1 00:00:00,000 --> 00:00:02,973 Two, what? If we can see both members of a 2 00:00:02,973 --> 00:00:06,972 binary pair, and if we can resolve them, and if they are near enough, that we can 3 00:00:06,972 --> 00:00:10,769 actually distinguish their motion, then we can use astrometry to measure the 4 00:00:10,769 --> 00:00:13,250 period and the size of the orbit and learn a lot. 5 00:00:13,250 --> 00:00:17,152 We're interested in binary stars. Most binary stars, like most everything, 6 00:00:17,152 --> 00:00:21,542 is not going to be near enough for us to measure their motion, and often, we won't 7 00:00:21,542 --> 00:00:25,824 be able to even resolve the two stars. We'll see the light of the two as one 8 00:00:25,824 --> 00:00:30,322 star just as our eye unaided, does not distinguish the two members of Albireo. 9 00:00:30,322 --> 00:00:34,062 We see one star, even with a big telescope, the stars are far enough or 10 00:00:34,062 --> 00:00:36,610 one of them is dim enough, we'll only see one star. 11 00:00:36,610 --> 00:00:40,910 How do we know by looking at a star that it might actually be a binary? 12 00:00:40,910 --> 00:00:45,189 Well, one way is to measure its spectrum, that's what we do and what will we 13 00:00:45,189 --> 00:00:47,658 recognize when we're looking at the spectrum? 14 00:00:47,658 --> 00:00:51,388 What about the spectrum will tell us that we're looking at a binary? 15 00:00:51,388 --> 00:00:56,051 Well, if we have a binary system that is not oriented such that its orbital motion 16 00:00:56,051 --> 00:00:59,727 of the two members is completely tangential. If there's some radial 17 00:00:59,727 --> 00:01:04,061 component, then as we saw, that means that some of the time the stars will be 18 00:01:04,061 --> 00:01:08,450 moving towards us and sometimes away from us. And furthermore, when one is moving 19 00:01:08,450 --> 00:01:10,700 towards us, the other will be moving away. 20 00:01:10,700 --> 00:01:13,529 What this will give us is a periodic Doppler shift. 21 00:01:13,529 --> 00:01:17,690 We'll see that if you look at the spectrum, there will be some of the lines 22 00:01:17,690 --> 00:01:21,130 that will be blue shifted and some others will be red shifted. 23 00:01:21,130 --> 00:01:25,181 And then, the rows will exchange, so we'll find a periodically shifting 24 00:01:25,181 --> 00:01:29,092 Doppler shift, a really characterically look simple version of 25 00:01:29,092 --> 00:01:34,485 this is two stars with one spectral line. And what we see here is two copies of 26 00:01:34,485 --> 00:01:39,221 this spectral line, moving greatly exaggerated amount to the right and to 27 00:01:39,221 --> 00:01:43,663 the left, to the red and to the blue. And so, when one star is moving towards 28 00:01:43,663 --> 00:01:47,652 us the other is moving away from us, we can use the maximal amount of the 29 00:01:47,652 --> 00:01:51,695 Doppler shift to estimate the speed with which those two stars are moving. 30 00:01:51,695 --> 00:01:56,066 the example below shows the same thing, again, caricatured one spectra line. 31 00:01:56,066 --> 00:01:59,890 But here we see that one of the stars is moving faster than the other. 32 00:01:59,890 --> 00:02:04,261 One of those lines shifts more than the other line does that tells us that the 33 00:02:04,261 --> 00:02:07,540 two stars are moving, orbiting each other, but one moves more. 34 00:02:07,540 --> 00:02:10,548 that means that, that star is lighter than the other stars. 35 00:02:10,548 --> 00:02:15,010 And in many cases, one of those stars is so dim that we only see one, then this is 36 00:02:15,010 --> 00:02:18,070 very much like discovering a planet through Doppler shifts. 37 00:02:18,070 --> 00:02:22,168 You see a star orbiting something, you can try to conclude what it is that, that 38 00:02:22,168 --> 00:02:24,970 star is orbiting if you know something about the star. 39 00:02:24,970 --> 00:02:28,290 Of, obviously, the more you observe, the more information you can get. 40 00:02:28,290 --> 00:02:32,754 So we learn by measuring the spectrum. We look at a star, we notice that it has 41 00:02:32,754 --> 00:02:37,219 sets of spectral lines that move are Doppler shifted in oposite directions and 42 00:02:37,219 --> 00:02:40,077 they move back and forth. Well, what can we measure? 43 00:02:40,077 --> 00:02:44,131 By measuring how long it takes for the lines to move back and forth, we 44 00:02:44,131 --> 00:02:48,755 certainly measure the period with which the two stars that we're actually seeing 45 00:02:48,755 --> 00:02:52,295 as one are orbiting each other. We can use the Doppler formula, 46 00:02:52,295 --> 00:02:56,577 we measure the maximal shift of some spectral line and we measure if it's a 47 00:02:56,577 --> 00:03:00,459 double line binary star, we can measure the speeds of each of the 48 00:03:00,459 --> 00:03:03,542 individual stars. Now, you may remember that we're only 49 00:03:03,542 --> 00:03:07,710 measuring the radial component of the speed, not the tangential component. 50 00:03:07,710 --> 00:03:11,878 So I'm going to assume that the system is aligned if you sort of imagine 51 00:03:11,878 --> 00:03:16,041 [SOUND] a set of axises like these. I'm going to put the, the orbit in the 52 00:03:16,041 --> 00:03:20,983 horizontal plane in this picture, so that the maximum value of radial velocity 53 00:03:20,983 --> 00:03:26,115 towards us or away from us is equal to the actual speed with which the stars are 54 00:03:26,115 --> 00:03:29,410 orbiting each other if the orbit is tilted like this, 55 00:03:29,410 --> 00:03:33,748 then this will be false. We'll talk about how we deal with that 56 00:03:33,748 --> 00:03:36,915 later. For now, let us assume for simplicity, we 57 00:03:36,915 --> 00:03:41,433 have circular orbits and no tilt and we'll have to do some algebra, 58 00:03:41,433 --> 00:03:43,730 so bear with me for a bit. So what do we do? 59 00:03:43,730 --> 00:03:47,416 We've measured the two speeds. If I know how fast something is moving 60 00:03:47,416 --> 00:03:50,355 and I know the period, how long it takes to go around the 61 00:03:50,355 --> 00:03:52,866 circle, I certainly can figure it out, the radii 62 00:03:52,866 --> 00:03:55,377 of the two circles in which the two stars move, 63 00:03:55,377 --> 00:03:58,957 this already tells me something. Remember that the radii of the two 64 00:03:58,957 --> 00:04:02,750 circles about which each of the stars move are, in fact, since these are 65 00:04:02,750 --> 00:04:06,543 circles about their center, common center of mass are related by this equation. 66 00:04:06,543 --> 00:04:10,871 so if I measure the two velocities, I already have the ratio of the two masses. 67 00:04:10,871 --> 00:04:17,496 I can write that M2 = * V1 / V2 as we predict it from guessing, by just by 68 00:04:17,496 --> 00:04:21,930 looking at the spectrum, the lighter star move faster than the heavier star. 69 00:04:21,930 --> 00:04:24,000 So far, so good. What else can we do? 70 00:04:24,000 --> 00:04:27,075 Lets collect some more information. In terms of that. 71 00:04:27,075 --> 00:04:31,273 we can write the total mass of the system, because remember that's what 72 00:04:31,273 --> 00:04:34,525 enters into Newton's expression for Keplers law M1 + M2. 73 00:04:34,525 --> 00:04:39,551 I write the expression I had for M2. I'll make a common denominator over here and 74 00:04:39,551 --> 00:04:44,104 add this relation between M1 and M. So if you tell me M, I can figure out the 75 00:04:44,104 --> 00:04:47,382 mass of the individual star. That's what I've done so far. 76 00:04:47,382 --> 00:04:51,971 Furthermore, the other parameter that enters into Kepler's formula is the total 77 00:04:51,971 --> 00:04:55,720 distance between the two stars. Remember, that's just the sum of the 78 00:04:55,720 --> 00:04:59,805 radii of the two circle,s because of the way they orbit, they're always on the 79 00:04:59,805 --> 00:05:02,212 opposite sides of their respective circles. 80 00:05:02,212 --> 00:05:06,688 I plug in the expression that I had for the radii of the two circles and I find 81 00:05:06,688 --> 00:05:10,829 that the radii, total distance between the stars is related to the period and 82 00:05:10,829 --> 00:05:13,291 the velocities or the speeds that I measure. 83 00:05:13,291 --> 00:05:17,153 In this way, remember, periods and V, period and V's is what we measure, 84 00:05:17,153 --> 00:05:19,880 this allows me to compute r. So, 85 00:05:19,880 --> 00:05:24,439 now, we're going to take all this information that we've accumulated here 86 00:05:24,439 --> 00:05:27,375 and plug it into Newton's form of Kepler's Law. 87 00:05:27,375 --> 00:05:31,622 Remember that P, R and the total mass are related by this expression. 88 00:05:31,622 --> 00:05:36,556 I remind you, on the previous slide, we found this relation between R and the 89 00:05:36,556 --> 00:05:41,622 observed quantities, and this relation between M, the observed quantities, and 90 00:05:41,622 --> 00:05:44,842 M1. Remember, the mass of one of the stars is 91 00:05:44,842 --> 00:05:48,267 what I am after. So, what I do, first, is I plug this 92 00:05:48,267 --> 00:05:53,611 expression for R into that expression. I solve this equation for M, multiplying 93 00:05:53,611 --> 00:05:58,133 by M dividing P^2. And then, I, for this R^3, I plug the 94 00:05:58,133 --> 00:06:03,614 cube of this expression and then I see some handy dandy cancellations going on 95 00:06:03,614 --> 00:06:06,971 here. pi is cancel all over the place and I get 96 00:06:06,971 --> 00:06:12,920 left with one power of P up top. And I can write an equation that says, that M 97 00:06:12,920 --> 00:06:23,996 is P / 2pi G * (V1 + V2)^3. But I also have this other expression for 98 00:06:23,996 --> 00:06:29,935 M in terms of my friend M1 and that's (V1 + V2) M1 / V2. 99 00:06:29,935 --> 00:06:38,149 And so, I see that one power of V1 + V2 cancels and multiplying both sides by V2, 100 00:06:38,149 --> 00:06:42,900 I finally have an equation for M1 in terms of P, 101 00:06:42,900 --> 00:06:46,286 the speeds, get a measure and Newton's constant. 102 00:06:46,286 --> 00:06:51,733 I am done by measuring this Doppler shift, which gave me the speeds, and the 103 00:06:51,733 --> 00:06:55,709 period, I can actually measure the mass of of a star. 104 00:06:55,709 --> 00:07:01,892 Of course, it's easiest to assess this by scaling it relative to we know that for 105 00:07:01,892 --> 00:07:07,708 the earth and the sun, I have that the mass of the sun is equal to the speed of 106 00:07:07,708 --> 00:07:12,455 earth times one year times nah, the speed of the sun is negligible. 107 00:07:12,455 --> 00:07:16,975 Let's write the speed of the earth squared divided by 2pi G. 108 00:07:16,975 --> 00:07:22,500 And then dividing one expression by the other. I get my happy scaling formula 109 00:07:22,500 --> 00:07:28,241 that says, since I know the speed of earth and it's orbit is 29.78 kilometers 110 00:07:28,241 --> 00:07:31,972 per second, I can get the mass of each star in units 111 00:07:31,972 --> 00:07:35,560 of the solar mass this way and so, we can find masses. 112 00:07:35,560 --> 00:07:40,612 spectroscopic binaries or other binaries stars are a way to measure masses, 113 00:07:40,612 --> 00:07:44,792 this is why it's so useful. The two caveats, of course, if the orbits 114 00:07:44,792 --> 00:07:48,659 tilt, we're only measuring the radial component of the speed. 115 00:07:48,659 --> 00:07:53,338 How would we know if the orbit is tilted? Well, if we're lucky, we would have 116 00:07:53,338 --> 00:07:57,580 astrometric measurements and we can make both as in the alpha 117 00:07:57,580 --> 00:08:00,702 Centauri version. We can make measurements both in the 118 00:08:00,702 --> 00:08:03,535 tangential direction and in the radial direction. 119 00:08:03,535 --> 00:08:07,929 And otherwise, we have to infer what really we're measuring is one component 120 00:08:07,929 --> 00:08:11,283 of the velocity, so the inclination shows up as an overall 121 00:08:11,283 --> 00:08:14,405 factor here. and often, we only see one of the stars 122 00:08:14,405 --> 00:08:17,296 and then we're only measuring V1. We don't know v2, 123 00:08:17,296 --> 00:08:21,633 we can't get complete information, but we can learn a lot of things from 124 00:08:21,633 --> 00:08:24,293 this. Binary stars are cool and useful objects.