Optical and X-ray luminosities of expanding nebulae around ultraluminous X-ray sources

We have performed a set of simulations of expanding, spherically symmetric nebulae inflated by winds from accreting black holes in ultraluminous X-ray sources (ULXs). We implemented a realistic cooling function to account for free-free and bound-free cooling. For all model parameters we considered, the forward shock in the interstellar medium becomes radiative at a radius similar to 100 pc. The emission is primarily in optical and UV, and the radiative luminosity is about 50 per cent of the total kinetic luminosity of the wind. In contrast, the reverse shock in the wind is adiabatic so long as the terminal outflow velocity of the wind v(w) greater than or similar to 0.003c. The shocked wind in these models radiates in X-rays, but with a luminosity of only similar to 10(35) erg s(-1). For wind velocities v(w) less than or similar to 0.001c, the shocked wind becomes radiative, but it is no longer hot enough to produce X-rays. Instead it emits in optical and UV, and the radiative luminosity is comparable to 100 per cent of the wind kinetic luminosity. We suggest that measuring the optical luminosities and putting limits on the X-ray and radio emission from shock-ionized ULX bubbles may help in estimating the mass outflow rate of the central accretion disc and the velocity of the outflow.