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Astrophysical Sources of Statistical Uncertainty in Precision Radial Velocities and Their Approximations

We investigate astrophysical contributions to the statistical uncertainty of precision radial velocity measurements of stellar spectra. We analytically determine the uncertainty in centroiding isolated spectral lines broadened by Gaussian, Lorentzian, Voigt, and rotational profiles, finding that for all cases and assuming weak lines, the uncertainty is the line centroid is $σ_V\approx C\,Θ^{3/2}/(W I_0^{1/2})$, where $Θ$ is the full-width at half-maximum of the line, $W$ is the equivalent width, and $I_0$ is the continuum signal-to-noise ratio, with $C$ a constant of order unity that depends on the specific line profile. We use this result to motivate approximate analytic expressions to the total radial velocity uncertainty for a stellar spectrum with a given photon noise, resolution, wavelength, effective temperature, surface gravity, metallicity, macroturbulence, and stellar rotation. We use these relations to determine the dominant contributions to the statistical uncertainties in precision radial velocity measurements as a function of effective temperature and mass for main-sequence stars. For stars more than $\sim1.1\,M_\odot$ we find that stellar rotation dominates the velocity uncertainties for moderate and high resolution spectra ($R\gtrsim30,000$). For less massive stars, a variety of sources contribute depending on the spectral resolution and wavelength, with photon noise due to decreasing bolometric luminosity generally becoming increasingly important for low-mass stars at fixed exposure time and distance. In most cases, resolutions greater than 60,000 provide little benefit in terms of statistical precision. We determine the optimal wavelength range for stars of various spectral types, finding that the optimal region depends on the stellar effective temperature, but for mid M-dwarfs and earlier the most efficient wavelength region is from 6000A to 9000A.