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On the basis of volume we expect 1000x as many stars encompassed, and this seems to me likely true. The nearer majority look brighter than those on the fringes that barely qualify as visible. The new marginal stars, of 100x greater abundance, are all in the outermost rind of the sphere encompassed, the nearer ones of same intrinsic Magnitude having been counted already. We are just talking about the area of a sphere being the radius-derivative of its volume, thus proportional to R² rather than to R³. So we might expect the count to rise by powers not of sqrt(100^2/5) but by the 3/2 power more: about 4. Does it?

Slopes 1 and 3/2 are drawn over this plot of reported star counts against the dimmest naked-eye magnitude made visible by telescopes

A log plot of the four star counts as powers of 100^1/5 against limiting magnitude shows we are getting more than 3 times as many stars each time we capture the next dimmer magnitude, not 2.5 times as many, nor 4 times as many: the trend seems to have slope less than 3/2, maybe in part because the biggest sample is a bit deficient and/or the outer limits of that big sphere may be protruding beyond the "plane" of our galaxy. If you have now incorporated the idea of "magnitudes" into your working toolkit --- which was the purpose of these two "Discovery" columns --- then you should be able to figure out this big sphere’s radius, given (last week) that Alpha Centauri is like the Sun and looks like magnitude 0 at distance 4 light years (or that the Sun is like the Sun and looks like magnitude -27 at distance 8 light minutes.) There seems to be something more to Discover here, e.g., maybe* the nearby count of 500 is a bit inflated by an accident density in our near neighborhood?

But all this is only talk since we have no direct observation of distances except by parallax at relatively short distances encompassing relatively few stars.

More importantly: all this could be nonsense. I am no astronomer, and these exercises are doubtless just working through things that were contentious and finally got settled by intelligent people a century ago. It doesn’t matter for our purposes of coming up to speed with ideas new to us. We can do that only by bravely making our own mistakes. I expect you to correct mine.

In this and my prior "Stellar Magnitudes" column you saw the word "intensity" a dozen times. In all cases it means the amount of light per unit area of some receiving surface, e.g., the pupil of your eye. In a couple of instances it could alternatively mean per steradian of view angle. I have since learned that "per steradian" is the official usage of "intensity" in the photometry business. For "per unit area" the right term is "irradiance".

 

* As this goes to press I have compulsively sought out better data than given in this plot. They suggest a more interesting hypothesis for the deficit, based on an obvious consideration totally overlooked in this "inadequate explanation". Can you guess what before I report back in later column?