In yesterday’s math forum meeting some interesting questions regarding orbits of planets were raised. Since these would interest everyone in the VERITAS group I thought I should post the answers as a VERITAS article:
1) How do we know that earth orbits the sun and not the other way?
This is actually a very valid question. We take it for granted that earth orbits the sun. But why? The Sun is not God! Why should earth orbit the Sun? When we look at the Earth-Sun system we realize that these are 2 bodies moving under the gravity of each other. Sun pulls Earth. Earth pulls Sun. SUch a system is called a two body system. In a two body system the bodies move around their centre of mass. So earth does not orbit the Sun. And the Sun does not orbit the earth. They both orbit their common centre of mass. But the sun is SOOOOOOOOOO massive compared to the earth. The centre of mass of the earth-sun system lies within the body of the Sun. See that! So the earth orbits the Centre of mass which is within Sun. It cannot be the other way round. No need to even observe. All this can be shown mathematically. The solution of the two body problem is well known and is textbook material.
I would also like to mention that the three body problem is still unsolved in Physics. Physicists dont have an equation to describe how three bodies move under mutual attraction!!!! They have to do this approximately and numerically! Physicists talk about the universe but they cannot solve a 3-body problem. What to talk of the 10 body solar system!
Remember Einstein’s words:
“Whoever undertakes to set himself up as judge in the field of truth and knowledge is shipwrecked by the laughter of the Gods.”
2) Why are the orbits of Planets elliptical and not round?
I have struggled with this question a lot in my younger days.
It made me feel so miserable. Only after reading Herbert Goldstein’s “Classical Mechanics” did I felt better.
Why do we have a desire for round orbits? Lets first look deeper at this question? This desire comes from our stupid educational system that gives us analogies to help our learning but does not warn us when we go too far with these analogies. In our school days we were told that earth orbits sun much like a stone tied to a thread can be rotated in a circular motion! And electrons also orbit the nucleus the same way! Nice for learning but these kind of analogies hamper any deeper reflections on these subjects.
Lets understand the stone tied to a string rotation in detail. A stone tied to a string is rotated. It moves in a circular fashion. The stone’s “desire to move outward”(centrifugal force) is exactly balanced by the pull of the thread. Now rotate the stone faster. The stone’s velocity increases. Its centrifugal force increases. But it is exactly balanced by the increased tension in the string. Now increase the velocity further. More tension! …. it keeps going round and round till the string breaks and the stone goes far away!
Now lets look at the earth sun example. first thing: there is no string. So there is no balance! It is something like this: The earth moves towards the sun… in this motion it gains a lot of velocity( think of it like falling toward the sun). When it comes near the sun the velocity is so huge that it whizzes past the sun. Now it is moving away from the sun.
In this motion it keeps getting slower( its direction is opposite to sun’s pull). It reaches a point where its “desire to move away” is exactly balanced by the sun’s attraction. Its velocity in a direction away from the sun becomes zero. And it falls again.
Here are the major differences between the earth-sun orbit and the stone -string analogy:
a) In the earth sun system the balance between centrifugal force and pull happens only at the end points. In a stone with string thing every point is balanced.
b)If the velocity of the earth is increased it will make a bigger orbit around the sun. But if the velocity of the stone tied to a string is increased the orbit stays circular. This is because the thread can take more tension. And you have not let the stone move to a larger orbital.
You have taken a unstretcheble string. You have not allowed the stone to move away!
c)The pull of sun on earth decreases as earth moves away. The pull of the thread on the stone always adjusts to exactly match the pull of the stone.
So the analogy on which we base our desire to give a circular orbit to earth is not a valid one. A better analogy would be to imagine a stone tied to a flexible elastic. The elastic allows the stone to move away.
However even this analogy is not perfect.
There is a point around the sun where if earth is placed with a particular velocity a perfect balance between opposing forces will be achieved. And a circular motion will follow! But the natural motion of planets around a star is not circular. It can be elliptical(least velocity of planet), parabolic( more velocity) or hyperbolic( highest velocity! The planet will not come back!)
And, if each system in gradation roll
Alike essential to th’ amazing whole,
The least confusion but in one, not all
That system only, but the whole must fall.
Let earth unbalanc’d from her orbit fly,
Planets and suns run lawless through the sky;
Let ruling angels from their spheres be hurl’d,
Being on being wreck’d, and world on world;
Heav’n’s whole foundations to their centre nod,
And nature trembles to the throne of God.
( Alexander Pope)
Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;