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# h2s - Homework 2 Solutions 1 Suppose the initial mass...

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Homework # 2 Solutions 1. Suppose the initial mass function is the Salpeter mass function with a low- mass cutoff of 0 . 1 M and a high-mass cutoff of 100 M . Determine the normalization from the local disk column mass density (i.e., the number density integrated vertically through the Galactic plane in units pc 2 ). What is the average stellar mass? Where does the mass function reach a maximum? If L ∝ M 3 . 5 for all masses, what mass star has the average luminosity? From these results, what is the column density of stars in the solar neighborhood and what is the mean distance between stars? The local disk column mass density is about 55 M pc 2 . The total mass column density of stars is found from integraldisplay M u M M ξ ( M ) d M = ξ o M 0 . 35 parenleftbigg M M parenrightbigg 0 . 35 bracketleftBigg 1 parenleftbigg M M u parenrightbigg 0 . 35 bracketrightBigg ξ o 10 0 . 35 0 . 35 [1 0 . 09] M , where the normalization is ξ o . Setting these results equal, we find ξ o = 9 . 4 pc 2 . The average stellar mass is M M = integraltext M u M M ξ ( M ) d M M integraltext M u M ξ ( M ) d M = 1 . 35 M 0 . 35 M bracketleftBig 1 ( M / M u ) 0 . 35 bracketrightBig bracketleftBig 1 ( M / M u ) 1 . 35 bracketrightBig = 0 . 35 The mass function reaches a maximum at M .

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