Astronomy Problems (Homework) 4 Solutions

# Astronomy Problems (Homework) 4 Solutions - Homework 4...

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Homework 4 University Astronomy 5/10/10 1 University Astronomy 1017-301 Homework Assignment 4 Instructor: Dr Andrew Robinson Due Date: Friday, 4 May 2010. To tackle some of the following problems you will need to do some research, using either the textbook or other resources, to find various pieces astronomical data and the values of certain physical constants. For calculations, show all your workings in order to receive full credit . 1. Calculate the main sequence lifetimes of the following stars a. An O star that has a luminosity 6 × 10 4 L (solar luminosities). b. An F star that has a luminosity 3 L . c. An M star that has a luminosity 0.03 L . Use the fact that the Sun has an estimated main sequence lifetime of 12 billion years. 2. Following the arguments given in Lecture 15, estimate the length of time that each of the stars in problem 2 could shine by their gravitational energy alone. At what stage in a star’s life does it shine mainly by the release of gravitational energy? Hint: to do the calculation you will first need to calculate values for the radius of each star. The surface temperatures given in Lecture 11 or in Table 15.1 in the textbook will help here. From Lecture 15, the main sequence lifetime is t ~ E/L, where E = energy content and L = luminosity. Since the energy comes from nuclear fusion E Mc 2 , where M = mass of star, so t M/L. We can make use of the mass-luminosity relation, L M 3.5 , to eliminate M (or L), to get t L -5/7 . Scaling to the Sun, t = (L /L) 5/7 t , where t is the sun’s main- sequence lifetime. Therefore, a) O star t = (1/6x10 4 ) 5/7 x1.2x10 10 y 4.6x10 6 y b) F star t = (1/3) 5/7 x1.2x10 10 y 5.5x10 9 y c) M star t = (1/0.03) 5/7 x1.2x10 10 y 1.5x10 11 y From Lecture 15, t grav ~ 1/2|U|/L where |U|~3GM 2 /5R, and R is the radius of the star. We know L from the information given in question 1, and we can get M from the mass-luminosity relation, but we don’t know R. Continued…

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Homework 4 University Astronomy 5/10/10 2 3. We saw in Lecture 15 that there is a maximum mass for main sequence stars, which is set by the condition that the Eddington limit is not exceeded. Show that if main sequence stars follow a mass-luminosity relationship of the form L " M 3.5 , the maximum mass is 60 M . 4. Spectroscopic observations of an A star reveal that it is in an eclipsing binary system with an unseen companion. Measurements of the Doppler shifts of certain absorption lines in the star show cyclical variations with a period of 3.65 days. The extreme red- and blue-shifts are both equal to 300 km/s. Estimate the total mass of the system and
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