CO-to-H$_2$ Conversion and Spectral Column Density in Molecular Clouds:
  The Vriability of $X_{\rm CO}$ Factor
  Analyzing the Galactic plane CO survey with the Nobeyama 45-m telescope, we
compared the spectral column density (SCD) of H$_2$ calculated for $^{12}$CO
line using the current conversion factor $X_{\rm CO}$ to that for $^{13}$CO
line under LTE in M16 and W43 regions. Here, SCD is defined by $dN_{\rm
H_2}/dv$ with $N_{\rm H_2}$ and $v$ being the column density and radial
velocity, respectively. It is found that the $X_{\rm CO}$ method significantly
under-estimates the H$_2$ density in a cloud or region, where SCD exceeds a
critical value ($ \sim 3\times 10^{21}\ [{\rm H_2 \ cm^{-2} \ (km \
s^{-1})^{-1}}]$), but over-estimates in lower SCD regions. We point out that
the actual CO-to-H$_2$ conversion factor varies with the H$_2$ column density
or with the CO-line intensity: It increases in the inner and opaque parts of
molecular clouds, whereas it decreases in the low-density envelopes. However,
in so far as the current $X_{\rm CO}$ is used combined with the integrated
$^{12}$CO intensity averaged over an entire cloud, it yields a consistent value
with that calculated using the $^{13}$CO intensity by LTE. Based on the
analysis, we propose a new CO-to-\Htwo conversion relation, $N_{\rm H_2}^* =
\int X_{\rm CO}^*(T_{\rm B}) T_{\rm B} dv$, where $X_{\rm CO}^*=(T_{\rm
B}/T_{\rm B}^*)^\beta X_{\rm CO}$ is the modified spectral conversion factor as
a function of the brightness temperature, $T_{\rm B}$, of the ${12}$CO
($J=1-0$) line, and $\beta\sim 1-2$ and $T_{\rm B}^*=12-16$ K are empirical
constants obtained by fitting to the observed data. The formula corrects for
the over/under estimation of the column density at low/high-CO line
intensities, and is applicable to molecular clouds with $T_{\rm B} \ge 1$ K
(rms noise in the data) from envelope to cores at sub-parsec scales
(resolution). (Full resolution copy available at
http://www.ioa.s.u-tokyo.ac.jp/~sofue/news/2020_mn_Xco12co13_fugin.pdf)
