Constraining the magnetic field strength of a flaring radio core in the compact steep spectrum source 3C 138
Compact steep spectrum (CSS) sources generally show weak Doppler boosting, yet some exceptions show multi-year-scale radio flux variability and high-energy activity. Since 2022, the CSS quasar 3C 138 has been in a radio high state accompanied by multiple gamma-ray outbursts, offering unique opportunities to study changes in jet physical conditions. We estimated the synchrotron self-absorption (SSA) magnetic field ($B_{\rm SSA}$) in the SSA core of 3C 138 during its high state and compared it with the equipartition magnetic field ($B_{\rm eq}$) to assess the core field environment. Using extended Korean Very long-baseline interferometry Network (KVN) data at 22, 43, 86, and 129 GHz (2024-2025), we calibrated the visibilities and modeled resolved components with circular Gaussians. A single-zone SSA model fitted to the core spectrum provided the turnover frequency and peak flux density, from which we estimated the $B_{\rm SSA}$ and $B_{\rm eq}$. We used Very Large Array and Atacama Large Millimeter/submillimeter Array data to constrain the broadband spectra with the same model. The KVN SSA core shows a turnover at about 33 GHz and a peak flux of about 1.45 Jy. The inferred $B_{\rm SSA}$ is far below equipartition, with $B_{\rm SSA}/B_{\rm eq}\approx0.05$. The flux variability of 3C 138 is driven by a compact, particle-dominated core. Shock-driven particle injection in the inner jet could account for the core brightening and the production of X-ray/gamma-ray emissions through an inverse-Compton process without requiring extreme relativistic beaming effects.