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 $sigma_Vapprox C,Theta^{3/2}/(W I_0^{1/2})$, where $Theta$ 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 ($Rgtrsim30,000$). 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.
We analyze a high-resolution spectrum of a microlensed G-dwarf in the Galactic bulge, acquired when the star was magnified by a factor of 110. We measure a spectroscopic temperature, derived from the wings of the Balmer lines, that is the same as the
photometric temperature, derived using the color determined by standard microlensing techniques. We measure [Fe/H]=0.36 +/-0.18, which places this star at the upper end of the Bulge giant metallicity distribution. In particular, this star is more metal-rich than any bulge M giant with high-resolution abundances. We find that the abundance ratios of alpha and iron-peak elements are similar to those of Bulge giants with the same metallicity. For the first time, we measure the abundances of K and Zn for a star in the Bulge. The [K/Mg] ratio is similar to the value measured in the halo and the disk, suggesting that K production closely tracks alpha production. The [Cu/Fe] and [Zn/Fe] ratios support the theory that those elements are produced in Type II SNe, rather than Type Ia SNe. We also measured the first C and N abundances in the Bulge that have not been affected by first dredge-up. The [C/Fe] and [N/Fe] ratios are close to solar, in agreement with the hypothesis that giants experience only canonical mixing.
