The Tantalizing Case of the Quasar J0950+5128 -- I. Presentation of the Data and Detailed Exploration of the Binary Supermassive Black Hole Scenario

Spectroscopic observations of the quasar J0950$+$5128 spanning 22 years reveal monotonic radial velocity variations in its broad H$β$ emission line. Moreover, the line profile becomes broader over time, necessitating careful measurements. We present robust H$β$ velocity shift measurements obtained via cross correlation, applied to both the full spectra and to isolated broad H$β$ components derived from spectral decomposition. We also examine the light curves for variability consistent with the spectroscopic trends. Using Lomb-Scargle periodogram analysis we find no significant periodic signal. We consider several interpretations for the observed changes, including a binary supermassive black hole, dust-cloud obscuration, outflows, a recoiling black hole, and a single perturbed, disk-like broad-line region. We deem the binary and perturbed broad-line region scenarios to be physically plausible. The binary interpretation is the only one for which we can immediately compare a physical model to the available data. Thus, we incorporate radial velocity ''jitter'' to emulate typical quasar variability and fit the radial velocity curve with a Keplerian model to examine whether it can reproduce the observations. In this context, the available observations trace only a segment of the putative orbit. The fit yields a period of 33 years (observed frame) and an eccentricity of 0.65, with lower limits on the semi-major axis and black hole mass of $10^{-2}\;$pc and $10^7\;{\rm M}_\odot$, respectively. Thus, J0950$+$5128 is a binary candidate deserving further study. The single, perturbed broad-line region interpretation remains viable but requires additional observations and modeling for further evaluation. Continued monitoring is, therefore, essential.