Incorporating Wavefront Error, Wavefront Sensing and Control, and Sensitivities into Exposure Time Calculations for Future Space Missions with the Error Budget Software (EBS)

A primary goal of NASA’s Habitable Worlds Observatory (HWO) mission concept is to explore the Habitable Zones (HZ) of ~100 stellar systems and acquire spectra of ~25 terrestrial-type planets (with planet/star flux ratios on the order of 1E-10) which places tight constraints on the performance of observatory systems. In particular, coronagraph instrumentation needs to be matured for higher throughput, deeper contrasts, and better broadband performance, while also considering their sensitivity and ability to mitigate the impact of telescope instability and wavefront error (WFE), which can have a profound impact on exo-Earth imaging. The success of various proposed HWO mission architectures is often represented by the estimated exo-Earth candidate yield. Computation of the minimum exposure time to achieve the required signal-to-noise on a given target, using an exposure time calculator (ETC), is a key part of yield estimation. The impacts of coronagraph sensitivity, WFE, and wavefront sensing and control (WFS&C) have been well studied in the context of developing error budgets for missions and instruments such as the Roman Coronagraph Instrument, but there is currently no easily accessible way to incorporate the effects of these key parameters into calculating exposure times for HWO. To address this, we developed the Error Budget Software (EBS) - an open-source tool that synthesizes sensitivity, WFE, and WFS&C information for a variety of temporal and spatial scales and directly interfaces with the open-source yield code EXOSIMS to produce exposure times. We demonstrate how EBS can be used for mission error budgeting using the example of the Ultrastable Observatory Roadmap Team (USORT) observatory design. This includes both single and multi-variate parameter explorations using EBS where we identify trends between raw contrast and wavefront error, and detector noise and energy resolution.

💡 Research Summary

The paper presents the Error Budget Software (EBS), an open‑source Python package that integrates wavefront error (WFE), wavefront sensing and control (WFS&C), and coronagraph sensitivity into exposure‑time calculations for future space‑based direct‑imaging missions such as NASA’s Habitable Worlds Observatory (HWO). The authors begin by outlining HWO’s scientific ambition: to achieve high‑completeness surveys of ~100 nearby stars and obtain spectra of ~25 Earth‑like planets, which requires detecting planet‑to‑star flux ratios on the order of 10⁻¹⁰. This extreme contrast places stringent demands on telescope stability, coronagraph throughput, raw contrast, and detector performance.

The methodology follows the error‑budget approach used for the Roman Coronagraph Instrument (Roman CGI). The required signal‑to‑noise ratio (SNR) and the Earth‑equivalent planet‑star flux ratio (E_EPSR) define an allowable total flux‑ratio noise (F_RN = E_EPSR / SNR). The total noise is split into a random component (r_n t) that integrates down with time and a systematic component (r_ΔI t²) that does not. Equation (1) gives the required integration time t_req = (r_n / r_pl SNR) √(r_n² + r_ΔI² SNR²). The systematic term r_ΔI represents residual speckle noise that arises from wavefront instability and cannot be reduced by longer integration.

Differential imaging techniques (ADI, RDI, CDI) are formalized by subtracting a reference electric‑field image from a target image. The authors show that any change in the wavefront between reference and target (ΔE) produces a cross‑term that adds systematic noise. They introduce two key factors: speckle‑stability f_ΔC (the ratio of post‑subtraction contrast standard deviation to the mean raw speckle) and post‑processing gain f_pp. These combine to give r_ΔI = f_ΔC f_pp r_sp, where r_sp is the raw speckle count rate.

The novel contribution is the translation of WFE, WFS&C performance, and coronagraph sensitivity β into the speckle‑stability term. Assuming the heterodyne (cross‑term) dominates, the variance of the contrast stability is σ_ΔC² = ¯C σ_ΔE², with σ_ΔE = β δ, where δ is the WFE amplitude. This yields σ_ΔC² = ¯C C₀ s² δ², where s = ∂¯C/∂δ evaluated at a reference contrast C₀. The residual WFE after control is modeled as r_ij = γ_ij δ_ij, where γ_ij (0–1) is the “instability‑transmission factor” describing how effectively the WFS&C subsystem damps a particular temporal/spatial mode (i,j). By summing over all modes, EBS computes the total systematic speckle noise r_ΔI, which is then fed directly into the EXOSIMS exposure‑time calculator.

Implementation details describe a modular Python framework where users supply WFE power‑spectral‑density files, WFS&C gain matrices, coronagraph sensitivity maps, detector noise parameters, and spectral resolution. EBS can perform single‑parameter sweeps, multi‑parameter Markov‑Chain Monte‑Carlo (MCMC) explorations, or custom Monte‑Carlo simulations. Results are automatically passed to EXOSIMS, which returns exposure times, detection probabilities, and expected exo‑Earth candidate (EEC) yields.

The authors demonstrate EBS on the Ultrastable Observatory Roadmap Team (USORT) design. In a raw‑contrast versus WFE sweep, they identify a “bifurcation point” where modest increases in WFE cause the required exposure time to diverge, effectively rendering the mission incapable of detecting Earth analogs. A detector‑noise versus spectral‑resolution trade shows that higher energy resolution can mitigate detector noise but at the cost of longer integrations, revealing an optimal operating region. Finally, a 23‑parameter MCMC study maps the high‑dimensional design space, highlighting parameter combinations that satisfy mission yield goals and those that lead to failure modes.

In conclusion, EBS provides a transparent, reproducible, and extensible tool for integrating detailed wavefront error budgets into mission‑level performance predictions. By linking hardware‑level specifications directly to scientific yield, it enables more informed technology‑development priorities and risk‑mitigation strategies for HWO and other future high‑contrast imaging missions. The open‑source nature encourages community contributions and adaptation to other concepts such as LUVOIR or HabEx.