Volume 528, April 2011
|Number of page(s)||23|
|Section||Interstellar and circumstellar matter|
|Published online||16 February 2011|
In images extracted from each data cube the nominal “source” position was found to shift slightly with wavelength. This shift is largely due to extinction, although differential refraction and/or imperfect cube construction from the 2-dimensional spectral images obtained with the IFS could also contribute.
To investigate this possibility, we examined images extracted from our standard star observations. Six images spread across the H and K-bands were extracted from seven standard star data cubes. We found that the x and y positions of the standard (measured from 2-dimensional Gaussian fits) shifted very slightly with wavelength; a tight linear correlation was identified for each standard star: Δx = Sx × δλ and Δy = Sy × δλ (where Δx and Δy are the shift in pixels, λ is the wavelength in microns, and S is the slope of the linear fit). However, Sx and Sy were not the same for all standard stars. Rather, these proved to be a function of airmass, AM, as expected for differential refraction, with Sx = 3.5 − 2.38 × AM and Sy = 1.0 − 0.55 × AM. Overall, this effect was mild: we found that at low airmass, between [Fe ii] and Brγ (increasing wavelength), the source shifted down and to the right by only 0.45 pix, 0.21 pix (0.022″, 0.011″), at intermediate airmass the shift is insignificant in both x and y, while at high airmass the shift is to the left by 0.41 pix, 0.01 pix (0.021″, 0″). These shifts are very small when compared to the offsets measured in Table 2 and essentially only apply to the [Fe ii] offsets. They have therefore not been applied to the data.
Molecular hydrogen excitation diagrams may in principal be used to simultaneously estimate the gas excitation temperature and the extinction towards the line-emitting region (e.g. Nisini et al. 2002; Caratti o Garatti et al. 2006).
The left-hand panel shows an H2 excitation diagram for HH 26-IRS. The data, corrected for extinction (Av = 27 in this case), have been fitted with a straight line and a second-order polynomial (dashed line). The equations describing each fit are displayed; the R2 values represent the square of the correlation coefficient associated with each fit. Plots over a range of Av values were generated and R2 measured for the two fits in each case. The right-hand panel shows a plot of R2 against Av. Data from the linear and polynomial fits are represented by a cross and an open square, respectively; the full and dashed lines represent third-order polynomial fits to these data. The peak in each curve in the right-hand plot marks the extinction associated with the greatest value of R2, which in turn is associated with the least scatter about the linear and polynomial fits in the excitation diagram.
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In thermalised gas at a single temperature, all of the points in the H2 excitation diagram will lie on a single straight line, the slope of which is a measure of the temperature. In reality, when emission lines from a range of ro-vibrational states are detected, the full diagram is better fit with a curve (Schwartz et al. 1995; Gredel 2006, 2007), or by a separate straight line for each vibrational state (e.g. Giannini et al. 2002; Caratti o Garatti et al. 2006), since lines from higher vibrational states – which have higher critical densities for collisional excitation by hydrogen atoms – may not be fully thermalised. Moreover, the shocks in outflows are rarely planar so the gas in unresolved bow shocks will not be heated to a single temperature; nor will it be maintained at a single temperature as it flows through the shock front and post-shock cooling zone. However, if one plots column densities derived from lines from only the first few vibrational states, a linear fit is usually quite adequate (e.g. Lorenzetti et al. 2002; Giannini et al. 2004; Davis et al. 2004; Gredel 2006; Caratti o Garatti et al. 2008; Martín-Hernández et al. 2008). By minimising the scatter of points about this line, one can then estimate the extinction.
The column density of a given v,J ro-vibrational level, Nv,J, is related to the line intensity, I(vJ → v′J′) (measured in W m-2), the wavelength of the transition, λ (in microns), and the v,J → v′,J′ transition probability, A(vJ → v′J′), by: (C.1)In thermodynamic equilibrium, the ratio of column densities between two levels, Nv,J and Nv′,J′, is given by the Boltzmann equation:
If we adopt an extinction law of the form Aλ/Av = (λv/λ)1.6 (Rieke & Lebofsky 1985), then the column density may be corrected for extinction:
(C.3)For a given value of Av, the (negative) reciprocal of the slope of a linear fit to a plot of ln(Nv,J/gv,J) against Ev,J (the energy of the v,J ro-vibrational level) yields an estimate of Tex. Increasing Av will increase the value of Nv,J(corr) and alter the distribution of points around the fit. Minimising this scatter should then yield an estimate of the extinction towards the line-emission region.
In Fig C.1 we present an excitation diagram for HH 26-IRS; similar plots were generated for the other five H2 line-emitters in our sample. Over a range of extinction values (Av = 0 − 100) we plot corrected column densities (divided by the statistical weights listed in Table C.1) against upper level energies. An ortho-para H2 ratio of 3 is assumed throughout (note that a ratio that deviates from the statistical value of 3 would offset the ortho-H2 and para-H2 data points with respect to each other; an ortho-para ratio of 3 is in any case usually observed in YSO outflows, e.g. Smith et al. 1997). In each plot we fit the data with a single straight line and a second-order polynomial. We measure the extinction values that yield the least scatter about each fit; from the best linear fit we also measure the excitation temperature (see Fig. C.1 for further details).
Our results are listed in Table C.2. In SVS 13 and HH 999-IRS we had insufficient data to measure Av (although from the two observed lines we could at least estimate Tex assuming Av = 0; note that Tex is relatively insensitive to Av). Towards the other four sources a range of values were measured. Excluding the Q-branch emission lines, which are suspect due to their location at the edge of the K-band, had little effect on the results. Excluding the higher-energy lines (lines from the v > 1 levels and the 1–0 S(7)–S(9) transitions in the H-band) also had only a subtle effect on extinction estimates, although excitation temperatures were consistently lower, as expected.
© ESO, 2011
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