 MARK HARTMAN: We call this a spectrum. 54 00:02:39,340 --> 00:02:46,800 We observed the spectrum, spectrum of a neutron star. 55 00:02:51,780 --> 00:02:56,446 Was the spectrum different when we did the two different fits? 56 00:02:56,446 --> 00:02:57,310 AUDIENCE: No. 57 00:02:57,310 --> 00:02:58,180 MARK HARTMAN: No. 58 00:02:58,180 --> 00:03:00,040 We're fitting to the same data. 59 00:03:00,040 --> 00:03:03,280 We always look at that same spectrum. 60 00:03:03,280 --> 00:03:05,800 So our observation is the data that we collect 61 00:03:05,800 --> 00:03:07,420 from the object, that spectrum. 62 00:03:10,040 --> 00:03:14,260 Now, what, in this case, is our model? 63 00:03:14,260 --> 00:03:16,210 What model or models did we use? 64 00:03:24,652 --> 00:03:26,110 AUDIENCE: Black body and power law. 65 00:03:26,110 --> 00:03:26,605 MARK HARTMAN: OK. 66 00:03:26,605 --> 00:03:27,100 AUDIENCE: Histogram. 67 00:03:27,100 --> 00:03:28,266 MARK HARTMAN: Nice and loud. 68 00:03:28,266 --> 00:03:30,080 AUDIENCE: Black-body and power-law models. 69 00:03:30,080 --> 00:03:30,788 MARK HARTMAN: OK. 70 00:03:30,788 --> 00:03:34,570 We do have, Juan, a histogram, but a histogram is both-- 71 00:03:34,570 --> 00:03:35,800 it's the observation. 72 00:03:35,800 --> 00:03:37,450 It's the spectrum here. 73 00:03:37,450 --> 00:03:40,090 But Steve just said we looked at two different models. 74 00:03:40,090 --> 00:03:43,870 We looked at a black-body model, which is a mathematical way 75 00:03:43,870 --> 00:03:45,590 to describe that curve. 76 00:03:45,590 --> 00:03:48,190 We also looked at a power-law model. 77 00:03:48,190 --> 00:03:49,690 So our model, we had two. 78 00:03:52,360 --> 00:03:59,920 Let me just say we had a black-body model, 79 00:03:59,920 --> 00:04:01,380 and we had a power-law model. 80 00:04:05,090 --> 00:04:09,940 So each one of those, we tried to get the model-- 81 00:04:09,940 --> 00:04:12,010 and it usually showed up as green. 82 00:04:12,010 --> 00:04:19,269 We tried to get the model to go as close to this as we could. 83 00:04:19,269 --> 00:04:23,860 This model allows us to make some predictions. 84 00:04:23,860 --> 00:04:31,722 What are the predictions that we were able to make? 85 00:04:31,722 --> 00:04:33,680 Let's just take for the black-body [? model. ?] 86 00:04:33,680 --> 00:04:38,710 What predictions did it allow us to make? 87 00:04:38,710 --> 00:04:43,422 What values did we get because we fit the black body? 88 00:04:43,422 --> 00:04:46,398 AUDIENCE: The green was more precise 89 00:04:46,398 --> 00:04:48,890 for the black-body [? scale. ?] 90 00:04:48,890 --> 00:04:50,060 MARK HARTMAN: OK. 91 00:04:50,060 --> 00:04:53,210 So maybe one model fit better, but what 92 00:04:53,210 --> 00:04:57,011 was in the output file for each one of these models? 93 00:04:57,011 --> 00:04:57,510 Remember? 94 00:04:57,510 --> 00:04:59,180 Think back to what we did this morning. 95 00:04:59,180 --> 00:05:01,670 We changed a property of the black body 96 00:05:01,670 --> 00:05:04,970 to get it to fit the [? sun. ?] What property of the model 97 00:05:04,970 --> 00:05:07,514 did we have to change? 98 00:05:07,514 --> 00:05:09,244 AUDIENCE: The [INTERPOSING VOICES] 99 00:05:09,244 --> 00:05:10,160 MARK HARTMAN: Hang on. 100 00:05:10,160 --> 00:05:13,070 Let's go back to raising our hands, OK? 101 00:05:13,070 --> 00:05:15,980 So what property did we predict? 102 00:05:15,980 --> 00:05:17,624 OK, I saw Chris, and then Steve. 103 00:05:17,624 --> 00:05:18,290 Go ahead, Chris. 104 00:05:18,290 --> 00:05:21,170 AUDIENCE: [? We ?] changed the temperature [INAUDIBLE].. 105 00:05:21,170 --> 00:05:23,940 MARK HARTMAN: OK, we had to change the temperature. 106 00:05:23,940 --> 00:05:28,580 So a black-body model allows us to make 107 00:05:28,580 --> 00:05:30,980 a prediction of temperature. 108 00:05:35,340 --> 00:05:37,040 Anything else? 109 00:05:37,040 --> 00:05:38,900 Steve, was that what you were going to say? 110 00:05:38,900 --> 00:05:40,070 OK. 111 00:05:40,070 --> 00:05:42,320 What other information did we get 112 00:05:42,320 --> 00:05:44,120 from fitting the black-body model? 113 00:05:44,120 --> 00:05:46,820 What else was in that text file? 114 00:05:46,820 --> 00:05:48,460 Bianca? 115 00:05:48,460 --> 00:05:49,160 Nice and loud. 116 00:05:49,160 --> 00:05:50,402 AUDIENCE: Flux. 117 00:05:50,402 --> 00:05:51,110 MARK HARTMAN: OK. 118 00:05:51,110 --> 00:05:55,970 We predicted the temperature, and we also predicted the flux. 119 00:05:55,970 --> 00:05:56,660 That's useful. 120 00:05:56,660 --> 00:05:57,320 That's interesting. 121 00:05:57,320 --> 00:05:58,570 We can do something with that. 122 00:05:58,570 --> 00:06:00,076 Go ahead, Juan. 123 00:06:00,076 --> 00:06:00,950 AUDIENCE: Energy. 124 00:06:00,950 --> 00:06:03,680 MARK HARTMAN: Energy. 125 00:06:03,680 --> 00:06:06,100 Where did it predict the energy? 126 00:06:06,100 --> 00:06:07,820 AUDIENCE: Oh. 127 00:06:07,820 --> 00:06:08,360 Wrong one. 128 00:06:08,360 --> 00:06:08,860 Sorry. 129 00:06:11,390 --> 00:06:13,400 MARK HARTMAN: Unfortunately, in this case, 130 00:06:13,400 --> 00:06:16,010 we measure temperature in units of keV. 131 00:06:16,010 --> 00:06:18,080 We're going to show you how to switch that back. 132 00:06:18,080 --> 00:06:24,470 But for the power law, what predictions 133 00:06:24,470 --> 00:06:25,480 were we allowed to make? 134 00:06:25,480 --> 00:06:26,813 You should be writing this down. 135 00:06:29,960 --> 00:06:32,945 Were we able to predict a temperature? 136 00:06:32,945 --> 00:06:35,558 What were we able to predict, Nikki? 137 00:06:35,558 --> 00:06:36,800 AUDIENCE: For the power law? 138 00:06:36,800 --> 00:06:39,790 MARK HARTMAN: For the power-law model. 139 00:06:39,790 --> 00:06:40,770 Hang on [? one sec. ?] 140 00:06:40,770 --> 00:06:43,220 AUDIENCE: Yeah. [? We got ?] magnetic field 141 00:06:43,220 --> 00:06:45,242 of the [? sun. ?] 142 00:06:45,242 --> 00:06:45,950 MARK HARTMAN: OK. 143 00:06:45,950 --> 00:06:49,524 Did we get a number that said magnetic field equals? 144 00:06:49,524 --> 00:06:50,190 What did we get? 145 00:06:57,190 --> 00:06:59,590 Did we get a temperature? 146 00:06:59,590 --> 00:07:00,520 No? 147 00:07:00,520 --> 00:07:02,320 What did we get in that text file? 148 00:07:02,320 --> 00:07:03,720 Juan. 149 00:07:03,720 --> 00:07:05,122 AUDIENCE: Power-law index. 150 00:07:05,122 --> 00:07:06,580 MARK HARTMAN: OK, we got the index. 151 00:07:06,580 --> 00:07:08,890 Remember, when we fit the power law this morning, 152 00:07:08,890 --> 00:07:10,600 we changed the power-law index, and that 153 00:07:10,600 --> 00:07:13,060 changed the shape of the model. 154 00:07:13,060 --> 00:07:20,000 So we got a power-law index. 155 00:07:20,000 --> 00:07:21,450 And what else did we get? 156 00:07:21,450 --> 00:07:22,684 Lauren? 157 00:07:22,684 --> 00:07:23,890 AUDIENCE: Oh, I was-- 158 00:07:23,890 --> 00:07:24,580 MARK HARTMAN: Oh, you going to say-- 159 00:07:24,580 --> 00:07:26,290 AUDIENCE: Power-law index, yeah. 160 00:07:26,290 --> 00:07:27,040 MARK HARTMAN: OK. 161 00:07:27,040 --> 00:07:31,780 Was there anything else in that text file for the power law? 162 00:07:34,990 --> 00:07:36,073 AUDIENCE: Power-law index. 163 00:07:36,073 --> 00:07:39,730 MARK HARTMAN: Yeah, power-law index. 164 00:07:39,730 --> 00:07:43,630 Whenever you get a piece of text that is tossed out at you, 165 00:07:43,630 --> 00:07:46,960 you always need to read it very, very carefully. 166 00:07:46,960 --> 00:07:51,130 You also got a prediction of the flux from the power-law model. 167 00:07:53,890 --> 00:07:56,720 Every time you fit a model to a spectrum, 168 00:07:56,720 --> 00:08:00,850 it's going to give you the flux, and then some other parameters 169 00:08:00,850 --> 00:08:05,080 that describe that model. 170 00:08:05,080 --> 00:08:07,090 For anybody who is taking calculus 171 00:08:07,090 --> 00:08:11,740 or may soon be taking calculus, the flux 172 00:08:11,740 --> 00:08:14,890 is simply the area underneath this curve. 173 00:08:18,100 --> 00:08:22,020 So what we do for each case is if we have a model, what 174 00:08:22,020 --> 00:08:23,520 the computer does is it figures out 175 00:08:23,520 --> 00:08:25,717 the area underneath the curve of the model. 176 00:08:25,717 --> 00:08:27,675 And that's what it gives you back for the flux. 177 00:08:31,480 --> 00:08:34,630 But there's actually a couple of different predictions of flux, 178 00:08:34,630 --> 00:08:36,935 and we're going to get to that in just a minute. 179 00:08:36,935 --> 00:08:39,309 But here's the way that we've been thinking about things. 180 00:08:39,309 --> 00:08:41,080 We have an observation. 181 00:08:41,080 --> 00:08:44,650 We have a model that we now want to fit to that observation, 182 00:08:44,650 --> 00:08:48,780 and it allows us to make some predictions. 183 00:08:48,780 --> 00:08:51,020 Now, just because we fit a power law-- 184 00:08:51,020 --> 00:08:54,770 I think everybody agreed that the black-body model fit 185 00:08:54,770 --> 00:08:56,930 better. 186 00:08:56,930 --> 00:08:58,640 So let's take a look at what do we mean 187 00:08:58,640 --> 00:09:00,040 by if something fits better. 188 00:09:00,040 --> 00:09:02,600 And I think most people have this already. 189 00:09:05,150 --> 00:09:09,230 When you're looking at data like this, what we would like to do 190 00:09:09,230 --> 00:09:11,630 is we want to look at a couple of different ways 191 00:09:11,630 --> 00:09:13,620 for us to figure out which fit is better. 192 00:09:13,620 --> 00:09:16,640 There's three different ways that we can figure it out, 193 00:09:16,640 --> 00:09:22,205 three ways to tell a good fit. 194 00:09:25,850 --> 00:09:36,830 The first is to check the fit by eye. 195 00:09:36,830 --> 00:09:39,110 What does it mean to check the fit by eye? 196 00:09:39,110 --> 00:09:40,084 What do you think? 197 00:09:40,084 --> 00:09:41,814 AUDIENCE: See if the lines match up/ 198 00:09:41,814 --> 00:09:43,480 MARK HARTMAN: See if the lines match up. 199 00:09:43,480 --> 00:09:44,240 AUDIENCE: Reading-- 200 00:09:44,240 --> 00:09:44,840 MARK HARTMAN: Reading. 201 00:09:44,840 --> 00:09:45,715 AUDIENCE: --the data. 202 00:09:45,715 --> 00:09:47,210 MARK HARTMAN: Oh, reading the data? 203 00:09:47,210 --> 00:09:48,092 OK. 204 00:09:48,092 --> 00:09:48,800 What did you say? 205 00:09:48,800 --> 00:09:49,700 AUDIENCE: Comparing. 206 00:09:49,700 --> 00:09:50,810 MARK HARTMAN: Comparing. 207 00:09:50,810 --> 00:09:52,700 You want to compare. 208 00:09:52,700 --> 00:09:56,150 Now, we don't necessarily always have two models 209 00:09:56,150 --> 00:09:57,840 that we want to pick. 210 00:09:57,840 --> 00:10:02,400 But we want to compare the model to the data. 211 00:10:02,400 --> 00:10:04,400 So if we check the fit by eye, we 212 00:10:04,400 --> 00:10:08,210 see if something is a good fit. 213 00:10:08,210 --> 00:10:24,930 Good equals the model passes near most data points, 214 00:10:24,930 --> 00:10:26,180 or rather-- 215 00:10:26,180 --> 00:10:28,610 I don't like to use data points-- 216 00:10:28,610 --> 00:10:30,635 most observation points. 217 00:10:33,760 --> 00:10:35,567 AUDIENCE: What's wrong with data? 218 00:10:35,567 --> 00:10:37,400 MARK HARTMAN: It's just not specific enough, 219 00:10:37,400 --> 00:10:39,830 because data, we could say, is numbers 220 00:10:39,830 --> 00:10:42,330 that we figured out from something. 221 00:10:42,330 --> 00:10:45,170 So these measurements of temperature flux, those 222 00:10:45,170 --> 00:10:46,890 can be data. 223 00:10:46,890 --> 00:10:49,730 But we want to say that the model passes 224 00:10:49,730 --> 00:10:51,850 near most observation points. 225 00:10:51,850 --> 00:10:56,660 Again, the observation is the spectrum itself, 226 00:10:56,660 --> 00:11:04,610 and the model is the line or the mathematical shape 227 00:11:04,610 --> 00:11:08,850 that we try to get close to those observations. 228 00:11:08,850 --> 00:11:10,842 So that's the first way to do it, 229 00:11:10,842 --> 00:11:12,050 which I think everybody said. 230 00:11:12,050 --> 00:11:13,550 Everybody said, if you look at the model 231 00:11:13,550 --> 00:11:16,175 and the green line passes pretty close, then that's a good fit. 232 00:11:16,175 --> 00:11:19,700 And we saw, in our case, the black-body model line 233 00:11:19,700 --> 00:11:21,530 went pretty close. 234 00:11:21,530 --> 00:11:23,990 The second way that you can do it is you 235 00:11:23,990 --> 00:11:38,070 can check what we call the reduced chi-squared statistic. 236 00:11:40,960 --> 00:11:44,070 Now, I'm going to put this in brackets 237 00:11:44,070 --> 00:11:46,110 because when you get your data out, 238 00:11:46,110 --> 00:11:49,800 it's just going to say the reduced statistic. 239 00:11:49,800 --> 00:11:54,620 What this is, this chi-squared-- chi is a Greek letter. 240 00:11:54,620 --> 00:11:56,510 That's chi-squared. 241 00:11:56,510 --> 00:11:59,340 It is a measurement that tells you 242 00:11:59,340 --> 00:12:06,790 how close is your model to the observation. 243 00:12:06,790 --> 00:12:11,150 And a very, very simple way to think about it, 244 00:12:11,150 --> 00:12:12,360 it is one number. 245 00:12:15,030 --> 00:12:25,650 So we'll say that this is a number which 246 00:12:25,650 --> 00:12:41,490 is the average number of error bars away. 247 00:12:46,370 --> 00:12:50,200 Well, let's just say the number of error bars 248 00:12:50,200 --> 00:12:59,485 between observation and model. 249 00:13:06,070 --> 00:13:08,370 Let's just do a quick example. 250 00:13:08,370 --> 00:13:11,220 So it's the average number of error bars 251 00:13:11,220 --> 00:13:14,520 between the observation and the model, average meaning averaged 252 00:13:14,520 --> 00:13:18,010 over all points in your spectrum. 253 00:13:18,010 --> 00:13:21,420 So say I had intensity versus energy, 254 00:13:21,420 --> 00:13:25,260 and I had this data point, that data point, and this data 255 00:13:25,260 --> 00:13:27,060 point. 256 00:13:27,060 --> 00:13:29,880 And I had error bars, which are indications 257 00:13:29,880 --> 00:13:33,580 of how well do I know really where that value is. 258 00:13:33,580 --> 00:13:40,830 Say I had error bars, which we normally show like that. 259 00:13:40,830 --> 00:13:44,216 That means this value is probably right about here 260 00:13:44,216 --> 00:13:46,340 at this intensity, but it could be a little higher. 261 00:13:46,340 --> 00:13:49,160 It could be a little bit lower. 262 00:13:49,160 --> 00:13:58,770 If I had a model that went like this, 263 00:13:58,770 --> 00:14:01,830 that's not a very good fit, right? 264 00:14:01,830 --> 00:14:06,330 So let's say at this point of energy, 265 00:14:06,330 --> 00:14:10,830 my observation is down here, but my prediction 266 00:14:10,830 --> 00:14:14,665 is actually up there. 267 00:14:14,665 --> 00:14:16,290 At this energy, I'm predicting that I'm 268 00:14:16,290 --> 00:14:19,130 going to get that much flux or that much intensity. 269 00:14:19,130 --> 00:14:22,970 At this energy, I'm predicting I'm going to get that much. 270 00:14:22,970 --> 00:14:24,810 And at this energy, I'm predicting 271 00:14:24,810 --> 00:14:27,570 I'm going to get that much. 272 00:14:27,570 --> 00:14:33,270 In this case, my prediction is different from my model by-- 273 00:14:33,270 --> 00:14:36,190 here's the width of one error bar. 274 00:14:36,190 --> 00:14:42,339 So I'm going to say 1, 2, 3. 275 00:14:42,339 --> 00:14:43,880 So here's the width of my error bar-- 276 00:14:43,880 --> 00:14:45,420 1, 2. 277 00:14:45,420 --> 00:14:47,530 That's three error bars away. 278 00:14:47,530 --> 00:14:50,130 In this case, I've got-- there's the size of my error bar-- 279 00:14:50,130 --> 00:14:54,540 1, 2, maybe 2.8. 280 00:14:54,540 --> 00:14:56,340 And in this case, I've got-- 281 00:14:56,340 --> 00:14:57,930 here's the size of my error bars-- 282 00:14:57,930 --> 00:15:00,660 1, 2. 283 00:15:00,660 --> 00:15:03,150 So what I would do is I'd take the average of how far 284 00:15:03,150 --> 00:15:05,890 away am I in both cases. 285 00:15:05,890 --> 00:15:10,560 So in this case, my chi-squared is 286 00:15:10,560 --> 00:15:16,830 going to be 3 plus 2.8 plus 2. 287 00:15:16,830 --> 00:15:19,650 If I wanted to take the average of how far away that is-- 288 00:15:19,650 --> 00:15:22,050 that's reduced chi-squared, and so they 289 00:15:22,050 --> 00:15:25,300 put this little V or this nu down here. 290 00:15:25,300 --> 00:15:30,108 That's 3 plus 2.8 plus 2 divided by 3. 291 00:15:30,108 --> 00:15:31,042 AUDIENCE: Why 3? 292 00:15:31,042 --> 00:15:33,000 MARK HARTMAN: Because that's the number of data 293 00:15:33,000 --> 00:15:36,030 points that I have. 294 00:15:36,030 --> 00:15:42,850 So on average, this is maybe 2.8, something like that. 295 00:15:42,850 --> 00:15:45,480 So my reduced chi-squared statistic 296 00:15:45,480 --> 00:15:48,340 here is going to be about 2.8. 297 00:15:48,340 --> 00:15:51,150 So it's not a very good fit. 298 00:15:51,150 --> 00:15:55,410 What I want is that instead of that, 299 00:15:55,410 --> 00:15:56,730 I want to be able to say-- 300 00:16:00,650 --> 00:16:05,260 I want my model to kind to do this, right? 301 00:16:05,260 --> 00:16:07,700 Maybe it doesn't quite fit all of them, 302 00:16:07,700 --> 00:16:10,400 but the reduced chi-squared-- 303 00:16:10,400 --> 00:16:14,210 here, that's only maybe half an error bar away. 304 00:16:14,210 --> 00:16:16,250 Here, that's right on. 305 00:16:16,250 --> 00:16:19,610 And then here, maybe that's one error bar away. 306 00:16:19,610 --> 00:16:25,000 So a good chi-squared, which means you've got a good fit-- 307 00:16:25,000 --> 00:16:31,820 so we're going to say good equals close to 1. 308 00:16:31,820 --> 00:16:35,090 If that number is close to 1, that means, on average, 309 00:16:35,090 --> 00:16:37,880 you're only about one error bar away 310 00:16:37,880 --> 00:16:40,640 when you predict using your model to what you actually 311 00:16:40,640 --> 00:16:42,960 see with the data. 312 00:16:42,960 --> 00:16:43,820 Yeah, Bianca? 313 00:16:43,820 --> 00:16:46,255 AUDIENCE: So in the blue region of the nebula 314 00:16:46,255 --> 00:16:49,664 that we saw before, if it matched it completely 315 00:16:49,664 --> 00:16:54,047 on every single point, is that 1 or 0 because there's no error 316 00:16:54,047 --> 00:16:54,550 bar? 317 00:16:54,550 --> 00:16:57,940 MARK HARTMAN: In this case, we didn't show the error bars 318 00:16:57,940 --> 00:17:02,470 when we looked at the Crab Nebula spectrum. 319 00:17:02,470 --> 00:17:04,569 So it's kind of hard to tell unless you 320 00:17:04,569 --> 00:17:06,246 know how big the error bar is. 321 00:17:06,246 --> 00:17:08,079 Typically, you're not going to get something 322 00:17:08,079 --> 00:17:09,700 that fits that perfectly. 323 00:17:09,700 --> 00:17:11,170 And I think in that case, the data 324 00:17:11,170 --> 00:17:12,836 was actually fudged a little bit so that 325 00:17:12,836 --> 00:17:14,920 would be a perfect power law. 326 00:17:14,920 --> 00:17:19,246 But on average, if it's less than 1, 327 00:17:19,246 --> 00:17:20,829 what that means is probably your error 328 00:17:20,829 --> 00:17:23,410 bars are a little bit too big. 329 00:17:23,410 --> 00:17:28,089 [? Because ?] then, if you're so close to the actual value-- 330 00:17:28,089 --> 00:17:29,584 so it should be around 1. 331 00:17:29,584 --> 00:17:32,000 And I will say that this is the last time that we're going 332 00:17:32,000 --> 00:17:34,492 to get models that fit great. 333 00:17:34,492 --> 00:17:36,700 Because you will have [? seen, ?] for the black body, 334 00:17:36,700 --> 00:17:39,790 what was the reduced statistic that we got? 335 00:17:44,520 --> 00:17:45,634 Anybody? 336 00:17:45,634 --> 00:17:46,800 AUDIENCE: What's [? that? ?] 337 00:17:46,800 --> 00:17:48,633 MARK HARTMAN: What was the reduced statistic 338 00:17:48,633 --> 00:17:52,716 that we got when we fit the black-body model here? 339 00:17:52,716 --> 00:17:54,612 AUDIENCE: 12.58. 340 00:17:54,612 --> 00:17:56,510 AUDIENCE: I got 4.07. 341 00:17:56,510 --> 00:17:57,890 MARK HARTMAN: OK, 4.07. 342 00:17:57,890 --> 00:17:59,020 It should be around 4. 343 00:18:04,510 --> 00:18:05,710 You got 12? 344 00:18:05,710 --> 00:18:06,690 AUDIENCE: [INAUDIBLE] 345 00:18:06,690 --> 00:18:07,424 MARK HARTMAN: OK. 346 00:18:07,424 --> 00:18:09,280 AUDIENCE: 9.915. 347 00:18:09,280 --> 00:18:11,600 AUDIENCE: Yeah, around [? 4. ?] 348 00:18:11,600 --> 00:18:12,540 MARK HARTMAN: OK. 349 00:18:12,540 --> 00:18:17,580 So in our case, we had the black body was around 4. 350 00:18:17,580 --> 00:18:20,010 The power law was around what? 351 00:18:25,279 --> 00:18:26,720 AUDIENCE: 12. 352 00:18:26,720 --> 00:18:28,920 MARK HARTMAN: Around 12. 353 00:18:28,920 --> 00:18:31,910 So this is another way to check. 354 00:18:31,910 --> 00:18:34,700 OK, yeah, it looks good by eye, but this is another way. 355 00:18:34,700 --> 00:18:36,920 You don't want to count on just one of these ways 356 00:18:36,920 --> 00:18:38,210 to figure it out. 357 00:18:38,210 --> 00:18:39,890 Because eventually, we're going to have 358 00:18:39,890 --> 00:18:44,240 you fit different spectra models to the same source 359 00:18:44,240 --> 00:18:45,976 that we don't know what it is. 360 00:18:45,976 --> 00:18:47,350 In this case, we knew that it was 361 00:18:47,350 --> 00:18:48,800 supposed to be a black body. 362 00:18:48,800 --> 00:18:51,042 In other cases, we don't necessarily know. 363 00:18:51,042 --> 00:18:53,000 So you'll have to figure out which one of these 364 00:18:53,000 --> 00:18:54,950 fits the best. 365 00:18:54,950 --> 00:19:00,370 Now, the last thing that we can look at-- 366 00:19:00,370 --> 00:19:02,360 and this is probably the most important one-- 367 00:19:05,270 --> 00:19:07,210 is we want to look at a-- 368 00:19:07,210 --> 00:19:20,280 so three is the physical interpretation of-- what did I 369 00:19:20,280 --> 00:19:21,090 say-- 370 00:19:21,090 --> 00:19:28,170 model parameters. 371 00:19:28,170 --> 00:19:30,185 We're going to get some predictions about this. 372 00:19:30,185 --> 00:19:32,310 We're going to get a prediction of the temperature. 373 00:19:32,310 --> 00:19:33,960 We're going to get a prediction of the power-law index. 374 00:19:33,960 --> 00:19:35,970 We're going to get a prediction of the flux. 375 00:19:35,970 --> 00:19:38,100 We want to look at those values and see, 376 00:19:38,100 --> 00:19:40,680 do those actually make sense? 377 00:19:40,680 --> 00:19:46,800 Because if we get a measurement of the temperature 378 00:19:46,800 --> 00:19:50,940 of our neutron star, if we come out with a temperature of 100 379 00:19:50,940 --> 00:19:54,330 kelvin, that's really cold. 380 00:19:54,330 --> 00:19:56,840 Would a really cold object be producing X-rays? 381 00:19:56,840 --> 00:19:57,780 AUDIENCE: No. 382 00:19:57,780 --> 00:19:59,940 MARK HARTMAN: Probably not, right? 383 00:19:59,940 --> 00:20:04,910 So we want to look physically at our model parameters, 384 00:20:04,910 --> 00:20:08,230 see if they make sense. 385 00:20:08,230 --> 00:20:11,520 So what I would like to do is I want to ask you 386 00:20:11,520 --> 00:20:13,260 another, just a quick question. 387 00:20:13,260 --> 00:20:15,450 Because what was the-- 388 00:20:15,450 --> 00:20:17,430 so we know that our temperature should come out 389 00:20:17,430 --> 00:20:18,270 to be about right. 390 00:20:18,270 --> 00:20:20,790 It's about 0.1 keV. 391 00:20:20,790 --> 00:20:23,890 Each keV of energy-- 392 00:20:23,890 --> 00:20:25,800 so this [? is ?] we're going to look 393 00:20:25,800 --> 00:20:30,630 at temperature of black body-- 394 00:20:30,630 --> 00:20:34,740 and I'll just shorten that to BB, Black Body-- 395 00:20:34,740 --> 00:20:38,930 fit to neutron star. 396 00:20:38,930 --> 00:20:39,710 What did we get? 397 00:20:39,710 --> 00:20:41,539 What was your value for the temperature? 398 00:20:48,694 --> 00:20:49,360 Go ahead, Steve. 399 00:20:49,360 --> 00:20:50,220 AUDIENCE: [? 0.1 ?] 400 00:20:50,220 --> 00:20:57,917 AUDIENCE: 0.0971122 keV. 401 00:20:57,917 --> 00:20:59,375 MARK HARTMAN: That's what you said? 402 00:20:59,375 --> 00:21:00,620 AUDIENCE: Yeah. 403 00:21:00,620 --> 00:21:01,840 MARK HARTMAN: OK. 404 00:21:01,840 --> 00:21:04,570 So this is actually an interesting point for us 405 00:21:04,570 --> 00:21:05,530 to bring out. 406 00:21:05,530 --> 00:21:08,390 We're going to start measuring a lot more numbers. 407 00:21:08,390 --> 00:21:11,290 Whenever we have numbers that have so many decimals 408 00:21:11,290 --> 00:21:15,430 after them, typically, a good rule of thumb 409 00:21:15,430 --> 00:21:19,930 is to keep maybe two decimal places or two 410 00:21:19,930 --> 00:21:21,730 significant figures. 411 00:21:21,730 --> 00:21:25,810 Any number that's not 0 is significant. 412 00:21:25,810 --> 00:21:28,420 So when we get a number like this, what we actually 413 00:21:28,420 --> 00:21:30,550 want to say is, well, it's not that we 414 00:21:30,550 --> 00:21:33,460 know these numbers for sure-- because remember, 415 00:21:33,460 --> 00:21:34,790 we're adjusting our model. 416 00:21:34,790 --> 00:21:37,180 And if we adjust it by just a little tiny bit, 417 00:21:37,180 --> 00:21:40,672 it's not really going to change how it looks that much. 418 00:21:40,672 --> 00:21:42,130 So we're going round off, and we're 419 00:21:42,130 --> 00:21:48,333 going to say that the temperature is about 0.097 keV. 420 00:21:48,333 --> 00:21:50,291 AUDIENCE: Can you do it in scientific notation? 421 00:21:50,291 --> 00:21:51,190 MARK HARTMAN: Yep, we could write 422 00:21:51,190 --> 00:21:52,360 that as scientific notation. 423 00:21:52,360 --> 00:21:57,980 That's 9.7 times 10 to the minus second keV. 424 00:22:00,520 --> 00:22:03,880 Or I think a couple of other people got about-- 425 00:22:03,880 --> 00:22:07,745 and that's about 0.1 keV. 426 00:22:10,990 --> 00:22:17,500 A way to convert this between units, a temperature of about 427 00:22:17,500 --> 00:22:26,560 1 keV is going to be equal to about 1 times 10 428 00:22:26,560 --> 00:22:32,530 to the sixth kelvin, so 1 million kelvin. 429 00:22:32,530 --> 00:22:35,430 So let's think about this. 430 00:22:35,430 --> 00:22:40,030 KeV is just a nice way to measure this. 431 00:22:40,030 --> 00:22:41,530 It's just a unit. 432 00:22:41,530 --> 00:22:44,680 So this is for temperature. 433 00:22:48,784 --> 00:22:52,130 AUDIENCE: How much is 1 keV in Celsius? 434 00:22:52,130 --> 00:22:56,630 MARK HARTMAN: So 1 keV is about 1 times 10 to the sixth kelvin, 435 00:22:56,630 --> 00:22:58,010 so 1 million kelvin. 436 00:22:58,010 --> 00:23:00,320 The kelvin and Celsius scales are only 437 00:23:00,320 --> 00:23:03,530 offset by a couple of hundred degrees each. 438 00:23:03,530 --> 00:23:11,710 So that's about 1 million degrees Celsius as well. 439 00:23:11,710 --> 00:23:16,372 So if we have 0.1 keV, how hot our we? 440 00:23:16,372 --> 00:23:17,860 AUDIENCE: What's that? 441 00:23:17,860 --> 00:23:20,330 MARK HARTMAN: So if we have 0.1 keV 442 00:23:20,330 --> 00:23:23,650 as our temperature for our neutron star, 443 00:23:23,650 --> 00:23:27,050 and we know that 1 keV is a million-- 444 00:23:27,050 --> 00:23:28,400 AUDIENCE: [? 100,000. ?] 445 00:23:28,400 --> 00:23:29,150 MARK HARTMAN: Yep. 446 00:23:29,150 --> 00:23:37,220 We can say temperature equals 0.1 keV in temperature times 447 00:23:37,220 --> 00:23:42,230 1 keV per 1 times 10 to the sixth kelvin. 448 00:23:42,230 --> 00:23:52,030 That gives us 1 times 10 to the fifth kelvin, or 100,000. 449 00:23:52,030 --> 00:23:55,070 Now, anybody who's doing the X-ray binaries project 450 00:23:55,070 --> 00:23:57,370 or the active stars project, do you 451 00:23:57,370 --> 00:24:00,790 remember an estimate for how hot the surface of a neutron star 452 00:24:00,790 --> 00:24:01,700 might be? 453 00:24:01,700 --> 00:24:04,902 AUDIENCE: [INAUDIBLE] [? give me ?] a second. 454 00:24:04,902 --> 00:24:06,860 MARK HARTMAN: No, not off the top of your head. 455 00:24:06,860 --> 00:24:08,026 But what do you think, Juan? 456 00:24:11,312 --> 00:24:17,884 AUDIENCE: [INAUDIBLE] 457 00:24:17,884 --> 00:24:19,550 MARK HARTMAN: Go ahead and keep looking, 458 00:24:19,550 --> 00:24:23,790 but 100,000 kelvin is pretty reasonable. 459 00:24:23,790 --> 00:24:24,870 It seems really high. 460 00:24:24,870 --> 00:24:27,570 But for objects that emit X-rays, 461 00:24:27,570 --> 00:24:30,270 most objects are around a million kelvin-- so 462 00:24:30,270 --> 00:24:33,851 maybe like 100,000 kelvin, all the way up to 10 or 20 million 463 00:24:33,851 --> 00:24:34,350 kelvin. 464 00:24:34,350 --> 00:24:35,400 What do you think, Steve? 465 00:24:35,400 --> 00:24:36,406 You found something? 466 00:24:36,406 --> 00:24:38,270 AUDIENCE: Yeah, something like that. 467 00:24:38,270 --> 00:24:42,508 AUDIENCE: [INAUDIBLE] to temperatures of 10 million 468 00:24:42,508 --> 00:24:43,795 to 100 million kelvins. 469 00:24:43,795 --> 00:24:46,170 MARK HARTMAN: OK, so somewhere between 10 million and 100 470 00:24:46,170 --> 00:24:49,150 million kelvins for other stars. 471 00:24:49,150 --> 00:24:52,410 So yeah, this one is actually fairly cool. 472 00:24:52,410 --> 00:24:56,490 That's why this particular star was observed, I think, 473 00:24:56,490 --> 00:24:58,740 because it is fairly cool. 474 00:24:58,740 --> 00:25:00,960 And we can see that because the black-body shape 475 00:25:00,960 --> 00:25:04,110 was shifted to the left. 476 00:25:04,110 --> 00:25:06,180 So that makes sense. 477 00:25:06,180 --> 00:25:12,470 So then black body makes sense.