Statistical Modeling of an astro-comb for high precision radial velocity observation

Abstract: The advent of the laser frequency comb as the wavelength calibration unit allows us to measure the radial velocity at $cm\ s^{-1}$ precision level with high stability in long-term, which enable the possibility of the detection of Earth-twins around solar-like stars. Recent study shows that the laser frequency comb can also be used to measure and study the precision of the instrumental system including the variations of line profile and the systematic uncertainty and instrumental drift. In this paper, we present the stringent analysis of a laser frequency comb(LFC) system with 25GHz repetition frequency on a R$\sim$50,000 spectrograph with the wavelength spanning from 5085\AA \ to 7380\AA. We report a novel fitting model optimized for the comb line profile, the constrained double Gaussian. The constraint condition is set as $\left|\mu_{1,2} - \mu \right| <\sqrt{2ln2}\sigma$. We introduce Bayesian information criterion to test various models. Compared to the traditional Gaussian model, the CDG(Constrained Double Gaussians) model provides much better goodness of fit. We apply the CDG model to the observed comb data to demonstrate the improvement of RV precision with CDG model. We find that the improvement of CDG model is about 40\%$\sim$60\% for wavelength calibration precision. We also consider the application to use the LFC and CDG model as a tool to characterize the line shape variation across the detector. The motivation of this work is to measure and understand the details of the comb lines including their asymmetry and behaviors under various conditions, which plays a significant role in the simultaneous calibration process and cross-correlation function method to determine the Doppler shift at high precision level.