The Ratio of Velocities at Perigee and Apogee

    Kepler's 2nd law prescribes the way the velocity V of an object orbiting a single center varies around its orbit:

As noted in section #12, this means that the object's velocity V increases when its distance r decreases, and vice versa.

    Imagine an Earth satellite, located at perigee P (see drawing). In one second it will have advanced a distance V₁ meters (or whatever units are used). Since that is just a tiny part of the orbit, no great error occurs if, in calculating the area swept by the radius r₁, we replace it with a straight line. The swept area is then a long and skinny right-angled triangle with height r₁ and base V₁ (skinnier than we could draw here, so the drawing is just to give the labeling).

    The area A₁ of such a triangle, by the formula for areas of right-angled triangles, is one half base time height, or

    Similarly, the area A₂ covered in one second after passing apogee A equals

However, by Kepler's 2nd law A₁ = A₂ so

or, multiplying everything by 2

A more useful form of that relation appears if both sides are divided by V₂r₁ :

    The ratio of velocities equals the inverse of the ratio of distances. The smaller the distance, the faster the motion. If perigee distance is half of the apogee distance, the velocity there is twice as large. (But please remember--this proportionality only holds with P and A, not with other points along the orbit).

Why are Seasons Unequally Spaced?

    But are they?

    Spring equinox in 2003 was on March 21, while fall equinox was on September 22, 184 days later. In 2004, a leap year, spring equinox occurs March 20, and if you count the days, you will see that to get there only takes 181 days more. The two intervals are not equal.

    How can that be? Equinox positions are on opposite sides of the Earth's orbit, 180° apart, yet getting through (northern) winter takes Earth 3 days less than getting through summer.

    Because by Kepler's 2nd law, the Earth moves in its orbit a little faster in winter. As noted at the end of "Seasons of the Year" and also in connection with the Milankovich Theory of ice ages, Earth is closest to the Sun--at perihelion--around January 4. By the equations derived here, that's when it moves fastest. Furthermore (see drawing), the half of the ellipse closest to the Sun is also shorter. With Earth the difference is small, since the orbit is quite close to a circle; but it exists, nevertheless, and is the reason for that difference of 3 days.

Note: According to item on Hipparchus in "Wikipedia" (section #4.4) the above asymmetry was known to the ancient Babylonians (Chaldeans), was measured by Hipparchus around 140-130 BC, and was quoted by Ptolemy.