A systematic TMRT observational study of Galactic $^{12}$C/$^{13}$C
  ratios from Formaldehyde
  We present observations of the C-band $1_{10}-1_{11}$ (4.8 GHz) and Ku-band
$2_{11}-2_{12}$ (14.5 GHz) K-doublet lines of H$_2$CO and the C-band
$1_{10}-1_{11}$ (4.6 GHz) line of H$_2$$^{13}$CO toward a large sample of
Galactic molecular clouds, through the Shanghai Tianma 65-m radio telescope
(TMRT). Our sample with 112 sources includes strong H$_2$CO sources from the
TMRT molecular line survey at C-band and other known H$_2$CO sources. All three
lines are detected toward 38 objects (43 radial velocity components) yielding a
detection rate of 34\%. Complementary observations of their continuum emission
at both C- and Ku-bands were performed. Combining spectral line parameters and
continuum data, we calculate the column densities, the optical depths and the
isotope ratio H$_2$$^{12}$CO/H$_2$$^{13}$CO for each source. To evaluate photon
trapping caused by sometimes significant opacities in the main isotopologue's
rotational mm-wave lines connecting our measured K-doublets, and to obtain
$^{12}$C/$^{13}$C abundance ratios, we used the RADEX non-LTE model accounting
for radiative transfer effects. This implied the use of the new collision rates
from \citet{Wiesenfeld2013}. Also implementing distance values from
trigonometric parallax measurements for our sources, we obtain a linear fit of
$^{12}$C/$^{13}$C = (5.08$\pm$1.10)D$_{GC}$ + (11.86$\pm$6.60), with a
correlation coefficient of 0.58. D$_{GC}$ refers to Galactocentric distances.
Our $^{12}$C/$^{13}$C ratios agree very well with the ones deduced from CN and
C$^{18}$O but are lower than those previously reported on the basis of H$_2$CO,
tending to suggest that the bulk of the H$_2$CO in our sources was formed on
dust grain mantles and not in the gas phase.
