3.3.1 Rotation Diagrams
The rotation diagram approach can be used to determine the rotational temperature and column density of a species in the limit of local thermal equilibrium (LTE), optically thin emission, and negligible background radiation brightness. Under these simplest of possible conditions the integrated intensity of a transition, R−∞∞ Tbdv, is:
Z ∞
−∞Tbdv = hc3
8πkν2Agu NT
Q(Trot) e−Eu/kTrot (3.1)
whereν is the transition frequency,NT is the beam averaged total column density,Ais the transition EinsteinA-coefficient, gu is the upper state degeneracy, Q(Trot) is the partition
function,Trot is the molecular rotational temperature, andEu is the transition upper state energy [29]. A rotation diagram is then the plot of lnh(8πkν2)/(hc3Agu)R−∞∞ Tbdvi versus Eu, which gives a line with slope inversely proportional to Trot and with intercept equal to ln(NT/Q(Trot)).
The mechanisms for determining the integrated intensity (R−∞∞ Tbdv), the Einstein A- coefficients times the upper state degeneracy (Agu), and the molecular partition function (Q(Trot)) are outlined below.
3.3.2 Integrated Intensities
The integrated intensity of a transition can be calculated by:
Z ∞
−∞Tbdv = 1.064Tb∆v (3.2)
whereTbis the peak brightness temperature of the line and ∆vis its full width half maximum (FWHM). Tb can be approximated as TM B, the peak line intensity (TA∗) corrected to the main beam temperature scale by the relationship TM B = TA∗/η, where η is the aperture efficiency.
Beam dilution effects must be considered when the beam and source sizes are unequal.
In this case, Tb = BTM B, where the beam filling factor, B, can be calculated from the relationship between the source size,θs, and the beam size,θb, by:
B = θ2s
θ2s+θb2 (3.3)
For interferometric observations, TA∗ (in K) can be determined by:
TA∗ = 1.22×106Int
θAθBν2 (3.4)
whereθA andθB are the beam FWHMs, Intis the peak intensity in Jy/Beam, andν is in GHz.
3.3.3 Line Strengths
The line strengths in terms of the Einstein A-coefficients times the upper state degeneracy (Agu) for a given transition can be calculated from the information given in the .cat files associated with the JPL CALPGM program. These files are generated by SPCAT (see Appendix C), and such files are available for a wide variety of species in the submillimeter and microwave spectral line catalog available at http://spec.jpl.nasa.gov. The line strengths are given here as thelog of the intensity, I. For a given transition, then:
I = Xj
1
10logIj (3.5)
which accounts for multiple transitions contributing to the integrated intensity of the line, and the resultant intensity is in units of nm2MHz. The line strength, Agu, can then be calculated by the relationship:
Agu= 2.7964×10−16Iν2Q(T)
e−El/kT −e−Eu/kT (3.6)
when ν is in MHz and I is in nm2MHz [30].
3.3.4 Molecular Partition Functions
Neglecting centrifugal distortion, the partition function for an asymmetric rotor is given by:
Q(T) =
"3N−6 Y
i=0
à e−Ei/2kT 1−e−Ei/kT
!#vu ut π
ABC ÃkT
h
!3
(3.7)
where Q Ã
e−Ei/2kT 1−e−Ei/kT
!
is the total vibrational state partition function, the Ei’s are the
energies of the normal modes of vibration, vu ut π
ABC
Ã
kTh
!3
is the rotational partition function, andA,B, andC are the rotational constants. The rotational constants determined for the ground state and thenvibrational states populated at T are used such that the molecular partition function is approximated as:
Q(T)≈ Xn i=0
e−Ei/kT vu ut π
AiBiCi ÃkT
h
!3
(3.8)
Chapter 4
1,3-Dihydroxyacetone
4.1 Introduction
One of the sugars detected in carbonaceous chondrites is 1,3-dihydroxyacetone, or CO(CH2OH)2. Dihydroxyacetone is a monosaccharide commonly used as the active ingredient in sunless tanning products. Monosaccharides are polyhydroxylated aldehydes (or aldoses, whose chemical formula is H-[CHOH]k-CHO) and ketones (or ketoses, whose general structure is given by H-[CHOH]l-CO-[CHOH]m-H) with the general formula [C(H2O)]n, where n≥3. Dihydroxyacetone and glyceraldehyde (CH2OHCHOHCHO) are thus the simplest ketose and aldose monosaccharides, respectively. Dihydroxyacetone is used in numerous biological pathways, including the production of ATP, and is synthesized via glycolysis.
The 2C species glycolaldehyde is less stable than its structural isomers acetic acid (CH3COOH) and methyl formate (HCOOCH3), with methyl formate being the most stable.
Similar patterns are observed with the 3C analogs, with sugars being more energetic than esters and acids (hence their biological utility), while ketoses are more stable than aldoses.
The relative energies for the 2C and 3C structural isomers were calculated from Gaussian 98 MP2 6-311G++(d,p) geometry optimizations [31], and these results are discussed in
Appendix D.
The relative energies of the 3C sugars indicate that dihydroxyacetone would be more likely to survive under hot core conditions than glyceraldehyde. Indeed, in the laboratory glyceraldehyde is seen to isomerize to dihydroxyacetone [32]. Lovas et al. recently characterized the rotational spectrum of dihydroxyacetone to 40 GHz, fitting rotational and quartic centrifugal distortion constants to FTMW data [32]. We simultaneously began our study of dihydroxyacetone in a similar manner, using ab initio studies to guide initial FTMW experiments to 18 GHz. We then conducted millimeter and submillimeter direct absorption experiments to provide more accurate spectral predictions for observational searches.
The laboratory studies of glyceraldehyde and dihydroxyacetone provided the necessary basis for deep observational studies [32, 33]. The C2v symmetry of dihydroxyacetone leads to somewhat stronger millimeter-wave emission features compared to glyceraldehyde. A glyceraldehyde K-band search with the GBT to an RMS of∼5 mK revealed no transitions [17].
The ab initio studies of dihydroxyacetone are presented in Section 4.2, while the laboratory studies of dihydroxyacetone are presented in Section 4.3. The results of observational searches for dihydroxyacetone are given in Section 4.4.