††thanks: Equal contributions††thanks: Equal contributions

# Universal anomalous turbulent diffusion in quantum fluids

Satoshi Yui  Research and Education Center for Natural Sciences, Keio
University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan  Yuan Tang
National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive,
Tallahassee, Florida 32310, USA  Mechanical Engineering Department, FAMU-FSU
College of Engineering, Florida State University, Tallahassee, Florida 32310,
USA  Wei Guo<EMAIL_ADDRESS>National High Magnetic Field Laboratory,
1800 East Paul Dirac Drive, Tallahassee, Florida 32310, USA  Mechanical
Engineering Department, FAMU-FSU College of Engineering, Florida State
University, Tallahassee, Florida 32310, USA  Hiromichi Kobayashi
<EMAIL_ADDRESS>Research and Education Center for Natural Sciences, Keio
University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan  Department of
Physics, Keio University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8521, Japan
Makoto Tsubota<EMAIL_ADDRESS>Department of Physics & Nambu Yoichiro
Institute of Theoretical and Experimental Physics (NITEP) & The OCU Advanced
Research Institute for Natural Science and Technology (OCARINA), Osaka City
University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

###### Abstract

In classical viscous fluids, turbulent eddies are known to be responsible for
the rapid spreading of embedded particles. But in an inviscid quantum fluid
where the turbulence is induced by a chaotic tangle of quantized vortices,
dispersion of the particles is achieved via a non-classical mechanism, i.e.,
their binding to the evolving quantized vortices. However, there is limited
existing knowledge on how the vortices diffuse and spread in turbulent quantum
fluids. Here we report a systematic numerical study of the apparent diffusion
of vortices in a random vortex tangle in superfluid helium-4 using full Biot-
Savart simulation. We reveal that the vortices in pure superfluid exhibit a
universal anomalous diffusion (superdiffusion) at small times, which transits
to normal diffusion at large times. This behavior is found to be caused by a
generic scaling property of the vortex velocity, which should exist in all
quantum fluids where the Biot-Savart law governs the vortex motion. Our
simulation at finite temperatures also nicely reproduces recent experimental
observations. The knowledge obtained from this study may form the foundation
for understanding turbulent transport and universal vortex dynamics in various
condensed-matter and cosmic quantum fluids.

Turbulent diffusion in classical fluids has been studied extensively due to
its wide range of applications such as chemical mixing in star formation [1],
pollution migration [2], and airborne virus transmission [3]. It has been
known that the turbulent eddies can carry embedded particles and facilitate
their spreading [4]. However, this knowledge does not apply to inviscid
quantum fluids such as superfluid helium-4 (He II) and atomic Bose-Einstein
condensates (BECs), since the injected particles are not entrained by the
superfluid flow at low temperatures. Instead, a distinct transport mechanism
exists.

In a quantum fluid, turbulence can be induced by a chaotic tangle of quantized
vortex lines [5], which are line-shaped topological defects featured by a
circulating flow with a fixed circulation $\kappa=h/m$, where $h$ is Plank’s
constant and $m$ is the mass of the bosons constituting the superfluid [6].
The vortices evolve with time chaotically, and they also undergo reconnections
when they move across each other [7]. Impurity particles in the quantum fluid
can bind to the vortex cores and subsequently move together with the vortices
[8, 9, 10, 11]. Knowing how vortices diffuse in space is crucial for
understanding turbulent transport in quantum fluids. Such knowledge could
potentially benefit various studies such as chemical dopant behaviors in
superfluid nanodroplets [12], vortex-aided nanowire fabrication [13], and
possibly matter aggregation around cosmic strings [14].

So far, there have been very limited studies on the apparent diffusion of
vortices in a quantum-fluid turbulence. On the theoretical side, the overall
expansion of a decaying random vortex tangle near a solid wall [15] and in
bulk He II [16] was simulated. But these studies only provide limited insights
into the diffusion behavior of individual vortices in a fully developed
tangle. In a recent experiment, Tang _et al._ decorated a random tangle of
vortices in He II with solidified deuterium tracer particles and observed that
the vortices undergo anomalous diffusion (superdiffusion) at small times [17].
Their measured diffusion time exponent appears to be insensitive to variations
in both the temperature and the vortex-line density, suggesting possible
generic nature of this vortex-diffusion behavior. However, since the
experiment was conducted only in a narrow temperature range (i.e., 1.7 K to
2.0 K), it is not clear whether the observed phenomenon holds at other
temperatures and whether it is truly generic for all quantum fluids.

Figure 1: (a) Snapshots of an evolving random vortex tangle with $t_{in}=0.08$
s. The red dots represents the vortex-filament points tracked for diffusion
analysis. (b) Variation of the corresponding vortex-line density $L(t)$. The
blow-up in the inset shows fluctuations in $L$ due to ring injections. (c)
Representative data showing the mean square displacement of the vortices
$\langle\Delta x^{2}(t)\rangle$ in the $x$ direction. The solid and dashed
lines are power-law fits to the data in the shaded regions. (d) Diffusion
exponents $\gamma$ averaged over $x$, $y$, and $z$ directions versus the mean
vortex-line spacing $\ell=L^{-1/2}$. The error bars (barely visible for
$\gamma_{1}$) represent the standard deviations of the $\gamma$ values for
diffusion in the three axial directions.

In this paper, we report a systematic numerical study of the apparent
diffusion of individual vortices in a random vortex tangle in He II using the
full Biot-Savart simulation [18]. A random tangle does not generate any large-
scale flows to advect the vortices [19, 20], which makes it feasible to
observe intrinsic vortex diffusion behaviors that are characteristics of the
Biot-Savart law. We reveal that in pure superfluid the vortices indeed undergo
superdiffusion at small times with a _universal_ diffusion exponent regardless
how dense the tangle is. At large times, the superdiffusion transits to normal
diffusion due to vortex reconnections. Our analysis shows that this universal
diffusion behavior is caused by a generic temporal correlation of the vortex
velocity, which should exist in all quantum fluids where the Biot-Savart law
applies. At finite temperatures, the viscous effect is found to only mildly
affect the vortex diffusion, which nicely explains the experimental
observations.

Note that random vortex tangles are indeed ubiquitous: they can be produced by
counterflow in quantum two-fluid systems such as He II [21, 22], atomic BECs
[23], superfluid neutron stars [24, 25], and dark matter BECs [26, 27]; and
they can also spontaneously emerge following a second-order phase transition
in quantum fluids via the Kibble-Zurek mechanism [28, 29]. Therefore, the
knowledge obtained in our study may also offer valuable insights into the
evolution and quenching dynamics of these diverse quantum-fluid systems.

_Vortex diffusion in a pure superfluid._ –Like many other quantum fluids, He
II can be considered as a mixture of two miscible fluid components, i.e., an
inviscid superfluid (the condensate) and a viscous normal fluid (collection of
thermal quasiparticles) [30]. The normal-fluid fraction in He II drops with
decreasing temperature and becomes negligible below 1 K. Experimentally, a
random vortex tangle can be produced in pure superfluid helium at zero-
temperature limit by injecting small vortex rings [20]. Here, we adopt a
similar method numerically to study vortex diffusion in pure superfluid. As
shown in Fig. 1 (a), we first place 64 randomly oriented vortex rings (radius:
$R_{in}=0.11$ mm) in a cubical computational box (side length: $D=1$ mm) with
periodic boundary conditions in all three axial directions. These vortice are
described by the vortex filament model [31], and each vortex filament is
discretized into a series of points. In the absence of the normal fluid, a
vortex-filament point at $\mathbf{s}$ moves at the local superfluid velocity
$\mathbf{v}_{s}(\mathbf{s})$ induced by all the vortices as given by the Boit-
Savart law [31, 32]:

$\frac{d\mathbf{s}}{dt}=\mathbf{v}_{s}(\mathbf{s})=\frac{\kappa}{4\pi}\int\frac{(\mathbf{s_{1}}-\mathbf{s})\times
d\mathbf{s_{1}}}{|\mathbf{s_{1}}-\mathbf{s}|^{3}}.$ (1)

The time evolution of the vortices can be obtained through a temporal
integration of equation (1) (see details and movies in Supplemental
Materials). When two vortex filaments approach to have a minimum separation
less than 0.008 mm, we reconnect them at the location of the minimum
separation following the procedures as detailed in references [32, 33]. We
also inject a randomly oriented vortex ring of radius $R_{in}$ in the
computational box with a repetition time $t_{in}$ to balance the cascade loss
of the vortices [32]. Fig. 1 (a) shows the evolution of the vortex tangle with
$t_{in}=0.08$ s. The variation of the vortex-line density $L$ (i.e., length of
the vortices per unit volume) is shown in Fig. 1 (b). It is clear that after
about 10 s, $L$ settles to a nearly constant level. This steady-state $L$
level can be tuned by varying $t_{in}$.

To study vortex diffusion, we track randomly chosen vortex-filament points and
analyze their mean square displacement (MSD) along each axis (see Supplemental
Materials). Fig. 1 (c) shows the MSD of the vortices in the $x$ direction
$\langle\Delta x^{2}(t)\rangle=\langle[x(t_{0}+t)-x(t_{0})]^{2}\rangle$ in a
representative case, where the angle brackets denote an ensemble average of
all the tracked vortex-filament points in the steady-state time window.
Usually, a power-law scaling $\langle\Delta x^{2}(t)\rangle\propto t^{\gamma}$
is expected, where the exponent $\gamma$ defines different diffusion regimes,
i.e., normal diffusion ($\gamma=1$) and anomalous diffusion (superdiffusion at
$\gamma>$1 and subdiffusion at $\gamma<$1) [34]. Our data exhibit a clear
superdiffusion regime ($\gamma_{1}=1.58$) at small $t$ and a normal diffusion
regime ($\gamma_{2}=0.97$) at large $t$.

Simulations conducted at other $L$ also show similar behaviors. The derived
$\gamma_{1}$ and $\gamma_{2}$ are plotted in Fig. 1 (d) as a function of the
mean vortex-line spacing $\ell=L^{-1/2}$. It is clear that $\gamma_{1}\simeq
1.6$ is universal regardless how dense the tangle is. $\gamma_{2}$ is around 1
but has sizable variations in the three axial directions. These variations are
caused by the reduced sample numbers at large $t$ as well as the non-
uniformity of the tangle at large length scales due to the limited size of the
computational box (see Supplemental Materials for details).

Figure 2: (a) Vortex-velocity temporal correlation function
$C_{x}(t^{\prime})$ for a representative tangle with $\ell=0.16$ mm exhibiting
a power-law scaling that leads to the observed superdiffusion. (b)
$\langle\Delta x^{2}(t)\rangle$ data of two selected groups of vortex-filament
points (i.e., blue circles and red crosses) whose first reconnection event
occurred in the shaded time intervals of the respective colors.

_Mechanism of the universal vortex diffusion._ –To explain the physical
mechanism of the observed vortex diffusion behavior, we first note that
superdiffusion has been observed in various physical and non-physical systems
[35, 36, 37, 38]. For systems involving random walkers, the appearance of
superdiffusion is usually attributed to the so-called Lévy flights, i.e.,
occasional long-distance hops of the random walkers [39]. These flights lead
to a power-law tail of the walker’s displacement distribution $P(\Delta
x)\propto|\Delta x|^{-\alpha}$, which is flat enough (i.e., $\alpha<3$) to
cause superdiffusion [39]. In He II, large displacements of the vortex-
filament points over a short time can occur at the locations where the
vortices reconnect [40, 41]. However, we find that these reconnections always
result in a tail of the vortex displacement distribution $P(\Delta x)$ steeper
than $|\Delta x|^{-3}$ (see Supplemental Materials). Therefore, they cannot
account for the observed vortex superdiffusion. On the other hand,
superdiffusion may emerge if the motion of the walkers is not completely
random but has extended temporal correlations [39, 42]. To see this, we write
the MSD of a vortex-filament point $\langle\Delta x^{2}(t)\rangle$ in terms of
its velocity $v_{x}(t)$ as [43]:

$\langle\Delta
x^{2}(t)\rangle=2\int_{0}^{t}dt_{0}\int_{0}^{t-t_{0}}dt^{\prime}\langle
v_{x}(t_{0})v_{x}(t_{0}+t^{\prime})\rangle.$ (2)

For a fully developed random tangle, the vortex-velocity temporal correlation
function $C_{x}(t^{\prime},t_{0})=\langle
v_{x}(t_{0})v_{x}(t_{0}+t^{\prime})\rangle$ only depends on the lapse time
$t^{\prime}$, i.e., $C_{x}(t^{\prime},t_{0})=C_{x}(t^{\prime})$. If
$C_{x}(t^{\prime})$ shows a power-law scaling $C_{x}(t^{\prime})\propto
t^{\prime-\beta}$ over a large time interval, $\langle\Delta x^{2}(t)\rangle$
then scales as $\langle\Delta x^{2}(t)\rangle\propto t^{2-\beta}$ in the same
time interval and can exhibit superdiffusion when $\beta<1$. In Fig. 2 (a), we
show the calculated $C_{x}(t^{\prime})$ for a representative tangle with
$\ell=0.16$ mm. There is a clear power-law scaling with a fitted exponent
$\beta=0.42$, which leads to $\langle\Delta x^{2}(t)\rangle\propto t^{1.58}$,
matching nicely the superdiffusion exponent reported in Fig. 1. Similar
results are obtained for other cases at different $\ell$, which reveals that
the universal vortex superdiffusion at small $t$ is caused by the temporal
correlation of the vortex velocity. This correlation should be an intrinsic
property of the Biot-Savart law applicable to all superfluids.

The transition to the normal diffusion at large $t$ was also observed
experimentally by Tang _et al._ [17]. They proposed that this transition is
caused by vortex reconnections, which effectively randomize the motion of the
participating vortex-filament points. This randomization suppresses the
velocity temporal correlation and results in the normal diffusion at large $t$
according to equation (2). To verify this view, we analyze the MSD of two
representative groups of vortex-filament points whose first reconnection event
occurred in the diffusion time interval $0.1-0.15$ s and $0.6-0.65$ s,
respectively. Relevant data for the case with $\ell=0.195$ mm are shown in
Fig. 2 (b). Obvious deviation from the superdiffusion scaling is observed for
each group only after their first reconnection event. This observation clearly
proves the causality between vortex reconnections and the transition towards
the normal diffusion.

Figure 3: (a) A representative snapshot of the vortex tangle produced by
counterflow at $T=1.6$ K with $U_{n}=5$ mm/s. The red dots represent the
tracked vortex-filament points. (b) A representative vortex-line density
$L(t)$ curve obtained in counterflow. (c) MSD of the vortices $\langle\Delta
x^{2}(t)\rangle$ in the $x$ direction in counterflow. The solid and dashed
lines are power-law fits to the data in the shaded regions.

_Finite-temperature effect._ –At finite temperatures, the vortices experience
a drag force as they move through the normal fluid due to scattering of the
thermal quasiparticles in He II [44]. The velocity of a vortex-filament point
at $\mathbf{s}$ is now given by [31, 32]:

$d\mathbf{s}/dt=\mathbf{v}_{s}(\mathbf{s})+\alpha\mathbf{s}^{\prime}\times(\mathbf{v}_{n}-\mathbf{v}_{s})-\alpha^{\prime}\mathbf{s}^{\prime}\times[\mathbf{s}^{\prime}\times(\mathbf{v}_{n}-\mathbf{v}_{s})],$
(3)

where $\alpha$ and $\alpha^{\prime}$ are temperature dependant mutual friction
coefficients for He II [45], $\mathbf{s}^{\prime}$ is the unit tangent vector
along the filament, and $\mathbf{v}_{n}$ is the normal-fluid velocity. We then
generate a steady-state vortex tangle using two distinct methods. The first
method is similar to the one adopted at 0 K, i.e., by injecting small vortex
rings in the computational box with static normal fluid (i.e.,
$\mathbf{v}_{n}=0$). The second method is via thermal counterflow as adopted
in the experiment conducted by Tang _et al_ [17].

Figure 4: (a) Diffusion exponent $\gamma_{1}$ versus $\ell$ for vortex tangles
produced by counterflow at various $T$. (e) $\gamma_{1}$ versus $\ell$ for
tangles produced by vortex-ring injection.

In He II, a counterflow can be generated by an applied heat flux $q$, where
the normal fluid moves in the heat flow direction at a mean velocity
$U_{n}=q/\rho sT$ while the superfluid moves oppositely at
$U_{s}=(\rho_{n}/\rho_{s})U_{n}$ [30]. Here, $\rho=\rho_{n}+\rho_{s}$ is the
total density, and $s$ is the He II specific entropy [46]. To compare with the
experiment where the normal-fluid flow is laminar, we set
$\mathbf{v}_{n}=U_{n}\mathbf{\hat{e}}_{z}$ and $\mathbf{v}_{s}$ as the sum of
$-U_{s}\mathbf{\hat{e}}_{z}$ and the induced velocity given in equation (1)
(see Supplemental Materials). We then place a few randomly oriented seed
vortex rings in the computational box. These rings can grow and reconnect,
eventually leading to the formation of a steady vortex tangle [18]. A snapshot
of such a tangle at $T=1.6$ K and $U_{n}=5$ mm/s is shown in Fig. 3 (a), and
the line-density evolution is given in Fig. 3 (b). In the steady-state time
window (i.e., $5-20$ s), we track randomly selected vortex-filament points and
analyze their MSD in the directions perpendicular to the counterflow.
Representative data for $\langle\Delta x^{2}(t)\rangle$ are shown in Fig. 3
(c). Again, a superdiffusion regime is observed at small $t$, which transits
to normal diffusion at large $t$.

In Fig. 4 (a) and (b), we collect the derived superdiffusion exponent
$\gamma_{1}$ for tangles generated respectively by counterflow and ring
injections at various $\ell$ and $T$. It is clear that $\gamma_{1}$ is around
1.6 and is nearly independent of $\ell$, which is in good agreement with the
experimental observations [17]. We also see that $\gamma_{1}$ increases by
less than 0.3 from 1.1 K to 1.9 K, which makes it hardly resolvable in the
narrow temperature range examined in the experiment. At a given $T$, the
$\gamma_{1}$ value for tangles produced by counterflow is slightly larger than
that for random tangles produced by ring injections. This difference becomes
more visible as $T$ increases. Interestingly, it has been known that the
vortex tangle produced by counterflow becomes increasingly anisotropic as $T$
increases [18]. This tangle anisotropy could be the origin of the observed
difference, which is a topic for future research.

Figure 5: Variation of the vortex-velocity temporal correlation function
$C_{x}(t^{\prime})$ with $T$. The data shown are for tangles with $\ell$ of
about $0.14-0.16$ mm. The solid lines represent power-law fits to the data in
the shaded region.

_Discussion._ –The weak temperature dependence of $\gamma_{1}$ indeed involves
interesting physics. To explain it, we again refer to the vortex-velocity
temporal correlation $C_{x}(t^{\prime})$. In Fig. 5, we show the normalized
$C_{x}(t^{\prime})/C_{x}(0)$ curves for vortex tangles produced by counterflow
at various $T$ with $\ell$ in the range of $0.14-0.16$ mm. The
$C_{x}(t^{\prime})/C_{x}(0)$ curve for the random tangle produced by ring
injections at 0 K with $\ell=0.16$ mm is also included as a reference.
Compared to the $T=0$ curve, a major difference of the curves at finite $T$ is
that they saturate to 1 at larger $t^{\prime}$. This saturation, which
corresponds to ballistic motion of the vortices (i.e., $\langle\Delta
x^{2}(t)\rangle\propto t^{2}$) according to equation 2, is controlled by how
the turbulent energy decays. At 0 K, the turbulent energy can cascade down to
scales smaller than $\ell$ by exciting Kelvin waves on the vortex lines [47].
These waves result in small-scale deformations and kinks on the vortices (see
movies in the Supplemental Materials), which can cause rapid turning of the
vortex-velocity vector [32]. This process suppresses the velocity temporal
correlation and is the reason why $C_{x}(t^{\prime})$ at 0 K remains far from
saturation even down to $t^{\prime}=10^{-3}$ s. At finite $T$, the mutual
friction from the normal fluid damps out the Kelvin waves and terminates the
energy cascade at scales comparable to $\ell$ [48, 49]. Therefore, small-scale
deformations on the vortices are all smoothed out, which helps extending the
ballistic motion of the vortices to larger $t^{\prime}$. This extension first
makes the $C_{x}(t^{\prime})/C_{x}(0)$ curve steeper in the power-law region,
thereby leading to a larger exponent $\beta$ as compared to the 0 K case. As
$T$ further increases, the power-law region gradually shrinks and levels off,
which reduces $\beta$. Although $\beta$ and $\gamma_{1}$ are no longer
correlated exactly as $\gamma_{1}=2-\beta$ due to the presence of the
saturation regime, the decrease in $\beta$ as $T$ increases still causes
$\gamma_{1}$ to gradually rise as observed.

In summary, our numerical study has produced a comprehensive understanding of
the intriguing diffusion behaviors of the quantized vortices in a random
vortex tangle. This study may stimulate future research on hidden universal
scaling laws in quantum-fluid turbulence pertinent to trajectory statistics of
individual vortices.

###### Acknowledgements.

Y.T. and W.G. are supported by the National Science Foundation under Grant
DMR-2100790 and the US Department of Energy under Grant DE-SC0020113. They
also acknowledge the support and resources provided by the National High
Magnetic Field Laboratory at Florida State University, which is supported by
the National Science Foundation Cooperative Agreement No. DMR-1644779 and the
state of Florida. S.Y. is supported by the Grant-in-Aid for JSPS Fellow
program under Grant No. JP19J00967. M.T. acknowledges the support by the JSPS
KAKENHI program under Grant No. JP20H01855.

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# The JWST Early Release Science Program for Direct Observations of
Exoplanetary Systems: Recommendations for Best Practices for Data Collection
in Cycle 2 and Beyond

Sasha Hinkley University of Exeter, Astrophysics Group, Physics Building,
Stocker Road, Exeter, EX4 4QL, UK.  Beth Biller  SUPA, Institute for
Astronomy, The University of Edinburgh, Royal Observatory, Blackford Hill,
Edinburgh, EH9 3HJ, UK  Andrew Skemer  Department of Astronomy & Astrophysics,
University of California, Santa Cruz, CA 95064, USA  Aarynn L. Carter
Department of Astronomy & Astrophysics, University of California, Santa Cruz,
CA 95064, USA  Julien Girard  Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21218, USA  Dean Hines  Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Jens Kammerer
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218,
USA  Jarron Leisenring Steward Observatory, University of Arizona, 933 N.
Cherry Ave, Tucson, AZ 85721-0065 USA  William Balmer  Center for
Astrophysical Sciences, The William H. Miller III Department of Physics and
Astronomy, JHU, Baltimore, MD 21218, USA  Elodie Choquet  Aix Marseille Univ.,
CNRS, CNES, LAM, Marseille, France  Maxwell A. Millar-Blanchaer Department of
Physics, University of California, Santa Barbara, Santa Barbara, CA, USA
Marshall Perrin  Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  Laurent Pueyo  Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA  Jason Wang  Northwestern
University  Kimberly Ward-Duong Department of Astronomy, Smith College,
Northampton MA 01063, USA  Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218, USA  Anthony Boccaletti  LESIA, Observatoire de
Paris, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, 5
place Jules Janssen, 92195 Meudon, France  Brittany Miles  Department of
Astronomy & Astrophysics, University of California, Santa Cruz, CA 95064, USA
Polychronis Patapis Institute for Particle Physics & Astrophysics, ETH Zurich,
8092 Zurich, Switzerland  Isabel Rebollido Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA  Emily Rickman European Space
Agency (ESA), ESA Office, Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218, USA  B. Sargent Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21218, USA  Center for Astrophysical
Sciences, The William H. Miller III Department of Physics and Astronomy, JHU,
Baltimore, MD 21218, USA  Kadin Worthen  Center for Astrophysical Sciences,
The William H. Miller III Department of Physics and Astronomy, JHU, Baltimore,
MD 21218, USA  Kielan Hoch  Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218, USA  Christine Chen  Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Stephanie Sallum
Department of Physics and Astronomy, University of California, Irvine, Irvine,
CA, USA  Shrishmoy Ray University of Exeter, Astrophysics Group, Physics
Building, Stocker Road, Exeter, EX4 4QL, UK.  Karl Stapelfeldt Jet Propulsion
Laboratory, California Institute of Technology, M/S 321-100, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Yifan Zhou  Department of Astronomy, The
University of Texas at Austin, 2515 Speedway Boulevard Stop C1400, Austin, TX
78712, USA  Michael Meyer  Department of Astronomy, University of Michigan,
Ann Arbor, MI 48109, USA  Mickael Bonnefoy  Universitè Grenoble Alpes / CNRS,
Institut de Planètologie et d’Astrophysique de Grenoble, 38000 Grenoble,
France  Thayne Currie  University of Texas, San Antonio  Camilla Danielski
Instituto de Astrofísica de Andalucía, CSIC, Glorieta de la Astronomía, 18008,
Granada, Spain  AIM, CEA, CNRS, Universitè Paris-Saclay, Universitè Paris
Diderot, Sorbonne Paris Citè, F-91191 Gif-sur-Yvette, France  Elisabeth C.
Matthews Observatoire Astronomique de l’Université de Genève, 51 Ch. Pegasi,
1290 Versoix, Switzerland  Anand Sivaramakrishnan Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Rachel A. Cooper
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218,
USA  Deepashri Thatte Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218, USA  Jordan Stone US Naval Research Laboratory,
Remote Sensing Division, 4555 Overlook Ave SW, Washington, DC 20375  Malavika
Vasist  Space Sciences, Technologies, and Astrophysics Research (STAR)
Institute, University of Liège, B-4000 Liège Belgium

###### Abstract

We present a set of recommended best practices for JWST data collection for
members of the community focussed on the direct imaging and spectroscopy of
exoplanetary systems. These findings and recommendations are based on the
early analysis of the JWST Early Release Science Program 1386, “High-Contrast
Imaging of Exoplanets and Exoplanetary Systems with JWST.” Our goal is for
this information to be useful for observers in preparation of JWST proposals
for Cycle 2 and beyond. In addition to compiling a set of best practices from
our ERS program, in a few cases we also draw on the expertise gained within
the instrument commissioning programs, as well as include a handful of data
processing best practices. We anticipate that this document will be regularly
updated and resubmitted to the arXiv to ensure that we have distributed our
knowledge of best-practices for data collection as widely and efficiently as
possible.

##  1 Introduction

With a combination of unprecedented sensitivity and wavelength coverage JWST
has already demonstrated its potential for carrying out potentially
transformative science related to the detection and characterization of
exoplanetary systems. The exquisite wave front stability, as well as the
capability to observe exoplanet atmospheres near the peak of their thermal
emission at 3-5 $\mu$m and beyond, means that JWST will have extraordinary
power going forward for characterizing wide-separation exoplanets through
direct imaging. The precise calibration of the observatory Point Spread
Function (PSF) that is afforded by this spacecraft stability (e.g., Perrin et
al., 2018), combined with sensitivity in the near- and mid-infrared that is in
some cases hundreds of times greater than that for ground-based instruments
means that JWST will likely be sensitive to an entirely new class of sub-
Jupiter mass planets at wide orbital separations (e.g., Carter et al., 2021).
This stability and wavelength coverage, combined with wide fields of view in
JWST imagers, will also allow resolved imaging of circumstellar disks at
wavelengths largely out-of-reach to ground-based observatories. The broad
wavelength coverage will also allow the JWST spectrographs within NIRSpec and
MIRI to cover multiple spectroscopic features of exoplanet atmospheres, giving
precise measurements of atmospheric compositions and atmospheric chemistries.

In 2017, the Space Telescope Science Institute (STScI) awarded roughly 500
hours of Director’s Discretionary Time to 13 community-driven Early Release
Science (ERS) programs with the goals of: 1) testing the observatory in the
modes expected to be commonly used by that community; and 2) to make clear
recommendations to the community on best-practices to be used in future cycles
as well as distribute a set of Science Enabling Products (SEPs). Our program
“High-Contrast Imaging of Exoplanets and Exoplanetary Systems with JWST”
(Hinkley et al., 2022), was ultimately awarded 55 hours of DDT time to utilize
all four JWST instruments, and assess the performance of the observatory in
these representative modes. The first science products from this program have
already been presented in two initial publications. Carter et al. (2022)
showcases the first-ever direct images of an exoplanet with JWST, as well as
the first images of an exoplanet at wavelengths longer than 5 $\mu$m. Miles et
al. (2022) shows the first direct spectrum of a planetary mass comapanion with
JWST, and the first spectrum of such an object covering its full luminous
range from 1 to $\sim$20 $\mu$m, clearly showing evidence for disequilibrium
chemistry and the first definitive detection of silicate clouds in an
atmosphere of a planetary mass companion.

In this brief document our aim is to compile a set of data collection best
practices for members of the community focussed on the direct detection and
spectroscopy of exoplanetary systems. Our goal is that this information will
be useful for preparation of proposals for Cycle 2 and beyond. In a few
instances we also make recommendations for data processing best practices. In
addition to compiling a set of best practices from our ERS program, in a few
cases we also draw on the expertise gained within the JWST commissioning
programs (e.g., Girard et al., 2022; Kammerer et al., 2022a, b;
Sivaramakrishnan et al., 2022). We anticipate that this document will be
updated and resubmitted to the arXiv, perhaps multiple times, to ensure that
we have distributed our knowledge of best-practices for data collection, as
efficiently as possible. In §2 and 3, we outline our recommendations for best
practices for carrying out NIRCam and MIRI coronagraphy of point sources and
extended (e.g., disk) structures, respectively. In §4 we outline our
recommendations for spectroscopy of point sources using NIRSpec and the MIRI
Medium Resolution Spectrgraph (MRS), and in §5 we highlight some best
practices for employing Aperture Masking Interferometry (AMI) with the NIRISS
instrument.

##  2 Recommendations for Point Source Coronagraphy

In this section, we outline our recommendations for the best practices for
carrying out point source coronagraphy (e.g., exoplanets and substellar
companions to bright stars) with both NIRCam and MIRI, as well as some
recommendations that can apply to both instruments.

###  2.1 NIRCam Coronagraphy

  * •

Dual-band Imaging: Unlike the first observation cycle, dual-band imaging will
be enabled for NIRCam Coronagraphic imaging for Cycle 2. This will allow for
simultaneous observations in the Short Wavelength (SW) and Long Wavelength
(LW) channels and the filters in each. Only one coronagraphic mask can be used
for both channels.

  * •

Inner Working Angle: Because the NIRCam round coronagraphic masks have a
smooth transmission function, the “inner working angle” (IWA) of the
coronagraph is stated to be where the coronagraph has 50% transmission.
However, as discussed in Kammerer et al. (2022a) and Girard et al. (2022), the
stability of the JWST wavefront means that objects interior to this IWA can be
detected at high confidence, depending on the particulars of the target.

  * •

Choice of Coronagraphic Mask: The MASK335R coronagraphic mask currently has
the best Target Acquisition (TA) performance (e.g., Girard et al., 2022),
which translates directly into somewhat superior coronagraphic suppression. If
given the choice between the MASK335R and MASK430R, proposers should choose
the former.

  * •

Small Grid Dithers (SGD): The use of SGDs is still highly recommended to
compensate for TA errors, at least when searching for faint sources within
$\sim$2 arcsec.

###  2.2 MIRI Coronagraphy

  * •

Stray-light Artifacts: Data obtained during commissioning and the ERS period
revealed that MIRI coronagraphic images are significantly impacted by
scattered light, producing bright linear features at the boundaries of the
Four Quadrant Phase Mask (4QPM). This scattered stray-light, sometimes
referred to informally as the “glow stick” feature and shown in Figure 2 of
Carter et al. (2022), significantly impacts all MIRI coronagraphic
observations. To mitigate this very serious systematic, dedicated background
observations are required for Cycle 2, with exposure times matching those of
the target, as well as any reference observations. It is also highly
recommended that observers use the automatic extra observation feature in APT
to obtain two backgrounds, one each in opposing quadrants. This enables a
basic dither and mitigates against having a background astrophysical source
fall on the glow stick that could severely hamper the glow-stick correction.

  * •

Target Acquisition: TA is currently only available in Quadrant 1 of each of
the MIRI coronagraphs. The MIRI Instrument Team is commissioning TAs for
Quadrant 4 during Cycle 1 observations. Observers who may need TA in this
quadrant should contact the JWST Help Desk.

  * •

Neutral Density Filters for Target Acquisitions: TA is currently only
available to be used with the Neutral Density Filter (FND). The other three TA
filters (F560W, F1000W and F1500W) are being commissioned during Cycle 1,
starting with the F560W. Observers needing one of these filters for TA should
contact the JWST Help Desk.

  * •

Special Considerations for Cycle 2: Additional information on MIRI
coronagraphic imaging, including some considerations that should be taken into
account for Cycle 2 can be found in the on the JWST User Documentation
(“JDox.”).

###  2.3 Considerations for Both NIRCam and MIRI Point Source Coronagraphy

  * •

Reference Star Vetting: We urge future observers to exercise caution in their
selection of reference stars for coronagraphy using all possible archival
information (e.g., previous AO imaging, Gaia RUWE statistic), as well as
additional ground-based supporting observations, to ensure that reference
stars are indeed single stars. This applies, of course, also to coronagraphy
of circumstellar disks as well discussed in §3. Ground-based observations such
as speckle-imaging, AO Imaging, or Aperture Masking Interferometry (AMI)
observations can be highly effective for eliminating reference stars that
might have contaminating additional objects within $\sim$0.5 - 1.0′′.

  * •

ADI+RDI versus RDI Alone: As discussed in Carter et al. (2022), the combined
ADI+RDI KLIP PSF subtraction did not provide significantly improved contrast
performance over a solely RDI-based subtraction for the ERS datasets. As a
result, users may be able to improve observational efficiency by opting for
observations that facilitate an RDI-based subtraction alone, especially for
targets within $\sim$500 mas to $\sim$1 arcsec. However, at large angular
separations from the star, users may find that using ADI alone will be
adequate to remove any residual scattered starlight.

  * •

Additional Tools: The default ETC offers a means to simulate coronagraphic
performance, but only for simplistic RDI subtractions with no small-grid
dithers. Users should consider investigating additional tools such as PanCAKE,
pyNRC, MIRISim, and NIRCCoS to produce more accurate estimates of the
achievable performance across a wider range of observational strategies.

  * •

Detailed information on a variety of observational considerations and
recommendations can be found in the JWST Documentation. The specific links
might be helpful related to

    * –

NIRCam Coronagraphic Imaging,

    * –

MIRI Coronagraphic Imaging, as well as

    * –

MIRI Cross-Mode Strategies

##  3 Best Practices for Coronagraphy of Circumstellar Disks

The ERS program includes coronagraphic observations with both NIRCam and MIRI
of a benchmark debris disk system, HD 141569A (e.g., Weinberger et al., 2000),
which has been well-studied from both ground-based (e.g., Biller et al., 2015;
Bruzzone et al., 2020; Singh et al., 2021) as well as space-based imaging
(Clampin et al., 2003; Weinberger et al., 1999). This target is comprised of
at least three concentric disk rings, and is therefore also an ideal target to
test the sensitivity of JWST to extended structures as a function of angular
distance from the host star, as well as study the impact of various post-
processing algorithms on extended structures.

While the NIRCam coronagraphic data for this object have yet to be gathered at
the time of this writing, the MIRI data were successfully obtained on 2022
August 02. The processed and combined data show an image in all three filters
dominated by the thermal emission of the inner ring ($r{\sim}$0.4′′). The
emission of the outer rings ($r{\sim}$1.7′′ and $r{\sim}$3.2′′) are hidden by
the complex response of the bright inner disk, which extends out in a diffuse
halo out to ${\sim}6^{\prime\prime}$. This response is difficult to simulate
and subtract off from the images because the ring lies near the inner working
angle of the 4QPM mask and is neither separated from, or fully masked by, the
coronagraph. Thus, the poor knowledge of the spatial distribution of the dust
emission makes it difficult to disentangle the complex instrument response
from possible asymmetries in the disk surface brightness.

Reference Star Vetting: As emphasized in §2.3, as a blanket statement covering
both NIRCam and MIRI disk coronagraphy, we urge future observers to carefully
select reference stars for MIRI coronagraphy taking advantage of all possible
archival information, as well as potential ground-based supporting high-
resolution observations (e.g., speckle, AO, AMI) to ensure that reference
stars are indeed single stars.

###  3.1 NIRCam coronagraphy Best-Practices for Disk Observations

Multiple Systems: Observers should exercise caution in observing targets that
are members of multiple systems, even if the angular distance of any
additional companions initially seems too large to be significant, due to the
risk of unexpected scattered light or diffraction effects from neighboring
stars interfering with the primary science observations. HD 141569A (A2V) has
two M-star companions, closely positioned on the sky, and both at a separation
of ${\sim}8^{\prime\prime}$. Since the HD141569A primary star is much brighter
than these M-dwarf companions, these two objects have not typically posed a
problem for disk observations in previous high contrast imaging with HST/STIS
and ground-based instruments. However, at the longer wavelengths probed by
JWST these objects are much more comparable in brightness. Simulations of our
upcoming NIRCam observations indicate that the diffraction patterns of these
stars at the location of the disk are of comparable (or greater) brightness to
the disk. As a result, we have added offset observations of our reference star
to our program at a position such that we plan to use it to subtract off these
diffraction patterns. This strategy will be tested once the observations are
obtained later in 2023.

###  3.2 MIRI Coronagraphy Best-Practices for Circumstellar Disk Observations

  * •

Bright Dust Emission Close to Central Star: Some systems may exhibit bright
emission of circumstellar dust close to the star (with an angular extent still
inside the inner working angle of the coronagraphs). As was seen in our ERS
program, dust that is resolved (or partially resolved) inside of the 4QPM
inner working angle may deviate sufficiently from a point-like structure,
introducing significant diffraction patterns such that the coronagraphic
suppression may be compromised, and thereby limiting sensitivity. This has so
far proven difficult to forward model and subtract, but is an active area of
research in the ERS team.

  * •

Ground-based Supporting Observations: One potential protection against this
would be to obtain supporting ground-based observations (e.g., VISIR at VLT)
to test for such resolved emission from inner dust structures.

  * •

JWST Direct Images: If there is clear evidence (perhaps from supporting
ground-based observations) for a bright, spatially-extended inner disk, then
direct (i.e. non-coronagraphic) images using MIRI may prove advantageous to
help forward-model the disk signal in coronagraphic modes. Indeed, with
appropriate forward modeling a good inner disk model may enable subtraction of
the extra diffraction introduced. Our team is currently exploring this.

  * •

A Warning About Saturation: Bright targets may saturate normal direct imaging
mode MIRI filters, which may complicate the direct imaging approach discussed
above. One potential option is to use coronagraphic TA images with the FND
filter (not available to direct imaging observations). However the wavelength
range is extremely broad and the data requires significant forward modeling to
interpret.

##  4 NIRSpec & MIRI MRS Spectroscopy of Point Sources

Below we highlight some aspects of data collection, as well as a few for data
processing, that have arisen as part of this ERS program. More details can be
found in Miles et al. (2022), and the fully processed spectrum of the wide-
separation brown dwarf VHS 1256b, a relatively bright point source, can be
delivered upon request.

###  4.1 Best Practices for Point Source Spectroscopy Data Collection using
the NIRSpec IFS

The ERS observations of VHS 1256b using the NIRSpec IFS are photon noise
limited. Currently, any limiting factor for the NIRSpec IFS data quality for
this source is the data processing carried out by the JWST pipeline. This is
expected to be improved by further Cycle 1 calibration observations and
improvements to the IFS cube building algorithm.

####  4.1.1 NIRSpec IFS Best Practices for Data Collection

  * •

NIRSpec Dithering: The option employing the four-point nod was used to ideally
improve spatial and spectral sampling which is consistent with the
recommendations contained within the NIRSpec Dithering decision flowchart.

####  4.1.2 NIRSpec IFS Post-Processing Considerations for Point Source
Spectroscopy

  * •

Hot pixels and other sources: Currently the data quality flags of the JWST
Pipeline do not adequately flag and remove hot pixels from detector images.
These extremely high values are propagated through the image cubes where
spectra are extracted. In the work presented in Miles et al. (2022) this was
addressed by reducing dithers individually and taking a weighted mean to
reduce the likelihood of hot pixels influencing the final spectra.

  * •

Flux Calibration and Spectral Response: The flux calibration and spectral
response correction produced by the default JWST pipeline may not give
reliable results even for standard stars. When flux calibrating and stitching
together observation modes of your target use a correction factor based on
standard targets (e.g., A-stars) with a known spectral response.

  * •

Oscillations within Spectra \- An oscillating structure is present in spectra
extracted from spectral cubes. Some of these oscillations result from the cube
building algorithm itself or potentially arise from the undersampled PSF of
NIRSpec. These features were not addressed in Miles et al. (2022) but need
more work to understand why they are created in the Stage 3 cube building step
of the pipeline.

  * •

Spectral Extraction: \- The extraction step of the JWST pipeline for NIRSpec
does not let the user define extraction widths and no aperture correction
exists for NIRSpec at the moment.

###  4.2 MIRI MRS Spectroscopy of Point Sources

The ERS MIRI observations for VHS 1256b are background limited compared to the
NIRSpec observations. A significant source of noise in the observation of
faint, isolated point sources is prominent detector noise (vertical striping
on the detector), as well as cosmic ray showers. Mitigations steps for these
are described below:

  * •

Background observations: Background observations were not used directly in the
processing carried out by the ERS team, since the contamination of cosmic ray
showers introduced artefacts in the science images. The background was instead
estimated with a reference aperture off source in the reconstructed data
cubes. An annulus background estimation would also have been appropriate.

  * •

Vertical Detector Striping: The vertical detector stripes are in the same
direction of the dispersion, therefore very difficult to disentangle from the
spectrum. The background observations can be used to estimate this artifact
and remove it from the science frames (this is relevant mainly for the
IFUSHORT detector). Background observations can also detect remaining hot/warm
pixels that are not flagged by the pipeline.

  * •

Cosmic Rays: Cosmic ray showers appear as diffuse structures at similar levels
as the point source. The pipeline is not able to flag these during the ramp
fitting step. Therefore the best solution at the moment is to rely on the
information of the different dithers (4 being ideal) to average out the effect
of cosmic rays.

Thus, our summary recommendation for MIRI MRS observations of point sources is
to 1) obtain background observations, assuming time allows for this; 2) hope
for the absence of high-intensity cosmic ray showers; and 3) use the four-
point dither pattern. All other calibration steps are handled very well by the
STScI pipeline. For any remaining features in the spectra, residual fringes
may be the cause.

##  5 Best Practices for Aperture Masking Interferometry with NIRISS

Observations of the exoplanet host star HIP65426 using Aperture Masking
Interferometry (e.g., Tuthill et al., 2000; Ireland, 2013; Sallum & Skemer,
2019) with the NIRISS instrument were carried out within this program, as
described in Hinkley et al. (2022). Observations were obtained on 2022 July
30. Below we highlight some findings and recommendations from our ERS program,
but much more information on NIRISS AMI and commissioning can be found in
Sivaramakrishnan et al. (2022).

  * •

Achievable Contrast (Compared to Coronagraphy): NIRISS operating in the AMI
mode outperforms NIRCam coronagraphy within $\sim$300-500 mas, with a contrast
curve that is essentially flat between this region and $\sim$100 mas.

  * •

Achievable Contrast (Compared to Theoretical Photon Noise Limits): Initial
analysis of both ERS and commissioning data has shown that the achieved
contrast with AMI is at least one magnitude poorer than the theoretical
prediction based on photon noise that can be found in the AMI Recommended
Strategies in JDox.

  * •

Reference PSF Brightnesses and Detector Settings: Observations of the target
and any reference calibration stars should have a similar number of groups and
integrations so that the photon noise limit is reached in the analysis of the
target. In other words, reference and science should have equal brightness in
the observed filter, with the goal of achieving a similar level of systematic
effects such as charge migration.

  * •

Reference PSF Spectral Types: Initial analysis of ERS observations of
HIP65426, as well as commissioning observations (Kammerer et al., 2022b;
Sivaramakrishnan et al., 2022) indicate that matching the number of groups and
integrations has a greater impact on the overall calibrations, than matching
the spectral types between target and calibrator.

  * •

Reference PSF Vetting: In general it is good practice to select well-vetted
reference PSF calibrators when possible, to avoid potential contamination by
binary calibrators. When this is not possible, selecting multiple calibrator
stars can help to mitigate this risk.

  * •

Optimizing Target Placement: Target acquisition during AMI commissioning
showed highly repeatable filter-dependent systematic placement errors of up to
a third of a pixel. Improved centering within a pixel may well reduce AMI
sensitivity to flat-fielding error, jitter, and other effects. An offset for
an observation should be specified (as a Special Requirement) in AMI’s APT
template to place targets close to the pixel center. Given the observation’s
filters, offsets can be chosen after an inspection of Figure 5 in
Sivaramakrishnan et al. (2022).

  * •

Frame Rejection: Commissioning and ERS data analysis often excluded 10% of
integrations (stringent data cleaning rejects 30%). A tool to identify and
ignore or repair affected parts of the 2-dimensional images in cal_ints data
cubes is under development.

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