The data presented in the article are necessary, first, for astrophysical applications, in particular, when modeling the spectra of low-temperature stars, where it is necessary to know the shock parameters of the H2O lines contour by the helium and hydrogen pressures in a wide temperature range. Hydrogen and helium are the most common substances in the universe. They make up most of the atmospheres of cold stars, giant planets, and exoplanets (planets outside the Solar system). The broadening of the water vapor vibrational-rotational lines by the pressure of hydrogen and helium at high temperatures has been poorly studied. In this paper, the broadening of the water molecule vibrational-rotational lines by helium pressure at high temperatures up to 2000 K.

The data available in the literature are clearly insufficient: single measurements of the H2O–He broadening coefficients are presented in [1–16] and the most extensive experimental values are given in [17–23]. The calculated data are presented in [23] together with the measured coefficients of the broadening of water vapor lines by helium pressure for an extensive list of lines on 11 absorption bands. The coefficients of temperature dependence in the literature are obtained only for the case of broadening of two lines (31 3 ← 22 0 and 41 4 ← 32 1) at temperatures of 80–600 K [1–3] by helium pressure, as well as by hydrogen pressure. In [12], the low-temperature measurements were carried out for the 11 0 ← 10 1 line in the range of 17–200 K. A very weak temperature dependence was found for this line.

To calculate the lines contour shock parameters, several methods are currently used, among which are various modifications of the Robert–Bonami method [24], the modified Robert–Bonami complex formalism [25], and the semi-empirical method [26]. Recently, a method of medium frequencies has been proposed [27], which makes it possible to quickly calculate the coefficients of broadening of vibrational-rotational lines of the asymmetric top molecules on the basis of a small amount of empirical data. The idea of the method is that a part of the values that reflect the influence of the buffer molecule on the internal state of the absorbing molecule is isolated from the entire calculation scheme. These values strongly depend on the absorbing molecule quantum indices. Of these, the value that is called the medium frequency of collision transitions is compiled. By virtual transitions, we mean all possible transitions from a certain energy level caused by collisions. The medium frequency of virtual transitions for the line i → f is determined by the following expression:

$$\begin{gathered} {{{\bar {\omega }}}_{{if}}} = \frac{1}{W}\left\{ {\sum\limits_{i'} {(2i{\kern 1pt} '\; + 1){{D}^{2}}\left( {ii{\kern 1pt} '\left| l \right.} \right)f({{\omega }_{{ii'}}}){{\omega }_{{ii'}}}} } \right. \\ \left. { + \;\sum\limits_{f'} {(2f{\kern 1pt} '\; + 1){{D}^{2}}\left( {ff{\kern 1pt} '\left| l \right.} \right)f({{\omega }_{{ff'}}}){{\omega }_{{ff'}}}} } \right\}, \\ \end{gathered} $$
(1)

where W = \(\sum\nolimits_{i'} {(2i{\kern 1pt} '\; + 1){{D}^{2}}(ii{\kern 1pt} '{\kern 1pt} \left| l \right.)f({{\omega }_{{ii'}}})} \) + \(\sum\nolimits_{f{\kern 1pt} '} {(2f{\kern 1pt} '} \) + \(1){{D}^{2}}(ff{\kern 1pt} '{\kern 1pt} \left| l \right.)f({{\omega }_{{ff{\kern 1pt} '}}})\).

Here, \({{\omega }_{{ii'}}}\) and \({{\omega }_{{ff'}}}\) are the collision (virtual) transitions frequencies, \({{D}^{2}}\left( {ii{\kern 1pt} '\left| l \right.} \right)\) and \({{D}^{2}}\left( {ff{\kern 1pt} '\left| l \right.} \right)\) are the matrix elements of the dipole (l = 1) and quadrupole (l = 2) moments of virtual transitions, and \(f({{\omega }_{{ii'}}})\) and \(f({{\omega }_{{ff'}}})\) are resonance functions. It can be seen that the averaging takes place over all virtual transitions, as well as over the lower and upper energy states forming this transition i → f.

Knowing the correspondence between the values \({{\bar {\omega }}_{{if}}}\) (calculated in advance for all possible transitions up to J = 20) and the lines quantum identification, we compare the collision coefficients of the broadening of the lines with the \({{\bar {\omega }}_{{if}}}\) values. According to the latter dependence, we obtain an approximation in the form of some simple expression. Usually this is a rectilinear or quadratic dependence, according to which it is possible to restore the lines half-widths.

When applying the medium frequencies method, first, it is necessary to identify the most verified literary data. Since the experimental half-widths of water vapor lines induced by helium pressure in [20–22] are in good agreement with each other, we used them to implement the medium frequencies method. Figure 1 shows the dependence of the broadening coefficients on the medium frequency of collision transitions for the case of H2O–He, as well as the corresponding approximation depending on the medium frequency of collision transitions. Here, the most extensive measurements in [20–22] are shown in black symbols and all single data in [3–6, 9, 10, 12]—by gray squares.

Fig. 1.
figure 1

Experimental half-widths of the H2O–He lines depending on the medium frequency of the collision transitions. The squares represent data in [3–6, 9, 10, 12], triangles—in [20], circles—in [21], stars—in [22], line—approximation.

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The following approximation by three straight line segments is obtained:

$${{\gamma }_{{if}}} = - 0.0000615{{\bar {\omega }}_{{if}}} + 0.02830$$
$${\text{at}}\quad 0 < {{\bar {\omega }}_{{if}}} \leqslant 105,$$
$${{\gamma }_{{if}}} = - 0.0001302{{\bar {\omega }}_{{if}}} + 0.03552$$
(2)
$${\text{at}}\quad 105 < {{\bar {\omega }}_{{if}}} \leqslant 190,$$
$${{\gamma }_{{if}}} = - 0.0000364{{\bar {\omega }}_{{if}}} + 0.01769\quad {\text{at}}\quad {{\bar {\omega }}_{{if}}} > 190.$$

Then the line broadening coefficients were calculated using Eq. (2) for the values of the rotational quantum number J ≤ 20. A part of the data obtained in this way is presented in [28]. There are no experimental data for larger values of 20 < J ≤ 50, therefore, an approximation of the values γ on J was used to obtain them. For this purpose, the most verified data for the lower vibration bands were collected [3–5, 9, 10, 12, 20–22] and the H2O–He broadening coefficients were plotted depending on the rotational quantum number J and then averaged over the quantum numbers Ka and Kc. The measured coefficients of broadening of the lines of the rotational band, the ν2 band, and 2ν1 band depending on J are shown in Fig. 2. Also, the averaged values of the lines half-widths for all quantum n-umbers Ka and the approximation are plotted in the figure.

Fig. 2.
figure 2

Rotational dependence of the half-widths of the lines for the case of H2O–He. Black squares, triangles, inverted triangles, stars indicate data in [3], [4], [5], [9], respectively; gray squares, triangles, inverted triangles, stars indicate data in [10], [12], [20], [21], respectively; the black circles indicates data in [22], the line with hollow squares indicates the average values for a certain rotational quantum number J, and the line indicates an approximation.

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The rotational dependence of the lines half-widths for the case of helium broadening in comparison with the case of hydrogen broadening [29] constructed according to the literature data is shown in Fig. 3. It can be seen that the half-widths of the H2O–H2 lines are, on average, 2.5 times larger than the half-widths of the H2O–He lines, which is due to the large quadrupole moment of the H2 molecule compared to that of the He atom.

Fig. 3.
figure 3

The broadening coefficients of the water vapor lines by hydrogen and helium pressure depending on the rotational quantum number J. Gray triangles and squares denote sets of measured values and a line with hollow triangles and squares—the average values for a certain rotational quantum number J for the H2O–H2 and H2O–He mixtures, respectively.

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The lines half-widths for the H2O–H2 system both in the purely rotational band and in other vibrational absorption bands are calculated using the medium frequencies obtained for the rotational band and are in good agreement with experimental data because of the weak vibrational dependence of the lines half-widths. However, as shown in [23], the measured lines half-widths for the H2O–He gas mixtures cannot change by more than 10% when switching from one absorption band to another. Therefore, for such a buffer gas as helium, the medium frequencies for excited states were changed to obtain more accurate the lines half-widths. Figure 4 shows a comparison of the half-widths of the lines calculated by us without considering the oscillatory dependence with the experimental and calculated data obtained in [1]. The maximum mismatch in the data is observed for small frequencies of collision transitions.

Fig. 4.
figure 4

The broadening coefficients of the H2O lines by He pressure calculated by the medium frequency method without considering Eq. (4) (triangle) and using Eq. (4) (square), in comparison with (line with stars) experimental and (circles) theoretical data [23]. The line number n corresponds to an increase in the transition frequency.

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In [27], to clarify the vibrational dependence of the half-widths of the lines for the H2O–N2 system, a formula for calculating the medium frequencies of the collision transitions in the vibrational band \(\bar {\omega }({{\nu }_{1}}{{\nu }_{2}}{{\nu }_{3}})\) as the sum of the medium frequency for the rotational band and the medium frequency differences for the first excited state ν1, ν2, and ν3 multiplied by the number of vibrational quanta corresponding to a certain vibration was proposed:

$$\begin{gathered} \bar {\omega }({{\nu }_{1}}{{\nu }_{2}}{{\nu }_{3}}) = \bar {\omega }(000) + {{\nu }_{1}}\left[ {\bar {\omega }(100) - \bar {\omega }(000)} \right] \\ + \;{{\nu }_{2}}\left[ {\bar {\omega }(010) - \bar {\omega }(000)} \right] + {{\nu }_{3}}\left[ {\bar {\omega }(001) - \bar {\omega }(000)} \right]. \\ \end{gathered} $$
(3)

The equation adequately describes the behavior of line broadening with a weak dependence on vibrational quantum numbers (e.g., in the case of H2O–N2). However, in the case of broadening with atomic helium, when a strong vibrational dependence of the half-widths is manifested, its use is unacceptable. Therefore, this equation was modified by introducing multipliers before the differences that were obtained from experimental the half-widths of the lines in the bands: 2ν1, 2ν2, and 2ν2 + ν3 [23] (one transition was used in each band):

$$\begin{gathered} \bar {\omega }({{\nu }_{1}}{{\nu }_{2}}{{\nu }_{3}}) = \bar {\omega }(000) + {{a}_{1}}{{\nu }_{1}}\left[ {\bar {\omega }(100) - \bar {\omega }(000)} \right] \\ + \;{{a}_{2}}{{\nu }_{2}}\left[ {\bar {\omega }(010) - \bar {\omega }(000)} \right] + {{a}_{3}}{{\nu }_{3}}\left[ {\bar {\omega }(001) - \bar {\omega }(000)} \right], \\ \end{gathered} $$

where

$${{a}_{1}} = 0.14954\bar {\omega } - 22.431,$$
$${{a}_{2}} = - 0.06477\bar {\omega } + 9.715,$$
(4)
$${{a}_{3}} = 0.08684\bar {\omega } - 14.527.$$

Figure 5 shows the dependence of the experimental the lines half-width [3–6, 9, 10, 12, 20–23] on the medium frequency of collision transitions obtained by Eq. (4). The measured values in [23] are sketched separately for each vibrational band. After recalculating the parameters \(\bar {\omega }\) according to Eq. (4), the approximation coefficients in Eq. (2) obtained for low vibrational bands will be the same. The standard deviations of the calculated half-widths of the lines without using the modification in Eq. (4) and using modification in Eq. (4) from the experimental data in [23] will be 0.0029 cm–1 atm-1 and 0.0017 cm–1 atm–1, respectively.

Fig. 5.
figure 5

The broadening coefficients of the H2O lines by He pressure depending on the frequencies of collision transitions. Black squares, circles, triangles, stars denote the measured values in [23] in the bands 2ν2, ν1, ν1 + ν2, and ν2 + ν3, respectively; gray squares, circles, triangles, stars denote the measured values in [23] in the bands 2ν2 + ν3, 2ν1, ν1 + ν3, and in the rotational band, respectively; hollow triangles—in [3–6, 9, 10, 12, 20–22]; and the line is an approximation for the medium frequency method.

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For the applications in astrophysics, in particular, when modeling the spectra of low-temperature stars, it is necessary to know the lines contour shock parameters in a wide temperature range. The temperature dependence coefficients N for the broadening of two water vapor lines (31 3 ← 22 0 and 41 4 ← 32 1) by helium pressure determined in the temperature range of 80‒600 K are given in [1–3]. In this paper, the temperature index for the case of H2O–He was defined as the ratio to the values of N for the case of H2O–H2:

$${{N}^{{{{{\text{H}}}_{{\text{2}}}}{\text{O}} - {\text{He}}}}} = 0.576{{N}^{{{{{\text{H}}}_{{\text{2}}}}{\text{O}} - {{{\text{H}}}_{{\text{2}}}}}}},$$
(5)

where \({{N}^{{{{{\text{H}}}_{{\text{2}}}}{\text{O}} - {{{\text{H}}}_{{\text{2}}}}}}}\) is obtained by the medium frequency method in [29]. The multiplier in Eq. (5) was obtained from the measured data [3].

The calculations of the line broadening coefficients and their temperature parameters for the H2O–He mixture are carried out in the work. The quantum number of the total angular momentum varies in the range from 0 to 50. It is shown that when studying the broadening of water vapor lines by helium pressure in a wide spectral range, it is necessary to consider the dependence on vibrational excitation. The approximation expression for the vibrational dependence that allows us to adequately describe the available experimental data is obtained. Thus, the medium frequency method developed by the authors on the basis of the assessment of the contributions of collision transitions to the half-width of the spectral line makes it possible to calculate the half-width of the lines of the asymmetric top molecules with an accuracy close to the one of modern computational and experimental methods, without resorting to a complex computational scheme. The half-widths of the lines of the H2O–He molecules calculated using the proposed method are in good agreement with various experimental and calculated data.