1. Practical applications: CCD spectroscopy Tracing path of 2-d spectrum across detector –Measuring position of spectrum on detector –Fitting a polynomial to measured spectrum positions Optimal extraction of spectra from CCD images with simultaneous sky background subtraction: –Scaling a profile + constant background Wavelength calibration of 1-d spectra –Measurement of positions of arc-line images –Fitting a polynomial to measured positions of images of arc lines with known wavelengths

2. Observing hints Rotate detector so that arc lines are parallel to columns: To minimise slit losses due to differential refraction, rotate slit to “parallactic angle” –i.e. keep it vertical: Spectra are then tilted or curved due to –camera distortions –Differential refraction To zenith x x Night sky lines Target spectrum (Reference spectrum)

3. After bias subtraction and flat fielding Recall Lecture 3 for subtraction of B(x, ), construction of flat field F(x, ) and measurement of gain factor G. Corrected image values are

4. Tracing the spectrum Spectra may be tilted, curved or S-distorted. Trace spectrum via a sequence of operations: –divide into -blocks –measure centroid of spectrum in each block (fit gaussian) –fit polynomial in to calibrate x 0 ( ). Once this is done, use x 0 ( ) to select object/sky regions on subsequent steps. x0x0 x x0x0 x

5. Sky subtraction Alignment (rotation) of CCD detector relative to grating aims to make ~const along columns. Imperfect alignment gives slow change in along columns. This causes gradient, curvature of sky background when is close to a night- sky line. Solution: fit low-order polynomial in x to sky background data. Alternative: fit linear function to interpolate sky from “sky regions” symmetric on either side of object spectrum: on edge of night-sky emission line away from night-sky emission line x Target Ref star Ref Target Slices across spectrum at  =const:

6. “Normal” extraction Subtract sky fit, and sum the counts between object limits: Dilemma: How do we pick x 1, x 2 ? –too wide: too much noise –too narrow: lose counts S(x) x x1x1 x2x2 Slice across spectrum at  =const:

7. Optimal extraction 1) Scale profile to fit the data: 2) Compute  (x) from the model: 3)  -clip to “zap” cosmic-ray hits. Iterate 1 to 3, since  (x) depends on A: S(x) x x1x1 x2x2 Slice across spectrum at  =const: Starlight profile P(x)

8. Estimating the profile P(x) The fraction of the starlight that falls in row x varies along the spectrum and is given by: This is an unbiased but noisy estimator of P(x). It varies as a slow function of wavelength. Plot against and fit polynomials in at each x. Column 20 Column 60

9. Optimal vs. normal extraction Pros: –Optimal extraction gives lower statistical noise. –Equivalent to longer exposure time –Incorporates cosmic-ray rejection Cons: –Requires P(x, ) slowly varying in (point sources). Essential papers: –Horne, K., 1986. PASP 98, 609 –Marsh, T. and Horne, K.

10. Wavelength calibration Select lines using peak threshold. Measure pixel centroid x i by computing x or fitting a gaussian Identify wavelengths i Fit polynomial (x) to i, i=1,...,N. Reject outliers (usually close blends) Adjust order of polynomial to follow structure without too much “wiggling”. threshold level

11. Dealing with flexure Flexure of spectrograph causes position x i of a given wavelength to drift with time. Measure new arcs at every new telescope position. Interpolate arcs taken every 1/2 to 1 hour when observing at same position. Master arcfit: Use a long-exposure arc (or sum of many short arcs) to measure faint lines and fit high-order polynomial. Then during night take short arcs to “tweak” the low-order polynomial coefficients.

12. Statistical issues raised Outlier rejection: what causes outliers, and how do we deal with them? –Robust statistics. Polynomial fitting: how many polynomial terms should we use? –Too few will under-fit the data. –Too many can introduce “flailing” at ends of range. We’ll deal with these issues in the next lecture.