Fitting the dispersion curve

Now you have several possibilities to perform your wavelength calibration. At first there are three different modes to identify the calibration lines ( (F)):

Identify:

You identify at least 2 arc lines in one slitlet with the command IDENTIFY/MOS. The command CALIBRATE/MOS then performs a first fit for the CCD row with the identified lines.

Linear:

You know the central wavelength and the mean linear dispersion of your grism. These values are used as first fit of the first selected CCD row. You have to correct the value of this central wavelength if you used the command OFFSET/MOS to determine the offsets of your slitlets:

Example:

Your reference slitlet has an offset of -100 relative to the center of the CCD in x-direction and you have a mean dispersion of 2Å/pixel and a central wavelength of 5500Å. This central wavelength will always lie at the x-position of the respective slitlet, which is in this case -100 pixels (i.e. -200Å) from the center of the CCD. This means that you have a wavelength of 5700Å at the center of the CCD within the reference slitlet. This wavelength should be used as wcenter, since the program assumes that :xoffset = 0 means that the slitlet is at the center of the CCD in x-direction.

Recall:

Method Linear is performed in the first slitlet. The dispersion coefficients of the this slitlet is recalled to calibrate the remaining slitlets. This identification is more stable in this case than for method Linear and the convergence of the fit is reached faster.

The first fit is used to identify as many lines as possible in the corresponding CCD row by comparing the fitted line positions to wavelength catalogue .tbl (hear). For the identified lines a polynomial fit of chosen order is performed (using Legendre or Chebyshev polynomials - the type being selected with the keyword (CHEBYSHEV)). The line identification criterion will identify a computed wavelength ($\lambda_c$) with a catalog wavelength ( $\lambda_{cat}$), if the residual

$$\delta \lambda = \vert\lambda_c - \lambda_{cat}\vert$$

is small compared to the distances to the next neighbours (in the arc spectrum as well as in the catalog):

$$\delta \lambda < min(\delta \lambda_c, \delta \lambda_{cat})*\alpha$$

where $\delta\lambda_{cat}$ ( $\delta\lambda_c)$ is the distance to the next neighbour in the catalog (arc spectrum) and $\alpha$ is the tolerance parameter (0 ...0.5) given by (0.2). The automatic line identification is repeated with this polynomial fit in order to identify additional lines to further improve the dispersion curve.

Note

For very low dispersion spektroscopy one would expect that a linear guess will cause line mismatchs at the edge of the detector. One can avoid this, if more then two lines are identified with method Identify.

After the polynomial fit the residual of each line are checked and the line is thrown out, if it the residual exceeds the tolerance parameter (2) (> 0 - in pixels; < 0 - in units of the wavelength). One of three fitting methods can be selected by keyword (C):

Constant fit in spatial direction:

the dispersion coeficiants are constant for the whole slitlet. This method is typically appropriate for small slits.

Variable fit in spatial direction:

bad lines are thrown out. Dispersion coeficiants are calculated for any row. The dispersion relation of the first fitted row is used as estimate for all following rows. A large number of arc-lines is required for this method. If there are only a few lines identified at the edge of the detector, small oscillations at the edge of the detector may occur in spatial direction.

Two dimensional fit over the slitlet:

A two-dimensional fit is performed in spatial and dispersion direction over the slitlet. In spatial direction a ``normal polynom'' is fitted but a polynom as specified in keyword in dispersion direction. The dispersion coeficiants may smoothly evolve over the slitlet. This method is the most accurate for most applications, although the resulting residuals are typically larger than for a variable fit.

The iteration is repeated until a stable solution is obtained (and the minimum number of iterations (3) is exceeded) or the maximum number of cycles ( (20)) is reached. The resulting dispersion coefficients are stored in table .tbl (coerbr), together with the r.m.s. error of the fit, the slitlet and the y-coordinates (world and pixel coordinates). Also a plot option for the resulting residuals ( , (N)) and various degrees of display ( , (0)) are available.

After fitting all rows of the respective slitlet with polynomials of the chosen order the program performs at last a linear fit to get the central wavelength and the mean linear dispersion necessary to derive a starting wavelength for the next slitlet from its known offset (modes Linear/Recall). In the mode Ident it tries to match the manually identified lines in the next slitlet, using the known offsets and a maximum allowed shifting tolerance stored in (10). Rows where no fit could be achieved are stored in the table .tbl with the slit number -1.

Any selection of slitlets made in the table .tbl will be taken into account, but all selections of the table .tbl will be ignored. If you want those to be respected, too, redo the search for the wavelength calibration lines with the chosen selection in .tbl.

After the wavelength calibration you may rebin your frame two-dimensionally to constant wavelength steps with REBIN/MOS. Point sources are normally wavelength calibrated after extraction (see below).