Multi-Wavelength Constraints on the Outflow Properties of the Extremely Bright Millisecond Radio Bursts from the Galactic Magnetar SGR 1935+2154

Recently, a bright coherent radio burst with millisecond duration, reminiscent of cosmological fast radio bursts (FRBs), was co-detected with an anomalously-hard X-ray burst from a Galactic magnetar SGR 1935$+$2154. We investigate the possibility that the event was triggered by a deposition of a magnetic energy in a localized region of the magnetosphere, thereby producing a so-called trapped fireball (FB) and simultaneously launching relativistic outflows. We show that the thermal component of the X-ray burst spectrum is consistent with a trapped FB with an average temperature of a few hundred keV and a size of $\sim10^5$ cm. Meanwhile, the non-thermal component of the X-ray burst and the coherent radio burst may arise from relativistic outflows. We calculate the dynamical evolution of the outflow, launched with an energy budget $\sim10^{39}\mbox{-}10^{40}$ erg comparable to that of the trapped FB, for a variety of baryon load $\eta$ and initial magnetization $\sigma_0$ parameters. If both the hard X-ray and radio bursts are produced by the energy dissipation of the outflow, the properties can be constrained by the conditions for photon escape and the intrinsic timing offset of $\lesssim 10$ ms among the radio and X-ray burst spikes. We show that the hard X-ray bursts need to be generated at $r_{\rm X}\gtrsim10^{8}$ cm from the stellar surface, irrespective of the emission mechanism. Particularly in the case of shock dissipation, the outflow should accelerate up to a Lorentz factor of $\Gamma \gtrsim 10^3$ by the time it reaches the outer edge of the magnetosphere and the shock dissipation should take place at $10^{12}\,\mathrm{cm} \lesssim r_{\rm radio, X} \lesssim 10^{14}\,\mathrm{cm}$. In this case, extremely clean ($\eta\gtrsim10^4$) and/or highly magnetized ($\sigma_0\gtrsim10^3$) outflows are implied, which may be consistent with the rarity of this phenomenon.