Wednesday, 6 February 2008

The Origin of Newton’s Gravitational Force

(Continued from The Final Theory)
Author:Mark McCutcheon

Add And Edited by:
Arip Nurahman
Department of Physics, Faculty of Sciences and Mathematics
Indonesia University of Education

The discussions so far have largely taken for granted that we are all very familiar with the Newtonian explanation of gravity as an attracting force that somehow emanates from matter; as such, the details and origin of this theory have not yet been addressed. If we could examine the progression of ideas that led to Newton’s theory of gravity, perhaps we could identify once and for all either where the overlooked power source may be for this force, or alternatively, how this fictitious force came to be invented.

The first publication of Newton’s Law of Universal Gravitation appeared in his famous work, widely known as “The Principia” today, published in 1687. In this publication Newton describes his proposed new force, showing how it explains our observations of falling objects and orbiting bodies, and even providing a simple and intuitive mathematical formula for calculating the strength of this gravitational force between any two objects. To arrive at this equation Newton would have had to follow the clues available to him at the time, both from his own experience and education as well as from the available astronomical data of his day. Let’s now follow the type of thought processes that would have led to Newton’s formal theory of a gravitational force.
At the time, a formal mathematical description of the orbits of moons and planets was already in existence – provided by Johannes Kepler (1571-1630) – based on the astronomical data of the day. In fact, Kepler’s three laws of planetary motion are very accurate and useful indeed, still remaining as some of the most important tools used in our space programs. Yet, despite this great achievement by Kepler, these laws only provided a mathematical description of planetary motion without explaining why and how this motion occurs. In essence, Kepler’s Laws described only the geometry of planetary motion, but not the physical reason for this geometry.
Prior to Newton’s Law of Universal Gravitation there were suspicions that some type of attracting force might be at work, but no one had managed to arrive at a solid theory or justification for such a force. Newton’s well-developed theory of a gravitational force finally managed to achieve this convincingly, bridging the gap between Kepler’s purely geometric laws of planetary motion and the strong suspicion that some type of attracting force in nature may underlie them. Newton’s Law of Universal Gravitation is now presented, followed by a consideration of its origins to see what insights can be gained into the source of the familiar, yet still quite mysterious gravitational force that we believe in today.

Newton’s Law of Universal Gravitation
There is an attracting force in nature emanating from all
objects, pulling them toward one another with a strength
that increases with their masses and decreases with the
distance between them squared.

According to this claim made by Newton, now considered a law of nature, the greater an object’s mass the greater its gravitational field strength, and this gravitational field diminishes rapidly in strength the further it extends out into space away from the object. Specifically, the strength of this gravitational force between any two objects is calculated by multiplying their masses together then dividing by the square of the distance between their centers. Finally, this result is multiplied by a constant, known as the gravitational constant, to present it in standard units of force. The resulting equation of the strength of the gravitational force, F, between two objects is written as:

where m1 and m2 are the masses of the two objects.
R is the distance (radius) between their centers.
G is a constant, called the gravitational constant.
This equation is known as the Law of Universal Gravitation. Yet this represented much more than just another equation when Newton introduced it. It ushered a completely new force of nature into our awareness and our science. It was not merely an abstract model of observations, but a statement of an actual force in nature emanating from objects – varying in strength with their mass, which we can lift, and their distance, which we can measure. This is a concept that we are now taught as children and have grown accustomed to, but it would have been truly revolutionary when it was first introduced in Newton’s day. Some had suspected that something of this nature might exist to explain falling objects and orbiting bodies, but Newton was the first to actually show that this force apparently did exist, and to describe it in very definite, concrete terms. Further, it is fairly straightforward to derive today’s Newtonian Orbit Equation from Newton’s Law of Universal Gravitation, as will be shown shortly, which very accurately predicts the motions of the planets and plays a central role in our space programs even today. All of this made Newton’s theory of gravity a revolutionary discovery, as well as apparently irrefutable proof of the existence of such a force in nature.

But where did this revelation come from? Somehow we went from a vague suspicion that an attracting force might be operating in the world around us, to a definite statement of its existence, its source in all material objects, and its precise behavior captured in an equation. How does something like this happen? The following investigation into this issue will help to clear up this mystery, showing that Newton’s gravitational theory is actually a completely superfluous and unnecessary invention that is based on a logically and scientifically flawed assumption. As a result of this invention, a crucially important equation for the orbits of planets was overlooked, then recast in Newtonian gravitational terms and presented as an entirely new equation – the Newtonian Orbit Equation that is currently in use today. The story of how this occurred and its enormous implications follows, showing surprising revelations about Newtonian gravitational theory.

An Alternate Origin
Although Newton provided a mathematical derivation for his law of gravity based on Kepler’s laws of planetary motion, the somewhat different derivation below provides a clearer picture of the origin of the gravitational force in our science, addressing the issues that still remain a mystery even today.
● Kepler developed three purely geometric equations of planetary motion involving no gravitational force, which described the heavens extremely well prior to Newton, and still do even today.
● A fourth purely geometric orbit equation of great importance
is easily identifiable in the astronomical data available at the
time, yet no formal record of this Geometric Orbit Equation
● Newton’s gravitational force equation can be easily arrived at
by equating the Geometric Orbit Equation to the equation for a
rock swung by a string, thereby inventing Newton’s force by
making the same rock-and-string assumption made by Newton.
● This assumed equality between swinging rocks and orbiting
planets is seriously flawed, leading to the unexplainable
mysteries and violations still present in Newtonian
gravitational theory today.
● The Newtonian Orbit Equation widely used today is derived
from Newton’s gravitational theory; however, this only
appears to give an entirely new and important orbit equation,
but is actually merely a disguised return to the original
Geometric Orbit Equation that pre-dated Newton.
● In actuality, Newton’s whole theory of gravity is a pure
invention with no scientific support, based on the pre-existing
Geometric Orbit Equation combined with a flawed rock-and-
string equality to orbits.

The Orbit Equation Actually Existed Prior to Newton
The analysis of the origin of Newton’s proposed gravitational force begins with Kepler’s three laws of planetary motion. Unlike Newton’s Law of Universal Gravitation and the Newtonian orbit equation that follows from it, Kepler’s laws are purely geometric descriptions of planetary motion based on observations of the heavens. They were arrived at prior to Newton’s theory of gravity, and make no reference to a gravitational force. These laws are as follows:
Kepler’s Laws of Planetary Motion
● Kepler’s First Law states that the planets move in oval-shaped
ellipses around the sun, with the sun at one end of the ellipse.
● Kepler’s Second Law states that as a planet proceeds in its
elliptical orbit, an imaginary line joining the sun and the planet
would always sweep out the same area in a given time period
regardless of where the planet is along its elliptical path.
● Kepler’s Third Law provides an equation that calculates the
average distance of a planet from the sun simply by measuring
the time it takes to make a complete orbit.

These three laws are very accurate, reliable, and central to our space programs today. However, an additional and very important geometric relationship regarding orbits can be readily seen in the astronomical data that would have been available to Kepler and Newton, yet it is missing from both Kepler’s Laws and Newton’s gravitational theory. In fact, there is no formal record of it at all in our scientific history. This purely geometric relationship is so “Kepler-ian” in nature that it is tempting to call it Kepler’s Fourth Law, but since this would obviously be inappropriate, we’ll call it the Geometric Orbit Equation:

The Geometric Orbit Equation
The Geometric Orbit Equation is a previously unrecognized, purely geometric equation embodying a relationship in the standard astronomical data showing that the orbital radius of any planet in our solar system (i.e. its distance from the sun) multiplied by the square of its velocity always gives the same constant value. This would be written as:

v2R = K, where K is a constant with the unchanging
value of 1.325 x 1020 [m3/s2]
R is the orbital radius of the planet
(distance from the sun)
v is the velocity of the planet

This relationship can be readily deduced from any standard table of planetary data that can be found in most introductory physics textbooks. The constant, K, is the same for all planets orbiting the sun, but differs for other orbital systems. For instance, the value of K for objects orbiting the Earth rather than the sun can be readily calculated as 3.7 x 1014 by referring to these same tables of planetary data. This value of K for our Earth-based orbital system would apply to the orbit of the moon, for instance, as well as the orbits of the various satellites and spacecraft about our planet.

This geometric orbit equation allows the distance of orbiting objects to be calculated if their speed is known. Perhaps more importantly, it allows for the planning or alteration of satellite and spacecraft orbits by indicating the speed required to achieve a given orbit, and the required speed change to transfer from one orbital trajectory to another. This type of calculation would underlie everything from fuel requirement planning for space shuttle missions to orbital insertion of satellites around Mars. Notably, the Geometric Orbit Equation pre-dates Newton and achieves these results in a purely geometric fashion, as its name implies, without any reference to masses or gravitational forces.

The Geometric Orbit Equation is the type of important astronomical observation that we might expect to be noticed and identified in the time of Kepler and Newton. Although there is no clear record of this occurring, the existence of this earlier geometric relationship provides an intriguing alternate derivation for Newton’s gravitational force and the final form of his Law of Universal Gravitation. To see this, we turn to the common analogy for planetary orbits taught in all elementary physics courses – the presumably equivalent scenario of a rock swung in a circle at the end of a string, as assumed by Newton.
The Rock-And-String Assumption

The idea of the moon being forcefully constrained by gravity to circle the Earth seems very reasonable at first, since we are all familiar with the seemingly similar concept of swinging a rock on the end of a string, causing it to “orbit” about us. Of course, this is not truly an orbit since it involves a physical length of string with clear physical tension throughout it as our muscles strain to keep the rock from flying off. This leads to the mysterious concept that the orbit of our moon involves a mysterious attracting force acting across space in a manner that is still unexplained by science, apparently forcefully keeping the moon from flying off without drawing on any power source. However, since this is the equivalence made by Newton and widely accepted today, we will follow this same assumed rock-and-string equivalence in this alternate derivation of Newton’s gravitational force.

Once this assumption is made, it may then seem reasonable to equate the force required to constrain the rock in a circular path about us with the gravitational force said to constrain the moon in its orbit about the Earth. The Centripetal Force Equation for calculating the force, F, required to constrain a rock swung by a string is well known, as it was in Newton’s day:
Centripetal Force Equation (“rock-and-string”)

F = mv2/R where m is the mass of the rock
v is the velocity of the rock
R is the radius of swing (string length)
Equating this with the scenario of gravitational orbits gives the picture of equivalence between all elements involved, as shown in Figure 1-2.

Fig. 1-2 Assumed Equivalence between Rock-and-String and Orbits
At this point, we have an equation for orbits (the Geometric Orbit Equation), an equation for a rock swung by a string (the Centripetal Force Equation), and an assumed equivalence between them. So then, it should be valid to combine these two separate equations to create one single equation that embodies this equivalence. This can be done by first rearranging the Geometric Orbit Equation in terms of its velocity parameter ( ), then substituting this velocity expression into the Centripetal Force Equation, resulting in the equation:

Hypothetical Gravitational Force Equation
F = mK/R2 where m is the mass of the orbiting body
K is the constant from the Geometric
Orbit Equation
R is the orbital radius, also from the

Geometric Orbit Equation

This new equation is a hybrid of the Geometric Orbit Equation and the Centripetal Force Equation, obtained by making the completely arbitrary assumption that swinging rocks are physically equivalent to orbiting objects – and not simply similar in appearance. This would mean that there must somehow be an actual physical force pulling on objects to constrain them in orbit, just as there is a physical tension force in the rock-and-string equivalent as shown in Figure 1-2. As we will see soon, this new equation forms the foundation of Newton’s Law of Universal Gravitation, and the force, F, is the first-ever occurrence of a hypothetical “gravitational force.”
This new hybrid equation marks the first appearance of an attracting gravitational force in our science.

As noted above, this new hybrid equation is no mere mathematical exercise, but the literal creation point for the supposed “gravitational force,” and the first point where a force of any kind appears in relation to orbits. Prior to this a description of orbits was already available, provided by the Geometric Orbit Equation, but in completely geometric fashion involving only velocity and distance, with no mention of an attracting force emanating from the mass of the orbiting body. Now we have an equation that implies a gravitational force may be at work, which is somehow directly related to the mass of the orbiting body, m, and diminishes with the square of its orbital radius, R.

While this would be an exciting result for a scientist in Newton’s day when this issue was a deep mystery and a very hot topic in science, we must keep in mind that this is still an unsupported hypothesis in the derivation so far. We went from a fully functional, purely geometric orbit equation to an equation implying that forces and masses are involved in orbits merely by making a few simple assumptions and mathematical manipulations. This hypothetical force is still just as mysterious as it always was in scientific circles, with no scientific explanation for why it should spring forth from matter and pull on other objects. However, this new equation does give form to this proposed force. Instead of being just a vague suspicion, now it has an equation describing it, an identifiable material source (presumably the mass, m, of the orbiting object), and the characteristic that it diminishes in strength with the square of the distance between the object and the orbited body. Whether or not this is based on pure assumption, it is certainly a very compelling result.

To review, at this point we have a hybrid equation involving mass and a force, resulting from the assumption that a rock swung forcefully by a string is equivalent to the otherwise purely geometric orbits in the heavens. This hypothetical gravitational force equation has the form:
F = mK/R2 – Hypothetical Gravitational Force Equation

This equation claims that there is an attracting force holding objects in orbit, whose strength varies directly with the mass of the orbiting object, diminishes with distance squared, and is also dependent on a mysterious constant, K, that differs from one orbital system to another. But what could this constant refer to?

Since this new, hypothesized gravitational force presumably emanates from the orbiting object, m, it stands to reason that it should also emanate from the object that is being orbited; therefore, we would expect the mass of the orbited body to appear in this equation as well. So then, if we assume that the constant, K, is actually the mass of the orbited body, we have a viable explanation. We know that the mass of the sun is a constant factor in all planetary orbits, but not in the orbit of our moon; the mass of the Earth is the constant factor in the orbit of our moon (and all man-made satellites) in our separate Earth-based orbital system. Therefore, it seems quite reasonable that this constant that differs between orbital systems may well be the mass of the orbited body, which is also a constant that differs between orbital systems. So then, replacing K by this second mass, m2, now gives our hypothetical gravitational force equation the form:

F = m1 m2 /R2 – Hypothetical Gravitational Force Equation
with K replaced by m2

The only remaining step is to make sure the results from this calculation are expressed in the units of force, and are reasonable values. Currently this equation multiplies two masses and divides by a distance squared, giving the units of [kg2/m2] – that is, kilograms squared per meter squared. These are not the proper units for a force, and the values that result when using reasonable estimates for the mass of the Earth or the sun as the larger mass, m2, are also millions of times too large to be sensible. However, this problem is easily solved by multiplying our equation by a value that reduces the results to within a reasonable range and alters the units into those of a force. This simply involves the arbitrary introduction of a constant of proportionality that has these qualities. However, if we now assume that our hypothetical gravitational force equation truly describes an actual attracting force in nature, then this arbitrarily invented constant of proportionality would have to be a true natural constant. Although all of this is still only an assumption, if true, this constant would become what is known as the gravitational constant, G, today, giving the final form:

F = G(m1 m2 )/R2 – Newton’s Law of Universal Gravitation
This is precisely the form of Newton’s Law of Universal Gravitation shown earlier and presented in his Principia.

As noted above, this final result is precisely the equation for the gravitational force that Newton presented in his Principia in 1687. Although this alternate derivation differs somewhat from that provided by Newton, it shows that the origin for his gravitational force can be clearly found in the Geometric Orbit Equation. Given this, we can now evaluate where our current belief in this force comes from, and the firmness of the foundation for this belief. We now know, for example, that there was no advanced knowledge or understanding of a hidden power source that led Newton to this belief. Instead, it is simply based on the assumption that the scenario of a rock swung by a string is the literal physical equivalent to that of objects in orbit. Yet the rock-and-string scenario does have an identifiable power source – our muscles, while the gravitational force maintaining orbits does not. Also, the rock-and-string scenario does have a physical explanation for the attracting force constraining the rock – the tension in the string, while Newton’s proposed gravitational force has no clear physical explanation. In short, the assumption that these two scenarios are equivalent is based more on their similarities in appearance as systems involving circling objects than on any verified physical equivalence.

Further, there are other physical systems that may have even more similarities to orbiting objects than a rock swung by a string; consider a rock swung by a spring, for example. One of the problems with the rock-and-string equivalence assumption is that the rock can be swung faster and faster while remaining the same distance away at the end of the string – the tension in the string simply increases. If this were a true physical equivalence to orbits then gravity would have to increase its attracting force to constrain a faster moving object at the same orbital distance. However, this does not happen, either in theory or in practice. Instead, orbiting objects that are given more forward thrust move further out into space, much the way the rock would if it were swung faster at the end of a stretchable spring instead of a rigid string.

So, as long as we’re making arbitrary intuitive guesses at familiar mechanisms that might possibly be a literal physical equivalent to orbiting objects, we would have to seriously consider abandoning the rock-and-string idea for that of a rock-and-spring. This is not to say that orbits are the physical equivalent of a rock-and-spring either – this model also has its limitations and problems, and is just as arbitrarily chosen since we are merely going on superficial similarities in appearance. Still, as an educated guess it is perhaps more functionally similar to orbits than the rock-and-string scenario upon which today’s gravitational theory is built, exposing the weak and arbitrary foundation of Newtonian gravitational theory.

Interestingly, if we used the rock-and-spring model, we would end up with an entirely different version of Newton’s Law of Universal Gravitation since the centripetal force equation for the rock-and-spring is different than for the rock-and-string. That is, this difference in the centripetal force equation for circling rocks using springs means that when we substitute the velocity from the Geometric Orbit Equation into the Centripetal Force Equation as we did before, the resulting expression for the gravitational force must also differ. Yet this resulting spring-based gravitational force equation would still give us a numeric value for the gravitational force, just as Newton’s current equation does. And although this numeric value is not directly measurable – even from Newton’s current equation – it gives the appearance of an actual force in nature; one whose strength we can even calculate, using the concrete attributes of mass and distance.

Therefore, the familiar form of Newton’s Law of Universal Gravitation is not a true law of nature, but merely a flawed invention based on superficial similarities in appearance between orbits and the very different scenario of a rock-and-string.

The preceding alternate origin for Newton’s gravitational force shows that the introduction of an attracting gravitational force in orbits was completely arbitrary and unnecessary, considering the contributions by the already existing body of purely geometric equations, i.e. Kepler’s three laws plus the Geometric Orbit Equation. But this is a fact that could not have been realized without this alternate derivation since the Geometric Orbit Equation is unknown to science, at least in the formal manner presented in this discussion. Instead, we have the Newtonian Orbit Equation today, derived from Newton’s Law of Universal Gravitation. Since this Newtonian orbit equation is central to our science of astronomy and our space programs, Newton’s theory of gravity is considered to be of immense importance as the origin of this equation. However, it is now possible to show that the Newtonian Orbit Equation is simply the pre-existing Geometric Orbit Equation in disguise. To see this, let’s take a closer look at the origin of the Newtonian Orbit Equation in use today.

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