Abstract

In this paper, we will deal with the problem of calculating the average velocity of a celestial object revolving around another celestial object in an elliptical orbit. After proving our main theorem to this effect, we will give some alternate forms of the formula for the average velocity, and show that this average value is in fact, attained at certain points of the orbit. We will conclude the paper by providing an intuitively natural and straightforward amendment of this formula.

1. Introduction

Human awareness of planetary motions dates back to prehistoric times. It is well-known that, scholars from the Mesopotamian, Greek and Egyptian civilizations, for reasons ranging from scientific to spiritual to paranormal, tried to observe the motions of celestial objects, their efforts culminating in the Ptolemaic model.

Eventually, medieval scientists, in particular, Nicolas Oresme (1320-1382) and Jean Buridan (1300-1361), paved the way to Johannes Kepler’s (1571-1630) invention of a system that correctly described the major aspects of the motions of the planets around the sun. The crucially needed mathematical support of the theory was later furnished by Sir Isaac Newton (1643-1727), mainly through his law of gravitation. For a more detailed discussion of this fascinating historic development, see Thurston ¹.

Our principal goal in this paper is to compute the average velocity of a planet rotating around the sun in the Keplerian model. For physical and astronomical background Dilgan ², Gamow ³, and Curtis ⁴.

Suppose that a planet of mass is moving in an elliptical orbit about an object of mass located at tone of the foci of the ellipse at distance from the center. Let the semi-major and semi-minor axes of the orbit be and respectively. Let be the velocity of when it is at a distance from Let the average velocity be denoted as

Setting the kinetic energy

equal to the gravitational potential energy

we get

Solving this equation for the velocity we obtain

the so-called vis-viva equation. Here is the standard gravitational parameter.

2. The Main Theorem

Before we state and prove our main theorem, let us prove a simple algebraic identity:

Lemma 1. For any two real numbers and

Proof. Clearly, the left hand-side can be written as

Now for the main result we want to prove:

Theorem 1. Suppose a planet of mass is moving in an elliptical orbit about an object of mass located at one of the foci of the ellipse at distance from the center. Let the semi-major and semi-minor axes of the orbit be and respectively. Then, the average velocity is given as

Proof.

Our goal is to compute the average value of the function namely the integral

Let us first compute the indefinite integral by making the change of variables

Then,

Thus, our integral becomes

Since

integrating we obtain

Consequently,

Invoking Lemma 1, we obtain

and since

we get,

Using the fact that the eccentricity of an ellipse is

one can also write

Recalling that

for we have

and subsequently,

proving our theorem.

Lemma 2. (An Alternate Formulation) Let be the radius of curvature of the ellipse. Then

Proof. Suppose we parametrize the equation of an ellipse using

and

Then, using the formula

for the radius of curvature, we get at

and consequently,

3. Some Additional Results

Here are some additional results of interest:

Lemma 2. The average velocity is the arithmetic mean of the maximum velocity attained at the pericenter and the minimum velocity attained at the apocenter.

Proof. This follows immediately from the formulas of the maximum and minimum velocity ⁵. Indeed,

Lemma 3. Every planet attains its average velocity

Proof. Solving the equation

we get that these velocities are equal whenever

4. A Modification of the Formula for the Average Velocity

The results of the previous sections were obtained using the first degree of approximation to the inverse tangent function. If we now take two terms in the series, and use the approximation

for the equality

yields

and subsequently,

Implying the formula

would yield a more accurate approximation to the average velocity without introducing any additional computational complexities. Of course, if then this second-degree approximation will be close to the first-degree approximation obtained in Section 2.

References