We present new results of a numerical study of the pumping of 4.8 and 14.5 GHz maser of oinlineformulatexmath idTM0001 notationLaTeXmathrmH2COtexmathinlineformula in the presence of a freefree radiation field. It is shown that in the presence of a freefree radiation field inversion of not only the inlineformulatexmath idTM0002 notationLaTeX110111texmathinlineformula but also the inlineformulatexmath idTM0003 notationLaTeX211212texmathinlineformula and other doublet state transitions occur. Further results are presented to illustrate how as a consequence of the pumping scheme the inversion of the inlineformulatexmath idTM0004 notationLaTeX110111texmathinlineformula and inlineformulatexmath idTM0005 notationLaTeX211212texmathinlineformula transitions respond to the freefree radiation fields associated with H II regions with different emission measures and levels of geometric dilution with respect to the masing region. We also discuss the criticism raised in the past by various authors against the pumping of the 4.8 GHz inlineformulatexmath idTM0006 notationLaTeXmathrmH2COtexmathinlineformula masers by a freefree radiation field. It is argued that the rarity of the inlineformulatexmath idTM0007 notationLaTeXmathrmH2COtexmathinlineformula masers is not to be ascribed to the pumping scheme but to other factors such as e.g. the evolution of the associated H II region or the chemical evolution of the starforming region that determines the inlineformulatexmath idTM0008 notationLaTeXmathrmH2COtexmathinlineformula abundance or a combination of both.
['Astrophysics - Astrophysics of Galaxies']
Revisiting the formaldehyde masers  II. Effects of an H II region and beaming
{'D.J. van der Walt /uni2605': 'Centre for Space Research, North-West University, Hoffman Street, 2520, Potchefstroom, South Africa  \nAccepted 2024 September 16. Received 2024 September 13; in original form 2024 June 19', 'ABSTRACT': 'We present new results of a numerical study of the pumping of 4.8 GHz and 14.5 GHz maser of o-H2CO in the presence of a free-free radiation field. It is shown that in the presence of a free-free radiation field inversion of not only the 110 -111, but also the 211 -212 and other doublet state transitions occur. Further results are presented to illustrate how, as a consequence of the pumping scheme, the inversion of the 110 -111 and 211 -212 transitions respond to the free-free radiation fields associated with H /i.pc/i.pc regions with different emission measures and levels of geometric dilution with respect to the masing region. We also discuss the criticism raised in the past by various authors against the pumping of the 4.8 GHz H2CO masers by a free-free radiation field. It is argued that the rarity of the H2CO masers is not to be ascribed to the pumping scheme but to other factors such as, e.g., the evolution of the associated H /i.pc/i.pc region or the chemical evolution of the star forming region which determines the H2CO abundance or a combination of both.  \nKey words: masers - stars:formation - ISM:molecules - radio lines:ISM - H /i.pc/i.pc regions', '1 INTRODUCTION': 'Following the discovery of the first 4.8 GHz o-H 2 CO (henceforth H 2 CO) maser in NGC 7538 by Downes & Wilson (1974), Boland & de Jong (1981) presented the first model to explain such maser emission. Boland & de Jong (1981) showed that the 1 10 -1 11 transition can be inverted by the free-free radiation field of a compact H /i.pc/i.pc region. Since the model by Boland & de Jong (1981) was the only one for many years, various authors tested the model against their observations. Authors such as, for example, Gardner et al. (1986), Mehringer et al. (1994), Martín-Pintado et al. (1999), Hoffman et al. (2003) and Araya et al. (2007) concluded that the H 2 CO masers known at the time these papers were published, did not meet the conditions required by the model of Boland & de Jong (1981) and, therefore, that the pumping of the 4.8 GHz H 2 CO masers by a freefree radiation field is not a viable mechanism to explain the masers. Some of the reasons for dismissing the model of Boland & de Jong (1981) are, that the emission measures (EM) of the H /i.pc/i.pc regions with which some of the 4.8 GHz H 2 COmasers are associated, are smaller than the lower limit of 10 8 pc cm -6 for which Boland & de Jong (1981) found an inversion, and that some of the masers are, according to the results of Boland & de Jong (1981), offset too far from the H /i.pc/i.pc regions to explain the inversion by a free-free radiation field. The apparent failure of the Boland & de Jong (1981) model to explain the H 2 COmasers prompted Hoffman et al. (2007) to postulate that some rare collisional process may be responsible for the pumping of the masers. Based on their observations of G339.88-1.26, Chen et al. (2017b) argued in favour of the collisional excitation in a jet/outflow driven shock of the 1 10 -1 11 H 2 CO masers. Still more recently, McCarthy et al. (2022) also express the opinion that the pumping  \nmechanism for the population inversion of the 1 10 -1 11 transition of H 2 CO is poorly understood.  \nAnother outstanding question to be addressed is the apparent absence of the 14.5 GHz (2 11 -2 12 ) H 2 CO maser which, according to the model of Boland & de Jong (1981), should be associated with the 4.8 GHz maser if the pumping is through a free-free radiation field. Hoffman et al. (2003) concluded that the non-detection or absence of the 14.5 GHz maser possibly indicates particular physical conditions for the excitation of the 4.8 GHz masers. Recently, Chen et al. (2017a) reported the detection of 14.5 GHz emission from three high mass star forming regions which have associated 4.8 GHz H 2 CO masers. These authors consider the 14.5 GHz emission associated with NGC 7538 to be maser emission, although Shuvo et al. (2021) argues that it might be thermal rather than maser emission. Independent of whether the 14.5 GHz emission in NGC7538 is maser or thermal emission, it is clear that, apart from understanding the inversion of the 1 10 -1 11 transition, it is also necessary to understand the behaviour of the 2 11 -2 12 transition.  \nA further question related to the H 2 CO 1 10 -1 11 masers is the rarity of these masers. Counting individual masers, e.g. two components associated with NGC 7538, two with G29.96-0.02, nine with Sgr B2, to date only 21 such masers have been detected in the Galaxy (Downes & Wilson 1974; Whiteoak & Gardner 1983; Mehringer et al. 1994; Pratap et al. 1994; Araya et al. 2005, 2006, 2008, 2015; Ginsburg et al. 2015; Chen et al. 2017b; McCarthy et al. 2022). While the rarity of the masers may be related to the pumping mechanism, as proposed by Hoffman et al. (2007), there are also other possibilities which can explain the rarity of the masers. As one of a number of possibilities, Ginsburg et al. (2015) also consider a short life-time of the masers as a possible explanation of the rarity of the 1 10 -1 11 masers. Factors that might determine the life-time of the masers are, for example, the chemical, dynamical and radiative (dust  \nand free-free) evolution of the star forming environment. The pumping scheme, however, is determined exclusively by the molecular structure and not any external macroscopic factors. Whether or not a certain transition is inverted depends on its interaction (collisional and radiative) with its environment.  \nIn a recent paper, van der Walt & Mfulwane (2022) presented the results of a non-LTE study on the inversion of the 1 10 -1 11 transition of H 2 CO. The focus of these authors was primarily on understanding the claim by Baan et al. (2017) that the extragalactic megamasers associated with three ultraluminous infrared (starburst) galaxies are radiatively pumped by dust emission, as well as expanding on the results of van der Walt (2014) by increasing the parameter space in kinetic temperature and H 2 density. These authors showed that inversion of the 1 10 -1 11 transition can be achieved from collisions alone, and that, in the presence of a far-infrared dust radiation field, inversion does not take place when collisions are excluded. van der Walt & Mfulwane (2022) also proposed a pumping scheme which can explain why, in the case where the external radiation field is only that of dust, collisions are required for the inversion of the 1 10 -1 11 transition.  \nThe context of the present calculations is a pumping calculation with the aim to, first, establish whether the pumping scheme proposed by van der Walt & Mfulwane (2022) applies to the case of excitation by a free-free radiation field. It is further investigated to what extent the 1 10 -1 11 and 2 11 -2 12 transitions are inverted by different free-free radiation fields and what the effect of beaming is. We also comment on past critique against the viability of the pumping of the masers by a free-free radiation field. Although it is necessary to try to reproduce the optical depths of the 1 10 -1 11 transition as derived from observations (for example in Hoffman et al. 2003), the aim of the present calculations is not to explain individual sources.', '2 NUMERICAL METHOD': 'The numerical method used for the present calculations is the same as that described in detail by van der Walt et al. (2021) and van der Walt & Mfulwane (2022) and will therefore not be repeated here. The molecular data used is also the same as that described in van der Walt & Mfulwane (2022). We assumed a constant density and temperature H /i.pc/i.pc region to calculate the free-free continuum radiation field given by  \n/u1D439 /u1D708 ( /u1D447 /u1D452 ) = /u1D435 /u1D708 ( /u1D447 /u1D452 )( 1 -/u1D452 -/u1D70F /u1D708 ) . (1)  \n/u1D435 /u1D708 ( /u1D447 /u1D452 ) is the Planck function, /u1D447 /u1D452 the electron temperature, and /u1D70F /u1D708 the frequency dependent optical depth given by  \n/u1D70F /u1D708 = 8 . 235 × 10 -2 ( /u1D447 /u1D452 K ) -1 . 35 ( /u1D708 GHz ) -2 . 1 ( EM pc cm -6 ) (2)  \n(Wilson et al. 2009). EM is the emission measure and is given by  \nEM = /uni222B.dsp /u1D45F outer /u1D45F inner ( /u1D45B /u1D452 ( /u1D45F ) cm -3 ) 2 /u1D451 ( /u1D45F pc ) . (3)  \nwith /u1D45B /u1D452 the electron density.  \nAn important factor in the present calculations is the geometric dilution of the free-free radiation field. The geometric dilution factor, /u1D44A H /i.pc/i.pc is defined as the ratio of the solid angle, /u1D714 , subtended by the H /i.pc/i.pc region at the location of the masing gas, relative to the total solid angle 4 /u1D70B . Assuming a spherical H /i.pc/i.pc region and expressing /u1D714 in terms of the radius, /u1D445 H /i.pc/i.pc , of the H /i.pc/i.pc region and the distance, /u1D45F , from  \nFigure 1. Schematic illustration of two possible positions of the masing gas relative to the H /i.pc/i.pc region with radius /u1D445 . For case A the centre of the masing region, with linear dimension of 1.5R, is at a distance 2R from the centre of the H /i.pc/i.pc region which corresponds to /u1D44A H /i.pc/i.pc = 0.07. At the right hand end of the masing volume /u1D44A H /i.pc/i.pc = 0.06. In case B the masing region is projected against the H /i.pc/i.pc region such that the left hand end is at distance 2R from the centre of the H /i.pc/i.pc region. The right hand end is at 3.5R from the centre of the H /i.pc/i.pc region where /u1D44A H /i.pc/i.pc = 0.02.  \n<!-- image -->  \nthe centre of the H /i.pc/i.pc region, the distance dependent dilution factor is given by  \n/u1D44A H /i.pc/i.pc ( /u1D45F ) = 1 2        1 -√ 1 -( /u1D445 H /i.pc/i.pc /u1D45F ) 2        . (4) For the case when /u1D45F /greatermuch /u1D445 H /i.pc/i.pc it follows that  \n/u1D44A H /i.pc/i.pc ( /u1D45F ) ∼ 1 4 ( /u1D445 H /i.pc/i.pc /u1D45F ) 2 . (5)  \nFrom Eq. 4 it follows that /u1D44A H /i.pc/i.pc = 0 . 5 when /u1D45F = /u1D445 H /i.pc/i.pc , and that /u1D44A H /i.pc/i.pc is therefore restricted to values ≤ 0 . 5. For all practical purposes one can say that Eq. 5 applies for /u1D45F > 5 /u1D445 H /i.pc/i.pc in which case /u1D44A H /i.pc/i.pc < 0 . 01.  \nWithin the framework of the escape probability method, and for an inverted transition, beaming is taken into account by modifying the escape probability  \n/u1D6FD /u1D708 = /u1D452 /u1D70F /u1D708 -1 /u1D70F /u1D708 (6)  \nto become  \n/u1D6FD /u1D708 ( /u1D70F ) = /uni03A9 /u1D708 ( /u1D70F ) 4 /u1D70B /u1D452 /u1D70F /u1D708 -1 /u1D70F /u1D708 (7)  \nwhere /uni03A9 /u1D708 is the frequency dependent beaming angle (see Section 5.3 of Elitzur 1992). According to Elitzur (1992, Section 5.1.3), the beaming solid angle varies as /uni2113 -1 for unsaturated masers and as /uni2113 -2 for saturated masers, where /uni2113 is the length along the ray path. Since /u1D70F ∝ /uni2113 , it was assumed that  \n/uni03A9 /u1D708 ( /u1D70F /u1D708 ) 4 /u1D70B = 1 ( /u1D6FC /u1D708 /u1D70F /u1D708 + 1 ) 2 (8)  \nas an approximation of such behaviour. Thus,  \n/u1D6FD /u1D708 ( /u1D70F ) = 1 ( /u1D6FC /u1D708 /u1D70F /u1D708 + 1 ) 2 /u1D452 /u1D70F /u1D708 -1 /u1D70F /u1D708 . (9)  \nIt should be noted, however, that /u1D452 /u1D70F increases faster than what /u1D70F -2 decreases for /u1D70F /greaterorsimilar 1 and, therefore, that /u1D6FD /u1D708 ( /u1D70F ) increases toward larger values of /u1D70F rather than to decrease. Inspection of the behaviour of /u1D6FD /u1D708 ( /u1D70F ) (as in Eq. 9) has further shown that /u1D6FD /u1D708 ( /u1D70F ) has a power law behaviour of the form /u1D44E /u1D70F -/u1D6FE for 0 . 4 < /u1D70F ≤ 1 for /u1D6FC ≥ 1. To avoid the escapeprobability to increase for /u1D70F > 1, it was assumed that the power law behaviour continues for /u1D70F > 1. The power law index, /u1D6FE , depends on /u1D6FC /u1D708 ; for example, /u1D6FE = -0 . 41 , -1 . 15, and -1 . 36 respectively for /u1D6FC = 1 , 5 and 10.  \nAlthough it is not the explicit intention of the calculations to model specific maser sources, it is necessary to use beaming angles that correspond to observed 4.8 GHz maser spot sizes. Consider therefore a maser spot at a distance /u1D451 from the observer and which has an angular size of /u1D703 radians. The surface area of the spot is then /u1D434 = /u1D70B /u1D451 2 /u1D703 2 / 4. On the other hand, for a maser path length /uni2113 /u1D45A and a beaming angle /uni03A9 given by Eq. 8, the cross sectional area of the cone is /u1D434 = 4 /u1D70B /uni2113 2 /u1D45A /( /u1D6FC /u1D70F + 1 ) 2 . Equating the two areas, it follows that  \n/u1D6FC /u1D70F = 4 /uni2113 /u1D45A /u1D703 /u1D451 -1 (10)  \nTo illustrate the application of Eq. 10 consider, for example, the case of G29.96-0.02. Hoffman et al. (2003) measured the angular size of the two maser components to be ∼ 19 milli-arcseconds at a distance of 6.5 kpc. For a maser path length of 7 × 10 16 cm (to be discussed later) we have /u1D6FC /u1D70F = 150. Choosing, for example, /u1D6FC = 15 implies that for /u1D70F = 10 the angular diameter of the spot will agree with the observed angular diameter.  \nThe present calculations are done within the framework of the escape probability method which requires that the physical conditions, e.g. density, temperature and the free-free radiation field, be uniform over the length of the masing column. To meet this requirement all parts of the masing volume of gas should be almost equidistant from the H /i.pc/i.pc region. The implied geometry is therefore a volume of gas with length /uni2113 /u1D45A illuminated from the side with the free-free emission from a hyper- or ultra-compact H /i.pc/i.pc region, schematically illustrated as case A in Fig. 1. The masing region can also be located as in case B. In this scenario there is a gradient in the geometric dilution of the free-free radiation field across the length of the masing region which means non-uniform pumping rates between the two end points of the masing region. Case B is certainly the more realistic scenario but cannot be dealt with using the escape probability method. Case A is, however, sufficient to investigate to what extent the free-free radiation field can lead to the inversion of the 1 10 -1 11 and 2 11 -2 12 transitions. Case A is qualitatively similar to the LVG method for a uniform source as was used by Cragg et al. (2002) to study the pumping of CH 3 OH masers.', '3 PUMPING SCHEME': "In brief, the pumping scheme proposed by van der Walt & Mfulwane (2022) is as follows: In the presence of a far-infrared radiation field radiative excitation out of 1 11 into the ladders of lower and upper doublet states is asymmetric. This follows from the fact that although the Einstein B-coefficients for the transitions 1 11 → 1 10 (4.8 GHz) and 1 11 → 2 12 (140.8 GHz) are almost equal, the spectral energy density at 140.8 GHz is significantly larger than at 4.8 GHz for a typical far-infrared radiation field of the form /u1D439 /u1D708 ( /u1D447 /u1D451 ) = ( 1 -exp (-( /u1D708 / /u1D708 0 ) /u1D45D )) /u1D435 /u1D708 ( /u1D447 /u1D451 ) with e.g. /u1D447 /u1D451 ∼ 50 to 100 K, /u1D45D = 1 . 8, and /u1D708 0 ∼ 10 12 Hz. Thus, radiative excitation out of 1 11 will therefore populate the ladder of lower double states faster than the ladder of upper doublet states. Using the collision coefficients  \nof the Leiden Atomic and Molecular Database (Schöier et al. 2005) it is rather straight forward to show, for collisions with both o -H 2 and p -H 2 , that there is a slight asymmetry which favours collisional excitation from 1 11 into the ladder of lower doublet states and that the asymmetry increases with increasing kinetic temperature. To create an inversion of the 1 10 -1 11 transition, it is necessary to transfer molecules from the ladder of lower doublet states to the ladder of upper doublet states. For the case where excitation is through a farinfrared radiation field and collisions, the transfer of molecules from the ladder of lower doublet states to the ladder of upper doublet states is collisional as was shown by van der Walt & Mfulwane (2022).  \nTo illustrate the pumping scheme in the presence of a free-free radiation field (but also for later examples) four emission measures were chosen such that the turnover frequencies of the free-free radiation fields correspond to the frequencies for the 3 12 -3 13 (28.9 GHz), 4 13 -4 14 (48.3 GHz), 5 14 -5 15 (72.4 GHz) and, 6 15 -6 16 (101.3 GHz) doublet transitions. The turnover frequencies therefore correspond to transitions where molecules can be transfered from the ladder of lower doublet states to the ladder of upper doublet states. The associated emission measures (with units of pc cm -6 ) are respectively 2 . 46 × 10 9 , 7 . 19 × 10 9 , 1 . 69 × 10 10 and 3 . 41 × 10 10 . The respective spectral energy density distributions are shown in Fig. 2 where, in the upper panel /u1D44A H /i.pc/i.pc = 0.5 for all four emission measures and in the lower panel for different dilution factors such that the spectral energy densities at 1000 GHz are the same as for EM = 2 . 46 × 10 9 pc cm -6 with /u1D44A H /i.pc/i.pc =0.5. This means that /u1D44A H /i.pc/i.pc =0.171 for EM=7 . 19 × 10 9 pc cm -6 , /u1D44A H /i.pc/i.pc =0.073for EM=1 . 69 × 10 10 pc cm -6 and, /u1D44A H /i.pc/i.pc =0.036for EM=3 . 41 × 10 10 pc cm -6 . The upper and lower panels of Fig. 2 therefore represent two special cases where, for the upper panel, the excitation rates (which are directly proportional to the spectral energy density, Eq. 11) for /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = 1 transitions within the ladder of lower doublet states (lines 1 to 5) range approximately over an order of magnitude from EM = 2 . 46 × 10 9 pc cm -6 to EM=3 . 41 × 10 10 pc cm -6 . On the other hand, the /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = -1 transition rates for the three lowest doublet states (lines 6, 7, 8) are approximately the same for the four emission measures. In the lower panel the situation is reversed, that is, the excitation rates within the ladder of lower doublet states (lines 1 to 5) are practically the same for the four emission measures while there are marked differences for the /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = -1 transitions for the transfer of molecules from the ladder of lower doublet states to the ladder of upper doublet states. Also shown in both panels is the spectral energy density distribution for a far-infrared radiation field for /u1D447 /u1D451 = 100 K and a dilution factor of 0.5.  \nFor radiative excitation out of 1 11 there are, as mentioned above, two possibilities, viz. 1 11 → 1 10 (4.8 GHz - dashed blue vertical line 6) and 1 11 → 2 12 (140.8 GHz - dashed red vertical line 1). Due to the significant difference in the spectral energy densities at these two frequencies, radiative excitation rate from 1 11 → 2 12 is higher than from 1 11 → 1 10 , resulting in a faster population of the ladder of lower doublet states compared to the ladder of upper doublet states. Further excitation within the ladder of lower doublet states, for example, 2 12 → 3 13 → 4 14 → 5 15 → 6 16 are facilitated through absorption of photons respectively at frequencies indicated by the vertical dashed red lines 2, 3, 4, 5. The transfer of molecules from the ladder of lower doublet states to the ladder of upper doublet states, for example 2 12 → 2 11 , 3 13 → 3 12 etc, is through the absorption of photons at frequencies indicated by the vertical dashed blue lines 7 to 11.  \nThe important point to be established is whether the transfer of molecules from the ladder of lower doublet states to the ladder of upper doublet states is essential for an inversion of the 1 10 -1 11  \ntransition. The radiative transition rate between a lower and an upper state is given by  \n/uni0393 /u1D459 /u1D462 = /u1D435 /u1D459 /u1D462 /u1D448 /u1D462/u1D459 = /u1D454 /u1D462 /u1D454 /u1D459 /u1D450 3 8 /u1D70B /uni210E/u1D708 3 /u1D462/u1D459 /u1D434 /u1D462/u1D459 /u1D448 /u1D462/u1D459 . (11)  \nwhere /u1D448 /u1D462/u1D459 is the spectral energy density of the free-free radiation field at the frequency /u1D708 /u1D462/u1D459 . /uni0393 /u1D459 /u1D462 can be changed by changing either /u1D434 /u1D462/u1D459 or /u1D448 /u1D462/u1D459 . To test whether the inversion of the 1 10 -1 11 transition depends on the rate at which molecules are transfered from the ladder of lower doublet states to the ladder of upper doublet states ( /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = -1), it is necessary to only change the rates of these transitions while keeping the rates for all other transitions the same. The simplest way to do this is by adjusting, with the exception of the 1 11 → 1 10 transition (through which molecules are radiatively excited into the ladder of upper doublet states), the Einstein A-coefficients for the /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = ± 1 transitions rather than modifying the radiation field.  \nFigure 3 shows an example of the variation of /u1D70F 4 . 8 with specific column density when (a) the 'standard' A-coefficients for the /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = ± 1 transitions are used (black line), (b) when all the coefficients are set to be of the order of 10 -15 s -1 (dashed blue line) (c) when the standard coefficients are reduced by a factor of 100 (solid blue line), (d) when the standard coefficients are reduced by a factor of 50 (dashed black line), (e) when standard coefficients are increased by one and two orders of magnitude above their standard values (red and green lines respectively). The A-coefficients for all the other allowed transitions were kept at their 'standard' values. Reducing the A-coefficients for the /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = ± 1 transitions to be of the order of 10 -15 s -1 , implying that there is virtually no exchange of molecules between the ladder of lower doublet states and the ladder of upper doublet states, has as consequence that there is no inversion of the 1 10 -1 11 transition for specific column densities less than ∼ 8 × 10 11 cm -3 s. The small inversion that follows can be shown to be due to collisional transfer of molecules between the ladders of lower and upper doublet states. The trends in the behaviour of /u1D70F 4 . 8 for the different cases shown in Fig. 3 clearly show that the inversion of the 1 10 -1 11 transition depends on the transfer of molecules from the ladder of lower doublet states to the ladder of upper doublet states. The pumping scheme proposed by van der Walt & Mfulwane (2022), as explained above, therefore also applies when excitation is due to a free-free radiation field, the difference being that in the presence of a free-free radiation field the pumping is dominantly radiative.  \nComparing the spectral energy density distributions in both panels of Fig. 2 clearly shows that for the far-infrared radiation field the spectral energy densities responsible for the excitation from 1 11 into the ladder of lower doublet states (dashed vertical red lines 1 to 5 ) are significantly smaller (in some cases more than an order of magnitude) than for the free-free radiation fields. It is also seen for the far-infrared radiation field, that the spectral energy densities at the frequencies responsible for the transfer of molecules from the ladder of lower doublet states to the ladder of upper doublet states (dashed vertical blue lines 6 to 11) are many orders of magnitude smaller than for the free-free radiation fields. The dust radiation field therefore plays a completely insignificant role in the pumping of the masers in regions where the free-free radiation field dominates.  \nFigure 2. Upper panel: Theoretical spectral energy density distributions for four H /i.pc/i.pc regions with emission measures and dilution factor as shown. The vertical dashed red lines numbered 1 to 5 correspond respectively to the first five /uni0394 /u1D43D = 1 , /u1D43E /u1D44E = 1 , /uni0394 /u1D43E /u1D450 = 1 transitions in the ladder of lower doublet states with frequencies 140.84, 211.21, 281.53, 351.77 and, 421.92 GHz. The vertical dashed blue lines numbered 6 to 11 correspond respectively to the first five /uni0394 /u1D43D = 0 , /u1D43E /u1D44E = 1 , /uni0394 /u1D43E /u1D450 = -1 transitions from the ladder of lower doublet states to the ladder of upper doublet states with frequencies 4.83, 14.49, 28.97, 48.28 and, 72.41 GHz. Bottom panel: Same as for the upper panel but with the spectral energy densities normalized at 10 3 GHz to that for EM = 2 . 46 × 10 9 pc cm -6 with /u1D44A H /i.pc/i.pc = 0.5. The resulting dilution factors associated with each emission measure are shown on the graph.  \n<!-- image -->", '4.1 Basic results without beaming': "The basic result of the calculational procedure to solve for the level populations is the variation of the level populations, and therefore also quantities such as the optical depth and the excitation temperature, as a function of the H 2 CO specific column density as described in van der Walt et al. (2021). As a first illustrative result, the variation of the optical depth for the 1 10 -1 11 ( /u1D70F 4 . 8 ) and 2 11 -2 12 ( /u1D70F 14 . 5 ) transitions in the presence and absence of a free-free radiation field are compared in the upper panel of Fig. 4. The results were obtained for /u1D45B H2 = 10 5 cm -3 , /u1D447 /u1D458 = 180 K, an H 2 CO abundance ( /u1D44B H2CO ) of 5 . 0 × 10 -6 , a maser linewidth of 0 . 3 km s -1 , EM = 3 × 10 10 pc cm -6 and /u1D44A H /i.pc/i.pc = 0.5. The lower panel shows the corresponding variation of the excitation temperatures for the two transitions where negative excitation temperatures indicate a population inversion. It is seen that in the absence of a free-free radiation field, the 1 10 -1 11 transition is not inverted up to a specific column density of ∼ 4 × 10 11 cm -3 s.  \nFigure 3. Comparison of the variation of the optical depth of the 110 -111 transition as function of specific column density where the Einstein Acoefficients for the doublet state transitions ( /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = ± 1) have been adjusted to illustrate the effect of changing the rate of transfer of molecules from the ladder of lower doublet to the ladder of upper doublet states.  \n<!-- image -->  \nThe 2 11 -2 12 transition is not inverted, which was more generally found to hold for all combinations of /u1D45B H2 and /u1D447 /u1D458 where the 1 10 -1 11 transition is inverted in the absence of a free-free radiation field. In the presence of a strong free-free radiation field (minimum geometric dilution) both transitions are seen to be inverted, even at those specific column densities where the 1 10 -1 11 transition is not inverted in the absence of a free-free radiation field.  \nSince, as was shown in Section 3, the inversion of the 1 10 -1 11 transition can be achieved through the pumping by a free-free radiation field, it can be expected that /u1D70F 4 . 8 and /u1D70F 14 . 5 depend on the emission measure and the dilution factor, /u1D44A H /i.pc/i.pc . The variation of /u1D70F 4 . 8 and /u1D70F 14 . 5 as a function of the specific column density for the two scenarios shown in the upper and lower panels of Fig. 2 is shown respectively in Figs. 5 and 6 with /u1D45B H2 = 10 4 . 2 cm -3 , /u1D447 /u1D458 = 100 K and, /u1D447 /u1D452 = 7500 K. It is worth keeping in mind that, from Eq. 4, the dilution factors of 0.5, 0.171, 0.073 and 0.036 correspond respectively to distances of 1.0 /u1D445 H /i.pc/i.pc , 1.3 /u1D445 H /i.pc/i.pc , 1.9 /u1D445 H /i.pc/i.pc and, 2.7 /u1D445 H /i.pc/i.pc from the centre of the H /i.pc/i.pc region.  \nComparison of Figs. 5 and 6 shows some obvious differences. For example, in Fig. 5 it is seen that there is very little difference in /u1D70F 4 . 8 for the different emission measures for specific column densities up to ∼ 3 × 10 11 cm -3 s. This can partly be ascribed to the transition rates being equal (or approximately equal) for the first three /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = -1 transitions (lines 6, 7 and 8 in the upper panel of Fig. 2) although the excitation rates into and within the ladder of lower doublet states (lines 1 to 5 in the upper panel of Fig. 2) differ significantly. On the other hand, when the dilution factors for the four emission measures are different, as in the lower panel of Fig. 2, the variation of /u1D70F 4 . 8 with specific column density as shown in Fig. 6 is quite different from that in Fig. 5. In fact, it is seen in Fig. 6 that the maximum of /u1D70F 4 . 8 for EM = 3 . 41 × 10 10 pc cm -6 at a distance of 2 . 7 /u1D445 H /i.pc/i.pc is greater than for EM = 2 . 46 × 10 9 pc cm -6 with the masing region located at the edge of the H /i.pc/i.pc region. Also, closer inspection of the upper panel of Figs. 5 shows that /u1D70F 4 . 8 is greater for EM = 2 . 46 × 10 9 pc cm -6 than for EM = 3 . 41 × 10 10 pc cm -6 for specific column densities less than ∼ 3 × 10 11 cm -3 s. At first sight, this behaviour appears to be counter intuitive since, if the masing region is closer to the H /i.pc/i.pc region, pumping might be expected to be more effective to establish a larger inversion of the 1 10 -1 11 transition.  \n5  \nFigure 4. Upper panel: Comparison of the variation of the optical depth as function of specific column density for the 110 -111 and 211 -212 transitions in the absence of a free-free radiation field with the case when a free-free radiation field is present. Bottom panel: Comparison of the variation of the excitation temperature for the 110 -111 and 211 -212 transitions.  \n<!-- image -->  \nSome insight into this behaviour can be gained by considering the radiative excitation rates between states within the ladders of lower and upper doublet states ( /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = ± 1) as well as between the lower to the upper doublet states ( /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = ∓ 1). To illustrate this we show the variation of /u1D70F 4 . 8 on the specific column density for /u1D44A H /i.pc/i.pc = 0.5, 0.1, and 0.05 for a free-free radiation field with EM = 3 . 41 × 10 10 pc cm -6 in the upper panel of Fig. 7. The values of the other parameters are as shown in the Figure. For a maser pathlength of 7 × 10 16 cm (at the specific column density indicated by vertical dashed line in the upper panel; see Section 4.2) it is seen that | /u1D70F 4 . 8 | WHii = 0 . 5 < | /u1D70F 4 . 8 | WHii = 0 . 1 < | /u1D70F 4 . 8 | WHii = 0 . 05 . Calculation of the radiative excitation rates for the /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = 1 transitions when /u1D44A H /i.pc/i.pc = 0.5, gives rates between ∼ 0 . 019 s -1 and ∼ 0 . 023 s -1 . This translates to characteristic 'lifetimes' for a H 2 CO molecule in the lower states of these transitions to be between 42 and 51 seconds. The corresponding transition rates for a molecule in the ladder of lower doublet states to make a transition to the associated upper doublet state ( /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = -1) fall between 6 × 10 -5 s -1 and 8 × 10 -4 s -1 , which translates to characteristic lifetimes between 1250 and 16700 seconds. The effect of the much higher transition rates (shorter lifetimes) for /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = 1 transitions compared to the /uni0394 /u1D43D = 0 , /uni0394 /u1D43E /u1D450 = -1 transitions is, therefore, that molecules excited into the ladder of lower doublet states will tend to stay in the ladder of lower doublet states with only a small probability of making a transition to the ladder of upper doublet states. Even if there is any population inversion for the 1 10 -1 11 transition, the high radiative excitation rate for 1 10 → 2 11 will further tend to reduce the magnitude of the inversion. On the other hand, for /u1D44A H /i.pc/i.pc = 0.05, the transition rates are an order of magnitude smaller with the characterisc lifetimes  \n6  \nD.J. van der Walt  \nFigure 5. Variation of the optical depths for the 110 -111 (upper panel) and 211 -212 (lower panel) transitions as a function specific column density for the four emission measures and a dilution factor of 0.5 as shown in the upper panel of Fig. 2.  \n<!-- image -->  \nfor the /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = 1 transitions, ranging between 430 and 513 seconds which increases the probability for a molecule in the ladder of lower doublet states to be transfered to the ladder of upper doublet states. The resulting level populations for the two dilution factors for a maser pathlength of 7 × 10 16 cm are compared in the bottom panel of Fig. 7. The effect of the higher radiative transition rates for /u1D44A H /i.pc/i.pc = 0.5 is evidenced in the larger level populations for the higher energy states compared to the case of /u1D44A H /i.pc/i.pc = 0.05.", '4.2 Beaming': "Inspection of Figs. 4, 5 and 6 shows that the maximum negative optical depth for the 1 10 -1 11 transition is less than -3, which is too small to account for observed maser brightness temperatures of the order of 10 6 to 10 8 K, even if the masing region is projected against a background source with a brightness temperature of 10 4 K. Fig. 8 compares the variation of /u1D70F 4 . 8 (upper panel) and /u1D70F 14 . 5 (lower panel) with specific column density when beaming is implemented, according to Eq. 9, for /u1D6FC =0(nobeaming), 1, 5, and 10. The first point to be noted is that, for /u1D6FC = 5 and /u1D6FC = 10, /u1D70F 4 . 8 can be as large as ∼ -23 and /u1D70F 14 . 5 ∼ -9. However, these values occur at specific column densities which correspond to maser pathlengths ∼ 8 × 10 17 cm, which is unrealistically large. Following Boland & de Jong (1981) and Hoffman et al. (2003), the maser pathlength is assumed to be geometrically constrained by the radius of the H /i.pc/i.pc region. Using the data of Meng et al. (2022), an average radius, 〈 /u1D445 H /i.pc/i.pc 〉 , of 4 . 8 × 10 16 cm and with a standard deviation, /u1D70E , of 3 . 0 × 10 16 cm, is found for 39 ultra-compact H /i.pc/i.pc regions in Sgr B2. For the present calculations a maser pathlength of 7 × 10 16 cm was used which is slightly smaller  \nFigure 6. Variation of the optical depths for the 110 -111 (upper panel) and 211 -212 (lower panel) transitions as a function specific column density for the four emission measures and dilution factors as shown in the lower panel of Fig. 2.  \n<!-- image -->  \nthan 〈 /u1D445 H /i.pc/i.pc 〉 + /u1D70E . In Fig. 8 such a maser pathlength corresponds to a specific column density of 1 . 85 × 10 11 cm -3 s and is indicated by the black dashed vertical line. At this specific column density /u1D70F 4 . 8 has the values -1 . 86 , -3 . 81 , -7 . 56 , -9 . 10 respectively for /u1D6FC = 0 , 1 , 5 , 10. The corresponding values for /u1D70F 14 . 5 are -0 . 77 , -1 . 79 , -3 . 85 , -4 . 69. These numbers clearly illustrate the effect of beaming.  \nA striking feature in the behaviour of the optical depths in Fig. 8 is the small but rapid changes in the optical depth with increasing specific column density for /u1D6FC = 5 and /u1D6FC = 10. For example, it is seen that, for /u1D6FC = 10, both the 1 10 -1 11 and 2 11 -2 12 transitions show a small decrease (becoming less negative) in optical depth at a specific column density of 10 12 cm -3 s, after which it further increases (becoming more negative) again. While this behaviour might at first appears odd when compared with the smooth variation of the optical depth as shown in Figs. 5 and 6, in fact, it reflects the coupling between the doublet states of o-H 2 CO.Thisbehaviour is further illustrated in Fig. 9 where the variation of the optical depth as a function of specific column density is shown for the first five doublet transitions as indicated in the graph. All five transitions are inverted at very small specific column densities. The 5 14 -5 15 transition (dashed black line) is the first to switch from inversion to non-inversion, followed by 4 13 -4 14 (green) and 3 12 -3 13 (red) at larger specific column densities. It is seen that when a higher /u1D43D doublet switches from being inverted to non-inverted it has the effect to create a small 'bump' in the optical depths of the inverted lower /u1D43D doublets. To understand the origin of this effect, we show in Fig. 10 the behaviour of n lower / nupper for the 1 10 -1 11 , 2 11 -2 12 and 3 12 -3 13 transitions for specific column densities between 3 . 7 -3 . 9 × 10 12 cm -3 s. A transition is inverted when n lower / nupper < g lower / gupper, while keeping in mind that gupper = g lower for the doublet transitions. The  \nFigure 7. Upper panel: Variation of the optical depth of the 110 -111 transition to illustrate the behaviour of /u1D70F 4 . 8 for different dilution factors. The vertical dashed line is located at a specific column density of 1 . 8 × 10 11 cm 3 s -1 which corresponds to a maser pathlength of 7 × 10 16 cm for nH 2 = 10 4 . 2 cm -3 , a H2CO abundance of 5 × 10 -6 and, a velocity width of 300 ms -1 . Lower panel: Solution for the level populations at a maser pathlength of 7 × 10 16 cm for the dilution factors as shown on the graph.  \n<!-- image -->  \n3 12 -3 13 transition switches from inversion to non-inversion at a specific column density of 3 . 807 × 10 12 cm -3 s, indicated by the vertical dashed black line. It is seen that there is a rapid increase in n 313 / n 312 after switching from inversion to non-inversion which is due to n 313 increasing faster than n 312 as the specific column density increases. Since the 4 13 -4 14 transition is not inverted anymore at this specific column density, it means that n 414 > n 413 . Furthermore, since the spontaneous decay rates between the lower levels of the doublet states are greater than between the corresponding upper levels, it follows that 3 13 is populated faster than 3 12 which explains the rapid increase in n 313 / n 312 after inversion. Similarly, the spontaneous decay rate for 3 13 → 2 12 is greater than that for 3 12 → 2 11 , which means that 2 12 is populated slightly faster than 2 11 . This gives rise to a slight decrease in the inversion of the 2 11 -2 12 transition and therefore to the decrease of /u1D70F 14 . 5 associated with the switching from inversion to non-inversion of the 3 12 -3 13 transition. The behaviour of /u1D70F 4 . 8 follows directly from similar reasons as for /u1D70F 14 . 5 . The important point of this behaviour is that it illustrates the coupling between the doublet states and how changes from inversion to non-inversion, and vice versa, of a higher /u1D43D doublet state affects doublet states at lower /u1D43D . The reason why this behaviour is present only for stronger beaming can easily be argued from the effect that beaming has on the escape probability for inverted transitions and therefore on the level populations which, in turn, determines the source function and therefore the internal radiation field.  \n7  \nFigure 8. Illustration of the effect of beaming on the optical depth of the 110 -111 and 211 -212 transitions. The parameters for the calculation are shown in the upper panel. The red dashed vertical line indicates the specific column density where the 413 -414 transition (48 GHz) switches from being inverted to non-inverted. The black dashed vertical line indicates the specific column density where the maser pathlength equals 7 × 10 16 cm.  \n<!-- image -->", '4.3 Variation of /u1D70F 4 . 8 and /u1D70F 14 . 5 on the n H2 -T k plane': 'Figures 11-14 show a small selection of examples of the variation of /u1D70F 4 . 8 and /u1D70F 14 . 5 on the /u1D45B H2 -/u1D447 /u1D458 plane that illustrate the typical variation of these two quantities. The maser pathlength used for these results was 7 × 10 16 cm. It is seen that, in the presence of a free-free radiation field, inversion of the 1 10 -1 11 and 2 11 -2 12 transitions can occur over almost the whole of the /u1D45B H2 -/u1D447 /u1D458 plane for 50 ≤ /u1D447 /u1D458 ≤ 300 K and 10 4 ≤ /u1D45B /u1D43B 2 ≤ 10 6 cm -3 . The regions in white on the /u1D45B H2 -/u1D447 /u1D458 plane should not be interpreted as implying that there is no inversion; in these regions inversion occurs at maser pathlengths less than 7 × 10 16 cm with /u1D70F 4 . 8 and /u1D70F 14 . 5 being less than 1. It is interesting to note that for EM = 2 . 46 × 10 9 pc cm -6 the contours for /u1D70F 4 . 8 > 5 (upper panel of Fig. 11) are almost independent of the gas kinetic temperature and that the largest values of /u1D70F 4 . 8 occur at lower H 2 densities. The pattern changes significantly when EM = 3 . 41 × 10 10 pc cm -6 (Figs. 12 and 13). The largest optical depths for /u1D70F 4 . 8 now are from -9 to -15 and are found at /u1D447 /u1D458 /lessorsimilar 100 K and /u1D45B H2 /lessorsimilar 6 × 10 4 cm -3 . As a further comparison with the upper panel of Fig. 11, the /u1D70F 4 . 8 = 5 contour, for example, has shifted to higher H 2 densities. An important aspect to note in the bottom panels of Figs. 12 and 13 is that /u1D70F 14 . 5 > -2 over most of the region where the 2 11 -2 12 transition is inverted and even have values > -6.  \nThe behaviour of /u1D70F 4 . 8 and /u1D70F 14 . 5 as shown in Figs. 11-13 is typical for other values of the emission measure, beaming parameter, dilution factor and X H2CO = 5 × 10 -6 . Recent estimates of the H 2 CO abundance relative to H 2 for 36 hot molecular cores (Taniguchi et al.  \n8  \nD.J. van der Walt  \nFigure 9. Illustration of the effect of beaming on variation of the optical depth of the first five doublet states. The parameters for the calculation are shown in the graph. From left to right the dashed vertical lines indicate the specific column densities where, respectively, the 413 -414, 312 -313, and 211 -212 transitions switch from being inverted to not inverted.  \n<!-- image -->  \nFigure 10. Illustration of the change in the ratio, n lower / nupper, of populations of the lower to upper levels for the 110 -111, 211 -212 and, 312 -313 transitions around the specific column density where the 312 -313 transition switches from inversion to non-inversion at the position indicated by the dashed vertical line.  \n<!-- image -->  \n2023), points to abundances ranging between log X H2CO = -8 . 68 and log X H2CO = -6 . 02 with log X H2CO = -7 . 23, which translates to a value of 5 . 8 × 10 -8 . In Fig. 14 we show the variation of /u1D70F 4 . 8 on the /u1D45B H2 -/u1D447 /u1D458 plane for H 2 CO abundances of 5 × 10 -7 and 5 × 10 -8 to illustrate the effect of having one and two orders of magnitude smaller abundances. Even with a beaming parameter of /u1D6FC = 20 and a dilution factor of 0.3, is it seen that /u1D70F 4 . 8 is significantly smaller than when /u1D44B H2CO = 5 × 10 -6 .', '5 COMMENTS ON CRITIQUE AGAINST FREE-FREE RADIATIVE PUMPING': "The results presented above are in principle the same as that of Boland & de Jong (1981), i.e. that the 1 10 -1 11 transition can be inverted by a free-free radiation field. Although the results presented above are a significant extension beyond the work of Boland & de Jong (1981), the same criticism raised against the model  \nFigure 11. Example of the variation of | /u1D70F 4 . 8 | (upper panel) and | /u1D70F 14 . 5 | (bottom panel) on the /u1D45B H 2 -/u1D447 /u1D458 plane for EM = 2 . 46 × 10 9 pc cm -6 and /u1D6FC = 20. The case for /u1D6FC = 10 does not differ substantially from what is shown.  \n<!-- image -->  \nof Boland & de Jong (1981), may therefore be raised against the results presented here. We, therefore, comment here on a number of points of critique which have been raised against the free-free radiative pumping of the H 2 CO masers.  \n(i) The emission measures of the associated H /i.pc/i.pc regions are too small: The validity of the model of Boland & de Jong (1981) has been questioned by e.g. Pratap et al. (1994), Mehringer et al. (1994) and Araya et al. (2005) on the basis of the emission measures of the associated H /i.pc/i.pc regions being too small. The fact that a significant number of H 2 COmasers are found in the vicinity of H /i.pc/i.pc regions, and that the results of the numerical calculations presented above suggest that these masers may be pumped radiatively by a free-free radiation field, raises the question of whether 'the problem' is with the pumping scheme or with estimating the emission measures of the H /i.pc/i.pc regions associated with 4.8 GHz H 2 CO masers. To illustrate that the observationally derived emission measure of an H /i.pc/i.pc region can differ  \nFigure 13. Same as for Fig. 12 but for /u1D6FC = 20.  \n<!-- image -->  \nFigure 12. Example of the variation of | /u1D70F 4 . 8 | (upper panel) and | /u1D70F 14 . 5 | (bottom panel) on the /u1D45B H 2 -/u1D447 /u1D458 plane for EM = 3 . 41 × 10 10 pc cm -6 and /u1D6FC = 10.  \n<!-- image -->  \nsignificantly from the real emission measure, the method followed by Meng et al. (2022) to determine the emission measures of H /i.pc/i.pc regions in Sgr B2, was applied to a synthetic H /i.pc/i.pc region calculated with the photo-ionization code Cloudy (Ferland et al. 2017). The following parameters were used in the Cloudy calculation: Blackbody temperature = 35900 K, total luminosity = 10 38 . 645 erg s -1 , inner radius = 10 12 . 845 cm, outer radius = 10 16 . 7 cm. The radial dependence of the H /i.pc density followed the power law  \n/u1D45B ( /u1D45F ) = /u1D45B /u1D45C ( /u1D45F /u1D45C ) ( 1 + /uni0394 /u1D45F /u1D445 /u1D460 ) /u1D6FC cm -3  \nwhere /uni0394 /u1D45F is the distance from the inner radius, and /u1D445 /u1D460 the scale depth. The H /i.pc density at the inner radius was 10 6 . 5 cm -3 , /u1D445 /u1D460 = 10 15 . 3 cm, and /u1D6FC = -0 . 7. The emission measure was calculated from the resulting radial electron density distribution and was found to be 1 . 2 × 10 10 pc cm -6 .  \nTo estimate the emission measures of the H /i.pc/i.pc regions in Sgr  \nB2, which contains roughly half of the known H 2 CO masers in the Galaxy, Meng et al. (2022) used the observed 6 GHz and 22.4 GHz flux densities. Following Meng et al. (2022), the observed flux density is related to the actual radius and emission measure of the H /i.pc/i.pc region according to  \n/u1D446 /u1D708 mJy = 8 . 183 × 10 -4 ( /u1D45F calc arcsec ) 2 ( /u1D708 GHz ) 2 ( /u1D447 /u1D452 K ) ( 1 -/u1D452 -/u1D70F /u1D708 ) (12)  \nwhere the optical depth /u1D70F /u1D708 is  \n/u1D70F /u1D708 = 8 . 235 × 10 -2 ( /u1D708 GHz ) -2 . 1 ( /u1D447 /u1D452 K ) -1 . 35 ( EM pc cm -6 ) (13)  \n(see equations 1 and 2 of Meng et al. 2022). With the observed flux densities at 6 GHz and 22.4 GHz, i.e. /u1D446 6 and /u1D446 22 . 4 , there are two equations with two unknowns which have to be solved for /u1D45F calc and EM.Thesolutionfor EMcanbeobtainedbytakingtheratio /u1D446 22 . 4 / /u1D446 6 ,  \nFigure 14. Sameas in the upper panel for Fig. 12 but for (a) XH2CO = 5 × 10 -7 in the upper panel and (b) XH2CO = 5 × 10 -8 in the lower panel.  \n<!-- image -->  \n6.0  \nwhich, after inserting the numerical values, gives  \n/u1D446 22 . 4 /u1D446 6 = 13 . 938 1 -exp (-/u1D458 1 EM ) 1 -exp (-/u1D458 2 EM ) (14)  \nwith /u1D458 1 = 4 . 7878 × 10 -10 and /u1D458 2 = 7 . 6128 × 10 -9 with units of pc -1 cm 6 .  \nIn the upper panel of Fig. 15, we show the variation of the optical depth at 6 GHz and 22.4 GHz, calculated from the outer boundary of the synthetic H /i.pc/i.pc region inwards as a function of the emission measure also calculated from the outer boundary inwards. For 6 GHz, the optical depth reaches a value of 1 at an emission measure of 1 . 53 × 10 8 pc cm -6 and for 22.4 GHz at an emission measure of 2 . 03 × 10 9 pc cm -6 . The fact that the emission at 6 GHz and 22.4 GHz originate from regions with different emission measures (both less than the true emission measure) is already an indication that an estimate of the true emission measure must result in an emission measure which is smaller than the true emission measure.  \nFigure 15. Upper panel: Optical depths at 6 GHz (black solid line) and 22.4 GHz (black dashed line) as a function of emission measure where both quantities are calculated inwards from the outer boudary. Lower panel: Variation of the ration /u1D446 22 . 4 / /u1D446 6 as a function the emission measure.  \n<!-- image -->  \nEstimating the emission measure using the method of Meng et al. (2022) is shown graphically in the lower panel of Fig. 15. The Sshaped solid line represents the ratio /u1D446 22 . 4 / /u1D446 6 as a function of the emission measure according to Eq. 14. From the emitted spectrum of the synthetic H /i.pc/i.pc region, it is found that /u1D446 22 . 4 / /u1D446 6 = 5 . 44, which corresponds to an emission measure ≈ 1 . 0 × 10 9 pc cm -6 as indicated on the figure. This is slightly more than an order of magnitude smaller than the true emission measure of the synthetic H /i.pc/i.pc region. The red dashed lines in Fig. 15 are for source number 5 in Tables B1 and B2 of Meng et al. (2022), which is used as verification. For this source /u1D446 22 . 4 / /u1D446 6 = 1 . 3 and the emission measure quoted by Meng et al. (2022) is 1 . 19 × 10 8 pc cm -6 . It is seen that the solution falls on the line representing Eq. 14.  \nThe implication of the above example for evaluating the current results, but also that of Boland & de Jong (1981), is that some of the H /i.pc/i.pc regions associated with H 2 CO masers may have emission measures significantly larger than what has been estimated. Rejection of the free-free pumping of the masers on the basis that the emission measures of the associated H /i.pc/i.pc regions are too small has, therefore, itself to be assessed more critically. Inspection of the emission measures derived by Meng et al. (2022) shows that there are five H /i.pc/i.pc regions with emission measures greater than 10 9 pc cm -6 . If, for all five of these cases, the emission measures were underestimated by one order of magnitude, it would mean that their sample contains five H /i.pc/i.pc regions with emission measures greater than 10 10 pc cm -6 which would significantly change the distribution of emission measures as shown in their Fig. 4. It is interesting to note that from 3-D modelling of the thermal dust and free-free emission in Sgr B2, Schmiedeke et al. (2016) found five H /i.pc/i.pc regions with emission  \nmeasures between 1 . 3 × 10 10 and 3 . 7 × 10 10 pc cm -6 . The reader is referred to Section 3.5 of Schmiedeke et al. (2016) where these authors also point out that the emission measures and electron densities are underestimated for optically thick H /i.pc/i.pc regions.  \n(ii) The masers are located too far from the nearest H /i.pc/i.pc region or has no associated continuum source: The model of Boland & de Jong (1981) requires that the masing region be located at the edge of an H /i.pc/i.pc region. Quite a number of detected H 2 CO masers are not projected against an H /i.pc/i.pc region but are offset from an H /i.pc/i.pc region e.g. in G29.96-0.02 (Pratap et al. 1994) and NGC 7538 (Andreev et al. 2017), or have no associated continuum source, for example, in Sgr B2 (Mehringer et al. 1994; Hoffman et al. 2007). Using the same values for /u1D45B H2 , /u1D447 /u1D458 and /u1D447 /u1D452 as in Fig. 8, the behaviour of the optical depth for the 1 10 -1 11 , 2 11 -2 12 , 3 12 -3 13 and, 4 13 -4 14 transitions as a function of radial distance from the centre of the H /i.pc/i.pc region was examined. This was done for /u1D6FC = 10, for EM = 3 . 41 × 10 10 pc cm -6 and EM = 2 . 46 × 10 9 pc cm -6 , and for a maser pathlength of 7 × 10 16 cm. The results are shown in Fig. 16. First, comparing the behaviour of /u1D70F 4 . 8 for the two emission measures, it is seen that for EM = 3 . 41 × 10 10 pc cm -6 , inversion occurs over the interval 1 ≤ /u1D45F / /u1D445 H /i.pc/i.pc /lessorsimilar 6 with /u1D70F 4 . 8 having its maximum value at /u1D45F / /u1D445 H /i.pc/i.pc /similarequal 1 . 3. For EM = 2 . 46 × 10 9 pc cm -6 , the maximum for /u1D70F 4 . 8 is at /u1D45F / /u1D445 H /i.pc/i.pc = 1 and inversion occurs only up to /u1D45F / /u1D445 H /i.pc/i.pc ∼ 3. Considering the 2 11 -2 12 , 3 12 -3 13 and 4 13 -4 14 transitions, it is seen, for EM = 3 . 41 × 10 10 pc cm -6 , that the maximum /u1D45F / /u1D445 H /i.pc/i.pc up to which there is inversion becomes progressively smaller from the 2 11 -2 12 to 3 12 -3 13 to 4 13 -4 14 transitions. For EM = 2 . 46 × 10 9 pc cm -6 it is only the 1 10 -1 11 and 2 11 -2 12 transitions that show significant inversion for /u1D45F / /u1D445 H /i.pc/i.pc < 3.  \nThe results presented in Fig. 16 show that inversion of the 1 10 -1 11 transition can occur for distances from the centre of the H /i.pc/i.pc region of up to six times the radius of the H /i.pc/i.pc region. That some H 2 COmasers are not projected onto or at the edge of an H /i.pc/i.pc region does not seem to be a reason to dismiss the pumping of the masers by the free-free continuum radiation field.  \nIt is also not necessary to invoke the possibility of some as yet unknown mechanism for the pumping of the H 2 COmasers in Sgr B2 which are not associated with a continuum source. Specific examples are those of masers B and E in Sgr B2, which appear to be 'located in regions devoid of continuum emission' (Mehringer et al. 1994) (see also Fig.1 of Hoffman et al. 2007). However, using the coordinates of the two masers as listed by Hoffman et al. (2007), it is found that maser B is located 0 '' . 72 from the methanol maser MMB G0.6720.031 (Caswell et al. 2010) and maser E is associated with the 6.7 GHz methanol maser MMB G0.645-00.042 with an angular separation of 0 '' . 36 (Caswell et al. 2010). For maser E the primary peak in the H 2 CO maser spectrum is at 49 km s -1 (Mehringer et al. 1994) and for the methanol maser it is at 49 . 5 kms -1 (Caswell et al. 2010). Both the class II methanol and 4.8 GHz H 2 COmasers are exclusively associated with high-mass star forming regions (Breen et al. 2013; Araya et al. 2007, 2015) which means that both these masers point to as yet undetected high mass star forming regions. Inspection of the 3mm maps of Sgr B2 (Ginsburg et al. 2018) shows that both masers B and E are associated with faint 3mm continuum sources.  \n(iii) The case of IRAS18566+0408 (G37.55+0.20)  \nAraya et al. (2007) considered the applicability of the Boland & de Jong (1981) model, i.e., whether population inversion by free-free emission, can explain the H 2 CO maser in IRAS 18566+0408 (G37.55+0.20). These authors observed this high mass star forming region with the VLA at 6, 3.6, 1.3, and 0.7 cm. The observed SED was fitted with a combination of optically thin thermal dust emission plus optically thin free-free emission from an ionized  \n11  \nFigure 16. Upper panel: Dependence of the optical depths of the first four doublet states as a function of /u1D45F / /u1D445 H /i.pc/i.pc for EM = 3 . 41 × 10 10 pc cm -6 . The three dashed horisontal lines indicate theoretical brightness temperatures assuming /u1D447 /u1D44F = 7500 exp | /u1D70F | . Lower panel: Same as for the upper panel but for EM = 2 . 69 × 10 9 pc cm -6 .  \n<!-- image -->  \njet (see Fig. 2 of Araya et al. 2007). Based on the observed SED, Araya et al. (2007) concluded that 'excitation by radio continuum is not a feasible mechanism with which to explain the maser in IRAS 18566+0.0408'.  \nIt is, however, necessary to keep in mind that, due to wavelength dependent optical depth effects, the SED of the radiation field at the location of the maser is not necessarily the same as the observed SED.Although there seems to be no detectable H /i.pc/i.pc region associated with IRAS 18566+0408, it does not mean that no highly compact H /i.pc/i.pc region is present. Keto & Klaassen (2008) concluded that 'an accreting star begins to produce an H /i.pc/i.pc region once the star reaches a mass, temperature, and luminosity equivalent to early B, about 15 20 /u1D440 /circledot '. If, as remarked by Araya et al. (2007), IRAS 18566+0408 is powered by a single O8 type star (30.8 M /circledot , Sternberg et al. 2003), it seems unlikely that no H /i.pc/i.pc region has been excited by the 30 M /circledot star. According to Keto (2007) an H /i.pc/i.pc region forms if the ionizing photon rate exceeds the hydrogen accretion rate. Consider then, for example, the simplified case of an O8 star spherically accreting matter at a rate of 10 -5 M /circledot yr -1 . This corresponds to a hydrogen accretion rate of 3 . 8 × 10 44 s -1 over the surface of the star. For an O8 star, the ionizing photon rate is 10 48 . 75 = 5 . 62 × 10 48 s -1 , which is a factor of 1 . 5 × 10 4 greater than the hydrogen accretion rate. A young hypercompact H /i.pc/i.pc region with, say, EM /greaterorsimilar 10 10 pc cm -6 , /u1D447 /u1D452 = 10 4 K, might thus exist but its mm/sub-mm wavelength emission may be undetectable due to the strong dust emission. The turnover frequency of the free-free emission for such an H /i.pc/i.pc region is ≈ 55 GHz, which implies it also is highly optically thick at centimetre wavelengths (see e.g. Pratap et al.  \n1992; Tanaka et al. 2016). However, the continuum radiation field of such an H /i.pc/i.pc region can still be the dominant radiation field within a couple of H /i.pc/i.pc region radii and be responsible for the pumping of the H 2 CO masers.  \nIt is also noted that using values of the electron density, electron temperature and physical dimension of the jet as given by Araya et al. (2007), the emission measure for the jet is found to be 2 . 6 × 10 6 pc cm -6 . This is about three orders of magnitude less than the case of EM = 2 . 46 × 10 9 pc cm -6 which we have considered above. The implication is that the radiative excitation rates from 1 11 into the ladder of lower doublet states as well as the transition rates within the ladder of lower doublet states is even less than that for the far-infrared dust radiation field (Fig. 2). It can therefore be concluded that the maser is not associated with the jet.  \n(iv) The inversion of the 1 10 -1 11 transition is due to a rare collisional excitation: Based on the work of Hill & Hollenbach (1978), Hoffman et al. (2003) proposed that the rarity of the H 2 CO masers mayberelated to a small number of H /i.pc/i.pc regions for which the dissociation wave and the ionization shock wave evolve separately to create a shock/dissociation transition region in which rare transient species are produced. Within this scenario, the inversion of the 1 10 -1 11 transition is the achieved through the collision of H 2 CO with such a rare species. Hoffman et al. (2003) did not state what this rare molecule or 'hot' atom might be, but if it is rare, its abundance in the transition region must be significantly less than that of H 2 and H /i.pc in the transition region. Furthermore, since the photodissociation of H 2 CO can be caused by photons with energies greater than about 3.4 eV ( /u1D706 < 360 nm) (Federman & Allen 1991), the possibility of the photodissociation of H 2 COmusttherefore also be taken into account.  \nEven if such a rare transient species is produced, it is reasonable to argue that for such a collisional process to produce an inversion of the 1 10 -1 11 transition, the collisional excitation rates between the levels of the lower doublet states ( /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = 1) should be comparable to or larger than the corresponding radiative rates to possibly be the dominant pumping mechanism. The collisional excitation rate (probability per unit time) is given by  \n/uni0393 /u1D450 /u1D459 /u1D462 = /u1D45B /u1D44D /u1D436 /u1D459 /u1D462 = /u1D45B /u1D44D /u1D454 /u1D462 /u1D454 /u1D459 /u1D436 /u1D462/u1D459 /u1D452 -/uni0394 /u1D438 /u1D462/u1D459 / /u1D458 /u1D447 /u1D458 (15)  \nwhere /u1D44D indicates the rare species, /u1D436 /u1D462/u1D459 is the collisional de-excitation coefficient and /u1D45B /u1D44D the number density of the rare species. Equating the right hand sides of Eqs. 11 and 15 it follows that the collisional de-excitation rate required to have the collisional and radiative rates the same is given by  \n/u1D436 /u1D462/u1D459 = 1 /u1D45B /u1D44D /u1D450 3 8 /u1D70B /uni210E/u1D708 3 /u1D462/u1D459 /u1D434 /u1D462/u1D459 /u1D448 /u1D462/u1D459 /u1D452 /uni0394 /u1D438 /u1D462/u1D459 / /u1D458 /u1D447 /u1D458 (16)  \nConsider, for example, the case where EM = 3 . 41 × 10 10 pc cm -6 with /u1D44A H /i.pc/i.pc =0.036, as in the upper panel of Fig. 6. Even if the 'rare species' has a number density as high as 10 3 cm -3 , it is found that /u1D436 /u1D462/u1D459 must have values ranging from 1 . 25 × 10 -6 cm 3 s -1 to 2 . 67 × 10 -6 cm 3 s -1 for the first eight /uni0394 /u1D43D = 1 , /uni0394 /u1D43E /u1D450 = 1 transitions in the lower ladder of doublet states. This is four orders of magnitude larger than the corresponding collisional de-excitation rates for collisions with H 2 which will indeed require a very special collision partner for H 2 CO.  \n(v) The non-detection of the 14.5 GHz maser As pointed out, the numerical results presented above suggest that under most physical conditions, the 14.5 GHz maser should accompany the detection of the 4.8 GHz maser. Authors such as, for example Hoffman et al. (2003), explicitly searched for 14.5 GHz maser emission toward NGC 7538 and G29.96-0.02. According to these authors their surveys of  \nFigure 17. Histograms to show the distribution of the ratio exp (-/u1D70F 4 . 8 )/ exp (-/u1D70F 14 . 5 ) . The upper panel is for the case shown in Fig. 12 and the lower panel for Fig. 13. The vertical dashed lines indicate the average values.  \n<!-- image -->  \nthese two star forming regions had a 3 /u1D70E brightness temperature sensitivity of 43 K and that a 14.5 GHz maser with /u1D70F = -1 would produce an emission line of brightness temperature greater than 100 K. On the basis of this non-detection, Hoffman et al. (2003) concluded that there is no inversion of the 2 11 -2 12 transition in these two sources. All the results presented above indicate that, in the presence of a free-free radiation field, the 2 11 -2 12 transition should be inverted, as was also found by Boland & de Jong (1981). It is difficult to see, for example, how to have /u1D70F 4 . 8 /greaterorsimilar -5 while there is no inversion of the 2 11 -2 12 transition, especially when beaming plays a role. Inversion of the 2 11 -2 12 and higher /u1D43D doublet transitions can be reduced by increasing the escape probability for these transitions. This can be done, for example, by also setting /uni03A9 /u1D708 ( /u1D70F /u1D708 ) ∝ ( /u1D708 / /u1D708 0 ) /u1D6FF , where /u1D708 0 = 4.8 GHz. However, due to the coupling of the doublet states as described above, the effect of doing this was found to also reduce the inversion of the 1 10 -1 11 transition.  \nIn spite of the fact that the 2 11 -2 12 transition is almost always inverted when the 1 10 -1 11 transition is inverted, is it worth noting that the brightness temperature of the 14.5 GHz maser can be considerably less than that of the 4.8 GHz maser. To illustrate this, we first show in Fig. 17 the distribution of exp (-/u1D70F 4 . 8 )/ exp (-/u1D70F 14 . 5 ) for the cases presented in Figs. 12 and 13 for exp (-/u1D70F 4 . 8 )/ exp (-/u1D70F 14 . 5 ) < 100. The average, standard deviation and median as shown in both panels were calculated only for exp (-/u1D70F 4 . 8 )/ exp (-/u1D70F 14 . 5 ) < 100, since the fewer very large values ( > 1000) considerably skew the average and standard deviation from the main distribution. For unsaturated gain, the distributions as shown in the upper and lower panels also reflect the ratio of the brightnesses of the two masers for equal background intensities or brightness temperatures. Comparison of the two  \ndistributions suggest that stronger beaming ( /u1D6FC = 20 (lower panel) compared to /u1D6FC = 10 (upper panel)) shifts the average 23 to 30, suggesting that stronger beaming acts in favour of the 4.8 GHz maser.  \nHowever, the non-detection of the 14.5 GHz maser cannot be explained even for exp (-/u1D70F 4 . 8 )/ exp (-/u1D70F 14 . 5 ) = 100. Apart from the fact that, within the framework of our calculations, exp (-/u1D70F 4 . 8 ) > exp (-/u1D70F 14 . 5 ) , projection of the masing region against an H /i.pc/i.pc region might be such that the background brightness temperature at 4.8 GHz ( /u1D447 4 . 8 ) is greater than at 14.5 GHz ( /u1D447 14 . 5 ). Quite generally the electron density is a decreasing function of the radial distance from the centre of the H /i.pc/i.pc region. The effective emission measure along a chord offset from the centre of the H /i.pc/i.pc region, decreases with increasing projected distance from the centre of the H /i.pc/i.pc region not only due to a decrease in the electron density but also due to the shorter chord length. The turnover frequency, /u1D708 0 , also shifts to lower frequencies since /u1D708 0 ∝ EM 0 . 476 . Now /u1D447 /u1D44F = /u1D447 /u1D452 for /u1D708 ≤ /u1D708 0 while, as a reasonable approximation, /u1D447 /u1D44F ( /u1D708 ) = /u1D447 /u1D452 ( /u1D708 0 / /u1D708 ) -2 when /u1D708 > /u1D708 0 . Thus, if /u1D708 0 /lessorsimilar 1 GHz,theratio between the background brightness temperatures at 4.8 and 14.5 GHz is such that /u1D447 4 . 8 / /u1D447 14 . 5 = 9 . 13. Considering therefore /u1D447 4 . 8 exp (-/u1D70F 4 . 8 )/ /u1D447 14 . 5 exp (-/u1D70F 14 . 5 ) = 9 . 13 exp (-/u1D70F 4 . 8 )/ exp (-/u1D70F 14 . 5 ) , it follows from Fig. 17 that the 4.8 GHz maser may be more than two orders of magnitude brighter than the 14.5 GHz maser.  \nTo illustrate this with a numerical example, consider the above results of the Cloudy calculations of a synthetic H /i.pc/i.pc region with EM = 1 . 2 × 10 10 pc cm -6 and assume the maser to be projected against the H /i.pc/i.pc region at a distance of 4 × 10 16 cm from the centre. From the Cloudy calculations it is found that the electron density at that distance from the centre of the H /i.pc/i.pc region is ∼ 5900 cm -3 and /u1D447 /u1D452 = 7165 K. For a spherical H /i.pc/i.pc region the corresponding chord length is 6 × 10 16 cm with an average electron density of 5515 cm -3 along the chord. Although the true emission measure of the H /i.pc/i.pc region is 1 . 2 × 10 10 pc cm -2 , the emission measure of the region against which the maser is projected is 5 . 9 × 10 5 pc cm -6 , assuming that /u1D45B /u1D452 = 5515 cm -3 and /u1D447 /u1D452 = 7165 K is constant along the length of the chord. The corresponding turnover frequency, /u1D708 0 , for the free-free emission from such a thermal hydrogen plasma is 565 MHz. Using /u1D447 /u1D44F ( /u1D708 0 ) = /u1D447 /u1D452 we have /u1D447 /u1D44F ( 4 . 8GHz ) ≈ 99 K and /u1D447 /u1D44F ( 14 . 5 GHz ) ≈ 10 K. With, for example, /u1D45B H2 = 10 4 . 5 cm -3 , /u1D447 /u1D458 = 100 K, EM = 1 . 2 × 10 10 pc cm -6 , beaming factor /u1D6FC = 10 and, /u1D44A H /i.pc/i.pc = 0.075, we find /u1D70F 4 . 8 = 4.9 and /u1D70F 14 . 5 = 1.3 for a maser length of 7 × 10 16 cm. This gives a 4.8 GHz maser brightness temperature of 13335 K and a 14.5 GHz maser brightness temperature of only 40 K. Although this example cannot be considered as presenting a full explanation of the (apparent) absence of the 14.5 GHz masers, it does illustrate that the brightness temperature of the 14.5 GHz maser can be significantly less than that of the 4.8 GHz maser and that the projection of the masing region against the background H /i.pc/i.pc may be a contributing factor.", '6 EVALUATION AND CONCLUSIONS': "As stated in the Introduction, the primary aim of this work is to first establish whether the pumping scheme proposed by van der Walt & Mfulwane (2022) is indeed also responsible for the inversion of the 1 10 -1 11 transition by a free-free radiation field. The results presented in Section 3 convincingly showed the transfer of molecules from the ladder of lower doublet states to the ladder of upper doublet states is essential to create an inversion of the 1 10 -1 11 transition. Since inversion of the 1 10 -1 11 transition can also be achieved through collisions alone through the same process (van der Walt & Mfulwane 2022), it follows that any other proposed  \npumping mechanism must operate in the same way, i.e., a faster excitation into the ladder of lower doublet states compared to the ladder of upper doublet states, as well transfering molecules from the ladder of lower doublet states to the ladder of upper doublet states at a rate that will result in an inversion of the 1 10 -1 11 transition.  \nThe results presented above clearly indicate that the typical freefree radiation field of an H /i.pc/i.pc region associated with young high mass stars is very effective to invert the 1 10 -1 11 and other doublet transitions over a wide range of H 2 densities and kinetic temperatures. Depending on the emission measure of the H /i.pc/i.pc region, inversion can be achieved at distances from the centre of the H /i.pc/i.pc region of up to six times the radius of the H /i.pc/i.pc region. Our results also showed that with beaming, sufficient amplification factors can be achieved. Considering these results, there is, therefore, no reason to question the viability of the pumping of the 4.8 GHz masers by a free-free radiation field. In fact, even though the 1 10 -1 11 transition can be inverted by collisions alone, the results clearly indicate that, closer to the H /i.pc/i.pc region, radiative pumping by the free-free radiation field, by far dominates the inversion. The implication is that any other postulated collisional process for the pumping of the masers, will have to compete with the radiative pumping by the free-free radiation field.  \nAs stated in the Introduction, to explain the rarity of the H 2 CO masers a distinction must be made between the pumping scheme and other possible factors that determine the number and life-time of the masers. The fact that not only the 1 10 -1 11 but also the 2 11 -2 12 transition can be inverted over most of the /u1D45B H2 -/u1D447 /u1D458 plane would suggest that there should be significantly more H 2 CO masers than what has been discovered up to now. There are at least two such external factors that can possibly contribute to the small number of detected H 2 CO masers. The first is the rate at which the H /i.pc/i.pc region, and therefore also its emission measure, evolves. As is shown in Fig. 2, excitation from the 1 11 state into the ladder of lower doublet states is through frequencies in the optically thin part of the spectral energy density distribution. Since for the optically thin part the spectral energy density is proportional to the emission measure, the excitation rates into the ladder of lower doublet states will decrease as the emission measure decreases due to the evolution of the H /i.pc/i.pc region. The evolution of the H /i.pc/i.pc region may therefore affect the life-time and thus the number of H 2 CO masers.  \nAnother factor that may play a role in the number of detected H 2 CO masers is the H 2 COabundance. It is clear from Fig. 14 that the H 2 CO abundance is relevant in determining whether the amplification factor is large enough to produce a detectable maser. Considering again the results of Taniguchi et al. (2023), it is found for the sample of 36 hot cores that logX H2CO = -7 . 23 and /u1D70E = 0 . 62. The skewness of the sample is ∼ -0 . 4 which indicates that it is slightly skew toward smaller abundances. Assuming that the distribution of log X H2CO is Gaussian (for which the skewness is 0), it is found that the probability is 2.4% to find a hot core for which logX H2CO > -6. If the H 2 CO and the class II CH 3 OH masers are associated with the same population of ∼ 1000 high mass star forming regions detected in the MMB survey (Green & the MMB Survey Collaboration 2012), it follows that there should be about 24 4.8 GHz H 2 CO masers if all masers for which X H2CO > 10 -6 are detectable. This example should not be interpreted as a prediction of the total number of 4.8 GHz H 2 CO masers but only to illustrate that external factors which are independent of the pumping scheme, as as well as of the pumping mechanism (the free-free radiation field), can lead to a relatively small number of detected maser sources. The number may be even lower if the duration of the masing phase is also affected by the evolution of the H /i.pc/i.pc region.  \nThere are a number of 4.8 GHz H 2 CO masers that also have associated 6.7 GHz CH 3 OHmasers for which there appears to be very good spatial and velocity agreement between the two maser species. Masers B and E in Sgr B2 already refered to above are such examples. A striking example is in the case of G339.98-1.26 where three of the 4.8 GHz H 2 CO features correspond in velocity to three of the 6.7 GHz CH 3 OH maser features (Chen et al. 2017b). Also in the case of G0.38+0.04 is there very good spatial and velocity agreement between the 4.8 GHz H 2 COand 6.7 GHz CH 3 OHmasers (Ginsburg et al. 2015). The question then is to what extent there is similarity in the conditions to excite the two maser species. We note the following differences which need to be resolved if these two maser species are to be associated with the same physical and kinematic structures. From Fig.7 of Cragg et al. (2002) it is seen that to explain the brightness temperatures of the order of 10 12 K of the 6.7 GHz CH 3 OH masers requires dilution factors, /u1D44A H /i.pc/i.pc , to be less than about 10 -4 . This implies /u1D45F /greaterorsimilar 50 /u1D445 H /i.pc/i.pc (Eq. 4) while /u1D45F /lessorsimilar 6 /u1D445 H /i.pc/i.pc seems to be required for the 4.8 GHz H 2 CO masers (Fig. 16). Cragg et al. (2002) also concluded that the brightest 6.7 GHz CH 3 OH masers are found for 10 5 < n H2 < 10 8 . 3 cm -3 with T k < T d > 100 K. For H 2 CO,theresults presented above suggest that to have a reasonable amplification factor, e.g. /u1D70F 4 . 8 < -5, requires n H2 < 10 5 . 5 cm -3 (Figs. 11 and 12). Although the calculations of Cragg et al. (2002) and those presented here used different numerical methods which may contribute to some of the differences, at face value it would seem, from a theoretical point of view, that the two maserspeciesarenot associated with the same physical and kinematic structure in the star formation environment.  \nWithin the context of this discussion, it is has to be pointed out that the 4.8 GHz H 2 CO masers which are normally associated with G29.96-0.02, are most likely not associated with the cometary H /i.pc/i.pc region but rather with a hot molecular core (HMC) hosting an object identified as a high mass protostellar object (Cesaroni et al. 1998; De Buizer et al. 2002). Spatially there are three water masers associated with this HMC (Hofner & Churchwell 1996), with one of the 4.8 GHz H 2 CO masers located only 0 '' . 18 from one of the water maser features. Although it is not clear what the exact location of the 4.8 GHz H 2 CO maser in the HMC is, the conclusions drawn of it being associated with, and located toward, the edge of the cometary H /i.pc/i.pc region might be different than if it is associated with a young background HMC. The first, obviously, is the different evolutionary states of the star forming region with which the maser is associated which also have consequences regarding the chemical evolution. Different conclusions can also be reached regarding the optical depth required to explain the observed brightness temperature of the maser. In the case of G29.96-0.02, Hoffman et al. (2003) concluded that to explain the observed brightness temperature of ∼ 10 7 Krequires /u1D70F 4 . 8 ≈ -12 based on a 6 cm background temperature of 50 K. From the upper panels of Figs. 12 and 13 it is seen that, within the framework of the current calculations, such optical depths occur only at lower gas kinetic temperatures. However, if the maser is projected against a hypercompact H /i.pc/i.pc region associated with an earlier phase of high mass star formation, the background temperature may be significantly higher. For example, if the background temperature is 5000 K, the optical depth required to explain a brightness temperature of the order of 10 7 K, is ∼ -7 . 6. It is obvious from Figs. 12 and 13 that in this case a significantly different conclusion is reached about the conditions under which the maser might be excited.  \nAs a last remark we note that, having established that a free-free radiation field is effective to pump the 4.8 GHz H 2 CO masers and a far-infrared dust radiation field is not, it follows that the underlying mechanism for the periodic variability of the H 2 CO and CH 3 OH  \nmasers in IRAS18566+0408 (G37.55+0.20) cannot be due to varying pumping conditions for both maser species. Since the pumping schemes for the two masers are completely different, it would require an impossible degree of fine tuning of variability between the freefree and dust radiation fields for the amplification factors to behave in exactly the same way to produce similar light curves. The flaring must therefore be associated with the source of seed photons, as was argued by van der Walt (2014) and as is strongly suggested by the recent results of Gray et al. (2020, see their Section 6).", 'ACKNOWLEDGEMENTS': 'I would like to thank Prof. David Modise, Dean of the Faculty of Natural and Agricultural Sciences, as well as Prof. Francois van der Westhuizen, Deputy Dean (Research) of the Faculty of Natural and Agricultural Sciences of the NWU, for financial support without which this work could not have been done. I also thank an anonymous reviewer for constructive comments to improve the paper.', 'DATA AVAILABILITY': 'No new data has been generated in this project.', 'REFERENCES': 'This paper has been typeset from a T E X/L A T E X file prepared by the author.'}