= Solution
An <ultraluminous X-ray source> is a compact, apparently off-nuclear source in another galaxy with an inferred isotropic X-ray <luminosity> typically above $10^{32}$ W, exceeding the classical <Eddington luminosity> of an ordinary stellar-mass accretor. Imaging X-ray surveys revealed unusually bright sources away from galactic nuclei; distance association with the host converted the observed fluxes into surprisingly high <luminosities>. Better angular resolution helps exclude blends and background active nuclei. The 2014 discovery of coherent pulsations established a <neutron star> in M82 X-2, so the class cannot be identified exclusively with <black holes>.<Ultraluminous X-ray Sources in M82 Galaxy by NASA JPL>[8 October 2014: NuSTAR pulsations identify X-2 as a pulsar and establish diversity among the ultraluminous sources]{ref}
An <intermediate-mass black hole> can radiate approximately isotropically below its own <Eddington luminosity>: this avoids extreme super-Eddington emission, but the inferred <mass> requires dynamical or suitable spectral/timing support and is not proved by <luminosity> alone. A stellar-mass accretor with a supercritical <accretion disk> and a wind funnel can combine genuinely super-Eddington output with geometrical beaming; this is compatible with a neutron-star accretor but requires a consistent radiation/flow/<torque> model. Relativistic beaming from a jet is another possibility for some objects, but orientation, spectral and variability evidence must support it. These mechanisms need not apply identically to every member of the observed class.
For spherical radiation with flux $F=L/(4\pi r^2)$, the outward acceleration is $\kappa F/c$. Equating it to $GM/r^2$ gives
$$
\boxed{L_{\rm Edd}=\frac{4\pi GMc}{\kappa}}.
$$
The limit assumes the stated <opacity>, approximate <spherical symmetry> and coupling between the radiatively accelerated plasma and the accreting matter. It is not a universal ceiling on isotropic-equivalent <luminosity> for an anisotropic magnetized flow.
Write the available <accretion> power as $L=\eta\dot M c^2$. Then
$$
\boxed{\dot M_{\rm Edd}=\frac{4\pi GM}{\eta\kappa c}}.
$$
<Accretion> as the <energy> source alone does not fix $\eta$. For radiatively efficient Newtonian <accretion> onto a slowly rotating stellar surface, $\eta\simeq GM/(Rc^2)$ and the corresponding <surface accretion Eddington rate> is
$$
\boxed{\dot M_{\rm Edd}\simeq\frac{4\pi Rc}{\kappa}}.
$$
This is the appropriate convention for the numerical neutron-star estimates below, rather than imposing a separate reference efficiency of $0.1$.
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