The energy a planet carries divides in two: the part from moving and the part from being near the Sun.
Approach the Sun and the planet speeds up, so the moving part grows. The near part shrinks by as much. The energy of position is fixed to be infinitely far away, so it goes deeper and deeper negative as you close in.
What is gained and what is lost are always equal, so the sum does not change. Everywhere on the orbit, that sum is one and the same number.
And that sum turns out to be the number that fixes the orbit.
Write it out. With speed and distance from the Sun , , where is the quantity that does not change.
Compute and it comes to : it depends on the semi-major axis alone, with no eccentricity in it. A long thin ellipse and a nearly circular one with the same carry the same energy.
Put the two together and solve for : . It is called the vis-viva equation, from the Latin for living force.
The present distance from the Sun, and the semi-major axis of the orbit. Those two give the speed. Which way it faces and what time it is are not needed.
Feed in some special values. On a circle the distance always equals the semi-major axis, so the bracket becomes : , the speed that holds a circle.
On a parabola the semi-major axis is infinite, so vanishes: , the speed that does not come back.
Compare the two and the second is times the first. To send something circling at a given height away for good, multiply its speed by .
On a hyperbola is negative and the bracket grows further still. The formula itself keeps the same form for all four kinds of orbit.
Round one lap of an ellipse the speed is greatest at perihelion and least at aphelion. What the second law of Kepler said now arrives as a formula.
At perihelion , giving . At aphelion the fraction is turned over.
The ratio of the speeds is . For a comet of eccentricity , perihelion is times as fast as aphelion.
Passing a given distance, the speed is the same going out as coming in, because only the distance enters the formula and not the direction of travel.
The eccentricity of the Earth is . To the eye the orbit is a circle, and perihelion is indistinguishable from aphelion.
The speed changes all the same: km per second at perihelion in January, at aphelion in July, averaging . The difference is about km per second, some percent.
That difference makes winter in the northern hemisphere shorter than winter in the southern, because the Earth passes through it while moving fast. The gap is about days.
The next article takes up finding the position from the time. The speed comes out this easily; the position does not.