Shape and Rotational Motion Models for Tumbling and Monolithic Asteroid 2012 $\mathrm{TC_4}$:High Time Resolution Lightcurve with the Tomo-e Gozen Camera

We present visible and near-infrared observations of a near-Earth object (NEO), 2012 $\mathrm{TC_4}$. The NEO 2012 $\mathrm{TC_4}$ approached close to the Earth at a distance of about 50,000 km in October 2017. This close approach provided a practical exercise for planetary defense. This apparition was also an appropriate opportunity to investigate 2012 $\mathrm{TC_4}$, which is a monolithic asteroid \citep{Polishook13}. We conducted the observation campaign of 2012 $\mathrm{TC_4}$ using six small- and medium-sized telescopes. The multiband photometry analysis showed that the taxonomic class of 2012 $\mathrm{TC_4}$ to be an X-type. In particular, we successfully obtained the high time resolution lightcurve of 2012 $\mathrm{TC_4}$ with the Tomo-e Gozen camera, which is the world's first wide-field CMOS camera, mounted on the 1.05 m Schmidt telescope at Kiso Observatory. The shape and rotational motion models of 2012 $\mathrm{TC_4}$ were derived from the lightcurve. When 2012 $\mathrm{TC_4}$ was assumed to be a triaxial ellipsoid, the rotational and precession periods were 8.47 $\pm$ 0.01 min and 12.25 $\pm$ 0.01 min, respectively, with the long axis mode. This indicates that 2012 $\mathrm{TC_4}$ is a tumbling and monolithic asteroid. The shape models showed that the plausible axial lengths to be 6.2 $\times$ 8.0 $\times$ 14.9~m or 3.3 $\times$ 8.0 $\times$ 14.3~m. The flattened and elongated shape indicates that 2012 $\mathrm{TC_4}$ is a fragment produced by a impact event. We also estimated the excitation timescale, which implied that the impact event happened within $\sim$3 $\times$ 10$^{5}$ yr and 2012 $\mathrm{TC_4}$ has a fresh surface.