2 Noise-limited precision

Random errors set a theoretical upper limit to the radial velocity precision attainable from a given spectrum. Quantifications of this minimum error as a function of the intrinsic morphology of the spectrum and of its random errors were computed by several authors (Connes 1985; Merline 1985; Brown 1989; Butler et al. 1996). These results all assume that the observed spectrum is compared with a noise-free reference spectrum. The latter is of course not the case when cross-correlating with observed templates. In addition, many different templates are required to closely match the different spectral types when dealing with early-type spectra, which makes it more difficult to acquire all of them with a very high S/N. We therefore derived a generalised expression for the minimum random error on a cross-correlation shift in the presence of random errors on both object and template. In order to be able to consider sub-pixel errors, we assume throughout that the spectra are oversampled which is a condition generally met for early-type spectra at intermediate and high resolution (see also Sect. 3).

Assume an object spectrum $S_{\rm O}(x)$ and an intrinsically identical template spectrum $S_{\rm T}(x)$ which have been sampled (i.e. integrated) in N pixels with constant pixel-size $\Delta x$ (the x-coordinate may in general be any function of wavelength $\lambda$). We assume both spectra to be multiplicatively normalised to an identical but otherwise arbitrary flux level. We use the dimensionless coordinate p defined by the transformation $x = x_{0} + p \Delta x$ so that its integer values $1..\, N$ correspond to the central positions of the pixels and we conveniently refer to it as the "pixel number". Let I(p) denote the sampled, normalised and noise-free spectrum, and $N_{\rm O}(p)$ and $N_{\rm T}(p)$ the random noise on object and template spectrum, respectively, at that same flux level. Following the argument of Brown (1989), we can calculate the apparent shift $\delta$, due to noise only, between these two spectra by minimizing  

$$f(\delta) = \sum_{p} W(p) [I(p) + N_{\rm O}(p) - I(p + \delta) - N_{\rm T} (p)]^{2}.$$ (1)

Note that maximizing the cross-correlation function is indeed equivalent to a least-squares minimization of the difference between the two spectra (see e.g. Furenlid & Furenlid 1990). The statistical weight W(p) must equal the inverse of the variance on the difference spectrum:  

$$W(p) = \frac{1}{\langle N_{\rm O}^{2}(p)\rangle + \langle N_{\rm T}^{2}(p)\rangle }$$ (2)

where $\langle \rangle$ denotes the expectation value. Approximating $I(p + \delta) \cong I(p) + \delta \frac{{\rm d}I}{{\rm d}p}$ for well sampled spectra and sufficiently small $\delta$, where the derivative is indeed the derivative of the sampling values with respect to the coordinate p, we obtain  

$$\delta = \frac{\sum_{p} W(p) [N_{\rm O}(p) - N_{\rm T}(p)]\... ...um_{p} W(p) \left( \frac{{\rm d}I}{{\rm d}p} \right)^{2}}\cdot$$ (3)

Denoting the variance on $\delta$ by $\sigma^{2}$ = $\langle \delta^{2}\rangle$,one finds

$$\begin{eqnarray} \sigma^{2} & = & \frac{ \sum_{p} W^{2}(p) \left( \frac{{\rm d}... ...2}(p)\rangle } \left( \frac{{\rm d}I}{{\rm d}p} \right)^{2}}\cdot\end{eqnarray}$$ (4)

For the sake of clarity, we repeat that $\langle N_{\rm O}^{2}(p)\rangle$ and $\langle N_{\rm T}^{2}(p)\rangle$ are the variances on both spectra after normalisation, and that $\sigma$ is expressed in units of pixel number p. If the spectra are binned into pixels of constant pixel-size in x = log($\lambda$), which is required for computing a Doppler shift, one must multiply $\sigma$ by the pixel-size in log($\lambda$) units times c to obtain a velocity. The standard deviation $\sigma$ represents a lower bound for the expected random error on the measured shift of one sampled and noisy spectrum with respect to an intrinsically identical one. In fact the derivation assumes that the underlying (noiseless) structure of the spectra is known with infinite resolution and that this information can be used to find the shift. Even though (4) will therefore not always provide a realistic error estimate, it is very important as a reference value.

A similar result has been obtained long before in signal-processing theory: if one replaces the statistical weight in (2) by a constant one describing additive, wavelength-independent noise, then the expression for the expected error on the time-delay of a radar signal is recovered exactly. This is particularly interesting since Woodward & Davies (1950) derived this result without making a first-order approximation like the one leading to (3) so that this approximation probably does not restrict the validity of (4) to spectra with high S/N. However, Woodward (1953) does point out that there is a S/N-level above which it certainly cannot be meaningful anymore, i.e. the so-called ambiguity limit which was estimated by Fellgett (1953) to be about 3 for astronomical spectra.

One can trivially rewrite the variances on object and template appearing in Eq. (4) in terms of their S/N:  

$$\langle N_{\rm O}^{2}(p)\rangle + \langle N_{\rm T}^{2}(p)\r... ...{\rm O}^{2}(p)} + \frac{1}{(S/N)_{\rm T}^{2}(p)} \right]\cdot$$ (5)

In the common case of spectra I(p) being normalised to their continuum level, one may straightforwardly express the variances on these spectra as a function of their original continuum level ($C_{\rm O}(p), C_{\rm T}(p)$) and of their original read-out noise ($R_{\rm O}, R_{\rm T}$), both expressed in photon units:  

$$\langle N_{\rm O}^{2}(p)\rangle + \langle N_{\rm T}^{2}(p)\r... ...{2}(p)} + \frac{R_{\rm T}^{2}}{C_{\rm T}^{2}(p)} \right)\cdot$$ (6)

Note that the continuum level and read-out noise in this expression refer to those of the one-dimensional spectrum, thus after a possible extraction.

In the case of pure photon noise, identical S/N of object and template spectrum, a series of L separated identical Gaussian absorption lines of width $\sigma_{\rm l}$, central depth d ($0\, <\, d\;{\leq}\;1$), and signal-to-noise in the continuum $(S/N)_{\rm c}$, Eq. (4) can be written as  

$$\sigma = \frac{\sqrt{2}\sqrt{\sigma_{\rm l}}} {(S/N)_{\rm c} \sqrt{L} d {\rm f}(d) }$$ (7)

where the only approximation made is replacing the sum by an integral. f(d) is a weak monotonous function of d, being 0.94 for $d \rightarrow 0$and 2.01 for d=1. $\sigma$ and $\sigma_{\rm l}$ are expressed in pixels. The factor $\sqrt{2}$ is absent in case of a noise-free template.

For Gaussian emission lines with a peak signal-to-noise in their centre equal to $(S/N)_{\rm p}$, a negligible continuum intensity, and otherwise similar parameters as above, one finds  

$$\sigma=\frac{\sqrt{2}\sqrt{\sigma_{\rm l}}} {(S/N)_{\rm p}\sqrt{L}}(2 \pi )^{-0.25}$$ (8)

where the last numerical factor is about 0.63. Both the latter expressions may be useful for quick estimates of the intrinsic radial velocity precision of a (portion of a) spectrum, and give insight in the dependence of that precision on the spectrum parameters.