The LWS diffraction grating was mounted in a scanning mechanism which rotated through $\pm 7^{\circ}$ , allowing to cover the extended range of wavelengths of each detector. The wavelength corresponding to each scanning position was determined by the grating angle $\theta_i$ (angle between the input aperture direction and the normal to the grating) and the detector angle $\theta_d$ (angle between the input aperture direction and the detector direction), which was a constant for a given detector.

where

is the groove density of the grating (7.9 lines per mm) and

is the order: the grating was used in first order for the wavelength range 84-197 $\mu$ m with the five long wavelength detectors LW1 to LW5, and in second order for the wavelength range 43-93 $\mu$ m with the five short wavelength detectors SW1 to SW5.

In operations, the grating position was actually monitored via the engineering unit called LVDT (linear variable differential transformer). Therefore, once the ten detector angles were known, the wavelength calibration consisted in finding the relationship between the engineering units LVDT and the actual grating angle $\theta_i$ .

This was done by fitting a third order polynomial to a large database consisting of the measurements of emission line centroids in terms of LVDT units associated with the expected wavelengths of the lines for a number of calibration sources observed throughout the ISO mission.

5.10.2 Calibration sources and types of observations

**Table 5.12:** *Lines from astronomical sources used for grating wavelength calibration.*
line id.	det.	LVDT	# obs	sources
$\lambda\, [\mu m]$		mean( $\sigma$ )
[O III]51.815	SW1	1327(3)	171	NGC 6543 NGC 6826 G298.228 IRAS 15408
				NGC 7027
[O III]51.815	SW2	2783(3)	196	NGC 6543 NGC 6826 G298.228 IRAS 15408
				NGC 7027 NGC 6302
[N III]57.330	SW2	1993(2)	161	NGC 6543 NGC 6826 G298.228 IRAS 15408
				NGC 6302
[N III]57.330	SW3	3376(5)	152	G298.228 IRAS 15408 NGC 6302
[O I]63.184	SW2	1124(3)	88	G298.228 IRAS 15408 NGC 7027 NGC 6302
				NGC 7023 IRAS 23133
[O I]63.184	SW3	2584(3)	97	G298.228 IRAS 15408 NGC 7027 NGC 6302
				NGC 7023 IRAS 23133
[O III]88.356	SW5	1579(3)	185	NGC 6543 NGC 6826 G298.228 IRAS 15408
				NGC 7027 NGC 6302
[O III]88.356	LW1	3142(4)	189	NGC 6543 NGC 6826 G298.228 IRAS 15408
				NGC 7027, NGC 6302
[N II]121.889	LW2	2176(2)	7	NGC 6302
[O I]145.525	LW3	1878(4)	76	G298.228 IRAS 15408 NGC 7027 NGC 7023
				IRAS 23133
[O I]145.525	LW4	3250(5)	80	G298.228 IRAS 15408 NGC 7027 NGC 6302
				NGC 7023 IRAS 23133
[C II]157.741	LW3	945(4)	90	G298.228 IRAS 15408 NGC 7027 NGC 6302
				NGC 7023 IRAS 23133
[C II]157.741	LW4	2374(3)	91	G298.228 IRAS 15408 NGC 7027 NGC 6302
				NGC 7023 IRAS 23133

The wavelength standards are mainly planetary nebulae and HII regions. They were chosen so as to provide the largest possible sample of lines and so that several of them were visible from ISO as much as possible during the mission (see Table 5.12 and Figure 5.22). The lines used had to be strong enough to give good signal-to-noise and to be unresolved by the grating.

**Figure 5.22:** Main LWS grating wavelength standards observed during the mission. The gap between revolutions 378 and 442 corresponds to the period when LWS was not used because of a problem with the FP interchange wheel.
$\rotatebox {90}{\resizebox{!}{11cm}{ \includegraphics[60,60][560,700]{source_rev.ps}}}$

The observations were performed weekly with end-to-end grating scans and provided measurements of seven different emission lines, spread between $51\, \mu$ m and $158\, \mu$ m, appearing on two detectors each.

Note that there were no measurements for SW4 and LW5, as no strong lines were found in their wavelength range. However, the relationship is in principle independent of the detector and all measurements of all lines were used together.

5.10.3 Detector angles

The wavelength calibration was first derived by adopting the detector angles measured before launch. Then the plot of the residuals (normalised differences between the expected wavelengths and the wavelengths derived from the LVDT with the polynomial relationship) showed systematic offsets for some detectors, suggesting that some of the detector angles had changed after launch. Therefore their values have been slightly adjusted until minimising the residual offsets for all detectors. The new angles used from OLP Version 6.0 onwards are listed in Table 5.13 together with the corresponding shifts relative to the pre-launch angles. The angle shifts for detectors SW4 and LW5 were adopted from the neighbouring detectors.

**Table 5.13:** Detector angles adopted for wavelength calibration. The second line lists the shift of the new angle respective to the angle measured on the ground. Both sets of numbers are in degrees.
	SW1	SW2	SW3	SW4	SW5	LW1	LW2	LW3	LW4	LW5
angle	67.80	58.74	49.71	40.73	31.72	63.26	54.29	45.27	36.275	27.32
shift	0.10	0.01	0.00	0.00	+0.02	+0.02	+0.01	+0.01	+0.04	+0.04

5.10.4 Time dependence

The stability of the system was checked by monitoring the measured LVDT at the line centres in the weekly observations. It is found to be remarkably stable for measurements performed close to the rest (central) position of the grating (LVDT $\sim$ 2100). But elsewhere, a little jump happened in revolution 346. The jump was bigger the farther away the grating was from its rest position, and the jump had opposite signs for opposite angles (see Figure 5.23). After the jump, only a very slow drift was observed in the LVDT measurements. This jump implied that the relationship between grating angle and LVDT reading had changed on revolution 346 for an unknown reason and it was decided to derive a time-dependent wavelength calibration, which considers two distinct periods, i.e. pre- and post-revolution 346.

**Figure 5.23:** Deviation of the measured line centre (in engineering units LVDT) from the average value as a function of time, for two lines recorded at opposite grating directions relative to the rest position. 40 LVDT units correspond to 1 resolution element of the grating.
$\rotatebox {90}{\resizebox{11cm}{!}{\includegraphics{famous.ps}}}$

**Table 5.14:** *Grating wavelength coefficients.*
revs	0th order	1st order	2nd order	3rd order
1-345	69.624422	5.16459527 10 $^{-3}$	5.02618935 10 $^{-7}$	8.20047303 10 $^{-11}$
346-875	69.554848	5.13190430 10 $^{-3}$	5.02794834 10 $^{-7}$	8.18631699 10 $^{-11}$

5.10.5 Assessment of the achieved wavelength accuracy

The accuracy of the grating wavelength calibration has been checked by measuring the central wavelengths of the lines observed in a large number (65) of Auto-Analysis results from observations of NGC 7027, NGC 6543, S106 and W Hya. This check has shown that in an individual observation the wavelength calibration is measured with an accuracy better than 1/4 of a resolution element (i.e. 0.07 $\mu$ m for SW detectors and $0.15\,\mu$ m or LW detectors). Only in one case the errors were slightly higher for an observation performed in a revolution just preceding the jump, when the noise on the LVDT reading was the highest, but in most of the cases the wavelength determination was better than 0.1 resolution elements.

It has to be mentioned that, because of the effect illustrated in Figure 5.23, the wavelength accuracy is higher near the centre of a detector. Therefore a slight wavelength error can be observed for a line detected at a detector edge. In this case, the measurement of the line should be performed on the adjacent detector, where it is likely to fall more near the centre.

**Table 5.15:** *Wavelength calibration accuracy for an LWS grating spectrum. These accuracies are based on actual measurements.*
Mode	Accuracy
grating	$\sim$ 25% of a resol. element
	0.07 $\mu$ m for SW detectors
	0.15 $\mu$ m for LW detectors

5.10 Grating Wavelength Calibration

5.10.1 Basic principles and calibration strategy

5.10.2 Calibration sources and types of observations

5.10.3 Detector angles

5.10.4 Time dependence

5.10.5 Assessment of the achieved wavelength accuracy