Evaluation of the Effective Temperature of Sunspots Using Molecular Parameters of AlF

b 3 Σ + − a 3 Π r , c3Σ − a3Π r , and f3Π − a3Π r band systems of AlF molecule. The intensity of various bands is discussed with the help of derived FC factors. The band degradation and the nature of potential energy curves are studied using r-centroid values. The vibrational temperature of sunspot is estimated to be around 1220± 130K which falls in the reported temperature range of cold sunspots.


Introduction
The existence of AlF molecule in various astrophysical sources has been confirmed by various researchers.For instance, Cernicharo [1] and Turner [2] identified the presence of AlF molecule in the envelope of brightest C-rich evolved object IRC + 10216.The AlF molecular species was also found in the gas or dust envelopes of asymptotic giant branch (AGB) stars [3].With reference to HF spectroscopy of the red giant stars, the presence of AlF in IRC + 10216 indicated that large quantities of fluorine were present in the inner stellar envelope and the element was produced in helium shell flashes and not in explosive nucleosynthesis [4].Turner [2] has predicted that the AlF molecule must have a significant presence in the region of thermochemical equilibrium occurring in the dense, hot, and innermost envelope of the stellar atmosphere.Sauval and Tatum [5] have reported that the AlF molecule may be present in the stellar and cometary spectra.
According to Joshi et al. [6], the AlF molecule is likely to be present in the sunspots umbral atmosphere.Wöhl [7] examined sunspots spectra towards the identification of various diatomic molecules and found 100 lines of AlF molecule.Bagare et al. [8] made an extensive search for AlF molecular lines in the spectra of sunspots and confirmed their presence.With the help of vibronic transition probability parameters such as Franck-Condon (FC) factors, -centroids, relative intensities, oscillator strength, and vibrational temperature of diatomic molecular species, the spectroscopic technique could be very useful in the identification of molecular lines and in the estimation of relative abundance of the species in astrophysical sources.A number of workers have therefore undertaken theoretical studies to provide those parameters for diatomic molecules which are of importance not only in astrophysics, but also in the fields of gas kinetics, combustion process, and so forth [9][10][11].
The literature on the reports of Franck-Condon factors and -centroids for the  3 Σ + −  3 Π  ,  3 Σ −  3 Π  and  3 Π −  3 Π  band systems of AlF molecule is not made.Murty [12] reported on a partial array of FC factors and -centroids for  1 Σ + −  the complete array of the FC factors and -centroids using experimental vibrational levels and vibrational temperature of the source using relative intensity of the bands.

Theory and Computational Procedure
2.1.Franck-Condon Factors and -Centroids.The intensity of a vibrational band within a band system of a diatomic molecule is controlled mainly by the population on the vibrational level from which the transition takes place and by the FC factor ( V  V  ) which is defined as the square modulus of the vibrational overlap integral that is [10] where V  and V  are the vibrational quantum numbers and  ]  and  ]  are the vibrational wave functions for the upper and lower states, respectively.The -centroids  ]  ]  are seen to be the weighted average with respect to  ]   ]  of the range of  values experienced by the molecule in both states of the ]  − ]  transition.The form of  ]  ]  can be expressed as [10] Using the molecular constants [13] mentioned in Table 1, the potential energy curves for the electronic states , , , ,  and  of AlF molecule have been constructed first using Morse [14] and Rydberg-Klein Rees (RKR) [15] functions.
The turning points of the potential energy curves are finally presented in the Tables 2-7, where the potential energy curves derived from Morse function coincide well with the RKR curves.The Morse potential can yield reliable FC factors and -centroids for the bands in an electronic transition involving low vibrational quantum numbers [16].The computation of FC factor is made using the Bates' method of numerical integration [16] and Ureña et al. 's detailed procedure [17].The Morse wave functions are calculated at the intervals of 0.01 Å for  ranging from 1.42 Å to 2.01 Å, from 1.42 Å to 2.01 Å, from 1.44 Å to 1.88 Å, and from

Variation of Electronic Transition Moment and Band
Strength.With the help of FC factors and -centroids, one can determine the band strength of the vibrational bands using the relation where  2  ( ]  ]  ) is the variation of electronic transition moment.Mathematically the intensity ( ]  ]  ) of a molecular band for an electronic transition in emission (V  − V  ) is written as [9] where  is a constant partly depending on the geometry of the apparatus and  ]  is the population of the level ]  and  ]  ]  the energy quantum.
In the present study, the intensities ( ]  ]  ) of (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (1, 3), (2, 1), and (3, 2) bands as reported by Naudé and Hugo [21] are used to evaluate the electronic transition moment variation with internuclear distance for the band system  −  of AlF.A plot of ( −4 /) 1/2 V  V  versus  ]  ]  yields the variation of   wtih  ]  ]  over a progression.To place all progression on the same ordinate, the rescaling procedure of Turner and Nicholls [22] was adopted.A plot of rescaled values of   = ( −4 /) 1/2 versus  ]  ]  is shown in Figure 1 for the  −  band system of AlF.A least square fit yields with standard deviation 0.63.The form of   () represented by ( 5) is adopted in conjunction with (3) to calculate the band strengths using the computed  V  V  values.The band strengths have been relatively scaled by assuming the value of most intense band (0, 0) as one.The relative band strengths are evaluated using the relation  V  V  =  V  V  / 00 .

Effective Vibrational
Temperature.The vibrational quanta (V  ) are calculated from Using the relative band strengths  V  V  =  V  V  / 00 , (4) becomes Since  V  =  0 exp[(−ℎ/)(V  )], (7) becomes [23] ln where ℎ is the Planks constant,  is the velocity of light,  is the Boltzmann's constant, and  is the effective vibrational temperature of the source.2 shows a linear dependence.By least square fitting, the slope is determined and vibrational temperature  is evaluated and discussed in the following section.

Results and Discussion
In the case of  −  band system the FC factors manifest that (0, 0), ( The -centroid value increases for the  − ,  −  and  −  band systems of AlF, since    <    with the decrease in wavelength which is expected in the violet degraded band system.For the  −  band system, the -centroids value increases with the increase in wavelength which is expected in the red degraded band system.The sequence differences are found to be constant nearly 0.01 Å for all the four band systems of AlF molecule.For − band system, the sequence difference is varying from 0.002 Å to 0.056 Å.This suggests that the potentials are not so wide.The -centroid value for the (0, 0) transition is slightly greater than (   +   )/2 for all the band systems of AlF molecule which implies that the potentials are not very anharmonic.
The vibrational temperature of the source of  −  band system is estimated as 1220 ± 130 K and is found in the temperature range of cold sunspots.To confirm the presence of AlF molecule in sunspots spectrum, a careful study of sunspot umbral spectra in the wavelength region of 4400-9000 Å was carried out to search for the presence of AlF molecular lines of different band systems [8].The presence of several transitions of AlF molecule in sunspot spectra was confirmed with a total of 602 rotational lines.The rotational temperature for the  −  band system was 1240 ± 120 K.     Thus it clears that the vibrational temperature evaluated in the present study coincides with the reported rotational temperature.

Conclusions
The present work evaluates the transition probability parameters FC factors and -centroids which are mainly influencing the intensity of vibrational bands.Using the derived transition probability parameters and reported wavelength of the bands, the vibrational temperature of a band system of AlF molecule is determined.Since the vibrational temperature of the AlF molecule is found to coincide well with the reported sunspot temperature, the present work acts as the additional support for the confirmation of AlF molecule in sunspots.
1.49 Å to 1.82 Å, for every observed vibrational level of each state of  − ,  − ,  − , and  −  of the AlF molecule.The FC factors ( ]  ]  ) and -centroids ( ]  ]  ) are computed numerically by integrating the integrals in (1) and (2) for the bands of  − ,  − ,  − , and  −  of the AlF molecule and the results are given in the Tables 8-11 with the available wavelengths ( ]  ]  ) [18-20] for all the band systems.

Figure 1 :
Figure 1: The variation of   with  for AlF ( − ) band system.

, and ( 5 , 8 )
bands are intense.In the case of the  −  band systems of the AlF molecule, the FC factors indicate that the Δ] = 0 sequence bands are more intense and all other bands are relatively weak.The FC factors of − and − band systems indicate that the Δ] = 0 sequence bands are significantly intense followed by the Δ] = ±1 sequence bands.

Table 2 :
Turning points for the molecular vibration in the -state of AlF.V (V) in cm −1 Morse RKR  max in Å  min in Å  max in Å  min in Å

Table 3 :
Turning points for the molecular vibration in the -state of AlF.V (V) in cm −1 Morse RKR  max in Å  min in Å  max in Å  min in Å

Table 4 :
Turning points for the molecular vibration in the -state of AlF.Morse RKR  max in Å  min in Å  max in Å  min in Å V (V) in cm−1

Table 5 :
Turning points for the molecular vibration in the -state of AlF.V (V) in cm −1Morse RKR  max in Å  min in Å  max in Å  min in Å

Table 6 :
Turning points for the molecular vibration in the -state of AlF.V (V) in cm −1 Morse RKR  max in Å  min in Å  max in Å  min in Å

Table 7 :
Turning points for the molecular vibration in the -state of AlF.Morse RKR  max in Å  min in Å  max in Å  min in Å V (V) in cm−1