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The apparent magnitude of a distant object in the sky is measured as m=(-)2.5 log (I/I0) Where I is the intensity of the object whose apparent magnitude is to be measured and I0 is the reference magnitude. My question is why we put the minus sign in the expression.

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So if I read your question correctly, you're asking why as stars get dimmer, they are given higher magnitudes?

The reason is purely historical. The ancient greeks assigned stars with 6 brightness levels or magnitudes. The brightest stars were of first magnitude and the least bright stars (to the naked eye) were of 6th magnitude.

In the 1800's this system was formalised by an astronomer named Norman Robert Pogson. By then they were (presumably) able to more accurately measure the brightness of a star, and he decided to use a system that fitted well with the ancient measurements, but could be used to interpolate and extrapolate to brighter and dimmer objects. In order to do this, the formula has to have a negative in front of the log.

More information here

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