3.1.

Star, Planet Types and Ranges

In this section, we explore which star and planet types and distances, are accessible for starshade-based missions. To establish representative exposure times for the case of an Earth-like planet and other Solar System planets, placed in their respective orbits of scaled to the star luminosity. We are interested in determining the relationship between exposure time and target distance such that an SNR of 7 can be obtained, for star distances of 3 to 25 pc and exposure times of up to 24 h, for different planet types and we present the results at Fig. 2.

Fig. 2

Exposure time required to reach an SNR of 7 as a function of distance for each mission and star type: The black, blue, green, red, and purple curves correspond to Mars, Earth, Venus, Jupiter, and Saturn, respectively. Each planet has their orbit scaled by the luminosity of each star type. The exozodi factor was set to 0 for these simulations, and the local zodiacal dust present within the solar system was applied to add an additional realistic background flux source. Solar glint was not used for these simulations. If a curve for one of the planets is not shown, then it is able to maintain an $\mathrm{SNR}>7$ for the all distances and exposure times displayed in the figure.

We extract from Fig. 2 representative exposure times for the three missions, such that 85% of target distance are covered for each star type: (1) 24 h for NGRST, (2) 10 h for HabEx, and (3) 30 min for R-O. The imaging bands for each mission are 615 to 800 nm for NGRST, and 600 to 800 nm for HabEx and R-O. From this section forward, these exposure times and bands will be used throughout the paper unless specifically stated otherwise.

In Fig. 3, we display the imaging simulations for F5V-G5V-K5V-M0V stars, for a Venus-Earth-Mars-Jupiter-Saturn system at a distance of 8 pc, with an exo-zodi level of 4.5, and an inclination of 0 deg. For each of the twelve images, we provide the exposure time and resultant SNR for every one of the planets. These analyses are conducted assuming an infinite outer working angle (OWA).

Fig. 3

This figure explores how the Solar System would appear through each starshade mission and with four different host stars that have an exozodiacal dust disk 4.5 times the brightness of the local zodiacal dust around the Sun. Each system possesses Venus, Mars, Earth, Jupiter, and Saturn; their relative positions are labeled in the top row and are held constant for all proceeding rows. The broadband SNR is provided for all planets at the top left of each image and the exposure time. If the inner solar system planets have an $\mathrm{SNR}>7$, we provide a closer look in the bottom-right corner of the image to clearly identify the planets. The top-right image provides a closer look for both Jupiter and Saturn because the strong angular resolution makes it difficult to identify the outer planets. Each column corresponds to one of the starshade missions: NGRST, HabEx, and R-O, respectively, and each row corresponds to a different star spectral type: F5V, G5V, K5V, and M0V, respectively. As the host star brightness decreases, the planets’ semi-major axis grows smaller due to the luminosity scaling $\sqrt{L_{\text{star}}}$. The horizontal and vertical axes are in units of Astronomical Units (AU). Solar glint was not included within these simulations.

We finalize this section by combining in Fig. 4, a map that accommodates all main sequence GAIA stars within 25 Pc, as a function of the target star distance and luminosity. We then project the starshade angular size for each mission, such that all observable target stars could accommodate viewing an Earth-like planet at 1 AU, scaled to the star luminosity. We then mark the distance at which each mission could reach an SNR of 7 as a function of a target star luminosity and distance. This identifies which stars could be observed by each mission given the allotted exposure time.

Fig. 4

Accessible target stars for each mission, for the allotted exposure time, $\mathrm{SNR}=7$, and starshade Inner Working Angle (IWA): The black, green and blue curves correspond to R-O, HabEx, and NGRST, respectively. Solid curves correspond to each mission starshade IWA for a planet positioned at 1 AU scaled to the star luminosity. The colored regions correspond to the distance at which an Earth-like planet has an SNR equal to 7 for each mission. We provide for context stars from the Gaia DR2 in gray to identify where stars are located, and the names of stars selected as potential targets for starshade missions, and the total number of stars that have habitable zones outside of the IWA of each starshade mission at the top of the image, and the number of stars within the $\mathrm{SNR}=7$ curve on the right side of the image.

Figure 4 will evolve as we account for additional parameters such as planet phases in Fig. 6, exo-zodiacal dust in Fig. 10, and starshade perturbations in Fig. 17 to show the change in accessible target stars as each parameter is accumulated.

3.2.

Planet Phase

In this section, we present the imaging detection sensitivity to the planet phase when observed, driven by the Keplerian motion of the planet and the system inclination relative to the line of sight. In Fig. 5, we show for four target stars at different distances, the regions in which SNR is smaller than 7 as a function of system inclination and the planet argument of periapsis (phase).

Fig. 5

Earth-like planet observability as a function of phase for the three missions for four target stars: Each map is constructed of $\sim 500$ independent imaging simulations. In each map, the vertical axis corresponds to the orbital inclination in degrees. 0 deg represents a system that is face-on with respect to the observer, and 90 deg is edge-on. The horizontal axis is the planet’s argument of periapsis in degrees. There are two regions present in every image; regions in gray identify where the planet reaches an $\mathrm{SNR}\ge 7$, and black regions identify where it cannot maintain an SNR of 7.

Phase completeness, which in this paper indicates the chance an observer could reach an SNR of 7 at the allotted exposure time for a random planet phase, is a useful quantity that is taken into account where the observer is interested in maximizing the chance to observe a target. Therefore, we repeat the analysis conducted in Fig. 4 for the relevant statistical distribution of system inclinations, but now project for each mission the curves for a phase completeness value of 0.35 as can be seen in Fig. 6.

The curves for each mission indicate which luminosity and distances are within reach if such a requirement is imposed; the completeness curves that are presented in this figure will evolve throughout the paper as more natural, and system parameters will be taken into account to reflect the impact each parameter will have on this map.

Fig. 6

Accessible target stars for each mission, for the allotted exposure time, where the phase completeness is larger than 0.35: The black, green and blue curves correspond to R-O, HabEx, NGRST, respectively. Solid curves correspond to each mission starshade IWA for a planet positioned at 1 AU scaled to the star luminosity. The dashed lines correspond to the distance at which an Earth-like planet has a completeness of 0.35 for each mission. We provide for context stars from the Gaia DR2 in gray to identify where stars are located, the names of stars selected as potential targets for starshade missions, the total number of stars that have habitable zones outside of the IWA of each starshade mission at the top of the image, and the number of stars within phase completeness requirement curve on the right side of the image. The semi-major axis remains constant at 1 AU scaled by luminosity, the planet radius is set to $1R_{\oplus }$, planetary albedo is set to 0.3, and the exozodi factor is set to 0.

3.3.

Exo-zodi Dust and Resonant Dust Rings

Exo-zodiacal dust could become a major noise source and interfere with the imaging of exoplanets.¹⁸ This could be further complicated by dust structures in the observed system^19,20 as is evident by both the leading and trailing dust clumps near Earth. We first investigate the case of a symmetric disk, with a half peak brightness at 0.1 AU, similar to our own solar system. We open by maintaining such favorable assumptions and examine how the regions in Fig. 7 will change for the nearby target star Zeta Tucanae, which has a corresponding Hipparcos catalog ID of HIP 1599 for exo-zodi levels of 0, 3, 11.8, and 21, which represent a no noise, optimistic median, pessimistic median, and the $3\sigma$ optimistic scenario value respectively from the latest HOSTS survey²¹ and present the results in Fig. 7.

Fig. 7.

For all three starshade missions, four different exozodi factors have been applied to the HIP 1599 system, a G0V star 8.6 pc away: 0, 3, 11.8, and 21. We identify how these exozodi levels effect the regions identified in Fig. 5. Each column corresponds to one of the four exozodi levels, and each row corresponds to one of the three starshade missions. Similar to Fig. 5, the horizontal axis is the argument of periapsis in degrees for the planet, and the vertical axis is the orientation of the system. The exozodiacal dust disk maintains the same inclination as the orbital plane. The local zodiacal dust remains constant for all simulations. The solar glint is not activated for these simulations. About $\sim 600$ individual imaging simulations were performed to build each map.

We then further explore symmetrically distributed dust cloud with exo-zodi half peak brightness distances spanning from 0.1 to 1.5 AU, and exo-zodi levels of 0 to 50 while indicating where an SNR of 7 could be obtained, for HIP 1599 seen in Fig. 8.

Fig. 8

This figure explores how the exozodi factor and exozodi dust distribution effects the SNR performance for an Earth-like planet orbiting the habitable zone of Zeta Tucanae, which is 8.6 pc away and is a G0V star. For other simulations that include exozodiacal dust, the distance from the star where half of the peak intensity occurs is 0.1 AU, and it is scaled by the luminosity of the star. This figure spans this distance from 0.1 to 1.5 AU, and it spans across the horizontal axis. The vertical axis is the exozodi factor. Similar to the figures above, we divide each map into the two regions. The gray regions correspond to areas that yield an SNR larger than 7 and the black regions yield SNR values below 7. The phase of the planet remains constant at 90 deg. local zodiacal dust is present in all simulations and remains constant. Solar glint was not activated for these simulation runs. The order of the three images from left to right are NGRST, HabEx, and R-O.

We continue to introduce a clump-like structure, located in close proximity to the trailing edge of earth and show for the three missions imaging simulations for a planet only, clump only, and the two combined in Fig. 9.

Fig. 9

This figure examines on how a trailing blob of dust can impact the detection of Earth-like exoplanets for all starshade missions. The top row displays simulations of the Sun 10 pc away with exozodiacal dust and Earth in orbit at 1 AU. The second row replaced Earth with a trailing blob of dust, and the third row adds both the Earth and the blob of dust. The fourth row plots each flux component, as seen previously in Fig. 1, when both the planet and the blob of dust are present, but it is centered on the location of the planet. The brightness of the trailing dust blob was set to 50% of the exozodiacal dust peak brightness.

Finally, in Fig. 10, we compile optimistic and pessimistic dust level scenarios and present their impact on Fig. 6. The optimistic and pessimistic dust levels are set to 3 and 11.8, respectively.

Fig. 10

Optimistic and pessimistic implications of exo-zodi levels for a symmetrical dust distribution: The black, green, and blue curves correspond to R-O, HabEx, and NGRST, respectively. Solid curves correspond to each mission starshade IWA for a planet positioned at 1 AU scaled to the star luminosity. The dashed lines correspond to the distance at which an Earth-like planet has a completeness of 0.35 for each mission. We provide for context stars from the Gaia DR2 in gray to identify where stars are located, and the names of stars selected as potential targets for starshade missions, and the total number of stars that have habitable zones outside of the IWA of each starshade mission at the top of the image, and the number of stars within phase completeness requirement curve on the right side of the image. The semi-major axis remains constant at 1 AU scaled by luminosity, the planet radius is set to $1R_{\oplus }$, and the planetary albedo is set to 0.2.

3.4.

Planet Radius, Albedo, and Semi-Major Axis

In this section, we consider how predictions for the expected planet radius, semi-major axis, and albedo of the observed planets could impact the detection of such planets. Figure 11 shows for the three missions for cross-correlations the regions in which $\mathrm{SNR}=7$ can be obtained. In the first row, we explore the planet radii and semi-major axis dependence. In the second row, we pair the semi-major axis with the planetary albedo, and the third row explores the dependence on planetary albedo and radius. Finally, the fourth row intertwines both the planetary radius and semi-major axis and identifies the required albedo to maintain an $\mathrm{SNR}=7$ for each pair.

Fig. 11

This figure contains several images that investigate how the semi-major axis, planetary radius, and planetary albedo effect image quality. To quantify the effects, we place an Earth-like planet in orbit around the star Zeta Tucanae, a G0V star located 8.6 pc away. The first three rows correspond to two of the parameters spanning across a set distance while keeping the third parameter constant. The first row has the semi-major axis on the horizontal axis (AU), the planetary radius on the vertical axis (${\mathrm{R}}_{\oplus }$), and a constant albedo. The second row of images has the semi-major axis on the horizontal axis, the albedo on the vertical axis, and a constant planetary radius. The third row of images has the planetary radius on the horizontal axis, the albedo on the vertical axis, and a semi-major axis held constant at 1 AU. The fourth row has the semi-major axis and planet radius spanning across the vertical and horizontal axes, respectively, but each combination of these parameters is assigned the minimum albedo required to maintain an SNR greater than or equal to 7. If the required albedo could not be established, the region was considered unreachable and denoted as black. The gray zones indicate areas where the SNR is above 7.

4.1.

Tilt

For each mission, we conduct an array of imaging in this section we conduct simulations spanning from 0 deg to 50 deg tilts, as can be seen in Fig. 12, and provide the resultant SNR for an Earth-like planet located 1 AU from the star, aligning the maximal light leakage such that it is placed as close as possible to the planet position, as a worst case scenario.

Fig. 12

An examination of how the starshade tilt effects imaging performance for all starshade missions. The first row displays how the Beta Canum Venaticorum system would look through each telescope-starshade configuration with a 0-deg tilt. The second row shows the same astrophysical scene but with a tilt that yields poor imaging performance. The third row displays the normalized SNR of an Earth-like planet orbiting at the scaled habitable zone as a function of the starshade tilt angle. The blue, green, and black curves correspond to NGRST, HabEx, and RO, respectively. Each curve transitions into a dashed line once the actual SNR drops below 7. The cartoon on the bottom-right illustrates how the starshade tilt, identified with red arrows, changes the starshade-telescope configuration.

4.2.

Off- Line of Sight Error

One of the challenging aspects of starshade-based missions is the tight formation flight requirement which have been identified as a key target for the S5 program. In Fig. 13, we evaluate for the three missions the impact of non-ideal flight formation where the starshade deviates from the line of sight (LOS).

Fig. 13

This figure investigates how the alignment of the starshade and telescope impacts the quality of images. To identify how the image quality changes, an Earth-like planet is placed in orbit around 40 Eridani A; it is a K0V star located 5 pc away. The orbit has a semi-major axis of 1 AU and it is scaled by the star’s luminosity of 0.46. The top row of images are the outputs from the simulations with the ideal starshade. The second row of images are the simulations with a misaligned starshade. A significant increase in star-leakage can be seen as a result. The curves at the bottom contain the normalized SNR as a function of the line-of-sight shift in meters. NGRST, HabEx, and R-O correspond to the blue, green, and black curves, respectively. The image in the bottom right corner is a simple visualization of a starshade with a line-of-sight misalignment. In the diagram, the telescope points directly toward the starshade face on. The starshade with a dashed outline represents correct alignment, and the black starshade represents a starshade with a line-of-sight shift. For this example, the starshade is taken from NGRST and a 5-m vertical shift is applied to illustrate how the shift alters the observation setup. The red arrow points in the direction of the shift, and its length corresponds to the magnitude of the shift. The two red crosses correspond to the center of both starshades. The dashed cross is at the center of the ideal starshade, and the cross above sits at the center of the misaligned starshade.

4.3.

Glint

During the starshade operation, sun rays interact with the starshade petal edges to produce glint, and increase both the noise and confusion when trying to determine the incoming signal from an Earth-like planet. In Fig. 14, the imaging performance in the presence of solar glint is examined.

Fig. 14

This figure returns to Tau Ceti, but with the addition of solar glint to investigate how it impacts the simulated data and resulting images. The star-starshade-sun angle was set to 65 deg for all three starshade missions. These simulations can be seen in the top row. The three curves in the bottom row come from a horizontal slice taken from the images in the top row capturing the star leakage, exozodiacal dust, solar glint, local zodiacal dust, noise, and the planet flux. The gray region is the combined background flux from the star and exozodi dust, the green region is the local zodiacal dust, and the blue region is the planet flux. The dashed red line is the combination of the planet and background flux. All exposure times and SNR values can be seen in the top left corner of each image.

We continue to explore the sun-starshade-star angle dependence, and the impact on the SNR of such glints in Fig. 15

Fig. 15

The three images above illustrate how solar glint effects the SNR of Earth-like planets. The effect on an Earth-like planet’s SNR as solar glint is applied to all three starshade missions. The brightness of the glint corresponds to the angle between the starshade and sun spans from 15 deg to 80 deg. For each mission, three stars are shown: Tau Ceti in green, Zeta Tucanae in black, Iota Persei in blue, and HIP 27435 in red. These are G8V, G0V, G0V, and G4V stars with a distance of 3.65, 8.6, 10.5, and 15.2 pc, respectively. These simulations did not contain any exozodiacal dust but did contain the local zodiacal dust from our solar system. For each star, the semi-major axis of Earth’s orbit was scaled by each star’s luminosity. For all stars, Earth was placed in a 1 AU orbit and scaled by the luminosity of each star.

4.4.

Petal Edge Error

Petal edge effects are among the worst sources of error that could affect these observations; micrometer size deviations across meters in length could reduce the resultant contrast levels. In Fig. 16, we explore the sensitivity of missions to such perturbations, display before and after images to indicate its impact, and examine the SNR as response a function of the perturbation size.

Fig. 16

This figure analyzes how imperfections present on the petals of each starshade effects the resulting SNR of a planet. In each simulation, 40 Eridani A was chosen as the target star, and it has a distance of 5 pc, has a luminosity of $0.46{\mathrm{L}}_{\odot }$, and it is a K0V star. An Earth-like planet was placed in an orbit with a semi-major axis of 1 AU scaled by the luminosity of 40 Eridani A. The exozodi factor was set to 0, and solar glint was turned off. The first row of images contains the final simulation for all three starshade missions with ideal starshades for NGRST, HabEx, and R-O, respectively. The second row contains the final simulated image using starshades that have a perturbation along the edges of all petals. An increase in star-leakage can be seen as a result. The curves at the bottom are the normalized SNR as a function of perturbation amplitude. The amplitude spans from 1 to $6000\mu \mathrm{m}$. NGRST, HabEx, and R-O are shown in blue, green, and black, respectively. Once the SNR drops below 7, the curve transforms into a dashed line. The cartoon on the bottom-right illustrates the edge-errors applied to the RO starshade

5.1.

Performance Envelopes

In this section, we present each mission concept’s cumulative range of optimistic and pessimistic observable targets as established by our sensitivity analyses. Mission deliverables are mapped according to the target star distance and luminosity in Fig. 17. The optimistic observation scenario assumes favorable conditions for imaging, exo-zodi (3 zodi), large radius planets (1.4 Earth radius) with high albedo (0.5), without starshade perturbations, etc. This optimistic scenario is indicated by the upper limit of each mission’s colored region. The semi-pessimistic scenario is represented by the lower bound of each colored region, for which we assume the smallest planet radius (0.6 Earth radius), exo-zodi at the three sigma limit indicated by the HOSTS survey (21 zodi), albedo of 0.1, with starshade perturbations, etc. The dashed lines represent a shift of the exo-zodi peak brightness to 0.5 AU for all missions.

Fig. 17

SISTER-simulated accessible ranges of mission-specific target stars. NGRST, HabEx, and R-O are marked in purple, green, and black, respectively. The top and bottom edge of each colored region correspond to the optimistic and semi-pessimistic scenario for each mission. The pessimistic scenario is marked by a dashed line.

We find that even if the background noise could be perfectly calibrated, for a Sun-like star luminosity, the ranges for both the optimistic and semi-pessimistic scenarios are between 20 and 25 pc for R-O, 4.5 to 13 pc for Habex and 3 to 9 pc for NGRST. Where for stars’ luminosity smaller than 1, that range decays exponentially as can be seen in Fig. 17. Needless to say, both the semi-pessimistic and optimistic predictions are in fact somewhat optimistic since perfect background calibration might not be achieved, exo-zodi levels larger than 21 could be encountered, and starshade perturbations might be larger.

5.2.

Detection and Confusion

Exoplanet detection is one of the primary, simplest deliverables offered by direct imaging missions. Throughout this paper, SNR values have been calculated with pre-existing knowledge of target exoplanet location, size and separation among an array of other parameters. This degree of certainty will likely not be the case in future mission scenarios.

First, even when reasonable SNR levels are achieved ($\mathrm{SNR}=7$), it is challenging to identify the location of many of the planets studied in this paper. An observer must understand the predominant ensemble and system-specific background signal structures to a degree where advanced astronomical extraction and processing models can extract meaningful results. This knowledge accuracy is highly dependent on the relative and total planetary and background signal’s time-varying structure. We conduct imaging simulations such that a background knowledge accuracy is assumed to be known to levels of both 1% and 5% and show in Fig. 18 at which distance and star luminosity’s resultant SNR will drop below 7.

Fig. 18

The maximum distance each mission can observe as a function of Luminosity for two different background knowledge levels. The three black and red curves correspond to 1% and 5%, respectively with an exozodi factor of 4.5. The magenta curves correspond to an exozodi level of 11.8 with a 1% background knowledge.

As can be seen, even if one would be able to gain a background knowledge accuracy of 1% (black), a majority of the targets will not achieve their expected SNR, which will require expanding the targets exposure time (or updating the SNR requirement) to compensate for such losses. This phenomenon becomes even more prominent at a knowledge level of 5% (red), making it one of the most dominant factors to consider when coming to estimate SNR requirements for explanatory targets. This finding will have an even larger impact on other target SNR requirements like $\mathrm{SNR}=20$ for detailed spectral characterization⁹ and will be driven by the available or desired integration time allocation for target stars for each mission.

We provide in Table 2 estimates for the updated required SNR for the detection of Earth-like planets with few prime targets.

Table 2

NGRST and HabEx SNR requirements with 1% background knowledge for three high priority target stars.

Second, in Fig. 19 we assess the missions ability to disentangle the signal of an earth-like planet from a nearby planet at different separations. We define planets being fully inseparable (90% of the signal of both planets overlaps), confused (10% to 90% of the signal overlaps), and separable (90% of the signal is not overlapping).

Fig. 19

Two combinations of planets are explored to identify how the angular separation of planets affects signal identification. The two combinations are Earth and Jupiter, and Earth and a super Earth where the radius is $2R_{\oplus }$. For each combination, there are three different angular separations depending on the starshade mission. The first column corresponds to an angular separation where the planets cannot be distinguished because the signals combine into a single peak. The second column identifies the angular separation where you can begin to distinguish the planets from one another. The third column corresponds to the separation that yields two distinguishable peaks for the planets. However, the signal for the Earth-like planet close to Jupiter remains washed out even at 100 mas.

We find that even under ideal conditions, small planets (Super Earth or smaller) separated by $<60$, 40, and 5 mas and large planets (Jupiter, etc.) separated by $<100$, 90, and 10 mas, will be confused or outright inseparable for NGRST, HabEx, and R-O, respectively. These findings, however, do not account for less favorable observation conditions and other systematic or background signal sources such as resonant dust bulbs and moons, which are expected to enlarge the required minimal separation range to avoid confusion.

5.3.

Toward a Realistic Exoplanet observation Scenario

A realistic model of starshade observation should account for a variety of possible starshade perturbations and exo-planetary environments. In Fig. 20, we show for 40 Eridani A, a K0V star located 5 pc away, for the three missions, the evolution of an ideal image as we introduce a starshade line of sight shift of 0.5 m, $70\mu \mathrm{m}$ edge error, and solar glint, along with exo-zodi level of 7, and our local zodi dust. We presented for each image the resultant SNR and required noise knowledge in percent.

Fig. 20

Three simulations of image weakening observational perturbations for 40 Eridani A, a K0V star located 5 pc away. We show first the simulations with ideal conditions; there is no exozodi within the system, no solar glint, and no local zodi dust. In the second row, we introduce starshade imperfections: 0.5 m LOS shift and $70\mu \mathrm{m}$ errors on all petal edges. In the bottom row, we further applied an exozodi factor of 7, turned on solar glint and local zodi dust. The SNR for an Earth-like planet orbiting at 1 AU scaled by 40 Eridani A’s luminosity of 0.45 assuming a 1% background knowledge level is shown in the top right corner of each image. The exposure time for NGRST, HabEx, and R-O are 24 h, 10 h, and 30 min, respectively.

By examining Fig. 20, one can easily see that the observations could become extremely challenging even for nearby stars. To assess the impact of these perturbing factors on target stars completeness values, we perform a statistical study of ensemble SNR 7 threshold observability based on full end-to-end optical simulations for various missions and target stars. These results are given in Table 3

Table 3

A list of completeness values from prime starshade mission targets. Each target possesses multiple completeness CN values, which correspond to a different completeness. C1 is the our standard completeness solution computed within SISTER. C2 is the completeness with an additional source of background flux coming from a line of sight error and petal edge error on the starshade. C3 is the completeness when we require the background knowledge equal to 1%. NGRST and HabEx both contain a C0 value, and it corresponds to completeness values from previous reports as a reference.

For each mission and star, C0 represents previously establish completeness values, C1 is produced by running 4000 end-to-end optical simulations by pulling system parameters from the observational priors and distributions described in Sec. 3 and Appendix B. Including all the natural parameters studied in this paper, C2 integrates starshade perturbations as described in Sec. 4. Finally, C3 accounts for a background knowledge accuracy of 1%.

As can be seen, even for short and moderate distances, the completeness values for each target and mission are negatively impacted. These results, however, do not indicate that starshade-based missions produce lower completeness values compared to a coronagraph for example. It is merely an indication of the maturity level of new end-to-end optical models, and a more complete understanding of the challenges facing future direct imaging missions.

5.4.

Future Work

Future work will include an investigation into starshade-based spectral characterization capabilities and an expansion on the study of resonant dust rings and structure and their impact on detection capabilities. A study of the impact of other systematic non-planetary noise and signal sources (expanded from photo-metric noise limitations provided in this paper) will follow. Additional work will be conducted to generate target lists and case by case mission and target star selections, and SISTER-based multi-parameter optimization of starshades for their specific paired observatory. Finally, we aim to utilize insights provided in this paper to better understand their impact on the scientific yield and quality of an array of starshade missions.

6.1.

Appendix A: Applying Starshade Perturbations

SISTER performs high fidelity simulations of directly imaged exoplanets by considering both the detector, telescope, and the design of the starshade to create a PSF spatial response. All of these considerations are stored within files that SISTER reads and compiles before creating a simulation. For starshades, the design is stored within three individual arrays corresponding to each dimension ($X$, $Y$, and $Z$).

The first perturbation we discuss is a starshade with a tilt, meaning that there is a relative angle between the starshade’s plane and the telescope. To create the necessary optical response of this new starshade configuration, a new file containing the starshade design is required. The new $X$, $Y$, and $Z$ arrays can be computed by performing a simple rotation about the $X$ or $Y$ axis. Then assign this file to be the new starshade you desire to use for the simulation, and a new PSF will be built.

Starshades that possess an off-line of sight error were simple to create. The values within the starshade design files have units in meters, so we simply add or subtract a constant value to either the $X$ or $Y$ array. This will force SISTER to build a new PSF because the starshade is no longer centered within the simulations.

Petal edge errors are the most complicated perturbations to apply. We applied sinusoids across the perimeter of each starshade corresponding to different frequencies, amplitudes, and phases. We elected to use an array of amplitudes, and a frequency of 1 amplitude per meter.

6.2.

Appendix B: System Parameters

When we establish system parameters for our simulations, we either select parameters that match the Solar System, or we select from a range of values to create synthetic systems containing Earth-like planets. These ranges will be briefly summarized. The range of planetary radii corresponds to rocky worlds capable of maintaining an atmosphere. The range of orbital radii corresponds with established inner²² and outer²³ habitable zone ranges. Each orbital radius drawn from this range scales as $\sqrt{L_{*}}$. The inclination of the orbits follows a normal cosine distribution. We assume all planets are spheres following a Lambertian phase function with an albedo ranging from 0.1 to 0.5 averaging around an Earth-like albedo of 0.3. Finally, the exozodi level present within the system corresponds to a factor that is multiplied to the standard zodi level of the system, which we assume has a surface brightness of $23\mathrm{mag}/{\mathrm{arcsecond}}^2$.¹⁶ The HOSTS²¹ survey provided optimistic and pessimistic exozodi levels, which we adopted when selecting our range. A summary of all values can be found in Table 1.