Density

A catchy title… This afternoon has been a bit slow, so I thought I would create a post with no discernible use or function. I am supplying no direct references to the volumetric content by not defining anything. However, as a teaser, this has real function (or at least parts may have some function, or not).

Firstly, let’s throw in some provoking formulae to get started.

\[H_{O} = \frac{c}{R_{G}}\]

Now, if we do some substitution, we may end with a constant as a function of something else.

\[H_{O} = \frac{\sqrt{4 \pi G \rho_{U}}}{c}\]

Further, we solve for a particular value.

\[\rho_{U} = \frac{H_{O}^{2} c^{2}}{4 \pi G}\]

And, as a further value to use for something, somewhere, we derive this: \(4.0(10)^{-10} J/m^{3}\) at a fixed value of \(60 Hz-km/Mpc\). I most likely gave away something, somewhere, sometime by that last. But, sometimes, some things have to be mentioned, somewhere, even in levity. Is it a crime to accidentally upset a popular food product at random upon a flat surface? Add chili and you may have a popular Mexican dish…

In any case, we might derive an average figure of 2.7 units per volume. And as there is some error involved in an included value, in the ballpark of 50 to 85 \(Hz-km/Mpc\), this particular number could be different, namely such that \(2.8(10)^{-10} J/m^{3} < \rho_{U} < 8.1 (10)^{-10} J/m^{3}\), of which luminosity is considered only a fraction of this value (1-3%). Wow, that’s really pushing it. Also, as compact objects have a bad habit of radiating radio and high-energy emissions, it follows the density is small. But, it could mean the undefined product could have a low average kinetic energy content, giving a relatively weak average gravitational potential.

It’s really difficult to pursue this without giving too much detail, and that would defeat the purpose of the post, which is nothing. In any case, let’s push on with this exercise… Just for fun, let’s define a concentration of something as a function of luminousity.

\[\rho_{bp} \approxeq \frac{\rho_{R}}{\varepsilon_{U} c^{2}} \approxeq \frac{\rho_{B\gamma} + \rho_{l\gamma} + \rho_{CMB}}{\varepsilon_{U} c^{2}} \]

This gives us a possible value of \(9(10)^{-29} kg/m^{3}\) if we use values shown.

• \(\rho_{R} = 5.9(10)^{-14} J/m^{3}\).
• \(\rho_{B\gamma} = 1.6(10)^{-15} J/m^{3}\).
• \(\rho_{l\gamma} = 1.6(10)^{-14} J/m^{3}\).
• \(\rho_{CMB} = 4.165(10)^{-14} J/m^{3}\).

Now, not to leave everything in the dark, let’s delve into some imprisoned fractions of a material.

\[f_{bp} \approxeq \frac{4 \pi G (\rho_{B\gamma} + \rho_{l\gamma} + \rho_{CMB})}{H_{O}^{2} \rho_{U} c^{2}}\]

Depending on a particular constant error, this means we could be in the dark for 97.3% to 99% of our volumetric time, be surrounded by the most abundant element in the universe, and wouldn’t starve for ~45 trillion years. How’s that for endurance? Wow! Except…

Except that individual concentrated volumes eat at a much higher rate, giving hunger pangs in only 1.1 trillion years. Time to diet?

Well, that’s enough fun for one afternoon, so I will throw in the towel on this exercise in indistinctness. Have a great day!