Classification of Kepler's Laws

 Kepler’s laws

Satellites (spacecraft) which orbit the earth follow the same laws that govern the motion of the planets around the sun.

Kepler’s laws apply quite generally to any two bodies in space which interact through gravitation. The more massive of the two bodies is referred to as the primary, the other, the secondary, or satellite.

• Kepler’s First Law

Kepler’s first law states that the path followed by a satellite around the primary will be an ellipse. An ellipse has two focal points shown as F1 and F2. The center of mass of the two-body system, termed the barycenter, is always centered on one of the foci. In our specific case, because of the enormous difference between the masses of the earth and the satellite, the center of mass coincides with the center of the earth, which is therefore always at one of the foci.

The semimajor axis of the ellipse is denoted by a, and the semi-minor axis, by b. The eccentricity e is given by 𝑒= √𝑎2− 𝑏2𝑎

The eccentricity and the semimajor axis are two of the orbital parameters specified for satellites (spacecraft) orbiting the earth.

For an elliptical orbit, 0 <e <1. When e = 0, the orbit becomes circular.

• Kepler’s Second Law

Kepler’s second law states that, for equal time intervals, a satellite will sweep out equal areas in its orbital plane, focused at the barycenter.

Referring to Fig above, assuming the satellite travels distances S1 and S2 meters in 1 s, then the areas A1and A2 will be equal. The average velocity in each case is S1 and S2 meters per second, and because of the equal area law, it follows that the velocity at S2 is less than that at S1. An important consequence of this is that the satellite takes longer to travel a given distance when it is farther away from earth. Use is made of this property to increase the length of time a satellite can be seen from particular geographic regions of the earth.

• Kepler’s Third Law

Kepler’s third law states that the square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies. The mean distance is equal to the semi-major axis a. For the artificial satellites orbiting the earth, Kepler’s third law can be written in the form,

𝑎3=𝜇/𝑛^2

Where n is the mean motion of the satellite in radians per second and 𝜇 is the earth’s geocentric gravitational constant.

Value of 𝜇 is,

𝜇 = 3.986005 *1014 𝑚3/𝑠2

Above equation applies only to the ideal situation of a satellite orbiting a perfectly spherical earth of uniform mass, with no perturbing forces acting, such as atmospheric drag.

With n in radians per second, the orbital period in seconds is given by

P= 2𝜋/𝑛

The importance of Kepler’s third law is that it shows there is a fixed relationship between period and semi-major axis.

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