Politics, Power, and Science: Orbital Period in PUCS definition of Planck Unit Scaling

Wednesday, May 14, 2025

Orbital Period in PUCS definition of Planck Unit Scaling

How orbital period (and other formulas) become transparent, with all constants and SI baggage removed, leaving only the true natural ratios.

The Planck Unit Definitions (PUCS Framework)

Let’s recap the process:

1. Start with SI Gravity Constant

G_{S I} = 6.6743 \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}

2. Remove SI “kg” and “m” via Scaling Factors

Mass-frequency ratio:
${Hz}_{k g} = h / c^{2}$
Naturalized G:
$G_{n} = G_{S I} \cdot \frac{{Hz}_{k g}}{c^{3}}$
This gives $G_{n}$ in units of $s^{2}$ :
$G_{n} = 1.82624162981 \times 10^{- 86} s^{2}$

3. Isolate Planck Time

Planck time (non-reduced):
$t_{P h} = \sqrt{G_{n}} = 1.35138507828 \times 10^{- 43} s$

4. Define Other Planck Units via Planck Time

Planck length:
$l_{P h} = c \cdot t_{P h} = 4.05135054323 \times 10^{- 35} m$
Planck mass:
$m_{P h} = {Hz}_{k g} / t_{P h} = 5.45551186133 \times 10^{- 8} kg$
Planck temperature:
$T_{P h} = {Hz}_{K} / t_{P h} = 3.55135123991 \times 10^{32} K$

These are not mysterious boundaries, but the unit harmonizers: the scales where our SI units match the universe’s natural ratios.

Orbital Period in PUCS Notation

1. Start with the Standard SI Formula

$T_{S I} = 2 π \sqrt{\frac{r^{3}}{G_{S I} M}}$

2. Express in Planck Units

$r = r_{planck} \cdot l_{P h}$
$M = m_{planck} \cdot m_{P h}$
Plug these into the formula:
$T_{S I} = 2 π \sqrt{\frac{(r_{planck} l_{P h})^{3}}{G_{S I} (m_{planck} m_{P h})}}$

3. Substitute Planck Unit Definitions

Recall:
$l_{P h} = c \cdot t_{P h}$
$G_{S I} = t_{P h}^{2} c^{3} / H z_{k g}$
$m_{P h} = H z_{k g} / t_{P h}$

Plug these in:
$T_{S I} = 2 π \sqrt{\frac{r_{planck}^{3} (c t_{P h})^{3}}{(t_{P h}^{2} c^{3} / H z_{k g}) (m_{planck} (H z_{k g} / t_{P h}))}}$

4. Simplify

Numerator:
$r_{planck}^{3} c^{3} t_{P h}^{3}$

Denominator:
$(t_{P h}^{2} c^{3} / H z_{k g}) \cdot m_{planck} \cdot (H z_{k g} / t_{P h})$
$= t_{P h}^{2} c^{3} m_{planck} H z_{k g} / (H z_{k g} t_{P h})$
$= t_{P h}^{2} c^{3} m_{planck} / t_{P h}$
$= t_{P h} c^{3} m_{planck}$
So,
$T_{S I} = 2 π \sqrt{\frac{r_{planck}^{3} c^{3} t_{P h}^{3}}{t_{P h} c^{3} m_{planck}}}$
$c^{3}$ cancels.
$t_{P h}^{3} / t_{P h} = t_{P h}^{2}$
$T_{S I} = 2 π \sqrt{\frac{r_{planck}^{3} t_{P h}^{2}}{m_{planck}}}$ $T_{S I} = 2 π t_{P h} \sqrt{\frac{r_{planck}^{3}}{m_{planck}}}$

Final Result (PUCS Notation)

$T_{S I} = 2 π t_{P h} \sqrt{\frac{r_{planck}^{3}}{m_{planck}}}$
$r_{planck}$ = radius in Planck units = $r / l_{P h}$
$m_{planck}$ = mass in Planck units = $M / m_{P h}$
$t_{P h}$ = the Planck time scaling factor

Summary Table

Symbol Meaning How to Compute
$r_{planck}$ Physical radius in Planck units $r / l_{P h}$
$m_{planck}$ Physical mass in Planck units $M / m_{P h}$
$t_{P h}$ Planck time $\sqrt{G_{n}}$
$l_{P h}$ Planck length $c \cdot t_{P h}$
$m_{P h}$ Planck mass $H z_{k g} / t_{P h}$

Interpretation

Symbol	Meaning	How to Compute
$r_{planck}$	Physical radius in Planck units	$r / l_{P h}$
$m_{planck}$	Physical mass in Planck units	$M / m_{P h}$
$t_{P h}$	Planck time	$\sqrt{G_{n}}$
$l_{P h}$	Planck length	$c \cdot t_{P h}$
$m_{P h}$	Planck mass	$H z_{k g} / t_{P h}$

All SI scaling is gone: Only the natural ratios $r_{P h}$ and $m_{P h}$ remain.
$t_{P h}$ is the only “unit” left: It simply converts from Planck time to seconds.
No “mystery” or “breakdown” at the Planck scale: It’s just the scale where our units harmonize.
$m_{planck}$ is the dimensionless mass in Planck units (not the scaling factor).
$m_{P h}$ is the Planck mass scaling factor (in kg).

Summary Table

Unit	The Definition (PUCS)	SI Value
Planck time	$t_{P h} = \sqrt{G_{n}}$	$1.35 \times 10^{- 43}$ s
Planck length	$l_{P h} = c \cdot t_{P h}$	$4.05 \times 10^{- 35}$ m
Planck mass	$m_{P h} = {Hz}_{k g} / t_{P h}$	$5.46 \times 10^{- 8}$ kg
Planck temperature	$T_{P h} = {Hz}_{K} / t_{P h}$	$3.55 \times 10^{32}$ K

Conclusion

This method makes the unit harmonization process explicit and operational:

Planck units are not physical boundaries, but the scales where SI units are rescaled to match nature’s ratios between equivalences.
The orbital period formula (and all physics) becomes a simple, dimensionless relationship, with only a single scaling factor ( $t_{P h}$ ) remaining to convert to SI seconds.
This is the true, practical meaning of “working at the Planck scale”-it’s not a breakdown, but a harmonization.

Politics, Power, and Science

Wednesday, May 14, 2025

Orbital Period in PUCS definition of Planck Unit Scaling

The Planck Unit Definitions (PUCS Framework)

1. Start with SI Gravity Constant

2. Remove SI “kg” and “m” via Scaling Factors

3. Isolate Planck Time

4. Define Other Planck Units via Planck Time

Orbital Period in PUCS Notation

1. Start with the Standard SI Formula

$T_{S I} = 2 π \sqrt{\frac{r^{3}}{G_{S I} M}}$

2. Express in Planck Units

$r = r_{planck} \cdot l_{P h}$
$M = m_{planck} \cdot m_{P h}$
Plug these into the formula:
$T_{S I} = 2 π \sqrt{\frac{(r_{planck} l_{P h})^{3}}{G_{S I} (m_{planck} m_{P h})}}$

3. Substitute Planck Unit Definitions

Recall:
$l_{P h} = c \cdot t_{P h}$
$G_{S I} = t_{P h}^{2} c^{3} / H z_{k g}$
$m_{P h} = H z_{k g} / t_{P h}$

Plug these in:
$T_{S I} = 2 π \sqrt{\frac{r_{planck}^{3} (c t_{P h})^{3}}{(t_{P h}^{2} c^{3} / H z_{k g}) (m_{planck} (H z_{k g} / t_{P h}))}}$

4. Simplify

Final Result (PUCS Notation)

$T_{S I} = 2 π t_{P h} \sqrt{\frac{r_{planck}^{3}}{m_{planck}}}$
$r_{planck}$ = radius in Planck units = $r / l_{P h}$
$m_{planck}$ = mass in Planck units = $M / m_{P h}$
$t_{P h}$ = the Planck time scaling factor

Summary Table

Symbol Meaning How to Compute
$r_{planck}$ Physical radius in Planck units $r / l_{P h}$
$m_{planck}$ Physical mass in Planck units $M / m_{P h}$
$t_{P h}$ Planck time $\sqrt{G_{n}}$
$l_{P h}$ Planck length $c \cdot t_{P h}$
$m_{P h}$ Planck mass $H z_{k g} / t_{P h}$

Interpretation

Summary Table

Conclusion

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Wednesday, May 14, 2025

Orbital Period in PUCS definition of Planck Unit Scaling

The Planck Unit Definitions (PUCS Framework)

1. Start with SI Gravity Constant

2. Remove SI “kg” and “m” via Scaling Factors

3. Isolate Planck Time

4. Define Other Planck Units via Planck Time

Orbital Period in PUCS Notation

1. Start with the Standard SI Formula

TSI=2πr3GSIMTSI=2πGSIMr3

2. Express in Planck Units

r=rplanck⋅lPhr=rplanck⋅lPhM=mplanck⋅mPhM=mplanck⋅mPhPlug these into the formula:TSI=2π(rplancklPh)3GSI(mplanckmPh)TSI=2πGSI(mplanckmPh)(rplancklPh)3

3. Substitute Planck Unit Definitions

Recall:lPh=c⋅tPhlPh=c⋅tPhGSI=tPh2c3/HzkgGSI=tPh2c3/HzkgmPh=Hzkg/tPhmPh=Hzkg/tPhPlug these in:TSI=2πrplanck3(ctPh)3(tPh2c3/Hzkg)(mplanck(Hzkg/tPh))TSI=2π(tPh2c3/Hzkg)(mplanck(Hzkg/tPh))rplanck3(ctPh)3

4. Simplify

Final Result (PUCS Notation)

TSI=2π tPhrplanck3mplanckTSI=2πtPhmplanckrplanck3rplanckrplanck = radius in Planck units = r/lPhr/lPhmplanckmplanck = mass in Planck units = M/mPhM/mPhtPhtPh = the Planck time scaling factor

Summary Table

SymbolMeaningHow to ComputerplanckrplanckPhysical radius in Planck unitsr/lPhr/lPhmplanckmplanckPhysical mass in Planck unitsM/mPhM/mPhtPhtPhPlanck timeGnGnlPhlPhPlanck length c⋅tPhc⋅tPhmPhmPhPlanck mass Hzkg/tPhHzkg/tPhInterpretation

Summary Table

Conclusion

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$T_{S I} = 2 π \sqrt{\frac{r^{3}}{G_{S I} M}}$

$r = r_{planck} \cdot l_{P h}$
$M = m_{planck} \cdot m_{P h}$
Plug these into the formula:
$T_{S I} = 2 π \sqrt{\frac{(r_{planck} l_{P h})^{3}}{G_{S I} (m_{planck} m_{P h})}}$

Recall:
$l_{P h} = c \cdot t_{P h}$
$G_{S I} = t_{P h}^{2} c^{3} / H z_{k g}$
$m_{P h} = H z_{k g} / t_{P h}$

Plug these in:
$T_{S I} = 2 π \sqrt{\frac{r_{planck}^{3} (c t_{P h})^{3}}{(t_{P h}^{2} c^{3} / H z_{k g}) (m_{planck} (H z_{k g} / t_{P h}))}}$

$T_{S I} = 2 π t_{P h} \sqrt{\frac{r_{planck}^{3}}{m_{planck}}}$
$r_{planck}$ = radius in Planck units = $r / l_{P h}$
$m_{planck}$ = mass in Planck units = $M / m_{P h}$
$t_{P h}$ = the Planck time scaling factor

Symbol Meaning How to Compute
$r_{planck}$ Physical radius in Planck units $r / l_{P h}$
$m_{planck}$ Physical mass in Planck units $M / m_{P h}$
$t_{P h}$ Planck time $\sqrt{G_{n}}$
$l_{P h}$ Planck length $c \cdot t_{P h}$
$m_{P h}$ Planck mass $H z_{k g} / t_{P h}$

Interpretation