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Fall 2004 Dr. Mike Fanelli Solutions to Assigned Problems Chapter 18 PROBLEM 18-1: Given the density of gas within the “Local Bubble”, a region of space surrounding the solar system, determine the total mass within a volume the size of Earth. ANSWER: You are given a density (the number of hydrogen atoms per volume) and asked to determine the total amount of mass within that volume. The volume is the volume of Earth. Given the definition of density, Density = total mass volume, rearrange this expression to get the total mass. Remember that the volume of a sphere = 4/3 R3, and the radius of Earth is 6400 kilometers. Multiplying each side of the expression by volume gives: Total mass = density volume = # of atoms mass of 1 atom volume of Earth = 103 atoms / meter3 1.7 10-27 kilogram per atom volume of Earth = 1.7 10-24 kilogram / meter3 4/3 R3 = 1.7 10-24 kilogram / meter3 13.51 (6.4 106 meters)3 = 1.7 10-24 kilogram / meter3 1.1 1021 meters3 = 0.0019 kilograms or 1.9 grams (!) A very small mass. PROBLEM 18-10: To ionize interstellar hydrogen, a photon must have a wavelength smaller than 91.2 nanometers (9.12 10-8 m). Assuming a star had its peak wavelength at this value, what is the surface temperature of this star ? ANSWER: Use Wien’s law from chapter 3, to relate peak wavelength to the peak temperature. Note that you will need to convert the wavelength to centimeters in order to use the expression in More Precisely 3-2. The result will then be in units of degrees Kelvin. Wien’s law states: (max) = 0.29 T, where (max) is the peak (maximum) wavelength, measured in centimeters, and T is the stellar temperature, expressed in K. Therefore: 9.12 10-8 m = 0.29 T, converting to centimeters, -6 9.12 10 cm = 0.29 T, rearranging, Temperature = 0.29 9.12 10-6 cm = 31,800 K A star of this temperature is a late O star, O7-O9 spectral class. .