



1. The shape of planetary orbits





The shape of planetary orbit is elliptical. Take a unit circle and stretch it horizontally and the resulting shape that you get will look more or less like an ellipse. For a horizontal ellipse, the line segment between two horizontal ends and passes through the center of an ellipse is the major axis and the line segment vertical between two opposite vertical points and passes through the center is the minor axis. The ellipses have two focii and Kepler(born 1571) assumed that the sun sits fixed in one of the focii (actually wrong assumption)

Sun is at a focus and earth is moving around the curve





Let's try find the distance of the semi major axis with this equation. Let's call the semi major axis length 'q' , then (q,0) is a point on the graph. Look carefully a distance of 'q' horizontally and distance of '0' vertically! Let's put those values into equation





$$ \frac{q^2 }{a^2 } + \frac{0}{b^2} = 1$$





or,

$$ q = {+a,-a}$$





so we get two solutions for 'q' that's because 'y' is zero at the right end and also left horizontal end.

Take difference between (a,0) and (-a,0) we get the length of major axis which is precisely 2a.





Onwards!





2. Straight line joining sun and planet sweeps equal areas in equal intervals of time

Basically if you draw a line from the sun to where the planet is and see how much area that line sweeps in some time 't' then you would find that wherever the planet was at starting the amount of area swiped under time 't' is a constant! We can prove this using conservation of angular momentum!

Let's try!



Consider a tiny area sweep in time 'dt'

Observe that, that the total area is sum of the blue area + red area

i.e:





$$dA = \frac{r^2d\theta}{2} + r\frac{drd\theta}{2}$$



The second term is a product of two infinitesimal which is infinitesimally smaller than an infinitesimal hence you can just neglect it ( hand wavy mathematics strikes again!)





Hence,

$$ dA = \frac{r^2d\theta}{2} $$





divide by 'dt'





$$ \frac{dA}{dt} = \frac{r^2d\theta}{2dt}=\frac{r^2w}{2}$$



using $$[ w=\frac{d\theta}{dt}]$$





waittt that seems familiar .. angular uhh momentum!?!





$$ L= Iw=mr^2 w $$ [ angular momentum cool]





or, $$ \frac{L}{m} = r^2 w$$





substitution a bit $$ \frac{dA}{dt} = \frac{L}{m}$$!!!









Since L is constant always, that means rate of increase of area is also constant! Astounding!









3. The time period law





Kepler also derived a law about the time period of elliptical orbits. However newton(born jan 4 1643 ) got the same exact result even when he took the orbital path as a circle instead of an ellispe (wrong method but right answer meme) Let's go over how he did that

here, M= mass of sun , m = mass of earth, w= angular velocity of earth around sun, r = seperation of earth to sun ( constant , since newton assumed circle) , G= gravitational constant



Since it's a circle he started by equating the centripetal force ( we have an equation for this) with the gravitational force ,



As in,





$$ mw^2 r = \frac{GmM}{r^2} $$



cancelling some terms and rearranging





$$ w^2 = \frac{GM}{r^3}$$

$$ \frac{(2\pi)^2}{T^2} = \frac{GM}{r^3}$$





$$ \frac{T^2}{(2\pi)^2} = \frac{r^3}{GM}$$



$$ T^2 = (2\pi)^2 \frac{r^3 }{GM}$$

Now, since $$\frac{(2\pi)^2}{GM} = Constant$$





$$ T^2 = C r^3$$







Now, How did Kepler do it? well that's where the other side of physics come in.. that is experiments. He took data of the period of revolution and distance of semi major axis of different planets eg: mercury, uranus, jupiter and mars and found that they all behaved this relation









Finding the pattern kepler wrote



" "I first believed I was dreaming… But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance.""





Well that's it for today fellas. I didn't have much to write on the historical views and theory cause I'm not much big in that department but all these equations are really cool . Like how we can derive time period and everything xD Right now we come across a problem the 'r' here is radius of circle but Kepler's stuff include ellipses. hmm how do we fix that ??! Easy replace 'r' by the length of semi-major axis ('a' in that ellipse before)Now, How did Kepler do it? well that's where the other side of physics come in.. that is experiments. He took data of the period of revolution and distance of semi major axis of different planets eg: mercury, uranus, jupiter and mars and found that they all behaved this relationFinding the pattern kepler wroteWell that's it for today fellas. I didn't have much to write on the historical views and theory cause I'm not much big in that department but all these equations are really cool . Like how we can derive time period and everything xD





Bonus: Why did Newton take the orbit as a circle?? he came much after kepler so surely he must have heard it is an ellipse??!?!?





Answer: Newton proved that the orbital paths should be conic sections and he just took the case where it would be a circle. Turns out that time period law should be same for any conic section.















I've finally decided to start writing here again and without a further ado the topic today will be about Keplers laws of planetary motion w/ newton stuff. This article will feature a lot of calculus and infinitesimals. So I'm hopin that my dear readers have some familiarity with that already :DThe equation of the ellipse is given as : $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Basically if you find all the (x,y) pairs which solve that equation and you darken those pairs on the x-y plane, you will get the shape of an ellipse.