Analyzing profiles

The CLASS user may analyse spectra by fitting profiles. The fitting commands are available from the FIT language. The minimization method is taken from the MINUIT system of CERN, modified and optimized for this purpose. Reliability proved to be good. Five types of profiles are presently available, and can be selected by the METHOD command:

• METHOD GAUSS This is the default type of profile. One may use up to five Gaussians, which might depend on each other as specified by a system of control codes associated with each variable. For each of these Gaussians, the primary parameters are: 1) Area, 2) Position, and 3) Width (FWHM). The current X unit (for the lower axis) is used. Code 0 means that the parameter is adjustable; 1 that it is fixed; 2 that the parameter (head of group) is adjustable and that another parameter, coded 3, is fixed with respect to it; 4 that the parameter is a fixed head of group.
• METHOD SHELL (see details below) Profiles are like those encountered in envelopes of stars. The primary parameters are Area, Position, Width and Horn to Center ratio. The aspect of the profile varies from parabola (as obtain in optically thick lines) for Horn/Center = -1 to flat-topped lines (unresolved optically thin lines) for Horn/Center = 0 and double peaked profiles (resolved optically thin lines) for Horn/Center > 0. The profile is symmetric. Presently only code 0 and 1 can be used, and up to 5 independent lines can be fitted in a single spectrum. The X unit must be frequency.
• METHOD NH3(1,1) or NH3(2,2) or NH3(3,3)
  Profiles taking into account hyperfine structure of ammonia with a Gaussian distribution of velocity are fitted. Primary variables are 1) The product (Main Group Opacity) times (Excitation Temperature minus Background Temperature) 2) Velocity 3) Line Width (FWHM) and 4) Main Group Opacity . Up to 3 independent lines can be fitted, and only codes 0 and 1 are allowed. The X unit must be Velocity.
• METHOD HFS FileName This method is similar to the previous one, but the HyperFine Structure parameters are read from a file instead of being known by CLASS . The first line of this file must contain the number of hyperfine components ($< 40$). The other lines must contain, for each component, the velocity offset and the relative intensity. The parameters are the same as for NH3 method.
• METHOD CONTINUUM This method is used for continuum drifts. It fits a Gaussian and a linear baseline in the drift. If beam-switching was used and the reference beam is along the drift direction, two dependent Gaussian are used to optimize signal to noise. The method does not require any user input.

METHOD SHELL in details. The fitted function is:

$$f(\nu) = \frac{\ensuremath{{\cal A}}}{\ensuremath{\Delta \nu}}\, \frac{1+4H\,[(\nu-\nu_0)/\ensuremath{\Delta \nu}]^2}{1+H/3}$$

where the fitted parameters are:

1. : the area under the profile (in K MHz),
2. $\nu_0$: the middle frequency (in MHz),
3. : the full width at zero level (in MHz),
4. $H$: the Horn/Center parameter (dimensionless, see below)

The central value is $f(\nu_0) = \frac{\ensuremath{{\cal A}}}{\ensuremath{\Delta \nu}}\frac{1}{1+H/3}$ whilst the value at the edge is $f(\nu_0+\ensuremath{\Delta \nu}/2) = \frac{\ensuremath{{\cal A}}}{\ensuremath{\Delta \nu}}\frac{1+H}{1+H/3}$. The edge-to-center intensity ratio value is thus dictated by the Horn/Center parameter $H$ according to

$$\frac{f(\ensuremath{\Delta \nu}/2)}{f(0)} = 1 + H$$

The center-to-edge frequency shift corresponds to an expanding velocity

$$v_{\rm exp} = c\,\frac{\ensuremath{\Delta \nu}/2}{\nu_0}$$

Figure  shows synthetic shell-like profiles, for which the area takes values $\ensuremath{{\cal A}}=5$ to 20 by 5 K MHz. The full width at zero level is $\ensuremath{\Delta \nu}=3.83$ MHz in all cases, which corresponds to $v_{\rm exp}=2.49$ km.s at 230.537 GHz. The Horn/Center parameter is $H=5$ (top) or $H=-1$ (bottom, for which the intensity at the edge is zero).

Figure: Four synthetic shell-like profiles, corresponding to increasing areas $\ensuremath{{\cal A}}=5$ to 20 by 5 K MHz, a of 3.83 MHz or $v_{\rm exp}=2.49$ km.s at 230.537 GHz. The results of fitted shell profiles are indicated at the top. The upper scale is km.s . Top: The Horn/Center parameter is $H=5$. Bottom: Same as above for $H=-1$.

The FIT commands are:

• LINES N defines the number of components and prompts for the initial values of the parameters for each component. This command has no effect for method CONTINUUM. Parameters are read in list directed format in the following order:

      Code, Intensity, Code, Position, Code, Width, [Code, Parameter 4]
  

  The code is an integer number between 0 and 4. Note that, though the program works on the area (or other quantities as for NH3 methods), you have to give the intensity, since this quantity is more intuitive than area. The use of the list directed format makes things easier when only one parameter has to be modified (cf Fortran norms). The number of lines N may be zero; in this case the program finds out reasonable starting values by itself.

  Values may be also entered graphically if the cursor is available. After entering LINES N, first point the cursor to one side of the line, strike one key, point the cursor the other side, strike another key. The program computes the moment of the spectrum between these boundaries and use it to set up starting values. Proceed like this for all components. One drawback of this way of entering values is that you cannot change the control codes. It should be used only for entirely independent and free lines.

• MINIMIZE activates minimization, then prints out the results after convergence. A Simplex method is first used to ensure convergence, then a Gradient method to refine the results, and compute the errors.
• ITERATE is similar to MINIMIZE, but starts from the previous minimization results. Only the Gradient method is used. Consequently, this command is only useful close to the minimum.
• VISUALIZE [N] [/PEN] plots the $N$th component obtained by fitting; if $N$ is not given, the sum of all components is plotted.
• RESIDUAL N subtracts the Nth component from the current spectrum, or the sum of all components is N is not given). In this process, the R spectrum is first copied into T, then the difference is done in R.
• DISPLAY Prints the results of fitting from the current spectrum, without recomputing it ...
• KEEP Saves the fit results in the input file, which must be opened also for output. KEEP is in fact a reduced version of UPDATE, and to be used with the same care as UPDATE.
• SET MASK ... Defines masks in the spectrum for the fit. This commands has the same syntax and behaviour as SET WINDOW. Masked regions will not be used for the fit.

Fit results are always saved by a WRITE command and made available through the corresponding variable section (see SET VARIABLE help).