Department of Computer
Science, University of Victoria
After reading the Wikipedia page for dBm, I was curious if the claim "The Earth receives one nanowatt per square meter from a single magnitude +3.5 star" was true. If we are comparing the light energy from the star^{1}, and if we ignore any light absorption by the Earth's atmosphere, then the claim is true.
The textbook An Introduction to Modern Astrophysics^{2} tells us that we're interested in radiant flux: the total amount of light energy of all wavelengths received from a source. This textbook gives a simple equation for computing ratios of radiant flux (F) and apparent magnitude (m):
F_{2}
F_{1}
=100^{(m1-m2)/5}
To make a comparison, we need to know the radiant flux and magnitude of another star. Fortunately, we know one star very well: the Sun. The radiant flux for the Sun (known as the solar constant) is 1360 watts per square metre^{3}, and its magnitude is -26.81. Knowing that we wish to find the radiant flux of a magnitude +3.5 star, we get the following values:
So, the amount of light received at the Earth from a magnitude +3.5 star, per square meter, is 1.022 nW. The definition of -60 dBm is 1.0 nW, and the two values are essentially the same.
Footnotes:
^{1}I personally like this as an energy comparison. After all, if you're standing in the starshine of a magnitude +3.5, what kind of shadow would you expect to see?
^{2}B. W. Carroll and D. A. Ostlie, An Introduction to Modern Astrophysics. Reading,
Mass: Addison-Wesley Pub, 1996.
^{3}Note that this is a conversion of the solar constant from the textbook's value of 1.360×10^{6}erg·s^{-1}·cm^{-2}.
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