3. Extracting the o/p ratio
The observations yield three line intensities , and , measured in a beam of fixed solid angle . The S(0) and S(2) transitions involve para H2 with mass fraction , and the S(1) transition is ortho with fraction , so that . The energies and rotational quantum numbers of the upper levels of the transitions are and . The transition wavelengths are , radiative de-excitation rated and we assume extinctions . This yields the extinction corrected line fluxes . The line fluxes are related to the intrinsic intensities , assuming optically thin isotropic radiation, by .
The intensities are given by
where is the total column density, is the appropriate partition function for ortho or para H2 and h and c are the standard physical constants. We therefore assume here that the gas is in LTE and at a single temperature T which depends only on position. The LTE assumption is clearly valid for OMC-1, even with vibrational transitions involving the second vibrational level (Brand et al 1988). Here, we employ strong rotational transitions only in the first vibrational level, two factors which ensure LTE.
All the gas observed within a pixel is unlikely to be at the same temperature. Each pointing measures gas with a range of temperatures and it follows that the excitation temperature depends on the upper energy levels of the transitions involved (Brand et al 1988). Note that the excitation temperature is defined between any two upper levels in terms of the column density ratios and generally requires knowledege of the ortho-para ratio. However for the two para transitions we have observed
Note that by plotting against , straight lines would denote a constant temperature and convex lines a range in temperatures.
Here, we have chosen three lines which are quite closely spaced in upper energy levels, so we do not expect a large difference in excitation temperature. We have tested how this assumption will effect the derived values of using the C-type bow shock which provides an overall fit to the OMC-1 Peak 1 data over a wide range of energy levels (Smith, Brand & Moorhouse 1991a) and with J-type shocks (Smith 1994) which also demonstrate strong curvature. The models presume , but yield slightly lower values when we apply the final formula presented below to the 1-0 S(0), S(1) and S(2) transitions. We find for the C-type and 2.989 for the J-type model. The fact that these are below 3 is expected since the convex curvature implies that will lie below the linear connection between the two neighbouring lines. Thus we conclude that the fixed temperature assumption may introduce an error of at most one per cent. (One means of overcoming this error would be to include a fouth line, the 1-0 S(3), to define the curvature. However, this line lies outside the K-band and cannot be reliably measured from the ground).
The partition functions are the sums of over all energy levels. They could be calculated implicitly for each temperature and H2 form. However, we have calulated the ortho and para partition functions and compared them to a single, straightforward approximation over the excitation temperature interval observed for OMC-1. We find that
is 0.2% accurate over the temperature range 850K - 1900K and remains 1.0% accurate for 600K - 2450K. We thus are able to employ this single partition function for both species below.
The expression for which we use is:
which is generally valid for any three rotationally-consecutive lines (where is derived for sequences containing two ortho transitions). This equation is derived from the expression for the ratio (from Eq. (2),
on making the substitution
and then using Eq. (3) to eliminate .
We therefore only need to derive dereddened columns from the data to extract the ortho and para fractions. The column density ratios are given from Eq. (1) in terms of the observed line strengths:
For the three lines chosen here, with K, 6951K & 7584K, , 3.47 & 3.98 this yields
The extinction dependence is very weak. For example, adopting the extinction curve of Mathis (1990) and the extinction to Peak 1 of OMC-1 of (Everett, DePoy & Pogge 1995), yields . That is, for OMC-1, ignoring extinction altogether introduces on average underestimates of the ortho-to-para ratio of under 1%. Hence, the three lines observed (1-0 S(0), S(1) & S(2)) permit an accurate estimate for .
The rotational excitation temperature derived from Eq. (3) has a much stronger extinction dependence, however, since
which yields a value of for OMC-1 Peak 1, as given in Table 2.
Table 2. The ortho-para ratio () and excitation in Herbig-Haro outflows
3.3. General results
The values for derived from Eq. (7) are listed in Table 2 and plotted in Fig. 1 for a number of Herbig-Haro objects. The extinctions are taken from the corresponding reference. The quoted error is simply the highest error quoted for the three lines involved. This may include calibration as well as continuum subtraction errors. Hence only a crude error analysis is feasible here. Note, however, that depends at most linearly on each line flux, so the quoted error can be taken as an upper limit. The ortho-para ratio is consistent with 3 in each object in which the errors are less than 25%.
We remark that the contradictory results for HH 7 reported by Fernandes & Brand (1995) and Everett (1996) have been resolved here. The observations are, after all, in agreement (upon removing an error in the conversion of fluxes to columns in the earlier publication). We also note that where the errors are large in Fig. 1, the ratio falls below 3 and the rotational excitation temperature is relatively low. This suggests that the 1-0 S(0) flux has not been accurately measured in these objects. An alternative is that weaker lines are produced in cooler objects where ortho-para conversion is slow, thus revealing the value acquired from post-formation processes in cool surroundings. This is less plausible on theoretical grounds, however, since the conversion rate coefficients are not particularly sensitive to the temperature in the range 1000K-1600K (Burton et al 1992).
Finally, note that in Table 1 the rotational excitation temperatures are consistently below the vibrational 2-1/1-0 S(1) excitation temperatures. This is expected from shock theory when either hot gas is produced and allowed to cool unhindered (J-type), or when a variety of shock strengths are involved (supersonic turbulence and/or curved shock fronts). The narrow range in vibrational temperature may result from the fixed manner in which warm gas cools as well as the ubiquitous presence of bow-type shock fronts (Smith et al 1991a).
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998