As demonstrated by Bouchy2001 The information content of a spectrum for radial velocity measurement can be determined through the following relations :
\(\delta V_{\rm RMS} = \frac{c}{Q\sqrt{N_{e^{-}}}}\),
and the mean radial-velocity accuracy being :
Equation (6) in :
\(\frac{\delta V_{RMS}(i)}{c} = \frac{[A(i)-A_0(i)]_{RMS}}{(\lambda(i)\delta A_0(i)/\delta\lambda(i)}\).
Expressing \([A(i)-A_0(i)]_{RMS}\) as a fractional variation relative to the flux, one get \(\sigma = [A(i)-A_0(i)]_{RMS}/\overline{A_0}\). Here \(1/\sigma\) corresponds to the signal-to-noise ratio. Normalizing \(A_0[i]\) by \(\overline{A_0}\), one get \(A_0n[i] = A_0[i]/\overline{A_0}\), where \(A_0n[i]\) is the flux normalized to a mean of 1 over the wavelength domain of interest.
And that the radial velocity accuracy attainable with a given wavelength domain is (Eq. 13 in Bouchy2001) :
\(\overline{\delta V_{\rm RMS} } = \frac{1}{\sum{(\frac{1}{ \delta V_{\rm RMS}(k) })^2}}\)
Assuming a constant SNR through the domain, then one gets the following.
\(\overline{\delta V_{\rm RMS} } = \frac{\sigma}{c}\frac{1}{ \sum{\left(\lambda(i) \delta A_0n(i)/\delta\lambda(i)\right)^2 } }\).
We define :
\(\rho_{\delta V^2} = \left(\lambda(i) \delta A_0n(i)/\delta\lambda(i)\right)^2\),
which corresponds to a local density of information content in the spectrum at unit signal-to-noise ratio. In a shot-noise-limited regime, a two-fold increase in the \(\rho_{\delta V^2}\) value is equivalent to a two-fold increase in flux.
Considering, for the sake of establishing a comparison metric between spectra, we assume that over the wavelength interval the signal-to-noise ratio (SNR) is constant, we define a local radial-velocity information metric that is the mean \(\delta V\) at unit signal-to-noise. This value, \(\rho_{\delta V^2}\) can readily be scaled for a given wavelength domain and SNR. The radial velocity accuracy achieved with scale as \((\Delta \lambda/\lambda)^-{\frac{1}{2}}\) and as the inverse of SNR. A spectrum with a \(\rho_{\delta V} = 1\)km/s observed over a \((\Delta \lambda/\lambda)=20\)% at a SNR=200 should lead to an intrinsic accuracy of \(1000\times20^{-\frac{1}{2}}\times200^{-1}\sim1.1\)m/s.