• Our goal is to find the real kinematical explanation of Kepler’s Rule.
  

1. If Newton’s voodoo force does not exist how would we explain orbits?
   
2. Which means what is the reason such a proportionality as Kepler’s Rule exist
   

\(R/T^2 = 1/R^2\) is an algebraic permutation of Kepler’s Rule usually written as \(R^3 \propto T^2\). I’d like to know if \(R/T^2 = 1/R^2\) an equality or a proportionality? Should I write it as R/T² ∝ 1/R² or as an equality R/T² = 1/R²? How can we tell. Explain.

[ DO THESE REQUIRE THE CONSTANT K ? ]
You’re right. Let me provide all formulas in proper LaTeX format throughout the entire response:

You’re asking about different algebraic forms of Kepler’s Third Law, and whether your manipulation is correct. Let me work through this.

First, let me verify your algebraic manipulation of \(R^3 = T^2\):

• Starting with \(R^3 = T^2\)
  
• Dividing both sides by \(T^2\): \(R^3/T^2 = 1\)
  
• Dividing both sides by \(R^2\): \(R/T^2 = 1/R^2\)
  

Yes, this is a correct algebraic manipulation, and both statements are equivalent.

You’ve also represented \(R^3 = T^2\) as RRR = TT as shorthand for \(R \times R \times R = T \times T\).

For the algebraic permutations of RRR = TT (or \(R^3 = T^2\)), here are all the equivalent forms:

first	second	comment
\[R^3\]	\[T^2\]
\[\frac{R^3}{T^2}\]	\[1\]
\[T^2\]	\[R^3\]
\[\frac{T^2}{R^3}\]	\[ 1\]
\[R\]	\[(T^2)^{1/3}\]
\[R\]	\[T^{2/3}\]
\[T\]	\[R^{3/2}\]
\[T\]	\[\sqrt{R^3}\]
\[\frac{R}{T^2}\]	\[\frac{1}{R^2}\]
\[\frac{T^2}{R}\]	\[R^2\]
\[\frac{1}{R^2}\]	\[\frac{R}{T^2}\]
\[R^2\]	\[\frac{T^2}{R}\]

\[R^3 = T^2\]
\[\frac{R^3}{T^2} = 1\]
\[T^2 = R^3\]
\[\frac{T^2}{R^3} = 1\]
\[R = (T^2)^{1/3}\]
\[R = T^{2/3}\]
\[T = R^{3/2}\]
\[T = \sqrt{R^3}\]
\[\frac{R}{T^2} = \frac{1}{R^2}\]
\[\frac{T^2}{R} = R^2\]
\[\frac{1}{R^2} = \frac{R}{T^2}\]
\[R^2 = \frac{T^2}{R}\]

These are all algebraically equivalent statements of Kepler’s Third Law, each highlighting different relationships between the variables \(R\) and \(T\).

\begin{align*} R^3 &= T^2 \\ \\ \\ \frac{R^3}{T^2} &= 1 \\ \\ \\ T^2 &= R^3 \\ \\ \\ \frac{T^2}{R^3} &= 1 \\ \\ \\ R &= (T^2)^{1/3} \\ \\ \\ R &= T^{2/3} \\ \\ \\ T &= R^{3/2} \\ \\ \\ T &= \sqrt{R^3} \\ \\ \\ \frac{R}{T^2} &= \frac{1}{R^2} \\ \\ \\ \frac{T^2}{R} &= R^2 \\ \\ \\ \frac{1}{R^2} &= \frac{R}{T^2} \\ \\ \\ R^2 &= \frac{T^2}{R} \end{align*}

• In older literature as in Newton’s Principia the \(3/2\) power is called sesquiplicate ratio.
  
• [From Densmore, p.309] The sesquiplicate ratio is the subduplicate of the triplicate. This is equivalent to taking the square root of the cube–that is, taking the \(\frac{3}{2}\) power.
  

I call “Kepler’s Rule” what is usually known as “Kepler’s Third Law”

Calling Kepler’s Rule, “Kepler’s Third Law” is a recent convention. Newton called Kepler’s Rule, “Kepler’s Rule”. This is from one of Newton’s early notebooks:

And the same year I began to think of gravity extending to the orb of the Moon,[1] and having found out how to estimate the force with which a globe revolving within a sphere presses the surface of the sphere,[2] from Kepler’s Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their Orbs I deduced that the forces which keep the Planets in their Orbs must be reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her Orb with the force of gravity the surface of the earth, and found them answer pretty nearly.

This is from Catalogue of Portsmouth Collection (Cambridge, 1888), Section 1, Division xi, number 41. Cited in Herivel, Background to Newton’s Principia, p.65.

1. It is important to distinguish this rule from the other two “laws” because Kepler’s Third Law is fundamental.
   
2. With the third law you can compute orbits, with other laws you cannot.
   
3. In the case of circular orbits the first two laws become redundant. If something is a true law of nature it cannot be made redundant. Law of nature do not have exceptions. Except Mercury, all other planetary orbits are practically circular.
   
4. Kepler’s Rule is a rule, a proportionality. Are Newton’s laws proportionalities. If not, they cannot be laws of nature but Newton’s wishful thinking.
   
5. This rule is also called harmonic law.
   

Kepler’s Third Law states that the square of the orbital period \( T \) of a planet is proportional to the cube of the semi-major axis \( r \) of its orbit:

\[ T^2 \propto r^3 \]

We can derive this relationship using Newton’s law of gravitation and centripetal force. Let’s start with the expression for the centripetal force that keeps a planet in orbit:

1. Centripetal Force and Gravitational Force:
   

   For a planet orbiting the Sun, the centripetal force required to keep the planet moving in a circular path is provided by the gravitational force between the Sun and the planet. The gravitational force \( F \) is given by:
   

   \[ F = \frac{G M m}{r^2} \]
   

   where:
   

   • \( G \) is the gravitational constant,
     
   • \( M \) is the mass of the Sun,
     
   • \( m \) is the mass of the planet,
     
   • \( r \) is the radius of the orbit (distance between the Sun and the planet).
     

   The centripetal force required for circular motion is also given by:
   

   \[ F = \frac{m v^2}{r} \]
   

   where \( v \) is the orbital speed of the planet.
   

2. Equating Gravitational and Centripetal Forces:
   

   Since the gravitational force provides the necessary centripetal force, we set the two expressions equal to each other:
   

   \[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]
   

   Canceling \( m \) from both sides:
   

   \[ \frac{G M}{r^2} = \frac{v^2}{r} \]
   

   Multiplying both sides by \( r \):
   

   \[ v^2 = \frac{G M}{r} \]
   

3. Expressing Orbital Speed in Terms of Period:
   

   The orbital speed \( v \) of a planet in a circular orbit is related to the orbital period \( T \) by the distance traveled in one orbit (the circumference of the orbit, \( 2 \pi r \)):
   

   \[ v = \frac{2 \pi r}{T} \]
   

   Substituting this into our previous equation for \( v^2 \):
   

   \[ \left( \frac{2 \pi r}{T} \right)^2 = \frac{G M}{r} \]
   

   Expanding and simplifying:
   

   \[ \frac{4 \pi^2 r^2}{T^2} = \frac{G M}{r} \]
   

   Multiplying both sides by \( T^2 r \):
   

   \[ T^2 = \frac{4 \pi^2}{G M} r^3 \]
   

4. Conclusion (Kepler’s Third Law):
   

   This shows that:
   

   \[ T^2 \propto r^3 \]
   

   which is Kepler’s Third Law. The square of the orbital period \( T \) is proportional to the cube of the orbital radius \( r \), with \( \frac{4 \pi^2}{G M} \) as the constant of proportionality for objects orbiting the Sun.
   

• comment here
  

• @zeynel, Kepler’s laws are for un-propelled bodies in orbit.
  
• They don’t have anything to do with bodies being accelerated by forces other than gravity.
  
• You can simply accelerate your body away from the Earth (and the Sun) on a trajectory that doesn’t follow a ballistic orbit, for as long as you want, if you want to approach light speed. – The Photon
  

• He seems to say that Kepler’s Rule applies only to bodies beign accelerated by gravity only
  
• They assume that orbits are Newtonian and are powered by Newton’s occult force which manifests itself as gravity
  
• To me orbits are inertial motion. You don’t need force to keep bodies in orbit.
  
• This is what I’m trying to prove
  

[chatgpt]

Yes, Kepler’s Third Law can indeed be derived from Kepler’s Second Law (the area law) along with the inverse-square law of gravity. Let me outline the process step by step:

[chatgpt]

I read in Cohen’s Guide to Newton’s Principia, p.14, this sentence: “It requires only the minimum skill in algebraic manipulation to combine the v²/r rule with Kepler’s Third Law in order to determine that in a system of bodies in uniform circular motion the force is proportional to 1/r² or as inversely proportional to the square of the distance.” Can you explain step by step the algebraic manipulation to achieve this?

Sure! The statement from Cohen’s Guide to Newton’s Principia refers to the derivation of the inverse-square law of gravitation by combining the formula for centripetal force with Kepler’s Third Law. Let’s break it down step by step:

—

[chatgpt]

I know that Huygens discovered the formula v²/r and later Newton also derived the same formula. And there are several modern derivations like vector derivation. Since v²/r does not include mass m of the body in circular motion, can we assume that centrifugal tendency is independent of the mass m of the body in circular motion?

Yes, you are correct to observe that the formula \( v^2/r \), which describes the centripetal acceleration of a body in circular motion, does not include the mass \( m \) of the body. This implies that the centripetal acceleration (and consequently the “centrifugal tendency,” if we consider the inertial reaction to centripetal force) is independent of the mass of the body in circular motion.

Let’s explore this idea in detail:

—

[1] And the same year I began to think of gravity extending to the orb of the Moon

1. Newton means “force” of gravity.
   
2. Yes, something extends to the moon but what? Not Newton’s occult force.
   
3. This quote explains the whole Newtonian scheme.
   
4. Here Newton talks about a force that varies as \(1/r^2\) and hi says he derived this Kepler’s Rule.
   

[2] a globe revolving within a sphere presses the surface of the sphere…

It seems Newton connects the spin rotation with orbital revolution but spin is not orbital motion. They obey different rules of motion.

• Newton did not find the underlying cause of orbital motion.
  
• There is no need for such a cause. Orbits are natural motion.
  
• If so, why do we have different constants for different systems? This must be related to density.
  
  • Does this imply that orbits are forceful?
    
  • Are satellites tied to the central body with an attractive force. In other words, does the Newtonian occult force hold the satellites in orbit the way the string of the sling hold the stone attached to it in orbit?
    
• For different systems, we use different constants for Kepler’s Rule for instance we use different constants for planets and for Jupiter’s satellites.
  
• We don’t even know if satellites are tied to the “central” body with an attractive force. This is what Newtonists assume. This idea is so entranched in human psyche we cannot even conceive such a system where there is no central body. But there are such systems. We observe system where there’s not central body. Since Newtonism is considered to be absolutely and unquestionably true, there must be some invisible “mass” loaded with gravitational force. This is how Newtonians reason.
  
  • I have a fundamental rule that I don’t want to break: A phenomenon is caused only by the terms in the mathematical expression used to compute the motions of the system. Invisible or implicit terms do not enter.
    
  • Scientific method forces us to uphold this principle.
    
  • So if I describe orbits perfectly with the ratio \(R^3/T^2\) I must conclude that orbits depend only on the radius and period.
    
  • But, we don’t make computations with \(R^3/T^2\) but with \(R_0^3/T_0^2 = R^3/T^2\). The constant term.
    
  • In order ta make computation with the proportional form of Kepler’s Rule we need to have 4 terms in the form of \(R_0^3/T_0^2=R^3/T^2\) where \(R_0\) and \(T_0\) are constants of the system.
    
  • Can we write Hook’s Law for the spring in this form as equality of ratios?
    
  • \(F_0/x_0 = F/x\) but \(F\) doesn’t make sense to me. We need to express force as something else.
    
• Newton did not discover a force of nature. He defined a supernatural cause that can set objects in motion from a distance. He then imposed that supernatural cause to nature. Newton’s Principia is a masterful attemp to make nature supernatural.
  

Ok, good. If what you said is true, taking Kepler’s Third law R³ :: T², we can only conclude from this law that orbits do not depend on Newtonian forces because Kepler’s Third Law does not include a force term. How can you explain this apparent paradox?

This is a great observation, and the resolution of this apparent paradox lies in understanding the scope and derivation of mathematical laws in physics.

• This may be an academic issue.
  
• The distances in geocentric model is relative
  

1. Our central problem is Newton’s branding of Kepler’s rule as Newton’s laws.
   
2. Is Kepler’s Rule forceful?
   
3. Does Kepler’s Rule define a forceful orbit?
   
   1. Here force is Newtonian supernatural force?
      
4. Kepler’s Rule is definition of density.
   
5. Why doesn’t we explore this further and in depth.
   
6. What is the geometrical representation of period \(T\) in Kepler’s Rule? Period must be inversely proportional to the arc length. Or directly proportional? The sballer the arc bigger the period. So it is inversely proportional.
   

1. Kepler Kuralı güçsüz yörüngeler mi gösteriyor?
   
2. Kepler kuralı’nın gerçek geometrik açıklamısı nedir?
   
3. Yoğunluk
   

I asked chatgpt if Kepler’s Rule was different than his first two laws:
Yes, that’s a good way to look at it. Kepler’s first two laws are descriptive because they tell us how planets move but don’t directly allow us to compute their orbits.

1. First Law (Elliptical Orbits): Planets move in ellipses with the Sun at one focus. This describes the shape of the orbit but doesn’t give a way to calculate positions over time.
   
2. Second Law (Equal Areas in Equal Time): A planet sweeps out equal areas in equal times. This describes variations in orbital speed but doesn’t give an explicit formula for predicting positions at arbitrary times.
   

However, Kepler’s Rule (Third Law) is different because it is quantitative and allows calculations. Since it states that

\[ T^2 \propto a^3 \]

it directly relates the orbital period \(T\) and the semi-major axis \(a\), making it useful for predicting and comparing orbits. With additional information (such as the gravitational constant and mass of the central body), Kepler’s Rule can be used to derive precise orbital parameters.

So, you could say the first two laws are mainly descriptive, while the third is computationally useful for determining orbits.

1. This is a good answer but chatgpt has Newtonian tendencies and I questions this sentence:
   

With additional information (such as the gravitational constant and mass of the central body), Kepler’s Rule can be used to derive precise orbital parameters.

2. This is physics propaganda or at least a misconception.
   
3. The so-called “gravitational constant” is nothing more than a unit conversion factor. Its purpose is to make the units work in the decorative flagship “equation” of the Cult of Newton: \(F=GMm/r^2\). This unit is strategically named to imply that Kepler’s Rule is Newtonian. But adding \(G\) to the Kepler’s Rule does not make Kepler’s Rule Newtonian, or forceful or occult.
   
4. What about the mass \(M\) of the central body?
   
5. \(M\) corresponds to the constant term in Kepler’s Rule, \(R_0^3/T_0^2\).
   
6. Is the constant term in Kepler’s Rule the Newtonian mass of the central body?
   
7. Obviously, here we need to define the word mass. What is mass in this context?
   
8. Yes, we need to limit our search to the Newton by saying “in this context” because the sophists and equivocators who call themselves “physicists” defined the word so many times that in physics effectively mass is a masterpiece of equivocation. There are as many definitions of mass as there are physicists.
   
9. Physicists may even talk about fields giving mass to particle. In this sentence only the word “to” has meaning.
   
10. This reminds me Dorothy Parker’s “Every word she writes is a lie, including ‘and’ and ‘the’.” Every word physicists write is a lie… but not true, the word “to” was not a lie in this case.
    

1. First we need to state explicitly what we mean by “Newton’s laws.
   
2. Kepler’s Rule is well defined but Newton’s laws are not.
   
3. In fact, in the context of Kepler’s Rule and computation of orbits only Newton’s force enters. Newton’s first law is just a fairy tale explanation of orbit formation. We already have orbits and we are computing some orbital data.
   
4. Kepler’s Rule is based on observations and it is necassarily true.
   
5. Newton’s laws are Newton’s absurd and supernatural suppositions. We cannot compute orbits by using Newton’s laws alone, whateved those laws are.
   
6. Does Newton derive Kepler’s Rule from his laws? We will be looking at this more carefully and strip all Newtonian propaganda from this so-called derivation.
   
7. The relationship betwenn Kepler’s Rule and Newton’s laws is part of the central argument.
   

1. Kepler’s Rule does not have a force term and it does not have a mass term. It only has two terms, radius and period.
   
2. This is the main problem we need to decipher: How do Newtonists insinuate force terms, implicitly of course, into Kepler’s Rule.
   
3. It looks like force is implied through mass. They assume that force is proportional to mass and if there is mass there must be force.
   
4. Then where is mass in Kepler’s Rule?
   
5. Mass in Kepler’s Rule enters through the \(G\) term. Mass is hidden inside \(G\). We see this only when deconstruct the units of \(G\).
   
6. [detailed analysis to be developed]
   
7. \(G\) is nothing more than the constant term of Kepler’s Rule expressed in Newtonian units and Newtonian jargon. So I need to again prove that these two are numerically equal.
   

1. Computing orbits with standard units named after Newton does not make Kepler’s Rule Newton’s Law.
   

Writing Kepler’s Rule like this \[\frac{r^2}{t^2}=\frac{1}{r}\] reminded me equality or conservation of kinetic and potential energies. Chatgpt clarifies, but I’ll have to think about this in more detail.
[chatgpt]

There is this statement in physics saying something like, velocity squared, or half of velocity squared, equals the potential or 1/r. Can you refresh my memory and state the correct law? Is this what is usually called the Lagrangian?

It sounds like you’re recalling a relation from orbital mechanics (or more generally from the analysis of central force problems), not the definition of the Lagrangian.

1. Each system has its own constant.
   
2. Newtonians explain this with their mass term.
   
3. Can this be density?