BLACKBODY FIT

Simple Usage Summary:

This routine operates on data (e.g., LWS scans, SWS detectors) that have been averaged with the AVERAGE routine.
Within the Blackbody Fit routine, mask out any bad data, spectral lines, etc. with right mouse using Mask Selected Region.
Select the Number of Components to fit (top left; one component is a good place to start).
Select an emissivity law tau(lambda, alpha, tau_0.55) (bottom left; 1/lambda^alpha is a good place to start).
Press Make Initial Guess. The software makes an intelligent first guess.
Press Fit Data...
Look at final values and the report to see the results. The lower plots show the successive iterations.
The routine gives [(1/N) sum (y_data - y_model) ^ 2], (the variance of the residual) as the measure of the goodness of fit. (The routine currently calls this value the Chi-square. It is the Chi-square in the sense of unity weighting, rather than 1/sigma^2 weighting
Save to Disk (the fit report), and Exit.

Summary:

The purpose of this routine is to enable the user to fit one, two, or three summed blackbody components to a data set. The components share a common emissivity law selected by the user, and may, as an option, be constrained to use of the same size emitting region, as given by the solid angle input value, Omega. Each component may have as free parameters independent temperatures and Omegas. The user may fix any of the parameters or may float the values, and have the blackbody routine return the best fit values after an iterative search. The values of the parameters are plotted as a function of iteration number up to the point where convergence is reached (this is when the chi^2 changes by less than a percent over an iteration.) and the final values are returned. A chi^2 value for the resulting fit is reported.

The program will return with the smooth fit to the data as a new AAR. This function will be sampled with the same lambda values as the original data set. The resulting fit function may be extended beyond the data lambda range as well. The sampling of the extended data is specified by a delta lambda parameter by the user. This delta lambda parameter may also be used to fill-in gaps in the data with fit values. The user may mask out portions of the data to be ignored by the fit. The user may plot the data and fit using any of several unit systems. The fit results are reported in a file that may be written to disk. The blackbody fit is returned in one data set and the data together with the fit are returned in a second data set.

Masking Data:

Use the left mouse button to zoom in on non thermal features in your data (such as spectral lines or bad detectors), and the right mouse button to select features to mask out. Then press the Mask Selected Regions button. Masked data may be made visible and plotted with a unique symbol using the Plot Style for Original Data options.

Setting up the Blackbody Function:

In the top left of the window select the number of blackbody components that should be summed (as per the formula in the top left). Usually one or two components are all that can be indicated by the data:

Flux Density = {1-exp{-tau} summation_i { Omega_i B(lambda, T_i) }

where there are i components. tau is the emissivity function as described below, Omega is the solid angle in steradians of the emissive region, T is the temperature of the region in Kelvin, and B is the Plank function:

                2 h c^2 lambda ^-5
B =   --------------------
         ( exp { h c / [k lambda T] } - 1 )

               1.1910 10^4 lambda ^-5
B =   -----------------------                      [W/cm^2/um/sr]
            (exp {14387.7 / [ lambda T ] - 1 )

where lambda is in microns.

Constraining Omega:

If more than one summed component is requested, and Omega is left as a free parameter, then the user has the option of using the Constrain Omega Same for each Blackbody button. In this case, the routine will fit a single Omega value that will be used by the different blackbody components. The idea here is that dust emission is characterized by two or three temperatures but is arising from the same region.

Selecting the Emissivity Function, tau(lambda):

Note in the discussion below, tau is a function of lambda and other parameters. tau_0.55 is one of those parameters: it is the opacity at 0.55 um, and it is a parameter for which the user in general would like to fit. One of three emissivity laws may be selected (lower left):

lambda^(-alpha function). More exactly, tau = tau_0.55 x (0.55/lambda)^alpha. Here the function has the value tau_0.55 at 0.55 um (5500 Angstroms) and the program will fit for a leading coefficient: tau_0.55 as well as the exponent alpha. Note that alpha for warm dust is typically thought to reside in the range 0 - 3.

standard function, which is anFIR emissivity function produced from the set of references:

Draine & Lee 1984 ApJ, 287, 89 (for lambda < 18 um)
Houck et al. 1984 ApJ, 287, L11, Herter et al. 1981, ApJ 250 186 (for 18 um < lambda < 33 um)
Duffy et al., 1987 ApJ, 315, 68 ( 1/lambda: 34 um < lambda < 57 um)
Klein, Wielebinski & Morsi 1988 (1/lambda^1.5) ( lambda > 57 um)

These functions have been normalized to be continuous and to have a value of of 1.0 at 0.55 um (5500 Angstroms), again, multiplied by the leading coefficient tau_0.55. The blackbody routine interpolates this function to produce emissivities at all wavelengths and fits for the leading coefficient, tau_0.55. The standard function may be found in the file: .../isap/data/emissivity/standard_law.dat

User supplied function (make your own!): The user may supply any function, following the same format of the "standard_law.dat" file above. The routine will interpolate and will fit for a leading coefficient, tau_0.55.

The emissivity function (with the tau_0.55 set to 1.0) can be viewed with the Plot Emissivity button.

Fixed, Free, and Guessed Parameters:

The user may fix any parameter listed to a specified value and these will be taken as fixed constants by the fit. The user may even fix all of the parameters so as to simply see what a specific model looks like plotted against the data. This is recommended! Before the fit starts, there must be initial guesses entered for all free parameters. Pressing Make Initial Guess accomplishes this automatically using the following algorithm:

The first blackbody component temperature is guessed by assuming the highest flux present is the Wein Peak, and the Wein Law: T = 2898 / lambda_max. (Thus, if there is no "turn-over" in the data, the routine will have trouble making a good guess.)
If tau_0.55 and alpha are free, they are each guessed to be 1.0
Omega for the first component is then determined by forcing the component to pass through the single data point: (lambda_max, flux_max).
This guessed blackbody component is subtracted from the data.
The steps above are repeated to get second and third guesses, if requested.

Alternative, one can make one's own guesses in the Guess/Fit text areas.

Fit Data:

Pressing the Fit Data button will run a modified form of the iterative IDL CURVEFIT routine (actually, an IPAC modified version) to minimize the reduced chi^2 by following its gradients. Fit convergence is achieved when chi^2 changes by less than 1% for two consecutive iterations. It fails when no convergence occurs in 200 iterations or when parameter constraints are violated: T < 0, Omega < 0, or tau_0.55 < 0. If the Wein Peak (turn-over) is not present in the data set, the user should consider that the data set might be equally well fit by a variety of (T, Omega) values. A good check for the uniqueness of a fit, is to alter and fix a parameter, and refit the data, and see how the goodness of fit measurement changes.

The fit will take from seconds to minutes. On a Sparc Ultra, fitting 4000 points with 2 components can take a few minutes.

One sigma uncertainties for the fitted parameters are reported based on the diagonal values of the covariant matrix of the fit, multiplied by the rms of the data.

Omega and tau dependence:

When tau << 1 in the wavelength range of the user's data, then the expression (1-exp{-tau}) approximates to tau. In the case where only one blackbody component is being fitted or the case where Omega is constrained identical for all components. the free parameters tau and Omega are degenerate, and the product (tau x Omega) is a single free parameter that has meaning in terms of a fit result. In such cases the user is informed of this with a pop-up window Warning: tau and Omega are not independent. This happens frequently and should be taken seriously.

T and alpha dependence:

When the user chooses an emissivity law 1/lambda^alpha, the user should beware that the effect choice

of alpha (by the user or by the fit routine) may yield a different T result. The reason for this is that the

optimum T is dependent on the "turnover" wavelength, which is directly influenced by the choice of alpha.

Extended Range:

By pressing the Extend Plot Range button, the user can extend the plot to other wavelength regimes. This is especially useful for predicting the spectral energy distribution implied by the data for filters outside of the data range (e.g. using the PHOTOMETRY routine in ISAP). This action will also fill-in the fit results into gaps in the data that are larger than the Lambda Increment which the user has specified.

Fit Report:

The Blackbody fit results are stored in an ascii file that can be written to disk at any time using the Save to Disk button. The results build-up in this file during a GUI section (i.e. the internal file is not lost when exiting the blackbody fit routine, only when exiting the ISAP GUI itself). The user can exit the routine saving (as stored data sets) the fit, or alternatively saving two files, the fit and the data with the fit extension combined. These files will be named dataset-BBfit and dataset-BBext.

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Last update: 20 Nov 1998