Kepler’s laws of orbital motion in natural units

  Let’s apply the same dimensionless scaling framework to Kepler’s laws of orbital motion to uncover their universal, unit-invariant core. We’ll focus on the third law (the harmonic law), as it’s the most quantitative, but the principles extend to all three laws.

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1. Kepler’s Third Law (Traditional SI Form)

The standard form relates the orbital period $T_{\mathrm{S}\mathrm{I}}$ of a planet to the semi-major axis $a_{\mathrm{S}\mathrm{I}}$ of its orbit around a mass $M_{\mathrm{S}\mathrm{I}}$:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}{\mathrm{<span\; style="background-color:\; white;">M</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}}$$

Our goal is to rewrite this in terms of dimensionless ratios scaled by fundamental constants.

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2. Natural Unit Scaling

Using the same framework as before:

• Mass: $M_{\mathrm{S}\mathrm{I}}=m_n\cdot H{z}_{\mathrm{k}\mathrm{g}}$
  (where $m_n$ is the natural mass in Hz, $H{z}_{\mathrm{k}\mathrm{g}}=\frac{c^2}{h}$ is the SI scaling factor).

• Length/Semi-major axis: $a_{\mathrm{S}\mathrm{I}}=a_n\cdot c$
  (where $a_n$ is the natural "time" or inverse frequency of the orbit).

• Gravitational constant: $G_{\mathrm{S}\mathrm{I}}=G_n\cdot \frac{c^3}{H{z}_{\mathrm{k}\mathrm{g}}}$
  (where $G_n$ is the natural dimensionless coupling, here in ${\mathrm{s}}^2$).

• Orbital period: $T_{\mathrm{S}\mathrm{I}}=T_n\cdot 1$
  (since time is already a natural dimension in this framework).

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3. Substituting into Kepler’s Law

Plug these into the SI form:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">c</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">)</span>}$$

Simplify the denominator:

$${\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}$$

So the equation becomes:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}}$$

The $c^3$ terms cancel out:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}}$$

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4. Identifying the Dimensionless Core

Let’s rearrange to isolate a dimensionless ratio:

$$\frac{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

Here:

• $G_n\cdot m_n$ has units ${\mathrm{s}}^2\cdot \mathrm{H}\mathrm{z}={\mathrm{s}}^2\cdot {\mathrm{s}}^{-1}=\mathrm{s}$.

• $T_n^2$ has units ${\mathrm{s}}^2$.

• $a_n^3$ has units ${\mathrm{s}}^3$.

Thus, the left-hand side is:

$$\frac{\mathrm{<span\; style="background-color:\; white;">s</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">s</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">s</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}$$

So the entire equation reduces to a pure, dimensionless statement:

$$\boxed{{\displaystyle \frac{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">4</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}}$$

This is Kepler’s third law in its unit-invariant form, independent of meters, kilograms, or seconds. The traditional SI version is just this core relationship scaled by $c$ and $H{z}_{\mathrm{k}\mathrm{g}}$.

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5. Interpretation and Significance

(a) Universality of $4{\pi }^2$

• The dimensionless constant $4{\pi }^2$ is the same for all orbits (around a given mass) in any unit system. It arises purely from geometry (the $\pi$) and the inverse-square law of gravity.

• In Planck units ($G_n=1$, $c=1$, $\mathrm{\hslash }=1$), this reduces further to $m_n{T}_n^2=4{\pi }^2{a}_n^3$.

(b) Role of $G_n$

• $G_n$ acts as a coupling constant between mass and spacetime curvature. Its value here is in natural units (${\mathrm{s}}^2$), but the ratio $G_n\cdot m_n$ absorbs any unit dependence.

(c) Orbital Frequency Relationship

• Since $T_n$ is the orbital period in natural time units, we can define an orbital angular frequency ${\omega }_n=2\pi \mathrm{/}{T}_n$. Rewriting:

  $${\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}}{{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}$$

  This mirrors the classical ${\omega }^2=GM\mathrm{/}{a}^3$, but now in dimensionless form.

(d) Connection to Escape Velocity

• Recall that escape velocity $v_e=\beta c$, where $\beta =\sqrt{2G_n{m}_n\mathrm{/}{r}_n}$. For a circular orbit ($r_n=a_n$), we can relate the two:

  $${\mathrm{<span\; style="background-color:\; white;">\beta </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">(</span>\frac{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}}{{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}}<span\; style="background-color:\; white;">)</span>\text{<span style="background-color: white;">(Escape velocity)</span>}$$$${\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}}{{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}\text{<span style="background-color: white;">(Orbital frequency)</span>}$$

  Thus:

  $$\mathrm{<span\; style="background-color:\; white;">\beta </span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}{\mathrm{<span\; style="background-color:\; white;">a</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}\sqrt{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

  This shows how the escape velocity ratio $\beta$ is tied to the orbital dynamics.

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6. Extending to Kepler’s First and Second Laws

First Law (Elliptical Orbits)

• The first law states orbits are ellipses with the central mass at one focus. In dimensionless form:

  • The semi-major axis $a_n$ and eccentricity $e$ define the ellipse purely geometrically. No units are needed.

Second Law (Equal Area in Equal Time)

• The area sweep rate $\frac{dA}{dt}$ is constant. In natural units:

  • $A_n\propto a_n^2$ (dimensionless area, since $a_n$ is in natural time).

  • The sweep rate $\frac{d{A}_n}{d{T}_n}$ is a dimensionless constant (conserved angular momentum).

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7. Broader Implications

• Unification of Scales: The same dimensionless law applies to:

  • Planets orbiting stars ($a_n\sim$ years).

  • Electrons in Bohr-like orbits (if gravity were dominant).

  • Galactic dynamics (if dark matter halos are modeled this way).

• Quantum Gravity Hint: If space-time is discrete at small scales, $a_n$ and $T_n$ may have minimum values (e.g., Planck time), but the ratio $\frac{G_n{m}_n{T}_n^2}{a_n^3}=4{\pi }^2$ remains exact.

• Experimental Tests: Precision astronomy (e.g., exoplanet studies) could search for deviations from $4{\pi }^2$ as tests of modified gravity.

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Conclusion

Just like escape velocity, Kepler’s laws reduce to dimensionless relationships when expressed in natural units. The traditional forms are simply these universal ratios scaled by $c$ and $H{z}_{\mathrm{k}\mathrm{g}}$. This reveals that:

1. Orbital mechanics is fundamentally about geometry and ratios, not arbitrary units.

2. The constants $G,c,h$ are conversion factors, not intrinsic to the physics.

3. A deeper theory of gravity (quantum or beyond) must preserve these dimensionless cores.

This approach demystifies the "coincidental" exactness of Kepler’s laws and ties them to the same unit-invariant framework as escape velocity. Next, we could explore general relativity’s Schwarzschild metric in this light—where $\beta =1$ defines the event horizon!