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# h6s - Homework 6 Solutions 1 Demonstrate that the...

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Homework # 6 Solutions 1. Demonstrate that the temperature applicable to a stellar system is T = m < v 2 ( x ) > 3 k where it is assumed that the stars have equal mass m , v ( x ) is the velocity of a star relative to the center-of-mass, and <> represents an average. On the assumption that the stellar velocities are distributed in the most probably way, i.e., a Boltzmann distribution, one has n ( x , v ) = n o exp ± - ( m Φ( x ) + mv 2 / 2 ) / ( kT ) ² . The quantity kT is simply a constant representing an average energy. To see how it is related to the average velocity, we compute the mean square stellar velocity at x , which is h v 2 ( x ) i = R 0 4 πn o v 4 exp ± - ( m Φ( x ) + mv 2 / 2 ) / ( kT ) ² dv R 0 4 πn o v 2 exp[ - ( m Φ( x ) + mv 2 / 2) / ( kT )] dv = R 0 v 4 exp ± - mv 2 / (2 kT ) ² dv R 0 v 2 exp[ - mv 2 / (2 kT )] dv = Γ(5 / 2)(2 kT/m ) 5 / 2 Γ(3 / 2)(2 kT/m ) 3 / 2 = 3 kT m . 2. Calculate the gravitational potential a distance

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h6s - Homework 6 Solutions 1 Demonstrate that the...

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