There is growing evidence of a higher intrinsic fraction of broad absorption line quasars (BALQSOs) than that obtained in optical surveys, on which most previous X-ray studies of BALQSOs have focused. Here we present Chandra observations of 18 BALQSOs at $z\sim2$, selected from a near-infrared (2MASS) sample, where the BALQSO fraction is likely to be close to the intrinsic fraction. We measure photon indices using the stacked spectra of the optically-faint ($i-K_s\geq 2.3$ mag) and optically-bright ($i-K_s < 2.3$ mag) samples to be $\Gamma \simeq 1.5$--$2.1$. We constrain their intrinsic column density by modelling the X-ray fractional hardness ratio, finding a mean column density of $3.5\times10^{22}$ \cmsq\ assuming neutral absorption. We incorporate SDSS optical measurements (rest frame UV) to study the broadband spectral index between the X-ray and UV bands, and compare this to a large sample of normal quasars. We estimate that the optically-faint BALQSOs are X-ray weaker than the optically-bright ones, and the entire sample of BALQSOs are intrinsically X-ray weak when compared to normal AGN. Correcting for magnification of X-ray emission via gravitational lensing by the central black hole viewed at large inclination angles makes these BALQSOs even more intrinsically X-ray weak. Finally, we estimate AGN kinetic feedback efficiencies of a few percent for an X-ray wind of $0.3c$ in high-ionization BALQSOs. Combined with energy carried by low-ionization BALQSOs and UV winds, the total kinetic energy in BALQSOs can be sufficient to provide AGN kinetic feedback required to explain the co-evolution between black holes and host galaxies.

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