Odin Lines

The Odin satellite will not be the first to detect water emission -- that goes to the Infrared Space Observatory (ISO) which has detected various water lines in several objects. The best early detection was in W Hya; subsequently many other detections have been made, although Mira seems to have NO water lines. In the case of W Hya, which was already known to be a prime target for ODIN observations, a large number of water lines have been observed as can be seen in the figure which comes from the ISO press release.

The ODIN mission will be much superior to ISO for the study of the water and oxygen in the interstellar medium, and even for stars it will provide profile information for the 556.945 GHz line and possibly for other water lines. ISO cannot provide profile information except for cases where the expansion speed is much larger than usual for circumstellar shells. Still it does appear that ISO has an advantage over ODIN for detecting water in circumstellar sources, mostly due to the much smaller beam size. For most circumstellar sources the ODIN beam dilution will be fairly severe.

Rotational/Vibrational Lines of Water

An energy level diagram for the first 63 rotational levels can be viewed here for ortho-water or for para-water. The Figures show the levels and their transitions in each case. The simulations we are doing for the level populations and the line strengths which result use three sets of these rotational levels for different vibrational states of the molecule. The system is much more complex than those we use for the CO, Hydrogen, or Oxygen molecules. In the Figures the lines observable by ODIN are in red and the lines that have been observed from the ground, which are nearly all maser lines, are marked in blue or green. The X-axis quantity in these plots is a pseudo quantum number which is used to differentiate different levels of the same rotational quantum number.

For those who know the system of quantum numbers for such molecules, the K quantity in the plots is given by K = K+ - K- where K+ and K- are the more usual quantum numbers used for the rotational levels, along with the J quantum number which has its usual meaning. The K pseudo quantum number is also denoted as tau in some papers. Note also that which quantum number is called K+ and which is called K- differs from paper to paper in the literature. Another common notation is to call the quantum numbers K_A and K_C where "_" denotes a subscript. These are the K quantum numbers for the oblate and prolate top cases of an axi-symmetric rotator. Then K in the plot is K_A - K_C. A level is denoted by quantum numbers J, K+, and K- [or K_A and K_C] in that order. The 22 GHz water maser line is due to the transition from the (0616) level to the (0523) level, where the first quantum number in the () denotes the ground vibrational level.

The energy level diagrams for 183 vibrational/rotational levels of ortho-water and para-water, in two separate plots, can be viewed in a postscript file which shows only the rotational transitions. Note that the file probably requires ghostview to be displayed; the size is 281 KBytes. (If the transitions between vibrational levels are shown the plot becomes too messy to be useful.) The postscript file has two pages in it, one for the ortho- form and one for the para- form of the water molecule. This set of levels is currently used in our level population simulations.

The main transition of interest for the ODIN project from water is the 556.95 GHz line from the (0110) level [v,J,K+,K- quantum numbers] to the (0101) level. This is a transition from the first excited state to the ground state of ortho-water. Other transitions which may be observable for ortho-water are at 426.18 [(1854) to (1761)] and 578.05 GHz [(1743) to (1652)]. For para-water the potentially observable transitions are at 424.06 [(2414) to (2321)], 488.60 [(0624) to (0717)], 546.50 [(1524) to (1431)], and 575.20 GHz [(2423) to (2330)] lines. Here the vibrational levels 1 and 2 denote the higher vibrational levels shown in the postscript plot. These are the (010) and (001) vibrational modes in the standard notation.

With the water molecule the additional degrees of freedom caused by having three atoms in the molecule cause a significant increase in complexity compared to diatomic molecules. The molecule no longer has to be linear and there are additional rotational and vibrational degrees of freedom as a result. However it turns out that the water molecule has 2-fold rotational symmetry about the molecular axis of symmetry, and since the two H atoms are fermions and are in equivalent positions this leads to ortho- and para- forms of the molecule. Ideally the ortho to para ratio will be 3 to 1 but this is not likely to be always true in the non-equilibrium conditions of the interstellar medium or circumstellar envelopes.

In addition to the normal v and J quantum numbers for the levels there is another quantum number, K, or equivalently two quantum numbers K+ and K- which label the various levels. The K quantum number is related to the molecular angular momentum projected onto the axis of the primary moment of inertia of the molecule. For a given J there is not just one energy level with (2J+1) degeneracy but a set of levels with different K values from -J to +J each of which has (2J+1) degenerate levels. Everything becomes much more complex because the level energy depends upon K.

Then things become more complicated yet, calculationally, because there are maser transitions between some of the levels. In the numerical calculation of the level populations this tends to produce two distinct solutions for the level populations, and the simulation program oscillates between them. This can [usually] be corrected by using a more sophisticated method for finding the level population solution than is needed for the CO or oxygen molecules.

Kevin Volk, October 2001
last update 2006 January 17, 19:14 UTC by Steve Torchinsky. See changelog.