"Ahad's constant" - the official definition

Search the internet for "Ahad's constant" and you'll get an inordinate amount of discussion and yet an equal amount of contention about how this number ought to be calculated and what it should be called. Two years ago, when I first calculated a one number solution for the universe's total background illumination, it never occurred to me that the net value of such a logarithmic series could become the subject of so much controversy even to this day. It is understandable, however, on the grounds that the exact value cannot be computed, for reasons of "asymptoticness".

What is "Ahad's constant"?

If we exclude all light coming directly and indirectly from the nearby Sun, the rest of the universe collectively supplies us with a mere 1/300th of a Full Moon worth of light. That is "Ahad's constant"

Astronomers measure the brightnesses of stars across the night sky using something called the magnitude scale, first introduced by the Greek astronomer Hipparchus in the second century B.C. The scale in itself can be somewhat confusing, since a first magnitude star is actually brighter than a second magnitude star...opposite to what you'd expect going purely by every day common sense. You can learn more by searching Google for 'magnitude scale for star brightnesses' Here's one site that goes into much elaborate detail.Back to the definition of "Ahad's constant".
Suppose we have two stars of magnitudes M1 and M2. Then their luminosities L1 and L2 are related by the formula:-

L2/L1 = 10^[0.4*(M1-M2)]

The luminosity of the pair of stars is L1 + L2 = L1(1 + L2/L1), and their combined magnitude is then given by:-

Mc = M1 - 2.5*log10 (1 + L2/L1)

For the general case, where the magnitudes of "n" stars need to be aggregated, we can generalise this by computing all the ratios:-

Li/L1 = 10^[0.4*(M1-Mi)]

for all stars i from 2 through n. The combined magnitude is then:-

Mc = M1 - 2.5*log10 (1 + L2/L1 + L3/L1 + ... + Ln/L1)

"Ahad's constant" is simply defined to be the sum of all the individual magnitudes of every single star across the entire night sky, right down to the faintest star that could ever be seen with the most powerful telescope ever invented or is likely to be invented in the future.
In other words, the value of "n" in the above formulae (i.e. the star count) will tend to infinity. Based on my own numerical integrations, I have found that as n tends to infinity, the variable Mc in the above equation tends to a net figure of -6.5 magnitudes (1/300th of a full moon equivalent worth of light). That is what some are calling "Ahad's constant".

It can be appreciated by someone sailing more than a couple of light years beyond the neighborhood of the Sun in any direction. When you're that far out, you'd want certain physical barometers to pinpoint your overall "existence". One of them might be knowing whether the environment your ship is sailing through is a complete vacuum. Another might be knowing your distance from the next nearest planet or star. Yet another might be to know how much net starlight the sky is providing (i.e. "Ahad's constant").

It will remain an invariant celestial constant to a traveller located in deep interstellar space within several hundreds - if not thousands - of light years from here. All that humanity can ever hope to physically experience or meaningfully contemplate over within the foreseeable future of our species...

Stars in the neighborhood of the Sun are extremely feeble in their intrinsic brilliance - most of them being tiny red dwarves of < 0.01 x Sun power - and the average spacing between them is approximately 5 light years. Hence, 99.99% of the time during an interstellar voyage between two stars in this part of the galaxy, you will be travelling under the feeble illumination quantified by "Ahad's constant":

Above: Stars in the neighborhood of our Solar System, going 20-light years out in all directions. If you place your finger randomly anywhere on this map, 99.99% of the time when you're physically there, you are going to be surrounded by a perpetual cosmic night. The net illumination from all the pinpricks of starry light in the 360-degree celestial sphere around you will then equate to "Ahad's constant". FACT, not fiction!

To make my work somewhat "official", preliminary results of the above integrations were published in Journal of the British Astronomical Association, Volume 115, No. 5, October 2005 edition, page 297.

(Note: "Ahad's constant" is not to be confused with something called "Olbers Paradox")

Example of another asymptotic measure that is widely debated

Just because a physical quantity is "asymptotic" by calculation, it does not mean that it cannot be estimated or is not significant. Here is a more Earthly example: the air pressure in our atmosphere tails off "asymptotically" as you go up in altitude. If you're located at mean sea level, you experience a much higher atmospheric pressure than you would when you go up to the top of a mountain or climb up to 37,000 feet inside a jumbo jet. The higher up you go, the thinner the air and the lower the density of air molecules. See this site for more details. Atmospheric density falls off with altitude, following an "asymptotic" curve like this:-In other words, it's a very gradual thinning out of air and there is no clear boundary to say when you're 50-km up or 100-km up or 200-km up into the sky, you've reached the edge of space. Yet for managing rocket re-entries for things like the Space Shuttle, NASA engineers have set 100-km as an arbitrary altitude boundary that defines "where our upper atmosphere comes to an end and the vacuum of space officially begins".
So there you have it. Ultimately, the expanse of our upper atmosphere and the total flux of our night skies both have *finite* values, even though they are asymptotic...