Introduction
A satellite is an object that is orbiting a planet or other huge body. Satellites can
be classified as natural satellites or man-made satellites. The moons and the planets classified as natural satellites. The satellites launched from earth for aims of scientific research, communication, weather predict, intelligence, etc. classified as man-made satellites. Whether planets, moons or artificial satellite, all satellite’s motion is subject to the same physical laws and mathematical equations.
In the 1500s, Nicholas Copernicus of Poland presented the heliocentric theory the belief that the earth revolves around the sun as it rotates on its axis, this aspect of astronomy evolved into an intricate study of planetary motion known as orbital mechanics. Today orbital mechanics is applied to spaceflight and satellites that orbit the earth or travel beyond our solar system.
In the early 1600s, Johann Kepler a German mathematician by using the data on planetary observations collected by the Danish scientist Tycho Brahe, he developed three laws of planetary motion.
Kepler's Three Laws
‘ Kepler’s First Law of Motion – Elliptical Orbits
There were three paradigms of the solar system at that time, but none of them
worked very well for Mars. First, the Ptolemaic model set the earth at the center, with the sun and planets orbiting it in circles. There was also Copernicus's heliocentric paradigm, which put the earth among the planets, revolving around the Sun. And finally, Tycho had his own model to suggest, which joint aspects of both, he put the earth at the center with the Sun and Moon orbiting it, but let the other planets orbit the Sun. all three systems relied upon circular orbits, because a circle was accepted as an ideal shape. Copernicus, Tycho and Galileo all have confidence that planets should move in circular paths, but the data just didn't fit. Instead, Kepler found that another form, the ellipse, works better. An ellipse is sort of like flattened circle, and it has some special properties, that the sum of the distances from every point on the curve to two other points is a constant. The two other points (represented in Fig. 1) are known as the foci of the ellipse. The further apart the two foci are the longer and skinnier the ellipse, and this ‘skinniness’ parameter is called ‘eccentricity’. Comets can have very irregular orbits, coming in highly close to the Sun before traveling back to the outer reaches of the solar system. On the other hand, in a perfect circle, the two foci would lie right on top of each other at the same location.
Kepler's first law is rather simple ‘ The orbit of a planet follows an ellipse with the sun at one focus.
‘ Kepler’s Second Law of Motion ‘ Equal Areas
Kepler's first law states simply that Mars rounds in an elliptical orbit, with the Sun at one foci of the ellipse. although he chose to list it first, Kepler only came to this conclusion after figuring out his second law, which says that if you draw a line from the Sun to Mars, and wait a fixed period of time, that line will sweep out a specific area as Mars moves along its orbit. what Kepler noticed was that this area is exactly the same no matter where in the orbit you are. this is often phrased as Kepler's "equal area in equal time" law, and this law works because Mars doesn't move at a constant velocity – it speeds up the closer gets to the Sun. So, if Mars is approaching perihelion, (the point in the orbit nearest to the Sun), it's traveling faster than if it's at aphelion, (the point that's farthest away) (Fig. 2). In the first case, the line connecting Mars to the Sun is very short, but because the planet is moving faster, it covers a lot of distance. In the second case, the line segment is much longer, but Mars also moves more slowly. Either way, the area swept out in a fixed amount of time is the same.
Kepler and his contemporaries could see that Mars doesn't move at a constant rate, but they didn't know why. This inverse relationship that Kepler proposed between distance from the Sun and orbital velocity could explain the puzzling observations of Mars' movements, but only if the orbit is an ellipse. A circular orbit would mean no change in distance from the Sun with time, and thus the velocity would be constant as well. these two statements – that Mars travels in an elliptical orbit and that its speed varies so that the Mars Sun line sweeps out equal areas in equal time – were generalized to include all planets in 1621, and they constitute Kepler's first and second laws of planetary motion. The second law, it turns out, is also a consequence of the conservation of angular momentum (which was not a concept
known to Kepler in the 17th century). Angular momentum is a measure of the amount of rotational motion in a body or system of bodies, like Mars and the Sun, and in the absence of outside forces, it's a fixed quantity. This implies a tradeoff between the distance at which Mars orbits and its velocity – like Kepler noticed. Just as an ice skater spins faster after pulling his arms closer to his body, Mars has to move faster when it gets closer to the Sun. Kepler statement that the area swept out by the Mars-Sun line is constant is equivalent to the statement that angular momentum is a constant as well – that is to say, that it's conserved.
‘ Kepler's Third Law of Motion – Law of Harmonies
Kepler published his third law ten years later, in his 1619 book admonishes Mundi, or the harmony of the world. By sifting through observations of how the planets move in their orbits, he noticed something curious. The time it takes for a planet to orbit the Sun once (its orbital period, P) is related to its average distance from the Sun (its semi-major axis, a) in a very specific way. The square of the period is proportional to the semi-major axis cubed. In fact, if we choose our units carefully, with a period in years (the time it takes for earth to go once around the Sun) and the semi-major axis in astronomical units, or AU ‘Astronomical Units’ (the distance between the Earth and the Sun, about 150 million kilometers), the proportionality becomes an exact equation. The square of the period is equal to the semi-major axis cubed. Earth's semi-major axis is 1AU (because that's the definition of an astronomical unit). Mars is farther from the Sun, it has a semi-major axis of 1.524 AU. We can use Kepler's third law to figure out how long it takes Mars to complete one orbit around the Sun ‘ that is, how many earth years make up one Mars year. the semi-major axis, 1.524 AU, cubed equals 3.54, and the square root of that gives us Mars orbital period. 1.88 earth years (Fig. 3).
Find Mars orbit around the Sun in years, using Kepler’s third law:
P^2=a^3
P ‘mars’^2='(1.524)’^3=3.54
P mars=1.88 Earth years
Sure enough, Mars take 687 Earth days to orbit the Sun, which is 1.88 of years.
Kepler's Laws are wonderful as a description of the motions of the planets. However,
they provide no explanation of why the planets move in this way. Moreover, Kepler's Third Law only works for planets around the Sun and does not apply to the Moon's orbit around the Earth or the moons of Jupiter. Isaac Newton (1642-1727) provided a more general explanation of the motions of the planets through the development of Newton's Laws of Motion and Newton's Universal Law of Gravitation.
Newton's Version of Kepler's Third Law
Using his laws of motion and his universal law of gravitation, Isaac Newton was able
to derive all of Kepler's laws. This gave us a deeper understanding of what made the planets orbit in the way they do. Newton also revealed the universality of Kepler's laws, that is, Kepler's laws not only applied to planets orbiting the Sun, but equally well four moons orbiting planets, stars orbiting galaxies centers, galaxies orbiting each other, artificial satellites orbiting the Earth, etc.
Newton's derivation of Kepler's third law was more general and more complete. Newton showed that masses were involved. The equation shown have the familiar cube of the semi-major axis ‘a’ and the square the orbital period ‘P’, but it also has the sum of the masses of the two objects as shown in (Fig. 4). That is, if one mass is orbiting another, the sum of their masses is equal to the semi-major axis distance cubed divided by the orbital period squared. There are some additional constants involved, but if we always have the period in units of years and the semi-major axis distance in astronomical units, then those constants all come out to equaling 1 and do not need to be included. this simplifies the equation.
Satellite Motion and Newton's Mountain Thought Experiment
A satellite is often thought of as being a projectile which is orbiting the Earth. But
how can a projectile orbit the Earth? Doesn't a projectile accelerate towards the Earth under the influence of gravity? And as such, wouldn't any projectile ultimately fall towards the Earth and collide with the Earth, thus ceasing its orbit?
If a man stands on a mountain and fires a projectile horizontally gravity will cause the path of the projectile to curve downward and it will strike the earth. However, if the man fires the projectile fast enough at a specific speed the curvature of its path due to gravity will match the curvature of the earth under it the projectile will then fall around the earth becoming an earth orbiting satellite. A projectile fired even faster will have a flight path away from the earth, but gravity will act to slow the projectile down change its flight path and pull it back toward Earth. If the projectiles velocity increased enough a velocity sufficient to escape the Earth's gravitational pull will be reached this velocity is known as the escape velocity, it is equal to about seven miles per second at the Earth's surface.