Kepler's First Law

To perform a circular orbit round a planet, the object needs to attain a certain velocity at a particular distance from the planet. This is the reason why a spacecraft, after launching, will attain an elliptical orbit at the first instance. The spacecraft then needs to fire its rocket engine, usually at the apogee (遠地點), to boost it to the required velocity for a circular orbit.

Planets were formed by natural mechaisms. It was only that their velocities were not high enough for them to escape from the gravitational attraction of the sun. The chance for each of them to attain a given velocity at a certain distance from the sun is, though not impossible, quite remote.

Without further ado, here is a rigorous proof of the theorem.


圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701006210149513873425270.jpg


2010-06-22 23:07:01 補充：
Here energy E is negative whenever the object is undergoing a periodic trajectory, i.e. circular or elliptic.

exact circular motion
=> velocity always perpendoicular to the line joining the sun and thew planet.

change of the distance between plants
=>change in r

How can we change r suddenly with an velocity always perpendoicular to the line joining the sun and the planet?(imagine it, change in r requires a velocity NOT perpendoicular to the line joining the sun and the planet)
=> the statement is wrong

In reality, planet moves in ellipse, the velocity is not always perpendoicular to the line joining the sun and the planet. Therefore, its distances between the sun always changes.

2010-06-21 21:19:55 補充：
perpendicular not perpendoicular