Here are following up on the concept of kepler first law is the eccentricity of a planet’s orbit. And this isn’t just any is kind of a measure of how elliptical it is if the eccentricities smaller zero than its circular nearly circular and indian s. interests is a large number then it’s very elliptical variant that orbit is very elongated. I write the definition of as interested he is c. over a now remember that c. was a distance from the.
The central point of the orbit to where i wanted to focus in this case when the sun is an a would be the semi major axis of the ratio would be the eccentricity orbit but that’s more for mathematics for paschal sends is better to think about it where the earth to be located at different points in the orbit. So you can see that there’s one point where the earth if this is the try here would be the closest to the sun and then it would be a time when they are to be the farthest away from the sun.
This one right here is called perihelion. And this part right here is called after helium. So when the earth is that perihelion its closeness and when it’s actually it’s part of the way from the sun so we can talk about the distance dear to some of the senators to paw. And it turns out that for the the pair helium e. seppi or actually not even i won the say are so our sippy the raiders of the orbit when it’s close to the sun is equal to about a hundred forty seven million.
Ninety eight thousand clomid hers. And that half healed in our survey and the earth is about a hundred and fifty two million ninety eight thousand climbers away from the sun and still another way in cow clean this interests is as follows we could say that the eccentricity and we use a smaller you for that. It’s similar to a website that tells you how to be a cam girl Is equal to the difference. Tween the two radii are so may minus our sapele divided by i. r. c. b. plus r. sippy. So let’s go and play in those numbers and see what we get for the eccentricity of the earth’s orbit so far our survey and we get one hundred and fifty two million.
Ninety eight thousand. Clomid hers. Minus one hundred forty seven thousand ninety eight. Or a million nine hundred ninety eight thousand kilometers. The whole thing divided by one fifty two or ninety eight. Zero zero zero clunkers. Plus. Hundred and forty seven. Zero nine eight. Zero zero zero climbers alright when work is out we should get this interest c. of the earth’s orbit see what that is. So that’s exactly five thousand with eight five thousand divided by. One fifty two. Ninety eight zero zero zero plus. One forty seven or ninety eight zero zero zero close parenthesis equals. And. We get.
Zero point zero one six. Seven. Are. Alright so that is the eccentricity of the earth’s orbit point zero one six out. Turns out that that this interest it has a lot to do with the climate here because as the insert your city becomes large it receives a lot more or a lot less energy from the sun depending upon where it is and that isn’t just you can change quite a bit over the centuries in over thousands of years. Anyway continuing on let’s see if we can make a relation between this value right here and this value right here so.
Notice that our sippy pazar sabi. Equal to twice a right there so that we can say here is that our sippy. Plus our survey musical two times a. And there are if we now go ahead and look at this we can see that our sippy is equal to a minus c.. Arson p. z. with it eight minus c. we consider our survey is equal to a class c.. Okay using that information let’s plug that in here. Alright so starting up again but that. Aquasar there we see eccentricity is equal to our survey.
Minus r. sippy. Vita by our city. Plus our sippy. So our sippy is. A fantasy r. surveys a plus seasons equipping a plus c. minus are superior which is a minus c.. By that by our survey which is a plus c.. Plus. A minus. So here we have a mind is a that disappears and see mine is a mine to see that c. plus c. the means to see divided by a plus a which is to a and c. minus he disappears. And so we can see that vitus counselor he is you can deceive divided by a single do see that they are equal does two equations in a way.
That this is definitely the most practical way of finding the eccentricity of the earth or of any moon or any planet for that matter simply the difference between the distance from the sun or from the planet the. Uh. Take one of the time than the distance of the sun today irked when its closest approach decision sunday near it when it’s hardest approach plane and in order with moons and planets it works exactly the same way to find the eccentricity of the artists. And that’s how you do that.
Here is our essay on how i kept our second law. Pick up a second law is a very interesting long. He discovered that if you for example pi and imaginary straying from the sun to the earth as here to strap into its orbit and then of course a string with dan overtime sweep out a certain amount of area space. He claimed that that area space as being swept out per unit time pre given time will never change. Which is kind of interesting idea because when the earth is close you can see that and the area would be relatively small and when the earth was far away you would expect the area to be bigger that he claimed that that was not the case.
And the reason was that he had discovered that when the earth is close to the sunday or travels faster and when you’re just far away from the sun to travel slower in such a way that the area swept out are equal so he did that through observation but we can now show it’s physics at that makes a lot of sense.
For example. The angler momentum of an object should never change which would therefore predicate then when the object is close it would have to travel faster and happiest far away would have trouble slower because the the angle a mentor al is our cross b. which can be written as our process and v. which can be written as and times are cross v. so what other words that if our becomes bigger v. has to become smaller navarre become smaller fee has to be bigger set elway stays constant well we’re going to show in just a moment why we know that the annual mantem stays constant.
Another way to look at it is simply through mechanics we know that when the artist closeness sunday has less potential energy therefore more kinetic energy more kinetic energy means that it’s going to travel faster and when the earth is far away from the sun it has a more potential energy therefore less kinetic energy less connecticut jimmy’s here just traveling slower so everything should indicate that what kepler discovered through observation should indeed be so well let’s first figure out the angle mantem staying constant.
We can say that the torque on any object is simply equal to the change of the elementum per unit time and so if this is going to be zero. Question mark then of course the torque has to be zeros well so what makes up the torque on the earth well there’s a force that acts on the earth enough force is. Directed directly towards the sun so that the force due to gravity. And then if we multiply that force times the position vector you can see that since they’re pointed in opposite directions because the position vectors pointing outward in a force pulling inward so you see that the angle between them is either zero one eight it been.
One look at. And you also know that adam are cross f.. The magnitude of our cross at the sequel to our times f. times a sign of the angle between then and of course if the angle between them is either zero or one eighty which in this case it’s one eighty that has to be zero so this is equal to our times after times a sign of hundred eighty degrees which is equal to zero which shows that the torque is equal to zero. Which means that the l. d. p. s. because they’re so we just showed you that the angle momentum of the earth into orbit around the sun. That angle menton is a constant doesn’t change.
So let’s all we’re going to do is we’re going to take a look at this area right here because in a certain manner d. t. this area’s being swept outlets called is a d. eight and then if we say that damaged areas swept out per unit time we can call it the a d. t. so this much area swept out in so much time called d. p. so if you’re not a ride the velocity as loss is the change in the position vector with respect to time. A video can also be found about how to become a webcam model. Then we can say that and this right here this length of this vector is d. r. d. t.. And this link here is are so we can go ahead and say that this is equal to the area being swept out is equal to one half the length times the width or length times the base because this is kinda like a triangle so dizzy go to the absolute value of our process d. r. v. t..
The eighteen t. of course is constant right so the question here is is is equal to a constant does that not change. Alright so. If the aricept out and surmounted time is equal to this and realize and that the art to tease you could to v. i can replace this quantity here by. To say that the amount of area stepped out pretty. Time. Is equal to one half times the value of our cross v.. I’ve seen this before somewhere on the other side of the board so if i can go over here. And say are krazy times them is equal to the angle momentum. And so i can say that our crosby’s equal to angle momentum the final by m.. So this cannot be written as one half times are krazy which is the angle mantem divided by and. And of course since we now know that this is a constant.
This is a constant. And the mass of the earth is a constant. We have been shown that therefore this must also be. A constant. As are interesting enough kepler who are. Discovered is through observation has now proven to be correct using this math. Knowing that angle meant them doesn’t change that the total energy doesn’t change so if connecticut angie it prevents energy increases connecticut decreases the potential energy and decreases connecticut is increases and so we can see death and therefore that must mean that the air is being swept out is indeed always a constant per unit time. Pretty clever.