TKS X: Confirmation of TOI-1444b and a Comparative Analysis of the Ultra-short-period Planets with Hot Neptunes

We report the discovery of TOI-1444b, a 1.4-$R_\oplus$ super-Earth on a 0.47-day orbit around a Sun-like star discovered by {\it TESS}. Precise radial velocities from Keck/HIRES confirmed the planet and constrained the mass to be $3.87 \pm 0.71 M_\oplus$. The RV dataset also indicates a possible non-transiting, 16-day planet ($11.8\pm2.9M_\oplus$). We report a tentative detection of phase curve variation and secondary eclipse of TOI-1444b in the {\it TESS} bandpass. TOI-1444b joins the growing sample of 17 ultra-short-period planets with well-measured masses and sizes, most of which are compatible with an Earth-like composition. We take this opportunity to examine the expanding sample of ultra-short-period planets ($<2R_\oplus$) and contrast them with the newly discovered sub-day ultra-hot Neptunes ($>3R_\oplus$, $>2000F_\oplus$ TOI-849 b, LTT9779 b and K2-100). We find that 1) USPs have predominately Earth-like compositions with inferred iron core mass fractions of 0.32$\pm$0.04; and have masses below the threshold of runaway accretion ($\sim 10M_\oplus$), while ultra-hot Neptunes are above the threshold and have H/He or other volatile envelope. 2) USPs are almost always found in multi-planet system consistent with a secular interaction formation scenario; ultra-hot Neptunes ($P_{\rm orb} \lesssim$1 day) tend to be ``lonely' similar to longer-period hot Neptunes($P_{\rm orb}$1-10 days) and hot Jupiters. 3) USPs occur around solar-metallicity stars while hot Neptunes prefer higher metallicity hosts. 4) In all these respects, the ultra-hot Neptunes show more resemblance to hot Jupiters than the smaller USP planets, although ultra-hot Neptunes are rarer than both USP and hot Jupiters by 1-2 orders of magnitude.