= Solution
For an adiabatic <Bondi accretion> flow, $\dot M\propto M^2\rho_\infty c_{s,\infty}^{-3}$. Inside the Bondi radius the speed and ion temperature are approximately virial, so $v\propto(M/r)^{1/2}$ and $T\propto M/r$, while <mass conservation> gives $n\propto\dot M M^{-1/2}r^{-3/2}$. The frequency-integrated <thermal bremsstrahlung> emissivity is proportional to $n^2T^{1/2}$. Its volume integral is dominated by the inner flow and consequently scales as
$$
L\propto\frac{\dot M^2}{M}.
$$
Since the <Eddington luminosity> is proportional to $M$, this may be written $L/L_{\rm Edd}=C(\dot Mc^2/L_{\rm Edd})^2$. With the standard fully ionized-plasma constants and the Bondi profiles, $\sqrt C=9\times10^{-3}$. Eliminating $\dot M$ from $\eta=L/(\dot Mc^2)$ then gives
$$
\boxed{\eta=9\times10^{-3}
\left(\frac{L}{L_{\rm Edd}}\right)^{1/2}.}
$$
The cancellation of $M$, $\rho_\infty$, and $c_{s,\infty}$ expresses the scale-free character of the ideal flow. More physically, two-body emission scales as density squared, so an increasingly dilute flow radiates a progressively smaller fraction of its available accretion power.
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