3.1.

Exoplanet Sampling Uncertainty

The most fundamental source of uncertainty for a blind survey is due to exoplanet sampling or the uncertainty that results from “luck of the draw.” Occurrence rates describe the mean rate of planets per star. Even if we knew the occurrence rates perfectly, we are not guaranteed to find the exact expectation value of planets in our population of target stars.⁴ In addition, the chances of a planet occurring around one star versus another can affect yields.

Exoplanet sampling uncertainty is a fundamental limit of a blind survey—the only way to fully retire it is to have perfect prior knowledge of every EEC. Precursor extreme-precision radial velocity (EPRV) surveys could identify, which stars host EECs and help reduce exoplanet sampling uncertainty while improving the efficiency of HWO.³ Simulations of EPRV surveys suggest exoEarths could be detectable around nearly $\sim 100$ high-priority stars for a range of future dedicated EPRV telescope architectures.³⁰ However, these simulations represent best-case scenarios and would take at least a decade after EPRV instrument commissioning. While such precursor information will be useful when conducting HWO’s EEC survey, allowing it to achieve faster EEC yields,³ it will come too late to affect the early stages of HWO design when the scale of the mission and key telescope/instrument trades will ultimately dictate the range of accessible targets (addressed in Sec. 4).

We note that a Poisson draw treatment for exoplanet sampling uncertainty is not strictly correct, as it would assume the presence of one planet in a given system does not affect the presence of another planet, which Newton would roundly reject. On the one hand, the presence of a planet should rule out nearby planets that would be gravitationally unstable, such that a Poisson draw would tend to concentrate planets around fewer stars and thus be a conservative choice. On the other hand, exoplanets may “flock together,” such that the presence of a planet implies a higher likelihood of another planet, making a Poisson draw an optimistic choice. Ideally, we would distribute planets consistent with known multiplicity rates and check for orbital stability, but empirical multiplicity rates in the HZs of FGK stars are unknown. We therefore proceed with Poisson draws and note the possibility that we will underestimate the exoplanet sampling uncertainty. We note that compared to the simple numerical alternative of a Monte Carlo success-based draw, in which a maximum of one planet per star is assigned, our method is conservative.

To estimate the exoplanet sampling uncertainty for an HWO blind survey, we first ran a single AYO calculation using our baseline mission and astrophysical parameters. This resulted in an expected yield of 22.5 EECs, in agreement with Ref. 2. As part of the calculation, we saved many properties of the ${10}^5$ EECs injected around each star: their orbital elements, fluxes, positions, exposure times, and critically, the visit during which they were detected. With this information in hand, we then performed a Poisson draw on each individual star using an occurrence rate equal to ${\eta }_{\oplus }$. We then randomly selected the appropriate number of planets from the star’s population of ${10}^5$ EECs. Randomly selected planets with a valid visit record were counted as detected, whereas those without a valid visit record were treated as undetected. We repeated this Poisson draw 10 k times, building up a distribution of yields.

Figure 4 shows the impact of sampling noise for our baseline mission. Given our assumption of Poisson draws that ignore multiplicity, our estimate for exoplanet sampling uncertainty should be considered a lower limit. With a mean-normalized standard deviation of 0.21, the spread in yields is substantial. HWO will need to contend with relatively large uncertainties inherent to a blind exoplanet survey.

Fig. 4

EEC yield distribution of our baseline mission considering only “luck of the draw” exoplanet sampling uncertainty. Without prior knowledge of which stars host EECs, there is no way to reduce this fundamental uncertainty. This distribution assumes Poisson draws ignoring exoplanet multiplicity and should be regarded as a lower limit on the amount of uncertainty.

3.2.

Exoplanet Albedo Uncertainty

The yield distribution above assumes all EECs have a uniform geometric albedo $A_G=0.2$, equivalent to that of an Earth-twin.³¹ In reality, EEC albedos will vary. Unfortunately, we have no way of knowing the distribution of EEC albedos in advance. Given an expected sample size of $\sim 25$ EECs, it is likely that we would not understand the distribution of albedos until well into the survey. We therefore treat albedo uncertainty as a static uncertainty.

To constrain the effect of a static albedo uncertainty, we implement a more detailed exoplanet sampling treatment than previously described, allowing for randomly assigned albedos among the drawn EECs. To do so, we first use AYO to optimize observations under the assumption that all EECs have $A_G=0.2$. We then perform a Poisson draw on each star and randomly select the appropriate number of random EEC orbits and phases, such as in Sec. 3.1. Next, we assign each randomly drawn planet an albedo that differs from the AYO-assumed $A_G=0.2$. With this difference in albedo and the saved planet fluxes from AYO, we can determine the adjusted flux of every randomly selected EEC under the assumption of Lambertian phase functions. For each visit to the star, we advance the randomly drawn planet along its orbit based on the orbital properties saved by AYO, determine its visit-updated separation and albedo-adjusted flux, and determine if it would have been detected during the visit.

To determine if the randomly drawn planet would have been detected during a visit, we take advantage of the fact that AYO resolves every orbit into 100 evenly spaced mean anomalies. Each AYO observation effectively detects planets along a segment of an orbit, such that any planets detected along the orbit segment occupy a finite range of fluxes. The faintest detected planet flux along the orbit segment (usually corresponding to the crescent phase) is limited by the exposure time, whereas the brightest planet flux along the segment (usually corresponding to the gibbous phase) is constrained by the IWA of the coronagraph. Therefore, we can approximately determine whether a randomly drawn EEC with differing albedo is detectable during a given visit by requiring (1) its albedo-adjusted flux to be greater than the minimum exoplanet flux detected during that visit by AYO along the same orbit and (2) its stellar separation to be greater than the minimum separation of any EEC on the same orbit detected by AYO during that visit. To verify that this new approximate sampling treatment produced adequate results, we first randomly drew planets, all with $A_G=0.2$, and compared results with the simple procedure described in Sec. 3.1. After 10 k random draws, the mean and standard deviation of the distributions were statistically identical, validating our method.

We note one significant limitation of our method for estimating the impact of albedo uncertainties. AYO “designs” the survey under the assumption that $A_G=0.2$. Detection times are used to determine whether a planet of differing albedo would have been detected, but characterization times are not considered. Characterization times are budgeted for by AYO under the assumption that the planets have a single $A_G=0.2$. Obviously, darker planets would require longer characterization times while brighter planets would require less. However, because the observation plan is already “set in stone” by AYO when doing the albedo draw, we cannot adjust the characterization times after the fact. To first order, we do not expect this issue to be a significant driver of yields as long as the albedo distribution is roughly symmetric about $A_G=0.2$ (which will be true for our final preferred albedo distribution). We do expect this limitation to overestimate the yield of faint planets in systems that are already near the 2-month characterization limit—planets with $A_G<0.2$ in these systems could require characterization times in excess of the limit and should not count toward yield. However, planets with $A_G<0.2$ and characterization times close to the 2-month limit will represent a minority of the planets contributing to the yield (see right panel of Fig. 6). We leave refining this method to future work and note that we may be underestimating the yield degradation due to the exoplanet albedo distribution.

We consider two extreme scenarios to constrain the impact of exoplanet albedo: relatively dark, completely cloud-free water worlds, and relatively bright, completely cloud-covered water worlds. Both models were 100% ocean-covered. The dark extreme imagines a cloudless ocean world whose full-phase brightness would be relatively small, owing to the low reflectivity of deep ocean water seen in backscatter. At the opposite (but still habitable) extreme, a completely cloud-covered ocean is relatively bright at full phase and has a phase function that is distinct from that generated by ocean glint. These extremes bound the expected reflectance behaviors for ocean worlds that, more realistically, would present cloud-covered and cloud-free scenes across the disk. Phase-dependent reflectance models were computed using an existing 3D tool for producing disk-integrated synthetic observations of a pixelated planetary disk.^31–33 We assume Earth-like liquid water clouds and ocean wind speeds (which are a necessary input to the ocean specular reflectance model³⁴).

The phase functions of these models are relatively close to Lambertian up to phase angles of $\sim 100\mathrm{deg}$. Given that the majority of detections occur near phase angles of 90 deg (i.e., quadrature), we choose to treat these models as Lambertian spheres and calculate albedos that reproduce the proper reflectance at quadrature. This results in $A_G=0.08$ for the cloud-free model and $A_G=0.56$ for the cloud-covered model.

The left panel of Fig. 5 shows the yield distributions that result when observations are tuned to $A_G=0.2$. The solid line shows a benchmark: the results when 100% of EECs are drawn with $A_G=0.2$. The dotted and dashed lines show the results when 100% of EECs are drawn with $A_G=0.08$ and $A_G=0.56$, respectively. While the width of the distribution does decrease in the case of $A_G=0.08$, to first order, the dominant effect of changing the exoplanet albedo is that the peak of the distribution shifts. Given the limitations of our method discussed above, we note that we are likely overestimating the yield in the case $A_G=0.08$. The right panel shows the albedo distribution of injected planets (dotted and dashed lines), as well as the albedo distribution of detected planets (solid lines). These distributions are effectively Dirac functions, broadened by our choice of histogram binning. The distributions show that while the detection rate for bright cloud-covered worlds is $\sim 55\%$, the detection rate for dark water worlds is $<20\%$. This highlights the impact of an observational bias: planets brighter than expected do not substantially help yields, but planets fainter than expected can go undetected. This sort of observational bias will creep up again in subsequent sections as we consider additional sources of astrophysical uncertainty.

Fig. 5

Left: EEC yield distribution of our baseline mission with exoplanet sampling and albedo uncertainties for the extreme scenarios in which all EECs turn out to be water-covered cloudless planets (dotted line) or cloud-covered water worlds (dashed line), compared to Earth-twins (solid line). Right: Geometric albedo distributions of injected planets (dotted and dashed) compared to detected planets (solid). The detection rate for dark water worlds is $<20\%$, explaining the shift in the yield distribution to lower values.

The dotted and dashed lines shown in Fig. 5 are extremes that set rough constraints on the impact of albedo uncertainty on exoplanet yield. In reality, we expect EEC albedos to occupy a distribution of values, but we do not yet know the nature of that distribution. Despite our ignorance, we look into the effects of adopting three different uniform distributions. First, we adopt our full range of water worlds with $0.08<A_G<0.56$. Second, we adopt an even broader range of $0.03<A_G<0.58$, roughly representing the range of reflectances one might expect near quadrature for rocky worlds as dark as Mercury and as bright as Venus. Both of these uniform distributions have $⟨A_G⟩>0.2$, so we adopt a third distribution with $0.08<A_G<0.32$, which has a mean geometric albedo equal to that of an Earth-twin. The left panel in Fig. 6 shows the results of these uniform distributions, with the blue, gray, and green curves corresponding to our full range of water worlds, full range of rocky worlds, and narrow range of water worlds, respectively. The $A_G=0.2$ benchmark is shown in black. The blue curve coincidently mirrors that of the benchmark, whereas the gray and green curves are shifted to lower yield values.

Fig. 6

Left: EEC yield distribution of our baseline mission with exoplanet sampling and albedo uncertainties assuming a distribution of albedo values. Yields are shown for a uniform distribution of water worlds (blue), a uniform distribution of rocky worlds (gray), a uniform distribution of water worlds with $⟨A_G⟩=0.2$ (green), and Earth-twins for comparison (black). Right: Geometric albedo distributions of injected planets (dotted) compared to detected planets (solid) for each of the three albedo distributions assumed. Observations are “tuned” to $A_G=0.2$, resulting in lower detection rates for $A_G<0.2$. We adopt the green curves as our fiducial albedo distribution.

The dotted lines in the panels on the right show the distribution of injected planets as a function of albedo for each of our three assumed distributions. Solid lines show the distribution of detected planets. The detection rate of planets with $A_G>0.2$ is relatively flat, but decreases linearly with albedo for $A_G<0.2$. This explains the shift in the yield distribution: many planets at the faint end of the albedo distribution will go undetected. In all scenarios, a minority of the yield is comprised of planets with $A_G<0.2$, and only a fraction of those would exceed the 2-month exposure time limit, suggesting that the limitations of our albedo draw method would have a relatively small effect on the estimated yields.

We experimented with “tuning” the observations to different exoplanet albedo. First, we ran AYO with $A_G=0.15$ for all EECs (tuning observations to fainter-than-Earth-twin planets), then drew the same three distributions. EEC yields were lower in all three cases, as AYO devoted more time to searching for fainter planets and ended up observing fewer stars during the 2-year time budget. We note that the limitations of our albedo draw method should have an even smaller impact in this case, as our observations are tuned closer to the faint edge of the albedo distribution. Next, we ran AYO with $A_G=0.25$ for all EECS (tuning observations to brighter-than-Earth-twin planets). EEC yields were slightly larger for the black, blue, and gray curves, as AYO opted to “pick off” the brighter portions of the distribution, whereas the green curve with $⟨A_G⟩=0.2$ remained approximately the same. However, we have less confidence in these results as the limitations of our albedo draw method should become more pronounced as observations are tuned to brighter planets, which should tend to overestimate yields to a greater degree. It is possible that if we are willing to assume that there are EECs with albedo greater than that of Earth, we could gain some yield at the expense of finding fewer planets fainter than the Earth. However, this assumption seems both poorly founded and risky. We conclude that observation optimization cannot significantly improve the yields of planets fainter than the Earth—we would need to improve the mission performance parameters to accomplish this.

None of the yield distributions shown in Fig. 6 are correct. Completely cloud-free water worlds are probably unlikely, as are completely cloud-covered water worlds. This suggests that our uniform albedo distributions are pessimistic. However, some EECs may in fact be as dark as Mercury or as bright as Venus. The actual albedo distribution may even be multi-modal. Given a need to budget for albedo uncertainty at some level, we choose to move forward with the most pessimistic uniform albedo distribution ($0.08<A_G<0.32$), which produces the green yield distribution shown in Fig. 6. The green yield distribution in Fig. 6 has a mean value of 19.8 EECS; budgeting for albedo uncertainty decreases the expected yields by $\sim 12\%$. Notably, the standard deviation of this green distribution is 22% of the mean; the albedo uncertainty did not significantly increase the fractional width of the yield distribution, i.e., exoplanet sampling dominates the uncertainty in yield. We note that the general resilience of EEC detections against albedo uncertainties does not imply that the characterization time for the EECs is also resilient against uncertainties in target spectra. Characterization time—in terms of, e.g., exposure time required to detect key atmospheric species—will depend strongly on the details of atmospheric composition, cloud distributions, and surface reflectivity.

3.3.

Exozodi Sampling Uncertainty

So far we have combined exoplanet sampling uncertainty with albedo uncertainty, but we have assumed all stars are assigned the same amount of exozodiacal dust. In reality, each star will have a different brightness of exozodiacal dust. While we may know some individual exozodi levels in advance of the HWO survey, we will not know all of them. However, we can learn the rest of the individual exozodi levels “on the fly” and adapt to them as the survey progresses. Reference 11 showed that the EEC yield when adapting to exozodi levels after the first observation is nearly equal to the yield if they were all known in advance. Real-time adaptation to exozodi levels should lead to even higher yields. Exozodi sampling uncertainty is therefore an actionable uncertainty.

To estimate the impact of exozodi sampling uncertainty on yield distributions, we adopt the best-fit exozodi distribution from the LBTI HOSTS survey, which has a median exozodi level of three zodis and is multi-modal, with several peaks at higher zodi levels.²³ From this distribution, we randomly draw exozodi levels, assign them to individual stars, provide that information to AYO, and then calculate an optimized yield. We repeat this process 500 times. We include the exoplanet sampling and albedo uncertainties previously discussed by performing 1000 draws of random EECs with $0.08<A_G<0.32$ for each of the 500 exozodi draws. The LBTI HOSTS survey detected dust around four potential HWO targets: $297\pm 56$ zodis around Eps Eri, $148\pm 28$ zodis around Tet Boo, $588\pm 121$ zodis around 72 Her, and $235\pm 45$ zodis around 110 Her; for yield calculations, we assigned these stars their LBTI-measured nominal exozodi levels.²³

Figure 7 shows the new EEC yield distribution including exozodi sampling uncertainty in orange, along with the previous green distribution from Fig. 6. The yield distribution again maintains roughly the same width but shifts to the left, with a mean of 17.6 EECs. This is due to another observational bias analogous to the albedo distribution discussed in Sec. 3.2. If a high-priority target is assigned a zodi value less than three zodis, there is little yield to be gained, as detection exposure times were already short. However, randomly assigning a larger exozodi value to a high-priority star can significantly extend exposure times. Even if the yield code replaces these high-zodi stars with other low-zodi stars, the limited pool of targets means the code must replace a previously productive target with a lower-productivity target, driving the yield distribution systematically to lower values. Roughly one third of this shift can be explained by the four specific stars discussed above being assigned high zodi values—these otherwise high-priority stars are effectively scrubbed from the target list, reducing the expected EEC yield by one.

Fig. 7

EEC yield distribution of our baseline mission with exoplanet sampling, albedo, and exozodi sampling uncertainties (orange), compared with the green yield distribution calculated in Sec. 3.2. Drawing exozodi values from a distribution (as opposed to assigning all stars the same median value) shifts the yield distribution to lower values, as some high priority targets are assigned higher exozodi values.

We note that the mean-normalized standard deviation of the orange curve in Fig. 7 is 0.24, not too dissimilar from the 0.22 mean-normalized standard deviation of the green curve. This shows that exozodi sampling uncertainty is a minor contribution to the total uncertainty budget, which at this point remains dominated by exoplanet sampling uncertainty.

3.4.

Exozodi Distribution Uncertainty

Not only is the individual exozodi level of each star unknown, but our understanding of the exozodi level distribution is uncertain. While the LBTI HOSTS survey fit the data to derive a single maximum likelihood distribution,²³ many other distributions are also consistent with the data, albeit at lower likelihoods. Here, we add exozodi distribution uncertainty to the planet sampling, albedo, and exozodi sampling uncertainties already estimated.

To estimate the impact of exozodi distribution uncertainty, we must first form a set of possible exozodi distributions and calculate their likelihoods. We start by considering the maximum likelihood methods used in the LBTI HOSTS analysis, which do not formally rely on Bayesian priors (discussed later). To do this, we generate a series of 300 k exozodi values from the maximum likelihood LBTI HOSTS exozodi distribution²³ following the iterative approach described in Sec. 4.6.3 of Ref. 35. We then generate 30 k “perturbed” distributions that differ from the best-fit distributions. We note that in practice, the method for perturbing the maximum likelihood distribution is not well defined, but we have examined multiple methods, all producing similar results. We then calculate the likelihood $L$ of having observed the data from each of those distributions using Eqs. (15) and (16) from Ref. 35 and compare with the maximum likelihood $L_{\max }$. For a given likelihood ratio, there are many possible perturbed distributions, and the perturbed distributions we generated do not follow a normal distribution. We therefore enforce $X=(-2\ln (L/L_{\max }){)}^{1/2}$ to follow a unit normal distribution by defining 31 bins ranging from $-3\le X\le 3$ and randomly drawing the correct number of distributions with the proper likelihood value. The end result is a “set” of $\sim 20\mathrm{k}$ distributions that follow the LBTI HOSTS maximum likelihood approach, with the statistics for the set following a normal distribution based on likelihood.

With these distributions in hand, we perform 498 distinct yield calculations. For each yield calculation, we draw a random exozodi distribution and then assign each star a random exozodi level drawn from that distribution. We continue to include exoplanet albedo and exoplanet sampling uncertainties following the methods previously described. Figure 8 shows the results of including the uncertainty in the exozodi distribution as a red line, compared with our previous yield distribution without it in orange. While the red curve with exozodi distribution uncertainty is slightly shifted to lower values and slightly broader (mean-normalized standard deviation of 0.26 compared with 0.24), there is remarkably little difference in the two yield distributions. This suggests that following the LBTI HOSTS formalism and assuming the LBTI HOSTS results are not systematically biased, any remaining uncertainty in the exozodiacal dust distribution has a smaller impact than the inherent exoplanet sampling uncertainty.

Fig. 8

EEC yield distribution of our baseline mission with exoplanet sampling, albedo, exozodi sampling, and exozodi distribution uncertainties (red), compared with the orange yield distribution calculated in Sec. 3.3. Drawing exozodi values from all distributions consistent with the LBTI HOSTS data negligibly impacts yield uncertainty.

While it has been previously shown that EEC yield is a weak function of median exozodi level,⁵ this is the first estimate suggesting that the uncertainty in the distribution has a negligible impact on an HWO blind survey. As such, we briefly consider alternative approaches to deriving a set of exozodi distributions consistent with the LBTI data set. To do so, we adopt a Bayesian approach to fitting the LBTI data set. We consider two possible functional forms: a log-normal distribution and a non-parametric 40th degree Bernstein polynomial basis distribution. For the non-parametric approach, we adopt two different priors described by a Dirichlet distribution with parameter $\alpha =[0.1,0.25]$, where $\alpha =0.1$ weights the result toward a smooth fit and $\alpha =0.25$ allows for a higher degree of modality in the distribution (i.e., multi-peaked).

The left panel of Fig. 9 shows the result of 10k exozodi draws from 500 randomly drawn distributions for all four of our approaches. The red curve shows the LBTI HOSTS maximum likelihood approach, whereas the black curves show the log-normal and non-parametric approaches. The right panel of Fig. 9 shows the yield distribution for each of these approaches, calculated in a similar fashion as previously described. The log-normal approach, which does not accommodate multi-modality, and the $\alpha =0.1$ prior approach, which allows only for modest multi-modality, predict higher exozodi levels on average and significantly shift the yield curve to lower numbers. The $\alpha =0.25$ prior approach, which does accommodate some degree of multi-modality, produces results that are very similar to the LBTI HOSTS maximum likelihood approach. We note that this is despite the median exozodi level for the $\alpha =0.25$ prior approach being twice that of the maximum likelihood approach—the reason for this is that while the approaches have different medians, both have similar fractions of the distribution $\lesssim 3$ zodis, as shown by the inset panel in Fig. 9.

Fig. 9

Left: 10 k randomly drawn exozodi levels from 500 randomly drawn distributions when fitting the LBTI HOSTS data with maximum likelihood (solid red), log-normal (dotted black), non-parametric with $\alpha =0.1$ (dashed black), and non-parametric with $\alpha =0.25$ (solid black) approaches. Right: Yield distributions including exozodi distribution uncertainty for each of these approaches; the red curve is identical to the red curve in Fig. 8. Adopting a multi-modal non-parametric fit ($\alpha =0.25$) produces similar yields to the LBTI HOSTS maximum likelihood approach²³ because the fraction of distributions $\lesssim 3$ zodis is similar.

We caution that the choice of priors appears to significantly affect the implied exozodi distribution and ultimately the yield distribution, and the yield distribution is most sensitive to the fraction of the exozodi distribution at very low exozodi levels. Determining which approach is best is difficult without more/better data and is therefore beyond the scope of this paper. However, given that the LBTI data set appears to clearly be multi-modal²³ and that the $\alpha =0.25$ and maximum likelihood approaches largely agree, we proceed with the LBTI HOSTS maximum likelihood approach to estimate exozodi distribution uncertainty, i.e., the solid red curve shown in Fig. 8.

We note that all of the yield distribution curves shown in Figs. 8 and 9 come with several major caveats. For example, our calculations explicitly assume that we can subtract exozodi to the Poisson noise limit without impacting the planet’s signal. While this has been shown to be true for inclined, smooth exozodi less dense than a few tens of zodis,³⁶ it may be difficult for smooth edge-on disks³⁶ as well as edge-on disks with structure.^37,38 In addition, the presence of hot dust near the star^39,40 or cold pseudo-zodi at small projected distances in edge-on disks⁴¹ may cause contrast degradation and make PSF subtraction difficult. We leave these issues for future investigations.

Of all the sources of astrophysical uncertainty we have considered thus far, the exoplanet sampling uncertainty inherent to a blind survey appears to be the dominant term. Our estimate of the impact of this source of uncertainty, which should be regarded as a lower limit, resulted in a mean-normalized standard deviation of 0.21, whereas all other terms combined only increased the mean-normalized standard deviation to 0.26. Assuming uncertainties add in quadrature, this suggests all other terms combined result in a mean-normalized standard deviation of 0.15. The dominant effect of these other sources of uncertainty has been to reduce the expectation value of the yield by nearly 25%, from 22.5 EECs to 17.3 EECs due to observational biases. Assuming we do not find the majority of EECS or measure the majority of target star’s exozodi levels in advance of the HWO mission, most of these uncertainties cannot be mitigated prior to the mission and thus must be budgeted for with yield margin. Next, we consider the one remaining source of astrophysical uncertainty, which will dominate over all other terms, but also has the potential to be somewhat mitigated prior to launch.

3.5.

Uncertainty in η_⊕

The final source of astrophysical uncertainty we consider is the occurrence rate of EECs, ${\eta }_{\oplus }$. While ${\eta }_{\oplus }$ uncertainty may seem like a static source of uncertainty, for a fixed mission lifetime we must budget spectral characterization time for each detected EEC. For example, in the event we detect more planets than expected, we must devote time to characterize them, reducing the time we can spend searching for EECs. We therefore treat ${\eta }_{\oplus }$ as an actionable source of uncertainty.

To incorporate the uncertainty in ${\eta }_{\oplus }$, which we refer to as ${\sigma }_{{\eta }_{\oplus }}$, we use the ${\eta }_{\oplus }$ values from Ref. 22. Reference 22 calculated occurrence rates using the Kepler DR25 exoplanet data catalog⁴² supplemented by Gaia-based stellar and exoplanet properties^43,44 and corrected for catalog completeness and reliability. Reference 22 used a power-law population model with a rate that depended on exoplanet radius, exoplanet insolation flux, and host star effective temperature, with the power law parameters inferred using a Poisson likelihood. To compute ${\eta }_{\oplus }$, we integrate the power law model using the posterior power law parameter values from their analysis of planets in the conservative habitable zone of quiet, isolated FGK main sequence dwarfs. Our domain of integration, defining our rocky habitable zone population, is

Reference 22 accounted for the lack of information about DR25 catalog completeness beyond 500-day orbital periods by computing power law posteriors for two bounding cases: case 1 assumed completeness was zero beyond 500 days, and case 2 assumed the completeness beyond 500 days was equal to the completeness at 500 days. We computed the ${\eta }_{\oplus }$ distribution for each of these cases and created our final ${\eta }_{\oplus }$ distribution by uniformly randomly drawing from both cases.

Performing this computation for each element of our posterior, we get an ${\eta }_{\oplus }$ with mean and 86% confidence interval ${\eta }_{\oplus }={0.26}_{-0.14}^{+0.29}$. The large uncertainties are primarily due to the very small number of detections of habitable zone planets orbiting FGK stars whose planet radius is within our desired range.

To include ${\sigma }_{{\eta }_{\oplus }}$ in our yield calculations, we repeat the 498 calculations performed in Sec. 3.4, but draw a unique value of ${\eta }_{\oplus }$ for each one using the methods described above. Figure 10 shows the results in purple compared to our previous results from Sec. 10 shown in red. While the median of the purple distribution is similar to that of the red distribution, the mean (shown as a vertical purple dotted line) is substantially higher due to a tail of large ${\eta }_{\oplus }$ values, including uncertainty in ${\eta }_{\oplus }$ leads to higher mean yields. Our uncertainty in ${\eta }_{\oplus }$ has a major impact on the breadth of the expected yield distribution. In short, even for missions with an expectation value close to two dozen, the uncertainties in ${\eta }_{\oplus }$ are large enough to produce non-negligible chances of single-digit EEC yields.

Fig. 10

EEC yield distribution of our baseline mission with all known sources of astrophysical uncertainty: exoplanet sampling, albedo, exozodi sampling, exozodi distribution, and ${\eta }_{\oplus }$ uncertainties (purple). The red yield distribution is the same as was calculated in Sec. 3.4, which excludes uncertainty in ${\eta }_{\oplus }$. The dotted purple line marks the mean of the purple distribution. Uncertainties in ${\eta }_{\oplus }$ substantially broaden the EEC yield distribution.

We note that the mean of the red distribution shown in Fig. 10 is $\sim 20\%$ lower than the yield predicted for a 6 m ID telescope by Ref. 2. Although we made several changes to assumed inputs, notably $R=140$ instead of $R=70$ to detect ${\mathrm{H}}_2\mathrm{O}$ and an updated LBTI best-fit exozodi distribution, these changes mostly offset. The majority of the decrease in expected yields is due to the geometric albedo distribution (which decreases yield by $\sim 15\%$) and exozodi distribution uncertainty (which decreases yield by $\sim 4\%$), which were not included in Ref. 2.

In summary, assuming plausible distributions in exoplanet albedo, uncertainty in the HWO EEC yield appears to be dominated by two sources of astrophysical uncertainty. The first is simply exoplanet sampling, which is inherent to a blind survey and may only be partially overcome by precursor detection of the EECs. To be maximally useful, this must happen prior to the design of the mission. The second and most significant source of astrophysical uncertainty is ${\eta }_{\oplus }$. We note that a goal of 25 EECs ignoring uncertainties in ${\eta }_{\oplus }$ is therefore not equivalent to a goal of 100 cumulative HZs. A goal of 100 cumulative HZs implicitly ignores both dominant sources of uncertainty, ${\sigma }_{{\eta }_{\oplus }}$ and exoplanet sampling uncertainties.

The Astro2020 Decadal Survey asserted that a sample size of 25 EECs “provides robustness against the uncertainties in the occurrence rate of Earth-sized worlds and against the vagaries associated with the particular systems near Earth.”¹ Here, robustness can be defined as the probability of achieving a given yield goal. With some minimum yield goal defined, we can use the distributions shown in Fig. 10 to calculate this probability. As the Astro2020 Decadal Report did not define the minimum acceptable yield goal, the meaning of spectral characterization, nor the vagaries of particular systems, we must define them here. We therefore adopt two minimum yield goals throughout the rest of this study: the detection of 25 EECs and subsequent search for water vapor including all sources of astrophysical uncertainty, and the detection of 25 EECs and subsequent search for water vapor including all sources of astrophysical uncertainty except ${\eta }_{\oplus }$ uncertainties. These two goals correspond to the two yield distribution curves shown in Fig. 10.

We define the probability of detecting and searching 25 EECs for water vapor, $P_{25}$, as the fraction of the yield distribution that exceeds 25 EECs. We calculate this quantity for each of the distributions shown in Fig. 10. For our baseline mission parameters with a 6 m inscribed diameter, we find $P_{25}$ is just 6% when ignoring ${\eta }_{\oplus }$ uncertainties and 32% when including ${\eta }_{\oplus }$ uncertainties. In the following section, we investigate modifications to the LUVOIR-B design that could improve scientific performance.

4.1.

Build a Bigger Telescope

As shown by Ref. 5, EEC yield is most sensitive to telescope diameter. Therefore, we study how the distributions shown in Fig. 10 vary with telescope diameter. To do so, we repeat the calculations from Secs. 3.4 and 3.5 for inscribed diameters, IDs, ranging from 6 to 9 m. Figure 12 shows the resulting EEC yield distributions, with solid, dashed, dotted, and dash-dotted lines corresponding to 6, 7, 8, and 9 m IDs, respectively. Purple distributions correspond to those with all known astrophysical uncertainties included and red lines correspond to those without ${\sigma }_{{\eta }_{\oplus }}$ included. Excluding ${\sigma }_{{\eta }_{\oplus }}$, a 9 m ID telescope increases yield by a factor of 2.2, in agreement with power law relationships established by previous works.^5,11

Fig. 12

EEC yield distributions for four different telescope diameters when adopting our baseline mission parameters including ${\eta }_{\oplus }$ uncertainty (purple) and excluding it (red). Solid, dotted, dashed, and dot-dashed lines correspond to IDs of 6, 7, 8, and 9 m, respectively.

Figure 13 shows the targets selected for observation as a function of telescope diameter. These plots assume the same single representative exozodi draw as in Fig. 11. The black dashed line roughly marks the working angle limit of a 6 m ID telescope. As the telescope diameter increases, the working angle limit shifts downward, as shown by the curved red line. Larger telescope diameters allow targets with smaller angular HZs to be selected, whether they are at larger distances or lower luminosities. Notably, none of the scenarios in Fig. 13 “use up” all of the potentially accessible targets. Regardless of aperture size, the LUVOIR-B baseline parameters therefore lead to missions that are limited by exposure times. Only by reducing exposure times can a mission take advantage of the full extent of the target list, a concept we explore in Sec. 4.2.

Fig. 13

Targets selected for observation for one simulation of the 7, 8, and 9 m ID telescope scenarios, color-coded by total completeness, assuming the same single representative exozodi draw as in Fig. 11. The red dashed lines indicate the boundaries of the accessible targets—the horizontal line roughly marks the assumed astrophysical noise floor while the curved line roughly indicates the working angle limit. The black dashed line indicates the working angle limit of a 6 m ID telescope for reference. Larger telescopes expand the range of accessible targets.

Figure 14 shows the distribution of possible spectral characterization times to detect water vapor absorption for all 498 yield calculations performed. We only show the scenario in which ${\eta }_{\oplus }$ uncertainty is excluded. Of course, as we shorten exposure times by going to larger telescopes, more observations can be included that will necessarily be around more challenging targets and extend the distributions, making it difficult to see the exposure time impacts for the highest priority stars. Therefore, to make the distributions shown in Fig. 14, we consider only the first 18 EECs of any simulation.

Fig. 14

Possible spectral characterization time distributions of the first 18 EECS for our baseline mission parameters with a 6, 7, 8, and 9 m ID telescope shown in solid, dotted, dashed, and dash-dotted lines, respectively. A 9 m ID telescope reduces spectral characterization times by a factor of 6.

Not surprisingly, the spectral characterization time distribution of the telescope with a 9 m ID is more sharply peaked toward shorter exposure times. In the non-background-limited regime, exposure times should scale as $D^{-2}$, whereas in the background-limited regime they should scale as $D^{-4}$. If all else were equal, we would therefore expect exposure times for our highest priority targets to be reduced by a factor of 2.3 to 5.1 when increasing the telescope diameter from 6 to 9 m. However, all else being equal, larger telescopes also provide higher coronagraphic throughput at a fixed angular separation for the DMVC, as well as an expanded target list to choose from, further reducing exposure times. We find the mean exposure time decreases by a factor of 6.2 going from 6 to 9 m ID; the mean spectral characterization time of the first 18 EECs is 22 days for a 6 m ID telescope but can be shortened to just 3.5 days for a 9 m ID telescope.

We calculate $P_{25}$ for each of the distributions shown in Fig. 12 and plot it as a function of telescope diameter in Fig. 15. For our baseline mission parameters, an $\sim 8.2\mathrm{m}$ ID telescope can achieve $P_{25}>90\%$ when ignoring uncertainties in ${\eta }_{\oplus }$, but even a 9 m ID telescope cannot achieve $P_{25}\gtrsim 80\%$ when budgeting for ${\eta }_{\oplus }$ uncertainties. Unless we consider diameters significantly larger than 6 m, telescope diameter alone cannot build in robust science margin for uncertainty in ${\eta }_{\oplus }$ under our baseline mission parameters. In the following section, we investigate modifications to the LUVOIR-B design that could improve scientific performance without increasing the primary mirror diameter.

Fig. 15

Fraction of yield distribution $>25$ EECs, $P_{25}$, when budgeting for detection and water vapor characterization, as a function of telescope diameter for our baseline mission parameters. The purple line includes ${\eta }_{\oplus }$ uncertainty and the red line excludes it.

4.2.

Improve the Mission Design

There are a number of changes to the LUVOIR-B baseline design that could improve the quantity and quality of data. Some of these are relatively straightforward changes, whereas others require the development of new technologies. Here, we highlight six possible improvements, noting that many others likely exist. We examine each design change one at a time, starting from the simplest and adding them up as we go. All of these changes will focus on reductions in exposure time. While previous works have shown that improvements to parameters controlling exposure time have only a modest impact on the EEC yield,^5,11 these impacts will ultimately compile, resulting in significant changes to the expected EEC yield. The end result demonstrates that some design changes are truly synergistic.

4.2.1.

Scenario A: minimize aluminum reflections

The LUVOIR-B design adopted a three-mirror anastigmat optical telescope assembly (OTA) with a fourth fast-steering mirror. Three additional pre-coronagraph optics were necessary prior to the UV-VIS channel dichroic. All seven of these mirrors were aluminum coated. Typical reflectivities for protected aluminum-coated mirrors are $\sim 90\%$, 87%, and 92% at 500, 760, and 1000 nm, respectively. Silver-coated mirrors have reflectivities of $\sim 98\%$, 97%, and 96% at the same three wavelengths. Just five aluminum-coated mirrors will reduce throughput at 760 nm, a key wavelength for detection of molecular oxygen (a biosignature gas), by almost a factor of two compared to silver. As every exoplanet photon is precious, we should strive to reduce the number of aluminum-coated mirrors. We note that Ref. 2 estimated the impact of some of these changes already; here, we break these choices down in detail.

Our first potential design change is to adopt a Cassegrain telescope with only two aluminum-coated telescope mirrors. These two mirrors preserve UV science for any other instrument in the observatory, including a UV coronagraph. While it may still be possible to operate a UV coronagraph in parallel with a VIS coronagraph under this assumption, here we make the conservative assumption that the UV coronagraph cannot be parallelized, simply to illustrate a point: in terms of EEC yield, the increased throughput will more than make up for the lack of a parallelized UV channel. Figure 16 shows the optical layout adopted for this design change and includes both IFS and broadband imaging modes for the single VIS channel. A possible UV channel is not shown. We assume dual polarization operation, but do not show the potential split needed for separate polarization channels; a detailed study of the need for separate polarization channels is beyond the scope of this paper.

Fig. 16

Optical layout for a Cassegrain design with a single VIS channel. We do not explicitly show dual parallel polarization channels, which we assume for the baseline coronagraph design. See Fig. 1 for legend.

Without a static dichroic present to split the parallel UV and VIS wavelengths, we are now able to implement the detection bandpass optimization feature within AYO. As shown by Ref. 8, this usually results in V band detections for the majority of stars but can select longer wavelengths for nearby and late-type stars. We carry this feature forward through the rest of the analyses in this study.

Figure 17 shows the end-to-end optical throughput (not including the coronagraph’s core throughput) of this Cassegrain VIS channel (red) compared with the LUVOIR-B baseline (black). At 500 nm, where most detections will be performed, the optical throughput is now $\sim 1.7\times$ that of our baseline design’s VIS coronagraph and is $1.2\times$ the effective throughput of the baseline design’s combined parallel UV and VIS coronagraphs.

Fig. 17

Wavelength-dependent optical throughput for a Cassegrain design with a single VIS channel imager and IFS (red), an ERD (green), and the LUVOIR-B baseline (black). Minimizing aluminum reflections can increase the throughput by 50% at 1000 nm and nearly double it at $\sim 700\mathrm{nm}$. Opting for an ERD can increase the throughput by an additional 40% compared with an IFS.

Figure 18 shows the impact of this change on EEC yield. The red curve shows the estimated yield distribution for scenario A compared to the LUVOIR-B baseline yield distribution in black. Dashed and solid lines correspond to including and excluding ${\eta }_{\oplus }$ uncertainties, respectively. Despite assuming only a single coronagraph channel, by minimizing the number of aluminum reflections, the EEC yield increases by 15%. The bar plots on the right in Fig. 18 show $P_{25}$ increases as well to 0.18 and 0.41 when excluding and including ${\sigma }_{{\eta }_{\oplus }}$, respectively.

Fig. 18

Left: Yield distributions for all design change scenarios considered, with dotted and solid lines including and excluding ${\eta }_{\oplus }$ uncertainty, respectively. Table 4 summarizes the design changes included in each scenario. Right: Fraction of yield distribution $>25$ EECs, $P_{25}$, when budgeting for detection and water vapor characterization. Dotted and solid bar plots correspond to including and excluding ${\eta }_{\oplus }$ uncertainty, respectively.

This change also improves exposure times in the visible channel. Figure 19 shows the spectral characterization time distribution for scenario A (red) compared to the LUVOIR-B baseline (black). The throughput at $\sim 700\mathrm{nm}$ is nearly twice that of our baseline design and is 50% greater at 1000 nm (cf. Fig. 17). Ultimately, this translates into mean characterization exposure times $1.4\times$ shorter.

Fig. 19

Spectral characterization time distributions for all design change scenarios considered when excluding ${\eta }_{\oplus }$ uncertainty. Table 4 summarizes the design changes included in each scenario.

The top-left panel of Fig. 20 plots each target selected for observation in scenario A, color-coded by the total completeness achieved on each target. The horizontal dashed line indicates the stellar luminosity at which a $1.4\hspace{0.17em}{\mathrm{R}}_{\oplus }$ planet at quadrature has a $\mathrm{\Delta }\mathrm{mag}$ equal to the assumed astrophysical noise floor, $\mathrm{\Delta }{\mathrm{mag}}_{\text{floor}}$. The curved dashed line indicates the luminosity at which the outer edge of the HZ (1.67 AU) is located at $1.5\lambda /D$ for $\lambda =1000\mathrm{nm}$. These two dashed lines roughly indicate the boundaries of the accessible target list. The stars at the top-right corner of this “wedge” of targets require longer exposure times. Therefore, expanding the yield without reducing the IWA requires shorter detection times.

Fig. 20

The targets selected for each scenario color-coded by total completeness. The scenarios are cumulative, in that we compile multiple design improvements. The dashed lines indicate the boundaries of the accessible targets—the horizontal line roughly marks the assumed astrophysical noise floor while the curved line roughly indicates the working angle limit.

The top-left panel of Fig. 21 shows the incremental change in the completeness of each star for scenario A compared with the baseline LUVOIR-B scenario. Blue indicates an increase in completeness, whereas red indicates a decline in completeness. The scale has been stretched to emphasize the sign of the change, as shown by the color bar on the right. As shown in Fig. 21, as the mission becomes more capable, the optimal redistribution of exposure time toward what were previously more challenging stars slightly reduces the completeness of nearby stars. While it may seem counter-intuitive to reduce exposure times of nearby stars in favor of more distant stars, this effect is the result of the less capable missions having “over-invested” time in nearby stars due to exposure time limitations.

Fig. 21

The incremental change in completeness between successive scenarios. Each design improvement allows the mission to access more distant stars.

4.2.4.

Scenario D: improve detector performance

The LUVOIR and HabEx studies baselined an EMCCD as the visible wavelength coronagraph detector. The adopted parameters for this EMCCD were optimistic, based on assumed future improvements to the Roman Coronagraph EMCCD. We have carried these assumptions through to this study, as shown in Table 2. Some of those assumptions have proven true. Roman’s EMCCD dark current values are on par with the LUVOIR-B baseline assumption of $3\times {10}^{-5}\text{counts}{\mathrm{pix}}^{-1}{\mathrm{s}}^{-1}$ and the dQE terms budgeting for photon counting efficiency, cosmic ray efficiency, hot pixel efficiency factors, etc., as defined by Ref. 52, are estimated to be $\sim 0.77$ at end of life,⁵² consistent with LUVOIR assumptions. Other assumptions remain optimistic. These include CIC, which remains a factor of 10 higher than the LUVOIR assumptions, and most notably, the raw QE near 1000 nm, which is only a few percent for Roman’s EMCCD²⁷ but was assumed to be 90% in the LUVOIR study. Despite this, there remain paths forward using an EMCCD. The desire for high QE near 1000 nm was motivated by the desire for efficient detection of water vapor, but Ref. 8 showed that low QE near 1000 nm can be partially mitigated by searching for water at shorter wavelengths. Alternatively, a dedicated ultra-low-noise NIR detector could be more appropriate for the detection of water vapor.

Several alternative detector technologies may improve performance beyond the LUVOIR-B assumptions. Here, we examine the potential benefits of such detectors. A thorough examination of the impact of different detector technologies, critical to the success of HWO, would require an exhaustive study comparing all detector options, which is beyond the scope of this paper. We therefore choose to adopt parameters consistent with two possible detector options. We start with performance parameters that may be possible with a photon-counting Skipper CCD. Table 3 summarizes the performance parameters we adopted compared to the LUVOIR-B baseline. We adopt a dark current one order of magnitude lower than the LUVOIR assumptions, which has been demonstrated for the Skipper CCD.^53,54 We note that this reduction in dark current stems from a combination of a high degree of shielding and cosmic ray identification and removal, which may be possible for a traditional EMCCD as well. However, the Skipper CCD also has a clock-induced charge $10\times$ better than LUVOIR assumptions and has negligible dQE. As a result, the parameters we adopt provide $\sim 30\%$ higher effective throughput, as well as noise properties that are effectively negligible.

Table 3

Detector parameters.

Parameter	Units	EMCCD	Skipper	ERD
QE	—	0.9	See Fig. 23	0.9
dQE	—	0.75	0.99	1.0
DC	counts ${\mathrm{pix}}^{-1}{\mathrm{s}}^{-1}$	$3\times {10}^{-5}$	$6.8\times {10}^{-9}$	0
CIC	counts ${\mathrm{pix}}^{-1}{\text{frame}}^{-1}$	$1.3\times {10}^{-3}$	$1.5\times {10}^{-4}$	0
RN	counts ${\mathrm{pix}}^{-1}{\text{read}}^{-1}$	0	0	0
Scenarios	—	0, A, B, C	D	E, F

Skipper CCDs use multiple non-destructive reads to average read noise down to deeply sub-electron levels and thereby count photons.⁵⁵ Recent advances in semiconductor fabrication technology have made possible thick, fully depleted, photon-counting p-channel Skipper CCDs such as those used by Ref. 55. These p-channel Skippers, developed at the Lawrence Berkeley National Laboratory (LBNL), have important advantages for space astrophysics including excellent radiation tolerance and QE greater than 80% at 940 nm. However, the primary challenge for using LBNL’s Skipper CCDs in space is radiation nonetheless. Although p-channel Skipper CCDs do not degrade in space like n-channel CCDs, they require short exposure times to minimize cosmic ray disturbance. This is on account of the $\approx 200\mu \mathrm{m}$ thick silicon that is used to achieve good near-IR QE. As a practical matter, p-channel Skipper exposure time will need to be on the order of 1 mi to limit cosmic ray disturbance to about 10% of pixels, requiring additional amplifier outputs—the recently developed Multi-Amplifier Sensing CCD⁵⁶ is a step toward this.

We define scenario D as scenario C with the EMCCD’s performance parameters replaced with the Skipper CCD parameters listed in Table 3. Here, we define QE as the traditional QE, i.e., the number of photoelectron groups created per photon received. We define dQE as the “detective” QE, a factor specific to EMCCDs discussed in Ref. 52.

Figure 22 shows the optical layout of scenario D and Fig. 23 shows our adopted raw QE curve for the Skipper CCD as a black solid line. For yield calculations, we are interested in a bandpass-integrated QE. For exoplanet detections, we assume the detection wavelength is centered in the bandpass as usual and integrated over a bandwidth of 20%, resulting in the blue dashed line that we adopt for detection raw QE. For spectral characterizations, we work with the longest wavelength of the bandpass to ensure the exoplanet is exterior to the coronagraph IWA over the whole bandpass. Using the single QE value at this wavelength would be doubly conservative, as the QE curve drops rapidly near 1000 nm. Thus, for spectral characterizations, we integrate the QE over a 20% bandpass with the longest wavelength given by the $x$-axis of Fig. 23, resulting in the red dashed line. We note that this detail is critical: the bandpass-integrated QE over the water vapor absorption feature near 1000 nm is twice that of the QE at 1000 nm. Future studies should investigate the impact of realistic QE curves on water vapor retrievals near 1000 nm.

Fig. 22

Optical layout for a Cassegrain design with two parallel VIS channels. We do not explicitly show dual parallel polarization channels for each wavelength channel, which we assume for the baseline coronagraph design. See Fig. 1 for legend.

Fig. 23

Skipper CCD raw QE (solid line) and the raw QE values adopted for the yield code when integrating over a 20% bandpass for detection (blue dashed) and spectral characterization (red dashed). The raw QE for the LUVOIR-B baseline EMCCD and ERD (scenario E) is shown for comparison (dotted line).

Figure 18 shows the resulting yield distribution for scenario D in green. EEC yield increases by 17%, and $P_{25}$ increases to 0.94 and 0.71 when excluding and including ${\sigma }_{{\eta }_{\oplus }}$, respectively. Figure 19 shows that the characterization times decrease as well by an additional factor of 1.5.

4.2.5.

Scenario E: adopt an energy-resolving detector

Next, we show the impact of swapping the Skipper CCD and IFS with a noiseless ERD. An IFS uses a lenslet at each “pixel” in the image plane to focus light onto a dispersing element, ultimately producing spectra of each image plane pixel in the final detector plane. While this instrument provides spatially resolved spectra over the entire field of view, there are some disadvantages. First, the additional IFS optics have a throughput estimated to be a factor of 0.7 (c.f., Table 2). Second, the PSF’s core is spread over potentially hundreds of pixels at the long wavelength end of the channel, effectively amplifying the detector’s per-pixel noise properties.

With an ERD, read noise manifests as part of the energy resolution budget⁵⁷ and there is no need for an IFS, eliminating throughput-reducing optics. Reference 10 showed that this, combined with negligible dQE, could result in a 30% increase in EEC yield compared with the LUVOIR-B baseline. Here, we examine the incremental improvement of an ERD compared with a Skipper CCD-based IFS, which also has negligible dQE and noise properties, so we will not see the same 30% increase in EEC yield.

Figure 24 illustrates the instrument layout with an ERD. We note that Fig. 24 does not show any optics to thermally isolate the detector from the rest of the instrument, which will be required for an ERD. These optics will reduce the throughput of the system, but a detailed assessment of thermal isolation and the transmissivities of the necessary optics is beyond the scope of this paper. Our estimate of the impact of an ERD should therefore be considered an upper limit.

Fig. 24

Optical layout for a Cassegrain design with two parallel VIS channels with ERD. See Fig. 1 for legend.

Figure 17 shows the assumed optical throughput using an ERD (green). Because there is no separate imaging mode for an ERD, the green line illustrates the throughput for detections and for spectroscopy. An ERD can potentially increase spectroscopic throughput by 40% compared with an IFS. Table 3 summarizes the adopted performance parameters for our ERD, based on the expected performance of a transition edge sensor.⁵⁷ To date, TES arrays have demonstrated $R=90$ at 485 nm⁵⁸ and MKIDS have demonstrated $R=52$ at 402 nm.⁵⁹ We assume that a future ERD can achieve the $R=140$ requirement that we adopt.

Figure 18 shows the resulting yield distribution for scenario E in blue. The ERD increases EEC yield by 9% compared with the Skipper CCD scenario and $P_{25}$ increases to 0.98 and 0.75 when excluding and including ${\sigma }_{{\eta }_{\oplus }}$, respectively. The ERD also reduces spectral characterization times compared with the Skipper CCD scenario by another factor of $1.4\times$, as shown in Fig. 19. Most of these improvements are the result of the increased throughput due to lack of IFS optics, not reduced detector noise, as the adopted noise parameters for the Skipper were already very low. We note that when compared with the LUVOIR-B baseline detector assumptions, the ERD gains are larger, with a $2.2\times$ reduction in characterization times and a $1.3\times$ gain in EEC yield.

4.2.6.

Scenario F: adopt a high-throughput coronagraph

The LUVOIR-B study baselined a DM-assisted charge six vortex coronagraph (DMVC6). Figure 25 shows the azimuthally averaged contrast for a $0.1\lambda /D$ star as a dotted black line over a 20% bandwidth for a single polarization. We note that while the DMVC only works for a single polarization, we have implicitly assumed so far that the design allows for parallel polarization channels. The solid black line shows the core throughput of this coronagraph. The IWA for the DMVC6 is $\sim 3.5\lambda /D$, where the core throughput reaches half its maximum value. We remind the reader that $D$ in this study and that adopted for the $x$-axis of Fig. 25, is the circumscribed diameter of the telescope. The DMVC6 is limited to using the inscribed diameter of the telescope, making its IWA in Fig. 25 larger than that of a circular aperture. Notably, there is a useful throughput interior to the IWA, which the yield code takes advantage of for nearby, later-type stars (see Fig. 11). While the core throughput reaches a relatively high maximum value of $\sim 0.45$, it rises fairly slowly with working angle, such that it is $\sim 5\%$ at $2\lambda /D$.

Fig. 25

Azimuthally averaged contrast for a star with diameter $0.1\lambda /D$ (dotted line) and core throughput (solid line) for the two coronagraphs included in this study. The $x$-axis is in units of $\lambda /D$, where $D$ is the circumscribed diameter of the telescope. The DMVC6 and PIAA-FPM2.5 are shown in black and red, respectively.

Here, we examine one possible alternative coronagraph design that has higher throughput at small working angles: the phase-induced amplitude apodizer (PIAA) coronagraph. We adopt a PIAA design created for the LUVOIR-B aperture, which incorporated a focal plane mask with radius $2.5\lambda /D$ and a DM-assisted solution robust to stellar diameters as large as $0.1\lambda /D$ (we note that this DM solution may help other coronagraph designs as well). We refer to this design as PIAA-FPM2.5. Figure 25 shows the azimuthally averaged performance of the PIAA-FPM2.5 in red. In comparison with the DMVC6, the contrast near $\sim 4\lambda /D$ is a factor of $\sim 4$ worse and the effective OWA is notably limited by contrast degradations to $\sim 20\lambda /D$. Notably, we maintain the same uniform noise floor used throughout this study, which is not proportional to the raw contrast. We choose this single, non-ideal coronagraph design as an example to illustrate a point: despite the degraded contrast, if the noise floor remains the same, the increase in core throughput at small working angles will more than compensate on a survey scale, resulting in significantly improved yields. This is because exposure times are reduced for targets in which the leaked starlight does not dominate the background count rate (e.g., distant stars).

Unlike the DMVC6, the PIAA-FPM2.5 works for both polarizations simultaneously. This has the potential to reduce the complexity of the instrument by eliminating parallel polarization channels. In principle, the complexity could be maintained and the parallel polarization channels could be replaced by another dichroic split, doubling the total instrument bandwidth. However, here, we make the conservative assumption that dual polarization channels are still required to minimize polarization cross-talk and do not adopt any increase in total instrument bandpass.

The purple curve in Fig. 18 shows the yield distribution for Scenario F. In spite of the degraded contrast, the higher throughput of the PIAA-FPM2.5 coronagraph design increases EEC yield by 30%. Combined with all previously discussed changes, $P_{25}$ is now estimated to be 0.99 and 0.85 when excluding and including ${\sigma }_{{\eta }_{\oplus }}$, respectively. The PIAA-FPM2.5 reduces spectral characterization times compared with scenario E by another factor of $1.6\times$, as shown in Fig. 19.

We note that many other coronagraph designs with smaller IWA and higher core throughput exist in addition to the PIAA-FPM2.5 examined here. Such coronagraphs may also lead to higher yields to varying degrees. A simple example that we do not consider in this paper is combining our baseline charge 6 DMVC with a charge 4 design. A future trade study building off of the Coronagraph Design Survey⁶⁰ and examining a broad range of coronagraphs designed specifically for HWO would be highly valuable.

5.1.

Budgeting for Astrophysical Uncertainty

Precisely how much science margin is required for HWO to budget for astrophysical uncertainty? This depends in large part on HWO’s risk posture, formalized minimum yield goal, whether HWO’s formal science goals should account for uncertainty in ${\eta }_{\oplus }$, the magnitude of that uncertainty, and whether the uncertainty can be reduced with precursor science observations or analyses. These decisions will likely come out of future formalized HWO modeling efforts and are beyond the scope of this paper. However, we can use the results of Secs. 4.1 and 4.2 to provide basic guidance for these future decisions by relating the expectation value of EEC yield to $P_{25}$. The solid lines in Fig. 26 show $P_{25}$ as a function of mean EEC yield excluding and including uncertainty in ${\eta }_{\oplus }$ in red and purple, respectively. The yields obtained by increasing telescope diameter (Sec. 4.1) are shown as filled circles, whereas the yields from improving mission design (Sec. 4.2) are shown as filled triangles with a connecting line. The agreement between the lines and circles suggests that $P_{25}$ is independent of the specific means used to obtain higher yields. Figure 26 can therefore be used to estimate $P_{25}$ for a broad range of HWO trade studies by calculating the expectation value of the EEC yield distribution.

Fig. 26

Fraction of yield distributions $>25$ EECS, $P_{25}$, as a function of expected EEC yield excluding (solid red) and including (solid purple) uncertainty in ${\eta }_{\oplus }$. Filled circles indicate yields obtained by changes to telescope diameter while filled triangles with a connecting line indicate those due to design changes. The agreement between lines and circles suggests $P_{25}$ is independent of the means used to obtain higher yields. The unfilled symbols and dashed lines show the expectation value of the benchmark yield, $Y^{\prime }$, which can be used to estimate $P_{25}$ without calculating a yield distribution.

We note that calculating the yield distribution including all astrophysical noise sources is critical, as it includes shifts in the expectation value of the yield due to observational biases induced by exoplanet albedo and exozodi uncertainties. However, in practice, calculating a yield distribution is numerically taxing, as it requires hundreds of independent yield calculations to sample the range of possible exozodi and ${\eta }_{\oplus }$ values, and calculating yield distributions including all sources of astrophysical noise could slow trade studies. To aid with this, we “translate” each of the filled points in Fig. 26 to a much simpler quantity that only requires a single yield calculation, a benchmark yield, $Y^{\prime }$. We define $Y^{\prime }$ as the yield assuming 3 zodis of dust around all stars, ${\eta }_{\oplus }=0.24$, and no albedo or exozodi observational biases included. The empty circles and triangles in Fig. 26, along with the thin dashed line, show $Y^{\prime }$ for each of the scenarios shown as filled symbols. Using the dashed curves in Fig. 26, one can design a mission to achieve a given $P_{25}$ via a single yield calculation, knowing that when astrophysical uncertainties are included the mean yields will shift to the solid curves.

5.2.

Budgeting for Performance Uncertainty

Science margin can also help reduce risk by budgeting for performance uncertainties. There are unlimited potential causes of on-sky performance degradations, manifesting as, e.g., poor line of sight jitter, coating or detector degradations, and unexpected stray light. Here, we do not focus on root causes. Instead, we discuss performance degradation in terms of bulk parameters used in our yield analyses, such as raw contrast and throughput.

As originally shown by Ref. 5 and later verified by Ref. 2 with updated fidelity, EEC yield decreases relatively gracefully with degradations in most parameters. For the LUVOIR-B baseline parameters, yield is only moderately sensitive to throughput-related terms (scaling roughly to the $\sim 0.37{\text{power}}^2$); a relatively large factor of two reductions in effective throughput would only reduce yield by $\sim 25\%$, though it would increase exposure times by roughly a factor of two. Assuming a noise floor independent of raw contrast, LUVOIR-B was also very insensitive to raw contrast (scaling as raw contrast to the $\sim -0.07{\text{power}}^2$); a factor of two degradation in raw contrast would only reduce yield by $\sim 5\%$. To first order, assuming the noise floor is not coupled to raw contrast, degradations in these quantities slow the progress of observing the full target list, but do not limit the boundaries of accessible targets (indicated by the red dashed lines shown in Fig. 20). Because this can be partially mitigated by additional survey time, we do not consider these parameters as substantial drivers of performance risk.

We posit that the astrophysical noise floor, $\mathrm{\Delta }{\mathrm{mag}}_{\text{floor}}$, is a primary driver of performance risk. The noise floor is ultimately determined by one of the largest technological “tall poles:” the ability to precisely estimate the coronagraphic speckle pattern, which may require picometer stability,⁶ very precise wavefront sensing if model-based PSF subtraction is used, or a combination of the two. As shown by Fig. 8 in Ref. 11, for $\mathrm{\Delta }{\mathrm{mag}}_{\text{floor}}=26.5$ as assumed in this study, degradations of a factor of two in the flux associated with the noise floor could lead to reductions in EEC yield of $\sim 25\%$, on par with throughput factors. However, the impacts of the noise floor differ from throughput factors in two important ways. The first is that the yield’s sensitivity to the noise floor appears to be a “cliff” (c.f. Fig. 8 in Ref. 11), suggesting that additional degradations beyond the initial factor of two become significantly more costly—i.e., it is a slippery slope.

The second is more fundamental: degradations in the noise floor directly limit the range of accessible targets. The horizontal dashed lines in Fig. 20 mark the luminosity at which a $1.4R_{\oplus }$ planet at quadrature at the EEID has a flux equal to the astrophysical noise floor. This is a rough visual guide. In reality, because planets could occupy a range of phase angles, semi-major axes, and radii, the effects of the noise floor can be seen in Fig. 20 as a broad horizontal band that reduces the completeness of targets with luminosities $\gtrsim 2L_{\odot }$. Therefore, reductions in the noise floor will move this band to lower luminosities, directly removing targets that cannot be accessed in any other way. Further, reductions in the noise floor would begin to remove the most Sun-like stars, G-type stars, from the target list, substantially affecting the mission’s ability to survey for Earth-like planets around Sun-like stars.

One approach to mitigating the risk of degradation in the noise floor is to adopt technologies that relax the system level requirements needed to achieve the desired noise floor. Any technology that relaxes optical stability requirements would aid in this. Of particular note is the model-based PSF subtraction method we considered in scenario C. Model-based PSF subtraction monitors the wavefront error at a high cadence so that we can reconstruct the instantaneous science PSF in high fidelity. This would allow us to discard photons collected at times with poor WFE or at least accurately model how the science PSF varies with WFE to a level better than the Poisson noise. It also would remove the need to maintain an ultrastable WF in the presence of a spacecraft slew or roll, which would be required for the RDI and ADI PSF subtraction methods, respectively. Much work is needed to determine if model-based PSF subtraction is viable for HWO, but the benefit could be significant.

Another approach to mitigating the risk of noise floor degradation is to budget for it with a science margin. If degrading the noise floor moves the horizontal dashed lines in Fig. 20 downward, we could maintain our access to a large pool of stars by shifting the curved dashed line downward as well. In other words, the target list would shift toward later type stars. The curved dashed line in Fig. 20 marks where a working angle of $1.5\lambda /D$ is at the outer edge of the HZ. This working angle is admittedly fairly extreme already, and it is unlikely that alternative coronagraph designs could reduce it further in units of $\lambda /D$. In addition, as discussed in Sec. 3.4, there is likely a minimum “useful” working angle set by spatial resolution requirements interior to which planets blend together too often⁴⁵—this may be larger than the $1.5\lambda /D$ shown in Fig. 20. There are therefore only two ways of shifting the curved dashed line downward: by operating at shorter $\lambda$ or larger $D$. Reference 8 showed that detecting water at shorter $\lambda$ is possible, but not preferable, as it requires significantly longer exposure times. Thus, we conclude that an increase in telescope diameter should be considered as an option to budget for noise floor degradation.