SFSU Physics and AStronomy Department

Ph 490 Problem Set 2

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Problem 1. Consider the effect of errors on determining the magnitude of supernovae. The relation between luminous intensity I and magnitude m is based on an ancient classification of stars by apparent brightness. The (visible) stars ranged from first-magnitude stars, the brightest, to fifth or sixth magnitude stars, at the limit of the sensitivity of the human eye. Roughly speaking, five magnitudes of difference corresponded to a difference in intensity (measured in watts/m²) of a factor of 100. This corresponds to the relation

,

where I₀ is about 10^-7 W/m².

• (a) Find an expression for the error on the magnitude, s_m, in terms of the fractional error in the intensity, .
• (b) If the star's intensity is measured to 10% accuracy, what is the corresponding error sigma-m on the magnitude?

Problem 2. Certain cosmic-ray experiments are carried out in the deep ocean, where there is very little sunlight, due to absorption by the water. These experiments detect cosmic rays by light which they produce via Cherenkov radiation. The sensitivity of these experiments is limited by background light due to the decay of the radioactive isotope potassium-40. Sea salt is about 1% potassium iodide by weight (the rest is mostly sodium iodide), and the radioactive isotope ⁴⁰K constitutes about 0.01% of natural potassium. The half-life of ⁴⁰K is 1.3 x 10⁹ years.

(a) What is the activity, in bequerels and curies, of one gram of sea salt, due to ⁴⁰K decay?

(b) What would this activity have been shortly after the big bang, where much of the ⁴⁰K was produced? (Take the age of the universe to be about 1.4 x 10¹⁰ years.)