Astronomical Spectroscopy - Physics - University of Cincinnati

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that it degraded the signal-to-noise ratio. Consider the case of a faint source with lots of sky
background. A large extraction aperture in which the data were summed in an unweighted
manner would be far noisier than one in which the extraction aperture was small and excluded
more sky: the sky adds only noise, but no new information. The mathematically correct
way to extract the data is to construct the sum by weighting each point by the inverse of
the variance. (Recall that the variance is the square of the expected standard deviation.)
The assumption in the Horne (1985) algorithm is that the spatial profile P(λ) varies
slowly and smoothly with wavelength λ. At each point along the spatial direction i at a
given wavelength λ, C i (λ) photons are measured. This value is the sum of the number of
photons from the star A i (λ) and of the sky background B(λ), where the latter is assumed
to be a constant at a given λ. In the absence of optimal-weighting, one would determine the
total sum T(λ) of the sky-subtracted object by:
T(λ) = ∑ i
A i (λ) = ∑ i
(C i (λ) − B(λ))
as A i (λ) = C i (λ)−B(λ). In practice the summation would be performed over some “sensible”
range to include most of the spatial profile. For optimal weighting one would instead weight
each point in the sum by W i (λ):
T(λ) = ∑ i
W i (λ)(C i (λ) − B(λ))/ ∑ i
W i (λ)
The weighting function W i (λ) is taken to be P 2 (λ)/σi 2 (λ), where P(λ) is the intrinsic
spatial profile of the spectrum at wavelength λ. P is usually forced to vary smoothly and
slowly with wavelength by using a series of low order functions to represent the profile as
a function of wavelength, and is normalized in such a way that the integral of P(λ) at a
particular λ is always unity. (For the more complicated situation that corresponds to echelle
or other highly distorted spectra, see Marsh 1989.) The variance σi 2 (λ) is readily determined
if one assumes that the statistical uncertainty of the data is described by simple Poisson
statistics plus read-noise. The errors add in quadrature, and will include the read-noise R.
The variance will also include the photon-noise due to the object √ A i (λ) and the photonnoise
due to the sky √ B(λ). Finally, the variance will also include a term that represents
the uncertainty in the background determination. Typically one determines the sky value
by averaging over a number N of pixels located far from the star, and hence the uncertainty
in the sky determination is √ B(λ)/(N − 1). Thus
σ 2 i (λ) = R2 + A i (λ) + B(λ) + B(λ)/(N − 1).
One can eliminate cosmic-rays by substituting T(λ)P i (λ) for A i (λ):
σ 2 i (λ) = R 2 + T(λ)P i (λ) + B(λ) + B(λ)/(N − 1),