1 00:00:02,510 --> 00:00:06,478 So, I hope you realize not only the binary stars are cool but that our 2 00:00:06,478 --> 00:00:11,183 understanding of the physics allows us to understand how you learned so much about 3 00:00:11,183 --> 00:00:13,904 them. as this is the case with the planets, if 4 00:00:13,904 --> 00:00:18,439 the inclination of the orbit is indeed zero, in the sense that the planets orbit 5 00:00:18,439 --> 00:00:22,804 in a plane that includes our line of sight, then something even more exciting 6 00:00:22,804 --> 00:00:26,319 is going to happen. not only will it measure the full radio 7 00:00:26,319 --> 00:00:30,973 velocities but halfway between those points where you measured the full radial 8 00:00:30,973 --> 00:00:34,660 velocities just as with planets, one star will transit the other. 9 00:00:34,660 --> 00:00:39,889 And so, there will be times during the evolution, the, the history of transiting 10 00:00:39,889 --> 00:00:44,829 binary like this where we see both stars. We can't resolve them, but we see and we 11 00:00:44,829 --> 00:00:47,531 see both sets of spectroscopic absorption lines. 12 00:00:47,531 --> 00:00:51,611 And then, there will be times when we only see one of the two stars or we see 13 00:00:51,611 --> 00:00:55,373 parts of one and all of the other. We see a non-trivial behavior of the 14 00:00:55,373 --> 00:00:58,075 light curve. And the light curve, as I promised with 15 00:00:58,075 --> 00:01:02,261 planets, will tell us things about the periods, the sizes, the temperatures of 16 00:01:02,261 --> 00:01:05,122 the two stars. And if we can combine this with Doppler 17 00:01:05,122 --> 00:01:07,718 measurements, we can learn a lot about the system. 18 00:01:07,718 --> 00:01:12,116 I want to spend some time looking at what we learn from a light curve because I 19 00:01:12,116 --> 00:01:15,745 think it will interest you and I promised that when we get to planets, 20 00:01:15,745 --> 00:01:18,660 let's take a look at an eclipsing binary simulator. 21 00:01:20,220 --> 00:01:25,762 This is our eclipsing binary simulator. It has many features, I hope you'll play 22 00:01:25,762 --> 00:01:31,164 with it, let's see what we can learn from an eclipsing binary, so here is our 23 00:01:31,164 --> 00:01:36,047 system over here on the left. we can change its parameters if we want, 24 00:01:36,047 --> 00:01:39,583 but I've had set it up so we have star number one, this is a hot blue star. Its 25 00:01:39,583 --> 00:01:43,301 temperature is 7850 K, its radius is five solar radii. 26 00:01:43,301 --> 00:01:48,553 Star number two is a smaller reddish star, it's only got 2.8 so the radii and 27 00:01:48,553 --> 00:01:53,440 the temperature of 6,200 Kelvin. And [COUGH] I've chosen to view the system 28 00:01:53,440 --> 00:01:58,938 [UNKNOWN] let me try to show you what it looks like if ypu turn it around. 29 00:01:58,938 --> 00:02:03,057 this is our a system. Notice that astronomers call inclination 30 00:02:03,057 --> 00:02:07,887 the angle that is zero when you're seeing the system face on and 90, when you're 31 00:02:07,887 --> 00:02:11,449 seeing it edge on. We're using a complementary notation for 32 00:02:11,449 --> 00:02:16,218 silly historical reasons, but I wanted you to know that other conventions exist. 33 00:02:16,218 --> 00:02:19,818 And we can't, we're imagining that we cannot resolve these two stars. 34 00:02:19,818 --> 00:02:23,071 So, what we see is the combined light of both in our telescope. 35 00:02:23,071 --> 00:02:26,743 And we can make photometric measurements. We can measure the intensity. 36 00:02:26,743 --> 00:02:29,367 And over here on the right, we see the light curve. 37 00:02:29,367 --> 00:02:33,460 In other words, the time dependence of the combined intensity of the two stars. 38 00:02:33,460 --> 00:02:36,411 This will be a periodic periodically changing. 39 00:02:36,411 --> 00:02:39,423 And we folded it back, so we see a complete period. 40 00:02:39,423 --> 00:02:43,940 And this red cursor shows us the current situation at any given instant. 41 00:02:43,940 --> 00:02:47,614 While on the left, we can see the configuration of the system. 42 00:02:47,614 --> 00:02:51,590 And so, as I start the animation, the stars are orbiting each other. 43 00:02:51,590 --> 00:02:56,396 And if I've stopped it at an interesting point, this is the point at which the red 44 00:02:56,396 --> 00:03:01,145 dim star is moving away from us, the blue more luminous star is moving towards us. 45 00:03:01,145 --> 00:03:05,160 So, if we measure the Doppler shifts, this is the moment when you'd find 46 00:03:05,160 --> 00:03:09,796 maximal red shift in the spectral lines of the red star and maximal blue shift in 47 00:03:09,796 --> 00:03:13,924 the spectral lines of the blue star. And we could use that, for example, to 48 00:03:13,924 --> 00:03:17,034 measure their speeds. And that will be useful, of course, 49 00:03:17,034 --> 00:03:21,444 because we'll have some information from the Doppler shift, from the radial 50 00:03:21,444 --> 00:03:23,480 velocity curves. And as time goes by, 51 00:03:23,480 --> 00:03:26,303 well, the Doppler shifts will re, will decrease. 52 00:03:26,303 --> 00:03:31,583 And if I stop the animation at this point we're seeing that as the red star begins 53 00:03:31,583 --> 00:03:36,556 to be eclipsed by the blue star the total combined intensity begins to dip, of 54 00:03:36,556 --> 00:03:39,810 course, because I'm losing the light from the red star. 55 00:03:39,810 --> 00:03:44,880 And so, just looking at the depth of the dip that is being created, 56 00:03:44,880 --> 00:03:50,597 when the difference between full intensity and intensity during the full 57 00:03:50,597 --> 00:03:55,615 eclipse of the red star by the blue star, this difference is precisely a 58 00:03:55,615 --> 00:03:59,136 measurement of b2. And, of course, if I know the brightness 59 00:03:59,136 --> 00:04:03,784 of star number two, then by subtraction, I know the brightness of star number one. 60 00:04:03,784 --> 00:04:08,607 So, I have a measurement independently of the brightnesses of the two stars just by 61 00:04:08,607 --> 00:04:11,280 looking at the light curve. Step number one, 62 00:04:11,280 --> 00:04:16,414 I'm going to have to erase this because I'm going to annotate this diagram very 63 00:04:16,414 --> 00:04:19,860 heavily. another thing we note is that, at this 64 00:04:19,860 --> 00:04:23,377 point, when they're about to eclipse each other 65 00:04:23,377 --> 00:04:28,301 the red star, star number 2, is moving precisely to our left with a speed, v2, 66 00:04:28,301 --> 00:04:31,958 which presumably may be measured by a Doppler shift. 67 00:04:31,958 --> 00:04:37,022 And at the same time star number one is moving to the right at its speed, v1. 68 00:04:37,022 --> 00:04:40,610 And so, the speed of relative motion is just v1 + v2. 69 00:04:40,610 --> 00:04:45,776 And this tells us something because what we are seeing is, over here at this time, 70 00:04:45,776 --> 00:04:49,540 we are seeing the beginning of the eclipse right over here. 71 00:04:49,540 --> 00:04:55,491 And, of course, if we let time pass, we'll see that as star number two is 72 00:04:55,491 --> 00:05:01,944 increasingly eclipsed by start number one, the light curve yet deepens, and it 73 00:05:01,944 --> 00:05:08,734 reaches it's maximum depth at this time right over here, and if we call this time 74 00:05:08,734 --> 00:05:12,530 interval, t2, then we can describe what happened during 75 00:05:12,530 --> 00:05:16,449 this time interval. What happened during this time interval 76 00:05:16,449 --> 00:05:21,830 is that the combined motion of the two planets carried them through the diameter 77 00:05:21,830 --> 00:05:27,543 of star two from the point when star two was just touching the outside of star one 78 00:05:27,543 --> 00:05:31,861 with its left edge until it just disappeared around, behind it. 79 00:05:31,861 --> 00:05:35,951 So, v1v2. + v2 * t2 is twice R2. If we measure the 80 00:05:35,951 --> 00:05:41,801 speeds using Doppler shift we, now have an independent measurement of the radius 81 00:05:41,801 --> 00:05:47,505 of star number two and, of course, if I let the animation run a little bit more 82 00:05:47,505 --> 00:05:50,890 then we can see that when I get, 83 00:05:50,890 --> 00:05:57,072 I when the red star reemerges, I now have a measurement of the combined 84 00:05:57,072 --> 00:06:02,620 intensity again, and about this point, I get maximal blue shift on the red star's 85 00:06:02,620 --> 00:06:06,707 spectral lines, maximum red shift and the blue star's spectral lines. 86 00:06:06,707 --> 00:06:11,526 And when the next eclipse starts, first of all, the directions of the speeds are 87 00:06:11,526 --> 00:06:14,454 of course reversed. But I'll leave them there. 88 00:06:14,454 --> 00:06:19,151 Please do not get confused by this. But now I can measure a different time 89 00:06:19,151 --> 00:06:21,774 interval. Of course, the length of this time 90 00:06:21,774 --> 00:06:25,678 interval during which the dip develops, is again the same length. 91 00:06:25,678 --> 00:06:30,619 It's the time that it takes the red star to move relative to the blue star twice 92 00:06:30,619 --> 00:06:34,020 the red diameter. But if instead, I start my clock now and 93 00:06:34,020 --> 00:06:39,980 measure the time it takes until this, until the following happens, 94 00:06:39,980 --> 00:06:45,941 until the eclipse begins to, to be decreasing until this point. 95 00:06:45,941 --> 00:06:53,041 This time interval over here is the time that it took the relative motion of the 96 00:06:53,041 --> 00:06:58,738 two stars to carry them through the diameter of the blue star. 97 00:06:58,738 --> 00:07:03,394 So, v1 + v2 times this time interval, t1, is just twice R1. 98 00:07:03,394 --> 00:07:07,980 So, we have a measurement of the radii of the two stars and at least their ratio, 99 00:07:07,980 --> 00:07:10,502 if we don't have Doppler shift measurements, 100 00:07:10,502 --> 00:07:14,400 if we do have Doppler shift measurements, we can get the exact radii. 101 00:07:14,400 --> 00:07:17,037 But certainly, we can get a ratio of the radii. 102 00:07:17,037 --> 00:07:20,305 We found the brightnesses of the two stars independently. 103 00:07:20,305 --> 00:07:23,630 We can also get an independent measure of the temperature. 104 00:07:23,630 --> 00:07:26,934 Notice that the dip created when star number one, 105 00:07:26,934 --> 00:07:31,920 when the blue star is occluded, is deeper than the dip created when the red star is 106 00:07:31,920 --> 00:07:34,923 occluded. This has to do simply with the fact that 107 00:07:34,923 --> 00:07:38,708 the blue star is harder. Notice we don't get a total eclipse of 108 00:07:38,708 --> 00:07:41,952 the blue star. It's not measuring its total luminosity. 109 00:07:41,952 --> 00:07:46,818 What I'm missing here is a disk out of the surface of the blue star equal to the 110 00:07:46,818 --> 00:07:50,714 size of the red star. what this means of course, is that the 111 00:07:50,714 --> 00:07:55,384 reason this deep is, this dip is deeper is because this corresponds to loosing a 112 00:07:55,384 --> 00:08:00,233 disk with a temperature t1 to the fourth and this corresponds to loosing the disk 113 00:08:00,233 --> 00:08:04,903 with the temperature t2 to the fourth, the ratios of the depths of the two dips 114 00:08:04,903 --> 00:08:08,915 tell me about the ratio of the temperature of the two stars to the 115 00:08:08,915 --> 00:08:11,583 fourth. And you can see that as I adjust the 116 00:08:11,583 --> 00:08:16,873 relative temperatures, I will find that this will the relative 117 00:08:16,873 --> 00:08:22,775 depth of the two dips will change when I set the temperatures equal the dips are 118 00:08:22,775 --> 00:08:26,923 equal in depth as long as the smaller star is 119 00:08:26,923 --> 00:08:30,405 cooler, the dip went, it is obscured as less deep. 120 00:08:30,405 --> 00:08:36,084 And if the, I make the smaller star hotter, then the arrangement is reversed. 121 00:08:36,084 --> 00:08:39,450 So, we can read lots of things off of this figure. 122 00:08:39,450 --> 00:08:44,809 I hope you'll play with the simulator. I should show you two other things if I 123 00:08:44,809 --> 00:08:49,549 can erase all my notations. One is that if the orbit happens to be 124 00:08:49,549 --> 00:08:52,504 elliptical, I can give it some eccentricity. 125 00:08:52,504 --> 00:08:57,863 And what happens then is that depending on the orientation of the ellipse, the 126 00:08:57,863 --> 00:09:02,260 two dips might not be symmetrically positioned within the 127 00:09:02,260 --> 00:09:05,831 orbit, within the period, because it might take the two stars. 128 00:09:05,831 --> 00:09:10,118 what we see is that this in between this eclipse and this eclipse 129 00:09:10,118 --> 00:09:13,392 falls perihelion. When the two stars are close together, 130 00:09:13,392 --> 00:09:16,666 they go fast. In between this eclipse and the following 131 00:09:16,666 --> 00:09:21,310 eclipse this way, lies aphelion that takes longer so we can get the a measure, 132 00:09:21,310 --> 00:09:26,072 but the asymmetry gives us a measure of the electricity, the eccentricity of the 133 00:09:26,072 --> 00:09:30,223 orbit. And furthermore if we have stars whose 134 00:09:30,223 --> 00:09:36,966 radii are comparable then, of course, when the radii become close we never get 135 00:09:36,966 --> 00:09:43,709 this flat region on the bottom because total eclipse occurs only for an instant. 136 00:09:43,709 --> 00:09:49,070 But even more fun if, as the stars 137 00:09:49,070 --> 00:09:55,714 combined radii become closer and closer to the radius, we get what we call a 138 00:09:55,714 --> 00:09:58,550 contact binary. A contact binary, 139 00:09:58,550 --> 00:10:02,004 such that the stars are essentially touching each other. 140 00:10:02,004 --> 00:10:06,904 That's the situation I have here. What a contact binary does is it causes a 141 00:10:06,904 --> 00:10:11,364 flat region where we get constant full luminosity to disappear because 142 00:10:11,364 --> 00:10:16,327 essentially the stars will eclipse each other and then immediately start the 143 00:10:16,327 --> 00:10:21,101 other eclipse and we never get this constant period of a full illumination. 144 00:10:21,101 --> 00:10:24,744 Lots and lots of information to be read off a light curve, 145 00:10:24,744 --> 00:10:29,016 the same applied to planets. I promised that, that I would show it to 146 00:10:29,016 --> 00:10:32,720 you at some point. I hope this has been instructive. 147 00:10:32,720 --> 00:10:35,865 So, eclipsing binaries are a rich source of information. 148 00:10:35,865 --> 00:10:40,040 In general, we can learn a lot from binaries. The most famous example of 149 00:10:40,040 --> 00:10:46,307 binaries, eclipsing binaries, of course, as we call it Algol, or perhaps Alghoul 150 00:10:46,307 --> 00:10:51,772 which is the ghoulish looking star in Perseus that dims for a few hours every 151 00:10:51,772 --> 00:10:54,879 few days because it turns out, it's an eclipsing binary. 152 00:10:54,879 --> 00:10:58,899 As we'll see later, there is yet more information that we can extract from 153 00:10:58,899 --> 00:11:01,833 binaries. Binary stars as I like to call them are a 154 00:11:01,833 --> 00:11:05,202 star with a built-in probe, and you can learn a lot about both 155 00:11:05,202 --> 00:11:09,114 members of the pair because there's something for them to interact with. 156 00:11:09,114 --> 00:11:12,972 So, we'll come to binary stars again and again as the class, class goes on. 157 00:11:12,972 --> 00:11:15,580 For now, let's observe that our friend, Alphecca. 158 00:11:15,580 --> 00:11:20,178 Alpha Coronae Borealis happens to be a double line eclipsing binary with a 159 00:11:20,178 --> 00:11:24,657 period of about 17.4 days and we can do the Doppler measurements. 160 00:11:24,657 --> 00:11:29,674 again, I quote in the credits section, the paper from which I lifted these data, 161 00:11:29,674 --> 00:11:32,362 and it would be fun to go and read the paper. 162 00:11:32,362 --> 00:11:35,109 And you can see you can learn a lot from that. 163 00:11:35,109 --> 00:11:39,170 Doppler measurements give me the radial velocities of the two stars. 164 00:11:39,170 --> 00:11:41,917 And I give here the speeds as we measure them. 165 00:11:41,917 --> 00:11:46,576 I plug into our formula for the masses of the two members of the binary. 166 00:11:46,576 --> 00:11:51,850 Of course, this was our formula for M1 replacing everyone here with a two, I get 167 00:11:51,850 --> 00:11:54,760 the formula for M2. And I predict that 168 00:11:54,760 --> 00:12:00,065 Alphecca is a pair of stars, one with a mass three times the solar mass, one with 169 00:12:00,065 --> 00:12:04,640 a mass approximately a solar mass. I can compare this with my spectro 170 00:12:04,640 --> 00:12:09,746 measurements because it's a double line binary, I see the spectra of both stars. 171 00:12:09,746 --> 00:12:13,080 Indeed we talked about Alpha Coronae Borealis A 172 00:12:13,080 --> 00:12:15,690 last time. We said, there was an A-type star. 173 00:12:15,690 --> 00:12:20,607 that mass of three solar masses and the radius, we said then, about three solar 174 00:12:20,607 --> 00:12:25,584 radii, is about right for A star. Alpha Coronae Borealis B, turns out to be G5V, 175 00:12:25,584 --> 00:12:28,558 which makes it a little bit cooler than the star. 176 00:12:28,558 --> 00:12:33,293 It's radius should, than the sun I mean, it's radius should perhaps be a little 177 00:12:33,293 --> 00:12:38,574 smaller, but another main sequence star. you can read about the life curve fits in 178 00:12:38,574 --> 00:12:42,459 the paper that I quote. You can also see how life is never quite 179 00:12:42,459 --> 00:12:46,887 as simple as what I presented in this idealized scenario in particular, for 180 00:12:46,887 --> 00:12:51,350 example, the eccentricity of the orbit is very high and so our circular orbit 181 00:12:51,350 --> 00:12:55,019 calculation is bound to be a little bit inaccurate. In fact, from the paper 182 00:12:55,019 --> 00:13:00,361 including both the eclipse data and the radial velocity data, the best estimate 183 00:13:00,361 --> 00:13:06,362 they find is about 2.6 solar masses for A and 0.9 solar masses for Alpha Coronae 184 00:13:06,362 --> 00:13:10,160 Borealis B. A radius of about three hm, 185 00:13:10,160 --> 00:13:14,403 we got that right because temperature and radius don't care for radial 186 00:13:14,403 --> 00:13:17,232 measurements. so indeed, the radius of, of a Alpha 187 00:13:17,232 --> 00:13:20,592 Coronae Borealis A is three solar radii and of secondary 0.9. 188 00:13:20,592 --> 00:13:25,367 The temperature we had right for the primary the secondary is about the solar 189 00:13:25,367 --> 00:13:28,550 temperature and the [UNKNOWN] ratio of luminosities. 190 00:13:28,550 --> 00:13:33,501 the luminosity in units of the solar luminosity is what you would expect from 191 00:13:33,501 --> 00:13:36,442 their spectral type. The beautiful thing is, yeah, these 192 00:13:36,442 --> 00:13:40,225 people did some very careful numerics, but we understand and can actually 193 00:13:40,225 --> 00:13:44,319 emulate at least the physical principles, if not all the details, that went into 194 00:13:44,319 --> 00:13:47,170 this calculation. I hope you appreciate how much you've 195 00:13:47,170 --> 00:13:47,585 learned.