figure table

# Guest Editorial: Overview of the Special Issue and a Dialog on Starshades

Jonathan W. Arenberg  Northrop Grumman Space Systems, One Space Park Drive,
Redondo Beach California, USA, 90278  Anthony D. Harness  Princeton
University, Princeton NJ, USA, 08544  Rebecca M. Jensen-Clem  University of
California, Santa Cruz California, USA, 95064

###### Abstract

This special issue is dedicated to starshades: science, engineering,
technology and programmatics. Our reasons for organizing this special issue
are several fold. First as a new technology and with research accomplished in
many institutions, recent results are widely scattered in the literature. As
such, we see great value in co-locating many of the most recent results. This
guest editorial summarizes the 19 contributed papers as the result of a
special call for papers. Since this is a rapidly maturing technology, we
wanted to co-locate a primer with the most current work in the field. It is
hoped that this primer will provide a tutorial to the starshade concept and
pathway to the literature not in this issue. In doing so, we hope to widen the
starshade community in terms of engineering and scientific engagements. This
tutorial takes the form of a dialog, where frequently asked questions are
answered.

###### keywords:

Starshades, exoplanets, starlight suppression, high contrast imaging,
technology development

* Corresponding author: Jonathan W. Arenberg<EMAIL_ADDRESS>
##  1 Overview of Special Issue on Starshades

This special issue is dedicated to the subject of starshades. The starshade is
a technology that has seen rapid development and wide interest at many
institutions. Many of the advances in this field are spread over many journals
and meetings. As a result they are difficult to collect in a single location
in order to get a good view of the state of starshades. By gathering in one
issue many papers representing the most recent research, we hope that this
special issue is a useful overview as of 2021. The editors of this special
issue are members of the NASA chartered Starshade Technology and Science
Working Group (TSWG). The idea for this special issue arose at a meeting and
was endorsed by this body.

Figure 1 is a plot of the number of papers about starshades on a year by year
basis. The papers include both refereed and non-refereed contributions. The
formal designation of the start of the starshade is 2006 when the first paper
showed that a starshade can meet the basic requirements to detect extra-solar
planets, especially an exo-Earth[1]. The figure also shows that most of the
contributions are non-refereed papers contributed at meetings and conferences.
By our count the refereed contributions in this special issue constitute
almost 50 percent of all of the refereed starshade literature.

Figure 1: Published starshade papers by year. Those published before 2006 are
those that are commonly recognized as being precursor papers. The data for
this figure was gathered from NASA’s Astrophysics Data System.

As interest in developing starshade-based missions grows, we hope that this
special issue will serve as a tutorial, providing enough of a background for
potential investigators who are not familiar with starshades to have a current
overview of the field in one location. This editorial has the traditional
summary of the contributed articles in Section 2. The co-editors have also
created a tutorial of starshades in Section 3, which has been written in the
form of a dialog between a hypothetical student and starshade “expert”. We
hope this format of “things I am afraid to ask” will allow those new to the
field to easily “get up to speed” quickly and maybe have a little fun as well.

##  2 Summary of Contributions

This special issue is comprised of 19 contributed papers, representing the
wide variety of research and programmatic advancement of the starshade
community. The summary of papers is organized by the area of research starting
with programmatics, technology (formation flying, deployment, modeling,
starlight suppression), science, and concludes with mission related papers.

###  2.1 Starshade Program

Willems and Lisman describe the goals and plans of NASA’s Starshade Technology
Development Activity to TRL 5 (S5). The S5 Project is the name given to the
effort to close the outstanding technology gaps and raise their Technology
Readiness Levels to TRL 5. The ultimate goal of this effort is to make
Starshade technology ready enough when potential exoplanet missions might
enter formulation. The paper describes the technology gaps and the planned
efforts to close them. Attention is paid to to the consistency from which the
gaps are closed.

Hu, R. et al. – Data Challenge describe the Starshade Exoplanet Data
Challenge: an effort to connect the science requirements of future starshade
missions with specific performance parameters by way of synthetic images. The
Data Challenge produces these synthetic images using the Starshade Imaging
Simulation Toolkit for Exoplanet Reconnaissance (SISTER; Hildebrandt et al. )
and takes into account the spatio-temporal variability of astrophysical,
starshade, and telescope sources of background noise in synthetic science
images. In contrast to other space-based exoplanet imaging data challenges,
the Starshade Exoplanet Data Challenge includes both “smooth” and “clumpy”
exozodiacal disks so that the influence of exozodiacal disks structures on
exoplanet detection can be assessed. The Data Challenge is expected to
conclude in September of 2021.

Turnbull et al. report on the in-house analysis of simulated starshade
observations for the Roman Exoplanet Imaging Data Challenge, a community
engagement effort that examined extracting exoplanets and their orbits from
simulated images for the Nancy Grace Roman Space Telescope mission. Along with
data from the baselined coronagraph, the Roman Data Challenge also included
simulations for the potential Starshade Rendezvous mission. In their analysis,
Turnbull et al. find that the starshade’s higher throughput and lower noise
resulted in a 4$\times$ improvement in signal-to-noise over the Hybrid-Lyot
Coronagraph (HLC). Furthermore, the combination of starshade images with
radial velocity (RV) and HLC data resulted in a 2$\times$ improvement of mass
and orbit constraints compared to RV+HLC data alone. This work demonstrates
the enhancement in science enabled by the potential Starshade Rendezvous
mission with the Roman telescope.

###  2.2 Deployment and Formation Flying

Arya et al. report on experiments on the repeatability of the deployment of a
starshade. A fundamental aspect of current starshade designs is being launched
in a stowed configuration and then deploying to final shape in space. This
paper describes the measurement of the two key mechanical subsystems, the
inner disk and the petals. This work shows that the design currently in
development can deploy with sufficient accuracy and repeatability to meet
requirements.

Flinois et al. focus on the retargeting phase of acquisition and propose an
efficient guidance, navigation, and control architecture that is based solely
on chemical propulsion and does not require ground tracking or interactions
with the telescope during the retargeting cruise. The proposed architecture
relies on commercial-off-the-shelf (COTS) accelerometers to enable actively-
controlled retargeting maneuvers and a laser beacon on the starshade to
initiate the acquisition phase. In their analysis, the authors use a
covariance propagation framework to demonstrate this concept is feasible for
the Starshade Rendezvous and HabEx mission concepts and identify the driving
sources of uncertainty. The proposed retargeting architecture could help
reduce the cost and complexity of the retargeting phase of a starshade
mission.

Soto et al. focus on the science phase of acquisition and present an
analytical model for starshade-telescope relative dynamics to include the cost
of station-keeping in observation scheduling. The dynamical model includes a
deadbanding strategy to minimize maneuvers (which require an interruption in
observations) and incorporates keepout angle constraints set by light from the
Sun, Earth, and Moon scattering off the starshade edges. The authors present
simulation results that capture the station-keeping costs as a function of
target star ecliptic coordinates, time, and halo orbit phasing parameter, and
find that gains can be made by optimizing the halo orbit phasing. This
analytical model allows for station-keeping costs to easily be integrated into
design reference mission studies.

###  2.3 Starlight Suppression and Performance Modeling

Harness et al. report on optical verification and model validation experiments
of sub-scale starshades. The experiments were conducted at the Princeton
starshade testbed, an 80 m long enclosure testing 1/1000th scale starshades at
a flight-like Fresnel number. They demonstrate 10-10 contrast at the
starshade’s inner working angle over a 10% bandpass in the visible spectrum,
with a contrast floor of 2$\times 10^{-11}$. Experiments of intentionally
flawed starshades validate diffraction models to better than 35% and set an
upper limit on the effect of unknowns in the diffraction models. The
experiments reveal a deviation from scalar diffraction theory due to light
propagating through narrow gaps between petals, but the authors provide a
model that accurately captures the effect and argue this deviation is
negligible at flight scales. The experiments show no optical impediments to
building a starshade capable of imaging exo-Earths.

Barnett presents an efficient algorithm for accurately solving the Fresnel
diffraction equation for Fresnel numbers relevant to a starshade mission. The
dynamic range in sizes over the starshade (millimeter-wide petal gaps over a
diameter of 10’s meters) make diffraction calculations difficult. The proposed
algorithm utilizes areal quadrature and efficient nonuniform FFT algorithms to
simultaneously achieve the accuracy of state-of-the-art 1D edge algorithms and
the speed of FFT algorithms. The new method is shown to be in agreement with
previous methods through simulations of starshades with numerically determined
and hypergaussian apodization functions, but is an astonishing
10${}^{4}\times$ faster. The author also proves the equivalence of different
edge integral methods currently used in starshade modeling[2, 3]. Starshade
calculations with this speed and accuracy could open the door to new
strategies for designing and tolerancing a starshade mission.

###  2.4 Solar Glint

Shaklan et al. use scatterometry results to present estimates of the
brightness distribution of solar glint from uncoated metallic edges. Using
custom-built scatterometers, the authors test 50 cm coupons of uncoated
amorphous metal edges before and after environmental testing to produce a 2D
scatter distribution function (SDF) over mission-relevant solar angles. The
SDF results are incorporated into the SISTER simulation (Hildebrandt et al. )
to estimate the solar glint brightness for the Starshade Rendezvous and HabEx
mission concepts. This study finds the integrated brightness at the IWA to be
$\sim$V=25 for Starshade Rendezvous and $\sim$V=27 for HabEx and find no
significant change in optical performance as a result of environmental
testing.

McKeithen et al. report on the development of thin, conformal, multi-layer
anti-reflection (AR) coatings to reduce solar glint brightness relative to
uncoated metallic edges. Commercial finite-difference time-domain (FDTD)
software was used to simulate starshade edges coated with thin-film designs
that have low reflectivity at mission-relevant wavelengths. FDTD simulations
show these coatings reduce not only the reflection component of scattering but
also the diffraction component. The total scattering brightness was estimated
to be at least an order of magnitude lower than that of uncoated metal,
representing a substantial improvement over state-of-the-art uncoated edges.
Experimental scatterometry measurements of a representative 50 cm long coupon
were consistent with simulation results and environmental testing showed no
significant change in optical performance.

###  2.5 Exoplanet Detection

Hildebrandt et al. present the Starshade Imaging Simulation Toolkit for
Exoplanet Reconnaissance (SISTER), an open-source, Matlab-based software tool
for simulating images of exoplanetary systems obtained with a starshade and
space telescope. SISTER models the astrophysical scene (e.g. exoplanets,
exozodiacal dust, host stars, and background objects), starshade properties
(e.g. the design and glint parameters), and telescope properties (e.g. the
optical aberrations, bandpass, and detector noise characteristics). This code-
base will be a valuable resource for planning future starshade missions.

Hu, R. et al. – Noise Budget evaluate the noise budget for starshade-assisted
observations with the Nancy Grace Roman Space Telescope and HabEx. The authors
include a solar glint term in their noise budget that is informed by recent
lab-based demonstrations of the starshade’s optical performance, an effort led
by NASA’s Starshade Technology Development Activity to TRL5 (S5). By
simulating a range of astrophysical scenes and their associated background
noise contributions, the authors find that the solar glint is the dominant
error term in the search of Earth-sized planets in the habitable zone of
later-type (K and M) stars, while exozodiacal light is the dominant error term
for Sun-like and earlier-type stars. The authors discuss the science yield of
a starshade with Roman or HabEx under the assumption that these background
terms can be calibrated to the photon-noise limit.

Hu, M. et al. proposes the generalized likelihood ratio test (GLRT) as a
method for detecting faint exoplanets in starshade-based high contrast imaging
data. The GLRT is designed to detect faint exoplanets in the presence of
starshade-induced image artifacts and signal from exozodiacal dust. The
authors show that the GLRT performs favorably compared to existing detection
methods, on the basis of their receiver operator characteristic (ROC) curves.
They also discuss the technique’s astrometric and photometric precision.

Romero-Wolf et al. – Detection Sensitivity consider a starshade rendezvous
with the Nancy Grace Roman Space Telescope. This paper calculates the
sensitivity of Roman to faint exoplanets in the Starshade Rendezvous Mission
architecture, focusing on three primary science cases: 1) searching for rocky
planets in the habitable zones of the nearest stars and searching their
atmospheres for biosignature gases; 2) detecting zodiacal dust disks around
nearby sun-like stars; 3) investigating the atmospheres of known gas-giant
planets at visible wavelengths.

###  2.6 Observations

Morgan et al. compare the exo-Earth yield of the HabEx and LUVOIR mission
concepts, given 1) a blind planet search, 2) partial knowledge of exoplanetary
candidates and their orbits based on precursor ground-based extreme precision
radial velocity observations (EPRV), and 3) perfect knowledge of all possible
exoplanet systems. The authors further consider starshade only, coronagraph
only, and hybrid mission architectures. The authors conclude that prior EPRV
knowledge both improves the exo-Earth yield and the observing efficiency of
all mission concepts, highlighting the important of EPRV precursor
observations.

Romero-Wolf et al. – Orbit Constraints consider the science yield of the
Starshade Rendezvous Mission, designed to work in tandem with the Nancy Grace
Roman Space Telescope. They specifically address the ability of the Starshade
Rendezvous Mission to constrain the orbits of Earth-like planets orbiting the
16 nearby stars that have been identified as the optimal targets for the
mission, finding that rocky planets in the habitable zones of those stars
would be detectable in at least three out of four available observing windows.
They further constrain the astrometric precision and planet classification
true and false positive rates.

Peretz et al. – Performance Envelopes use the SISTER simulation tool to
examine the effects of various observational conditions on the ability to
detect and spectrally characterize temperate planets in reflected light. They
examine a slew of different planet types, host star types, levels of
exozodiacal dust, and various starshade perturbations to set performance
envelopes on a number of starshade-based missions. They find that signal-to-
noise ratio requirements under-predict the performance achieved under
realistic observational conditions. They also use a statistical analysis of
simulated observations to set observational completeness values for a number
of accessible target stars.

Peretz et al. – Remote Occulter Observing Constraints discuss the
observational constraints of the Remote Occulter. The Remote Occulter is a
mission for an Earth-orbiting starshade working with a ground-based telescope,
designed to take advantage of the inner working angle and light collecting
area of the upcoming 30-meter class ground-based telescopes. This work shows
that nearly the entire sky could be observed for up to 8 hours a night. The
authors investigate the sensitivity of observational windows to the
constraints and discuss the implications for future studies.

Peretz et al. – Remote Occulter Observing Schedule consider optimized
observing scheduling with a Remote Occulter mission that includes refueling,
finding that 80 exoplanet targets could be observed in the mission lifetime.
The authors present a method for optimizing the science yield of such a
mission, consider exposure time and observability windows, as well as time
given to station-keeping and retargeting moves.

##  3 A Dialog on Starshades

As was stated earlier, in addition to the contemporary compilation of recent
work in starshades, which are of course the research articles we have
summarized above, the editors would like to provide a resource to the
community in the form of a review. Rather than a formal review article, the
editors have reached back into astronomical history and chosen to present this
review as a dialog. The use of this literary form is intended to make it easy
to get a quick answer to the likely questions of one new to starshades. It is
also an homage to the Galilean original [4] and reminds us that like the
original Dialogo, this science is relatively new. The editors also thought it
would be a bit of fun, and a gift from the starshade special issue editors at
the end of a trying period for us all.

For our dialog, we will organize the discussion over four days, like the
original. In the present case, each day will cover various aspects of the
subject. Day One is for the overall basics and some history. Day Two will
discuss questions regarding engineering and technology. Day Three will cover
questions of science and the final day, Day Four, will address questions of a
programmatic nature. Unlike the original, our dialog involves a student of
starshades, given the moniker Morgan Nemandi. Like all students of a new
subject, Morgan has many questions and asks them over the course of the four
day dialog. Answering Morgan’s questions is our wise and experienced starshade
community acting as Morgan’s tutor. As a stand-in for the collective knowledge
of the starshade community and literature, we introduce Urania Sage as
Morgan’s guide.

###  3.1 Day One: Starshade Basics

Morgan: Hi, my name is Morgan, and I have some questions about starshades. I
hope they aren’t too simple.

Urania: Morgan, it’s nice to meet you and I’m glad you are interested in
starshades. They are an exciting development and should prove to be very
scientifically productive. I look forward to hearing your questions, and
besides, if you have a question, chances are others do too.

Morgan: Great, let’s start with the simplest question there is: very simply,
what is a starshade and how does it work?

Urania: Let’s take a look at Figure 2, which shows the architecture of a
starshade. Morgan, please note that this picture is not at all to scale, but
is drawn this way for its pedagogical purposes. From left to right, there is
the host star and its planet. The starshade is next: it is a shaped optic of
radius $R$, typically tens of meters. We also call it a binary optic, meaning
the transmission is either complete, unity or 1, or entirely opaque, null or
0. The starshade resides on the line connecting the telescope at right and the
target star at left. The spacing between the starshade and the telescope is
typically tens of thousands of kilometers and is called $Z$. As shown in the
picture, the starshade casts a dark shadow and the telescope sits in it, this
shadow effectively turns “off” the host star. The light from the target planet
which is off-axis relative to the star is not attenuated and is collected by
the telescope.

Figure 2: Cartoon of the starshade architecture. A starshade of radius $R$
located at a distance $Z$ from a telescope creates a dark shadow, which
suppresses the starlight, while allowing the faint exoplanet light to enter
the telescope unimpeded.

Morgan: So the telescope is looking over the edge of the shade, sort of like a
total solar eclipse. Is it really that simple?

Urania: Yes, conceptually it is. Implementation is a bit harder though. In
fact, if you look carefully in the literature, you can see that external
occulters have been around since the 1940’s for use in solar astronomy,[5]
apodized or “toothed” apertures enter that literature in the early 1960’s. [6]

Morgan: Speaking of “toothed” apertures, why are starshades shaped like
flowers?

Urania: Starshades are shaped like flowers to control the diffraction of
starlight around them and create a dark shadow in the image formed by the
telescope. To begin to understand this fundamental aspect, let’s consider
Figure 3, which shows a simple circular disk as the black region. You also see
what are called Fresnel zones, alternating gray to white, which are a
mathematical construct essentially counting the difference in path length (in
multiples of $\pm\lambda/2$), between the on-axis point and the radius of the
zone. This difference in path length causes phase changes that determine how
the electric field adds up.

Figure 3: Circular disk, with Fresnel number of 14, in front of alternating
gray and white Fresnel zones (shown out to the 26${}^{\text{th}}$ zone).
Figure 4: Sixteen petal starshade, with Fresnel number of 14 and hypergaussian
design, in front of alternating gray and white Fresnel zones (shown out to the
26${}^{\text{th}}$ zone).

Morgan: It looks to me like the distance is the same for the entire edge of
the circle. So light diffracted from the edge of the circle travels the same
distance?

Urania: Yes, exactly.

Morgan: So everything has the same phase and adds up to the bright on-axis
spot.

Urania: Yes, Morgan, you got it!

Morgan: So that is how we get the Spot of Arago! For any circle all the
distances and phases add up constructively[7].

Urania: Correct. Since we want to “turn off the star”, we want destructive
interference, so the diffracting screen needs to vary the distance and
therefore the phase, as is done in Figure 4. In early literature this was
called a toothed aperture.[8] Toothed apertures are well known to suppress
diffraction effects.[9]

Morgan: Let’s see if I understand this critical piece. Our goal is to get all
the light that diffracts from the starshade to the entrance aperture of the
telescope to be a dark null. This will happen when light with positive phase
exactly cancels the light with the opposite phase – total destructive
interference.

Urania: Exactly! Here’s another way to understand how it works. We can think
of the starshade as consisting of two components: (1) a central circular disk
and (2) radially tapered petals attached to the disk. The central disk serves
as the occulting spot that blocks the on-axis source and creates a geometric
shadow that covers the telescope. However, as we just discussed, light will
diffract around the sharp transition of the disk’s edge, spoiling the darkness
of the shadow. This is where the petals come in: to create a smooth reduction
in the transmission of the disk to reduce the diffractive ringing caused by
the sharp truncation of the electromagnetic field[10, 11]. This process is
called apodization and is a common problem in optical design and
engineering[12, 7].

Morgan: I see. This reminds me of my signal processing course where we used
specific window functions to reduce the spectral leakage in Fourier
transforms.

Urania: Great observation! Yes, this is the same concept applied to the
tangible world on a much larger scale. The petals are binary (either fully
transparent or fully opaque) and approximate the radial apodization function
that provides the critical 10 orders of magnitude of starlight
suppression[13].

Morgan: How does the number of petals come into this, that seems to add
complexity.

Urania: More parts is more complexity, so starshade designers want to get it
right. Given a sufficient number of petals[14, 2] ($n_{p}\gtrsim 12$), the
apodization function $A(\rho)$ of radius $\rho$ is well approximated by petals
whose edges are defined by the polar coordinate
$\theta(\rho)=A(\rho)\frac{\pi}{n_{p}}$. $A$ refers to the fraction of light
blocked by the starshade: $A=1$ is fully opaque, $A=0$ is fully transparent.
Too few petals and the approximation begins to break down.

Morgan: It sounds like the current petal shaped starshade was not born fully
formed like Minerva!

Urania: No Morgan, but good mythology reference.

Morgan: In my signal processing course, we used a Gaussian as our apodization
function. Will any function work here? What determines the shape of the petal?

Urania: A simple Gaussian would not work for our application, as it cannot
provide sufficient starlight suppression for imaging Earth-like exoplanets,
the standard of which is considered to be $10^{-10}$ fainter than and 100
milli-arcseconds angular separation from its host star. At least, it cannot
provide sufficient suppression with a reasonably sized starshade.

Morgan: Were there mission ideas that predate the starshade?

Urania: Yes Morgan, there were few such ideas. The first mention of an
occulter with a space telescope cited is due to Lyman Spitzer in the
1960’s.[10].

In the 1980’s, during the formulation of what is now called the Hubble Space
Telescope, a number of very creative mission ideas for exoplanet detection
were developed and bear some relevance to our story. One proposal was for an
apodized telescope[15]. Another concept for a co-orbital occulting sphere (co-
orbiting with Hubble) was also discussed[16]. A mission concept from the late
1990’s, called UMBRAS, was an occulting body, but was not apodized[17].

In the early 2000’s there was an occulter concept, called BOSS, which was to
work with the Next Generation Space Telescope (now called the James Webb Space
Telescope) which used an apodization on a transmissive film [18].

Morgan: A binary structure sounds much easier to build and maintain than a
transmissive film.

Urania: Yes Morgan, it is significantly less complicated and better performing
than a semi-transparent apodized optic. While all interesting ideas, none of
these concepts were able to demonstrate a feasible design that could achieve
the contrast needed for exoplanet detection.

Morgan: So the real breakthrough was the apodization that allowed a very deep
shadow?

Urania: Yes Morgan, that was the real change: the first high-performance
apodization solution that could be achieved with an opaque screen. This
discovery kicked off the modern era of the starshade. Cash[19] provided an
analytic solution, called the offset hypergaussian, that for the first time
provided sufficient suppression with a starshade of practical size.

Morgan: Offset hypergaussian? I haven’t heard that term before.

Urania: The offset hypergaussian is given by the apodization function

$A(\rho)=\begin{cases}1,&\rho<a\\\
e^{-\left(\frac{\rho-a}{b}\right)^{n}},&\rho\geq a\,,\\\ \end{cases}$ (1)

with tunable parameters $a,b,n$ (typically, $a=b=R/2$, $n=6$). The analytic
solution lends itself well to examining the sensitivities and tolerances of
the petal shapes, as done in Ref. 2. The hypergaussian design also has an
extremely wide wavelength coverage, as the performance improves with shorter
wavelengths – in fact suppression scales steeply as $N^{-2n}$, with
$N=R^{2}/(\lambda Z)$ being the Fresnel number of a starshade of radius $R$ at
a separation $Z$ from the telescope and at wavelength $\lambda$. The downside
to this design is that it achieves its broadband response by having very
narrow petal tips and valleys between petals. For a 30-m diameter starshade,
the tips and valleys must be on the order of 10s of microns, which make the
starshade hard to stow and deploy.

Vanderbei et al.[14] developed a method, through numerical optimization, to
achieve the same performance with a slightly smaller diameter starshade and
with wider tips and valleys (on the order of 100s of microns). The wider tips
come at the cost of a smaller bandpass, with performance degrading at shorter
wavelengths, but the numerical optimization scheme allows one to suit the
design to their needs and minimize the diameter and maximize tip widths while
achieving the target performance over a desired bandpass.

Morgan: How many petals are needed? What determines the length of the petals
versus the diameter of the inner disk?

Urania: The number of petals is set by how well they approximate the radial
apodization function, generally $\gtrsim 12$ petals are needed. Too few petals
and the approximation begins to break down; higher order Bessel terms in the
Jacobi-Anger expansion become relevant at larger radii[20], meaning the size
of the deep shadow begins to shrink from the outside in. The ratio of the
petal length to inner disk diameter is a design decision to be made for
specific missions. Shorter petals are easier to make sturdy and require less
stiffening structure per petal, but a larger number of petals are needed to
approximate the apodization function.

Morgan: You’ve mentioned the terms _contrast_ and _suppression_ a few times,
what do these mean in the context of starshades?

Urania: Suppression is the total amount of starlight blocked by the starshade
and is independent of the telescope’s optics; it is usually quoted as the
amount of residual light incident on the telescope’s aperture. Contrast is
measured in the image plane of the telescope and is thus dependent on the
telescope and starshade properties. At a given spot in an image, the contrast
is the amount of light there, relative to the on-axis peak brightness of the
unocculted star. Contrast is generally the performance metric used in science
contexts as it is tied to the ability to detect an object of a certain flux
ratio at a given spot in the image. Figure 5 shows a comparison between
suppression and contrast measurements for the Starshade Rendezvous
Mission[21].

Figure 5: Suppression (a) and Contrast (b) for the Starshade Rendezvous
Mission[21]. Suppression is measured at the aperture plane. The vertical
dotted lines in (a) mark the width of the telescope aperture. Contrast is
measured in the image plane. The vertical dotted lines in (b) mark the inner
working angle.

Morgan: I have another basic question: how big and how far in front of the
telescope should the starshade be?

Urania: There are three main constraints driving the design of a starshade
architecture[1, 22, 23]. To first order, you need a starshade to be large
enough to create a deep shadow that covers the entire telescope aperture, with
some additional margin. This additional shadow size (typically $\sim$2 m in
diameter) gives sufficient tolerance to allow a formation flying solution to
be found. In effect, the extra shadow size allows imperfect alignment between
the starshade, its shadow, and the entrance aperture of the telescope. As we
discussed earlier, the central disk of the starshade creates the geometric
shadow to cover the telescope and the petals work to keep the diffracted light
out of the center and create a deep shadow. The size of the deep shadow
depends on many parameters of the starshade design and sets the first design
constraint:

$R>\frac{D}{2}\frac{1}{\alpha(N,S,\Omega)}\,,$ (2)

where $R$ is the radius of the starshade; $D$ is the diameter of the
telescope; $\alpha(N,S,\Omega)$ is the ratio of the radius of the deep shadow
relative to the starshade radius, which depends to first order on the Fresnel
number $N$, the target suppression level $S$, and the starshade petal shape
$\Omega$. Typically, $\alpha$ is in the range of 0.05 - 0.15[23]. So, say we
want to build a starshade for a 2.4 m telescope and we think we can design a
shape with $\alpha$ = 0.1, let’s also add another 1 meter margin for formation
flying, then we would need a starshade of radius $R\sim$ 13 m.

The main driver to the separation (Z) of the telescope and starshade is the
inner working angle (IWA), which is the closest angular distance to the star
we want to observe and is set by our science goals. Unlike coronagraphs, for
the starshade the IWA is mostly decoupled from the size of the telescope and
is set by geometry: IWA = $R/Z$. There is a small increase to the angular
extent of the IWA due to the point spread function (PSF) of the telescope
blending in scattered and diffracted light from the starshade’s edge. This
gives the IWA as

$IWA=\frac{R}{Z}+\beta\frac{\lambda}{D}\,,$ (3)

where $\beta$ is the fraction of the PSF width that we must move away from the
starshade edge before we reach our target contrast[24]. $\beta$ is set by the
brightness of residual light at the edge. We can rewrite the second term of
this equation in terms of the number of resolution elements across the
starshade radius $n_{res}=(R/Z)/(\lambda/D)$; typically, $\beta<1$ and
$n_{res}\sim 2$. This yields our second design constraint:

$Z>\frac{R}{IWA}\left(1+\frac{\beta}{n_{res}}\right)\,.$ (4)

As we move the starshade farther away from the telescope to decrease the IWA,
the Fresnel number decreases and it becomes more difficult to control the
diffraction, so we must counteract this by increasing the radius, which
increases the IWA, and so on. This game is played until we reach a solution
that satisfies all the constraints. The Fresnel number sets our third and
final design constraint:

$\frac{R^{2}}{\lambda Z}>N_{0}(\Omega,S)\,.$ (5)

$N_{0}$ is the minimum Fresnel number for which the starshade can sufficiently
control the diffraction to provide the necessary suppression level $S$.
$N_{0}$ is a shallow function of contrast (conversely, the contrast is a steep
function of Fresnel number) and depends on the starshade petal shape. In
practice, this number can be difficult to determine for numerically designed
starshades, but typically $N_{0}\sim 8-10$. For a hypergaussian design, it can
be shown[2] that $N_{0}\sim 2S^{-1/12}$, so for a suppression of 10-10,
$N_{0}\sim 13.5$.

These various constraints can be laid out on a plane where starshade size is
one axis and the Z distance is the other, the (R, Z) plane[25]. An example is
shown as Figure 6. In this plot, the various constraints are shown and the
direction on the plane where they are satisfied are indicated by the arrows.
The hatching shows the region of the (R, Z) plane where all of the constraints
are satisfied.

Figure 6: A typical design space plot for a Starshade, Figure 5 of Ref. 25 The
equations referenced in the plot legend are the relevant references in the
original paper.

Morgan: So if I have a science goal that needs a certain contrast level at a
given IWA, I can make a plot similar to Figure 6 and get a quick estimate of
the starshade size and distance?

Urania: Yes, you have it. You seem to be getting the hang of this!

Morgan: What is the “shadow size constraint” in Figure 6? It doesn’t seem to
be a driving constraint.

Urania: In a word, it isn’t. For the case in Figure 6, the shadow constraint
of Equation 2 is not the driving or limiting constraint. In systems design
there are usually many factors that are related to a system parameter. The
term driving requirement is given to the requirement that is the most
stringent or sensitive. In most starshades for exoplanets Equation 4 and
Equation 5 are what set the limit. Fortunately, there is always a solution to
these constraints, at least mathematically. [22]

Morgan: If telescope size isn’t a main driver, how does the starshade size
scale with the telescope size? Is it 1:1? In other words, to go from a 4 meter
telescope to a 12 meter telescope, do I need a starshade that is 3$\times$
larger?

Urania: Not exactly. It helps to think of the extra telescope size as an
additive, rather than multiplicative, term, since most of the starshade size
is used to generate the shadow in the first place. Once we have a starshade
large enough to create a dark shadow, making the shadow larger requires (to
first order) a linear addition to the starshade size. As we saw in Figure 6,
the main driver to the starshade size is IWA. If we have designed a 50 meter
starshade to work at a given IWA for our 4 meter telescope, widening the
shadow by 8 meters would result in a $15-25\%$ increase in the starshade size.

If we keep the IWA constant, we get additional help in keeping the starshade
size reasonable when moving to a larger telescope. We can rewrite the Fresnel
number as $N=R\theta/\lambda$, where $\theta$ is the IWA. If we hold $\theta$
constant, the Fresnel number increases with increasing starshade radius. A
larger Fresnel number means it is easier to control the diffraction, so we can
design an apodization function that can support a larger telescope relative to
the starshade size.

Morgan: Earlier, you mentioned that the IWA is decoupled from the telescope
size and that telescope size is not a driving requirement. What role does the
telescope size play in the system design?

Urania: Of course the telescope size is driven by light collecting power
(exoplanets are very faint!), but in terms of high contrast, high image
resolution helps to separate diffracted starlight from the location of the
exoplanet in the image. All of the diffracted light in the image emanates from
the location of the starshade in the image (think of the Geometric Theory of
Diffraction[26]), which means the contrast rapidly improves as you move away
from the geometric IWA in the image. The amount to which we resolve the
starshade can be expressed as the number of resolution elements across the
starshade radius: $n_{res}=\frac{R/Z}{\lambda/D}=\frac{\theta D}{\lambda}$. An
increase in resolving power allows us to relax the starshade performance as we
can take advantage of the telescope’s PSF to separate the diffracted starlight
from the region of interest in the image.

As a result, we can allow tolerances on the starshade to be reduced by
designing a starshade with a smaller geometric IWA (requiring a larger,
farther starshade) and allowing the contrast to fall with the telescope’s PSF.
Or, if the IWA is held constant, the starshade is more and more resolved with
increasing telescope size. If we are able to achieve the same contrast with
poorer suppression, we can create a wider, shallower shadow in which to fit a
larger telescope. This behavior enables Peretz et al. to pair a starshade only
100 meters across with the 39 meter aperture of the next generation of
extremely large telescopes.

Morgan: Will there be an outer working angle?

Urania: The starshade has no outer working angle (OWA). A coronagraph’s OWA is
usually set by the number and density of actuators on the deformable mirror.
Because a starshade does not require wavefront control with a deformable
mirror, its OWA can be unlimited, though the instrument’s field of view may
play a role. However, the field of view constraints can be overcome by
offsetting the telescope’s pointing; unlike a coronagraph, there is no
requirement for the star to be on-axis.

Morgan: Why doesn’t the star need to be on-axis?

Urania: Since the starshade creates the shadow independent of the telescope,
it doesn’t matter where the telescope is pointed as long as it remains in the
shadow. As always, imagining yourself in an eclipse is helpful here. This
freedom enables a wide field of view by building a mosaic image and also opens
the door to multi-object spectroscopy.

Morgan: If the starshade is located quite far from the telescope, won’t it
take forever to slew the starshade to a new target? How can I do a survey of
many targets with a starshade?

Urania: Now, now Morgan, forever is not a very precise term. But your
intuition is good here. The starshade is of course far away, so moving between
targets takes a while. Moving between targets requires thrust to change the
velocity, $\Delta V$, of the starshade and consumes fuel. It is the finite
fuel capacity that is a strong limitation of starshade missions.

Morgan: Is there a solution to this problem?

Urania: This issue was recognized very early in the development of the
starshade idea. In this era, it was the fashion for missions to try and do it
all themselves. So early mission analysis studied this mission planning[27,
28, 29, 30]. Additionally, mission designs with multiple starshades were
studied. [31]

This is of course an area of great concern. The ability to move the starshade
to a target, make an observation and then return is key to establishing
orbits. There are papers in this issue (Flinois et al. and Morgan et al.) that
show the most recent thinking.

Morgan: Are you saying that a Starshade can be be used as a survey tool?

Urania: It can of course, but that is not its real strength. It is far better
at characterization.

Morgan: So it is a good idea to know about the targets ahead of time?

Urania: Yes, that is a main point of the paper by Morgan et al. in this issue.

Morgan: Earlier, when we were talking about outer working angles, you
mentioned wavefront control. Does a telescope working with a starshade need
active wavefront control?

Urania: No and this is the key strength of the starshade: the starlight has
already been suppressed before it even reaches the telescope. Any distortion
of the telescope’s PSF due to wavefront errors introduced by its optics will
occur at lower intensity levels in the wings of the PSF, which will have
minimal impact on the exoplanet image – the on-axis and off-axis PSFs are of
equal strength. The quality of telescope’s optics merely affect the image
quality (Strehl ratio) as it would in standard imaging applications. By
comparison, in a coronagraphic system, the full strength starlight enters the
telescope and wavefront-induced distortions in the wings of the on-axis PSF
can be many orders of magnitude brighter than the peak of the off-axis PSF.

Morgan: Ok, I think I am starting to understand this. Does this mean
starshades can work even outside a vacuum?

Urania: Yes, excellent thought! Starshades work outside a vacuum, which makes
it much easier to perform optical tests on the ground. The first starshade
experiments[32, 33] were conducted in vacuum and air and saw little difference
in performance between the two (except for light scattered by aerosols). This
revelation, with subsequent explanation[2], opened the door to experiments at
larger separations in order to test at flight-like Fresnel numbers. Starshades
have been tested over kilometers of separation on dry lakebeds[34, 35, 36] and
with a heliostat on Kitt Peak[37, 38]. Similarly, the optical experiments of
Harness, et al. were conducted in air.

Morgan: That’s all of the questions I have for today. I think I shall dream of
hypergaussians tonight.

###  3.2 Day Two: Starshade Technology and Engineering

Morgan: Yesterday was very helpful in getting me up to speed with how the
starshade operates, but now I have some questions on the technologies needed
to build and operate a starshade. First off, how do you get such a large
structure in space?

Urania: Starshades are large structures that must fit into the fairly small
volume of a fairing. Stowing and then deploying a starshade is a key element
of its design and has been addressed with various designs over time. [39]

Figure 7: Deployment of 10 meter Inner Disk Subsystem prototype hardware.
Photo credit: Jet Propulsion Laboratory.

Morgan: How can we be sure that it will unfold properly?

Urania: The development of dependable deployment mechanisms have been a major
focus of starshade engineering[40, 39, 41, 42, 43]. The S5 technology program
presented in Willems and Lisman has a number of technology milestones[44, 45]
that demonstrate the deployment of full-scale subsystems with sufficient
accuracy and repeatability, and Arya et al. present their solution to this
design problem. Figure 7 shows a sequence of such a deployment.

Morgan: That looks hard…

Urania: Of course, but there are many complex deployable systems that have
been built and successfully operated over the last 60 years. Current starshade
designs leverage much of this heritage.

Morgan: Once a starshade is deployed, do we have to worry about vibrations?

Urania: The starshade’s optical performance is only sensitive to in-plane
perturbations to its shape. Vibrations from thruster firings or low frequency
vibrations from thermal effects that cause in-plane shape deformations are
accounted for in the error budget[46], however, they are expected to be
minimal. Dynamic modeling shows that vibrations due to thruster firings dampen
to micron-levels within a few seconds of firing[47]. Ref. 48 performed a
dynamical and thermal analysis on the optical performance and found the
effects to be benign. Refs. 45, 44 have shown that the on-orbit thermal and
mechanical stability requirements of critical components can be met.

Morgan: What are the alignment tolerances between the starshade and telescope?
Is it similar to an interferometer?

Urania: No, it is much easier. Since the starshade suppresses starlight
independently of the telescope (you can even achieve high contrast with the
naked eye!), the alignment tolerances are measured in meters rather than
wavelengths. The lateral position requirement is just to keep the telescope
inside the deep shadow. The size of the deep shadow is designed to be larger
than the telescope ($\sim 2$ meters larger in diameter), which allows the
starshade to freely drift around the line of sight to the star and need
minimal corrections (every 100’s of seconds) to keep the shadow centered over
the telescope[49]. The axial positioning requirement is even less stringent,
as the shadow is created over many 100’s kilometers axial distance. In the
relatively benign gravitational environment of L2, it is understood to be
straightforward to maintain $\pm 1$ meter position control of the spacecraft
and many methods to doing so have been identified[50, 51, 52, 53, 54, 55].
Much more stringent formation flying is achieved in the more onerous gravity
environment of Low Earth Orbit with the International Space Station.

Morgan: Yes, I am familiar with formation flying with the ISS, but that is
done over a very short distance where you can easily see the target. How do
you know where the starshade is over such large separations?

Urania: Ah yes, you’ve highlighted the crux of the formation flying problem:
accurately sensing the relative position over very large distances. The
required $\sim 10$ cm positioning knowledge over 10,000’s of kilometers
separation corresponds to a sub-milliarcsecond angular measurement and would
push on the resolution capabilities of most instruments. However, with the
starshade we are able to exploit the light diffracted around the starshade to
provide high precision position information over arbitrarily large distances.

Morgan: But I thought we suppress all the diffracted light?

Urania: We suppress the diffracted light at wavelengths in the bandpass of our
design, but outside of that the starshade’s performance quickly deteriorates
and a strong signal of diffracted light emerges as the Spot of Arago. This
diffraction pattern provides a one-to-one mapping of the lateral position of
the starshade relative to the line of sight to the star[50]. A pupil imaging
sensor can image the diffraction pattern incident on the telescope’s aperture
and extract the lateral position of the starshade to high accuracy. This
method of position sensing and formation flying control has been demonstrated
to the necessary levels in multiple experiments[56, 49, 57].

Morgan: If the Spot of Arago reappears outside the operating bandpass, does
that mean light at that wavelength is constructively interfering? Like the
circular disk we discussed yesterday?

Urania: Exactly. It is easiest to think about this at long wavelengths – the
long wavelength light cannot discern the subtle shape of the petals, so the
starshade begins to resemble a circular disk where the diffracted light
reaches the telescope in phase.

Morgan: How do we know the starshade will perform this way? Isn’t the
mathematics of diffraction complicated?

Urania: There are a number of approximations we can employ to simplify the
math into a manageable problem, for example, assuming that the starshade is
large enough and far enough away that we can use scalar diffraction theory. We
have developed a number of methods to solve the diffraction equation for the
starshade[2, 3, 58, 59], most recently in this issue with the work of Barnett.

Morgan: How confident are we that the math is correct?

Figure 8: Left: A starshade is used to observe stars with the heliostat at the
McMath-Pierce Solar Telescope[37, 38]. Right: A starshade is tested over
kilometers of separation on a dry lake bed[34, 35, 36]. Photo credit: Robert
Brown (Northrop Grumman)

Urania: There have been numerous efforts to test scaled down versions of
starshades in order to validate the diffraction models. Although it is
difficult to test at the large separations needed to achieve a flight-like
Fresnel number, we have found a number of creative solutions, a couple are
shown in Figure 8. These tests have been done in a lab setting over tens of
meters[32, 33, 60, 61, 62, 63, 64], on a dry lake bed over hundreds of
meters[34, 35, 36], and using a heliostat over kilometers[37, 38]. The recent
lab experiments of Harness et al., performed at a flight-like Fresnel number,
show good agreement with the diffraction models.

Morgan: How sharp do the petals’ edges need to be?

Urania: The large perimeter of the starshade petals provides a large source of
scattered light[24] from the brightest object in the sky, the Sun, which is 56
magnitudes brighter than a 30${}^{\text{th}}$ magnitude exoplanet. With the
target star suppressed, sunlight scattered off the petal edges becomes the
dominant source of noise in the image[65] and can have an integrated
brightness of $V\sim 25$ magnitude[66]. To minimize the scattering cross-
section seen by the sun, the optical edges of the petals are designed to be as
sharp as possible. The technical difficulty here is to create an optical edge
that is as thin as possible, but is strong enough to maintain its shape and
survive launch stresses.

Morgan: If it is difficult to build a thin optical edge, why not reduce the
reflected light by painting it black?

Urania: The light scattered by sharp, uncoated edges is dominated by specular
reflection and diffraction. A black coating can decrease the specular
reflection component, but the coating increases the radius of curvature of the
edge and leads to a stronger diffuse reflection. The magnitude of scattered
sunlight from various edge geometries has been quantified through stray light
models and laboratory experiments[24, 67, 68, 69, 70, 71, 72, 73, 74]. In this
issue, McKeithen, et al. report on the promising development of anti-
reflection coatings that are thin and able to reduce the scattering brightness
by at least an order of magnitude.

Morgan: Do we have to worry about other sources of light (e. g. moonlight)
reflecting off of the telescope-facing part of the starshade?

Urania: Light from secondary sources such as moonshine, Earthshine, the Milky
Way, etc, can be reflected into the telescope[24]. For this reason, the opaque
optical shield on the telescope side is made from a diffuse scattering, low
reflectivity material such as Kapton. Ref. 47 estimates the worst case
incident angle for Jupiter reflecting off Kapton makes the starshade appear as
30${}^{\text{th}}$ magnitude in integrated brightness. However, the power is
spread over the size of the starshade and most of it comes from the central
disk of the starshade, which is a number of resolution elements away from the
IWA. The effect of secondary stray light sources are estimated in Hu, R. et
al. – Noise Budget.

Morgan: Do we have to worry that the plume from the starshade’s fuel will
reflect light (e. g. sunlight) and contribute to the background noise on the
science detector?

Urania: Yes, the exhaust plume from the starshade’s chemical thrusters creates
a source of scatterers that can reflect sunlight into the telescope and ruin
the image. To prevent this, the telescope and starshade are coordinated to
pause the observation while the starshade is thrusting to maintain alignment,
which occurs in $\sim$500 second intervals. Ref. 75 estimates the plume will
exit a 10 arcsecond diameter field of view in just 0.8 seconds. Ref. 49
developed a formation flying framework that aims to minimize the observing
time lost to closing the shutter by minimizing the number of thruster firings
and syncing the firings to occur at times when the camera electronics are
already in the process of being read out.

Morgan: What happens if a micrometeoroid hits the starshade?

Urania: The starshade can tolerate a significant number of micrometeoroid-
induced pinholes in its opaque optical shield before it begins to degrade in
contrast. A cumulative area of 1 cm2 of clear-through pinholes would result in
only 10-12 contrast[47]. Adding multiple layers to the optical shield will
prevent most micrometeoroid impacts from creating a pinhole with a direct line
of sight to the telescope[24], further reducing the effect of pinholes.
Similarly, sunlight entering the opaque shield through the pinholes will
require a significant number of bounces to reach the telescope and will be
substantially reduced. Ref. 47 conservatively estimates the starshade can
sustain 10’s to 100’s of square centimeters of pinholes while meeting its
contrast requirements.

Morgan: I’ve heard about proposals for spinning starshades – why would the
starshade spin during observations?

Urania: The starshade is not required to spin during observations, but that is
the design choice made for the potential Starshade Rendezvous Mission[21],
where the starshade spins at $\sim 0.33$ rpm. Spinning the starshade
accomplishes three goals. First, spinning the starshade stabilizes its
attitude and eliminates the need for reaction wheels for attitude control.
Second, spinning the starshade keeps it rotating in and out of direct
starlight and prevents thermal gradients from developing. The starshade is
designed to be thermally stable and distortions to its shape from thermal
gradients are expected to be low[48, 44], but spinning helps ease the thermal
requirements of materials. Finally, spinning the starshade at a rate faster
than the exposure time (which is the current baseline) will smear any defects
on the starshade into a uniform annulus interior to the inner working angle in
the image. A smooth, uniform annulus can more easily be subtracted from the
image in post-processing and will not be mistaken for an exoplanet. If the
starshade is spun at a rate slower than the exposure time, defects on the
starshade will not be smeared into an annulus, but will still move in
subsequent images while the astrophysical scene will remain stationary. This
opens the door for an angular differential imaging (ADI) like post-processing
method to distinguish defect from exoplanet.

Morgan: All of this engineering talk is making my head spin, so let’s break
for the day.

Urania: Agreed. Tomorrow we shall focus on observations and discuss the
exciting science the starshade can enable.

###  3.3 Day Three: Starshade Science

Urania: Good morning! So far we’ve learned the basics of how a starshade
operates, along with the various technologies and engineering solutions that
enable it. Today, let’s examine starshade observations and their scientific
output.

Morgan: I’m ready!

Urania: To start, what science questions would you like to answer?

Morgan: I would like to find Earth’s twin. A rocky, terrestrial exoplanet in
the habitable zone of its host star.

Urania: A pale blue dot, say?

Morgan: Yes, and search it for biosignatures – look for signs of life.

Urania: So you would like spectra of that terrestrial exoplanet?

Morgan: Yes! I’m searching for a pale _rainbow_ dot, if you will.

Urania: Well, you’ve come to the right place. Starshades excel at spectral
characterization of faint exoplanets.

Morgan: What makes a starshade better than a coronagraph?

Urania: To be fair, starshades and coronagraphs each have different strengths
and weaknesses. For example, by minimizing the amount of starlight reaching
the observatory, starshades are largely insensitive to optical aberrations
within the observatory. Coronagraphs, however, require exquisite wavefront
control to correct for the aberrations that give rise to speckles when the
starlight encounters imperfections in the observatory’s optical system. A
coronagraph’s performance is also impeded by any telescope obscurations, such
as the shadow of the secondary mirror and its support structure in the case of
an on-axis telescope, while a starshade’s performance is unaffected by such
obscurations.

Morgan: On Day One, you mentioned that the telescope’s optics affect the image
quality; I think a large obscuration would affect the telescope’s PSF and
reduce the image quality.

Urania: Yes, that’s correct. The obscurations do not affect the starshade
performance, but they do affect the quality of the image collected by the
telescope, although much less so than with a coronagraph.

Morgan: OK, thanks for clarifying. Please continue.

Urania: Without the need for a host of additional optics to control the
wavefront, the optical throughput of the starshade observatory can be much
higher than that with a coronagraph. In general, starshades’ insensitivity to
the details of the observatory mean that they allow for the detection of
fainter exoplanets than coronagraphs for a given observatory design.
Starshades offer another advantage in terms of exoplanet detection: they can
support an IWA smaller than $1\lambda/$D and they have no OWA. Coronagraphs,
however, typically have a minimum IWA of $1\lambda/$D, and an OWA that is
determined by the number of actuators spanning the wavefront control system’s
deformable mirror. Starshades therefore support exoplanet observations over a
greater area surrounding the host star.

Once a planet has been detected, starshades offer additional advantages
compared with coronagraphs: they operate over a wide bandpass. Assuming a
hypergaussian starshade design (see Ref. 2), the starlight suppression $S$ is
related to the wavelength $\lambda$, the starshade-telescope separation $Z$,
the radius of the inner disk component of the starshade $a$, and two
hypergaussian components ($b$ and $n$) as:

$S=\left[n!\left(\frac{\lambda Z}{ab}\right)^{n}\right]^{2}$ (6)

where this equation has been adapted from Ref. 22. It can be recognized from
this equation that the contrast is strongly dependent on the Fresnel number
($N$): $S\propto N^{-2n}$; typically $n=6$. Hence, the starlight suppression
improves for shorter wavelengths where all other parameters are fixed. Hence,
the starshade’s size and separation from the telescope are chosen on the basis
of the longest desired wavelength, and, in this ideal hypergaussian case,
there is no lower limit to the starshade’s bandpass.

Morgan: These are all wins for the starshade so far…

Urania: There are several programmatic considerations that tend to disfavor
the starshade. First, as we discussed on Day One, a starshade takes time and
fuel to slew between targets, reducing the total amount of time available for
exoplanet observations. Because a coronagraph is inside the telescope, it does
not require any additional slew time, so it can support a survey with a larger
number of target stars than a starshade system for a given total survey time.
Second, coronagraphs are a mature technology for ground-based exoplanet
imaging, and will soon be demonstrated in space as part of the Coronagraphic
Instrument (CGI) on the Nancy Grace Roman Space Telescope. Starshades, on the
other hand, have not yet been demonstrated in a scientific context, due to the
challenges associated with using starshades on the ground. Hence, there are
some technical considerations, such as the starshade’s deployment after
launch, that will be fully realized only in space. Other considerations, such
as maintaining the alignment between the starshade and telescope, can be at
least partially tested on the ground ahead of launch, as we discussed
yesterday.

Morgan: So the coronagraph is fast, but has lower throughput and the starshade
has higher throughput, but is slow – it sounds like these instruments are
complementary to each other.

Urania: I agree. Coronagraphs are quick and nimble and can survey many targets
in a short amount of time, but cannot support the spectral characterization as
well as the starshade can. Starshades are slow to move between targets, but
have excellent throughput and wavelength coverage. I think a coronagraph for
surveying plus a starshade for characterizing makes for a powerful
observatory.

Morgan: If I use a starshade, what sources of noise will ultimately limit my
ability to detect faint exoplanets outside the inner working angle?

Urania: The sources of noise that will impact observations of faint exoplanets
are summarized nicely in Hu, R. et al. – Noise Budget. The largest impediment
to detecting Earth-like exoplanets is likely to come from exozodiacal dust
around the target star, but for now I will focus on noise sources that are
specific to a starshade mission (as such, I will also neglect local zodiacal
dust and detector noise).

As we discussed yesterday, the largest source of noise from the starshade
comes from solar glint off the petal edges[24, 47]; In this issue, Shaklan et
al. discuss its expected effects and McKeithen et al. offer a possible
mitigation strategy. The solar glint appears as two bright lobes interior to
the IWA and should remain constant for the duration of the observation. The
ongoing Starshade Data Challenge summarized in Hu, R. et al. – Data Challenge
should provide insight into our ability to calibrate and subtract these lobes
from the image.

The next dominant source of noise comes from perturbations in the starshade’s
shape that degrade the suppression performance. Perturbations can occur during
manufacturing and assembly, in-space deployment, or due to distortions caused
by thermal gradients. The dominant types of perturbations have been identified
and are estimated in Refs. 76, 77, 46, 78, 2, 79. In this issue, Hu, R. et al.
– Noise Budget and Peretz et al. – Performance Envelopes estimate the effect
of these perturbations on exoplanet detection.

Morgan: I know that coronagraphic images suffer from residual “speckles”,
which result from uncorrected wavefront errors. Are these a problem for
starshades as well?

Urania: No. If you remember back in Day One, we learned how the starshade is
not sensitive to wavefront errors in the telescope. Any stray light that
enters the telescope has already been reduced by many orders of magnitude
($>9$) and speckles that arise from wavefront errors will be even further
reduced in the wings of the PSF. For coronagraphs, the speckles are generated
from the full strength starlight and can be as bright or brighter than the
planet light.

Figure 9: SISTER (see Hildebrandt et al.) generated image of the star $\beta$
CVn in the Roman telescope ‘Green band’ (615-800 nm). This fictional planetary
system includes Venus, Earth, a ‘super Mars’ (the diameter of Earth but at the
location of Mars), a Neptune-sized planet, Saturn, 3$\times$ the solar system
level of zodiacal dust, and a Kuiper belt analog debris disk. The system has a
visual inclination of $30^{\circ}$, is dynamically stable, and is consistent
with recent radial velocity analysis[80]. The instrumental effects include an
imperfect starshade with 10-10 contrast at the IWA, formation flying
misalignments, solar glint, and optical throughput[81] and detector noise[82]
consistent with the Roman telescope end of life. The integration time is less
than a week. A background galaxy can be seen in the lower right corner, along
with a V=25 background star in the upper left. Simulation credit: Sergi
Hildebrandt (JPL/Caltech).

Morgan: If we have not yet flown a starshade, how do we know what the images
will look like?

Urania: That’s a great point. While it is true that we have not observed an
astronomical object with a full scale starshade, the sub-scale experiments
done by Harness, et al. have validated diffraction models such that we are
confident in estimating the optical performance. These diffraction models are
incorporated into image simulations that allow us to explore what a real
starshade observation would look like. Hildebrandt et al. present the
Starshade Imaging Simulation Toolkit for Exoplanet Reconnaissance (SISTER),
which incorporates instrumental, astrophysical, and starshade noise sources
into the simulation software.

Figure 9 shows a SISTER-simulated broadband (615-800 nm) image of a fictional
planetary system around the star $\beta$ CVn (aka Chara). With an integration
time of less than a week, this image shows the power of the starshade, as
faint Earth-like planets are detectable even in the presence of astrophysical
and instrumental noise. If you would like to better understand what a
starshade image will look like and what the main limitations will be, you can
download SISTER from http://sister.caltech.edu and try it for yourself!

Morgan: Great! I will download that code immediately and start simulating
starshade images on my own. I will want to take spectra of exoplanets over a
wide range of wavelengths. What limits do starshades place on the spectral
coverage?

Urania: The starshade’s wavelength coverage can be made arbitrarily large, but
at the expense of a larger starshade at a larger separation. In general, a
wider bandpass, or a redder bandpass means a larger, farther away starshade.
On Day One, we discussed how the spectral coverage of the starshade is set by
the design of its apodization function. Remember that the hypergaussian
design[19, 2] has a wide bandpass, whose suppression scales with hypergaussian
parameter $n$ as $\lambda^{-2n}$ ($n\sim 6$). The size of the starshade is set
by the reddest wavelength and the suppression improves for all wavelengths
blue of that. This design however, results in narrow petal tips and valleys
that are more difficult to stow and deploy. A numerically determined
apodization function[14] can selectively choose the bandpass to operate over
and will result in a slightly smaller starshade with wider tips and valleys.
The width of the tips and valleys are set by the size of the bandpass, with a
wider bandpass requiring narrower tips and valleys.

Morgan: Are there restrictions on the angular distance between the starshade
and the Sun during science observations?

Urania: Yes, to prevent sunlight from scattering off the starshade and into
the telescope, the observatory has more stringent solar keepout zones than a
typical space observatory. The angle between the starshade normal axis and the
sun is kept between $40^{\circ}<\phi<83^{\circ}$ during science observations
(see Shaklan et al. in this issue). The lower bound is the standard
observatory keepout zone set by the telescope’s baffle, while the upper bound
is set by the starshade $-$ at angles $>83^{\circ}$, the telescope-facing side
of the starshade would be illuminated and scatter sunlight into the telescope.

Morgan: What if the star is resolved?

Urania: The angular size of the star has minimal impact on the starshade’s
performance; in fact, the starshade even works on resolved objects such as
Jupiter[37]. The main impact of a resolved source is to slightly reduce the
size of the shadow, but this can be compensated for by designing a slightly
larger starshade or tightening the formation flying performance. For a target
with angular extent $\theta$, observed with a starshade at separation $Z$, the
diameter of the shadow is reduced by $\theta Z$. For the HabEx starshade at
76,600 km, a 1 milli-arcsecond star reduces the size of the shadow by 0.37 m,
which represents only 0.7% of the starshade’s 52 m diameter.

Morgan: Starshades sound pretty great, but they also sound expensive and
technically challenging! What’s an example of a science case that’s only
possible to address with a starshade.

Urania: The advantages of starshades compared with coronagraphs that we
discussed earlier are such that a telescope would likely achieve a deeper
contrast over a larger field of view and larger bandpass with a starshade than
the same telescope would with a coronagraph alone. The Roman Space Telescope
illustrates this concept: because the Roman telescope itself was donated to
NASA by the National Reconnaissance Office, its secondary obscuration is less
favorable than would have been the case if the telescope was originally
designed with exoplanet imaging in mind. Hence, a “starshade rendezvous”
concept, in which a starshade is launched and aligned with the Roman
spacecraft after the launch of the telescope itself, has been proposed[21].
The starshade would improve Roman’s inner working angle by a factor of two or
more, and improve the limiting contrast by an order of magnitude. While
Roman’s coronagraph would enable direct images of well-separated Jupiter-mass
planets, the small IWA of the starshade would enable the direct imaging of a
large diversity of small planets (rocky planets, mini-Neptunes, etc.) in the
habitable zones of the nearest stars. The simultaneous large OWA also means
that well-separated planets down to Neptune’s mass will be accessible for
characterization. For larger telescopes that may be launched in the coming
decades (such as HabEx or LUVOIR), the starshade increases the number of
rocky, habitable-zone planets that smaller ($<8$-m) telescopes can detect and
characterize compared with a coronagraph[83].

Morgan: What about other types of science? Are starshades only useful for
exoplanet science?

Urania: The starshade is useful for any high-contrast science application, not
only exoplanets. For example, the starshade could be used to study the
material surrounding active galactic nuclei, the dust shells of AGB stars, the
Sun’s corona, and even the faint rings of Jupiter!

Morgan: What about the outflows of Wolf-Rayet stars?

Urania: Yes! Very good suggestion! I will have to write that one down.

Morgan: Here’s another – conduct a deep galaxy survey in regions of the sky
normally blocked by bright stars.

Urania: Great! Please feel free to keep thinking about different science
applications for the starshade. We are always looking to expand the membership
of the starshade community.

Morgan: What kind of telescopes can work with a starshade? Must they be of a
certain size or a certain quality?

Figure 10: Various mission concepts, with a large range in telescope size,
that propose using a starshade. mDOT[84] uses a 10 cm telescope; Starshade
Rendezvous[21] operates with the 2.4 m Roman Telescope; HabEx[75] proposes
telescopes from 2.4 m to 4 m; and Remote Occulter[85] operates with a 39 m
ground-based extremely large telescope. mDOT image credit: D’Amico, Macintosh,
Koening, and NASA Ames. Other images were adapted from their respective
citations.

Urania: Since the starshade creates a deep shadow independently of the
telescope, almost any telescope size or design will support high contrast
observations behind a starshade. However, in order to achieve high contrast at
a certain angular separation, the size and quality of the telescope’s PSF must
be taken into consideration. Figure 10 shows the wide range of missions that
are considering a starshade. The telescopes being considered for these
missions range from the very small $-$ a 10 cm telescope on a small
satellite[84], to the extremely large $-$ an orbiting starshade working with a
40-meter ground-based telescope[86]. A small telescope paired with a small,
close-in starshade could provide interesting disk science[84] with a fast,
nimble, and relatively inexpensive starshade. Larger telescopes require large
starshades, but the improved angular resolution more easily separates
diffracted light from the exoplanet of interest and allows us to relax the
tolerances and performance requirements of the starshade.

There is one other attribute that is needed from the telescope, namely some
sort of communication with the starshade. Since this is not an optical
property, one could say that it’s a property of the observatory itself,
whether that observatory is located on the ground or in space.

Morgan: To close the alignment loop?

Urania: That’s correct, you are really getting this stuff! Just to be clear, I
should mention that this kind of link is very hard to retrofit into a flying
space telescope.

Morgan: So the space telescope would have to be designed to work with a
starshade.

Urania: It would have to be designed into a space telescope, like Roman. But
it could be a retro-fit for a ground-based observatory.

Morgan: It sounds like the starshade is quite versatile.

Urania: Indeed! We have only begun to scratch the surface of different
starshade configurations. We could reduce the size and separation by observing
at very blue wavelengths, or working at a moderate IWA; put a starshade on a
heliostat to observe from the ground[37, 38]; fly multiple starshades to cut
down on slew time[31]; assemble starshades with…

Morgan: OK! OK! Maybe that’s enough for today.

###  3.4 Day Four: Starshade Programmatics

Morgan: In the past three days, you’ve made a true believer of me and I’m now
convinced I need a starshade.

Urania: Excellent! Welcome to the club.

Morgan: However, I’m still unsure of how we turn this into a reality. How can
we validate the starshade’s performance before launch? It sounds like it would
be difficult to do from the ground, given gravity and the distance between the
telescope and the starshade.

Urania: This is a very good question. This is the thinking behind the
starshade technology program (S5): to identify the necessary technologies and
show them to be well understood and of sufficient determinism to allow a
starshade mission. This is what is driving the technology development efforts
presented in Willems and Lisman.

Morgan: What is the best example of this?

Urania: In my mind the best example is the optical performance of the
starshade. Given the size and distances needed for a flight scale starshade,
the Earth is not large enough to do a full scale test. Therefore, we must use
scaled testing. In order for this to be validated and be convincing, analytic
work and experiments to validate the analysis are needed. Analyses of the
starshade’s shadow have been at the root of the planet-finding starshade from
inception[1, 19] to the present[64].

Morgan: Can we really expect to successfully launch a starshade to use with
one of these future observatories without an initial space-based starshade
technology demonstration?

Urania: The question of what is required to show technical readiness for a
starshade mission was addressed by a group assembled by NASA HQ in late 2015.
This group was called the “StarShade Readiness Working Group” (SSWG) and it
had a formal charter[87]. The deliverable item this group was to produce was
to “Develop and deliver to the NASA Astrophysics Director by July 2016 a
recommendation for a plan to validate starshade technology (to TRL 6/7) that
is both necessary and sufficient prior to building and flying a Starshade
Rendezvous science mission.” This group was made up of representatives from
various NASA Centers, JPL, academia and industry and considered many options
to answer the questions:

  1. 1.

How do we go from Technology Readiness Level (TRL) 5 to TRL 6?

  2. 2.

Imagine ourselves at Key Decision Point (KDP)-C for a possible starshade
science mission. Looking back, how did we convince all stakeholders to approve
the mission?

  3. 3.

Put another way: Is a flight tech demo required to prove TRL 6, and if so,
what is it?

Here is what the SSWG found: first, a ground-only development strategy exists
to enable a starshade science flight mission such as WFIRST (now Nancy Grace
Roman Space Telescope) Starshade Rendezvous. A prior flight technology
demonstration is not required prior to KDP-C of WFIRST Rendezvous. Development
solutions exist that support a WFIRST Starshade Rendezvous by Launch Readiness
Date fiscal year 2026-28. Additionally, the technology development for a
Starshade Rendezvous mission is likely to provide significant technology
benefits to both the HabEx and LUVOIR large mission studies The SSWG found
that there were two optional enhancements to the recommended development
approach. The first enhancement, a flight technology demonstration (mDOT[84]),
would enhance the ground development strategy for formation flying sensing and
control and optical performance with additional cost and technical risk. The
second enhancement was that long baseline ground demonstrations in air may
provide some additional benefit for optical verification but at medium-to-high
risk for interpretation of results[88]. Simply put, the SSWG convinced itself
and its built-in review committee that a scaled ground-only set of
demonstrations are necessary and sufficient to achieve maturity for a flight
mission.

Morgan: I again have to say that starshades sound expensive and technically
challenging.

Urania: Yes, any space-based observatory will be expensive and technically
challenging, and a starshade is not necessarily unique in those aspects.
Current coronagraph technologies are driving the flagship missions towards
large, expensive, ultra-stable telescopes[75, 89]. Since the starshade does
not put strict requirements on the telescope, the cost and complexity shifts
from the telescope to the starshade. Refs. 90, 75 outline a starshade-only
architecture for HabEx that forgoes the coronagraph and as a result has a
lighter, more compact telescope. The science yield is lower than the hybrid
architecture, but higher than that of the coronagraph-only architecture.
Starshades are too new a technology and have not seen enough dedicated effort
or funding to tell if they will be “harder” or “more expensive”.

Morgan: What are the prospects for using a starshade in conjunction with one
of the upcoming telescopes (e. g. JWST or Roman)? How about observatories
farther in the future like HabEx or LUVOIR?

Urania: We touched on this briefly in Day Three, but there’s a lot more to
say! Ref. 91 studied the possibility of a starshade rendezvous with JWST, but
that ship has likely sailed. In the near term, the miniaturized Distributed
Occulter Telescope (mDOT) proposes a small 3 meter starshade operating in low
Earth orbit with a small-sat telescope that could provide interesting disk
science and serve as a technology demonstration. In 2019, NASA released a
report describing a starshade that would work in tandem with the Roman Space
Telescope[21]. The goals of this rendezvous would include technology
demonstrations and science – for example, searching for planets around the
nearest 10-12 sunlike stars. Both the HabEx[75] and LUVOIR[89] concept studies
considered a starshade component to their mission, but their final
architectures are yet to be defined. HabEx has baselined a starshade in their
primary mission architecture to provide spectral characterization of rocky
planets in the habitable zone. Appendix I.1 of the LUVOIR final report
discusses the possibility of a starshade rendezvous, but they conclude that
the only starshades that would not be unfeasibly large would be near-UV/blue
starshades (as opposed to NIR or full-bandpass starshades). Further in the
future, the Remote Occulter[85] concept proposes an Earth-orbiting starshade
operating with a ground-based extremely large telescope.

Morgan: What are some open research questions that need to be worked on? How
can I get involved?

Urania: As starshades are a relatively new technology, there are many exciting
research problems prime for exploration. First, there has not been much
progress on the optimization of the apodization function for many years[14,
92, 93] and surely there is room for improvement; perhaps this could lead to
the design of smaller, closer-in starshades.

Additionally, the numerically determined apodization functions are a black box
of sorts and do not lend themselves to simple, analytic scaling relations. A
clear set of scaling relations for numerically determined starshades would be
greatly beneficial for those wanting to design a mission architecture for
their science application.

Image processing techniques and the extraction of science from starshade
images lag behind that of coronagraphic images and there is much to be done in
this space (though progress has started in Hu, M. et al. and Hu, R. et al. –
Data Challenge). The SISTER software of Hildebrandt et al. provides easy
access to simulating starshade images and is a great place to start exploring
the analysis of starshade observations.

There are too many other topics to list here, but I encourage you to read the
papers in this issue to understand the current state of the field and for
inspiration on where to take it next.

Morgan: Where can I learn more about starshades?

Urania: NASA’s Exoplanet Program maintains a websiteaaa
https://exoplanets.nasa.gov/exep/technology/starshade/ with information about
NASA’s starshade technology development program. I have created a public ADS
librarybbbhttps://ui.adsabs.harvard.edu/public-
libraries/L5KELKzcTAq5lNfb9nynaQ including as many starshade publication as
possible. I look forward to reading your contributions to the literature!

Morgan: Thank you for taking the time to enlighten me on such a fascinating
subject.

Urania: My pleasure! I appreciate your interest in starshades; welcome to the
community.

### Disclosures

The authors of this paper are members of the Starshade Science and Industry
Partnership’s Technology and Science Working Group. They have no relevant
financial interests in the manuscript and declare no conflicts of interest.

### Acknowledgments

The authors would like to thank the many reviewers who provided their time and
effort in carefully reviewing the papers in this special issue. We would also
like to call out Gary Blackwood, Renyu Hu, and Phil Willems for their helpful
comments on the editorial, and Sergi Hildebrandt for generating the simulated
starshade images specifically for our Dialog. Finally, a huge thank you to the
editorial staff of JATIS, you have been helpful and supportive at every step
of this journey.

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Author Biographies

Jonathan Arenberg is the Chief Mission Architect for Science and Robotic
Exploration at Northrop Grumman Space Systems. His degrees are in physics (BS
’83) and engineering (MS ’85, Ph.D. ’87) from the University of California,
Los Angeles. He has worked on the Chandra X-ray Observatory, the Starshade and
James Webb Space Telescope and many mission studies.

Anthony Harness is an Associate Research Scholar in the Mechanical and
Aerospace Engineering Department at Princeton University. He received his
Ph.D. in Astrophysics in 2016 from the University of Colorado Boulder. He
currently leads the experiments at Princeton validating starshade optical
technologies.

Rebecca Jensen-Clem is an Assistant Professor of Astronomy & Astrophysics at
the University of California Santa Cruz. She received her Ph.D. in
Astrophysics in 2017 from Caltech. Her research focuses on extreme adaptive
optics technology development and observational exoplanet science.