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  • On the optical and near-infrared radial velocity content of M dwarfs

    Introduction

    Exoplanet study

    Datasets

    While various very-high (\(\lambda/\delta\lambda>50\,000\)) resolution spectrographs covering most or all of the near-infrared domain at once are currently under design or construction (e.g., Carmenes, SPIRou, PRVS), none is currently in operation. As a result, very-high resolution spectra available in archives only cover relatively short wavelength intervals. The only publicly available spectrum of an M dwarf at a resolution sufficient to resolve the intrinsic stellar line width (\(\sim5\,km/s\) or \(\lambda/\delta\lambda>60\,000\)) is that available through the CRIRE-POP1 spectral library for the Barnard star ( refcrirespop). These observations cover the \(Y\), \(J\), \(H\), \(K_s\), \(L\) and \(M\) bands; of specific interest here being the \(\lt 2.38\mu\)m domain that is amenable to m/s-level accuracy velocimetry. As the sole M dwarf with complete near-infrared coverage at this resolution, Barnard’s Star is a relatively convenient choice as, with an M4.0V spectral type it is representative of field M dwarfs that will be targeted by upcoming near-infrared radial-velocity surveys. For earlier M dwarfs, optical spectrographs are competitive against near-infrared spectrographs. Vert late M dwarfs (\(\gt\)M7V) are even more promising targets for nIR velocimetry, but are generally fast rotators, even at ages of Gyr refxavier. Furthermore, the rotation period of Barnard’s star is known to be very long (130.4days; http://arxiv.org/pdf/astro-ph/9806276v1.pdf). With a interferometrically determined radius of \(0.196\pm0.008\) (http://arxiv.org/pdf/0906.0602v1.pdf), the rotational broadening is \(\lt76\)m/s. This value is about two orders of magnitudes smaller than the intrinsic line width of M dwarfs. The only peculiarity regarding Barnard’s star relative to the bulk of field star is that it is a metal-poor thick disk star. This affects the radial velocity content of its spectrum. This is addressed in section \ref{metallicityQ}


    1. http://www.univie.ac.at/crirespop/

    Results

    Analysis

    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.