1.

Introduction

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 $\sim 100$ mas and are extremely faint at 30th Mag, typically $\sim 25$ 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 $\sim 2M_{\mathrm{jup}}$.⁸ Proposed next generation space telescopes including HabEx⁹ and LUVOIR¹⁰ are one promising path toward expanding this catalog, by way of coupling with coronagraphs^11,12 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 ${10}^{-8}$ 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 perturbations^21,22 and are expected to deliver contrast ratios exceeding ${10}^{-10}$.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).

2.

Observational Requirements

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.

2.1.

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 ${\theta }_{S-\odot }$ 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 ${\theta }_{S-\odot }$, producing between 15 and 18 Mag. This may impact the detection of low-separation exoplanets, pending future performance simulations.

Fig. 1

Schematic of starshade plane-Sun angle ${\theta }_{S-\odot }$, where the Sun illumination direction is marked by arrows and the starshade plane normal is dashed. An angle of 90 deg corresponds to the Sun being perpendicular to the starshade and 180 deg to the Sun being behind the observer and fully illuminating the starshade surface.

Fig. 2

The starshade surface’s apparent magnitude plotted as a function of its surface material reflectance. Each tilt angle corresponds to the starshade plane-Sun angle.

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 ${\theta }_{S-\odot }$ of 89 deg. Following this constraint, we expect the primary solar effect on the starshade brightness to be edge glint.

2.2.

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 ${\theta }_{S-T}$,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.

Fig. 3

Schematic of the starshade plane-target angle ${\theta }_{S-T}$, where the line of sight is marked by the line connecting the telescope at Earth and the target star. The target-facing starshade plane is marked with a dashed arrow and a dotted surface.

Fig. 4

Remote Occulter sensitivity to ${\theta }_{S-T}$ deflection angles. (a) SISTER results for a 30-min exposure of a G2V solar system at a distance of 10 pc with 0 exozodi at various starshade deflection angles ${\theta }_{S-T}$. An exo-Earth is placed at 3 o’clock spaced to 1 AU. The $X$ and $Y$ axes define the window of the simulation in AU, and the colorbar corresponds to the total electron counts in log scale. (b) Simulated image SNRs normalized to 0 deg tilt plotted as a function of deflection angle.

2.3.

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 ${\theta }_{z-\odot }$ as shown in Fig. 5. The sky brightness has been shown to fall by three magnitudes as the ${\theta }_{Z-\odot }$ 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.

Fig. 5

A schematic of local zenith-Sun angle ${\theta }_{Z-\odot }$ and local zenith-target star angle ${\theta }_{Z-T}$ as measured from the observatory’s local zenith. We set ${\theta }_{Z-\odot }$ to be below the horizon (dashed) by at least 18 deg.

2.5.

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 ${\theta }_{z-\odot }>108\mathrm{deg}$. The available observation time is then the duration a given star’s position satisfies ${\theta }_{Z-T}$, bounded by ${\theta }_{z-\odot }$’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.

Table 1

A summary of all geometric requirements. Each row provides a description for an angle and the maximum/minimum value.

Angle	Description	Value (deg)
${\theta }_{Z-T}$	Local zenith-target star	$<60$
${\theta }_{S-\odot }$	Starshade plane-Sun	$<89$
${\theta }_{S-T}$	Starshade plane-target star	$<30$ to 40
${\theta }_{Z-\odot }$	Local zenith-Sun	$>108$

Fig. 6

Geometric schematic of the observable window sliced across the ecliptic plane. Each night’s observable sky region is centered on the Sun–Earth RA, where the Sun is marked in yellow and the Earth with an “E.” As the Earth rotates over a given night (as defined by the Sun being $>18\mathrm{deg}$ below the horizon), the conical regions of observable sky whose zenith-target angle ${\theta }_{Z-T}$ is $<60\mathrm{deg}$ and satisfy limiting starshade constraints ${\theta }_{S-\odot }$ and ${\theta }_{S-T}$ may be observed. These regions are marked for the start and end of night as blue and red arc-sectors. For a given target coordinate, the starshade surface is best deflected to ensure ${\theta }_{S-\odot }<89\mathrm{deg}$ by a rotation in the plane formed by starshade-target and starshade-Sun vectors up to ${\theta }_{S-T_{\max }}=30\mathrm{deg}$, when the angle between target star and the Sun is 119 deg, equal to ${\theta }_{S-T_{\max }}+{\theta }_{S-{\odot }_{\max }}$. These bounds are marked as “ss-Sun° limit.” The external arc-gradient indicates the observable duration of each potential target position, which linearly increases in direct proportion to RA up to a potential “max. $\mathrm{\Delta }{t}_{\mathrm{obs}}$” in the purple zenith cone overlap region. Targets cannot be observed in the starshade-Sun constrained zone marked by a black arc.

3.

Observable Sky

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, $\alpha$ and DEC, $\delta$) into geometric angles relative to the time-dependent Sun location (${\alpha }_{\odot }$, ${\delta }_{\odot }$) and projected local zenith (${\alpha }_z$, ${\delta }_z$), using the following equations based on polar reference frame conversions:

Fig. 7

Flowchart and calculation procedure for our numerical observable map generation method, which produces an observability contour for a given telescope latitude (${\lambda }_T$). Sun-zenith, telescope-zenith, combined target-Sun, and checked for suitability at each time-step (every minute of the day $0\to n$, $n=1440$ for each given day of the year). A minute of observation time is added if all geometric requirement statements resolve to “true” at a given coordinate.

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 $\sim -90\mathrm{deg}$ and 30 deg, constrained by the limiting boundary produced by ${\theta }_{Z-T}=60\mathrm{deg}$, 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 ${\theta }_{Z-\odot }>108\mathrm{deg}$, and the inner bound of maximal target-Sun angle, ${\theta }_{S-T}+{\theta }_{S-\odot }=30\mathrm{deg}+89\mathrm{deg}$. 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.

Fig. 8

Observable regions in the night sky for March 14, 2035, for observations from Cerro Armazones. The horizontal axis is 0 deg to 360 deg RA and the vertical axis is $-90\mathrm{deg}$ to $+90\mathrm{deg}$ DEC. The colors correspond to the observation time in hours.

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.

Fig. 9

Observable night sky, showing minutes available for observation each night as indicated by the color bar. The range of each image is 0 deg to 360 deg RA and $-90\mathrm{deg}$ to $+90\mathrm{deg}$ for DEC. Upper set is for Mauna Kea, lower set for Cerro Armazones. Dates are the first of each month, 2035. Each sky-map is colored according its total observation time in hours. Video 1, MOV, 4 MB [URL: https://doi.org/10.1117/1.JATIS.7.2.021212.1] shows a time lapse of the observable sky as seen from Cerro Armazones.

4.

Discussion

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 (${\theta }_{S-\odot }$) and starshade plane-target star angle (${\theta }_{S-T}$) is seen to inversely correlate with the central unobservable area’s size, and the available local zenith-Sun angle range (${\theta }_{Z-\odot }$) 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.

Fig. 10

Sky coverage changes (observable sky) due to the max starshade-target deflection angle, ${\theta }_{S-T}$, and ${\theta }_{Z-T}$, the maximum target zenith angle, for observation from Cerro Armazones in April 2035, the range of each image is 0 deg to 360 deg RA and $-90\mathrm{deg}$ to $+90\mathrm{deg}$ for DEC.

5.

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.

6.

Appendices