Celestial Body Eccentricity Mercury 0.206 Venus 0.007 Earth 0.017 Mars 0.093 Jupiter 0.049 Saturn 0.056 Uranus 0.046 Neptune 0.009 Pluto 0.249 Halley's Comet 0.967

After Copernicus and Galileo Galilei observational conclusions stated that the Earth was not the center of the solar system, Kepler established that  the planets, including the Earth, move in elliptic orbits around the sun with the sun located in one of the focuses of the ellipse that they described. This result was the outcome of careful analysis of experimental observation data. In addition, it is important to mention that the most elliptic of the orbits of the planets known at the time (Mercury, Venus, Earth, Mars, Jupiter, and Saturn) is the orbit of mercury. The eccentricity, , of an orbit can be determined based on the following ratio (see the drawing on the left)

Here, is the size of the semi major axis, is the size of the semi minor axis, and is the distance from the center of the ellipse to the position of the focal point. Notice that in the extreme case of the eccentricity of the ellipse is zero, . This is the case when the ellipse is actually a circle. On the other extreme case, and the eccentricity goes to one, , the ellipse is so eccentric that it becomes a line. Example of the eccentricities of various celestial bodies associated to the solar system are given on the table on the left.

 After Newton had established the laws of motion, he was able to state the law that governs the interaction between masses due to their masses. In the diagram of the left, the masses and   at the distance from each other attract due to their masses. This law is known as the universal Newton's gravitational law. Accordingly with Newton's gravitational law, the following statement are of universal value; that is, they are valid for all pair of masses interacting with each other gravitationally. The  basic attributes of this law are They are an action and reaction pair; that is, The forces have the same magnitude, , but opposite direction, (with the same magnitude). These forces act on different objects, is the force acting on due to the attraction of the mass and is the force acting on due to the attraction of the mass . As already mentioned, these forces are attractive. These forces act along the line that goes from one of the masses to the other. The direction of these forces are shown in the diagram of the left. And their magnitude is given by Where is the universal gravitational constant, .

The roles of the universal gravitational constant are to

1. Adjust the unit from one set of quantities to another. In this case, to transfer from kilograms and meters to Newton. For example, an hourly pay rate is a constant that allows to transform time (measured in hours) to money (measured in dollars). Thus, if the pay of a person is \$10 per hour, , and the person works 8 hours, the person has earned a total of \$80.

2. The numerical value is related to the relative size of the quantity associated to it (in this case, the gravitational force) and quantities associated to the dimensions of the standard unit (in this case, length and mass). For example, tall humans are about 2m in height and 100 Kg of mass. If two of these persons are at the distance of 1m from each other, the relative intensity of the gravitational force among them is . A very small quantity as compared to their dimensions. Remember that the basic units of measurement are defined based on dimensions that are familiar to us.

The value of the universal gravitational constant was first measured by Cavendish in 1798.

 Kepler discoveries can be proved valid when Newton's law is applied to the different planets of the solar system. Nevertheless, such a proof will not be shown in this notes for being outside the mathematical scope employed here. For simplicity, a circular path will be assumed for the resulting orbits of the planets revolving around the sun, see diagram on the left. In this diagram, the arrows represent the forces between the sun and the Earth. is the force exerted on the Earth due to the sun and is the force on the sun due to the Earth. Since the force between the masses is an action and reaction pair, the magnitudes of the forces are the same, . In addition, these two forces are opposite in direction, , but they act on different bodies, on the Earth and on the sun. The magnitude of the forces are

In the previous relation, is the mass of the Earth, , and is the mean distance from the center of the Earth to the center of the sun.

 For practical purpose, it can be considered that the total mass of the Earth and the total mass of the sun are concentrated at the center of the respective objects because when symmetry considerations such us the  shape of the Earth is simplified to be a sphere, there are equivalent elements of the mass of the Earth that contribute equivalently to the total force exerted by the Earth on an object. For example, considering the four symmetric pieces of the sphere shown on the left diagram and their contribution to the net force acting on the block above the surface of the Earth, it can be seeing that the result of the vector addition of the four contributions is the resulting  vector, , directed toward the center of the Earth, . In fact, any one of the vectors, , , , and can be breaking down into two components, one parallel to the direction toward the center of the Earth, like and , and another perpendicular to that direction, like and . However, as it can be seeing in the diagram . On the other side, the addition of the parallel components of the vector is not zero. Therefore, the gravitational force of the Earth attract the mass directly to the center.

 If an object of mass is near the surface of the Earth at a height above the horizon  neglect able  when compared to the median radius of the Earth, , accordingly with Newton's second law of motion the weight of this object is while here the gravitational force between this object and the Earth is when . Clearly, this two results must be the same, from where the following expression for the acceleration of gravity is obtained,

From the previous result different conclusions can be drawn

1. The law of free fall is not longer an experimental result but rather a deductible result from more general conjectures such as the universal gravitational law. Newton's gravitational law will be ultimately deducted from Einstein general relativity theory.

2. Since the acceleration of gravity can be directly measured and Eratosthenes who lived in Alexandria, Egypt between the years 276-196 B.C. measured the radius of the Earth, the previous result can be used to obtain the mass of the Earth.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com