ARVE: Analyzing RV Elements

• Analyzing Radial Velocity Elements (ARVE) is a comprehensive framework for extracting, characterizing, and modeling stellar radial velocity time series to uncover exoplanet signals.
• Its innovative methods—including Cross-Correlation Function and Line-by-Line extraction—provide high-precision RV measurements while effectively mitigating stellar noise.
• The modular, Python-based design integrates synthetic auxiliary data, automated mask generation, and injection-recovery routines to support robust analysis across multiple spectrographs.

Analyzing Radial Velocity Elements (ARVE) refers to a comprehensive approach to extracting, characterizing, and modeling radial velocity (RV) time series obtained from high-precision stellar spectroscopy, with the goal of disentangling planetary signals from complex stellar variability and instrumental noise. The ARVE framework—described in "ARVE: Analyzing Radial Velocity Elements. I. The Code" (Moulla, 25 Jul 2025)—embodies a modular, Python-based software system designed to serve as a unified toolkit for modern RV analysis, addressing both traditional challenges and current frontiers in extreme-precision radial velocity (EPRV) science.

1. Modular Software Architecture and Design

ARVE is architected as a class-based, object-oriented codebase, centralizing all data structures and algorithms within a primary “arve” class and four distinct subclasses: functions, data, star, and planets. Each subclass is responsible for a specific pipeline stage, reflecting common RV analysis workflows.

• functions: Core utilities for wavelength transformations (vacuum/air conversion per Eq. [5]–[8]), Doppler shifts (Eq. [9]–[10]), analytic Gaussian and periodogram calculations, and other shared operations.
• data: Manages ingestion of reduced high-resolution spectra or RV time series, loads and calibrates auxiliary synthetic data (stellar models, line masks, telluric templates), and provides two central RV extraction routines:
  • Cross-Correlation Function (CCF): Computes CCFs by sliding a weighted line mask across velocity space, fitting inverted Gaussians to yield RVs (Eq. [16]–[21]).
  • Line-by-Line (LBL): Constructs a high SNR reference spectrum, then performs template matching with a Taylor expansion (Eq. [22]–[25]) to derive local RVs on spectral chunks or individual lines.
• star: Handles stellar parameter management (either user-supplied or database-queried), selects appropriate auxiliary data, computes the RV Variability Power Spectral Density (VPSD) via unnormalized generalized Lomb-Scargle periodogram (Eq. [13]–[14]), and decomposes the VPSD into physical components (oscillations, granulation, supergranulation) using analytical models such as Harvey laws and Lorentzian profiles (Eq. [26], coefficients scaled per Eq. [27]–[32]).
• planets: Detects and characterizes planetary signals via normalized GLS periodograms, performs iterative multi-Keplerian (circular) fitting (Eq. [34]), and enables injection-recovery simulations for quantifying detection limits (Eqs. [35]–[36]).

This modular design allows each component to be independently maintained, expanded, and integrated with future functionalities or specialized pipelines for additional spectrographs.

2. Synthetic Auxiliary Data and Automated Mask Generation

A unique feature of ARVE is its curated library of synthetic stellar spectra and pre-generated line masks:

• Synthetic Spectra: For main-sequence stars (F0–M5), computed with PySME and VALD, covering 3000–23000 Å, incorporating physical diagnostics such as the average formation temperature at each wavelength.
• Telluric Models: A comprehensive telluric absorption spectrum based on HITRAN, for precise masking or simultaneous modeling of atmospheric lines.
• Line Masks: Automatically constructed by identifying local minima in synthetic spectra, assigning a velocity content weight per line:

  $$w = \sum_{i} \left(\frac{dF}{d\lambda}\right)^2 \lambda_i^2 F_i \quad [\text{see Eq. (4)}]$$

These weights are optimized for RV precision and are used in both the CCF and LBL extraction routines for robust and reproducible results.

Pre-computed auxiliary data are loaded as needed based on the user's target spectral type and wavelength coverage, minimizing manual input and ensuring rapid, low-latency processing.

3. Radial Velocity Extraction Methods

ARVE implements:

Cross-Correlation Function (CCF) Method

• Constructs a spectrum-to-template CCF using the pre-computed line mask and spectral weights.
• Computes the CCF over a velocity grid for each observation and fits an inverted Gaussian (Eq. [11]–[12]) to locate the RV centroid.
• Incorporates error propagation and uncertainty estimation per Eq. (17).

Line-by-Line (LBL) / Piece-by-Piece (PBP) Method

• Builds a high SNR reference template from co-added observations.
• For each segment (typically a spectral line), models the observed flux as a Taylor expansion of the template flux plus a Doppler-induced shift:

  $$\tilde{F} = A \left[F - \left(\frac{dF}{d\lambda}\right)\frac{v}{c}\lambda \right] \quad [\text{Eq. (23)}]$$

• Solves analytically for the velocity shift $v$, with proper uncertainty propagation (Eq. [24]–[25]).

Comparison of these methods on real datasets (see demonstration section) shows LBL outperforms CCF in sensitivity to low-mass planets and robustness to systematics near problematic frequencies (e.g., solar rotation, annual aliasing).

4. Stellar Variability Characterization via Power Spectral Decomposition

ARVE provides rigorous tools for dissecting the stochastic stellar noise that plagues precision RV measurements:

• Variability Power Spectral Density (VPSD):
  • Computed from the RV time series using an unnormalized GLS periodogram (Eq. [13]–[14]).

  • Decomposed analytically into:

    $$\text{VPSD}(f) = \text{VPSD}_{\text{osc}}(f) + \sum_i \text{VPSD}_{\text{gra},i}(f) + \text{VPSD}_{\text{noise}}$$

  where the oscillation (p-mode) signal is a Lorentzian and granulation/supergranulation components are Harvey-like power laws (Eq. [26], coefficients from Eq. [27]–[32]).

• Activity Tracking: By fitting the VPSD at different epochs or in activity-level bins (e.g., S-index intervals), ARVE facilitates empirical tracking of how activity processes (granulation, supergranulation) evolve with stellar magnetic cycle.

• Noise Simulation: Using the fitted models, ARVE can synthesize synthetic RV noise time series for sensitivity testing or injection-recovery analysis across a range of activity levels (Eq. [33]).

5. Planet Detection, Keplerian Fitting, and Injection-Recovery

ARVE integrates established and statistically rigorous planetary detection and characterization methodologies:

• GLS Periodogram: Locates periodic signals in RV data; significance assessed via user-specified false alarm probability, allowing automated extraction of candidate planetary periods.

• Keplerian Modeling: Fits circular orbits (sinusoidal model, Eq. [34]) iteratively, subtracting each detected signal from the residuals to search for additional modulations.

• Injection-Recovery Routines: Systematically injects synthetic planetary signals (period and mass determined by Eqs. [35]–[36]) into observed or synthetic RV series, rerunning the detection pipeline to assess recovery probability and thus quantify detection limits as a function of period and mass.

This unified workflow enables objective comparison of RV extraction techniques and their sensitivity to both stellar noise and planetary signatures.

6. Demonstration on Solar Data and Comparative Analyses

Demonstrations with multi-year HARPS-N and NEID solar data showcase the platform's ability to extract and interpret meaningful astrophysical trends:

• Stellar Noise Evolution:

  • VPSD fits in 10-day segments reveal that the solar granulation timescale is invariant with activity, whereas supergranulation timescale increases with chromospheric activity (Mount Wilson S-index).
• Detection Limits Comparison:
  • CCF with velocity-content weighting and LBL extraction are compared via injection-recovery tests.
  • LBL extraction—particularly when utilizing temperature-binned lines (e.g., 4000–4750 K)—achieves consistently superior detection capability, less susceptible to reduced sensitivity near the solar rotation or one-year signal frequencies.

7. Prospects and Future Expansion

ARVE is specifically designed for extensibility and wide compatibility:

• Instrument Support: Direct compatibility with ESPRESSO, HARPS, HARPS-N, NEID, and others is included, with further additions straightforward due to modular data ingestion and processing pipelines.
• Planned Enhancements: Incorporation of non-circular Keplerian fits, advanced stellar activity indicators derived directly from spectra, Gaussian process regression for correlated noise, and support for user-defined or additional synthetic auxiliary data sets.
• Community Utility: By standardizing pipelines from raw spectra to detection limits, ARVE aims to foster methodological consistency and reproducibility in the EPRV field, serving both as a practical analysis tool and a learning platform.

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In summary, ARVE provides an integrated environment for the extraction, characterization, and astrophysical interpretation of radial velocity elements in spectroscopic time series. Its modular approach, comprehensive synthetic auxiliary data, robust stellar activity modeling, and modern planet detection workflows position it as a central tool for state-of-the-art studies in exoplanet detection, stellar variability analysis, and precision RV methodology (Moulla, 25 Jul 2025).