Flat Fielding

In this section, we investigate three methods of determining the flat field. We want to eliminate the intrusion of spurious low spatial frequency effects that would be injected into the 2D spectra when dividing by the flat field. These low spatial frequency effects can cause changes in the shapes of the observed Balmer line profiles that lead to the determination of incorrect atmospheric parameters. We show this by comparing what we think are inferior methods to what we think works best, in part to quantify the importance of determining a good flat field. The three methods we consider are:

Method A: We fit a 4th_{-order polynomial to the median of 10 rows in the center of the}

frame. We divided each row by this polynomial to get our final flat.

Method B: We fit a 3rd_{-order polynomial to the median of 10 rows in the center of the}

Systematic Error Teff Error logg Units

(K) (dex)

Flat Fielding 200 0.1 1% change in flat field

Extinction 30 0.001

Flux Calibration 18 0.001

Relative flux calibration 20 0.005 1000 K change

with model in model

Pseudogaussian Fitting 51 0.007

Normalization Wavelengths 100 0.02 15 percent change

from Liebert et al. (2005)

Model Resolution 20 0.005 10 percent change

from true resolution

Night-to-Night 51 0.012

Variation with Choice Varies Varies see subsection 4.5.9 of Spectral Line

Variation with Choice Varies Varies see subsection 4.5.10 of Line Combinations

Table 4.1: A summarization of the error incurred by each systematic investigated. Flat fielding, pseudogaussian fitting, and choices of spectral line or line combinations lead to offsets. The other errors given are 1 sigma errors determined by averaging the standard deviation in atmospheric parameters found for each systematic.

smoothed the flat with a 2D boxcar of side 200 to remove low frequency structure. We then divided the flat divided by the polynomial by the boxcar smoothed flat to get our final flat.

Method C: This method is the one applied to our data as described in section 4.3. In Figure 4.4, we show a median of 10 rows from each flat showing the structure in the final flat field for each method. The vertical line at 3785 ˚A marks the blue edge of H 10, the bluest portion of the spectrum used in the determination of Teff and logg. Ignoring the

bluest portions of the spectrum, Method A leaves structure of order 150 pixels and amplitude 2 percent. Method B leaves structure of order 100 pixels and amplitude 1.5 percent. And Method C leave structure of order 50 pixels and amplitude 1 percent.

Comparing Method A to Method B, we find very little change in the results, considering both individual lines and combinations of lines. The changes in Teff are less than 30 K and

changes in logg are less than 0.03 dex.

Figure 4.4: Final flat field from each method discussed in the text. Each method shows the median of 10 rows from the final flat, taken in the region where the star spectrum typically falls. The methods have been offset by 0.1 for clarity and each red dashed line shows the 1.00 level for each flat. An approximate wavelength scale in shown on the top axis. The Littrow ghost is responsible for the sharp feature around pixel 1225. The vertical line at 3785 ˚A marks the bluest wavelength used in the determination of Teff and logg.

C has H β effective temperatures 100-200 K hotter and logg values 0.10-0.12 less massive than Method B. H γ can change by up to about 30 K and less than 0.01 dex in logg. The higher order lines through H 10 change insignificantly.

Finally, comparing Method A to Method C, we find similar results. Method C has H β

effective temperatures 100-200 K hotter and logg values 0.06 - 0.1 less massive than Method A. H γ can change by up to 30 K and less than 0.01 dex in logg. The higher order lines through H 10 similarly change insignificantly.

Inspecting the flat fields shown in Figure 4.4 makes it clear why H β is changing so much. The normalized H β line covers the wavelength range 4721 ˚A to 5001 ˚A. In Method A and B, there is significant structure in the flat that is imprinting itself onto the line. This structure is not nearly as prominent in Method C. We do not believe this structure to be real variations in the sensitivity of the CCD; we think that Method A and B do not completely

take out structure present in the illuminating light. For this reason, we adopted Method C as our approach for flat fielding for the entire sample. Additionally, this approach also helps minimize the night-to-night variations in different objects. Some of the higher order lines are not affected as much because they cover relatively fewer pixels on the CCD.

As a word of warning to future attempts at deriving atmospheric parameters from spectra of white dwarfs, this shows that the flat field must be accurate to variations less than 1 per cent on length scales similar to line widths. Otherwise, spurious low spatial frequency structure can alter lines by up to 200 K and 0.1 dex.