We present the optical requirement-driven observational constraints of the Remote Occulter, an orbiting starshade designed to work with ground-based telescopes to produce visible-band images and spectra of temperate planets around Sun-like stars. We then utilize these constraints to develop and present numerical simulations of time-dependent observable sky regions along with each region’s nightly available exposure duration and show that nearly the entire sky could be observed for up to 8 h a night. We further examine how changes introduced to our established constraints will impact such observational windows and discuss their implications, setting the ground for upcoming studies aiming to further investigate the Remote Occulter mission capabilities and architecture.
Measuring the reflected light spectra of temperate planets around Sun-like stars has been identified by the exoplanet science community as a high-priority objective.1,2 This is a major technological challenge: temperate exoplanets typically reside close to their parent star at mas and are extremely faint at 30th Mag, typically magnitudes fainter than their parent star.3 Parsing temperate exoplanetary signal from its parent star is further complicated by a variety of natural variables including exozodiacal dust4 and its planetary phase.
Although some 48 exoplanets have been directly imaged to date,5 including some around Sun-like stars,6,7 none are rocky worlds; the lowest mass planets yet imaged are .8 Proposed next generation space telescopes including HabEx9 and LUVOIR10 are one promising path toward expanding this catalog, by way of coupling with coronagraphs11,12 and/or starshades.13,14 However, space telescopes face many cost prohibitions and engineering challenges in exceeding their current sizes, including stability constraints15 and launch vehicle fairing diameters. Future extremely large ground-based telescopes with diameters exceeding 20 m, including the Extremely Large Telescope (ELT) (39 m), Thirty Meter Telescope (TMT) (30 m), and Giant Magellan Telescope (GMT) (24.5 m), will be the largest telescopes in the coming decades, with the first lights currently planned to occur between 2025 and 2030.16–18 However, due to engineering, programmatic, and fundamental atmospheric physical constraints, AO and coronagraph-assisted ground systems are likely limited to contrast for the foreseeable future.19,20
Another approach, marrying an extremely large ground telescope with an orbiting starshade may further enhance exoplanet imaging capabilities. Ground-coupled starshade systems like the Remote Occulter provide smaller inner working angles, better angular resolution, larger observable ranges, are less sensitive to exozodiacal dust, starshade perturbations21,22 and are expected to deliver contrast ratios exceeding .23 Where previous research on the Remote Occulter mission concept has resolved its imaging and spectral capabilities, ongoing efforts are focused on understanding and optimizing its challenging operational scenario, including spacecraft architecture, orbital configurations, and mission schedule, which will require well-defined models of target observability.24,25
In this paper, we derive the observational geometry constraints produced by an earth-orbiting starshade, including the starshade plane-Sun angle (Sec. 2.1), the starshade plane-target star angle (Sec. 2.2), observatory zenith-Sun angle (Sec. 2.3), and observatory zenith-target star angle (Sec. 2.4). We then incorporate all requirements alongside time-dependent Sun–Earth ephemerides to produce a sky map that reveals which regions of the sky may be observed with the Remote Occulter at a given date and time, as well as each celestial coordinate’s cumulative nightly exposure availability (Sec. 3). Finally, we discuss the operational implications of the mission (Sec. 4) and provide a road map for future work advancing this starshade mission concept (Sec. 5).
To acquire both images and spectra of temperate planets around Sun-like stars, we must mitigate conditions and effects that could interfere with observation. We would ideally like to observe for long periods anywhere in the night sky through most of the year to reduce target star scheduling complexity and provide flexibility in mission planning. However, an array of optical requirement-driven constraints limits the range of favorable operating conditions. For mission planning purposes, it is useful to describe these constraints in geometric terms that can directly translate to the ground telescope’s available sky domain and the orbiting starshade’s corresponding configuration and orbit. In this section, we establish these geometric requirements, enabling the numerical calculation of a well-defined observable sky region.
Starshade Plane-Sun Angle
We would like to minimize sunlight illumination of the Earth-facing starshade surface, as the reflected light could interfere with observations. We define the starshade plane-Sun angle in Fig. 1 and assess the apparent magnitude of the starshade surface at various relative starshade-Sun angles in Fig. 2. We find that direct illumination can produce a Mag. 8 to 14 starshade, restricting both image and spectral capabilities, additional information is provided in Appendix B. Starshade brightness can be further reduced using a low-reflectance carbon nanotube coating under reduced starshade-Sun angles , producing between 15 and 18 Mag. This may impact the detection of low-separation exoplanets, pending future performance simulations.
To avoid these effects, we constrain the Sun to be behind the earth-facing starshade surface by at least one degree, corresponding to a starshade-Sun tilt requirement of 89 deg. Following this constraint, we expect the primary solar effect on the starshade brightness to be edge glint.
Starshade Plane-Target Star Angle
The ability to deflect the starshade surface normal away from the telescope-target line of sight (LOS) could enable a wider coverage of targets across the sky by limiting Sun-surface illumination and loosening attitude tolerances, therefore, relaxing the Remote Occulter’s operational regime. However, as with other starshade missions, preliminary optical error budget analyses indicate that tilting the starshade plane reduces the contrast performance, negatively impacting the ability to obtain meaningful imaging and spectral information from a target.
Off-LOS tilting significantly complicates predicting starshade performance, as light diffracts around petals which no longer hold symmetry or ideal apodization profiles. We evaluate the effect of starshade deflection angles ,as seen in Fig. 3 on optical performance with the Starshade Imagining Simulation Toolkit for Exoplanet Reconnaissance (SISTER).26 Following after our previously developed approach for simulating the optical performance of a tilted starshade, we conduct high-fidelity imaging simulations (30-min exposure with 39-m aperture) of temperate planets around a Sun-like star (G2V) at 10 pc and record the resultant signal-to-noise-ratio (SNR) of an exo-Earth placed at 1 AU radius. As seen in Fig. 4, the starshade can be safely tilted up to 30 deg, after which a sigmoidal drop in performance is observed. Although acceptable SNRs could be extracted from larger tilts, we impose a strict requirement to accommodate for unknown loss effects, challenging targets, and spectral variation of target stars in a mission environment.
Local Zenith-Sun Angle
Observation shows that the night sky at high mountain observatories is fully dark at zenith when the Sun is at least 18 deg below the horizon, or equivalently 108 deg off-zenith, here defined as as shown in Fig. 5. The sky brightness has been shown to fall by three magnitudes as the increases from 102 deg to 108 deg.27 Though observations are possible before the sky is fully dark and scattered sunlight is not the dominant sky brightness at all wavelengths, we constrain the observing window by assuming this is a strict requirement.
Local Zenith-Target Star Angle
The ELT, TMT, and GMT permit observations from near- or at-zenith down to 20 deg,28 25 deg,16 and 30 deg17 above the horizon, corresponding to a angle of 70 deg, 65 deg, and 60 deg, respectively. We therefore set . Additional considerations relating to specific telescope configurations like zenith constraints produced by Alt-azimuth mounts and instruments are provided in Sec. 4.
Intuitive Geometric Model
The result of these combined geometric constraints is difficult to visualize in three dimensions. We present a model of the observable regions in a stretch of sky sliced across the ecliptic plane, driving an intuitive understanding of each geometric constraint’s, detailed in Table 1, effect on the observable sky window in Fig. 6. The constraint of the telescope-zenith angle projects an observable cone onto the celestial sphere that sweeps across the sky between Sun-down and Sun-up, while . The available observation time is then the duration a given star’s position satisfies , bounded by ’s definition of night. On the other hand, the unobservable bounds produced by the starshade plane-Sun and starshade plane-target star deflection constraints are effectively constant over a sidereal day, being entirely dependent on the relative Sun–Earth positioning. They produce an unobservable region centered on the hour angle of sidereal midnight, marked in black, when the sun directly faces the starshade.
A summary of all geometric requirements. Each row provides a description for an angle and the maximum/minimum value.
|Local zenith-target star|
|Starshade plane-target star||to 40|
To map the regions and duration over which all geometric requirements are met, we adopt a standard epoch (J2000), generate a right ascension (RA) and declination (DEC) grid to project results onto, and choose an observatory location, here Mauna Kea and Cerro Armazones, the locations of TMT and ELT, respectively, for the proposed Remote Occulter science operation period, starting January 2035. Through reference frame and coordinate conversions, we sum the total accumulated duration that these requirements can be kept for each celestial position on a given night and map the resulting observation window contour.
These numerical checks are iteratively applied in discrete time steps using Python. As per Fig. 7, the Sun’s ecliptic and topocentric coordinates are calculated for each day of the year, and the local zenith’s projected angle is calculated for each minute in a given sidereal day. As the program steps through the time of night that the solar-zenith constraint is satisfied, a minute of observation is added to each celestial coordinate that satisfies the remaining three geometric constraints. We translate the celestial target positions (RA, and DEC, ) into geometric angles relative to the time-dependent Sun location (, ) and projected local zenith (, ), using the following equations based on polar reference frame conversions:
Figure 8 shows the observable regions in the night sky for March 14, 2035, for observations from Cerro Armazones. As indicated by the red border, the maximum declinatory angles that can be observed are and 30 deg, constrained by the limiting boundary produced by , the maximum local zenith-target angle. Limits on RAs are set by the first and last RA that fulfill all four requirements and are primarily constrained by the length of the observable night, produced by , and the inner bound of maximal target-Sun angle, . The outermost red border reflects at least one of the four geometric constraints is at its limit, whereas different color curves indicate the regions where such conditions can be kept for different time periods. The blue line, for example, contains where observations can be maintained for at least 2 h over a night.
We then continue to dynamically map the observable night sky for the months of January, April, July, and October for both Mauna Kea and Cerro Armazones. As can be seen in Fig. 9, the observable region sweeps across the sky, covering more than 65% of the sky for each telescope and over 90% of the celestial sphere over a year for both the north and south hemisphere telescopes.
The length of the night directly drives the maximum available observation time and angular width of the overall window. The longest and shortest times to observe occur at the summer and winter solstices depending on the telescope hemisphere. To evaluate the sensitivity of our observable map to the geometric and optical mission constraints, we show in Fig. 10 the result of varied starshade plane-target star angle and local zenith-Sun angle requirements. The maximum combined starshade plane-Sun angle () and starshade plane-target star angle () is seen to inversely correlate with the central unobservable area’s size, and the available local zenith-Sun angle range () with the observable window width and maximum observation time. As presented in Sec. 2.3, the starshade plane-target star angle could be increased up to 40 deg while delivering a normalized SNR above 0.8. Reduced starshade plane-target star and starshade zenith-target star angles limit the observable window’s upper and lower declinatory range depending on the time of year. The observable window’s progression over a year is available as Video 1.
Summary and Future Work
In this paper, the observational constraints driven by potentially detrimental optical effects to direct exoplanet imaging with the Remote Occulter mission are reviewed, from which geometric operational requirements have been derived. We develop a numerical method for calculating the observable sky bounds and duration for an Earth-orbiting starshade-coupled ground telescope as constrained by these geometric constraints and present the effects of each geometric constraint on observability and mission architecture and planning. We find that about two-thirds of celestial DECs may be observed with the Remote Occulter at some point in the year for up to 9 h, and that a partnership between both north and southern hemisphere telescopes can enable full sky coverage.
Although the observable window sweeps across all RAs over a year, its time-dependent position and form must be carefully factored into future mission planning studies. Future work will include an array of follow-up studies, from orbital configurations to support minimized station-keeping costs for observations, retargeting strategies that minimize orbit transfer costs while maintaining observational spatial and time windows, to scheduling optimization for the mission which will inform design activities of mechanical configurations, and operational strategies ultimately leading to a full mission design reference. This could be further expanded by adding or modifying additional constraints; a variety of missions, including the orbiting configurable artificial star mission,29 could benefit from the mapping of observational regions in their mission planning activities.
Appendix A: SISTER
To create high-fidelity simulations of directly imaged exoplanetary systems, we use the SISTER. SISTER performs these simulations by considering the optical and telescope configurations, and the design of the starshade such as the size and distance from the telescope. A large collection of exoplanetary system parameters such as the spectral type of the host star, the number and type of planets within the system, exozodiacal dust disks, and any background sources such as galaxies or stars. SISTER derives the optical response of the starshade and telecope configuration by performing boundary diffraction modeling to calculate the PSF spatial response. Experimental results show that the resulting PSF’s agree to a level of contrast of , and it agrees with other simulation tools to a level of 1%. See the SISTER Handbook for further technical details on SISTER.
SISTER allows us to create the PSF response for the Remote Occulter mission. The telescope used is the ELT with an aperture of 39 m. The starshade has 48 petals with the length of each petal set to 24.5 m, and the total diameter is 99 m. The distance between the starshade and the telescope was set to 170,000 km. The PSF response received additional effects by the presence of emission lines in the upper atmosphere, moonlight, scattered starlight, and Earth shine.
To determine how a starshade with a deflection angle effects image quality, we create an astrophysical scene that will remain constant. We select the Sun to be the host star, and it is assigned a range of 10 pc which yields an apparent V magnitude of 4.83. We place Earth in an orbit of 1 AU (100 mas planet-star separation at 10 pc). Excluding the flux from star leakage, we do not turn on any other sources of background such as background galaxies or exozodiacal dust. The only parameter that remains variable is the starshade plane-target star angle . Within SISTER, the design of the starshade is stored as arrays, this allows the user to alter the design of the starshade such as perturbations. To apply a , we perform a simple rotation about the axis of the starshade. Once a new starshade design is built, SISTER creates a new optical response from the new starshade.
As the starshade receives new deflection angles, SISTER creates a new optical response to perform the simulation. Once a new response is created, the SNR21 of Earth is found by utilizing the simulation output data. Once several SNR values were extracted for 16 angles, a sigmoid function was used to fit the data.
Appendix B: Starshade Reflection
We estimate the illuminated starshade magnitude for various surface-Earth phase angles and reflectances by approximating the reflected flux at the starshade surface—to which Lambertian cosine losses may be applied for oblique tilting—and subsequently derive the magnitude at Earth :
We appreciate numerous discussions with the Goddard Space Flight Center Orbit Analysis Group, including David Folta, Rizwan Qureshi, Robert Pritchett, Donald Dichmann, Sun Hur-Diaz, and Steven Hughes. We thank the Team-X Laboratory at Jet Propulsion Laboratory for engineering analysis and cost estimation, and the Exoplanet Exploration Program (ExEP) Team for support and technical discussion, especially Gary Blackwood, Keith Warfield, Karl Stapelfeldt, and Erick Mamajek. This work was supported by Goddard Space Flight Center’s Internal Research and Development Funds (IRAD) at Goddard and NASA’s ExEP supported the Team-X study at JPL and scientific and engineering workshops at Goddard.
Eliad Peretz is a technology program manager and leads the smallsat development lab at Goddard Space Flight Center, He received his PhD in mechanical aerospace engineering with a minor in applied physics from Cornell University, and has been with GSFC since 2015.
John C. Mather is the senior project scientist on the James Webb Space Telescope. He is also a senior astrophysicist in the Observational Cosmology Laboratory located at NASA’s Goddard Space Flight Center.
Lucas Pabarcius is a NASA Pathways student trainee in attitude control systems at Goddard Space Flight Center and a junior undergraduate student studying physics and astronomy at Wesleyan University.
Sara Seager is the Class of 1941 Professor of Planetary Science, Professor of Physics, and Professor of Aeronautics and Astronautics at the Massachusetts Institute of Technology. She received her PhD from Harvard University.
Stuart Shaklan is the supervisor of the High Contrast Imaging Group in the Optics Section of the Jet Propulsion Laboratory. He received his PhD in optics at the University of Arizona in 1989 and has been with JPL since 1991.
Sergi Hildebrandt is a scientist at the Jet Propulsion Laboratory and lecturer at the California Institute of Technology. He received his PhD in theoretical physics at the University of Barcelona. He has worked in the data analysis of the cosmic microwave background from space and ground missions, adaptive optics in the visible, and more recently on leading the development of SISTER, a user friendly, open source, project that generates starshade simulations with very high fidelity.
Phil Willems is an optical engineer at the Jet Propulsion Laboratory, where he is the manager of the S5 Starshade Technology Development Activity. He received his BS degree in physics from the University of Wisconsin-Madison in 1988, and his PhD in physics from the California Institute of Technology in 1997.