Planet energy

The left hand diagram below is an attempt to represent a gravitational potential, but only two of the three dimensions can be included. The potential is inversely proportional to the separation of two attracting objects, and is considered to be negative; a closer object has a lower potential energy than a further one, or as we say in a local area on Earth, a lower object has less energy than an identical higher one. The reason that the potential energy is counted as negative is that potential energy is not an absolute value: it has to be defined relative to some point chosen by the observer, and it is usual to consider that zero energy occurs when the objects are completely free of one another’s influence, which happens only at an “infinite” distance. In addition to the potential energy (PE), an orbiting object has kinetic energy (KE), proportional to the square of its speed. In a circular orbit the numerical value of the potential energy is twice the value of the kinetic energy, that is, PE = – 2 x KE. Thus, when the negative PE is added to the positive KE, the total energy (TE) is exactly the negative of the KE, that is, TE = KE + PE = KE + (- 2 x KE) = – KE. When an object travels around an elliptical orbit, there is a continual interchange of kinetic energy and potential energy, KE being maximal at perihelion, and PE being maximal at aphelion, so a cometary orbit is rather like a roller-coaster. Since the KE in a circular orbit is proportional to the square of the speed, and the PE is inversely proportional to the orbital radius, we can see that the square of the speed is inversely proportional to the radius. But the period of the orbit is inversely proportional to the speed, and also proportional to the radius, and hence it is easy to see that Kepler’s third law is compatible with these observations.

This representation of a gravitational well must not be considered as an image of a model on which a small ball will roll. A ball will roll on such a shape, but that does not mean that its motion has any relevance for the motion of an object in a gravitational field. For example, in an eccentric orbit, the ball will have a vertical component of momentum, which does not correspond with anything that happens in a real orbit.
The small circles represent successive positions of an object separated by equal intervals of time, enabling relative speeds to be shown. The objects in the inner orbits move faster than those in the outer orbits, but they are deeper in the potential well, and their total energy is lower than that of the objects in the outer orbits.
In this perspective view of the orbits, they are displaced vertically in proportion to their total energy, which is measured from the short horizontal blue line at the top, which marks the level defined as the zero of energy, corresponding to an orbit of infinite radius.

Kinetic energy seems to be very peculiar. A fast moving object can “possess” vast amounts of kinetic energy, but someone travelling with the object will not be able to find any properties that can be attributed to the kinetic energy. This does not matter: quantities such as energy and momentum can be used as accounting quantities without any knowledge of what they “actually are”. Are all physical quantities like this? Some can at least be compared; it is possible to place two objects close to one another to compare their lengths. Using an interferometer, their lengths may be compared very accurately without them having to be very close to one another. But two time periods cannot be compared unless they are partially contemporary, although audio and video recordings can be compared after the event.