... bulge

Although it does not contribute to microlensing towards M 31, it contributes to the dynamical mass of the Galaxy.

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...$\Gamma_{\rm p} \propto L_{\rm s}$,

Indeed, the faintest detectable pixel lensing events require a threshold source magnification $A_{\rm T} \propto L_{\rm s}^{-1}$, to be seen against the local background. On the other hand, in pixel lensing the bulk of observed events involves highly magnified sources for which $A_{\rm T} \propto u_{\rm T}^{~-1}$. Therefore since $\Gamma_{\rm p} \propto u_{\rm T}$, it follows that $\Gamma_{\rm p} \propto L_{\rm s}$.

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