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Do my capstone church fort worth tx excel 2013 chapter 2 capstone exercise for money examples of goals for high school students okay so we can group different stars in three major groups based on their nuts the first group are so called low mass stars their mass is arranged between the threshold of 8% of the solar mass and about 0.5 solar mass so their masses range from 8% of the solar mass to about 1/2 of the solar mass okay so this here is the mass of the star and as we discussed before this denotes this symbol denotes solar mass it's about 10 to the 20 kilograms of or 300,000 times bigger than the mass of the earth so these stars produce a energy via nuclear fusion of hydrogen into helium using the proton-proton chain those of you who took astronomy 1 po1 will remember that process which we discussed when we talked about the way in which the Sun produces the energy so just to go through it quickly basically there are several steps in this proton-proton chain first the two protons fuse into the nucleus of deuterium the heavy hydrogen and basically in the process one of these two protons converts to a neutron and since the electric charge is conserved the a particle called positron which is identical in all properties to electron except that it has opposite charge instead of negative it has a positive charge also there are these ghostly particles neutrinos that have essentially zero mass very tiny mass not zero they are produced in the process and this is the slowest step in the proton-proton chain because one has to the odds of a proton one of these protons transforming into Neutron while they are close enough so that the strong nuclear force can bind them are very small and on average this reaction takes place once every 14 billion years which is the current age of the universe so you can say well how come this can happen it's so unlikely well the thing is that in ten to the twenty kilograms three-quarters of which is hydrogen there are so many nuclear hydrogen that there is enough of them so that even with that low probability these things happen all the time okay so once you have a deuterium then it fuses in the next step with another hydrogen forming the nucleus of helium-3 which has two protons just like helium four but only one Neutron and hence there are three nucleus and that's helium-3 and then it fuses that helium three fuses with another nucleus of hydrogen to form helium four in the process two hydrogen's are released back so that they can participate in the fusion reaction so look what has happened we have 1 2 3 4 plus 2 here plus 2 hydrogen coming in what is coming out is the nucleus of helium and the two protons are released so the net result of this proton-proton chain so 6 minus 2 which is 4 protons are fused into the nucleus of helium-4 with the release of energy okay so that is very quick summary of the proton-proton chain those of you who took astronomy one po1 will remember it those of you who have not taken the course and kind of are slightly puzzled as to what is going on here I advise you to go to the textbook look up in the index proton-proton chain and read more about it so why is the energy released well it turns out that the end product the nucleus of helium 4 has mass that is slightly smaller than the mass of 4 protons that went to it a make up right so and that difference in mass has been converted into energy via famous Einstein relation e equals to MC squared right so if I denote this mass difference with M and that is four times the mass of proton which is H one minus mass of helium four so that difference in mass has been converted into energy and the relationship is is that energy released turns out to be exactly equal to this mass difference times the speed of light in vacuum squared now because they don't have much mass they have enough mass to start fusing hydrogen into helium and they become the main sequence stars because of low mass their luminosity is low recall the edingtons a mask luminosity relationship for main-sequence stars the luminosity is within a constant given by the mass of the star to the power 3.5 right so the smaller the mass the smaller the luminosity and vice versa the larger the mass the larger the luminosity of the star and they end up at the bottom of the main sequence right where we have those are dim and cool stars right so if I try to reproduce here the HR diagram oops so temperature runs this way and luminosity runs this way and the main sequence is represented by this group of stars then these low mass stars will occupy the bottom because they have low luminosity and low temperature what are what we call red dwarfs right red because of their low surface temperature and therefore they emit most of their energy at the wavelengths corresponding to the red light and dwarfs because they are small in size if you recall the Stefan Boltzmann's law that relates the star's luminosity we have some universal constant here to its surface area which is proportional to the radius squared of the star and the temperature to the fourth power here because both the surface temperature is low and the radius is small they are small math objects they will result in a low luminosity now these red works could actually be the most numerous stars in the universe the difficulty that we have of course is that because of their luminosity they are hard to see right if I have a light bulb which emits a small amount of power if it is a sufficient distance from me I can't see it right it's not intrinsically bright enough there's an emit enough energy to reach me here some distance from it okay so but the you know if you look at this process statistically it is to be expected that most of the stars in the universe are these red dwarfs is just that we can't see them okay those of you who took us from a one po1 will remember one of the fundamental relations in astronomy that is being used all the time to measure the luminosity of a star provided that we know its energy output oh sorry provided we know its distance from us or it can be used the other way around to determine its distance if we somehow are able to determine its luminosity and that relationship is that the brightness which is the amount of energy per unit time and per unit area that reaches ours here on earth per unit area and the location of the observer okay and it's related to the luminosity of the star which is this first thing here which is the amount of energy emitted by the star per unit time okay so this what I've underlined with - line is basically the definition for luminosity okay so this is luminosity and I have to divide this by the surface area of a gigantic sphere whose radius is equal to the distance between us and the start and that is 4pi distance to the star squared okay so if the luminosity of the object is low like it is for a red dwarf stars then in particular if they are at great distance from us the their brightness is rather low and we cannot detect them and in fact it turns out that the star that is closest to us so called a Proxima Centauri which is a member of the triple star system called Alpha Centauri is a red red dwarf it is closest to us its distance from us is about four point two four light-years but in spite of the fact that it's closest plus that for that star the D is the smallest number its luminosity is so low that we can't see it with a naked eye you can't see a Proxima Centauri with the naked eye so it takes about four and a quarter years for its light to reach us here from the distance that's what the yardstick of a light here means right takes light that is emitted by Proxima Centauri 4.2 four years to reach us here and as I emphasized in astronomy one po1 the greater the distance the longer is the look-back time right when you see a star you see not how it looks now but how it looked years ago depending on the time required for its light to just here on earth chamberlain msn capstone Fordham University.