Basic Thermodynamics ~ J. Pohl ©

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Altitude of Geo-stationary Orbits

Modern rockets routinely place communications satellites in Earth orbit such that they maintain a constant position above and relative to Earth. Such orbits (which are possible only in the plane of the equator) are accomplished when the rocket "parks" the satellite at the proper altitude and with the proper velocity.

The plane of these orbits is the plane of the Earth equator. Once placed in that plane above Earth the proper distance and moving circularly at the proper speed, a geo-stationary satellite will remain stationary (relative to Earth) and directly above some point on the equator of earth. This permits Earth-surface communication satellite dishes to be "pointed" in a specific directions to constantly receive satellite telecasted information. The position of the satellite can be selected about the equator; its altitude cannot.

Calculate the altitude of a geostationary orbit.
♦  The figure depicts the satellite orbiting in the plane of Earth's equator. Coordinates and unit vectors have been included. Imagine an origin at the center of Earth and assume that center is moving in a straight line. (More on this later). Place the X axis locked (at mid-night) pointing at a distant star.

The time dependent position of the satellite is written as:

(1) 1

And the velocity is the derivative of the position:

(2) 2

And logically, "most" of the momentum of the satellite equals:

(3) 3

The above is the momentum of the satellite. we write Newton's Second Law:

(4) 4

Next the momentum is written as the mass times its velocity:

Now enter the momentum and write the gravity force.

(5) A longer equation needs more paper.

Perform the differentiation left of equality. Then gather terms.

(6) Write more compactly, as you go.

(7) Things got smaller? Is this right - check everything before this.

(8) 8

The above example describes position and velocity using sines and cosines with the usual, vector basis locked to Earth at its center. The z-axis aligns with the Earth-axis of rotation; that unit vector, K. The unit vector I is directed at a distant star (as is J). The only calculus needed is that one be able to differentiate the sine and cosine functions.

Altitude of Geo-stationary Orbits

Modern rockets routinely place communications satellites in Earth orbit such that they maintain a constant position above and relative to Earth. Such orbits (which are possible only in the plane of the equator) are accomplished when the rocket "parks" the satellite at the proper altitude and with the proper velocity.