A direct measurement of the electron density turbulence parameter $C_1$ and implications for the emission size of the magnetar XTE J1810-197

We report a direct measurement of the electron density turbulence parameter $C_1$, enabled by 550-750~MHz baseband observations with the upgraded Giant Metrewave Radio Telescope. The parameter $C_1$ depends on the power law index of the wavenumber spectrum of electron density inhomogeneities in the ionized interstellar medium. Radio waves propagating through the inhomogeneous ionized medium suffer multipath propagation, as a result of which the pulsed emission from a neutron star undergoes scatter broadening. Consequently, interference between the delayed copies of the scatter-broadened electric field manifests as scintillation. We measure a scintillation bandwidth \nud=$149\pm3$~Hz as well as a scatter-broadening timescale \taud=$1.22\pm0.09$~ms at 650~MHz. These two quantities are related through the uncertainty relation $C_1 = 2\pi$\nud\taud, using which we directly measure $C_1=1.2\pm0.1$. We describe the methods employed to obtain these results and discuss their implications in general, as well as for the magnetar XTE~J1810\textminus197, towards which the measurements have been made. We also discuss how such, effectively in-situ, measurements of $C_1$ can aid in inferring the wavenumber spectrum power law index and hence quantitatively discriminate between the various possible scattering scenarios in the ionized medium. Finally, using the fact $C_1 \sim 1$, we nominally constrain the emission size to less than a few 1000~km for a screen very close to the magnetar, and to within the magnetosphere for all screen distances.