The Sizes of Stars. The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume e = 1 for these surfaces. Find the radii of the following stars (assumed to be spherical): (a) Rigel, the bright blue star in the constellation Orion, which radiates energy at a rate of 2.7 X 1032 W and has surface temperature 11,000 K; (b) Procyon B (visible only using a telescope), which radiates energy at a rate of 2.1 X 1023 W and has surface temperature 10,000 K. (c) Compare your answers to the radius of the earth, the radius of the sun, and the distance between the earth and the sun. (Rigel is an example of a ?supergiant star, and Procyon B is an example of a w? hite? ?dwarf? star.)
Solution 75E Here again we can use the stefan-Boltzmann’s law for the output power of the stars. a) Rigel radiates at the rate of 2.7 × 10 watts The stefan-boltzmann’s equation says the power per unit area is, 4 j = T -------------------(1) 4 Where = 2 K = 5.670373 × 10 8 Wm K 2 4 15c h3 The surface temperature is 11000 K. Power is, power = j × area = 2.7 × 10 32 The surface area of the star is, 32 8 4 area = power / j = (2.7 × 10 ) / (5.670373 × 10 × 11000 ) area = 3.52 × 10 23 2 23 4 × × r = 3.52 × 10 r = (3.52 × 10 ) / 4 r = 2.588 × 10 22 22 r = 2.88 × 10 = 151261363210 meters = 151261363.210 km. Conclusion: The radius is 151261363 km. b) 23 Procyon radiates at the rate of 2.1 × 10 watts The stefan-boltzmann’s equation says the power per unit area is, 4 j = T -------------------(1) 4 Where = 2 K = 5.670373 × 10 8 Wm K2 4 15c h The surface temperature is 10000 K. Power is, 23 power = j × area = 2.1 × 10 The surface area of the star is, 23 8 4 area = power / j = (2.1 × 10 ) / (5.670373 × 10 × 10000 ) area = 3.703 × 10 14 2 14 4 × × r = 3.703 × 10 2 14 r = (3.703 × 10 ) / 4 2 13 r = 2.94 × 10 13 r = 294 × 10 = 5422176.68469 meters = 5422.17 km. Conclusion: The radius is 5422.17 km. c) The radius of earth is 6371 km . The radius of sun is 695700 km. The radius of Rigel star we found is 151261363 km. The radius of the white dwarf Procyon B we found is 5422 km approximately. So, the star’s radius and sun’s radius are of same order of magnitude. The radius of the white dwarf and earth are of same order of magnitude.
