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

TL;DR

The paper addresses the challenge of incorporating wavefront error (WFE), wavefront sensing and control (WFS&C), and coronagraph sensitivity into exposure-time calculations for future space missions like Habitable Worlds Observatory (HWO). It introduces the Error Budget Software (EBS), an open-source Python tool that wraps around EXOSIMS to compute exposure times while accounting for speckle stability ($f_{\Delta C}$) driven by WFE and WFS&C across spatial/temporal modes. Through single-parameter sweeps and high-dimensional Markov Chain Monte Carlo (MCMC) and Nested Sampling analyses using USORT parameters, the work reveals scaling relationships such as $t_{req}$ vs raw contrast and WFE (often a $\sim$quadratic dependence before single-exposure visibility) and the transition of detector-noise dominated regimes with energy resolution $R$ and exo-zodi levels. The results provide actionable insights for mission error budgeting, showing how stability, post-processing, and detector choices shape exposure-time requirements and the attainable exo-Earth yield, while offering a flexible, community-accessible platform for broader coronagraph-yield trade studies.

Abstract

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.

Figures (11)

• Figure 1: Flowchart depicting the basic operations of EBS as described in Section \ref{['sec:core_func']}.
• Figure 2: Required integration time for the detection of an Earth-like planet around two G-stars as a function of coronagraphic raw contrast. In each panel the two stars are represented by different colors with both the inner and outer habitable zones (HZ) displayed. The area between the inner and outer HZs is shaded. From top to bottom, the three panels represent increasing amounts of assumed WFE. The breakpoints where exposure times shoot to infinity indicate raw contrasts for which there is no possibility of detection, regardless of exposure time.
• Figure 3: Contrast at which the required exposure time for an $SNR=5$ exo-Earth detection goes to infinity (the raw contrast breakpoint) as a function of the multiplicative factor applied to the input WFE ($\sigma_{\Delta C}$). Here the solid line is the data from EBS and the dashed line is the best fit power law. The region to the left and above the solid line represents a regime of coronagraphic raw contrast and WFE where no Earth-like planet can be found around this star, regardless of exposure time. The crossed circle and dotted line represents the $EEPSR$ scaled by the SNR for this target. The breakpoints remain constant once the raw contrast is lower than the $EEPSR/SNR$ meaning that an Earth-like planet can be detected at the desired SNR in single exposures.
• Figure 4: Same plot as Figure \ref{['fig:contrast_breaks_single']} but including data for all 5 fiducial target stars studied and removing the colored shaded region for clarity. As in the previous figure, regions to the left and above these lines represent a regime of coronagraphic raw contrast and WFE where no Earth-like planet can be found around this star, regardless of exposure time.
• Figure 5: Sweeps of required integration time for $H_2O$ detection ($\lambda = 1000$ nm, SNR = 5) as a function of energy resolution ($R$) for a variety of values of dark current ($\xi$, counts/pix/s). Here all other detector noise terms are set to 0. As is also highlighted in Figure \ref{['fig:det_noise_powers']}, when dark current is increased, the relationship between $R$ and exposure time goes from linear to quadratic. The black dashed lines show the best fit power law following $y = \beta + \alpha x^k$.