the next property of stars that we're going to talk about our stellar sizes and you might imagine this is one of the most uncertain and most difficult properties to ascertain just because of the fact that stars are so far away even though they're humongous it's so far away and they take up basically no angular size all the stars for the most part look like points of light dots in the sky and therefore even our best telescopes can't really get any angular size to them or show an angle to use trigonometry to calculate their radii or their diameters for only the very nearest and largest stars like Betelgeuse in this slide that you see here which is the shoulder star of Orion it's only about what 600 light-years away and it's a star which is even bigger than the orbit of Mars around our Sun so it's an enormous star and galactically speaking just a stone's throw away very close to us so we're able to actually deduce some angular size to it we call this technique speckle interferometry nurse a fancy word and it just means that Ray of attacks DC multiple pixels on our detectors being taken up by the the size of this object and it's image that you see over here in frame a is truly a snapshot in visible light and this would be fluctuating all the time very quickly in real time Epping and flowing as the star appears to dynamically move due to turbulence due to all sorts of other factors and in a detector so this is just a snapshot of time but it looks so static and stationary like this so you know okay what about the other you know zillion stars out there that are not big and close enough to us to actually image an angular size for well that is the question of this topic we know that for most stars the vast majority you know that cannot be imaged directly that says they're right there that you can have to do some calculations educated guessing of a couple of properties and then of course using physics using mathematical formula to determine stellar radius we know this relationship in in in you know pretty obvious terms that luminosity of a star its total power is proportional to its radius raised to the second power meaning if you were to double the radius of a star well 2 squared is 4 and become 4 times as powerful just by doubling its size or the luminosity the power of a star is proportional to its temperature surface temperature raised to the fourth power if you double a star's surface temperature raise it to the power of 4 to the 4th power is 16 you'll increase its overall power by a factor of 16 so we know that these relationships and these proportionalities are true from physics so all right knowing that generalized relationship then we're gonna have giant stars which have radii between 10 and a hundred times the sun's diameter our radius and that's what it's gonna qualify you as a giant star somewhere between 10 to 100 times the size of our Sun and there's plenty of those they're not the majority but there's plenty of those in the sky dwarf stars we're gonna find out or by far the most common and they've read yeah that are smaller than our Sun so you know equal to or less than the Suns size do we call our Sun a dwarf star yeah you know only in a sense that if you get very literal about this definition do you say that the Sun is about the biggest type of a dwarf star there is but supergiant stars are even rarer still more rare and they have radii that are even more than other times the size of our Sun and even though they are very rare in their abundance of the top you know 40 stars top 40 brightest stars in our night sky I'm a good fourth of them are supergiant stars just because my goodness their luminosity is going to be enormous due to their both size as well as their proximity being fairly close to us so if you're a supergiant star in our neighborhood the galaxies more than likely we can see it without even telescopic aid just because of the enormous power output its luminosity so let's remind ourselves of a couple of things okay we've just discussed the fact that stars can take on many different sizes so let's get them out of my way me that is you're looking over here on the on the right we have you know one astronomical unit which is you know the Earth's Sun distance and we're gonna call that one well Earth's Sun distance but that's about 215 times the sun's radius yeah earth is about 215 times the sun's radius away from the sun's center it's bizarre way to put it but it's true Mars is about one and a half times as far away from the Sun as Earth is so it's about 325 solar radii away from the sun's center Antares which is the heart of the scorpion in the summertime sky is a huge red supergiant star with a radius its own radius of 500 solar radii which means that if you were to place in the middle of our solar system of course it would extend beyond the orbit of Mars wouldn't make it out to Jupiter Jupiter is about five times as far from the Sun as Earth is so you know five of these this is only about two of those but nevertheless you know it's in between about the Mars and Jupiter is distance in orbit away from the Sun down below here in the lower part of the graphic you see some stars that are very well-known to us their brightest stars in our sky because of their proximity but Aldebaran which is again the eyeball of Taurus the bulb is you know a good 40 solar radii in size we're gonna call it a giant star capella which is the force by two star in our sky up there now our idea very close to the zenith in the wintertime sky there's about 15 times the sun's radius Spica the bright blue-white star in Virgo the virgin is seven solar radii serious you know the brightest star in the sky in the winter time two solar radii so these are all bigger than us we're already above average you're gonna find out that that's true so you know these are really you know pretty spectacular stars above average stars and they're also nearby so they appear to be some of our brightest stars in the sky for a comparison purposes down here below you know you see the sun's yellow disc here one solar radii of course and it shows you some comparisons so Jupiter is about a tenth of the sun's radius not big enough to eat will be a star the planet were larger you have enough self gravity to crush itself to a high enough temperature to generate nuclear fusion it would have been a star if it were just bigger but a point one solar radii is not going to do it Barnard's star the star with the greatest proper emotion that we know of is only you know 0.2 solar radii so clearly you can be a star if you're that big and therefore it sounds like if Jupiter only about what twice it's its radius that it would have been in the star as well and it's basically true that's many more times the volume but it's about twice the solar radius that it currently is are just twice the radius and then now we have a Proxima Centauri down here which is a star and the Alpha Centauri system which is you know 8/100 of the sun's radius so it's not even as big as Jupiter's a size here to what you say but it got to be a star well it's true that there's all sorts of oddities about this system and Sirius B down here which is the small white dwarf dead by an area of Sirius the brightest star in our sky is only about 1/100 of the sun's radius so that's true of a dead white dwarf stars you'll see two other graphics are two of their links show up here on this slide and because of copyright concerns I can't just blast them for you here in my own personal video but I'll put links to them up on the canvas page so you can view them yourselves there they're pretty good and pretty short YouTube videos to kind of display just the overall range of stellar size so here's the real guts of it how did we get the stars radii to the best of our ability it's realizing the fact that well we had this lock on the stefan-boltzmann law way back in chapter 3 when it was the law that told us about surface flux from a blackbody this equation here flux equals Sigma times T to the fourth remember which you probably don't recall is the detail and that is that the flux of a black body and its surface is equal to this constant Sigma just a number times the surface temperature raised to the fourth power but we also have the inverse square law for light don't we that the overall power of a star its luminosity is equal to 4 PI d squared where D is the distance between you and the star of the source times the flux that you receive at your distance so the D and the F there are highly correlated yes at this distance D you're detecting this amount of flux well this again is a surface flux for a blackbody so you can't just plug this into the equation for N equals 4 PI d squared times F unless you do what to the D if this is the surface flux from the D that you're measuring the flux F becomes just the radius itself on the star well before you lose track of everything is there a way to get stellar surface temperature now yeah pretty easily right that's what stellar spectra tells us so actually coming up with the star's surface temperature is pretty trivial just by analyzing the light and if I just knew what a star's true power was what a luminosity was I could substitute this into my equation for F Sigma T to the fourth and then the D would become the R or the radius of the star so it sounds like problem solved I can always get a radius of a star if I just know its luminosity and its surface temperature well that's true but how many times and how easily can we determine a star's overall power or luminosity and the answer is yeah not that often but then again if I do measure a flux at earth so the distance between the flux detection and the source is the Earth's star distance then I do know its luminosity to you know there's going to be some uncertainty there but to a degree and then if I do know the star is true luminosity and it's gonna have some error to it maybe plus or minus 10% but then I can actually substitute in this replacement for F knowing its surface temperature as well from the spectrum and then I can solve for R and at R will be its radius but if the original value of L was maybe plus or minus 10% then you know your R is also going to be plus or minus 10% but my point is there really is one of the better methods we have for determining estimating keyword estimating but calculating ultimately the Stars radius so what this comes down to in the end is if we're you know comparing two stars making a ratio a lot of things if you measure luminosity radius and temperature and units of the sun's units we can write that the luminosity of a star meaning of the one solar luminosity is equal to its radius in solar units so if it had the sun's radius it'd just be 1 squared if it had a solar temperature you know 50 100 Kelvin set the surface that would be the T in this equation so if R and T are both 1 then R squared is 1 and T to the fourth is 1 and L is 1 as well so no one solar luminosity for a star of the sun's radius and a star of the sun's surface temperature but if you doubled the radius and you double the temperature of the star then okay it's got two solar radii don't forget to square it that's 4 and then it's temperature is doubled so that becomes to the fourth power or 16 times greater and 16 times for 64 very good a luminosity of a star is 64 times larger its power output if it is only double the size of our Sun and double its surface temperature so you're saying oh wow that's that's a lot of bang for your buck it is 64 times it was much power but think about it you're doubling the size of the star and you're therefore using its volume by a factor of four so it's got four times as much fuel but it's also burning through that fuel at sixty-four times the rate so mmm what's that gonna do that's gonna shorten the life isn't it yep the biggest stars you know they have more mass and they have more fuel burn through it it's such a higher rate that that's really gonna shorten their lifetime so Valles also be a conclusion after we're done with this chapter yeah the biggest stars have the shorter lives they're in an intense bright life but it's a very short life as a result [Music]