© ESO, 2012

1. Introduction

The molecular hydrogen is the most common molecule in the interstellar medium. The H₂ molecule in the ground electronic state does not have a permanent dipole moment, hence no dipole-allowed vibrational or rotational transitions. Consequently, the observations of H₂ emission lines from cold molecular clouds are difficult. The H₂ molecule can, however, be observed in the lowest rotational levels (up to J = 5–8) of the ground electronic and vibrational level (ν = 0) by its ultraviolet absorption spectrum. The H₂ molecules are formed on the surface of dust grains. However, the distribution of newly formed H₂ molecules on energy levels is not known exactly. The absorption lines of molecular hydrogen in the Lyman and Werner bands are in the UV and far-UV ranges, observed, for example, by the FUSE (Far Ultraviolet Spectroscopic Explorer) and HST (Hubble Space Telescope) satellites. The fluorescence that follows the absorption leads then to populating the ro-vibrational levels of the ground electronic level. The later decay by quadrupole transitions is seen as emission in the infrared spectral range. A detailed description of these processes was presented by Black & Dalgarno (1976). The H₂ IR and UV emission lines have been detected in various astronomical objects, such as planetary nebula, SNR, clusters of galaxies, AGN, starburst regions, T Tauri stars, circumstellar disks, and planetary atmospheres.

Observations of molecular hydrogen in vibrationally excited levels (ν ≥ 1) can be used to distinguish shock-heated gas from gas heated by the far-ultraviolet radiation field. First detection of interstellar H₂ absorption lines from vibrationally excited levels (H₂*) was performed by Federman et al. (1995) in an HST spectrum of ζ Oph. A rich spectrum of vibrationally excited hydrogen molecule was described by Meyer et al. (2001) in the HST/STIS (Space Telescope Imaging Spectrograph) spectrum of HD 37903. Based on the nonthermal population of H₂ energy states and the constancy of H₂ line velocities and line widths they concluded that the H₂ excitation arises through UV fluorescence and not through shock heating. Meyer et al. (2001) have also noticed that vibrationally excited H₂ is present in the cloud towards HD 147888. The presence of both FUSE and HST/STIS spectra for HD 147888 (ρ Oph D) gives us a rare opportunity to measure the column density of H₂ energy levels excited by fluorescence (ν ≫ 0) and by other processes (collisions, formation, and fluorescence) on the ν = 0 level. As a consequence, we can calculate a photon-dominated region (PDR) model that is well constrained by density and by the ultraviolet radiation flux (star-cloud distance).

In a previous paper (Gnaciński 2011), the H₂ observations from the HST and FUSE satellites toward HD 37903 were analyzed. The sightline toward HD 147888 is the second and last sightline where H₂ lines from ν = 0 (FUSE) and from vibrationally excited (HST) levels were observed. Their weakness meant that the H₂* absorption lines from vibrationally excited levels toward HD 147888 were not analyzed previously. We present the first determination of H₂* column densities toward HD 147888 and a PDR model based on H₂ column densities on the ν = 0 and on vibrationally excited levels. Our PDR model significantly differs from the previously calculated model and agrees with independent determinations of hydrogen density. The two other sightlines toward which Meyer et al. (2001) have detected H₂* absorption lines, namely HD 37021 and HD 37061, do not have FUSE spectra, therefore determination of H₂ column densities on the basic vibrational ν = 0 level is impossible for these two sightlines.

The interstellar H₂ plays a fundamental role in gas chemistry. The H₂ molecule is the key reagent in many chemical reactions leading to produce heavier molecules. Many chemical reactions need activation energy or are endothermic. The internal energy of H₂ on rotationally and vibrationally excited levels can lead to overcoming the energy barrier. In this way the presence of H₂ on excited states influences the reactions’ rate coefficients and the chemistry in interstellar clouds. For many reactions the H₂* on vibrationally excited levels has significantly higher reactions rate than on the basic ν = 0 level. The rate coefficients at 100 K for reaction O + H₂ →  OH + H increases by 11 orders of magnitude for the ν = 3 vibrational state as compared to the ν = 0 level (Sultanov & Balakrishnan 2005). Also the formation of CH⁺ in the reaction H₂ + C ⁺  →  CH⁺ + H is ten orders of magnitude higher for H₂* than for the H₂ ground level (at 200 K) (Agúndez et al. 2010). The methylidyne cation (CH⁺) formed easily in reactions with H₂* reacts again with H₂, forming CH$\begin{array}{}\mathrm{+}\\ \mathrm{2}\end{array}$ and CH$\begin{array}{}\mathrm{+}\\ \mathrm{3}\end{array}$ in exotermic reactions. In hot PDRs with large FUV exposition, as for HD 34078 or the Orion Bar, forming of CH⁺ in reactions with H₂* is the dominant path to producing the methylidyne cation (Agúndez et al. 2010).

Also the ortho-to-para (hereafter O/P) H₂ ratio influences the chemical reaction rate. The reactions involving deuterated species are particularly affected (Pineau des Forêts et al. 1991). The degree of deuteration of molecules in dense interstellar clouds is correlated with the O/P H₂ ratio. The correlation is caused by the proton-exchange reactions, such as HD + H ⁺  ⇄  H₂ + D⁺ or HD + H$\begin{array}{}\mathrm{+}\\ \mathrm{3}\end{array}\rightleftarrows$ H₂ + H₂D⁺, which produce H₂ in J = 0 and J = 1 states, depending on input energies (Pineau des Forêts et al. 1991).

The H₂ column densities in the direction of HD 147888 for the two lowest rotational levels, J = 0 and J = 1, have been derived by Rachford et al. (2009) using the FUSE spectra. These spectra of 22 stars were also used by Jensen et al. (2010) to derive column densities of rotational excited molecular hydrogen. Interstellar H₂ toward HD 147888 was detected by them up to rotational level J = 6. A Meudon PDR model for the cloud toward HD 147888 was also calculated by Jensen et al. (2010). The H₂ and CO column densities toward HD 147888 were derived by Sheffer et al. (2008) from the FUSE and HST/STIS spectra. The HST/STIS spectra of HD 147888 was also used by Sonnentrucker et al. (2007) to determine the column densities and rotational excitations of CO and C₂. The high-resolution optical spectra of CN, CH, CH⁺, Ca I, K I and Ca II absorption lines toward HD 147888 were analyzed by Pan et al. (2004).

Snow et al. (2008) have analyzed the spatial distribution of stars in the ρ Oph cloud complex. Based on the Hipparcos parallaxes and absorption lines of Na I, K I, and Ca II, they estimated the distance to the molecular cloud to be 122 ± 8 pc. The stars ρ Oph A (HD 147933) and ρ Oph D (HD 147888) have similar column densities for K I, CH, and CN. Both stars are located in front of the densest part of the cloud, which is seen as a reflection nebula associated with ρ Oph A. The star ρ Oph C is located behind this cloud.

In Sect. 2 we present the HST and FUSE observations of HD 147888. The details of H₂ column densities calculations are also described in Sect. 2. In Sect. 3 we analyze the noise robustness of the fit to the HST spectrum. Section 4 presents the derivation of ortho/para H₂ ratio and temperatures calculated from the H₂ molecule. The calculated PDR models are presented in Sect. 5 and discussed in Sect. 6.

Fig. 1 A fragment of the average HST spectrum. The synthetic spectrum (thick line) was fit to the whole HST spectrum. The Doppler broadening parameter was set to b = 2.5 km s-1. The two deep absorption lines are C i lines.

Fig. 2 A fragment of FUSE spectra with the B–X (3,0) transitions of H2. The upper thick line represents continuum, and the bottom thick line is the simulated spectrum fitted to the observed one. The upper thin line is the stellar synthetic spectrum with rotational and instrumental broadening. The absorption line at 1066.66 Å is an Ar i line.

Fig. 3 Logarithms of H2 column densities divided by statistical weight plotted versus energy (squares with error bar). The ν symbol denotes the vibrational quantum number. For ν = 0 rotational quantum numbers are J = 0–6, and J = 0–5 for vibrationally excited levels (ν > 0). The first PDR cloud model is shown as circles. The second model is shown as crosses.

2. Observations and data analysis

We have used HST/STIS spectrum o59s05010 made 17 Aug. 2000 and spectra made on 2 Oct. 2009 (ob2602010 and ob2602020) to obtain column densities on vibrationally excited H₂ levels. No systematic differences were found between the two observations. Therefore all STIS spectra were combined into a single spectrum ranging from 1145 Å to 1356.8 Å. The STIS spectra were coadded using linear interpolation and a fixed step in wavelength of 0.005 Å. The column densities on the ground vibrational level (ν = 0) were determined from the FUSE spectra P1161501016–19. These spectra were averaged using the IRAF tasks poffsets and specalign. The FUSE spectrum extends from 904 Å to 1188 Å. We used FUSE spectra from the detectors LiF 2a and LiF 1a that have the best quality.

The column densities were derived using a program developed by the author. It uses the profile fitting method. The amoeba procedure described in Press et al. (2007) was used for minimization of the multidimensional function  ∑ _i(F_obs(λ_i) − F_fit(λ_i;v,b,N₁,N₂,...))², where λ_i are wavelengths in the observed spectrum F_obs, v is the cloud velocity, b stands for doppler broadening, and N are column densities of various H₂ ro-vibrational levels. The Voigt profile was calculated using the procedure developed by Shippony & Read (1993). The convolution with instrumental PSF was calculated using the routines from Press et al. (2007). The stellar continuum was modeled independently from interstellar line fitting. The continuum was formed by cubic spline functions passing through manually placed continuum points.

The whole HST/STIS spectrum was fit at once with a synthetic spectrum that included 440 H₂ absorption lines. The cloud velocity and column densities of all vibrationally and rotationally excited levels were free parameters. The H₂ spectral line positions and oscillator strengths were adopted from the MOLAT Atomic and Molecular Data^a database. The PSF function for HST/STIS spectrum was interpolated from tables presented by Kim Quijano et al. (2003) at wavelength 1242 Å (middle of the STIS spectrum). A fragment of HST spectrum together with a fit to the whole HST spectrum is shown in Fig. 1. Fits to STIS spectra were done assuming b = 2.25, 2.5, 2.75, and 3.0 km s^-1. Fits done with lower or higher values of b do not match the observed profiles of H₂ absorption lines. Table 1 presents an average column density for four fits with various b values.

Each transition from ground ν = 0 J = 0,1 levels to vibrationally excited levels of the B electronic level were fit independently in the FUSE spectrum. We calculated column densities from the following transitions: (0,0), (1,0), (2,0), (3,0), and (4,0). The first digit in parenthesis denotes the upper vibrational level of the B electronic state. The second digit (always 0) is the vibrational level of the ground electronic level X. At each point of the synthetic spectrum optical depths of spectral lines lying closer than 30 Å were summed. The natural line widths were given by Abgrall et al. (2000). The point spread function (PSF) for the FUSE spectra was a Gauss function with FWHM = 15 km s^-1 (Jensen et al. 2010). A synthetic spectrum, together with the FUSE observation, is shown in Fig. 2.

Fits to the FUSE spectrum were done with Doppler broadenings fixed to b = 2.5 and 3.0 km s^-1. As with HST spectra fits with lower or higher values of b do not match the observed profiles of H₂ lines. The continuum level for the FUSE spectra was placed manually in an iterative procedure of fitting column densities and changing the continuum level. There was a slight systematic shift between the observed H₂ line positions and wavelengths from the catalog. Therefore the fitting was performed on small sections of the FUSE spectrum. The derived molecular hydrogen column densities are presented in Table 1 and shown in Fig. 3. Table 1 presents the average logarithms of column densities calculated from various Doppler broadenings and for vibrational transitions (0,0) to (4,0).

To confirm our continuum placement for the FUSE spectrum, we used a synthetic star spectrum from the uvblue library (Rodrígez-Merino et al. 2005). The synthetic stellar spectrum from the library was calculated for a star with T_eff = 19 000 K, log g = 4, and [M/H] = 0. The synthetic spectrum was convolved with rotational and instrumental (FUSE) broadening functions using the Rotin3^b program. The stellar rotation velocities catalog (Głȩbocki & Gnaciński 2005)^c gives vsini = 175 km s^-1 for HD 147888. This rotational velocity is evidently too high. We used the velocity of 100 km s^-1 to match the observed stellar lines (Fig. 2). Our continuum level, especially the deep absorption line at 1066 Å, is similar to the synthetic stellar spectrum.

Table 2 presents a comparison between derived column densities on the ν = 0 level and the values derived by Jensen et al. (2010). Our column densities on the J = 3 to J = 6 rotational levels agree within errors with the Jensen et al. (2010) values that were derived using curve-of-growth methods. The H₂ column density on the J = 2 level differs slightly. The spectral lines from the ν = 0 J = 2 levels are saturated, and all methods of column density determinations are inaccurate. The column densities for the J = 0 and J = 1 rotational levels presented in the Jensen et al. (2010) paper were taken from Rachford et al. (2009). Our H₂ column density on the ν = 0 J = 1 level agrees with the value derived by Rachford et al. (2009). There is, however, a slight difference in column densities for the J = 0 level. Rachford et al. (2009) assume a quadratic polynomial continuum, while our continuum uses cubic splines and is similar to the stellar spectrum.

3. Noise robustness

Fig. 4 Robustness of the synthetic spectrum fit to the noisy data. The dashed horizontal line shows errors introduced by changing the original column densities by 30%.

Some of the H₂ absorption lines in the HST spectrum are very shallow or even at the noise level (e.g. absorption lines from the ν = 1 level). Therefore we tested the robustness of the fit against noise. The noise observed on the averaged HST/STIS spectrum is about 2.3% at 1256 Å and 0.85% at 1307 Å. We added Gaussian noise of amplitude A to the best fit synthetic spectrum to simulate the noisy spectrum. The points of the generated noisy spectrum are at the same wavelengths as the HST spectrum used previously in the fitting procedure. The Gaussian noise has standard deviation σ = 1. The spectrum with noise was generated using the formula F_A = F_fit(1 + A ∗ (1.64312 × 10^-4 × λ² − 0.420740 × λ + 270.283) × 0.01 × Gaussian   Noise), where the factor (1.64312 × 10^-4 × λ² − 0.420740 × λ + 270.283)% reflects the changing noise amplitude in the avaraged HST spectrum. The amplitude A = 1 reflects the noise level in the original HST spectrum. The F_fit is a synthetic spectrum that represents the best fit to the observed HST/STIS spectra.

Next we performed a fit to the generated spectrum F_A and obtained new column densities by fitting a new synthetic spectrum to the noisy one. All fits to the noisy spectrum were performed with the Doppler broadening parameter fixed to b = 2.5 km s^-1. The cloud velocity and all column densities for ν ≥ 1 were free parameters. The continuum level was common for all fits. To evaluate the error introduced by noise of various magnitudes A, we calculated a sum of squared relative errors: $\sum _{\mathit{\nu }\mathrm{\ge }\mathrm{1}\mathit{,J}\mathrm{=}\mathrm{0}\mathrm{-}\mathrm{5}}{\left(\frac{{\mathit{N}}_{\mathit{A}}\mathrm{\left(}\mathit{\nu ,J}\mathrm{\right)}\mathrm{-}{\mathit{N}}_{\mathrm{0}}\mathrm{\left(}\mathit{\nu ,J}\mathrm{\right)}}{{\mathit{N}}_{\mathrm{0}}\mathrm{\left(}\mathit{\nu ,J}\mathrm{\right)}}\right)}^{\mathrm{2}}\mathrm{·}$(1)The symbol N_A represents column density calculated from the spectrum with noise amplitude A. The generation of a noisy spectrum and subsequent deriving of column densities was repeated three times for each A value. The results are shown in Fig. 4. The uncertainties do not include contribution from continuum placement because the continuum level was kept fixed. They only show the influence of noise on the derived column densities. The energy levels without evident absorption lines gives about 60% contribution to the sum 1. As seen in Fig. 4, even for a noise amplitude two times larger than in the HST spectra, the resulting column densities give the sum of relative errors that is lower than errors introduced by changing the best fit column densities by 30%, so the fitting procedure is robust against the noise.

4. Ortho to para H₂ and temperatures

The column density of atomic and molecular hydrogen toward HD 147888 was initially estimated by Cartledge et al. (2001) based on the hydrogen column densities toward ρ Oph A (HD 147933) determined from Copernicus and International Ultraviolet Explorer (IUE) data. We adopted the N(HI) = (5.1 ± 1.1) × 10²¹ cm^-2 value determined by Cartledge et al. (2004) from the Ly α line in the HST/STIS spectrum of HD 147888. This N(HI) value toward HD 147888 (spectral type B3/B4V) complies with the atomic hydrogen column density toward HD 147933 (B2/B3V). This is not surprising since Snow et al. (2008) shows that HD 147933 and HD 147888 have a similar absorption structure of K I, CH, and CN. Both stars are also at similar distances.

For stars with spectral type B3 or later, the Lyman α line may be contaminated by stellar absorption (Shull & van Steenberg 1985). The adopted hydrogen column density N(HI) is less than the upper limit N(HI)  <  5.5  ×  10²¹ cm^-2 derived by Shull & van Steenberg (1985) from IUE spectra. The estimation of hydrogen column density based on interstellar reddening E(B − V) given by Rachford et al. (2009) is, however, almost two times less N(HI) = 2.75 × 10²¹ cm^-2.

The total observed column density (see Table 1) of molecular hydrogen is equal to N(H₂) = (3.7 ± 0.3) × 10²⁰ cm^-2. The resulting molecular fraction of hydrogen f(H₂) = 2N(H₂)/(N(HI) + 2N(H₂)) = 0.13 ± 0.03. The ortho-para H₂ ratio for vibrational levels ν = 1–4 is (O/P)* = 1.13 ± 0.09.

The Wilgenbus et al. (20003) method allows calculating the O/P H₂ ratio from an ortho level (J_o) and two adjacent para H₂ levels: J_o − 1 and J_o + 1. First one should calculate the rotational temperature from the two para H₂ levels: ${\mathit{T}}_{\mathrm{rot}}\mathrm{=}\frac{\mathit{E}\mathrm{(}{\mathit{J}}_{\mathrm{o}}\mathrm{+}\mathrm{1}\mathrm{)}\mathrm{-}\mathit{E}\mathrm{(}{\mathit{J}}_{\mathrm{o}}\mathrm{-}\mathrm{1}\mathrm{)}}{\mathit{k}\ln \frac{\mathit{N}\mathrm{(}{\mathit{J}}_{\mathrm{o}}\mathrm{-}\mathrm{1}\mathrm{)}{\mathit{g}}_{\mathit{J}}\mathrm{(}{\mathit{J}}_{\mathrm{o}}\mathrm{+}\mathrm{1}\mathrm{)}}{{\mathit{g}}_{\mathit{J}}\mathrm{(}{\mathit{J}}_{\mathrm{o}}\mathrm{-}\mathrm{1}\mathrm{)}\mathit{N}\mathrm{(}{\mathit{J}}_{\mathrm{o}}\mathrm{+}\mathrm{1}\mathrm{)}}}\mathrm{·}$(2)Next we can calculate the O/P ratio: ${\left(\frac{\mathit{O}}{\mathit{P}}\right)}_{{\mathit{J}}_{\mathrm{o}}}\mathrm{=}\frac{{\mathit{Z}}_{\mathrm{o}}\mathrm{\left(}{\mathit{T}}_{\mathrm{rot}}\mathrm{\right)}}{{\mathit{Z}}_{\mathit{p}}\mathrm{\left(}{\mathit{T}}_{\mathrm{rot}}\mathrm{\right)}}\frac{\mathit{N}\mathrm{\left(}{\mathit{J}}_{\mathrm{o}}\mathrm{\right)}}{{\mathit{N}}^{\mathrm{interp}}\mathrm{\left(}{\mathit{J}}_{\mathrm{o}}\mathrm{\right)}}\mathit{,}$(3)where N^interp(J_o) is the column density of a ortho state interpolated from the two adjacent para states on an ln(N(J)/g_J(J)) against the E(J)/k diagram. The Z(T) are the partition functions for ortho and para H₂. The ortho-para H₂ ratios on the ν = 0 J = 1 and J = 3 levels calculated using the Wilgenbus et al. (20003) method are significantly lower than (O/P)* on vibrationally excited levels. They are equal to $\mathrm{(}\mathit{O}\mathit{/}\mathit{P}{\mathrm{)}}_{\mathit{J}\mathrm{=}\mathrm{1}}\mathrm{=}\mathrm{0.15}\stackrel{\mathrm{+}\mathrm{0.04}}{-0.03}$ and $\mathrm{(}\mathit{O}\mathit{/}\mathit{P}{\mathrm{)}}_{\mathit{J}\mathrm{=}\mathrm{3}}\mathrm{=}\mathrm{0.39}\stackrel{\mathrm{+}\mathrm{0.31}}{-0.17}$. The J = 5 ortho level presents O/P as similar to the (O/P)* on excited vibrationally H₂ levels. The radiative excitation to upper electronic levels is therefore the dominant process for J < 5.

Sternberg & Neufeld (1999) have presented a simple model that explains the (O/P)* H₂ ratio on vibrationally excited levels. They assumed that collisional excitation is negligible to FUV pumping and that collisional de-excitation and photodissociation are negligible to spontaneous radiative decay. Additionally, the H₂ absorption lines from J = 0,1 (ν = 0) should be on the square root part of the curve of growth, as in our case for HD 147888 (see Fig. 2). The damping wings of the H₂ lines lead to the differentiation of ortho and para H₂ on vibrationally excited states (ν > 0). Under the assumption that local densities of ortho and para H₂ on the ground level are in LTE (n_O/n_P = α(T_gas)) and the ratio of local densities is equal to the ratio of column densities, the model predicts (O/P)* = $\sqrt{\mathit{\alpha }\mathrm{\left(}{\mathit{T}}_{\mathrm{gas}}\mathrm{\right)}}\mathrm{=}\sqrt{\mathrm{(}\mathit{O}\mathit{/}\mathit{P}{\mathrm{)}}_{\mathit{J}\mathrm{=}\mathrm{1}}}$. If on the contrary, the collisions are too rare to maintain the (O/P)_J = 1 in LTE then (O/P)_J = 1 = α²(T_form) and (O/P)* = α(T_form), where α(T_form) is the O/P formation ratio on the dust grain surface. This leads again to (O/P)* = $\sqrt{\mathrm{(}\mathit{O}\mathit{/}\mathit{P}{\mathrm{)}}_{\mathit{J}\hspace{0.17em}\mathrm{=}\hspace{0.17em}\mathrm{1}}}$ as in previous case. The H₂ in the cloud towards HD 147888 does not satisfy this equation because $\mathrm{1.13}\ne \sqrt{\mathrm{0.15}}\mathrm{=}\mathrm{0.39}$. Also the cloud toward HD 37903 (see Gnaciński 2011 for O/P H₂ values) does not satisfy the (O/P)* = $\sqrt{\mathrm{(}\mathit{O}\mathit{/}\mathit{P}{\mathrm{)}}_{\mathit{J}\mathrm{=}\mathrm{1}}}$ equation. One of the reasons for the discrepancy is that the bulk of H₂* and H₂(J = 0,1) are located in different parts of the cloud, so the column densities ratio of H₂* and H₂(J = 0,1) are not equal to the local density ratio. In our PDR cloud model the n(H₂*) density is low where n(H₂) is large (observer side of the cloud) and vice versa (Fig. 6).

The temperature of ortho-para H₂ equilibrium T_OP was derived from the equation $\frac{\mathit{N}\mathrm{\left(}\mathrm{1}\mathrm{\right)}}{\mathit{N}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}\mathrm{=}\frac{\mathit{g}\mathrm{\left(}\mathrm{1}\mathrm{\right)}}{\mathit{g}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}\exp \left(\mathrm{-}\frac{\mathit{E}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{-}\mathit{E}\mathrm{\left(}\mathrm{0}\mathrm{\right)}}{\mathit{k}{\mathit{T}}_{\mathrm{OP}}}\right)\mathit{,}$(4)where N(J) are the column densities, E(J) is the energy on the J level (ν = 0), and g(J) is the statistical weight $\mathit{g}\mathrm{\left(}\mathit{J}\mathrm{\right)}\mathrm{=}\{\begin{array}{c}\\ \mathrm{(}\mathrm{2}\mathit{J}\mathrm{+}\mathrm{1}\mathrm{)}& \mathrm{forparaH}\mathrm{2}\mathrm{(}\mathrm{even}\mathit{J}\mathrm{)}\\ \mathrm{3}\mathrm{(}\mathrm{2}\mathit{J}\mathrm{+}\mathrm{1}\mathrm{)}& \mathrm{fororthoH}\mathrm{2}\mathrm{(}\mathrm{odd}\mathit{J}\mathrm{)}\mathit{.}\end{array}$(5)The temperature of ortho-para H₂ equilibrium T_OP = 42 ± 3 K is similar to the value T_OP = 45 ± 3 K derived by Jensen et al. (2010). The T₀₂ rotational temperature only involves para-H₂ levels and was calculated from an equation that is analogous to Eq. (4). The T₀₂ rotational temperature equals 90 ± 3 K (Jensen et al. 2010 gives T₀₂ = 88 ± 1 K).

The rotational temperature was also calculated for each vibrational level. The rotational temperature is the inverse of the line slope taken with the minus sign on a plot: ln(N/g) versus E/k (like Fig. 3 but using a natural logarithm instead of base 10 logarithm). The slope was calculated with the linear regression method separately for ortho and para H₂ spin isomers. The resulting temperatures are presented in Fig. 5. For each vibrational level, the points on the ln(N/g) versus E/k plot are perfectly placed on a straight line (separately for ortho and para states). The correlation coefficient varies between 0.88 and 1.00. The excellent agreement with the Boltzmann distribution confirms the correctness of the H₂ column densities. For the sightline towards HD 37903, the rotational temperatures fall with increased vibrational number ν (Gnaciński 2011). This behavior is not observed towards HD 147888.

Fig. 5 The rotational temperatures on vibrational levels 0–5. Squares presents rotational temperature of para H2, while the circles are for ortho–H2.

5. Models

We used the Meudon PDR code (Le Petit et al. 2006) to calculate the model of the interstellar cloud in front of HD 147888. We calculated isobaric models with exact radiative transfer in all H₂ spectral lines. We calculated 1529 PDR models. The initial hydrogen density in the models was varied from n_H = 110 cm^-3 to 3500 cm^-3. The star cloud distance was changed from d = 0.51–1.4 pc. The interstellar radiation field on the star side was varied from 1 to 1200 Draine units and was fixed to 1 Draine unit on the observer side. The turbulent velocity was set to v_turb = 2 km s^-1 and the star spectral type to B3V. The interstellar reddening E(B − V) = 0.47, R_V = 4.06, and A_V = 1.91 were adopted from Rachford et al. (2009).

The first PDR model was chosen by minimizing the difference in relative H₂ abundances observed in the cloud towards HD 147888 and calculated in a PDR model: $\sum _{\mathit{\nu ,J}}{\mathit{w}}_{\mathit{\nu }}{\left(\log \frac{{\mathit{N}}_{\mathrm{obs}}\mathrm{\left(}\mathit{\nu ,J}\mathrm{\right)}}{{\mathit{N}}_{\mathrm{obs}}\mathrm{\left(}{\mathit{H}}_{\mathrm{2}}\mathrm{\right)}}\mathrm{-}\log \frac{{\mathit{N}}_{\mathrm{model}}\mathrm{\left(}\mathit{\nu ,J}\mathrm{\right)}}{{\mathit{N}}_{\mathrm{model}}\mathrm{\left(}{\mathit{H}}_{\mathrm{2}}\mathrm{\right)}}\right)}^{\mathrm{2}}\mathrm{·}$(6)The weights w_ν were chosen such that the levels usually excited by fluorescence (ν > 0) have the same influence on the final sum as the ν = 0 level excited by collisions and by H₂ formation. The weights w_ν were equal to five for ν = 0 and one otherwise.

The first PDR model has a star-cloud distance of 0.62 pc. The interstellar radiation field on the star side is equal to 700 Draine units. The hydrogen density varies from 89 cm^-3 on the star side to 336 cm^-3 near the observer’s side. The gas kinetic temperature changes from 358 K on the star side to 98 K on the observer’s cloud side. The distance is better constrained than the density. Assuming 1% change of the sum (6) the star-cloud distance can vary up to 10%, while the average density can vary up to 31%. The interstellar radiation field can change by 100 Draine units. In the first cloud model, the ortho/para H₂ ratio for vibrationally excited states equals 1.56. Figure 6 shows that the (O/P)* ratio on vibrationally excited H₂ levels changes significantly in the cloud. The relative abundance of H₂ on the ν = 1 level is also shown.

The (O/P)* curve has a maximum at A_V = 0.229 (Fig. 6). Looking from the star side (height A_V), the (O/P)* rises because the H₂(J = 1)/H₂(J = 0) on the v = 0 level gets higher. The ortho H₂ lines are stronger than the para H₂ absorption lines, and at the maximum of (O/P)* at A_V = 0.229 selfshielding in the ortho H₂ lines becomes important. Less ortho H₂ is then transferred to the upper electronic levels and the (O/P)* decreases. At the minimum of the (O/P)* curve (A_V = 0.128) lines from J = 0 and J = 1 are both saturated. At the plateau around A_V = 0.05, the Lorentzian wings dominate the absorption spectrum.

Our second model was selected by minimizing the difference between the observed and calculated column densities: $\sum _{\mathit{\nu ,J}}{\mathit{w}}_{\mathit{\nu }}{\left(\log {\mathit{N}}_{\mathrm{obs}}\mathrm{\left(}\mathit{\nu ,J}\mathrm{\right)}\mathrm{-}\log {\mathit{N}}_{\mathrm{model}}\mathrm{\left(}\mathit{\nu ,J}\mathrm{\right)}\right)}^{\mathrm{2}}\mathit{.}$(7)A model chosen using the above formula has to reproduce the H₂ column densities. Our attempts to find a model that satisfies the formula led to an unphysical model for the cloud in front of HD 37903. The hydrogen density was higher than 10⁶ cm^-3. However, we decided to present this model of the cloud towards HD 147888 for comparison purposes, as it has the hydrogen density on the observer’s side close to the density presented by Jensen et al. (2010). The PDR model presented by Jensen et al. (2010) has n_H = 2000 cm^-3. However, their model did not take the H₂ molecules in excited vibrational levels (ν > 0) into account.

Fig. 6 First cloud model (at d = 0.62 pc). The dashed line scale is on the right y-axis. The marked AV’s are discussed in the context of the (O/P)* variability.

Fig. 7 Second cloud model (at d = 0.96 pc). The dashed line scale is on the right y-axis.

In the second model the hydrogen density varies from n_H = 22   000 cm^-3 near the observer’s side to n_H = 3200 cm^-3 on the star side. The temperature changes from T = 34 K to T = 129 K, and the interstellar radiation field on the star side is 1 Draine unit. The star-cloud distance in the second model is 0.96 pc. As in the case of the first model, the second model is also better constrained in the star-cloud distance than in density. The 1% change of sum (7) changes the distance up to 5%, while the average density changes by 24%. The interstellar radiation filed on the star side changes at most by 100 Drain units. The ortho/para H₂ ratio in the cloud model equals (O/P)* = 1.36 and the total H₂ column density equals to 2.5 × 10²¹ cm^-2, so the total H₂ column density in this model is about one order of magnitude greater than the observed one. The model H₂ excess is in the J = 0 and 1 rotational levels (ν = 0). These levels are high in order to maintain an adequate population of vibrationally excited states. Figure 7 shows the changes in hydrogen density, gas kinetic temperature, the (O/P)* H₂ ratio, and the relative abundance of H₂ at the ν = 1 level.

For both models, the (O/P)* H₂ ratio on excited levels show large spatial variations in the cloud. Column densities of H₂ energy levels in both models are shown in Fig. 3. The input parameters and results of both models are compared with observations in Table 3. Owing to the normalization used in model 1, the total N(H₂) column density is by definition equal to the observed one. However column densities of other species cannot be derived for model 1.

6. Discussion

The CO and C₂ molecules toward HD 147888 were analyzed by Sonnentrucker et al. (2007). The density of hydrogen ${\mathit{n}}_{\mathrm{H}}\mathrm{=}\mathrm{21}{\mathrm{5}}_{-30}^{\mathrm{+}\mathrm{55}}$ cm^-3 derived from the C₂ rotational levels was calculated by Sonnentrucker et al. (2007) under the assumption of the average Galactic radiation field. Since the radiation field in the cloud toward HD 147888 may be higher, the hydrogen density may also be higher (nσ₀/I ~  population of excited rotational levels; van Dishoeck & Black 1982). Nevertheless the derived hydrogen density is similar to hydrogen density in our first model. The C₂ rotational temperature T₀₂ = 38 ± 20 K is similar to T_OP = 42 ± 3 K as derived from the H₂ molecule.

The density of the interstellar cloud towards HD 147888 was estimated by Pan et al. (2005) on the basis of a chemical model involving CN, CH, C₂, O, C ii, and N. Their density n_H = 425 cm^-3 is close to the density derived in our first model. Two independent density estimations thus give densities close to our density in the first PDR model. Therefore we prefer the first model, where the relative H₂ abundances were compared. Pan et al. (2005) have also modeled the CN and C₂ column densities in the direction of ρ Oph D. Both the observed and modeled N(CN) = 2.1 × 10¹² cm^-2. Our second model gives the CN column density as one order of magnitude higher N(CN) = 3 × 10¹³ cm^-2. The C₂ column density in our second model N(C₂) = 2.4 × 10¹⁵ cm^-2 does not agree with the model by Pan et al. (2005), which give N(C₂) = 1.9 × 10¹³ cm^-2. We cannot derive the denormalized CN and C₂ column densities for our first model due to the normalization used to compute the model.

7. Conclusions

The interstellar cloud towards HD 147888 is one of the rare sightlines where H₂* absorption from vibrationally excited levels is observed. The main results are:

• We have determined H₂ column densities on ν = 0–5 and J = 0–6 from FUSE and HST spectra.

• It is possible to obtain column densities from large numbers of shallow absorption lines arising from the same ground level. The whole HST/STIS spectrum should be fitted at once with a synthetic spectrum to derive column densities from a noisy spectrum.

• The occupation of the vibrationally excited H₂ levels toward HD 147888 did not show any temporal variations over the period of 9 years.

• The ortho-para H₂ ratio on excited vibrational levels (O/P)* = 1.13, while the O/P H₂ on the ground vibrational level (O/P)_{ν    =    0,J    =    1} = 0.15. These ratios do not agree with the simple model by Sternberg & Neufeld (1999).

• The Meudon PDR model (our first model) locates the interstellar cloud 0.62 pc from the HD 147888 star. The hydrogen density in the cloud is 89–336 cm^-3 and the gas kinetic temperature changes from 98 to 358 K.

Independent density estimations from C₂ and chemical model agree with our model, where the model relative H₂ abundances were compared with relative H₂ abundances from observations (first PDR model).

Acknowledgments

This research was supported by the University of Gdańsk grant BW/5400-5-0336-0 and Polish National Center for Science grant nr 5820/B/H03/2011/40. I would like to thank Herve Abgrall for providing the natural line widths for the H₂ lines.

References

All Tables

All Figures

	Fig. 1 A fragment of the average HST spectrum. The synthetic spectrum (thick line) was fit to the whole HST spectrum. The Doppler broadening parameter was set to b = 2.5 km s-1. The two deep absorption lines are C i lines.
In the text

	Fig. 2 A fragment of FUSE spectra with the B–X (3,0) transitions of H2. The upper thick line represents continuum, and the bottom thick line is the simulated spectrum fitted to the observed one. The upper thin line is the stellar synthetic spectrum with rotational and instrumental broadening. The absorption line at 1066.66 Å is an Ar i line.
In the text

	Fig. 3 Logarithms of H2 column densities divided by statistical weight plotted versus energy (squares with error bar). The ν symbol denotes the vibrational quantum number. For ν = 0 rotational quantum numbers are J = 0–6, and J = 0–5 for vibrationally excited levels (ν > 0). The first PDR cloud model is shown as circles. The second model is shown as crosses.
In the text

	Fig. 4 Robustness of the synthetic spectrum fit to the noisy data. The dashed horizontal line shows errors introduced by changing the original column densities by 30%.
In the text

	Fig. 5 The rotational temperatures on vibrational levels 0–5. Squares presents rotational temperature of para H2, while the circles are for ortho–H2.
In the text

	Fig. 6 First cloud model (at d = 0.62 pc). The dashed line scale is on the right y-axis. The marked AV’s are discussed in the context of the (O/P)* variability.
In the text

	Fig. 7 Second cloud model (at d = 0.96 pc). The dashed line scale is on the right y-axis.
In the text