Around 400 years ago, Johannes Kepler studied massive amounts
of accumulated astronomical observations of the planets, and he
came up with several Laws (a couple of which turned out to be
not true, such as the planets making music due to their motion!)
But Kepler did not have any understanding of WHY things were as
he calculated them to be. It turned out that his equations were
all very slightly incorrect.|
Around 330 years ago, Newton determined the Law of Gravitation, which provided an actual basis for what Kepler had found, and also provided the corrections for the slight errors. The basic Law is given by the following (simplified) equation:
This equation gives the Force which applies on any object due to a gravitational attraction of two masses M and m, when they are at a distance apart of r. G is a constant which gives the Force in Newton's (called the Gravitational Constant). μ is commonly used to represent the product of the central mass M and G.
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The quantities F and r are both VECTORS and not simply numbers. That means that each has both an amplitude (size) and a direction. Because of this, the above equation is not simply two numbers multiplied by each other, but what is called a Dot Product. IF they have the exact same direction, then it is simple multiplication. But if they are in DIFFERENT directions, where their direction is different by an angle that we will call θ, then the multiplication requires the sine of that angle, as:
If the Force is allowed to constantly change in either or both of amplitude and direction, and if the motion is not limited to being a straight line segment, then we have both values which are constantly varying vector quantities. With that we need the Integral Calculus version of the same Equation:
The f quantity represents the force which applies at any instant in any specific location. The fact that f exists everywhere, and for all times, means that we have a Force Field. For gravitation or electrostatic effects, we have the following equation (for the simplified case of being in a single x-y plane, where the three-dimensional form is similar but more complicated.)
You may have found Newton's Gravitation or Coulomb's electrostatics presented in a much simpler form! The form above is the technically correct form, for ANY type of motion in a gravitational or electrostatic field. IF we assume that the affected object will move in a uniform circular motion, then the square root of (x2 + y2) is always the radius of that circular orbit, and this is easily simplified into the form:
We can see that for a target which has unity mass or charge, this is simply Newton's Gravitation equation with k = μ, or similarly, it is Coulomb's Law of electrostatics with k being the central charge.
Above, we noted that the definition of Work or Energy is the (dot) product of the external force and the instantaneous radial distance. For discussion reasons, we will use the simplified form here, and we then have:
We have chosen to use r = infinity as one limit of the Integral. This results in the denominator becoming immensely large and the total quantity therefore disappearing to zero. We are essentially defining a Work Potential at infinity as being zero.
When we solve this simple Integral for both limits, we get the following:
Notice that ALL Work Potentials are therefore negative.
This situation is identically true for both gravitational and
electrostatic energy fields, as both are inverse-square
These two fields are confirmed as being Conservative Fields.
One consequence of this is that there is Conservation of Energy,
which is also called the First Law of Thermodynamics,
meaning that the total of kinetic energy and potential energy
remains constant unless external energy is either provided or
removed from the system. As we chose to define the potential
energy at infinity as zero, we similarly define the kinetic
energy there, for a total of zero total energy. As an object
falls in toward the source of the force field (whether a gravitational
mass or an electric charge) the potential energy becomes negative
(as given by
) This quantity MUST then equal the kinetic energy of motion of the target particle, which is given by the standard kinetic energy formula of:
If we keep with the simplified view of a circular orbit, then the velocity of motion in the circular orbit is equal to the length of a radian times the angular velocity, or:
We therefore have two quantities which must always total zero, in order to comply with Conservation of Energy:
We can therefore equate the two terms:
We can then cross-multiply to get 2 * k = m * ω2r3.
This confirms Kepler's Third Law which states:
The square of the sidereal period is proportionate to the cube of the distance from the Sun.
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago