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

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.

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

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.

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.