Kepler's first law states that all planetary orbits are elliptical, with the sun located at one of the two foci of the ellipse. This means that even circular orbits are a special case of ellipses. The major axis of the ellipse is the longer axis, divided into two halves, each represented by the variable a, making the total length of the major axis equal to 2a.

In an elliptical orbit, the closest point to the sun is called the periapsis (or perihelion for orbits around the sun), denoted as R_P, while the farthest point is called the aphelion (or apoapsis), denoted as R_A. The relationship between these distances and the semi-major axis can be expressed as:

R_A + R_P = 2a

From this, the semi-major axis a can be calculated as:

a = \frac{R_A + R_P}{2}

The minor axis, which is the shorter axis of the ellipse, has a length of 2b, where b is half of the minor axis.

Kepler also introduced the concept of eccentricity, denoted as e, which measures the deviation of an orbit from being circular. Eccentricity values range from 0 to 1, where values close to 0 indicate nearly circular orbits (like those of most planets) and values close to 1 indicate highly elliptical orbits (like those of comets). The eccentricity can be calculated using the following relationships:

R_A = a(1 + e)

R_P = a(1 - e)

In circular orbits, the distances R_A and R_P are nearly equal, resulting in an eccentricity of approximately 0.

To illustrate these concepts, consider Earth's orbit. The closest distance to the sun (perihelion) is approximately 1.471 × 10¹¹ m, and the farthest distance (aphelion) is about 1.521 × 10¹¹ m. The semi-major axis can be calculated as:

a = \frac{1.471 × 10¹¹ + 1.521 × 10¹¹}{2} = 1.496 × 10¹¹ m

Next, to find the eccentricity, we can use the aphelion equation:

R_A = a(1 + e)

1.521 × 10¹¹ = 1.496 × 10¹¹(1 + e)

By simplifying, we find:

1.017 = 1 + e

e = 0.017

This small eccentricity indicates that Earth's orbit is very close to circular, which is consistent with the orbits of most planets in our solar system.

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