The planet itself is a souourner in airless space, a wet ball flung across nowhere... Annie Dillard, "Soujourner", Teaching a Stone to Talk
Satellites are constrained to move in prescribed and patterned motions which we refer to as orbits, and the orientation of the orbit to earth will determine how and when different locations on the surface are viewed.
Isaac Newton (1642-1727) was a 23-year old scholar studying natural philosophy at Cambridge in 1664 when the subject of the motion of celestial bodies was of great interest (Newton was born in the year of Galileo's death, but he was very influenced by the idea's of Galileo). In 1665 the plague broke out and students were sent home, but Newton continued to ponder these new ideas. As the story goes, he was inspired seeing an apple fall from a tree and conceived that the same force which attracts the apple might also hold the moon in its orbit around the earth. In his own words, he says:
"And in the same year (1666) I began to think of gravity extending to the orb of the moon... I deduced that the forces which keep the planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth and found them answer pretty nearly. " Fundamentals of Physics, David Halliday and Robert Resnick, John Wiley and Sons, New York, 1974, pg. 248.
The Principia Mathematica Philosophiae Naturalis was published in 1687. This included Newton's statement of the laws of motion which define classical mechanics.
NEWTON'S LAWS
- Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it. (law of inertia)
- The rate of change of momentum of a body is equal to the impressed force and takes place in the direction in which the force acts. (F=ma)
- To every action there is always opposed an equal reaction.
Newton's law of Universal Gravitation can be expressed as follows:
(1.1)
This describes the attractive force between two point masses m1 and m2 separated by a distance r, where G is the Newtonian (Universal) gravitation constant (G=6.67259x10-11 N m2 kg -2). [Note: do not confuse G and g (earths acceleration due to gravity), they are very different. G is a scalar constant and has the same value for all pairs of masses, it has dimensions L3/MT2. Earth's gravity is a vector expressed in dimensions of acceleration, L/T2.] (For an enlightening and humorous discussion of the famous Cavendish experiments illustrating that all things exert attractive forces follow this link to "matters of gravity").
In his formulation of his gravitational theory, Newton used Johannes Kepler's (1571-1630) three laws of planetary motion. These were derived from empirical data collected by the astronomer Tycho Brahe (1546-1601; the last astronomer to make observations without a telescope). Kepler was Brahe's assistant and found he could base the following laws on regularities in the data.
KEPLER'S LAWS
- All planets revolve in elliptical orbits with the sun at one focus. (law of orbits)
- The radius from the sun to a planet sweeps out equal areas in equal times. (law of areas)
- The ratio of the square of the period of revolution of a planet to the cube of its semimajor axis is the same for all planets revolving around the sun.
(image of 4 inner planet orbits in our solar system, from John Walker's Solar System Live, which can be linked from the earthview website discussed below)
If we simply substitute satellite for planet, and Earth for sun, these same laws can be used to describe satellite motion.
If we make the simplifying assumption of a circular orbit, and assume the earth can be considered a sphere which we can treat as a point mass (which is what Newton assumed), then we can equate the centripetal force required to keep a satellite in orbit with the gravitational attraction, to get:
(1.2)
In this equation, the lefthand-side is just the centripetal force describing the circular orbit of a satellite of mass m orbitting at a distance r above the center of the earth moving with velocity v. The gravitational attraction between the satellite and the earth (me) gives rise to this centripetal force. Since we can divide both sides by m, it is clear that the orbit of a satellite is independent of its mass. We can use this equation to determine the period of a satellite (the time of one complete revolution). This is just the orbit circumference divided by the velocity, and making this substitution in the equation above results in the following:
(1.3)
However, it is not generally this simple, while a circular orbit is the goal for most meteorological satellites, as Kepler discovered, satellites do not travel in perfect circles. The exact form of a satellites elliptical orbit can be determined from Newton's laws of motion and the law of universal gravitation.
Now that you have gone through the geometry of orbits and their definition, some figures which illustrate what these different orbits look like should be helpful. The following tutorial provides a nice introduction to the two main satellite orbits we will be concerned with, geostationary and polar.
Tutorial on satellite coverages and orbits, prepared by David Johnson of the National Center for Atmospheric Research Mesoscale and Microscale Meteorology (MMM) Division.
3. Perspective
A nice tutorial on the signficance of orbital eccentricity of Earth's one natural satellite from John Walker at earthview.
4. Satellite Tracking Tools
Nasa J-Track Java Spacecraft Tracker
Skip right to the Tracking Tool
John Walker Earthview Satellite Tracking
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