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RHK
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Homework Statement
Stars in a globular cluster are distributed as a mass function of: [itex]\phi(M)=K M^{-2}[/itex], such that [itex]dN=\phi(M) dM[/itex] is the stars number in the infinitesimal mass interval. Masses are between a lower limit [itex]M_{inf}=0.3 M_{sun}[/itex] and an upper limit [itex]M_{sup}[/itex], unknown.
The constant K is equal to [itex]200 M_{Sun}[/itex].
Let's assume also that the relation between the bolometric luminosity and the stars masses is [itex]L(M)=L_{Sun}(\frac{M}{M_{Sun}})^{3.5}[/itex]. Requests:
(i) to show the necessity of the existence for the upper limit in the mass distribution, to avoid that the cluster mass diverge toward infinite.
(ii) to calculate the value of such limit so that the total mass of the cluster is [itex]600 M_{Sun}[/itex];
(iii) to calculate the cluster bolometric luminosity in solar units, and the corresponding absolute magnitude.
EDIT: the absolute bolometric magnitude for the Sun is given in the exercise text: [itex]m_{sun}=4.75[/itex]
Homework Equations
[itex]M_{TOT}=\int_{M_{inf}}^{M_ {sup}} M \phi(M) dM[/itex]
The Attempt at a Solution
[itex]M_{TOT}=\int_{M_{inf}}^{M_ {sup}} K M^{-1} dM = K log M[/itex]
Is it the right way?
Can i have a hand please?
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