Appendix A: Precision of the spectral shift obtained by cross-correlation

Expressions for the precision of the correlation peak location have been derived by several authors (Connes 1985; Murdoch & Hearnshaw 1991; Butler et al. 1996 and others). The standard error of $\hat{u}$ from Eq. (5) is here derived taking into account the synthetic template and the resampling and conditioning of the observed spectrum.

Let I_x,y be the number of charges in pixel (x,y), where x is the pixel coordinate approximately aligned with the wavelength. We assume that the noise is uncorrelated between pixels and that the total variance per pixel is I_x,y+r². The first term represents the Poisson noise and the second the readout noise with rms value r (for ELODIE, r=8.5 e^-). One-dimensional spectral orders are extracted by means of a numerical slit of length $m\simeq 6$ pixels (Baranne et al. 1996). Resulting charge values E_x are effectively the sums of I_x,y across the spectral order, thus having uncorrelated noise with variance ${\rm Var}(E_x)\simeq E_x + mr^2$. The conditioning of the spectrum scales the noise by the known factor $h(\Lambda)={\rm d}s_i/{\rm d}E_x$, where $\Lambda$, i and x are corresponding coordinates.

All error computations are in practice made on the resampled data, i.e. by summing over index i. However, in the resampled data the errors are no longer uncorrelated between adjacent points. We can take this into account by applying the appropriate factor. Thus, whenever the error propagation requires a sum of the quantity q_x over the uncorrelated data points, it can be replaced by a sum over the resampled data (q_i) according to the approximation:

$$p \sum_x q_x \simeq \Delta\Lambda \sum_i q_i ~ .$$ (A.1)

Here $p=\langle{\rm d}\Lambda/{\rm d}x\rangle$ is the average pixel size expressed on the $\Lambda$ scale. For ELODIE, $p\simeq 1.0\times 10^{-5}$.

In Eq. (5) the main error comes from the noise in the numerator (the denominator is the CCF curvature which has a relatively high signal-to-noise ratio). With $\delta$ denoting the error in a quantity caused by the total (Poisson and readout) noise we have

$$\delta \hat{u} \simeq \frac{\frac{1}{2}(\delta g_{j+1}-\delta g_{j-1})} {2g_j-g_{j-1}-g_{j+1}}~\Delta\Lambda ~ .$$ (A.2)

We can write the numerator $Q\equiv(\delta g_{j+1}-\delta g_{j-1})/2= \sum_i\delta s_i(t_{i-j+1}-t_{i-j-1})... ...ambda)_{i-j}\simeq p\sum_x\delta E_x h(\Lambda_x)({\rm d}t/{\rm d}\Lambda)_{x'}$, where $x'=x-j\Delta\Lambda/p$. In the last approximation we have made use of Eq. (A.1). The variance of Q is now found to be ${\rm Var}(Q)\simeq p^2\sum_x(E_x+mr^2)h^2(\Lambda_x) ({\rm d}t/{\rm d}\Lambda)^2_{x'}$. Using Eq. (A.1) to re-write this as a sum over the resampled points we finally get

$$\sigma_{\hat{u}} \simeq \frac{\left[p\Delta\Lambda \sum_i (... ...1}-t_{i-j-1})^2\right]^{1/2}} {2(2g_j-g_{j-1}-g_{j+1})} \cdot$$ (A.3)

Here E_i and h_i stand for E_x and $h(\Lambda)$ interpolated to the appropriate coordinate. It is noted that the right-hand side of Eq. (A.3) is independent of $\Delta\Lambda$ as long as the latter is small enough that $t_{i+1}-t_{i-1}\simeq 2\Delta\Lambda({\rm d}t/{\rm d}\Lambda)_i$.