Visible Opacity of M Dwarfs and Hot Jupiters: The TiO B 3 Π − X 3 Δ Band System

The TiO B 3 Π − X 3 Δ electronic transition ( g ¢ system ) is an important opacity source in the atmospheres of M dwarfs and hot Jupiter exoplanets. The 0 – 0, 1 – 0, and 2 – 1 bands of the B 3 Π − X 3 Δ band system have been analyzed using a TiO emission spectrum recorded at the McMath-Pierce Solar Telescope, operated by the National Solar Observatory at Kitt Peak, Arizona. Improved spectroscopic and equilibrium constants were determined. Line strengths were calculated from an ab initio transition-dipole moment function scaled using an experimental lifetime. A new line list for ¢ = – v 0 2 and v ″ = 0 – 4 of the B 3 Π − X 3 Δ band system is provided. Uni ﬁ ed Astronomy Thesaurus concepts: Laboratory astrophysics ( 2004 ) ; Spectral line lists ( 2082 ) Supporting material: machine-readable tables


Introduction
The TiO molecule has a long and significant astronomical history.Its fluted spectral lines were noted by Fowler (1907) in the spectra of Antarian stars, and its spectrum has been used as part of the Morgan Keenan (MK) classification system (Morgan & Keenan 1973).For M dwarf stars, the most numerous star type in our galaxy, the spectra are marked with strong absorption features due to TiO (Bochanski et al. 2007).
The near-IR and visible electronic transition spectra of TiO have been explored in sunspots (Ram et al. 1999) and in embedded protostars (Hillenbrand et al. 2012).The pure rotational microwave transitions of TiO have been detected in red supergiants, supporting the concept that titanium oxides are formed in circumstellar envelopes and seed inorganic dust formation (Kamiński et al. 2013).Its importance continues to expand as we explore exoplanets, where it has been detected by its electronic spectra and shown to cause formation of stratospheres in hot Jupiters (Evans et al. 2016;Nugroho et al. 2017).
TiO is particularly important as a strong source of opacity in the visible and near-IR (McKemmish et al. 2017).Therefore, accurate TiO line lists are essential in successfully modeling the spectra of many astronomical objects.The current ExoMol line list for TiO (McKemmish et al. 2019) remains the most reliable overall.Instead of using a Hamiltonian approach based on spectroscopic constants as this research does, the ExoMol line list is generated from potential energy curves using the DUO program (Yurchenko et al. 2016) A good recent summary of the state of laboratory spectroscopy of TiO research is provided by McKemmish et al. (2017).Efforts taken by our group since that summary include a reanalysis of the TiO singlet transitions (Bittner & Bernath 2018), the C 3 Δ − X 3 Δ transition (Hodges & Bernath 2018), high-resolution absorption cross sections in the visible and near-IR (Bernath 2020a), and the E 3 Π − X 3 Δ transition in the near-IR (Bernath & Cameron 2020).
The B 3 Π − X 3 Δ transition (g¢ system) contains strong lines and is a dominant feature of late-type stars (Hocking et al. 1979).The B − X 0-0, 1-0, 0-1, and 1-1 bands were previously studied by emission spectroscopy, revealing significant lambda doubling in the B 3 Π state, and provided molecular constants for the v = 0 and v = 1 levels of the B 3 Π and X 3 Δ states (Hocking et al. 1979).The B − X 1-0 band was revisited and updated by the laser spectroscopy of a molecular beam (Amiot et al. 1995).Through Stark spectroscopy, the permanent electric dipole moments of the B 3 Π 0 and X 3 Δ 1 states along with the E 3 Π 0 and A 3 Φ 2 were measured (Steimle & Virgo 2003).The IR spectrum of 46−50 TiO was measured around 1000 cm −1 using a laser ablation source probed by IR radiation produced by quantum cascade lasers (Witsch et al. 2021).
The starting point for our B 3 Π − X 3 Δ analysis is the study of the A 3 Π − X 3 Δ system by Ram et al. (1999), which provided the equilibrium constants for the X 3 Π state, the spectroscopic constants for v = 0-4 of the X state, and the constants for v = 1 of the B state.The data were obtained from laboratory and sunspot spectra recorded using a Fourier transform spectrometer.

Method and Results
The TiO experimental cross sections (Bernath 2020a) used in this analysis are based on the same emission spectrum recorded at the McMath-Pierce Solar Telescope using the the 1 m Fourier transform spectrometer operated by the National Solar Observatory at Kitt Peak, Arizona that was used by Bernath & Cameron (2020) for their E 3 Π − X 3 Δ work.The source, a carbon tube furnace operating at about 2300 K, and the method of conversion of the emission spectrum to calibrated cross sections are described in detail by Bernath (2020a).The spectral resolution is about 0.05 cm −1 with wavenumber calibration accuracy ±0.002 cm −1 .Figure 1 shows the cross sections in red pointing upwards with the simulation pointing downwards showing the B 3 Π − X 3 Δ 0-0, 1-0, and 2-1 bands (green) and the A 3 Φ − X 3 Δ 2-0 band (blue).
The B 3 Π and X 3 Δ states obey Hund's case (a) coupling.The B − X transition has three spin components: The state labels of the spin components are for 3 Π: ).Each sub-band has P, Q, and R branches, and each branch has "e" and "f" parities due to lambda doubling.The R-branch bandheads for the 0-0, 1-0, and 2-1 bands of the The PGOPHER program (Western 2017) was used to perform the rotational analysis of the TiO B − X transition.The process started with B 3 Π equilibrium constants from Amiot et al. (2002), which were used to calculate case (a) v state constants for v = 0 through v = 4, using Ram et al. (1999) v = 1 constants as a benchmark.For the X 3 Δ constants, Ram et al. (1999) was again used.Ram et al. (1999) improved the existing ground state constants by combining sunspot and laboratory spectra with the pure rotational measurement of Steimle et al. (1990) and Namiki et al. (1998).The cross section file was used as an overlay in PGOPHER.Spectroscopic constants were updated as lines were fit up to J of at least 100.Attempts were made to fit lines in the 3-2 band, but the region of the spectrum was so congested the fit was not deemed reliable.In the rotational analysis, 5507 lines were fitted with an average error of 0.024 cm  49 Ti, and 50 Ti are 8.25%, 7.44%, 73.72%, 5.41%, and 5.18% respectively (Meija et 2016).The minor isotopologues are clearly present in the experimental spectrum from Kitt Peak but did not complicate the analysis of the most abundant isotopologue.Rotational analysis of the four minor isotopologues of TiO is a topic of our ongoing research.New spectroscopic constants resulting from Figure 1.TiO spectrum in red pointing upwards, simulation pointing downwards showing the 2-0 A 3 Φ − X 3 Δ band in blue and the 0-0, 1-0, and 2-1 B 3 Π − X 3 Δ bands in green.The B 3 Π − X 3 Δ 0-0 band is to the left, running from about 15800-16200 cm −1 ; the 1-0 and 2-1 bands are intermixed to the right, running from about 16200-17100 cm −1 .
2. TiO B 3 Π − X 3 Δ spectrum in red pointing upwards, simulation pointing downwards in green, showing the 0-0 R 1e bandhead at about 16233.2cm −1 and the 0-0 R 1f bandhead at about 16231.8cm −1 .New equilibrium constants are provided in Table 3.The equilibrium constants were derived by the exact fit of the PGOPHER generated spectroscopic constants.The number of decimal places shown in the equilibrium constants was determined by allowing a ±1 standard deviation in the value of the spectroscopic constants.The equilibrium constants were input into Le Roy's Rydberg-Klein-Rees (RKR) program (Le Roy 2017a) to generate the potential energy curves for the B and X states, which were then inserted into Le Roy's LEVEL program (Le Roy 2017b), along with the transition-dipole moment points, to generate transition-dipole moment matrix elements.The transition-dipole moment points for the B − X transition were obtained from McKemmish et al. (2019), who fit a curve to their own ab initio data as well as the ab initio data of Langhoff (1997).The functional form of the fit for the transition-dipole μ is in which c, a, and r m are fitting parameters; for the B − X transition, c = 5.013, a = 4.101 and r m = 1.667.Excel was used to calculate 317 transition-dipole moment points for r = 0.84-2.42Å.
The average radiative lifetime of the three spin states of the v = 0 B state was measured to be 65.4 ± 1.3 ns (Hedgecock et al. 1995).The Einstein A for the v = 0 B state can be calculated, with the calculated lifetime τ for the state being The Einstein A is linked to line strength through the following relationship: where n ˜is in cm −1 (Bernath 2020b).The line strength S is the square of the transition-dipole moment.The transition-dipole moment matrix elements were obtained from the LEVEL program.Comparing the calculated radiative lifetime of 122.3 ns to the measured radiative lifetime shows that a correction factor of 1.87 must be applied to the calculated Einstein A values.The calculated transition-dipole moment matrix elements obtained from the LEVEL program were corrected by applying a scaling factor of 1.87 to make this correction to the PGOPHER band strengths.Based on the experimental accuracy of 2.0% for the lifetime of v = 0 (Hedgecock et al. 1995), the minimum error for calculated Einstein A values is also about 2%.The scaled transition-dipole moment matrix elements are shown in Table 4.

Discussion
The spectroscopic constants (Table 2) extend the previously published ¢ = v 1 constants of Ram et al. (1999) to also include ¢ = v 0 and ¢ = v 2. A sample of the lines fitted and the observed-minus calculated values are given in Table 5.Table 5 uses the traditional spectroscopic line label: where N is the quantum number for the total angular momentum excluding electron spin, J is the quantum number for the total angular momentum, p is the e/f rotationless parity, and F i is the spin component label discussed in the previous section.In addition, the line assignment information in Table 5 uses the PGOPHER NameJNFnp format.Through this analysis, the equilibrium constants (Table 3) of the B 3 Π state (Amiot et al. 2002) have also been updated.McKemmish et al. (2019) noted that future improvements on the TiO line list should concentrate on wavelength regions such as 570-640 nm (15625-17544 cm −1 ), which is coincident with the B 3 Π − X 3 Δ 0-0, 1-0, and 2-1 bands that are the focus of this research.Using the X 3 Δ spectroscopic constants for v″ = 0 through v″ = 4 from Ram et al. (1999) and the new B 3 Π spectroscopic constants from Table 2, a new line list for the B 3 Π − X 3 Δ transition has been calculated; a sample of that line list is shown in Table 6.The same line and line assignment formats used in Table 5 are used in Table 6.
The transition-dipole moment matrix elements shown in Table 4 were used in PGOPHER as band strengths to obtain a more complete line list.In addition, the oscillator strength for each line, ¢¬  f J J , was calculated from the Einstein A values, which are provided for each line by PGOPHER, using the following: Note.J is total angular momentum; p is the e/f parity; Obs is the observed line position in cm −1 ; Calc is the calculated line position in cm −1 ; O-C is the observedminus-calculated line position in cm −1 ; Line is the spectroscopic line label; Line assignment contains additional information.
(This table is available in its entirety in machine-readable form.)Note.J is total angular momentum; Pos is the line position in vacuum cm −1 ; Eup and Elow are upper and lower energy levels in cm −1 ; A is Einstein A value in s −1 ; f is the oscillator strength; Line is the spectroscopic line label.
(This table is available in its entirety in machine-readable form.) . The line list contains the 46 Ti 16 O, 47 Ti 16 O, 48 Ti 16 O, 49 Ti 16 O, and 50 Ti 16 O isotopologues of TiO with 30 million transitions for 48 Ti 16 O.Laboratory spectroscopy is essential in the continuing effort to expand and improve the TiO line list.

Table 2 .
(Ram et al. 1999ote the H v value for v = 2 appears to be anomalous; thorough investigation of the constant through PGOPHER shows it to be stable with small relative standard deviation.Higher-order lambda-doubling centrifugal distortion terms were statistically determined for the B state due to large lambda doubling in the B − X transition.It is possible global perturbations exist, but no local perturbations were discovered.The standard N 2 Hamiltonian(Ram et al. 1999) was used for fitting.
the rotational analysis for v = 0 though v = 2 of the B state are provided in

Table 3
Equilibrium Constants for the B 3 Π State in cm −1

Table 4
Scaled Transition-dipole Moment Matrix Elements for the

Table 5
Sample of Fitted Lines for the B 3 Π − X 3 Δ Transition Based on PGOPHER Log File

Table 6
Sample Line List for the B 3 Π − X 3 Δ Transition of TiO