The Invention of the Newtonian Orbit Equation

The Invention of the Newtonian Orbit Equation
(The Last 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

Throughout the following discussion it is important to keep in mind that the progression from the Geometric Orbit Equation to Newton’s Law of Universal Gravitation that was just shown is unknown to science, just as the formal Geometric Orbit Equation itself is unknown. Therefore, the following derivation of today’s Newtonian orbit equation from Newton’s Law of Universal Gravitation is currently believed to be the sole origin and form of the orbit equation in our science. The fully equivalent, pre-existing, and in fact, more proper Geometric Orbit Equation is unknown today, as is the flawed foundation of Newton’s Law of Universal Gravitation itself. This gives the appearance that the existence of today’s Newtonian orbit equation, as well as its tremendous contributions to astronomy and our space programs, is owed entirely to Newtonian gravitational theory. In actuality, however, this homage that is commonly paid to Newtonian theory is quite unfounded, as will now be shown.

The standard derivation of the Newtonian Orbit Equation in use today begins with the assumption that the rock-and-string scenario is equivalent to orbiting bodies in the heavens – a centuries-old assumption that is simply accepted unquestioningly today. Therefore, since Newton’s gravitational force and the rock-and-string centripetal force shown earlier are considered equivalent physical concepts today, the derivation of the Newtonian Orbit Equation starts by simply equating these two forces:

Newton’s Equation ® GmM/R2 = mv2/R ¬ Rock-and-String Equation

Here, the two masses, m1 and m2, in Newton’s equation are named m and M to signify the smaller mass, m, of the orbiting object and the typically much larger mass, M, of the orbited body. The above equality immediately simplifies to the familiar form of the Newtonian Orbit Equation that exists in our science today:

v2R = GM – Newtonian Orbit Equation

Note that, although this appears to be a completely new and important equation derived from Newton’s law of gravity, in actuality it is merely a reversal of the steps performed earlier in the derivation of Newton’s Law of Universal Gravitation from the original Geometric Orbit Equation. That is, where we started with the Geometric Orbit Equation and arrived at Newton’s Law of Universal Gravitation by making the (flawed) rock-and-string assumption, we now have simply used this same flawed assumption to work backwards from Newton’s equation to the original Geometric Orbit Equation again. The Newtonian Orbit Equation above looks a bit different from the Geometric Orbit Equation, but as we’ll soon see, this is only a cosmetic difference in appearance.

Today this fact is not recognized since Newton’s derivation for his Law of Universal Gravitation does not show its origin in the Geometric Orbit Equation. Therefore, it appears as if the orbit equation we use today is a perfectly valid Newtonian result derived solely from “solid gravitational theory.” Today, this mere reversal from Newton’s gravitational force equation to a disguised version of the Geometric Orbit Equation is unknown, lending unwarranted credibility both to Newton’s gravitational theory and to the assumed physical equivalence of the rock-and-string analogy. In actuality, the flawed rock-and-string analogy was used to invent Newton’s equally flawed equation of a gravitational force in the first place, then used again to undo this logic, merely arriving at a slightly disguised version of the only correct equation in this whole process – the original Geometric Orbit Equation.

This discussion literally means that although the Newtonian orbit equation above appears to differ from the geometric orbit equation, this is only a superficial appearance. A review of the earlier derivation for Newton’s gravitational equation shows that the constant, K, was essentially arbitrarily replaced with the two multiplied constants, GM. Recall that this occurred after assuming that K must refer to the mass of the orbited body, then realizing that the “natural constant,” G, had to be introduced to alter the size and units of the final result. But this switch from K to GM earlier was merely based on an arbitrary and unsupported assumption; as such, it is not only valid but also more correct to return to the original constant, K. Therefore, if we simply continue with the step-reversals that were started above and that led from Newton’s gravitational equation to the Newtonian Orbit Equation:

v2R = GM – Newtonian Orbit Equation

the next step in the reversal is to replace GM with K, giving the original Geometric Orbit Equation:

v2R = K – Geometric Orbit Equation

This means the Newtonian Orbit Equation used today, based on the Newtonian theory of gravity, provides exactly the same function as the Geometric Orbit Equation, which can be derived purely from astronomical observations without appealing to a gravitational force at all. Indeed, they are the same equation. In fact, this explains why the geometric orbit equation is unknown today – we already believe we have the proper gravitational version, including its reference to mass, M, and the “gravitational constant of nature,” G. Given this, there is no need to even take notice of the obvious, simple, and entirely equivalent geometric form that pre-dates our familiar orbit equation today. Yet, it is this very fact – that a simple and fully functional geometric form already exists – which is of such great significance, especially since we also widely use Kepler’s three laws in our science and space programs, which also have nothing to do with a gravitational force. This means that even when we use our Newtonian orbit equation, we are actually unknowingly using the geometric orbit equation, and so, all of astronomy as well as our space programs are actually based solely on geometry – and not on Newton’s gravitational force at all. The apparently insignificant fact that a simple geometric orbit equation can be easily identified which parallels our gravitational version is actually not so insignificant at all, but of great significance indeed.

Though not recognized today, Newton’s gravitational force is a completely superfluous and redundant abstraction, both in theory and in practice.

The above statement may seem premature since the Newtonian orbit equation involves the mass of the orbited body, M, while the geometric orbit equation has only an arbitrary constant, K. It might seem that, if nothing else, Newton’s gravitational theory shows that this constant actually refers to the mass of the orbited body, which could prove to be a very useful realization. In fact, one very important result from today’s Newtonian Orbit Equation is that it apparently allows us to calculate the mass of distant bodies, such as the planets in our solar system. That is, if we know the speed, v, with which an object is orbiting and the radius of its orbit, R, we can use the Newtonian Orbit Equation to calculate the mass, M, of the larger body it is orbiting. This would tell us the mass of a distant planet simply by observing the motion of its moons, for example, which is precisely how we have arrived at the values we believe to be the masses of the planets today.

Yet, if we used the Geometric Orbit Equation, knowing the speed and orbital radius of orbiting objects would only allow us to calculate the constant, K, for that orbital system rather than the mass of the body they are orbiting. Knowing the value of this constant for a particular orbital system is still very useful for calculating the speed or orbital radius of other orbiting objects in that system, but it would not tell us the mass of the orbited body. Therefore, it would appear that if we had never known of Newton’s gravitational theory we would not have been able to determine the masses of the moons, planets, and sun of our solar system – at least not by using Kepler’s three laws and a purely geometric orbit equation. And so, it might appear that Newton’s gravitational theory somehow provides a deeper physical meaning and insight into nature. However, the following discussion shows that this is not the case at all, and that it is merely an illusion that Newton’s gravitational theory provides any additional insight or utility beyond what was already possible prior to its introduction.

Newtonian Theory Does Not Give Mass-At-A-Distance

Newton’s theory of gravity claims that a gravitational force emanates from planets (and all objects) to act across space and out to remote distances, allowing a planet’s mass to be determined remotely since its mass is claimed to be directly related to the strength of this force. In particular, referring to the Newtonian Orbit Equation, v2R = GM, it would appear that we only need to note the velocity and orbital radius of an object in order to determine the mass of the body it is orbiting. However, the following discussion shows that it is only an illusion that mass can be directly determined at a distance in this manner.

● The orbit equation expresses a relationship between the speed

and the orbital distance of an orbiting object; in this respect,

both the geometric and Newtonian versions function equally.

● The known masses of moons and planets are merely

approximations based on an unsupported assumption that is

built into Newtonian theory – they are not the literal, accurate

masses we believe them to be.

● The above-mentioned assumption is that mass is directly

related to orbits – an assumption that is neither scientifically

proven nor entirely correct as it turns out, giving arbitrary,

inaccurate mass values.

● We are still able to use these inaccurate mass values in other

calculations of orbital velocity and distance since these

mass values are typically not used alone, but as part of the

expression GM, which is entirely equivalent to using the

original constant, K, in the original Geometric Orbit Equation.

We first begin by noting that whether we use the geometric or the Newtonian form of the orbit equation, the function of the orbit equation is to describe the relationship between the velocity and the orbital radius of an orbiting object. This role is equally fulfilled by either orbit equation since the Newtonian “gravitational” version is merely the original geometric equation with an arbitrary cosmetic change in the appearance of its constant, K. That is, we can arbitrarily change the symbol of the constant K in the geometric orbit equation into the two multiplied constants GM if we wish, creating the appearance of a new “gravitational” orbital equation but not actually altering the function of the original equation at all. The orbit equation still provides the same relationship between velocity and orbital radius as always, regardless of this cosmetic change.

However, since the value of K is easily determined by remote observation of orbiting objects, then arbitrarily changing K to GM would allow us to calculate M (since G is a known constant value), creating the illusion that we can remotely determine the mass of the orbited body. The possibility that K may actually be a direct reference to the mass of the orbited body is merely an interesting conjecture of Newtonian theory, but one that is both scientifically unproven and also irrelevant to our orbital calculations. This is an important point to note, since today we are under the illusion that we use the masses of moons and planets in the orbital calculations of our space missions. In actuality, we typically do not use these supposed masses alone, but as part of the expression GM. And as we now know, this expression is nothing other than the original constant, K, in the original Geometric Orbit Equation. The Newtonian exercise of redefining K as GM, solving for M, then using M in the expression GM is merely a winding path of logic disguising the fact that we are still simply using the original constant, K. The implied existence of a “gravitational force” in this circular Newtonian logic, as well as the supposed remotely-determined mass, are only conjectures at best – and at worst, pure fictions.

It is a powerful illusion that our current Newtonian orbit equation, v2R = GM, is the true original orbit equation, and that it contains an actual physical mass. This illusion arises because its purely geometric origins are well hidden under a compelling gravitational overlay. All of the previous discussions comparing Newtonian theory with the original Geometric Orbit Equation are impossible today, since this equation is not formally known in our science; its existence and significance have been buried for centuries beneath our unwavering and largely unquestioned Newtonian beliefs. We simply accept the mass of the sun listed in our textbooks, overlooking the fact that it was arrived at by plugging the known velocities and orbital radii of the planets into our current Newtonian orbit equation, which actually calculates K but disguises it as GM. We unknowingly accept that this hidden redefinition from K to GM is correct, arbitrarily turning a purely geometric constant calculated from purely geometric observations of our planets, into the solid mass of the sun. Without benefit of the analysis given in the previous discussions, we could not even know that we are making such an unsupported and arbitrary assumption. We believe in Newtonian gravity … we believe today’s orbit equation is solely a product of Newtonian theory … we believe the mass in today’s orbit equation describes a real mass … and we are fundamentally unable to contemplate the geometric origins of it all since they are firmly buried beneath these beliefs and illusions.

But then, it is natural to wonder if there remains any significance to the values listed as masses in our textbooks. Even though we may have arrived at these values by making the unsupported assumption that K is actually GM, it still seems reasonable that K must correspond to some material aspect of the orbited body. And further, the value of K does vary between different orbital systems in a manner that seems to reasonably reflect the expected mass differences between the central orbited bodies in these separate orbital systems. So, what are we to make of this situation?

This issue of mass will be more fully understood once the new principle in nature is introduced in the next chapter; however, for now it can be said that today’s mass values represent approximate masses – essentially reasonable educated guesses. This is because the observed gravitational effect that we call orbits (which does not involve a confirmed gravitational force unless proven scientifically viable) does indeed turn out to be related to the mass of the orbited body – though not directly related as assumed today. Therefore, our assumption that it is valid to arbitrarily replace the constant, K, in the orbit equation with the expression GM, involving the mass of the orbited body, is somewhat justified but inaccurate. That is, despite the fact that Newton’s model of a gravitational force emanating from matter cannot describe the true physical reality – for all the reasons mentioned so far – it still is undeniable that our massive planets and sun somehow cause our observations of falling objects and orbiting bodies. So then, since we know that one of the main defining qualities of our sun and planets is their mass, it would be expected that mass would be involved in our observations of the heavens – if not directly then at least indirectly. And as we will see in the next chapter, mass is only indirectly involved.

As an example of how mass might be indirectly involved in observations, just for illustration purposes lets consider a hypothetical scenario where all bodies in the heavens have an attracting magnetic field, but where we also have not discovered magnetism yet. In this case, we might tend to think that the mass of an object somehow directly causes the attraction that we observe in orbits, which would mean that an object with double the observed attraction must have double the mass. However, unknown to us, the doubled attraction would actually be due to double the magnetic field, which may or may not correspond to double the mass depending on whether magnetic field strength is correlated with mass in a direct one-to-one relationship. If two objects with the same mass but different material composition could have different magnetic field strengths, then this direct relationship would not hold. An observation of double the orbital attraction may be caused by a planet with only 30% more mass than another (though mass of a different material), yet our assumption of a direct relationship between orbital observations and mass would cause us to incorrectly list that planet as having double the mass.

This is similar to today’s belief that mass is directly related to orbital observations. This direct mass relationship supposedly occurs via Newton’s mysterious “gravitational force” – a force that has never been felt or detected remotely, but whose strength is said to directly mirror any changes in mass. So, if our Newtonian calculations tell us that an orbital observation corresponds to double the gravitational pull, we note the orbited body to have double the mass. However, the new principle in the next chapter shows that orbits are not caused by a “gravitational force,” and that, although the actual cause is related to mass, the relationship is not strictly a direct one-to-one correspondence. It is a reasonable assumption that a larger planet with a greater effective gravitational influence on orbiting objects would also have a correspondingly greater mass, but this assumption cannot be verified with certainty from a distance. It would be necessary to physically analyze the material composition of the planet to know for sure. This is analogous to the hypothetical magnetic field scenario, where a stronger influence on orbits (a greater magnetic field in this case) would seem to imply a correspondingly greater planetary mass, but could simply be due to a different magnetic material regardless of mass.

It is for this reason that the accepted masses today of the sun, planets, and moons of our solar system were stated earlier to be only approximations – not true mass measurements. Some of these values may be very close to the actual mass of the body, while others may be far off the mark. This has not been a problem for most standard orbital calculations since, as mentioned earlier, we typically use these mass values in the expression GM, which simply returns us to the constant K in the original Geometric Orbit Equation, and makes the actual individual mass value irrelevant. However, it is important to understand this mass issue for other reasons. For example, planetary geologists cannot gather a proper understanding of planetary formation, composition, and geology if the assumed mass is far from the actual mass of the planet. Also, theoretical fusion reaction calculations for our sun include mass in their calculations, and it may well be crucial to have the correct mass value for our sun in order to properly understand the physics of fusion itself.

Despite all of the preceding discussions suggesting that orbits are not ruled by Newton’s mass-based gravitational force, there can still be some compelling illusions that appear to support Newton’s theory. One such example from our space programs is the need to include the mass of our spacecraft in all trajectory calculations – even down to the diminishing weight of the fuel as it is expended or the additional weight of any rock samples that may be carried back to Earth from a distant moon or planet. If the mass of our spacecraft is an important consideration in the accuracy of our current trajectory calculations, doesn’t the success of most missions validate our Newtonian calculations and beliefs?

The answer is that the mass of the spacecraft is only important to the inertial calculations of the mission – not the orbital calculations. Inertial calculations involve any attempt to forcefully alter the trajectory of the spacecraft using a fuel burn. Just as the mass of a football player is of crucial importance to any player attempting a tackle, the precise mass of the spacecraft is of crucial importance to know how much fuel to burn for a given maneuver. A more massive spacecraft requires a longer or more powerful fuel burn, just as a heavier football player is harder to tackle. This is merely a classical Newtonian inertial calculation (not a gravitational one), given by Newton’s equation F = ma (force equals mass times acceleration). The fact that such mass-based inertial calculations are crucial to any space mission lends unwarranted credibility to the illusion that mass is further useful and necessary in our current Newtonian “gravitational” orbit calculations. Orbits (which form the basis of all spacecraft trajectories) are still completely described by the purely geometric equations of Kepler and the Geometric Orbit Equation, which do not involve mass or force.

Does the Evidence Support a Gravitational Force?

Despite the fact that Newton’s concept of a gravitational force violates our laws of physics and is unnecessary to describe orbits and spacecraft trajectories, it is still credited with explaining many other facets of life on Earth. For example, the reason objects have weight here on Earth is supposedly because a gravitational force emanates from our planet and pulls them down, forcefully and continuously holding them in place in proportion to their mass and giving them their mass-dependent weight. Even though we have no scientifically viable explanation for this constant pulling force, it would certainly appear as if such a force existed, nonetheless.

Yet, we have always known that something creates this effect, even before Newton arrived on the scene, but it wasn’t necessarily considered to be an attracting gravitational force from within the planet. It could have been due to the Earth’s magnetic field, or some type of downward repelling force from the stars in the heavens above, or any manner of other ideas. The weight of objects was simply an experience that was undeniable and common sense – no one expected objects to fall up when they were dropped – but the underlying cause could have been almost anything; it was simply unknown. We design spring-loaded measuring scales that we deliberately calibrate to properly weigh objects, but this is merely a device that takes advantage of this obvious weight effect all around us. Our mechanical scales are not actually based on a gravitational force principle, but rather, on a spring principle that takes advantage of whatever is causing the weight effect around us.

Even the science of calculating how a projectile, such as a cannonball, flies through the air is not actually based on Newton’s gravitational force, though this is commonly thought to be the case today. The work of Galileo Galilei (1564-1642) provided a very useful constant-acceleration equation for falling bodies or flying cannonballs, but a quick look at this equation shows no particular reference to a gravitational force:

d = ½at2 – Constant-Acceleration Equation

This equation essentially states that the vertical distance, d, that an object falls as it is either dropped or shot through the air is determined by a constant downward acceleration upon it, a, multiplied by the square of the time, t, that it takes to hit the ground. It is worth noting that this equation is a purely geometric equation involving no physical masses or forces, merely embodying the obvious fact that objects in free-fall experience a constant downward acceleration effect. It does not state the cause of this effect any more than the cause for the weight of objects was universally settled upon prior to Newton. This observable and measurable downward acceleration effect on Earth is the same for all objects no matter how massive they are, and can easily be measured to be 9.8 m/s2 and substituted directly into the above equation to give:

d = ½(9.8)t2

We typically use the symbol, g, for this measured constant-acceleration effect upon earthbound objects, giving us:

d = ½gt2

The symbol, g, is taken to mean the acceleration due to gravity (9.8 m/s2), in reference to Newton’s proposed gravitational force; but that interpretation, of course, is only an assumption.

Equal Acceleration Regardless of Mass

As mentioned above, whatever the cause may be for the acceleration effect of falling objects, it manages to accelerate all objects with equal ease at the same rate and with no noticeable stresses upon them. This is true whether they are as light as a golf ball or as massive as an ocean-liner. If a force were at work here, it would have to be quite a mysterious and unprecedented force indeed to achieve such a feat.

The “Gravity Shield” Mystery

Another ongoing mystery surrounding gravity is the idea of a “gravity shield.” After all, by using various materials we are able to insulate against electricity, electric fields, magnetic fields, light, radio waves, and radioactivity, so why not the gravitational field as well? Since science has never had a clear understanding of gravity, it has been impossible to either conceive of or rule out the possibility of developing some material or device to shield us from gravity. Such an invention would allow an object to levitate in mid-air simply by inserting this gravity shield between the object and the ground. If the attracting force of gravity cannot reach up past the gravity shield, then any objects above the shield should float and not be pulled downward. Such ideas have surfaced repeatedly over the years (and continue still), being shrouded in secrecy and mystery, and drawing short-lived interest and funding until ultimately fizzling out.

The preceding discussions have shown that, while Newton’s proposed gravitational force is a very compelling and intuitive idea, it is rife with problems. As a model of the true, and as-yet-unknown, underlying cause for many observations it has proven very useful – which is the purpose of any model or equation – but things become very problematic and mysterious when the model is taken as the literal reality. And in fact, as was also shown, Newton’s model is not even strictly necessary, as everything from falling apples to orbiting moons can be dealt with equally well with purely geometric equations. This model is part of our scientific legacy from centuries past, and as such, it sits largely unquestioned in our science today despite the fact that it clearly is not a scientifically viable theory.

We have tried applying logical patches, such as the misapplied Work Function, and even invented entirely new theories, such as General Relativity Theory – but to no avail. We have been unable to find true scientific justification for Newton’s gravitational force, yet we also have been unable to develop a truly viable theory to completely replace it. As a result, Newtonian gravitational theory remains our main and most compelling explanation for falling objects and orbiting bodies, while also clearly being a fatally flawed theory in our science.

The reason Newton’s gravitational explanation was so revolutionary when it was proposed is that it was thought to have finally provided a physical understanding of the underlying cause for these observations – something mankind had wondered about through the ages. However, if a gravitational force is not a viable scientific explanation for the underlying cause, then what is? An answer to this question that provides a clear physical explanation for gravity and resolves all of the mysteries and violations mentioned so far is provided in the following chapter, where a new principle in nature is presented – one that has been overlooked so far in our science.