Viva ‘Vis-viva’

Extract

Long live the vis-viva equation. There is sometimes more than one way of telling a story with the same ending, and this is particularly true in the field of applied mathematics where there are often different ways of obtaining the same result to a given problem, so that what might be lost or not appreciated in one approach can be found and appreciated in another. It is the purpose here to illustrate this by presenting a well-known example drawn from the field of orbital dynamics, namely the development of what is called the Vis-viva equation. This equation is simply an expression relating the square of the velocity of an orbiting object, for example a planet orbiting a sun, to orbit parameters and scientific constants. It is a standard workhorse equation that is used extensively today by orbit control specialists wishing to determine and affect velocities of spacecraft orbiting significantly larger masses, and it was developed from work carried out centuries ago by Gottfried Leibniz (1646 – 1716). The first story will simply acknowledge the equation and how it arose. The second story will present an alternative approach based on calculus, trigonometry, algebra and computations set against the backdrop of an ellipse geometry. This second story leads to the vis-viva equation in disguise, so to speak, and examples of the speeds of two planets, Earth and Mercury, orbiting the Sun will be discussed. Where relevant, the nomenclature of orbit dynamics will be acknowledged. The discussion will involve primarily a planet’s orbit velocity of translation and detailed considerations will not be given to effects due to its speed of rotation.