Wavelength adjustment

The target and a reference star have not the same radial velocity with respect to the spectrograph, and the first thing to do is to shift the spectrum of the reference star to the radial velocity of the object in order to make the absorption lines coincide exactly. As the difference in radial velocity does not correspond to a whole number of pixels, the flux corresponding to a given pixel of the shifted spectrum is not known and has to be interpolated. We chose to shift the reference spectrum instead of the contrary in order to avoid an interpolation on a too noisy function. A quadratic Bessel's interpolation formula was employed. It makes use of two points before and two points after the considered pixel x :

$$f(x)=(1-p)f(x_0) + pf(x_1) + A{p(p-1) \over 4}$$

where:

$$p = {(x-x_0) \over (x_1-x_0)}$$

$$x_{-1} < x_0 \leq x < x_1 < x_2$$

and:

A = f(x₂) + f(x_-1) - f(x₀) - f(x₁)

This formula is nicely insensitive to noise in the data: if the individual values are affected by a noise $\epsilon$ the expected noise on the interpolated value is at the most $(9/8)\epsilon$.High order formulae would be amplifying spectrum noise.