## 3. Extracting the o/p ratio## 3.1. FrameworkThe observations yield three line intensities
, and
, measured in a beam of fixed solid angle
. The S(0) and S(2) transitions involve para
H The intensities are given by where where is the total column density,
is the appropriate partition function for ortho
or para H 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 H 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. ## 3.2. MethodThe 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 .
For the three lines chosen here, with K, 6951K & 7584K, , 3.47 & 3.98 this yields where 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.
## 3.3. General resultsThe 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 |