A milli-Tidal Disruption Event Model for GRB$\;$250702B: Main Sequence Star Disrupted by an IMBH

GRB$\;$250702B is the longest GRB recorded so far, with multiple gamma-ray emission episodes spread over a duration exceeding $25\;$ks and a weaker soft X-ray pre-peak $\sim1\;$day gradually rising emission. It is offset from its host galaxy center by $\sim5.7\;$kpc, and displays a long-lived afterglow emission in radio to X-ray. Its true nature is unclear, with the two leading candidate classes of objects being a peculiar type of ultra-long GRB and a tidal disruption event (TDE) by an intermediate mass black hole (IMBH). Here, we consider the latter, mTDE origin. We model the afterglow data, finding a stratified external density profile $\propto r^{-k}$ with $k=1.60\pm0.17$, consistent with Bondi accretion of the interstellar medium (of initial number density $n_{\rm ISM}=n_0\;{\rm cm^{-3}}$ and sound speed $c_s=c_{s,6}10^6\;{\rm cm\,s^{-1}}$) for which $n(r)\approx n_{\rm ISM}(r/R_{\rm B})^{-3/2}$ within the Bondi radius $R_{\rm B}$. Moreover, we use the implied density normalization to infer the IMBH mass within this model, finding $M_\bullet\approx\left(6.55^{+3.51}_{-2.29}\right)\times10^3\,n_0^{-2/3}\,c_{s,6}^{2}(1+\mathcal{M}^2)\;M_\odot$ where $\mathcal{M}\equiv v_{\rm BH}/c_s$ is the IMBH's Mach number relative to the ISM. Together with an upper limit on $M_\bullet<\frac{c^3}{G}\frac{t_{\rm MV}}{1+z} \lesssim5\times10^4\,M_\odot$ from the source-frame minimum variability time $t_{\rm MV,src}\!=\!\frac{t_{\rm MV}}{1+z}\!\approx\!0.5\;$s this implies $v_{\rm BH}\lesssim 28\,n_0^{1/3}\;{\rm km\;s^{-1}}$. We show that a mTDE of a main-sequence star (but not of a white dwarf) can explain the duration and energetics of GRB$\;$250702B. The gradual rise to the peak may be caused by gradual circularization and accretion disk buildup, leading to an increase in the jet's power and Lorentz factor.