in the previous video we looked at wind's displacement law and understood the relationship between wavelength of electromagnetic radiation from all of these stars to the surface temperature why do we care about the temperature are we packing for a trip to the star of course not we want to know the temperature because using this really cool law from stefan bozeman we can then estimate the size of the star so this stefan boltzmann law will help us link intensity to temperature so let's write down the law a body when heated will emit electromagnetic radiation over a range of wavelengths we already know this from wind's displacement law okay with a total intensity that is proportional to the fourth power of its absolute temperature so what we have now is intensity i is proportional to the fourth power of its absolute temperature t to the power of four wow yes it's the power of four we can't make sense right the effect of temperature has is very very strong on the intensity of your electromagnetic radiation so right now if you want to uh get rid of the proportionality signing and have this in an equation we will introduce the stefan boltzmann constant sigma all right so let me box up the equation okay and there's a sigma our little cute little signal here this is our stefan i mean most of the time we just mentioned stefan but you know not to forget boxman constant so the stefan boltzmann constant will also be given to you in your formula sheet so don't forget to flip to the front page of your question paper if you need to use this constant so it will be 5.67 times 10 to the power negative 8 intensity is watt per meter squared okay and then this would be k to the power kelvin to the power of negative all right so but whenever we see the word intensity okay so let's let's think a little bit about intensity intensity is just a way to talk about power you know intensity is power per unit area but you know for stars we come up with a very fancy term is there a power per unit area we call it [Music] per unit okay same thing you guys it's the same thing all right so right now we're going to start from that and try to approximate is there a relationship between the size of the star which is the radius size of the star which i can sort of link to area right and the temperature so my hypothesis is a bigger size or hotter star right it's all about the chunk okay so the more thick the more big larger radius larger intensity but let's see that's my guess what's yours all right so we're going to start with intensity intensity is power per unit area but as usual we're going to take out the power have luminosity here then this will be per unit game okay so right now this intensity is actually not i right remember we have intensity which is actually radiant this is also equal to radian flux density f okay so i'm gonna stick to this one first okay i'm not gonna bother too much about f because we've used it previously my bad not radiant flux density but radiant flux intensively it's the same thing like guys can be f can be i all right but the good thing here is we know that i is equal to sigma t power 4 and luminosity is l area is the area of a sphere so if you look at my my beautiful star at the back of my head and let's say the star here has a radius of r okay and let's say if i take i take a chunk you know of this this curved you know this is a sphere right so i just slice off a little bit all right so this is my surface area you know a so for the whole surface area of a sphere okay so we're going to assume that the stars are spherical with the radius r so then the area will become 4 pi r squared so the surface area i'm gonna write that down for you surface area of the sphere is equal to four pi r square in this case we're gonna use capital r okay all right so just imagine we're looking at the whole race all right so i'm going to rearrange the equation a bit because uh we know like fraction so luminosity or power star power is equal to 4 pi sigma r squared t to the power of 4. so i put all the constants in front and then i arrange my variables which is the radius of the star whether i keep using small r neverland radius of the star and the temperature of the star to the power of four so this is the stefan boltzmann equation okay but it's not a new equation you guys okay it's just uh i take luminosity definition or radiant flux intensity definition and i have and i replace the radiant flux intensity or intensity by sigma t to the power of okay so from here just a few things to add on okay this one is your stefan boltzmann constant good old sigma was 5.67 times ten to the power of negative eight you want the negative two k negative four this r here is the radius of the star aka stella radii such a cool name stella radii the radius of the star all right and this t here would be the temperature it's more like the absolute temperature of the star which we can estimate using means displacement if needed okay so this is why this subtopic is called stellar radii we take wind's displacement law to give us a reliable estimate about the temperature and then we take stefan boltzmann law which tells us that the temperature raised to the power of 4 is directly proportional to its intensity okay then we put them together and we can use this equation together with the stefan boltzmann constant to find the radius of a star let's try an example that showcased these two equations so in this question 3 we have a black body probably a star plot twist has a maximum intensity of wavelength one four five zero nanometer and the radiation is 200 kelvin okay let's gather the information we have lambda max and we have the surface temperature okay so we already have this data now we are going to look at a star called beta grease beta juice and it has been measured to have a wavelength of 850 nanometer at peak intensity n has a luminosity okay so we have lambda max this and we have this associated luminosity to be this okay so you are asked to estimate the radius so of course what immediately could come to mind is your stefan boltzmann equation okay so i'm going to write that down first l is equal to 4 pi remember we gather all the constants first 4 pi sigma and then r squared the radius squared t to the power of 4. if you forget this don't worry this is the only equation okay this is not the one equation but this equation is given in your formula sheet so you just copy it yeah thanks cambridge cambridge got your back they gave you this equation very nice okay so we're gonna check it out so right now um i have l which is okay i'm looking for r all these are constant i'm good the only thing that is missing is temperature so if i have temperature i can just plug everything into the equation to help me find my value of r so let's find temperature first and of course we're going to use our one and only wayne's law okay let me write that down displacement law where your lambda is i mean right now i can just take lambda 1 divided by lambda 2 is equal to t2 over t1 because lambda is inversely proportional to temperature nice all right so we have one four five zero divided by um 850 you know if you're wondering how on earth do we measure this wavelength ah you stay tuned stay tuned maybe you can think about it you have already learned how to measure the wavelength previously but let's say we have a reliable reliable way to measure the wavelength of any electromagnetic radiation that we see what do we use what kind of phenomena do we use all right but regardless let's assume we can measure and we got the answer to the 850 or the measurement to be 850 nanometers you can't think guys okay anyway the temperature t2 is what we're looking for because this is our beetle geese all right and we have the temperature of the black body that will be two thousand okay we can now find t2 so from here i'm going to press my calculator t2 is 3 4 1 0 kelvin i'm just going to take 3 as f because you know the numbers like 1 4 5 0 is 3sf okay all right so now with t2 i can then use my stefan boltzmann change background color boltzmann you know whenever you are using something that is new it is useful for your brain to repeat because it helps with memory right so l will be equal to 4 pi sigma r squared t to the power of 4. let's substitute 3.1 times 10 to the power of 31. for pi sigma is 5.67 please check your data sheet constant power negative 8 r square we have temperature three four one zero to the power of four okay remember prefix are important so from here you can calculate the value of r so by my calculator this is 5.67 or 5.7 times 10 to the power of 11 meter it's a very big star you guys all right so that's it for this kind of question so sometimes you mean you will have to use wien's displacement law to figure out what the temperature is so we're going to look at a star and compare with another star then we can find the temperature and once we find the temperature and we can measure the luminosity which is essentially power then we can actually calculate the radius okay so let's put all of this together in a flow chart or a summary let's quickly summarize everything we have learned so far for this entire astrophysics chapter okay so that we can orientate ourselves right so remember when we started this chapter we say we look at the night sky but what are we looking at we are observers right and unfortunately most of us we can only gather data on planet earth because we're kind of like earthbound so this is the earth observer but what will be looking at okay so first thing we looked at was radiant flux intensity aka wah so bright okay i'm gonna let you so bright the stars okay so radiant flux intensity starlight star bright of the star you see tonight okay so radiant flux intensity so we can measure basically the amount of luminosity over 4 pi but this radiant flux intensity is not alone we could also look at standard candles right so in this uh chapter we actually recorded a little bit more because we sort of like need to understand what standard candles are okay so we provided a few examples okay may or may not be part of your syllabus but it doesn't matter it doesn't hurt to learn a bit more isn't it the world is a fascinating place okay so we look at the see fit variable the stars where the intensity decrease increase decrease increase with time time okay and then we also looked at the type 1 supernova where there is a sudden explosion there's a peak intensity and then it drops okay either way based on what standard candles are name also standard candle already right so it is something of known luminosity so from here we actually can figure out what the luminosity is and figuring out luminosity is really helpful because we can then link luminosity to this equation so that we can figure out what remember we were inquiring stuff so we can actually calculate or estimate how far away these stars are distance from startups okay so we can calculate and then for stella ready eye we looked at wow so bright but also so colorful the twinkling lights as color okay so when you think about color we can actually think about the wavelength and not any wavelength okay we are looking at once again this is a graph of intensity but now this is a graph of intensity against lambda so we are looking at the peak wavelength so the peak wavelength of different different hot and cold i mean hot they're all hot they're all stars what am i talking about okay so at the end of the day what we're measuring is the lambda maximum of the light from stars okay so once we have lambda maximum we can then use the equation of remember uh your lambda is inversely proportional to t so i'm gonna write that in different color for you lambda is inversely proportional to t this is your means displacement law means displacement and from this value of t i mean we can use inversely proportional or we can say lambda is equal to b the proportionality constant over t okay regard either way okay it doesn't matter we can bring these two together using our good friend stefan boltzmann okay so stefan boltzmann basically just ties it all together it says that the intensity is proportional to temperature to the power of four but intensity is actually radiant flux intensity so star we can put f okay and from here we actually have the equation of luminosity is equal to 4 pi sigma r squared t to the power of 4 noise so we actually have quite a few equations this one this one okay but looking guys we can actually find temperature plug it in here okay so you can see using winds displacement law we can just find the temperature this is radius of the star and from here we can actually estimate so number one we can estimate distance of from the stars coming from the stars to the observable and this r here is our stellar radii or let me move this a bit radius of our star size of the star okay and you can also probably tell that there is also a value of l here luminosity just like you can also find temperature okay so you can already see that questions can be different format okay they can pick and choose but at the end of the day when we look at the star what we are measuring is actually the em radiation what we are actually observing is the em radiation on one side we looked at the intensity and then we use our understanding of standard candles to actually measure the distance of these stars from us from the earthdawn observer and then we looked at the wavelength and using winds displacement law we know that wavelength is inversely proportional to temperature and just based off this we also know that because it's inversely proportional to temperature then if we want to find the size or the stellar radius size of the star we can use i mean we can just link this temperature to here okay this is your key all right so that's it in a nutshell all right so we managed to answer the two questions you know when we started the chapter how far away are these stars and how big are they in the next sub topic we are going to look at are the stars moving that will be the next one here okay so stay tuned and i'll see you in the next video bye