## Radiant Flux of a Magnitude +3.5 Star

Christopher Pearson

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 star1, and if we ignore any light absorption by the Earth's atmosphere, then the claim is true.
The textbook An Introduction to Modern Astrophysics2 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):
 F2 F1 =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 metre3, 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:
 F1=1.360×103  W · m-2,    m1=-26.81,    m2=+3.500
We can now solve for the total radiant flux of a magnitude +3.5 star received at the Earth:
 F2=F1·100(m1-m2)/5 = (1.360×103)·(100(3.5+26.81)/5)  W  m-2 = 1.022×10-9 W  m-2 = 1.022 nW  m-2
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:

1I 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?
2B. W. Carroll and D. A. Ostlie, An Introduction to Modern Astrophysics. Reading, Mass: Addison-Wesley Pub, 1996.
3Note that this is a conversion of the solar constant from the textbook's value of 1.360×106 erg·s-1·cm-2.

File translated from TEX by TTH, version 3.80.
On 22 Jul 2009, 13:53.