A multi-wavelength investigation of PSR J2229+6114 and its pulsar wind nebula in the radio, X-ray, and gamma-ray bands
G106.3$+$2.7, commonly considered a composite supernova remnant (SNR), is characterized by a boomerang-shaped pulsar wind nebula (PWN) and two distinct ("head" & "tail") regions in the radio band. A discovery of very-high-energy (VHE) gamma-ray emission ($E_γ> 100$ GeV) followed by the recent detection of ultra-high-energy (UHE) gamma-ray emission ($E_γ> 100$ TeV) from the tail region suggests that G106.3$+$2.7 is a PeVatron candidate. We present a comprehensive multi-wavelength study of the Boomerang PWN (100" around PSR J2229+6114) using archival radio and Chandra data obtained from two decades ago, a new NuSTAR X-ray observation from 2020, and upper limits on gamma-ray fluxes obtained by Fermi and VERITAS observatories. The NuSTAR observation allowed us to detect a 51.67 ms spin period from the pulsar PSR J2229+6114 and the PWN emission characterized by a power-law model with $Γ= 1.52\pm0.06$ up to 20 keV. Contrary to the previous radio study by Kothes et al. 2006, we prefer a much lower PWN B-field ($B\sim3$ $μ$G) and larger distance ($d \sim 8$ kpc) based on (1) the non-varying X-ray flux over the last two decades, (2) the energy-dependent X-ray PWN size resulting from synchrotron burn-off and (3) the multi-wavelength spectral energy distribution (SED) data. Our SED model suggests that the PWN is currently re-expanding after being compressed by the SNR reverse shock $\sim 1000$ years ago. In this case, the head region should be formed by GeV--TeV electrons injected earlier by the pulsar propagating into the low density environment.