NICER, NuSTAR, and Swift follow-up observations of the $\gamma$-ray flaring blazar BL Lacertae in 2020 August-October

During a period of strong $\gamma$-ray flaring activity from BL Lacertae, we organized Swift, Neutron star Interior Composition Explorer (NICER), and Nuclear Spectroscopic Telescope Array (NuSTAR) follow-up observations. The source has been monitored by Swift-XRT (X-ray Telescope) between 2020 August 11 and October 16, showing a variability amplitude of 65, with a flux varying between 1.0 $\times$ 10$^{-11}$ and 65.3 $\times$ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$. On 2020 October 6, Swift-XRT has observed the source during its historical maximum X-ray flux. A softer-when-brighter behaviour has been observed by XRT, suggesting an increasing importance of the synchrotron emission in the X-ray part of the spectrum covered by XRT during this bright state. Rapid variability in soft X-rays has been observed with both the Swift-XRT and NICER observations with a minimum variability time-scale of 60 and 240 s, and a doubling time-scale of 274 and 1008 s, respectively, suggesting very compact emitting regions (1.1 $\times$ 10$^{14}$ and 4.0 $\times$ 10$^{14}$ cm). At hard X-rays, a minimum variability time-scale of $\sim$ 5.5 ks has been observed by NuSTAR. We report the first simultaneous NICER and NuSTAR observations of BL Lacertae during 2020 October 11-12. The joint NICER and NuSTAR spectra are well fitted by a broken power law with a significant difference of the photon index below (2.10) and above (1.60) an energy break at $\sim$ 2.7 keV, indicating the presence of two different emission components (i.e, synchrotron and inverse Compton) in the broad band X-ray spectrum. Leaving the total hydrogen column density toward BL Lacertae free to vary, a value of N$_{H,tot}$ = (2.58 $\pm$ 0.09) $\times$ 10$^{21}$ cm$^{-2}$ has been estimated.