Sat Comm Part 4. How Satellites Orbit Earth: Kepler’s Laws of Motion (Physics & Space Science).
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The orbital positions of spacecraft in a communications satellite system play a crucial role in determining the coverage and operational characteristics of the services provided. The same laws of motion that govern the movement of planets around the sun also apply to artificial satellites orbiting the Earth. Satellite orbit determination is based on the Laws of Motion first developed by Johannes Kepler and later refined by Newton in 1665 using his own Laws of Mechanics and Gravitation.
Competing forces act on a satellite: gravity pulls it toward the Earth, while its orbital velocity tends to pull it away.
The gravitational force, Fin, and the angular velocity force, Fout, can be represented as
where m = satellite mass; v = satellite velocity in the plane of orbit; r = distance from the center of the earth (orbit radius); and Kepler’s Constant μ (or Geocentric Gravitational Constant) = 3.986004 × 105 km3 /s2 .
Note that for Fin = Fout
This result gives the velocity required to maintain a satellite at the orbit radius r.
Kepler’s First Law, as it applies to artificial satellite orbits, can be simply stated as follows: The path followed by a satellite around the Earth is an ellipse, with the center of mass of the Earth located at one of the two foci of the ellipse.
If no other forces act on the satellite—whether intentionally through orbit control or unintentionally, such as gravitational forces from other celestial bodies—the satellite will eventually settle into an elliptical orbit with the Earth at one of the foci of the ellipse. The size of the ellipse will depend on the satellite's mass and its angular velocity.
Kepler’s First Law is also known as – Laws of Ellipses.
Illustration of Kepler's laws with two planetary orbits. The orbits are ellipses, with foci F1 and F2 for Planet 1, and F1 and F3 for Planet 2. The Sun is at F1.
Kepler’s Second Law can be simply stated as follows: ‘during equal time intervals, a satellite covers equal areas in its orbital plane.’ The shaded area A1 represents the area covered by the satellite in a one-hour period near Earth. Kepler’s second law states that any other one-hour interval in the orbit will also cover an area equal to A1.
For example, the area A2, swept out by the satellite in one hour at apogee (the farthest point from Earth), is equal to A1. Therefore, A1 = A2. This result also shows that the satellite’s orbital velocity is not constant; it moves faster near Earth and slows down as it approaches apogee.
The orbital radius and angular velocity of a planet in an elliptical orbit vary. The planet travels faster when it is closer to the Sun and slower when it is farther from the Sun. Kepler's second law states that the blue sector has a constant area.
The same (blue) area is swept out in a fixed time period. The green arrow represents velocity. The purple arrow pointing toward the Sun represents acceleration. The other two purple arrows are the components of acceleration, parallel and perpendicular to the velocity.
Kepler’s Second Law is also known as – Law of Equal areas.
The shaded areas A1 and A2 are equal and are swept out in equal times by Planet 1's orbit.
The square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies. This is quantified as follows:
Under this condition, a specific orbit period is determined only by proper selection of the orbit radius. This allows the satellite designer to select orbit periods that best meet particular application requirements by locating the satellite at the proper orbit altitude.