Dimensional analysis peculiarity

[av163]

TL;DR

(e^2*G)/(c^2*k^2) appears to be exactly 1x10^-19

Sorry if this is in the wrong place, but I noticed an earlier thread about coincidences with natural constants was posted here, and this one is bugging me.

I can't remember why I calculated this, but the (Planck time^6 * Planck current^2 * Planck temperature^2)/(Planck length^3 * Planck mass^3) gives the low energy fine structure constant (reciprocal) multiplied by 10^-19.

Doing a bit more work, I found this to be tied to the relation that the product of the speed of light squared and Boltzmann's constant squared is 1.713199627x10^-29, whereas the product of the elementary charge squared and the gravitational constant is 1.713199627E-43. The additional precision in the leading digits comes from the assumption that the measured low energy fine structure constant is correct.
This would give a theoretically derived value for G of 6.674015085E-11.

Given how unnatural this exact value seems, and the fact that I've only been performing basic dimensional analysis, I suspect I might have missed a regularization of some sort? I've tried googling these values but nothing pops up. Does anybody have an idea what's happening here?

 

[Nugatory]

Does anybody have an idea what's happening here?

Quick sanity check:
Redo your calculations, except use the numerical values you get when distances are measured in feet instead of meters. Do these interesting relationships go away? If so, your calculations may be telling you something about the shoe size of a long-dead English king, but they aren’t telling us anything about any physical truth.

 

[av163]

Quick sanity check:
Redo your calculations, except use the numerical values you get when distances are measured in feet instead of meters. Do these interesting relationships go away? If so, your calculations may be telling you something about the shoe size of a long-dead English king, but they aren’t telling us anything about any physical truth.

Yep. It disappears when I use feet. Thank you, that was costing me sleep.

 

[Vanadium 50]

Additionally, you are off by 47 ppm and G (the worst known number) is known to 22 ppm.

 

[robphy]

TL;DR Summary: (e^2*G)/(c^2*k^2) appears to be exactly 1x10^-19

A little late to the party...

but more completely, the ratio is
1×10^{-19}\ \rm s^6A^2K^2/(kg^3m^3)
where the 1×10^{-19} has an experimental uncertainly as @Vanadium 50 says.

((electron charge)^2*(Gravitational constant))/((speed of light)^2*(boltzmann constant)^2)
https://www.wolframalpha.com/input?...)/((speed+of+light)^2*(boltzmann+constant)^2)

The units are important and should never be neglected.
As @Nugatory suggests, the 1×10^{-19} isn't anything fundamental since the units were ignored.

(Furthermore, I regard the Boltzmann constant less fundamental than any of the other constants in this expression.)

 

[Frabjous]

(Furthermore, I regard the Boltzmann constant less fundamental than any of the other constants in this expression.)

Why?

 

[robphy]

(Furthermore, I regard the Boltzmann constant less fundamental than any of the other constants in this expression.)

Why?

I regard the Boltzmann constant k_B essentially as a
conversion factor between energy and temperature for accounting purposes
because of the way energy and temperature were historically defined.

I would argue that k_B T (the energy-equivalent of absolute temperature)
is more physical than either k_B or T.
In other words, we might have defined a quantity \tau=k_BT and write all of our equations with \tau (like PV=N\tau) and never have to see k_B.
In fact, using the notion of thermodynamic-beta (\beta=\frac{1}{k_B T}) we can already write the ideal gas law as PV=N/\beta or maybe PV\beta=N.

One might want to say similar things about the speed of light c.
But, if it were my choice to simplify things, I'd start with demoting the fundamental importance of k_B.