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.


INTRODUCTION
Isotope abundance ratios provide a powerful tool to trace stellar nucleosynthesis, to evaluate the enrichment of the interstellar medium (ISM) by stellar ejecta and to constrain the chemical evolution of the Milky Way (Wilson & Rood 1994). In particular, the 12 C/ 13 C ratio is one of the most useful tracers of the relative degree of primary to secondary processing. 12 C is known to be produced primarily via Helium burning in massive stars, on rapid time scales, whereas 13 C is thought to be formed primarily through CNO processing of 12 C seeds from earlier stellar generations. This occurs on a slower timescale during the red giant phase in low and intermediate-mass stars or novae (Henkel et al. 1994, Meyer 1994Wilson & Rood 1994). Thus the 12 C/ 13 C ratio is expected to decrease with time and a 12 C/ 13 C gradient should exist with Galactocentric distance, assuming an inside-out formation scenario of the Milky way (Pilkington et al. 2012).
Under certain conditions, isotope abundance ratios can be effectively determined from the strengths of corresponding molecular lines (e.g., H 2 CO, CO and CN). Measurements of absorption lines of H 2 CO and H 2 13 CO toward strong continuum sources are believed to be one of the best ways to determine the isotope ratio between 12 C and 13 C. Previous observations support the existence of a gradient of 12 C/ 13 C, i.e., the ratio increases as a function of Galactocentric distance (Wilson et al. 1976;Gardner & Whiteoak 1979;Henkel et al. 1980Henkel et al. ,1982Henkel et al. ,1983Henkel et al. ,1985. A 12 C/ 13 C ratio cannot be directly deduced from 12 CO (hereafter CO) and 13 CO, since CO (and sometimes 13 CO) is optically thick. Therefore, their isotopologues C 18 O and 13 C 18 O, exhibiting optically thin lines, are sometimes used to determine 12 C/ 13 C (Langer & Penzias 1990). However, this is time-consuming, since thermally excited emission lines with low optical depths can have extremely low intensities (Wilson & Rood 1994). CN is also a tracer of the 12 C/ 13 C ratio. Although CN has a distinct hyperfine structure, which can be used to directly evaluate opacities (Savage et al. 2002;Milam et al. 2005), it is still difficult to obtain a true 12 C/ 13 C value from 12 CN/ 13 CN because of sometimes large opacities of 12 CN. Milam et al. (2005) proposed a gradient of 12 C/ 13 C with Galactocentric distance, from a combination of all previously published measurements of CN, C 18 O and H 2 CO. The goal of our study is to critically review their results using new distances and H 2 CO collision rates as well as a larger sample of sources and to examine in how far previous results have to be modified.
Thus we have performed a more systematic study of H 2 CO and H 2 13 CO toward a big sample of Galactic molecular clouds to determine 12 C/ 13 C isotope ratios more accurately. Based on a TMRT formaldehyde survey at C-band (∼ 5 GHz; Li et al. 2019, in preparation), we chose sources with strong H 2 CO signals as our targets for deep integration. Some previously studied strong H 2 CO sources with absolute flux densities larger than 0.5 Jy (Wilson et al. 1976;Henkel et al. 1982Henkel et al. ,1983Henkel et al. ,1985Araya et al. 2007) are also included in our sample. H 2 CO collision rates were taken from Wiesenfeld & Faure (2013) who consider H 2 CO collisions with H 2 based on the high accuracy potential energy surface introduced by Troscompt et al. (2009). These rates differ significantly from the scaled rates of H 2 CO in collision with He (Green et al. 1991) used in previous analyses. At the same time we also took improved distances for our sources to evaluate more accurately the previously reported gradient of the carbon isotope ratio as a function of Galactocentric distance. In Section 2, the observations of our large sample are described, encompassing Galactocentric radii from 0 to 10.5 kpc. In Section 3, we first perform the analysis of spectra and the radio continuum, followed by modelling work based on radiative transfer calculations. Section 4 provides discussions on the isotope ratio gradient, derived from our observational data alone as well as also including previously obtained data. Our main results are summarized in Section 5.

OBSERVATIONS
The 1 10 − 1 11 and 2 11 − 2 12 transitions were observed toward our sources in March and October 2016 as well as in May and July 2017 with the Shanghai Tianma Radio Telescope (TMRT). We used two cryogenically cooled receivers covering the frequency ranges of 4.0-8.0 GHz (C-band) and 12.0-18.2 GHz (Ku-band). The rest frequencies of the 1 10 − 1 11 transitions of H 2 12 CO and H 2 13 CO were set to be 4.829660 and 4.593089 GHz, respectively (Tucker et al. 1971;Wilson et al. 1976). 14.48848 GHz was adopted for the 2 11 − 2 12 transition of H 2 12 CO. The beam sizes are ∼4 arcmin at 5 GHz and ∼1.3 arcmin at 14.5 GHz, respectively. A Digital Backend System (DIBAS) was used for data recording (see Li et al. 2016). The DIBAS mode 22 was adopted for our observations, with eight spectral windows, to cover the H 2 12 CO and H 2 13 CO lines simultaneously, each with 16384 channels and a bandwidth of 23.4 MHz, supplying channel widths of 0.09 km s −1 and 0.03 km s −1 at 5 GHz and 14.5 GHz, respectively. The system temperature was 20-30 and 30-80 K on a T * A scale for the 1 10 − 1 11 and 2 11 − 2 12 transition observations, respectively. The active surface system of the primary dish and a subreflector were used to improve the aperture efficiency during our Ku-band

Optical Depth and Column Density
We used the GILDAS/CLASS package to reduce the spectral line data. A first order polynomial was subtracted from each spectrum for baseline removal. Then we obtained the line parameters via Gaussian fitting for those 38 sources, which are presented in Table 1. The spectra of the 2 11 − 2 12 lines of H 2 12 CO, as well as the 1 10 − 1 11 lines of H 2 12 CO and H 2 13 CO, after subtracting baselines and applying Hanning smoothing, are shown in Figure 1. For those 46 sources with only a detection of the H 2 CO 1 10 − 1 11 line, spectra and line parameters derived from Gaussian fitting are presented in Appendices B and C, respectively.
The continuum data calibration was obtained with the following procedure: First, baselines were subtracted, being defined by the lengths of the continuum scans avoiding the source itself at ± 1.5×HPBW (half power beam width), relative to the center of the scan. Second, Gaussian profiles were fitted to obtain the position offset between the real position of the source and the center of the cross scans. Then, the obtained amplitude was corrected for the pointing error, adopting a two-dimensional Gaussian intensity profile, using the formula:   where P ointing = exp(−4 × ln2 × (of f set/HP BW ) 2 ). Finally, the antenna temperature has been corrected for the elevation-dependent gain, defined by the parabolic equation: where Gain = A × Elevation 2 + B × Elevation + C. A, B, C are the coefficients of a 2nd order polynomial fit, obtained from "T /T max − Elevation" plots of well-known stable calibrators (e.g. 3C286, 3C123 and 3C48). For Kuband observations the last step of the data reduction was omitted because of an absence of a gain curve equation for this band, which provides up to 10% of uncertainty. The resulting measured amplitude of each source is an average from all its scans, in units of antenna temperature in Kelvin (K). The measurement uncertainty is formed by the statistical errors of the Gaussian fits. Figure 2 shows one example characterizing our sample of observed sources. The line parameters and continuum temperatures are listed in Table 1. Then we used these parameters to calculate optical depths ) where T L is the observed line temperature, T C is the continuum temperature, T BB is the 2.7 K background radiation, and T ex is the excitation temperature. We have used the RADEX non-LTE model 1 ( Van der Tak et al. 2007) to provide excitation temperatures for our sources (see details in Section 3.2). The apparent maximum optical depths τ , which are the optical depths at the velocities with the most negative (absorption) line temperatures, in the 2 11 − 2 12 transition of H 2 12 CO are listed in Table 1. In Table 2, the apparent maximum optical depths of the 1 10 − 1 11 transitions are listed. Finally, we can get with the velocity integrated optical depth the column density, following Wilson et al.(1976): The radial velocity V is in km s −1 and the excitation temperature T ex in K. The numerical coefficient is valid for H 2 12 CO; the coefficient for H 2 13 CO is 1.32 × 10 13 . The column densities in the 1 11 level divided by the a priori unknown quantity T ex for our sources are listed in Columns 6 and 7 of Table 2. Used velocity ranges are given in Columns 4 and 5.
We derived the H 2 12 CO/H 2 13 CO isotope ratios in two different ways: (1) from Gaussian least square fits and (2) using planimetry to derive the ratio of their column densities (Col.(8) of Table 2; see details in Wilson et al. 1976). Since the ratios obtained by these two methods differ by only 10% within the permissible margin of error, we average these two ratios to get the final H 2 12 CO/H 2 13 CO ratios. The average ratios corrected for telescope gain (see details in Section 2) for our sample are listed in Col.(4) of Table 3.

Beam Size Effect
The beam sizes are ∼4 arcmin at C-band and ∼1.3 arcmin at Ku-band, respectively. The temperature difference between the microwave background (2.73 K) and the excitation temperature of the collisionally cooled (Evans et al. 1975) H 2 CO lines is about 1.5 K, as derived from our non-LTE calculations. For a cloud fully filling the beam, and being optically moderately thin, e.g. τ = 0.3, this would result in about 0.5 K across that part of the beam not covered by a background HII region. Assuming that the HII region has a diameter of 6 arcsec (e.g., Zapata et al. 2009) and a radiation temperature of 6000 K (e.g., Reifenstein et al. 1970), we obtain for T non−HII−region(H2CO) /T HII−region(H2CO) values of about 0.01 and 0.1 for a 1.3 arcmin beam and 4 arcmin beam, respectively. The result indicates that in this case the tiny spot in front of the HII region dominates the H 2 CO absorption budget and the difference in beam sizes can be neglected. If the respective clouds are not filling the entire 4 arcmin beam, the comparatively small area covered by the continuum emission becomes even more dominant and the difference in beam size can also be neglected.

Photon Trapping Corrections
Photon trapping in the millimeter H 2 CO rotation lines, connecting the J = 1 and 2 K-doublets (i.e. the 2 11 -1 10 and 2 12 -1 11 transitions), has also to be considered. In case the two millimeter lines (2 11 -1 10 and 2 12 -1 11 ) are not entirely optically thin, their excitation temperature values rise and are higher than in the optically thin case, represented by H 2 13 CO. Thus, more population ends up in the H 2 CO J = 2 doublet than in the entirely optically thin case represented by H 2 13 CO. The J = 1 H 2 CO doublet then gets a little depopulated and thus the C-band H 2 CO/H 2 13 CO line intensity ratio becomes smaller than if both H 2 CO and H 2 13 CO lines were all optically thin. Here, the RADEX non-LTE model 1 was used to correct the ratios for photon trapping. As already mentioned in Sect. 1, the collision rates of H 2 CO are taken from Wiesenfeld & Faure (2013), which are calculated for H 2 CO in collision with H 2 and the high accuracy potential energy surface of Troscompt et al. (2009). These are significantly different from the scaled rates of H 2 CO in collision with He (Green et al. 1991;see below). Correction factors f 12/13 calculated by the new collision rates tend to be larger than those derived by the old collision rates, especially for larger optical depths at 4.8 GHz, as shown in Figure 3. The correction factor f 12/13 is defined as in Henkel et al. (1980):   (2) and (3): the apparent optical depths of H2 12 CO and H2 13 CO 110 − 111 respectively; Columns (4) and (5): the velocity range used for integrated optical depth; Columns (6) and (7): the values of the column densities in the 111 level divided by Tex for H2 12 CO and H2 13 CO respectively; Column (8): the H2 12 CO H2 13 CO isotope ratios obtained with the planimetry method; Column (9): the distance from the Galactic center.
where the star indicates the RADEX model and the superscript zero refers to the model with the same H 2 density and temperature but with a formaldehyde column density which is a factor 50 times lower. The results are listed in Table 3. We ran the RADEX offline code to independently estimate the H 2 density for our sources, and find in good agreement that it ranges from 10 4.7 cm −3 to 10 5.3 cm −3 . An example of this fitting process (see details in Ginsburg et al. 2011) is shown in Figure 4. The H 2 density of the HII regions is ∼ 10 5 cm −3 (Henkel et al. 1980;Ginsburg et al. 2011;Tang et al. 2017). For the 6 sources for which we only derived the upper limit of the 2 11 -2 12 optical depths due to the non-detection of continuum at Ku-band, we assume that the H 2 density in these 6 sources is also ∼ 10 5 cm −3 . Thus we adopt n(H 2 ) = 10 5 cm −3 for all of our sources. In Figure 5, we show the dependence of f 12/13 on τ 4.8 for various molecular hydrogen densities and temperatures. As expected, the corrections to the measured isotope ratios become more important with larger optical depth. As for the kinetic temperatures, molecular clouds near ultracompact HII (UCHII) regions have a temperature of around 40 K, which is warmer than the molecular environment of more evolved HII regions (Rivera-Ingraham et al. 2010;Ginsburg et al. 2011). Therefore, we chose the kinetic temperature range of 20K-40K to analyze the effect of temperature on f 12/13 . The results show that there is little difference for f 12/13 in the kinetic temperature range of 20K-40K, as can be seen in Figure 5. Thus we adopt 30 K as the kinetic temperature for our sources.

Hyperfine Structure
Both the 1 10 − 1 11 H 2 12 CO and H 2 13 CO lines are split by hyperfine interactions (Tucker et al. 1971). The H 2 12 CO line consists of six hyperfine components extended over 30 kHz (Kukolich & Rubin 1971;Johnson et al. 1972;Gardner & Whiteoak 1979), corresponding to a velocity range between -1.13 km s −1 and +0.71 km s −1 relative to the line center (4829.66 MHz). For line widths of order 1 km s −1 , the line profile can be characterized by two components. 89% of the intensity go into features between -0.07 km s −1 and +0.71 km s −1 , 11% into the feature at -1.13 km s −1 . This implies that for wider lines, 100% of the intensity can be observed. While for very narrow lines from dark clouds, the -1.13 km s −1 feature may be seen separately (e.g., Henkel et al. 1981). However, this did not occur in our sources. The hfs slightly broadens the H 2 13 CO lines with respect to the H 2 12 CO lines (Zuckerman et al. 1974). This should be taken into account. For 21 hyperfine components in the H 2 13 CO 1 10 − 1 11 transition, 77% go into components between -1.15 km s −1 and +1.15 km s −1 relative to the line center (4593.089 MHz), while 11.5% go into -8.5 and -6.0 km s −1 and another 11.5% into +5.7 to +8.1 km s −1 features. Thus, in case of weakly detected H 2 13 CO lines, we obtain only 77% of the total intensity for most of our sources. However, in case of really strong H 2 13 CO lines, we can also see the outer features near -7 and +7 km s −1 . Based on the different S/N ratio of each source, we derived the factors of hyperfine splitting (f hf s ) individually for each source. The corrections for hyperfine splitting, also accounting for the fact that H 2 CO and H 2 13 CO are measured at slightly different frequencies ((ν 13 /ν 12 ) · f hf s ), are listed in Column 5 of Table 3.
For comparison, previously published ratios are also collected and are included in Table 4. There are 15 sources from H 2 CO/H 2 13 CO measurements (Henkel et al. 1980(Henkel et al. ,1982(Henkel et al. ,1983, 18 from CN/ 13 CN measurements (Savage et al. 2002;Milam et al. 2005) and 9 from C 18 O/ 13 C 18 O measurements (e.g., Langer & Penzias 1990;Wouterloot et al. 1996;Keene et al. 1998). With respect to these previous measurements, our sample of 38 sources and 43 velocity components covering a wide range of Galactocentric distances provides a significant sample for studying the radial gradient of the isotope ratio 12 C/ 13 C.

Distances
Therefore, we also need Galactocentric distances for our sources that are more accurate than those used in previous less extended surveys. The trigonometric parallax is a very accurate method to measure the distance of sources, which can directly and geometrically determine source distances from the Sun (Reid et al. 2009;. Based on trigonometric parallax data, Reid et al. (2014) provide an accurate method (see details in Reid et al. 2009) for estimating revised kinematic distances with improved Galactic constants and rotation curve. Thus we derived the heliocentric distance for 12 of our sources (marked in Table 2) from their trigonometric parallax data . For the other 26 sources without trigonometric parallax data, we estimated their heliocentric distance using the Revised Kinematic  Distance Calculator 2 , also based on the results of the parallax measurements. This also includes 7 sources, which are not part of our sample but which were included in previous measurements. A more accurate distance was adopted here (listed in Table 4), including 5 sources (W3-OH, W51M, Orion A, Orion Bar and NGC7538) with trigonometric parallax distances (Menten et al. 2007;Reid et al. 2014) and 2 sources (S156, WB 391) with new heliocentric distances from the Revised Kinematic Distance calculator. The latter were obtained with a solar rotational velocity of V 0 = 240 km s −1 , assuming the derivative of the rotation curve beyond R = R 0 (for the solar galactocentric value, we adopted 8.125 kpc, Gravity Collaboration et al. 2018) to be dV/dR = 0.2 km s −1 kpc −1 and the solar motion with respect to the LSR to be (U ⊙ , V ⊙ , W ⊙ ) = (10.7, 15.6, 8.9) km s −1 ). Then, we can determine the Galactocentric distance from the heliocentric distance (Roman-Duval et al. 2009): where l is the Galatic longitude of the source and d is the kinematic distance.
The new Galactocentric distances for our 38 detected sources and 7 sources not in our sample are listed in Table 2  and Table 4, respectively.

DISCUSSION
With the accurate Galactocentric distance and the corrected H 2 12 CO/H 2 13 CO ratios, we can study the variation of the 12 C/ 13 C isotope ratios as a function of Galactocentric distance. We show our results as filled black squares in Figure 6. Previous results of the ratio are also plotted, but against the new distance values (see details in Sect. 3.5), and new linear fitting lines are also presented in Figure 6. A comparison shows that using the new distances really affects the fitting results (see details in Table 5). E.g., for the gradient results from H 2 12 CO/H 2 13 CO in Henkel et al. (1980Henkel et al. ( ,1982Henkel et al. ( ,1983Henkel et al. ( ,1985, the slope/intercept becomes (8.66 ± 1.64)/(9.30 ± 10.37) from (7.60 ± 1.79)/(18.05 ± 10.88). The fitting for CN/ 13 CN in Milam et al. (2005) becomes (6.87 ± 1.46)/(5.06 ± 11.07) from (6.01 ± 1.19)/(12.28 ± 9.33). In addition we give in Table 5 the ratios that would be derived using the old distances and the collision rates of Green et al. (1991). It is clear that the 12 C/ 13 C gradient in the case of the collision rates taken from Wiesenfeld & Faure (2013) and new distances are closer to the gradient derived from CN and C 18 O. In order not to bias our results towards low values with small error bars, the gradient was calculated from an unweighted least-squares fit of our corrected H 2 12 CO/H 2 13 CO ratios with the collision rates taken from Wiesenfeld & Faure (2013) and the new, more reliable distances: 12 C/ 13 C = (5.08 ± 1.10)D GC + (11.86 ± 6.60).  (6): the correction factor f 12/13 obtained from RADEX; Column (7): the corrected H2 12 CO H2 13 CO isotope ratios.  Henkel et al.(1980Henkel et al.( , 1982Henkel et al.( , 1983Henkel et al.( , 1985, using state of the art distances, and are fitted by a dashed line. The empty triangles and stars are values from CN (Savage et al. 2002, Milam et al. 2005) and C 18 O (Langer & Penzias 1990, Wouterloot & Brand 1996, Keene et al. 1998, respectively, also using modified distances. The dotted line presents the linear fit from CN, and the dash-dotted line is the fit from C 18 O. The symbol ⊙ indicates the 12 C/ 13 C isotope ratio of the Sun. All of the values are also presented in Table 4. This gradient is flatter than the gradient derived from previous H 2 CO measurements (Henkel et al. 1980(Henkel et al. ,1982 but agrees very well with those from CN and C 18 O measurements (Milam et al. 2005).
Although our results are more reliable (due to our bigger sample, the new collision rates, more accurate distances and presumably a more realistic determination of the photon trapping effect), some other uncertainties (observational bias due to different distances, beam sizes, excitation temperatures, isotope selective photodissociation and chemical fractionation) should be mentioned.

Observational bias due to distance effects
The 12 C/ 13 C isotope ratios from our H 2 CO/H 2 13 CO measurements plotted as functions of the distance from the Sun are shown in Figure 7. No apparent gradient can be found, which suggests that any observational bias due to distance related effects is not significant for the 12 C/ 13 C gradient as a function of Galactocentric distance.

Excitation temperature
We assumed that the excitation temperature of the 1 10 − 1 11 transition for H 2 CO equals that for H 2 13 CO. It is generally accepted that these lines are affected by a collisional pumping mechanism (Evans et al. 1975;Wilson et al. 1976), which should produce nearly the same excitation temperature for the two species. Hence any differences in the excitation between the two species should be negligible.  Figure 7. The 12 C/ 13 C isotope ratios from our H2CO/H2 13 CO measurements plotted as functions of the distance from the Sun.

Isotope selective photodissociation
Isotope selective photodissociation can occur in a molecular cloud affected by high UV radiation, which increases the 12 C/ 13 C isotope ratio (Riquelme et al. 2010). This effect is particularly pronounced in photon-dominated regions (PDRs). The UV photons can photodissociate the rarer isotopologues and affect less the main isotopologues (the more abundant molecules) due to their more efficient self-shielding, which would lead to higher 12 C/ 13 C isotope ratios. However, isotope selective photodissociation hardly affects the 12 C/ 13 C ratio from high density tracers like CN (Milam et al. 2005). And since the 12 C/ 13 C ratios derived from H 2 CO, C 18 O and CN are quite similar in the average (Fig. 6), isotope selective photodissociation should not be a dominant effect.

Chemical fractionation
In order to investigate the effects of chemical fractionation, we must understand the formation of formaldehyde (H 2 CO). In classic gas-phase reaction networks, H 2 CO is formed from neutral-neutral reactions between CH 3 and atomic oxygen, where CH 3 is built from carbon ions reacting with molecular hydrogen (e.g., Wirström et al. 2012). Alternatively, formaldehyde is efficiently formed on the surface of icy dust grains via the hydrogenation of CO (e.g., Charnley et al. 1997;Watanabe & Kouchi 2002). Subsequently, it may be released into the gas phase by photoevaporation or shocks and then behave, in the first case, like a tracer of photon dominated regions (PDRs), and in the second like a shock tracer. The relative importance of the two main pathways of H 2 CO formation, gas-phase chemistry or dust grain mantle evaporation, is poorly constrained.
Small differences in the zero point energy between reactants and products of isotopically distinct species may cause fractionation. Due to the charge exchange reaction of CO with 13 C + (Watson et al. 1976), gas phase CO should have a tendency to be enriched in 13 CO. Other molecules, like perhaps H 2 CO, formed in the gas phase through different chemical pathways, should then be depleted in the 13 C bearing isotopologue (e.g., Langer et al. 1984). However, if most of the formaldehyde originates alternatively from dust grain mantles, the situation is different. In this case, H 2 CO formed by the hydrogenation of CO, would be similarly enriched in 13 C as CO.
As a result, the carbon isotope ratios obtained from formaldehyde could be similar or larger than that from C 18 O and could provide us with a useful hint for the predominant H 2 CO formation scenario in massive star forming regions. In previous studies (see, e.g., Milam et al. 2005), larger values were found, suggesting a predominantly gas-phase origin of formaldehyde. In this study, however, we find slightly smaller values, consistent with formaldehyde formation on dust grains and subsequent release into the interstellar medium. Within the range of uncertainty, it could even contradict both grain mantle and gas phase formation scenarios outlined above. First results from four sources, hinting at this latter possibility, i.e. suggesting significantly 13 C enhanced formaldehyde as compared to CO and methanol (CH 3 OH), were already published by Wirström et al. 2012. Constructing more extensive fractionation networks including both gas-phase, dust grain and gas-dust interactions might provide further insight into this interesting and quite basic astrochemical puzzle.

SUMMARY
With the Tianma Radio Telescope (TMRT), we performed observations of the 1 10 − 1 11 and 2 11 − 2 12 lines of H 2 CO and the 1 10 − 1 11 line of H 2 13 CO toward a big sample of 112 Galactic molecular lines-of-sight. All three lines are detected toward 38 targets (43 radial velocity components), with a detection rate of ∼34%. For these 38 sources, their continuum at C-(∼ 5 GHz) and  were also observed and detected, at C-band in all sources and, at Ku-band, in 32 objects. Spectral line and continuum data for those 38 sources were analyzed. Our main results are : 1. Based on spectral line parameters and continuum temperatures, we obtained column densities, optical depths and the molecular abundance ratios.
2. We used the RADEX non-LTE model for the radiative transfer and took the new collision rates from Wiesenfeld & Faure (2013) to determine the photon trapping effect in the mm-wave lines connecting the J = 1 and 2 K-doublets of ortho-H 2 CO.

We took reliable distance values from trigonometric parallax measurements and the Revised Kinematic Distance
Calculator ) for our sources. Thus the 12 C/ 13 C gradient along the Galactocentric distance is confirmed from a linear fit to the 12 C/ 13 C data resulting in (5.08±1.10)D GC +(11.86±6.60), with a correlation coefficient of 0.58. Measurements of more sources, especially those with large Galactocentric distance, are needed to further improve the statistical significance.
4. The gradient determined by us is flatter than that obtained from previous studies of formaldehyde, but is consistent, within the uncertainties, with those obtained from CN and C 18 O. While the previous results may suggest an H 2 CO formation mechanism dominated by gas-phase reactions, our new result tends to support a formation on dust grain mantles followed by evaporation.

APPENDIX
A.  Table 6 continued on next page  Table 6 continued on next page  Table 6 continued on next page   Table 6 continued on next page b The sources not detected in both the 110 − 111 lines of H2 12 CO and H2 13 CO.