5. Results and discussion
The continuum opacity and the temperature-pressure profile of the atmosphere are set by matching the synthetic continuum spectrum with the observed spectrum for the entire wavelength region. The modeling of individual lines needs the values of additional parameters. In the absence of observational data for individual lines, we set the value of these parameters by matching the calculated flux in the continuum with the observed continuum flux at 2.3 . Since the line is very weak in intensity, we choose the profile function as (Mihalas 1978)
where is the thermal Doppler width and is the line center. It should be worth mentioning that individual molecular lines are usually not saturated enough so that pressure broadening is less important for them as compared to strong atomic lines. Moreover, molecular lines often overlap so strongly that their wings are completely masked (Schweitzer et al. 1996) and only the Gaussian line cores of the strongest molecular transitions are observed. The atmosphere of a brown dwarf is therefore only weakly sensitive to the Van der Walls damping constant. Nothing is known, at present, about the rotation of the brown dwarf Gl 229B around its own axis of rotation. If the projected velocity `' of the object is greater than 2 to 5 then rotational broadening could be significant. However, in the present work we have neglected rotational broadening in order to make the results consistent with the calculation of the evolutionary sequences by Saumon et al. (1996) that constrains the surface gravity and the effective temperature of the object. The whole purpose of the present work is to show that with different values of the surface gravity and the metallicity, the flux at the line core is significantly different although it is the same in the continuum. Since rotational broadening would affect the spectrum equally for both the models, it is not important in the context of the present work. We assume complete frequency redistribution and use Rayleigh phase function for the angular redistribution.
The parameters that are to be set in order to model the line at 2.3 are , f, , , and the degeneracy factor . We define guided by equation (6) and set the values of s and such that the calculated value of the flux at the continuum matches with the observed continuum flux. We assume that is independent of the geometrical depth. This is valid if the line formation is confined to a narrow region in the atmosphere. After testing several empirical laws we adopt the usual inverse square law with respect to the geometrical depth for the variation of the number density of C and . This should be verified when observational data becomes available. Experimental determination of the oscillator strengths for different transitions at 2.3 and at the relevant temperature could further constrain the number density of C and and hence the abundance of methane in the atmosphere of Gl 229B provided chemical equilibrium exists between the molecule and the atoms. We have solved the radiative line transfer equations by using discrete space theory (Peraiah 1980). The theoretical models for the methane line at 2.3 are presented in Fig. 2. For the model (a) we find that the flux at the continuum matches with its observed value when and where and are the number densities of C and respectively at the bottom of the atmosphere. The moderately high value of is consistent with the temperature at the lower atmosphere where the lines are formed. For the model (b) the flux at the continuum matches with the observed flux when and . Since the temperature of the lower atmosphere for the two models does not differ much, the value of remains the same for both the models. However s differs substantially for the two models. This is because of the fact that for the model (a) the continuum opacity is less and therefore one has to reduce the line opacity in order to keep the right ratio () between them so that the calculated flux matches with the observed flux at the continuum. However, the higher value of s in the model (b) makes the line opacity higher than that for the model (a) and so substantial decrease in the calculated flux at the line core is obtained. The difference in the flux reduces as we go towards the wings. It is worth mentioning that the wings could be masked by other lines whereas the continuum is overlapped by several lines.
Fig. 2 shows that a spectral resolution as high as 200,000 at is needed in order to investigate the individual molecular lines. This may be possible with an appropriate combination of the telescope and the instrument. For example, the Cooled Grating Spectrometer 4 (CGS4) available in UKIRT (United Kingdome Infra-Red Telescope) has a spectral resolution upto 40,000. If observation of Gl 229B is possible at present by UKIRT with the maximum resolution power of CGS4 then keeping the signal to noise ratio (which is proportional to the diameter of the telescope and to the square root of the integration time of exposure) unaltered, a 10 m telescope (such as Keck I) can obtain the desired resolution by using a similar type of spectrometer provided the resolution of the instrument is increased by about five times and the integration time of exposure is increased by about 2.5 times.
It is found that the numerical values of and s are very much sensitive to the emergent flux. The difference in the flux at the line core is clearly due to the different values of the surface gravity and the metallicity. Theoretical modeling of the continuum flux provides a few possible combinations of the metallicity, surface gravity and the effective temperature that are appropriate in explaining the observed continuum flux. The observational fit of the flux at the line core would decide which one of these combinations should describe the physical properties of the atmosphere. The physical parameters that are needed to model the individual molecular line will also be fixed once observational data is available. Therefore, in conclusion we would like to emphasize that a theoretical fitting of the observed flux for the individual lines of any of the dominant molecules, especially methane, would not only provide a much better understanding on the abundance of that molecule and the temperature of the lower atmosphere but also improve the constraints on the value of the surface gravity of brown dwarfs.
© European Southern Observatory (ESO) 2000
Online publication: June 8, 2000