1.1.

Starshades

Starshades – large, precisely shaped, external occulters flying in formation with a telescope – offer a path to direct observation and spectroscopic characterization of Earth-like exoplanets (e.g. Refs. 12.3.–4). Historical perspectives on starshades are included in Refs. 3 and 5. In the 2000s, flagship-class starshade missions were proposed (e.g., Refs. 6 and 7). In this decade, NASA starshade studies included both a Probe-class mission with a 20- to 30-m-diameter starshade that could rendezvous and operate with the Nancy Grace Roman Space Telescope (“Roman,” see Refs. 1, 8, and 9) and a flagship-class mission, the Habitable Exoplanet Observatory (“HabEx,” see Refs. 10 and 11) with a 52-m-diameter starshade.

A proposed operational concept for these missions, similar to Refs. 9 and 1213.14.–15 and shown in Fig. 1, would proceed as follows. First, the space telescope and the starshade spacecraft are inserted in similar quasi-halo orbits around the Second Sun-Earth Lagrange point (SEL2) using ground-based orbit determination (OD). After this initialization phase, a series of observations is performed by cycling through acquisition, science, and retargeting phases. The purpose of the acquisition phase is to align the starshade with respect to the telescope-star axis with sufficient precision to begin science observations. It includes handovers between three coboresighted telescope sensors with an increasingly fine resolution and narrow field of view (FOV): a coarse sensor (CS), an intermediate sensor (IS), and a fine sensor (FS). During the science phase, the two spacecraft are separated by tens of thousands of kilometers. Using FS measurements, the starshade uses its thrusters to maintain its meter-level alignment between the telescope and the star, for instance, as demonstrated in Refs. 12 and 15. At the end of the science phase, the starshade performs a retargeting maneuver to roughly position itself at the desired starshade-telescope distance in line with the next target star, as shown in Fig. 2. A new acquisition phase begins once the telescope reacquires the starshade with the CS at the end of of the retargeting cruise.

Fig. 1

Overview of a proposed concept of operations (not to scale). Figure adapted from Ref. 12.

Fig. 2

Earth-relative visualization of the nominal scenario for lateral retargeting (defined in Sec. 4.3). Locations shown at 3-day intervals with initial locations in green and final in red. Right inset: visualization of initial relative position error $\delta {\mathit{\rho }}_{\mathbf{0}}$. Left inset: visualization of final retargeting error $\delta {\mathit{\rho }}_{\mathit{f}}$ (see Sec 4.4). Point clouds show Monte Carlo results; ellipsoids and projections onto coordinate planes show $3\sigma$ covariance analysis predictions. The following scalings were applied to aid visualization: Earth-Moon distance increased $20\times$; Earth and Moon diameters increased $300\times$; and telescope-starshade distance increased $100\times$.

1.2.

Starshade Retargeting

In this article, a guidance, navigation, and control achitecture is proposed for the retargeting phase, and its feasibility is demonstrated. For current starshade concepts (e.g., Refs. 9 and 11), the retargeting cruise can last several weeks, with initial $\mathrm{\Delta }vs$ as large as tens of meters-per-second. At the end of the cruise, position control of the starshade relative to the telescope must be sufficiently precise to ensure the starshade can be reacquired by the telescope’s CS. The CS must also have sufficient resolution to hand over to the IS with its arcsecond-level FOV. As a result, a key constraint is that retargeting must be consistent with a degree-level CS FOV.

Apart from preliminary analyses such as Refs. 16 and 17, most previous approaches for retargeting are based on starshades with solar-electric propulsion (SEP) in an effort to reduce propellant mass and increase mission duration (e.g., Refs. 7, 8, 18 to 22). However, chemical propulsion (CP) is still likely required for starshade science as light emitted by continuously firing electric thrusters or sunlight reflected off their plumes could degrade observations. Relying on SEP for retargeting is therefore likely to lead to spacecraft designs with dual propulsion systems. In addition, high-power SEP systems require the starshade spacecraft to be fitted with significantly larger solar panels than for a CP-only design. Finally, measuring and correcting the low thrust of SEP with commercial off-the-shelf (COTS) on-board accelerometers with sufficient precision for reacquisition could be challenging. It might instead require comparatively expensive science-grade accelerometers or intermittent measurements of the starshade position from the ground, as in the Telescope for Habitable Exoplanets and Interstellar/Intergalactic Astronomy flagship study (THEIA, e.g., see Ref. 7). Alternatively, telescope operations could be interrupted at regular intervals during the cruise to acquire the starshade and measure its relative position, for instance, as in the Exo-S study (Ref. 8) or the New Worlds Observer concept (NWO, e.g., see Ref. 23).

1.3.

Proposed Retargeting Architecture

In this article, a relatively less complex and inexpensive retargeting architecture is proposed, analyzed, and shown to be feasible. It consists of using only CP and a set of COTS accelerometers to perform an actively controlled retargeting $\mathrm{\Delta }v$ maneuver, followed by a passive cruise until reacquisition with the CS. Using an array of COTS accelerometers enables precise knowledge and therefore execution of the initial retargeting $\mathrm{\Delta }v$. No additional SEP system is needed. Although a CP-only design is expected to have a higher propellant mass, this increase is partially offset by avoiding the need to carry an SEP system and the solar arrays to provide power for it (e.g., see Ref. 11). In addition, no science-grade accelerometers or intermittent sensor contacts from either the ground or the telescope are needed. Finally, the telescope does not need to perform a search with the CS at the end of the cruise: it is able to remain pointed at the star while its CS reacquires a starshade laser beacon (see Sec. 2 and, e.g., Ref. 14 for further details regarding relative bearing sensor concepts). These operational features could significantly reduce mission cost and complexity.

The feasibility of the proposed concept is demonstrated with a computationally efficient covariance propagation framework. This analytical approach evaluates the statistics of the retargeting error as well as its sensitivity to both nominal model parameters and uncertainty assumptions. Further insights into the error growth mechanisms are also provided, confirming the robustness of the retargeting architecture. Similar covariance analyses have been used in many prior studies for applications such as mission planning (Ref. 24), navigation (Refs. 25 and 26), descent and landing (Refs. 27 and 28), powered ascent (Ref. 29), and rendezvous and proximity operations (Refs. 30 and 31). In this article, the analysis is applied to both Roman-starshade and HabEx-starshade scenarios as representative examples. The results of these analyses show that in both cases a CS with a degree-level FOV is sufficient for reacquisition when using this CP-only retargeting architecture.

This article is structured as follows. In Sec. 2, based on the CP-only retargeting architecture and study of a possible starshade for Roman, a representative concept of operations (ConOps) for retargeting is described in detail. In Sec. 3, the covariance propagation framework is introduced. In Sec. 4, the framework is applied to the proposed retargeting concept for a Roman-starshade scenario, and the predictions are verified with Monte Carlo simulations for two stressing reference trajectories. A preliminary assessment is also conducted for a HabEx-starshade scenario. In Sec. 5, the error growth mechanisms are analyzed, and sensitivities to both nominal model parameters and uncertainty assumptions are computed. Finally, the conclusions of the work are given in Sec. 6. Some of the results presented in this article were also published in conference proceedings (Ref. 32).

2.1.

Retargeting Initiation ($t_{\mathrm{sci}}⩽t⩽t_0$)

At the end of the science phase ($t=t_{\mathrm{sci}}$), the two spacecraft are separated by tens of thousands of kilometers. The telescope has ended its starshade science observations and is pointing to the target star. Its absolute position is tracked at regular time intervals from the ground, and it performs monthly trajectory correction maneuvers (TCMs) to maintain its trajectory along its quasi-halo orbit. Starshade TCMs, performed with the starshade reaction control system (RCS) thrusters, are nominally identical to and synchronized with the telescope TCMs.

The starshade is spin-stabilized during the science phase and remains spin-stabilized throughout the retargeting phase (although the approach described here is expected to also apply to a three-axis stabilized starshade). Internal disturbances, solar radiation pressure (SRP) torque, and gravity gradient torques are sufficiently small not to require pointing maintenance maneuvers during the retargeting cruise. The position of the starshade along the telescope-star axis, referred to as the relative longitudinal position, is measured by radio frequency ranging, using S-band systems installed on the two spacecraft. The FS is used to measure the relative lateral position (i.e., across the telescope-star axis) during the science phase and is still active at $t=t_{\mathrm{sci}}$ (e.g., see Refs. 12 and 15).

In this article, slews refer to pure spacecraft rotations. Between $t=t_{\mathrm{sci}}$ and $t=t_0$, the telescope slews with its reaction wheels and begins performing nonstarshade science. The starshade slews using its RCS thrusters to point its main thrusters in the retargeting burn direction, while remaining spin-stabilized and interrupting S-band and FS measurements. The main retargeting burn is performed with the starshade main thrusters to improve fuel efficiency by avoiding cosine losses. RCS thrusters and accelerometers are used in a closed loop during the main burn to reduce the execution error. A second slew is then performed to orient the starshade in the desired direction for the cruise. Finally, an open-loop burn is performed with RCS thrusters to correct remaining residuals measured with the accelerometers during the two slews and the main retargeting burn. Detailed modeling of these retargeting initiation maneuvers is included in Appendix A.

2.2.

Cruise ($t_0⩽t⩽t_f$)

At $t=t_0$, the telescope is performing nonstarshade science and the starshade is spin-stabilized and pointing in the chosen cruise direction. A maximum cruise duration of 3 weeks is considered representative, and some results are given for a cruise of up to 4 weeks. Nominally, no starshade thruster firings occur during the cruise except to potentially match a telescope TCM. For Roman, telescope reaction wheel desaturation maneuvers (desats) occur every 4 days (subject to change).

To prepare for reacquisition, the telescope ends nonstarshade science and slews with reaction wheels to point to the new starshade science target star. It then sets up its optical configuration for starshade acquisition and science. This procedure is done sufficiently early in the cruise to allow for thermal settling before reacquisition.

2.3.

Reacquisition ($t_f⩽t⩽t_{\mathrm{acq}}$)

At $t=t_f$, the telescope is pointing to the new target star. When the starshade reaches the desired position relative to the telescope to initiate reacquisition, it slews using RCS thrusters toward the deceleration burn direction and uses its main thrusters to perform its main deceleration burn. It then slews with RCS thrusters to point to the target star, thus reestablishing S-band communications and relative longitudinal position sensing. It also turns on its laser beacon to enable reacquisition.

The telescope turns on the CS, which finds both the target star and the starshade laser beacon. At $t=t_{\mathrm{acq}}$, the starshade starts receiving relative bearing measurements from the CS, and the coarse acquisition phase begins.

3.1.

Notation and Problem Definition

In the following paragraphs, the problem definition is stated and key notation, shown in Figure 4, is introduced. Initial time $t=t_0$ corresponds to the start of the cruise and is indicated with a “0” subscript. During reacquisition ($t_f⩽t⩽t_{\mathrm{acq}}$), the maneuvers that the starshade performs take place shortly before the CS finds the starshade beacon. Their influence on the relative position dispersion is therefore small. As a result, the final time $t=t_f$ corresponds to the end of the cruise phase ($t=t_f$) and is indicated with an “$f$” subscript.

Fig. 4

Illustration of the retargeting phase. Top: nominal cruise; bottom: reacquisition (not to scale).

The telescope and starshade absolute positions, expressed in an inertial reference frame, are $\mathit{r}(t)$ and $\mathit{s}(t)$, respectively, where $t$ represents time. The relative position of the starshade with respect to the telescope is $\mathit{\rho }(t)=\mathit{s}(t)-\mathit{r}(t)$.

Scalars such as vector norms are denoted in roman italic font; for instance, $\rho =\Vert \mathit{\rho }\Vert$. Unit vectors are denoted with a “hat”; for instance, $\widehat{\mathit{\rho }}=\mathit{\rho }/\rho$. Time derivatives are indicated with a “dot”; for instance, $\dot{\mathit{\rho }}=d\mathit{\rho }/dt$. The matrix $I$ is an identity matrix, and the matrix $0_{n\times m}$ is an $(n\times m)$ zero matrix. The subscript is omitted for square matrices ($n=m$).

The nominal value of a variable corresponds to its current best estimate (CBE). In some cases, the nominal values are obtained from a combination of measurements, estimation, and propagation; in others, they are known by design (e.g., commanded values of $\mathrm{\Delta }vs$). The error in a variable’s value, denoted with a “$\delta$,” is the difference between the true value, denoted with an asterisk, and the nominal value of the variable. For instance, the relative position error is $\delta \mathit{\rho }={\mathit{\rho }}^{*}-\mathit{\rho }$. As such, the true values are random variables, and unless specified otherwise, their expected value is equal to their nominal value; for instance, $E({\mathit{\rho }}^{*})=\mathit{\rho }$. Covariance matrices thus represent the spread of possible true values about the nominal value. For example, the covariance matrix of the relative position error is $\mathrm{Cov}(\delta \mathit{\rho },\delta \mathit{\rho })=\mathrm{Cov}(\delta \mathit{\rho })=E(\delta \mathit{\rho }\delta {\mathit{\rho }}^{†})$, where † is the Hermitian transpose operation.

For the proposed architecture to be feasible, it is not sufficient to bound the final relative position knowledge error: the final retargeting error itself must be sufficiently small. In addition, for infinitely far target stars, the ability to successfully reacquire only depends on the relative position error at the end of the cruise $\delta {\mathit{\rho }}_f$. It does not depend on the final absolute position of the telescope $\delta {\mathit{r}}_f$ and starshade $\delta {\mathit{s}}_f$ explicitly. However, they must be considered in the analysis because the growth of $\delta \mathit{\rho }$ during the cruise does depend on the absolute position errors, as discussed below.

3.2.

Sources of Uncertainty

In this section, the sources of uncertainty included in the analysis are described. At the start of the cruise, when $t=t_0$, the errors in the initial telescope, starshade, and relative states are ${[\delta {\mathit{r}}_0^{†},\delta {\dot{\mathit{r}}}_0^{†}]}^{†}$, ${[\delta {\mathit{s}}_0^{†},\delta {\dot{\mathit{s}}}_0^{†}]}^{†}$, and ${[\delta {\mathit{\rho }}_0^{†},\delta {\dot{\mathit{\rho }}}_0^{†}]}^{†}$. respectively. First, the initial state errors are subject to the guidance error. This error source appears because the commanded retargeting burn is designed based on the nominal system state at the end of the science phase (i.e., when $t=t_{\mathrm{sci}}$). However, because of the knowledge error in the true absolute and relative states at $t=t_{\mathrm{sci}}$, the designed burn is not perfect in general. Even in the absence of all other sources of uncertainty described in this section, there would still be a nonzero guidance error. Second, the initial state errors are affected by residuals from the main starshade retargeting burn. Third, a TCM is conservatively assumed to occur at $t=t_0$ to maximize the propagation time of associated burn residuals. Additional TCMs could also be addressed, but they are only expected at a monthly cadence.

During the cruise, when $t_0⩽t⩽t_f$, these initial state errors propagate through SEL2 gravity gradients. In addition, the growth of telescope, starshade, and relative state errors, given by ${[\delta {\mathit{r}}^{†}(t),\delta {\dot{\mathit{r}}}^{†}(t)]}^{†}$, ${[\delta {\mathit{s}}^{†}(t),\delta {\dot{\mathit{s}}}^{†}(t)]}^{†}$, and ${[\delta {\mathit{\rho }}^{†}(t),\delta {\dot{\mathit{\rho }}}^{†}(t)]}^{†}$, respectively, is further exacerbated by residuals from telescope desats and uncertainty in SRP acceleration. The relative SRP acceleration uncertainty is $\delta {\ddot{\mathit{\rho }}}_{\mathrm{SRP}}=\delta {\ddot{\mathit{s}}}_{\mathrm{SRP}}-\delta {\ddot{\mathit{r}}}_{\mathrm{SRP}}$.

All sources of uncertainty that contribute to the final relative position dispersion are listed in Table 3 and shown in Fig. 5. The numerical values in the table are representative of a Roman-starshade scenario and are discussed in Sec. 4.

Table 3

Retargeting sources of uncertainty.

Fig. 5

Illustration of retargeting sources of uncertainty. Thin lines: position vectors; thick lines: velocity vectors; very thick lines: acceleration vectors; dashed lines: error position vectors; blue: telescope quantities; purple: starshade quantities; black: relative quantities; gray: nominal quantities (not to scale).

3.3.

Error Dynamics Formulation

Having listed the relevant sources of uncertainty, the linearized error dynamics are derived in Sec. 3.4 using the approach described in this section. In particular, three distinct features of the error dynamics formulation are highlighted.

The first feature is that error dynamics are linearized about given absolute trajectories for both the telescope and the starshade, $\mathit{r}(t)$ and $\mathit{s}(t)$, respectively. Note that a given relative trajectory for the starshade, $\mathit{\rho }(t)$, such that $\mathit{s}(t)=\mathit{r}(t)+\mathit{\rho }(t)$ could also be used. An absolute trajectory is chosen here for clarity and convenience. Whereas absolute position errors of 100s of kilometers are expected, SEL2 is more than 1,000,000 km from the Earth and Moon. For both spacecraft, the choice to linearize about $\mathit{r}(t)$ and $\mathit{s}(t)$ is therefore justified, and the validity of the approach is further confirmed via Monte Carlo simulations in Sec. 4. The major advantage of this approach is that the accuracy is effectively independent of the telescope–starshade distance. As a result, this covariance propagation framework is broadly applicable for retargeting analyses. An alternative would be to linearize both the telescope true position and starshade true position about the telescope reference trajectory as in Ref. 36. However, the telescope–starshade distance can be close to a 100,000 km, so the linearization errors may not be negligible in this case.

Thus, starting from the nominal initial state vectors (at $t=t_0$), telescope and starshade absolute positions $\mathit{r}(t)$ and $\mathit{s}(t)$ are propagated and stored for $t_0⩽t⩽t_f$ using a fixed time-step fifth-order Runge–Kutta ordinary differential equation solver. This solver was validated against MONTE simulations³⁷ (MONTE is the JPL’s high-fidelity mission design tool, validated operationally on several NASA deep space missions, including Cassini, Mars Science Laboratory, and Juno) and used in Ref. 12. This MATLAB^®-based environment propagates the position of a telescope and a starshade spacecraft in three-dimensional space, under the influence of the gravity from the Earth, Sun, Moon, and SRP. Rather than exactly matching a particular Design Reference Mission (DRM), trajectories representative of a Roman-starshade scenario (Ref. 9) that stress the feasibility of reacquisition are sought. The chosen reference trajectories are described and discussed in Sec. 4.3.

The second feature is that after linearization about $\mathit{r}(t)$ and $\mathit{s}(t)$, the equations are reformulated to be explicit in the dynamics of the relative state error, $\delta \mathit{\rho }$. This reformulation has two objectives. First, the final relative retargeting error $\delta {\mathit{\rho }}_f$ is what determines the feasibility of the retargeting architecture. Therefore, it is sensible to propagate its covariance directly rather than that of $\delta \mathit{s}$. Second, integrating $\delta \mathit{\rho }$ directly instead of $\delta \mathit{s}$ takes advantage of the fact that at $t=t_{\mathrm{sci}}$, the relative position errors ($\sim 0.1\mathrm{km}$) are orders of magnitude smaller than the absolute position errors ($\sim 100\mathrm{km}$).

The final feature is that the chosen formulation shows that the growth in $\delta \mathit{\rho }$ is relatively insensitive to the absolute telescope position errors. Specifically, $\delta \mathit{r}$ only appears as a second-order error in the gravity-gradient: compare the absolute position errors of 100s of kilometers with a distance of more than 1,000,000 km from SEL2 to the Earth and Moon.

3.4.

Error Dynamics Derivation

The dynamics of the telescope position error $\delta \mathit{r}(t)$, once linearized about its reference trajectory $\mathit{r}(t)$, can be shown to be (e.g., Ref. 38):

In addition, the telescope performs desats at fixed time intervals $\mathrm{\Delta }{t}_{\text{desat}}$. The residuals from these maneuvers are modeled as acceleration error impulses, leading to the following state-space system:

Once discretized, the dynamics become

3.5.

Covariance Propagation

The system output is chosen to be $\mathit{y}(t)=C\mathit{x}(t)=\delta \mathit{\rho }(t)$, where $C=[I,{\begin{array}{c}\mathit{0}\end{array}}_{3\times 15}]$ is the output matrix. The statistics of the final relative position error are thus given by the final output covariance matrix:

In this study, the following quantity is used to evaluate retargeting performance:

Starting from the initial state covariance matrix $P_0$, the final state covariance matrix $P_f$ is evaluated by propagating the state covariance matrix $P_k=\mathrm{Cov}({\mathit{x}}_k)$ using

The matrix $P_0$ provides the statistics of the initial state ${\mathit{x}}_0$, which has the following components. Initial absolute state errors are

4.1.

Conservatism in Assumptions

This work focuses on introducing a covariance propagation framework for starshade retargeting that is capable of authoritatively evaluating retargeting error statistics. It also shows that, for stressing and representative parameters, including actuator and sensor performance, the proposed retargeting architecture is feasible.

One of the conservative assumptions made in this article is that, even though many sources of uncertainty are anisotropic (for instance, the retargeting burn execution error, discussed in Appendix A), the uncertainty associated with the worst-case direction is applied to every axis. Some of the nominal parameters chosen in the sections below are also more conservative than expected for either a Roman or a HabEx starshade scenario. Finally, ${\sigma }_f$ is conservatively used as the main retargeting performance metric.

These simplifications reduce the number of parameters in the analysis, which allows for making bounding arguments about the retargeting error and generates valuable intuition about the system. For any specific DRM, however, the covariance propagation tool is able to incorporate the directionality associated with the initial conditions of each retargeting burn in the sequence and provide tailored retargeting performance estimates. This step will be addressed as part of future work.

4.2.

Selection of Uncertainty Parameters

The standard deviations of the science-phase relative position ${\sigma }_{\delta \mathit{\rho },\mathrm{sci}}$ and velocity ${\sigma }_{\delta \dot{\mathit{\rho }},\mathrm{sci}}$ in Table 3 are based on the expected science sensing performance for a starshade operating with Roman (e.g., Refs. 9, 12, and 15). The science-phase relative lateral sensing performance is tens of centimeters in position and millimeters per second in velocity. By contrast, the relative longitudinal sensing performance is hundreds of meters in position and centimeters per second in velocity. As discussed in Sec. 4.1, the more conservative longitudinal values are used for all axes.

Based on communications with the Roman navigation team about the design of the telescope and its operations at the time of writing, the conservative estimates of ${\sigma }_{\delta \mathit{r},\mathrm{sci}}$, ${\sigma }_{\delta \dot{\mathit{r}},\mathrm{sci}}$, ${\sigma }_{\delta \dot{\mathit{r}},\mathrm{TCM}}$, ${\sigma }_{\delta \dot{\mathit{r}},\text{desat}}$, and ${\sigma }_{\delta \ddot{\mathit{r}},\mathrm{SRP}}$ were generated (see Table 3). These values do not explicitly reflect Roman telescope requirements and are subject to change.

In this analysis, the residuals from each desat are assumed to be uncorrelated. In practice, some correlation might exist if the telescope pointing direction remains fixed while performing nonstarshade-related science for a large portion of the cruise. The execution error corresponding to the starshade retargeting burns and slews $\delta {\dot{\mathit{s}}}_{\mathrm{ret}}$ is discussed in detail and evaluated in Appendix A. The starshade TCM residual uncertainty is estimated using the open-loop performance of the RCS thrusters, also discussed in Appendix A.

To select ${\sigma }_{\delta \ddot{\mathit{s}},\mathrm{SRP}}$, the following model for the SRP acceleration due to a Sun-facing specularly reflective plate of surface area error $\delta {\mathcal{S}}_{\mathrm{SRP}}$ is used (e.g., Refs. 39 and 40): $\delta {\ddot{\mathit{r}}}_{\mathrm{SRP}}=[2\varphi \delta {\mathcal{S}}_{\mathrm{SRP}}/(mc)]{\widehat{\mathit{r}}}_{\odot }$. Here, ${\widehat{\mathit{r}}}_{\odot }$ is the unit vector pointing from the Sun to the spacecraft, $\varphi$ is the solar flux at a given distance between the spacecraft and the Sun, $m$ is the spacecraft mass, and $c$ is the speed of light in vacuum. As $\varphi$, $m$, and $c$ are approximately constant during the cruise, $\delta {\mathcal{S}}_{\mathrm{SRP}}$ is the only independent variable. Its standard deviation, denoted ${\sigma }_{\mathcal{S}}$, thus provides an intuitive means of estimating representative SRP uncertainty magnitudes.

In this analysis, the cruise is performed with the starshade facing the Sun. In practice, the cruise attitude would be determined by a system trade including, e.g., power, thermal, SRP, micrometeorites, and communications considerations. For a Roman starshade (Ref. 9), the full starshade frontal area is $335{\mathrm{m}}^2$. Despite its size, the starshade frontal area is precisely characterized for science purposes. Extensive in-orbit calibrations may also be performed on the deployed starshade during commissioning as it travels from Earth to SEL2. Based on these considerations, the chosen value of ${\sigma }_{\delta \ddot{\mathit{s}},\mathrm{SRP}}=40\mathrm{nm}/{\mathrm{s}}^2$ corresponds to a nominal standard deviation of 0.5% of the total frontal area. This value of ${\sigma }_{\mathcal{S}}$ is conservative: for non-Sun-facing starshade cruise attitudes, it represents a larger percentage of the total Sun-facing area.

4.3.

Selection of Reference Trajectories

The linear dynamics operators ${\mathrm{\Psi }}_s$ and ${\mathrm{\Psi }}_r$ in Eq. (4) are functions of the reference trajectories as they depend on the distances between the two spacecraft and each gravitational body at every point during the cruise. In this section, two full retargeting trajectories designed to be representative and stress the feasibility of the proposed retargeting architecture are presented. The sensitivity of the retargeting error to the nominal trajectory parameters is further discussed in Sec. 5. Several methods exist for optimizing DRMs and starshade trajectories with given initial and final conditions (e.g., Refs. 41 to 43). Because this work focuses on feasibility rather than on a specific DRM, stressing initial and desired final conditions are selected and a retargeting burn that approximately satisfies these boundary conditions is found by trial-and-error.

For both reference trajectories, starting from a representative Roman trajectory provided by the Roman navigation team, the initial telescope–Earth distance is minimized ($r_{\oplus ,0}$ close to 1200 Mm), and the starshade is initially positioned along the telescope–Earth axis, as close to Earth as possible: ${\rho }_0=37.7\mathrm{Mm}$ (maximum distance mentioned in Ref. 9) and ${\theta }_{\oplus }=0^{°}$. The Earth formation angle ${\theta }_{\oplus }$ is defined as the angle between $-{\mathit{r}}_{\oplus }$ and $\mathit{\rho }$ (see Fig. 4). Although neither a Roman nor a HabEx scenario would allow for ${\theta }_{\oplus }=0°$, the relative position vector has a secondary influence on the error dynamics, as discussed in Sec. 5.2.2. For this reason, ${\theta }_{\oplus }=0°$ is chosen for conservatism and simplicity (see Sec. 4.1). Both retargeting cruises last 3 weeks (again, consistent with Ref. 9), and both trajectories remain approximately in the plane defined by the telescope position, the Earth position, and the telescope velocity vector relative to Earth over the entire cruise.

The first trajectory is a longitudinal retargeting maneuver. It is designed so that the inertial direction of the formation is approximately unchanged at the end of the cruise ${\widehat{\mathit{\rho }}}_f\approx {\widehat{\mathit{\rho }}}_0$ and the final telescope–starshade distance is ${\rho }_f\approx 21.3\mathrm{Mm}$. This trajectory is chosen because minimizing the final telescope–starshade distance leads to the most stringent retargeting error requirement (in kilometers) for a given CS angular FOV. The magnitude of this burn is $19.5\mathrm{m}/\mathrm{s}$. The starshade trajectory relative to the telescope is shown in Fig. 6(a).

Fig. 6

Reference retargeting trajectories relative to telescope position. (a) Longitudinal retargeting maneuver. (b) Lateral retargeting maneuver.

The second trajectory is a lateral retargeting maneuver. It is designed to lead to an $\sim 40°$ retargeting angle between ${\widehat{\mathit{\rho }}}_0$ and ${\widehat{\mathit{\rho }}}_f$, while keeping the initial and final longitudinal distance approximately constant: ${\rho }_f\approx {\rho }_0=37.7\mathrm{Mm}$. This trajectory is chosen because large values of $\rho$ increase the retargeting error, as shown in Secs. 5.2.2 and 5.2.3. The magnitude of this burn is $20\mathrm{m}/\mathrm{s}$. The starshade trajectory relative to the telescope is shown in Fig. 6(b), and both the telescope and starshade trajectories relative to Earth are shown in Fig. 2.

4.4.

Results

Figure 7(a) shows the evolution of ${\sigma }_f$ as a function of the cruise duration for the two time-varying reference trajectories. It also compares them with the bounding time-invariant system defined in Sec. 5.2.4. As expected, both reference trajectories lead to slightly smaller final retargeting errors than those of the bounding model. In addition, the value of ${\sigma }_f$ for the two reference trajectories is almost identical for all cruise durations. In Sec. 5, it is shown that $\mathit{\rho }(t)$ does not have a significant effect on the error dynamics. This finding is consistent with Fig. 7(a), in which the only difference between the longitudinal and lateral retargeting maneuvers is the evolution of $\mathit{\rho }(t)$ during the cruise.

Fig. 7

(a) Retargeting $3{\sigma }_f$ error as a function of $t_f$ for both the longitudinal and the lateral retargeting trajectories, as predicted by the covariance analysis. The evolution of $\delta \rho (t)$ for the longitudinal retargeting maneuver is also shown for 100 out of the 5000 Monte Carlo trajectories. The value of $3{\sigma }_f$ for the bounding model defined in Sec. 5.2.4 is also shown. (b) Comparison between ${\sigma }_f$ values predicted by the covariance analysis versus Monte Carlo simulations for the two trajectories.

These covariance propagation predictions were also verified by running 5000 Monte Carlo simulations for each trajectory. These simulations use the high-fidelity nonlinear solver described in Sec. 3.4 to propagate the telescope and starshade absolute positions. In the inset of Fig. 2, the cloud of $\delta {\mathit{\rho }}_f$ vectors computed with Monte Carlo simulations for lateral retargeting is compared with the $3\sigma$ retargeting error ellipsoid predicted by the covariance analysis. The value of $\delta {\rho }_f$ as a function of $t_f$ is shown for a subset of the longitudinal-retargeting-trajectory simulations in Fig. 7(a). The difference between the ${\sigma }_f$ value predicted by the covariance analysis and the ${\sigma }_f$ value computed from the Monte Carlo sample covariance matrix is shown in Fig. 7(b) for the two trajectories. It is found to remain within 3% for all $t_f$ values in both cases.

Based on these results, a CS FOV of $\pm 1.17°$ and $\pm 0.66°$ would be required to guarantee successful reacquisition at a $3{\sigma }_f$ level at the end of the longitudinal and lateral retargeting cruises, respectively. As mentioned in Sec. 4.1, however, the conservative combinations of anisotropic sources of uncertainty were considered in the analysis, and the choice of nominal parameters and trajectories do not conform to a specific DRM. For instance, no longitudinal retargeting maneuvers are expected in the Starshade Rendezvous Probe study (Ref. 9). Nonetheless, the results presented here show that a CS with an FOV under 2° would be sufficient to enable successful retargeting even with these conservative assumptions.

4.5.

Application to HabEx-Starshade Scenario

Going further, the covariance propagation framework is applied to a HabEx-starshade scenario, using the bounding model from Sec. 5.2.4 for simplicity. As HabEx would also be flying in a quasi-halo orbit near SEL2, the Roman nominal trajectory is maintained. Based on Ref. 11, the nominal telescope–starshade distance is increased to 76 Mm. A retargeting cruise duration of 30 days is considered. The starshade SRP uncertainty is increased by a factor of 4 to reflect the fact that the HabEx starshade would have a 52-m diameter rather than a 26-m diameter. HabEx retargeting burns are also larger, and a bounding value of $100\mathrm{m}/\mathrm{s}$ is considered. For a mass of 5230 kg and a main thruster force of 445 N, the analysis in Appendix A predicts an execution error standard deviation of $0.142\mathrm{m}/\mathrm{s}$.

With these parameters, the predicted $3{\sigma }_f$ value is 3195 km or 2.41°. This preliminary assessment shows that even for a HabEx-starshade scenario, the proposed retargeting architecture is feasible with a degree-level CS FOV.

5.1.

Retargeting Error with No Gravity Gradient

Gravity gradients at SEL2 are significantly shallower than in low Earth orbit. The double-integrator model below is therefore examined to quantify the effect of gravity gradients on the retargeting error. Denoting variables associated with this gravity-gradient-free model with an overbar, the state matrix simplifies to

5.2.

Retargeting Error with Constant Gravity Gradient

Next, in Sec. 5.2.1, the effect of Earth’s gravity gradient is added for constant telescope and starshade positions. As a result, the model produces the instantaneous error growth due to Earth gravity for the assumed positions. Constant positions keep the model time invariant and thus amenable to analysis. Further, as discussed in Ref. 12, Earth is the dominant influence on formation flying dynamics at a telescope-Earth distance of 1200 Mm: it is about six times more significant than the Sun and over 20 times more significant than the Moon. As the relative dominance of Earth gravity decreases for larger Earth-telescope distances, the effects of the Sun and the Moon are also included in Sec. 5.2.4.

5.2.2.

Sensitivity to nominal trajectory parameters

It can be shown that there does not exist a unique set of ${\mathrm{\Psi }}_{\oplus s}$ and ${\mathrm{\Psi }}_{\oplus r}$ matrices that maximizes ${\sigma }_{f,\oplus }$ for all $P_0$ matrices. Instead, contours of ${\sigma }_{f,\oplus }$ are shown in Fig. 8 as a function of nominal parameters and for the specific $P_0$ matrix defined above.

Fig. 8

Earth gravity-gradient model: contour plots of $3{\sigma }_{f,\oplus }$ in kilometers as a function of cruise duration $t_f$, nominal telescope distance to Earth $r_{\oplus }$, telescope-starshade nominal distance $\rho$, and Earth formation angle ${\theta }_{\oplus }$.

Four observations are made in this paragraph and are revisited in Sec. 5.2.3. First, ${\sigma }_{f,\oplus }$ is maximized by small telescope–Earth distances $r_{\oplus }$. Second, ${\sigma }_{f,\oplus }$ is also maximized by small starshade–Earth distances $s_{\oplus }$, i.e., when ${\theta }_{\oplus }=0°$ and $\rho$ is maximized. Third, without gravity gradients, $3{\overline{\sigma }}_f=210$, 348, and 520 km for $t_f=2$, 3, and 4 weeks, respectively. Comparing these values to Fig. 8 shows that the influence of gravity gradients increases with $t_f$, especially for cruises longer than 2 weeks. Fourth, for given values of $t_f$ and $r_{\oplus }$ (i.e., within a given contour plot in Fig. 8), ${\sigma }_{f,\oplus }$ changes by $<6\%$ for all combinations of ${\theta }_{\oplus }$ and $\rho$.

For reacquisition to be successful, the starshade must be in the FOV of the CS at $t=t_f$. In Fig. 9, the same results as in Fig. 8 are plotted in terms of the angular retargeting error rather than the relative position distance error. Although it was observed above that $\rho$ only has a secondary influence on ${\sigma }_{f,\oplus }$, the size of the FOV in kilometers scales with $\rho$. As an example, Fig. 9 (bottom) predicts that reacquisition is not guaranteed at a $3{\sigma }_f$ confidence level with a CS FOV of $\pm 3000\mathrm{arcsec}$ (0.83°) for cruises of 3 weeks if $\rho ⪅30\mathrm{Mm}$. For cruises of 4 weeks, this FOV is not sufficient to guarantee reacquisition at any of the $\rho$ values shown in the figure.

Fig. 9

Earth gravity-gradient model: contour plots of $3{\sigma }_{f,\oplus }$ retargeting angular error in degrees as a function of cruise duration $t_f$, nominal telescope distance to Earth $r_{\oplus }$, telescope-starshade nominal distance $\rho$, and Earth formation angle ${\theta }_{\oplus }$, for $r_{\oplus }=1200\mathrm{Mm}$. As an example, regions exceeding a $\pm 3000\mathrm{arcsec}$ (0.83°) CS FOV are grayed out.

However, conservative uncertainty assumptions have been made as mentioned in Sec. 4.1. Moreover, in Figs. 8 and 9 (bottom), the value of ${\sigma }_{\delta \dot{\mathit{s}},\mathrm{ret}}=40\mathrm{cm}/\mathrm{s}$ is kept constant. At close relative ranges, the starshade has to cover smaller distances for a given retargeting angle, so 3-week cruises with large retargeting burns are unlikely. Burns of smaller magnitude correspond to smaller execution errors, as discussed in Appendix A. Figure 9 (top) therefore shows the same results for ${\sigma }_{\delta \dot{\mathit{s}},\mathrm{ret}}=25\mathrm{cm}/\mathrm{s}$. In this case, the model predicts successful retargeting down to 26 Mm for 3-week cruises.

5.2.4.

Bounding model

Sections 5.2.2 and 5.2.3 show that ${\theta }_{\oplus }$ only has a secondary influence on the error dynamics and that ${\theta }_{\oplus }=0°$ maximizes ${\sigma }_{f,\oplus }$. If ${\theta }_{\oplus }=0°$ and the Sun and Moon are in line with the Earth, starshade, and telescope, then ${\widehat{\mathit{s}}}_{\oplus }={\widehat{\mathit{s}}}_{\odot }={\widehat{\mathit{s}}}_{☾}={\widehat{\mathit{r}}}_{\oplus }={\widehat{\mathit{r}}}_{\odot }={\widehat{\mathit{r}}}_{☾}$, where $\odot$ refers to the Sun and $☾$ refers to the Moon. As a result, the linear system in Eq. (21) is unmodified, except that ${\mu }_{\oplus }/r_{\oplus }^3$ becomes $({\mu }_{\oplus }/r_{\oplus }^3+{\mu }_{\odot }/r_{\odot }^3+{\mu }_{☾}/r_{☾}^3)$ and ${\mu }_{\oplus }/s_{\oplus }^3$ becomes $({\mu }_{\oplus }/s_{\oplus }^3+{\mu }_{\odot }/s_{\odot }^3+{\mu }_{☾}/s_{☾}^3)$, as summarized in Table 4. Using the findings from Sec. 5.2.2, if the largest representative value of $\rho$ and the smallest representative values of $r_{\oplus }$, $r_{\odot }$, and $r_{☾}$ are used, this system is expected to lead to bounding transient growth for all cruise durations.

This bounding model with $r_{\oplus }=\mathrm{1,200}\mathrm{Mm}$, ${\theta }_{\oplus }=0°$, and $\rho =37.7\mathrm{Mm}$ is compared with the gravity-gradient-free model in Fig. 10. While the gravity-gradient-free model is a decoupled isotropic double integrator, the bounding model is an unstable model where relative and absolute error dynamics are coupled and grow anisotropically. Thus, when $t_f=4$ weeks, the final relative position error $1\sigma$ covariance ellipsoid is a 173-km-radius sphere for the gravity-gradient-free model. By contrast, the ellipsoid dimensions are 269, 137, and 137 km for the bounding model: without gravity gradients, the retargeting error is 36% smaller along the Earth-telescope axis but 26% larger in the directions normal to it.

Fig. 10

Comparison of the worst-case growth of relative position error through the bounding model and gravity-gradient-free model. (a) ${\sigma }_f$ as a function of cruise time $t_f$. (b) Percentage error between ${\sigma }_f$ values, relative to bounding model. (c) Angle between the initial conditions and disturbances vectors ${\tilde{\mathit{z}}}_0$ corresponding to ${\sigma }_f$ for the two systems (see Appendix B).

Despite these differences, the angle between the two corresponding ${\tilde{\mathit{z}}}_0$ vectors is only 7.3° [see Fig. 10(c)]. This similarity shows that, although gravity gradients do measurably increase ${\sigma }_f$ for cruise durations between 2 and 4 weeks, their influence is not sufficient to fundamentally change the distribution of initial conditions and disturbances that maximize it. Due to the relatively benign environment at SEL2, it takes longer cruises for the unstable modes associated with gravity gradients to dominate the dynamics. As an example, consider a cruise duration of 100 days, when $t_f$ is 6.5 and 6.8 times as large as the two unstable mode time constants in the bounding system. In this case, the angle between the ${\tilde{\mathit{z}}}_0$ vector of the bounding and the gravity-gradient-free system is 29.8°, indicating that different error growth mechanisms are involved. Thus, during Roman-starshade retargeting cruises, ${\sigma }_f$ mostly grows through a double-integrator-like propagation of initial errors and disturbances, which is further exacerbated by gravity gradients.

5.2.5.

Sensitivity to sources of uncertainty

In this section, the sensitivity of ${\sigma }_f$ to the sources of uncertainty included in the analysis is evaluated. Appendix C shows that ${\sigma }_f$ can be formulated as the root-sum-square (RSS) of contributions due to each source of uncertainty, including desats. With this formulation, all of the contributions of ${\sigma }_f$ can be evaluated with a single covariance propagation simulation. The relative importance of the different contributions as a function of $t_f$ is shown in Fig. 11 for the bounding model. The distribution of contributions after 3 weeks is also shown in Fig. 12(a).

Fig. 11

(a) Normalized contribution to ${\sigma }_f$ from each source of uncertainty. Components are shown for (b) initial position uncertainty (the inset is a close-up of small contributions), (c) initial velocity uncertainty, (d) desat uncertainty, and (e) SRP uncertainty.

Fig. 12

(a) Absolute contributions to ${\sigma }_f$ at $t_f=3$ weeks. (b) Normalized sensitivities of ${\sigma }_f$ at $t_f=3$ weeks.

The value of ${\sigma }_f$ is dominated by the initial relative position error for very short cruise durations. As $t_f$ increases beyond a few hours, the velocity errors begin to dominate. In particular, the retargeting burn execution error is the largest contributor, followed by the initial relative velocity knowledge uncertainty from the science phase. Figure 11 also shows the influence of desat residuals: the later the maneuver is triggered during the cruise, the less time its residuals have to grow. Finally, for $t_f$ values beyond a week, the influence of the relative position SRP uncertainty becomes considerable. For a cruise time of 3 weeks and beyond, it becomes one of the dominant contributors to the retargeting error. For all $t_f$ values, as expected, ${\sigma }_{\delta \mathit{r},\mathrm{sci}}$ and ${\sigma }_{\delta \dot{\mathit{r}},\mathrm{sci}}$ each contribute $<5\%$ of the total error.

In Appendix C, it is also shown that the normalized sensitivity of ${\sigma }_f$ to small fractional changes in individual sources of uncertainty can be expressed as the square of the corresponding normalized contribution. Figure 12(b) shows these sensitivities after a cruise of 3 weeks. It illustrates that ${\sigma }_f$ is highly sensitive to small fractional changes in its larger contributors but is almost not sensitive at all to changes in nondominant contributors. In this case, it shows that efforts to reduce ${\sigma }_f$ are likely to be most effective if they focus on reducing ${\sigma }_{\delta \dot{\mathit{\rho }},\mathrm{sci}}$, ${\sigma }_{\delta \dot{\mathit{s}},\mathrm{ret}}$, and ${\sigma }_{\delta \ddot{\mathit{s}},\mathrm{SRP}}$.

7.1.

Burn Execution Error

In this section, an expression for ${}^{\mathcal{I}}\delta \mathrm{\Delta }{\mathit{v}}_{\mathrm{exec}}$, the inertial frame execution error due to a $\mathrm{\Delta }v$ performed by the starshade’s main thrusters, is derived:

First, an accelerometer error model is introduced, and the corresponding error vector in individual measurements is derived. The first-order effects of accelerometer noise, bias, scale factor errors, internal misalignments, known mounting misalignment, and mounting misalignment uncertainty are taken into account. Spacecraft pointing errors and spacecraft pointing uncertainty are also included.

Second, an expression for the full execution error vector ${}^{\mathcal{I}}\mathrm{\Delta }{\mathit{v}}_{\mathrm{exec}}$ is derived based on this accelerometer model. As described in Sec. 2.1, the retargeting $\mathrm{\Delta }v$ is performed with the starshade main thrusters, nominally aligned with the commanded burn direction. The RCS thrusters are used to control attitude during the burn, closing the loop on measurements from a set of accelerometers to reduce the execution error. The burn is terminated when the total estimated $\mathrm{\Delta }v$ vector matches the commanded $\mathrm{\Delta }v$ vector up to a given threshold.