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Current time:0:00Total duration:11:40

Video transcript

in the last video we talked about how parallax is the apparent change in position of something based on your line of sight and if you experience parallax kind of in your everyday life if you look outside of the car window while it's moving you see that nearby objects seem to be moving faster than faraway objects so in the last video we measured the apparent displacement of the star at different points in the year relative to straight up but you could also measure it relative to things in the night sky at that same time of year that same time of day when think that don't appear to be moving and they won't appear to be moving because they're going to be way way way farther away than this star over here there might be other galaxies or maybe even other clusters of galaxies or who knows things that are not changing in position so that's another option and that's how another way to make sure that you're looking at the right part of the universe so you could measure relative to straight up if you know based on the time of year and the time of day that you're looking at the same direction of the universe or you could just find things in the universe that are way far back that their apparent position isn't changing so just to visualize this again I'll visualize it in a slightly different way let's let's let's say this is our let's say this is the night field of vision this is the night let me scroll to the right a little bit let's say our night field of vision looks like this and I'll do it in a dark color because it's at night so our night field of vision looks like this and let's say that this right over here this right over here is straight up this right here is if we're just looking straight up in the night sky and just to make the convention and the last in the last video I kind of changed our orientation a little bit but I'll reorient us in kind of a in kind of a traditional orientation so if we make this North this is south this will be West and then this will be east so when we're looking at the star in the summer what will what will be look like well first of all the Sun is just beginning to rise so if you can think about it this North this north direction we're looking at the Sun from a we're looking at the Earth from above so the North will be the top of this sphere over here and and so and the south will be the bottom of this view the other side of the sphere that we're not seeing the East will be this side this side of the sphere where the Sun is just beginning to rise where the Sun is just beginning to rise right over there so what will be the apparent position of this star well it's going to be towards the east it's going to be towards the direction that the Sun is rising so this angle this angle right over here is going to be right right over there so this will be the angle this will be this will be the angle theta so this is in the summer and what about the winter well in the winter too in order to straight up in order for straight up to be that same point in time or that same direction of the universe I should say then the Sun the Sun will just be setting we're rotating around that way so we're just going to be clap capturing the last glimpse the last glimpse of sunlight so in that situation the Sun is going to be setting so this is in the winter Sun our winter Sun I'll do in a slightly different color will be setting on the west and now the apparent direction of that star is going to be in the direction of the Sun again but it's going to be shifted away from Center so it's going to be - it is going to be to the right of Center I'm sorry to the left of center so it's going to be right over here it is going to be right over here and it's a little bit unintuitive the way I drew it just just in the last video well I won't make any judgement on whether the last one is easier to visualize or this one is over here I just wanted to make the convention so that North is up South is down but I just want to be clear over here the Sun is well the Sun always sets to the west so in the winter the Sun will be right over there this will be shifted from Center in the direction of the Sun so it'll be at an angle at an angle theta just like that in the winter now that's all review from the last video I just reoriented how we visualize it what I want to do in this video given that we can measure theta how can we figure out how far this star actually is so let's just think about a little bit before I even give you a theta value if we know theta then we know we know what this angle is right over here because this is a right angle we're going to know that this angle right here is 90 minus theta we also know we also know the distance from the Sun to the earth and let's say we're just going to approximate here it's one astronomical unit it changes a little bit over the course of the year but the mean distance is one astronomical unit so how we know that we know that angle we know side we know a side adjacent to the angle and what we're trying to do is figure out a side opposite to the angle this distance right here the distance from the Sun to the star and this is of course a right triangle and you can see it right here here is the hypotenuse so now we just have to break out some relatively basic trigonometry so if we know this angle what trig ratio deals with an adjacent side and an opposite side so let's let me write down my famous sohcahtoa I didn't come up with it so the famous sohcahtoa so Chi Toa sine is opposite over hypotenuse those aren't the two we care about cosine is adjacent over hypotenuse we don't know what the hypotenuse is and we don't care about it just yet but the tangent is the opposite over the adjacent opposite over the adjacent so if we take the tangent of the angle if we take the tangent of 90 minus theta if we take the tangent of 90 minus theta this is going to be equal to the distance to the star this distance right over here the distance to the star or the distance from the Sun to the star we can later figure out the distance from the earth to the star it's not going to be too different because the star is so far away but the distance from the Sun to the star divided by the adjacent side divided by one astronomical unit I'm assuming everything is in astronomical units so you can multiply both sides by one and you'll get the distance in astronomical units the distance is equal to the tangent of 90 minus theta not too bad so let's actually figure out what a distance would be based on some actual measurements let's say you were to measure some star measure this apparent this chain right here and let's say you got the total change in angle right over here from six months apart the biggest spread and you're making sure you're looking at a point in the universe relative to straight up you can do it other ways but this is really just simplifies our visualization and simplifies our math and you get it to be one point five three seven for arc arc seconds and I want to be very clear this is a very very very very very small angle just to visualize it or another way to think about it is one point there are 60 arc seconds per hour committed and there are 60 arc minutes per degree or another way to think about it as a degree is like an arc hour so if you want to convert this to degrees you have 1.5 3 7 4 arc seconds arc seconds times 1 degree is equal to 3600 arc seconds the dimensions cancel out and you get this as being equal to the calculator out this is being equal to 1.5 3 7 4 divided by 3600 so it's four point two zero four point two zero six I'll round because we only want five significant digits this is of infinite precision right here because that's an absolute quantity it's a definition so let me write this down so this is going to be this is going to be four point two seven oh six times ten to the negative four degrees could write it just like that now let me be clear this is the total this is the total angle this angle that we care about is going to be half of this so we could divide this by two so let me just do our significant digits for point two seven zero six divided by divided by two or I could even say times 10 to the negative four times 10 to the negative four divided by two is going to be two point one three five three times ten to the negative four so that's this angle right over here this angle or the shift from Center we could visualize it is going to be two point one three five three times ten to the negative four more so now that we know that we already figure it out how to figure out the distance we can just apply this right over here so let's just take let's just take the tangent make sure your calculator is in degree mode I've made sure of that before I started this video of 90 - 90 - this angle right here so instead of retyping I'll just write the last answer so 90 minus this angle and we get this large number two hundred sixty-eight thousand three hundred and twenty-six now remember what were our units this distance right here this distance right here is two hundred and sixty eight thousand 326 I should just round because I only have five significant digits here three hundred and although with trig trig number with a trig when you're using involving trigonometry the significant digits get a little bit a shade here but I'll just write the whole number here three three two hundred sixty eight thousand three hundred and twenty two hundred sixty eight thousand three hundred twenty-six astronomic units so it's this many it's this many distances between the Sun and the earth now if we wanted to calculate that into into light-years we just have to know and you could calculate this multiple ways you could just figure out how far an au is versus a Lightyear but there are so this is a use there are one or one light year one light year is equivalent to sixty 3115 astronomical units give or take a little bit so this is going to be equal to the aus cancel out this quantity divided by that quantity in lightyears so let's do that so let's take this number that we just got divided by sixty three thousand one hundred fifteen and we have it in lightyears and so it's about four point two five light-years I'm messing with the significant digits here but just roundabout answer four point two five light-years four point two five lightyears four point two five light-years now remember that's about how far the closest star to the earth is and so the closest star to the earth has this very very very area parent a very small change in angle you can imagine as you go to further and further stars from this that angle this angle right here is going to get even smaller and smaller and smaller than all the way until you get really far stars and it would be even with our best instruments you won't be able to measure that angle anyway hopefully you found that cool because you just figured out a way to use trigonometry and a really good way to measure angles and the night skies do in the night sky to actually figure out how far we are from the nearest stars I think that's that's pretty neat