Standing Sausage Modes In Nonuniform Magnetic Tubes: An Inversion Scheme For Inferring Flare Loop Parameters

Standing sausage modes in flare loops are important for interpreting quasi-periodic pulsations (QPPs) in solar flare lightcurves. We propose an inversion scheme that consistently uses their periods $P$ and damping times $\tau$ to diagnose flare loop parameters. We derive a generic dispersion relation governing linear sausage waves in pressure-less straight tubes, for which the transverse density inhomogeneity takes place in a layer of arbitrary width $l$ and is of arbitrary form. We find that $P$ and $\tau$ depend on the combination of $[R/v_{\rm Ai}, L/R, l/R, \rho_{\rm i}/\rho_{\rm e}]$, where $R$ is the loop radius, $L$ is the looplength, $v_{\rm Ai}$ is the internal Alfv\'en speed, and $\rho_{\rm i}/\rho_{\rm e}$ is the density contrast. For all the density profiles examined, $P$ and $\tau$ experience saturation when $L/R \gg 1$, yielding an inversion curve in the $[R/v_{\rm Ai}, l/R, \rho_{\rm i}/\rho_{\rm e}]$ space with a specific density profile when $L/R$ is sufficiently large. When applied to a spatially unresolved QPP event, the scheme yields that $R/v_{\rm Ai}$ is the best constrained, whereas $l/R$ corresponds to the other extreme. For spatially resolved QPPs, while $L/R \gg 1$ cannot be assumed beforehand, an inversion curve remains possible due to additional geometrical constraints. When a spatially resolved QPP event involves another mode, as is the case for a recent event, the full set of $[v_{\rm Ai}, l, \rho_{\rm i}/\rho_{\rm e}]$ can be inferred. We conclude that the proposed scheme provides a useful tool for magneto-seismologically exploiting QPPs.