4 Errors in velocity measurements

When scanning a spectral line, the intensity varies across the passband of the filtergraph for any given setting of $\lambda_{\rm c}$. Depending on the position of the center wavelength of the filtergraph, either the blue or the red side of the transmission profile receives more light. We have calulated the effect for a spectral absorption line with a Gaussian profile of width ${\Delta \lambda}_{\rm L} = 6.5$ pm and core intensity $I_{{\rm core}} = 0.3$,

$$S(\lambda) = I_{{\rm core}} + \left( 1 - I_{{\rm core}} \righ... ...da - \lambda_0)^2 }{2 {\Delta \lambda}_{\rm L}^2 } \right) \ ,$$ (9)

where the filter was tuned to three positions $\lambda_{\rm c} = \lambda_0$, $\lambda_0 - {\Delta \lambda}_{\rm L}$, and $\lambda_0 + {\Delta \lambda}_{\rm L}$. The corresponding point spread functions are shown for both magnifications in the right column of Fig. 6.

There are evident differences for the low resolution mode. The line core PSF $I_{\rm C}(\rho)$ resembles the PSF for a flat spectrum. The PSF $I_{\rm B}(\rho)$ at the blue line wing has a relatively sharp core with extended wings while the red line wing PSF $I_{\rm R}(\rho)$ is broader in the center with less intense wings. The high resolution mode PSFs are essentially the same for all positions in the spectral line.

Pupil apodisation would not do any harm as long as the observed source was featureless. Spurious spectral features could be produced when there are fluctuations of intensity, and could be misinterpreted. In particular, structure-dependent displacements of a Fraunhofer line of a medium which is at rest relative to the observer could be interpreted as Doppler shifts. We studied the magnitude of such an error by including a sinusodial intensity variation whose period and contrast was varied into the source model. The intrinsic velocity of the modelled structure was zero. The spectra were analysed for Doppler shifts, any deviation from zero Doppler velocity was interpreted as a velocity error. It turns out that spurious Doppler signals also show sinusoidal modulation of the same period as the intensity modulation, but with a period-dependent phase relative to the intensity modulation.

Figure 9 shows the amplitude and phase variation of the velocity error with the period of intensity modulation for an rms contrast of 15%, corresponding to a high contrast structure such as penumbral filaments. The dotted lines represent the phase between intensity fluctuation and velocity error. The velocity error in the low resolution mode approaches 25 m/s at scales between 1 and 2 arcsec. Brighter regions appear blueshifted at these scales, which corresponds to a phase shift of $180^\circ$ between intensity and velocity error. A secondary maximum of 18 m/s at 0.25 arcsec shows the velocity error in phase with the intensity modulation, so that brighter regions appear now redshifted. The velocity error of the high resolution mode is less than 2 m/s for scales above 0.7 arcsec where bright regions experience blueshifts. There is a maximum of just above 6 m/s in the presence of a spatial modulation with a scale of 0.3 arcsec, or 250 km on the solar surface, where bright regions experience redshifts.

Figure 9: Velocity error as a function of intensity modulation period for the low resolution (left) and high resolution (right) modes. The solid line represents the velocity error (scale to the left) and the dotted line the phase of the velocity error relative to the intensity modulation (scales to the right). The rms intensity contrast is 15% at all scales

Figure 10 shows the amplitude of the velocity error as a function of sinusodial modulation period and an rms contrast between 0 and 15%. Velocity errors of typical granular structures with an rms contrast of some 10% and scales between 1 and 2 arcsec are of the order of 15 m/s in the low resolution mode while they amount to less than 1 m/s in the high resolution mode. Scales at the diffraction limit with contrasts of 10% suffer from velocity errors of less than 5 m/s.