Radio constraint on outflows from tidal disruption events

Radio flares from tidal disruption events (TDEs) are generally interpreted as synchrotron emission arising from the interaction of an outflow with the surrounding circumnuclear medium (CNM). We generalize the common equipartition analysis to be applicable in cases lacking a clear spectral peak or even with just an upper limit. We show that, for detected events, there is a lower limit on the combination of the outflow's velocity $v$ and solid angle $\Omega$, $\simeq v\Omega^{a}$ (with $a \simeq 0.5$) that constrains the outflow's properties. Considering several possible outflow components accompanying TDEs, we find that: Isotropic outflows such as disk winds with $v\sim10^4\,\rm km\,s^{-1}$ and $\Omega = 4 \pi$ can easily produces the observed flares; The bow shock of the unbound debris has a wedge-like geometry and it must be geometrically thick with $\Omega\gtrsim1$ and a fraction of its mass ($\gtrsim 0.01 \,M_{\odot}$), has to move at $v \gtrsim 2 \times 10^4\,\rm km\,s^{-1}$; Conical Newtonian outflows such as jets can also be a radio source but both their velocity and the CNM density should be larger than those of isotropic winds by a factor of $\sim(\Omega/4\pi)^{-0.5}$. Our limits on the CNM densities are typically 30-100 times larger than those found by previous analysis that ignored non-relativistic electrons that do not emit synchrotron radiation. We show that unless $v$ and $\Omega$ are known radio observation alone cannot determine the CNM density. We also find that late (a few years after the TDE) radio upper-limits rule out energetic, $\sim 10^{51-52}\,\rm erg$, relativistic jets like the one observed in TDE Sw J1644+57, implying that such jets are rare.