Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay
When it was announced in the 1950's that the Palomar telescope had photographed galaxies of Magnitude 23, it was incredible.
The magnitude system is defined so that five magnitudes equals a difference of 100 times in brightness. That is, one magnitude is a difference of 100^(1/5) = 10^0.4 = 2.512. Thus, a difference of M magnitudes is a difference of 10^0.4M
The visual magnitude of the sun is -26.7. Thus the sun is 10^0.4*26.7 = 10^10.68 times brighter than a zero magnitude star, or 48 billion times.
Sunlight on the earth delivers 1361 watts per square meter (joules per second per square meters). The total energy output of the sun is 3.86 x 10^26 watts.
The last piece of information we need is the energy of light. The energy of a photon of light is E = h*frequency = h*c/wavelength. The peak radiation of the sun is about in the green at about 500 nm = 5 x 10^-7 m. h is called Planck's Constant and equals 6.6 x 10^-34 kg-sq m/sec. How did kilograms get into it? Well, the basic units of energy are kilograms times meters squared divided by seconds squared.
So a photon with wavelength 500 nm has a frequency of 3 x 10^8 m/sec /5 x 10^-7 m/cycle = 6 x 10^14 cycles/sec or 600 THz. A single green photon has E = 6.6 x 10^-34 kg-sq m/sec * 6 x 10^14/sec = 39.6 x 10^-20 joules or about 4 x 10^-19 J.
So the photon flux of sunlight = 1361/39.6 x 10^-20 joules. = 34 x 10^20 photons/sec.
The sun is 48 billion times as bright as a zero magnitude star, so to find the relevant information for a zero magnitude star, we simply divide by 48 billion.
The energy flux from a zero-magnitude star is 1361/48 billion watts/sq-m = 2.8 x 10^-8 W.sw-m
The photon flux is 34 x 10^20 photons/m-sq-sec/4.8 x 10^10 = 7.1 x 10^10 photons/m-sq-sec
An interesting statistic is, what's the photon flux on your eye? Assume a dark adapted pupil is 5 cm in diamere, so it's area is 20 sq mm. A square millimeter is 10^-6 sq-m, so the flux on tour eye is 20 x 10^-6 of the flux per square meter, = 7.1 x 10^10 = 20 x 10^-6 = 142 x 10^4 = 1.4 million photons per second.
The faintest star we can see is typically considered to be sixth magnitude, 1/250 as bright as a zero-magnitude star, so the photon flux in your eye is 5600 photons per second. That's impressive, that your eye can detect that few photons.
How faint a star we can detect depends on the size of the detector and the length of the exposure. Let's assume our mirror is 10 meters across, can detect every photon (our detectors are almost that good), and our target flux is one photon per second.
A 10-meter mirror has an area of 100 square meters, so a zero-magnitude star would deliver 7.1 x 10^12 photons per second. a star with a flux of one photon per second would be 10^0.4M = 7.1 x 10^12 times fainter. Taking logs of both sides, we have 0.4M = 12.85, or M = 31.
The 8-meter Subaru Telescope in Hawaii has imaged objects as faint as 27.7, using 10 hours of exposure. The Hubble Deep Field exposures have seen magnitude, with a 2.4 meter mirror and 277 hours of exposure.
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Created 26 March 1998, Last Update 2 April 1998