if a satellite has just the right velocity then we can make sure that the force of gravity will always stay perpendicular to that velocity vector and in that case the satellite will go in a perfect circular orbit because the gravitational force will act like a centripetal force we've seen this before in our previous videos now the goal of this video is to use insights from this and understand how planetary orbits works and talk about Kepler's loss so let's begin first of all we can model our Earth and sun to be very similar to that satellite if the Earth has just the right velocity then the force of gravity due to the sun on the earth will now act as a cental force making sure the earth goes in a perfect circular path but we can have a couple of questions over here our first question over here is if Sun puts a force on Earth does the Earth put a force back on the Sun what do you think all the answer is yes Newton's third law equal and opposite forces right so shouldn't the Sun also accelerate well it does but because the sun is so much more massive compared to the earth its acceleration is very tiny but it's there and because of that the sun would actually wobble so if I were to exaggerate that a little bit it would look somewhat like this look at that the sun is wobbling a little bit as the Earth would go around it but this is exaggerated because the Sun's mass is so much that wobbling is so tiny that we can completely neglect it so while modeling this we can completely neglect the wobbling of the Sun and we can assume that it's only the Earth that's going around in the orbit okay but that now brings us to the next question this only works if the Earth has that particular specific velocity but now what happens if it doesn't what happens if it has slightly bigger velocity or slightly smaller velocity well then you can kind of see if it has a bigger velocity it'll go away from this circle the path will look somewhat like this if it has a smaller velocity also it'll it'll go away from that Circle it path's going to look somewhat like that and in general the orbit will not be a circle instead it will look like this this is called an ellipse you can think of an ellipse as a sort of squished out Circle but if you're wondering if there's a slightly better definition of an ellipse I'm glad you asked in fact we can have some fun by actually drawing an ellipse to do that take a couple of thumb tacks and then put a thread around it put your pen or a pencil and draw that shape in such a way that the thread always stays T the shape that you get is an ellipse if you keep the thumb tax farther away it becomes more elliptical more squished out on the other hand if they keep the thumb TXS closer to each other it'll look more like a circle now which means you can see that circle is basically a special case of an ellipse a case in which the two thumb tacks are pretty much at the same point and the more you squish it out the more elliptical it gets now there's a particular mathematical term that we introduce over here to talk talk about how elliptical how squished out it is we call that eccentricity it's a number between 0 to 1 if it's a circular path then we say it's not squished out at all and we say its eccentricity is zero on the other hand if you squish it out so much that it almost looks like a straight line then it's eccentricity is one and by the way these two Thumbtack points well they mathematically they're called the FI forai is plural focus is singular this is one Focus this is the other focus of the the ellipse and when it comes to a circle both the fori are at the same point which becomes the center of the circle okay now the cool thing about our orbits is that it turns out that the sun will always be at one of the fouri it will not be at the center of the ellipse but it'll be at one of the forai of the ellipse and the first time I heard of this I was like wait a second is this the reason why we have seasons when the earth is closer to the sun we get Summer and when it goes farther away we get winter makes sense right well actually no that's a big misconception because if that were true then everybody on the planet would get somewhat at the same time and the winter at the same time but that doesn't happen right and the reason it doesn't happen is because this is a highly exaggerated figure it turns out that the orbits of most planets around the sun is actually having an eccentricity very close to zero so it's almost circular orbits okay so the distance of the Earth from the sun pretty much stays the same the reason for the seasons has something to do with the tilt of the Earth's rotational axis this we'll not talk much about that over here anyways putting it all together we can now write down Kepler's first law which says that all planets revolve around the Sun in an elliptical orbit with the Sun at one of the fosi it's one of the three laws that we're going to study in this video made by johannas Kepler who spent years recording the positions of the planets again remember most planets have very circular orbits but but there are other things like for example comets they too you know orbit around the Sun and they have highly elliptical highly eccentric elliptical orbits okay moving on to the next part here's the next question for us can we now comment about what happens to the speed of the you know speed of the planet as it goes around the Sun if it was a circular orbit then we know the speed Remains the Same but what about over here well one way is to draw the velocity and force vectors so here are the velocity vectors velocity vectors are is tangential but I don't know the magnitude of the Velocity the speed I don't know that's what I'm trying to figure out so don't concentrate on how big the vectors are I've just drawn them randomly but I also know the force vectors the force will always be directed towards the Sun and when you're closer to the Sun you'll have a much larger force and when you're farther away from the Sun the force weakens okay okay so just from this how can we comment on the speed of the planet well here's how I like to think about it if I were to focus at this position what I would do is I'll just drop a perpendicular to my velocity vector and the reason I do that is because I know I remember that if a force is perpendicular to the velocity Vector it will not change the speed of the planet or the object it will only make it curve but over here we see that the force is slightly tilted backwards that means it's slightly pulling it back therefore because the force is slightly pulling it back I know that this is going to slow down so right at this point the force will slow down a little bit and we can do the same analysis everywhere and we can comment on what happens to the speed so it'll be a great idea to pause the video and see if you can do the analysis yourself all right again I'm going to drop a perpendicular to my velocity Vector over here and I see that my force is tilted slightly behind slightly back and therefore it's again going to slow down what happens over here hey here it's exactly perpendicular so at this position it doesn't neither slows down nor speeds it up what happens over here hey here you can see it's the other way around the force is tilted you know kind of slightly forward can you see that and therefore it's now going to speed it up what about here hey it's again similar the force is slightly tilted in the forward Direction speeding it up and finally over here again it's perfectly perpendicular so look at this from here I know here there's not going to be any change so if I start from here it slows down slows down slows down it goes slower slower slower and then it speeds up speeds up speeds up and then again slows down oh that means that I should get the fastest over here because it speeds up speeds up fastest slows down slows down slows down slows down and the slowest over here so if you could see it would look somewhat like this it slows down slows slows slows slowest and then increases speed like this and so on and so forth so now we can draw the correct velocity vectors you'll have the biggest Vector over here become smaller smallest Vector over here becomes bigger and so on and so forth okay so here's a question to check our understanding all right so in this orbit let's consider a point when Earth is somewhere over here and let's wait for some time let's say about 3 months for the earth to come from here to here okay then again we'll wait for some more time and let's say We'll again look at when Earth is over here and we'll wait for 3 months now my question to you is now if we wait for another 3 months the distance TR by the Earth over here in the orbit will it be the same as over here more than over here less than over here why don't you pause the video and think about that all right we just saw that far away the speeds are lower and closer to the Sun the speeds are much higher therefore over here because Earth is traveling much faster we should expect it to cover more distance in that same amount of time right okay now here's the cool thing if we were to join these lines and find this area we will find that this area is exactly the same as the area over here we can get some intuition behind it because although this thing is this link is small over here you can see this length is bigger over here this length is small but this length is bigger so it all just works out in a beautiful way that the area stays the same and this now brings us to Kepler's second law which says that a line joining the planet and the Sun like this line over here sweeps equal areas equal areas in equal time intervals the 3 months was just an example it doesn't matter what time interval you take but as long as you take equal time intervals you will find that regardless of where you take those timing rols in the orbit as long as it's equal the area swept by this line will always stay the same but I'm sure you're wondering why does it work out that way without getting into all the proofs and stuff but there are some beautiful geometrical ways to think about it but the main reason why this works out this way is because the force is directed towards the sun turns out that whenever the force is directed at a specific point you will always get this to be true and in fact this is how Newton was able to use Kepler's second law to actually realize that the force on all the planets must be towards the Sun and that actually helped him establish his law of universal gravitation Kepler's laws came before Newton's Laws of gravitation anyways on to the final law and for this now we'll zoom out and start looking at all the planets now and since we've seen that most planets have circular orbits we'll just take stick to Circular orbits now okay this time we want to look at how long different planets will take to complete their orbit and again we can try and think about this slightly intuitively so if you consider a planet which is very close to the Sun and the force of gravity acting on it would be very large and therefore the acceleration due to gravity would also be very large right so if you want to have a perfect circular orbit to make sure that it misses the sun then it just goes in a perfect circular orbit we would need pretty high velocity so over here when you're close to the Sun you need very high velocity but what happens if you go slightly farther away from the Sun well then the force of gravity on this planet would be vaker remember the inverse Square law the force weakens as you go farther away and therefore the acceleration due to gravity over here would also be much weaker than over here now immediately you might say well wait a second Mahesh what about the mass of the object doesn't the mass also matter well no remember one of the cool things about gravity is that the acceleration that you get due to gravity is independent of the mass mass of the planets over here it does depend on the mass of the Sun of course but not on the mass of the planets we've seen that before just like how when you drop a baseball basketball sorry a bowling ball in a feather both of them they have the same acceleration due to gravity on the Earth right same is the case over here okay okay anyways since the acceleration of too gravity is very weak over here it it requires a much smaller velocity to stay in circular orbit because it the force will this this force will not curve it that much the acceleration is very little based on this piece of information can you think about which of these two will take more time to complete their orbits all right let's see we know this is faster and it's traveling a smaller distance this is slower and it's traveling a bigger distance so clearly this should take longer isn't it if you could see it it would look somewhat like this clearly this would take much longer and the farther you go the longer it will take to finish that orbit this finally brings us to Kepler's third is law I say third is because Kepler's Third Law is actually mathematical we not get into the math part of it but the key part of that law is that the farther the planet is from the Sun the longer its time period and there you have it the Kepler's three laws that governs how the planets go around the Sun Kepler's and Newton's laws are incredible at predicting any orbits not just the Sun and our solar system but any orbits in general unless things are very massive if you're dealing with things like black holes and neutron stars and stuff well now these laws will not work anymore now we will need a much more accurate description of gravity that is given by the general theory of relativity but as long as you're not considering these extreme cases kep's laws and Newton's laws are are going to be super super useful for us