Cosmic rays from massive star clusters : A close look at Westerlund 1

We study the effect of cosmic ray (CR) acceleration in the massive compact star cluster Westerlund 1 in light of its recent detection in $\gamma$-rays. Recent observations reveal a $1/r$ radial distribution of the CR energy density. Here we theoretically investigate whether or not this profile can help to distinguish between (1) continuous CR acceleration in the star cluster stellar wind-driven shocks and (2) discrete CR acceleration in multiple supernovae shocks -- which are often debated in the literature. Using idealized two-fluid simulations and exploring different acceleration sites and diffusion coefficients, we obtain the CR energy density profile and luminosity to find the best match for the $\gamma$-ray observations. We find that the inferred CR energy density profiles from observations of $\gamma$-ray luminosity and mass can be much different from the true radial profile. CR acceleration at either the cluster core region or the wind termination shock can explain the observations, if the diffusion coefficient is $\kappa_{\rm cr}\sim 10^{27}$ cm$^2$ s$^{-1}$ and a fraction of $\approx 10\%-20\%$ of the shock power/post-shock pressure is deposited into the CR component. We also study the possibility of discrete supernovae (SN) explosions being responsible for CR acceleration and find that with an injection rate of 1 SN in every $\sim 0.03$ Myr, one can explain the observed $\gamma$-ray profile. This multiple SN scenario is consistent with X-ray observations only if the thermal conductivity is close to the Spitzer value.