11institutetext: Anton Pannekoek Institute for Astronomy, University of
Amsterdam, 1090 GE Amsterdam, Netherlands 22institutetext: Center for
Astrophysics, Harvard & Smithsonian 60 Garden Street, MS-16, Cambridge, MA
02138, USA

# The effects of planetary day-night temperature gradients on He 1083 nm
transit spectra

F. Nail 11 A. Oklopčić 11 M. MacLeod 22

(Received 2023 August 11; resubmitted 2023 December 7.)

A notable fraction of helium observations at 1083 nm, probing evaporating
atmospheres of short-period gas giants, exhibit a blueshift during transit,
which might be indicative of a day-to-night side flow. In this study, we
explore the gas dynamic effects of day-to-night temperature contrasts on the
escaping atmosphere of a tidally locked planet. Using a combination of 3D
hydrodynamic simulations and radiative transfer post-processing, we model
transmission spectra of the metastable helium triplet. The key findings are:
(1) Increasing day-night anisotropy leads to a narrowing of the helium line
and an increase in the blueshift of the line centroid of a few km s-1. (2) The
velocity shift of the line depends on the line forming altitude, with higher
planetary mass-loss rates causing the line to form at higher altitudes,
resulting in a more pronounced velocity shift. (3) A critical point of day-
night anisotropy exists at which the blueshift saturates due to turbulent
flows generated by outflow material falling back onto the planet’s night side.
(4) A strong stellar wind and the presence of turbulent flows may induce time
variations in the velocity shift. Assuming that the day-night temperature
gradient is the main cause of the observed blueshifts in the He-1083 nm
triplet, the correlation between the velocity shift and day-night anisotropy
offers the opportunity to constrain the temperature gradient of the line
forming region.

###### Key Words.:

Planets and satellites: atmospheres – hydrodynamics – radiative transfer

##  1 Introduction

The escape of atmospheres is thought to play a key role in shaping the
demographics of exoplanets. Significant loss of gas from planetary atmospheres
has been proposed as a potential explanation for two intriguing features
observed in the exoplanetary radius-orbital distance diagram: the so-called
“radius valley” and the “hot Neptune desert”. The radius valley refers to the
bimodal distribution of small planet sizes, suggesting a bifurcation between
planets that are stripped of and retain their gaseous atmospheres (Lopez &
Fortney 2013; Owen & Wu 2013, 2017; Kurokawa & Nakamoto 2014; Fulton et al.
2017; Van Eylen et al. 2018; Allan & Vidotto 2019; Hallatt & Lee 2022). The
hot Neptune desert denotes the lack of Neptune-sized planets in proximity to
their host stars, where stellar irradiation levels are high (Szabó & Kiss
2011; Lundkvist et al. 2016; Mazeh et al. 2016). By studying a diverse range
of exoplanetary systems, we can gain deeper insights into the process of
atmospheric escape and its role in shaping the evolution of exoplanets.

The absorption of helium at 1083 nm (e.g. Nortmann et al. 2018; Salz et al.
2018; Kirk et al. 2022), the hydrogen Lyman-$\alpha$ line (e.g. Vidal-Madjar
et al. 2003; Ehrenreich et al. 2015), and various UV lines of metal species
(e.g. Sing et al. 2019; Cubillos et al. 2020; Yan et al. 2021; Sreejith et al.
2023) observed during planetary transits serve as evidence of atmospheric
escape. These spectral features probe the low-density, outflowing upper
atmosphere of exoplanets. Among them, the helium feature at 1083 nm has
emerged as an excellent tracer due to its favorable properties, including
being less affected by the interstellar medium and being accessible with
ground-based instruments (e.g. Oklopčić & Hirata 2018).

For planets that are relatively close to their host stars, tides might act to
tidally-lock their rotation to their orbital motion. In this case, a single
side of the planet would face the star at all times. Over time, energy input
from the stellar continuum or the extreme ultraviolet (XUV) radiation of the
host star could shape a contrast between the day and night side temperatures
of the planet. The value of the temperature likely depends on whether the
atmospheric loss is powered by photoevaporation due to XUV irradiation (Lammer
et al. 2003; Murray-Clay et al. 2009), in which case the day-side temperature
might reach $\sim 10^{4}$ K for typical parameters, or by the gravitational
energy of the planetary core (Ginzburg et al. 2018; Gupta & Schlichting 2019),
in which case a lower temperature, more similar to the planet’s equilibrium
temperature given the stellar bolometric luminosity would be predicted (e.g.
Bean et al. 2021). Regardless of the mechanism, a temperature (and thus
pressure) gradient from day to night side of a planetary atmosphere would
drive hydrodynamic rearrangement of those layers in potentially observable
flows.

Indeed, some observations of the helium 1083 nm line have shown evidence for a
net blueshift of the line at mid-transit, which would correspond to a bulk
flow away from the star toward the observer (Nortmann et al. 2018; Salz et al.
2018; Alonso-Floriano et al. 2019; Allart et al. 2018; Kirk et al. 2020; Palle
et al. 2020; Zhang et al. 2023a). The goal of this work is to study the gas
dynamical effects of a day-to-night temperature contrast on flow in the
vicinity of a planet with an escaping atmosphere. We then model how various
atmospheric properties lead to observable differences in transit spectra by
generating synthetic helium 1083 nm spectra.

Several approaches have been employed so far to model planetary outflows, and
to predict and interpret their signatures in the helium line (e.g. Allart et
al. 2018; Shaikhislamov et al. 2020; Wang & Dai 2021a, b; Khodachenko et al.
2021; MacLeod & Oklopčić 2022; Schreyer et al. 2023). Among these, the
hydrodynamics of anisotropic escaping atmospheres has been previously studied
by Stone & Proga (2009); Villarreal D’Angelo et al. (2014); Carroll-Nellenback
et al. (2017); Wang & Dai (2021a, b). This work compliments these past studies
by adopting these parameterized 3D hydrodynamic approach of MacLeod & Oklopčić
(2022).

The paper is structured as follows. In Sect 2, we describe our numerical
models to simulate the anisotropic planetary wind. In Sect 3, we present the
results of the hydrodynamic simulations and their radiative transfer analysis.
In Sect 4, we discuss the implications of our results for the interpretation
of helium observations of short-period gas giants regarding their day-night
anisotropy and possible observing strategies. We conclude with a summary in
Sect 5.

##  2 Methods and models

We investigate the interaction between the thermal winds of a planet and its
host star, while varying the degree of day-night anisotropy of the planet. To
achieve this, we use the 3D Eulerian (magneto)hydrodynamic code
Athena++111https://github.com/PrincetonUniversity/athena, version 2021 (Stone
et al. 2020), which is a derivation of the Athena code (Stone et al. 2008).
Our objective is to obtain observational signatures of different wind
configurations in the helium triplet at 1083 nm, by calculating the radiative
transfer of the stellar light passing through the planetary outflow in a
transiting configuration.

We parameterize the conditions at the base of the planetary wind, as opposed
to self-consistently simulating atmospheric heating and wind launching. We
examine observational signatures arising from different geometries of the
planetary outflow. This approach allows us to remain agnostic to the actual
wind-driving mechanisms, while encompassing a broad range of wind properties
and focusing on their observable signatures.

This study is a continuation of the work by MacLeod & Oklopčić (2022);
however, instead of assuming an initially-isotropic planetary wind, we
consider the impact of a day-night anisotropy on the planet (or, more
precisely, at the base of the wind). Our focus is on identifying observational
features in the helium line that are indicative of an anisotropic day-night
side structure. Since we only alter the planetary wind and utilize the same
simulation setup and radiative transfer code presented in their work, in the
following section, we will outline the main simulation features and discuss
the changes made for this study. For a detailed description of the simulation,
we refer the readers to Section 2 of MacLeod & Oklopčić (2022).

###  2.1 Simulation

We run hydrodynamic simulations in which the equations for mass, momentum, and
energy conservation of an inviscid gas are solved. The simulations use a
spherical polar mesh with the host star at the center of the frame of
reference that co-rotates with the orbital motion of the planet. The
computational domain spans from the star’s surface to a distance of 0.3 au in
all directions. The base mesh of the simulation is made up of $9\times 6\times
12$ meshblocks, and each block consists of $16^{3}$ zones. These zones are
spaced logarithmically in the $r$ direction and evenly in the $\theta$ and
$\varphi$ directions to maintain nearly cubic zone shapes throughout the
volume. To avoid zones with extreme aspect ratios, the number of effective
zones in the $\varphi$ direction is reduced near the poles. Instead of
restructuring the mesh itself, this is accomplished by averaging conserved
quantities across these zones. An additional five levels of static mesh
refinement $N_{\mathrm{SMR}}=5$ are used around the planet, within a box
covering a range of $\pm 12$ planetary radii.

The planet in the simulation is modeled after the system WASP-107b (Anderson
et al. 2017; Spake et al. 2018; Piaulet et al. 2021), with its mass and radius
specified in Table 1. The planet is located at the negative x-axis at a
distance of $a$ (the semi-major axis) from the star, and both the planet and
star are assumed to be rotating with the orbital frequency of the planet
$\Omega=\Omega_{orb}=G(M_{p}+M_{*})a^{-3}$. The angular momenta of the planet
and star are both in the positive z-direction.

We employ the ideal gas equation of state with the gas adiabatic index
$\gamma=1.0001$. The behavior along adiabats is nearly isothermal, but gas in
the domain includes the planetary and stellar winds with widely different
temperatures. We adopt an isothermal equation of state in order to focus on
outflow dynamics rather than its thermodynamics. In particular, the details of
radiative heating and cooling are therefore ignored by adopting a single
temperature as a crude representation of the balance of these processes. A
more detailed approach could self-consistently treat this physics, but at the
expense of the simplified, parameterized approach that lets us change model
properties and observe the effects on the system.

We take into account the gravitational influence of the two bodies, while the
backreaction of the outflow on the planetary orbit and the effect of radiation
pressure at 1083 nm on the gas dynamics are neglected. Each simulation is run
for $3\times 10^{6}$ s, equivalent to slightly over 6 orbits of the planet
around the star.

The density and pressure of the stellar surface are constant and are related
by the hydrodynamic escape parameter $\lambda_{*}$, which denotes the ratio
between the gravitational potential $U(R_{*})$ and thermal energy
$k_{B}T(R_{*})$. Thus, the pressure on the stellar surface is determined by
the density of the stellar surface, $\rho_{*}$, and $\lambda_{*}$. All our
models use a value of $\lambda_{*}=15$. Similarly, the pressure on the
planetary surface is defined as

$P_{p}=\rho_{p}\frac{GM_{p}}{\gamma\lambda_{p}R_{p}}\leavevmode\nobreak\ ,$
(1)

where $\lambda_{p}=\frac{GM_{p}}{c_{s}^{2}R_{p}}$ denotes the hydrodynamic
escape parameter for the planet. The unitless parameter reflects the thermal
potential of the planet and thus, effectively determining the local gas
temperature

$\centering T=\frac{1}{\lambda_{p}}\frac{\mu
m_{H}}{k_{B}}\frac{GM_{p}}{R_{p}}.\@add@centering$ (2)

###  2.2 Day-night anisotropy

In order to generate an anisotropic planetary outflow, we introduce a pressure
scaling factor $f_{\rm pres}$ that scales the pressure across the planetary
surface. The pressure is scaled to reach its maximum value at the substellar
point and then decreases along the surface of the planet towards the
antistellar point, where it reaches the predefined minimum value
$\tilde{f}_{\rm pres}$. To achieve this pressure distribution, we calculate
$f_{\rm pres}$ as a function of the distance to the substellar point along the
planet’s surface

$\centering f_{\rm pres}=\frac{1-\tilde{f}_{\rm
pres}}{2}\left[\sin(\theta_{p})\cdot\cos(\varphi_{s}-\varphi_{p})+1\right]+\tilde{f}_{\rm
pres}\leavevmode\nobreak\ ,\@add@centering$ (3)

where $\varphi_{p}\in[0,2\pi]$ and $\theta_{p}\in[0,\pi]$ are the spherical
coordinates centered around the planet, and the substellar point is defined as
$S(\varphi_{p}=0,\theta_{p}=\pi/2)$.

We vary the degree of anisotropy by adjusting $\tilde{f}_{\rm pres}$, with the
maximum degree of anisotropy being achieved when the pressure at the
antistellar point is only 10% of that at the substellar point. Figure 1
displays the pressure scaling factor $f_{\rm pres}$ over the planet’s surface
and interior for this 10% model. In addition, we modify the pressure inside
the planet to initiate the wind while avoiding discontinuities. For the
interior, where $r\leq 0.3\leavevmode\nobreak\ R_{p}$, $f_{\rm pres}$ remains
constant at a value of 1. For $0.3\leavevmode\nobreak\ R_{p}<r\leq
1\leavevmode\nobreak\ R_{p}$, the pressure decreases linearly towards the
planetary surface to reach the pre-defined value of $f_{\rm pres}$.

Figure 1: Maps of the pressure scaling factor of the planetary boundary
condition. The upper panel shows the pressure scaling factor, $f_{\rm pres}$
along the planet surface, in the case where the value at the antistellar point
$(\varphi_{p}=180^{\circ},\theta_{p}=90^{\circ})$ is 10% of the value at the
substellar point $(\varphi_{p}=0,\theta_{p}=90^{\circ})$. The lower panel
shows a slice through the planet, in which the observer is located in the $-x$
and the star in the $+x$-direction. For models with a lower degree of
anisotropy, a similar distribution is used, only with a different minimum
value on the night side $\tilde{f}_{\rm pres}$.

###  2.3 Wind variations

To gain insight into the impact of each parameter and identify potential
degeneracies, we analyze a series of models. To start, we investigate a
fiducial model, denoted as model $A$, which serves to illuminate the
fundamental mechanisms of the anisotropic evaporative atmosphere. In this
model, we use the system parameters of WASP-107b, an intermediate hydrodynamic
escape parameter of $\lambda_{p}=2$, and a relatively weak stellar wind, with
the stellar mass-loss rate only twice that of the planet. We vary only the
degree of anisotropy, as indicated by the index of each model between the
limits of $\tilde{f}_{\rm pres}$$=100\%$ (isotropic), and $\tilde{f}_{\rm
pres}$$=10\%$ (anisotropic).

We then consider the effects of other model parameters on our results. We
examine how a modified stellar wind affects the anisotropic outflow of the
planet. This analysis is inspired by that of MacLeod & Oklopčić (2022), where
the impact of intermediate (their model $B$) and strong (their model $C$)
stellar winds on planetary atmospheric escape was studied. We label our models
$B$ and $C$ with modified stellar mass-loss rates accordingly. Furthermore, we
consider variations in the orbital distance of the planet, denoting the
corresponding models with $D$. We also investigate the effects of different
hydrodynamic escape parameters, modifying the value of $\lambda_{p}$ and
labeling the corresponding models as $L$. The relevant input parameters are
summarized in Table 1. It is important to highlight that the values for
$\lambda_{p}$ represent the maximum values on the day side of the planet.
However, when a day-night anisotropy is present, these maximum values
decrease, along with the local gas temperature, as we move towards the night
side along the planetary surface.

Table 1: Model overview of the input parameters.

model group | $\lambda_{p}$ |  $a$ [au] |  $\rho_{*}$ [g cm-3] |  $\rho_{p}$ [g cm-3]
---|---|---|---|---
$A$ | 2 | 0.05 | $4.16\times 10^{-15}$ | $1.53\times 10^{-16}$
$B$ | 2 | 0.05 | $4.16\times 10^{-14}$ | $1.53\times 10^{-16}$
$C$ | 2 | 0.05 | $4.16\times 10^{-13}$ | $5.21\times 10^{-16}$
$D$ | 2 | 0.03 | $4.16\times 10^{-15}$ | $1.53\times 10^{-16}$
$L1.5$ | 1.5 | 0.05 | $4.16\times 10^{-15}$ | $1.43\times 10^{-16}$
$L3$ | 3 | 0.05 | $4.16\times 10^{-15}$ | $2.27\times 10^{-16}$

Notes. We present four groups of simulation models, categorized by the
strength of the stellar wind and the planetary parameters. The system
parameters are identical for all simulations, following those of WASP-107b,
$M_{*}=0.68\leavevmode\nobreak\ M_{\astrosun}=1.36\times 10^{33}$ g,
$R_{*}=0.67\leavevmode\nobreak\ R_{\astrosun}=4.67\times 10^{10}$ cm,
$M_{p}=0.096\leavevmode\nobreak\ M_{\jupiter}=1.82\times 10^{29}$ g (Piaulet
et al. 2021); $R_{p}=0.94\leavevmode\nobreak\ R_{\jupiter}=6.71\times 10^{9}$
cm (Anderson et al. 2017).

###  2.4 Radiative transfer and synthetic spectra

For the radiative transfer analysis, we follow the methodology of MacLeod &
Oklopčić (2022). We make the assumption that the gas composition throughout
the computational domain is solar, with mass fractions of hydrogen, helium,
and metals being X = 0.738, Y = 0.248, Z = 0.014, respectively. Moreover, we
consider that detailed equilibrium is reached in each cell of the simulation
grid at a steady state. The unattenuated photoionization rate $\Phi$ is
derived from the stellar flux and, in combination with the optical depth
$\tau$, provides the photoionization rate $\Phi e^{-\tau}$. The optical depth
to hydrogen-ionizing photons is calculated iteratively for each cell in the
simulation grid. The assumed stellar spectral energy distribution used in
calculating the photoionization rates is that of a late F or early G star
after Zhang et al. (2023b).

To generate synthetic helium spectra, we calculate the integrated optical
depth as a function of wavelength along the rays that propagate from the
stellar surface to the distant observer. We impose a Voigt line profile, with
a Gaussian component dependent on the local gas temperature, as given by Eq 2.
We do not consider rays that cross the planet interior. Additionally, the
contribution of the rays is weighted according to the stellar quadratic limb
darkening law.

##  3 Results

###  3.1 Fiducial model group

Figure 2: The gas density in the orbital midplane, showing the hydrodynamic
winds generated by a model exoplanet interacting with the stellar wind
emanating from the central star. We assume that the observer is located in the
direction -x. The white square marks the region around the planet shown in the
following figures. The top figure illustrates a planetary wind with isotropic
boundary conditions ($A_{100\%}$), while the bottom figure incorporates day-
night anisotropy ($A_{10\%}$). In the anisotropic scenario, the decreased
pressure on the night side leads to a reduction in the mass-loss rate of the
planet. As a result, the outflow becomes less extended compared to the
isotropic case.

We start our analysis by investigating the fiducial model group, $A$. Figure 2
shows a slice of gas density through the orbital midplane of the simulations
$A_{100\%}$ and $A_{10\%}$, which correspond to $\tilde{f}_{\rm pres}$=1 and
$\tilde{f}_{\rm pres}$=0.1, respectively. If the planet is transiting in front
of the star, the observer is positioned in the negative $x$-direction. The
snapshots shown in Fig. 2 capture the simulations in approximately steady
state, after roughly 6 orbits.

At large scales, the flow in these two simulations is broadly similar. Because
the planets are exposed to a relatively weak stellar wind (see Table 2 and
Appendix B), the planetary outflow forms a toroidal structure that surrounds
the star (e.g., as described by MacLeod & Oklopčić 2022). In the presence of a
day-night temperature gradient, the pressure on the night side decreases,
leading to a reduction in the planetary mass-loss rate. As a result, the
outflow becomes less extended compared the isotropic scenario.

One difference with varying anisotropy factor, $\tilde{f}_{\rm pres}$, is the
overall planetary mass-loss rate. With lower $\tilde{f}_{\rm pres}$, the
overall energy of the planetary wind is lower, and the degree of mass loss is
also reduced for a given density at the wind-launching height in the
atmosphere. We plot and tabulate the model mass-loss rates given our parameter
settings in Appendix B, in Table 2 and Fig. 11.This difference is reflected in
a slight decrease in the size of the cavity surrounding the planet with
increasing degree of anisotropy.

Our subsequent investigation focuses on the immediate vicinity of the planet,
indicated by a white square. This immediate planetary vicinity is primarily
responsible for the line-forming region in helium 1083 nm transit spectroscopy
and bears the primary signatures of any asymmetry at the planetary outflow-
launching scale.

####  3.1.1 Anisotropy

Figure 3: Temperature _(top)_ , number density of metastable helium _(middle)_
, and line-of-sight velocity _(bottom)_ in the vicinity of the planet for
increasing day-night side anisotropy from left to right. The contours in the
top row are indicating the sonic surface, while the middle row displays
contours of the number density of metastable helium. These contours range from
light to dark and correspond to values of $n_{\rm
Hef_{3}}=1,10^{-1},10^{-2},10^{-3}$, and $10^{-4}$, respectively. Velocity
quivers are added in the bottom panels. The observer is located in the $-x$
direction and the star in the $+x$-direction. The plot displays the equatorial
slice from a top-down perspective. By introducing anisotropy, the gas density
increases toward the second quadrant (i.e., upper left) and the line-of-sight
velocity becomes asymmetric around $x_{p}=0$.

Figure 3 illustrates the impact of day-night side anisotropy within the
unshocked region in the immediate vicinity of the planet. Starting from the
left side of the figure, the model $A_{100\%}$ has an isotropic boundary
condition, where the local gas temperature (top panel) remains constant
throughout the atmosphere. In this isotropic state, the outflow reaches the
approximately spherical sonic surface at a radius $\sim
1.5\leavevmode\nobreak\ R_{p}$. The number density of metastable helium
(middle panel) uniformly decreases radially, adhering to a Parker (spherical,
isothermal) wind profile. As a result of the nearly-radial flow, gas at
$x_{p}>0$ generally flows toward the star (is redshifted for a mid-transit
observer), while gas at $x_{p}<0$ generally flows away from the star (is blue-
shifted). Due to the Coriolis force, the outflow undergoes a slight clockwise
rotation, as indicated by the velocity quivers.

When introducing anisotropy, we lower the pressure on the night side compared
to the day side, thus reducing the gas’ radial acceleration. The pressure
gradient arising from this anisotropic pressure distribution acts as the
driving force, causing the gas to converge toward the lower pressure region on
the night side of the planet. On the night side, where the gas has
accumulated, lowered pressure therefore leads to enhanced density.
Additionally, as a result of the planet’s orbital motion, the gas experiences
a rotational component and tends to accumulate between the anti-stellar point
and the evening terminator of the planet (i.e., upper left quadrant in Fig.
3). Applying the same principle, the sonic surface expands in this direction,
becoming nonspherical.

The decreased thermal energy of the night side in the anisotropic scenario
causes the line-of-sight velocity to trend toward blueshift along a ray at
$x_{p}=0$. Material flowing toward the star (redshifted) is increasingly
confined to a conical region in the $+x_{p}$-direction. This deviation from
radial outflow reflects a net flux of gas from the day side to the night side
of the planet, even as the flow remains largely outflowing.

In 2D simulations of anisotropically-launched planetary outflows, Stone &
Proga (2009) also found a distorted sonic surface. They additionally identify
a shock at $\theta_{p}=3\pi/4$, formed by the compression of the wind as it
moves from the day side to the night side of the planet. In our three-
dimensional models, the additional spatial degree of freedom appears to
suppress this discontinuity to some extent (though see the discussion of
3.1.2). We observe a continuous, but distorted sonic surface in our
asymmetric-heating models. Similarly, in three-dimensional hydrodynamic
simulations by Carroll-Nellenback et al. (2017); Debrecht et al. (2019)
outflows at the planet scale are distorted by net flux from day to night side,
but do not show the broad night-side shock cone observed in Stone & Proga
(2009)’s 2D result.

####  3.1.2 Fallback

Figure 4: _Top:_ Line-of-sight velocity map in the close vicinity of the
planet. The observer is located in the $-x$ and the star in the
$+x$-direction. Overplotted streamlines illustrate the flow patterns and show
the fallback of material on the planet’s night side. The white outline
indicates the radius ($r=1.2\leavevmode\nobreak\ R_{p}$) of the sphere whose
projected surface is shown in the maps below. Due to the incident material,
the pressure on the night side in model $A_{10\%}$ reaches only 40% instead of
the expected 10% of the pressure on the day side in the evolved simulation.

In cases where the pressure on the night side becomes very low compared to the
day side, the outflowing gas on the night side stagnates and may even fall
back onto the planet, as illustrated in the top right panel of Fig. 4. This
fallback represents a saturation of the degree of day–night asymmetry
supported by our models. For example, in the model $A_{10\%}$ in Fig. 4,
higher pressure material from nearer to the day side of the planet converges
on the night-side low pressure region. The occurrence of falling back material
on the night side effectively reduces the degree of anisotropy between the day
and night side.

This saturation of a minimum night-side pressure is visualized most clearly in
the pressure maps of Fig. 4. To create these surfaces, we evaluate the
pressure of zones near a radius of 1.2 planetary radii, as shown by the white
line in the upper panels of Fig. 4. In the $A_{100\%}$ model, the pressure is
uniform, regardless of direction. In the $A_{50\%}$ and $A_{10\%}$ models, the
pressure is highest in the substellar (day-side) direction, which matches
conditions at the boundary condition. But the antistellar (night-side)
pressure is not as low as that of the boundary. In the initial condition of
model $A_{50\%}$, the pressure on the night side is set to be 50% of the
pressure on the day side. However, in the evolved steady state of the
simulation, the night-side pressure increases to $\sim$80% of the day side
value. While in model $A_{10\%}$, the visible flow convergence yields an
antistellar pressure of approximately $40\%$ of the substellar pressure. In
each case, the net flux of material away from the day side of the planet is
acting to reduce the degree of the asymmetry that has been imposed by our
planetary boundary condition.

We observe a relationship between the degree of day-night anisotropy and the
opening angle of the region where the fallback occurs. We measure this angle
as angular separation between points on the planet’s equator where material
becomes trapped and cannot escape. As the degree of day-night anisotropy
increases, this fallback angular spread expands correspondingly. The
occurrence of incident material is limited to a region of about $r\leq
2\leavevmode\nobreak\ R_{p}$. When zoomed out of the snapshot, see for example
Fig. 14, the planetary outflow appears nearly radial, due to the partial
equilibration of the temperature at smaller scales.

This finding suggests that extreme temperature gradients in exoplanetary upper
atmospheres are subject to some degree of rearrangement as the material
outflows from the planet. Higher degrees of anisotropy (for example at a lower
atmospheric level) appear to partially equilibrate as material flows away from
the planet and there is a net flux from the planet’s day side toward the night
side.

We caution, however, that the exact flow morphologies we observe are not
generic, but instead are likely reflections of the boundary condition we
assign in our simulations (e.g. as shown in Fig. 1). For example, in the more
self-consistent photoionization heating models of Tripathi et al. (2015),
outflow cools day-side material, while fallback and flow convergence on the
night side heat the flow in that region. These effects are not captured by our
prescriptive boundary condition and simplified equation of state. For a
comparison to our Fig. 4, see Figures 3, 4, and 6 of Tripathi et al. (2015).
By comparison, Wang & Dai (2021a, b) model the exoplanets WASP-69b and
WASP-107b, respectively, and observe markedly cooler night-side material along
with signatures of flow convergence and fallback in their models (see Figure 1
of each paper).

The comparison of these results suggests that the exact flow morphologies,
resulting temperatures, and degree of fallback to the night side are all
dependent on the detailed thermodynamics and perhaps the particular planet in
question. However, a general conclusion is that when the night side
temperature (or pressure) becomes very low compared to the day side, there
will be flow from the day side and fallback to the night side.

####  3.1.3 Synthetic Spectra

Figure 5: Excess absorption of metastable helium in the stellar frame. The
degree of day-night anisotropy is increasing from left to right. The
horizontal lines indicate the extent of the optical transit, while the tilted
lines show the Doppler-shifted wavelengths of the helium triplet in the
planetary rest frame. The vertical brown lines correspond to the stellar rest
frame. The red cross indicates the wavelength of the mid-transit maximum
absorption; the deviation from the vacuum wavelength is converted to a shift
velocity. With an increasing day-night anisotropy, the line profiles become
blue-shifted and narrower. There is only minor excess absorption before and
after the ingress and egress phases.

Figure 5 presents a time series of the metastable helium excess absorption in
the stellar rest frame. The synthetic spectra result from the radiative
transfer analysis of simulation snapshots. To illustrate the transit time
evolution, we post-process the snapshot for various observing angles. The
optical transit, which is observed in the broadband spectrum, occurs within a
phase range of $-0.01$ to $0.01$, equivalent to approximately 2.37 hours in
total duration.

Nonetheless, Fig. 5 encompasses phases preceding and following the optical
transit, serving the purpose of investigating the impact of extended planetary
wind material along the entire orbital trajectory. Although the outflow
material occupies significant portions of the orbit, as shown in Fig. 2, we
observe only minor excess absorption pre-ingress and post-egress. However,
this scenario changes for a higher density scaling (see Appendix A), where the
extended outflow leads to a notable pre- and post-transit excess absorption
(i.e., Fig. 13).

Within the helium triplet, we can differentiate between its blue and red
components, with the latter consisting of two closely spaced but unresolved
lines. In observational studies, it is common to solely measure the stronger
(red) component, which is why we focus on analyzing this component. In the
isotropic case, the helium line is relatively broad, causing the blue and red
components’ wings to partially overlap. As the degree of day-night anisotropy
increases, the spectral lines narrow, while their centroids shift towards
shorter wavelengths. This can also be seen in the mid-transit spectra in the
bottom panel of Fig. 6.

####  3.1.4 Blueshift

Figure 6: _Top:_ Zoomed in on the planet, a 2D slice of the orbital midplane,
with cells indicated by optical depths greater than $10^{-3}$, $10^{-2}$, and
$10^{-1}$ (from dark to light). The observer is located in the $-x$ and the
star in the $+x$-direction. _Bottom:_ The red component of the metastable
helium triplet, calculated only from cells that reach a certain threshold in
optical depth. These spectra were computed using the full 3D snapshot. The
velocity axis is centered around the peak of the red component in the stellar
rest frame. This red component comprises two unresolved spectral lines,
indicated here by vertical brown lines. Most of the blueshift is contributed
by cells with $10^{-3}<\Delta\tau<10^{-2}$, which corresponds to a line
forming region of $r\approx 1-2\leavevmode\nobreak\ R_{p}$.

As the night-side pressure is reduced, the helium line progressively becomes
blue-shifted, reaching a value of $\Delta v=-1.7$ km s-1 for model $A_{10\%}$.
Qualitatively, this blueshift comes from the net flow of material from the
planetary day side to the night side. However, in order to investigate where
in the outflow the blueshift is produced, we select individual cells close to
the planet with high density in which most of the helium line is formed. To
calculate the mid-transit spectra, we progressively incorporate cells with
increasing optical depth $\Delta\tau$, as illustrated in the upper panels of
Fig. 6. The resulting spectra are presented in the lower panels.

In the isotropic boundary condition, velocities are symmetrically arranged
around the $x_{p}=0$ axis. Even so, the helium line is slightly red-shifted by
$\Delta v=+0.3$ km s-1 in the isotropic case $A_{100\%}$. We find that this is
due to the finite spatial resolution of the grid and the imperfect alignment
of the cells with the spherical outflow. Imperfect cancelation of positive and
negative line-of-sight velocities means that a slight redshift of the
resulting helium line emerges. In Appendix C, we explore how line velocity
shift changes with spatial resolution during the post-processing step. Fig. 12
shows that the velocity shift for the isotropic boundary condition converges
towards zero with increasing resolution level. Consequently, the redshift is
due to numerical effects and the absolute numerical accuracy of the velocity
shift is in the range of $\lesssim 0.5$ km s-1.

As the anisotropy increases, the helium line forms more in the $-x$-direction.
This shift is influenced by the asymmetric density distribution caused by the
net day-to-night side wind, as revealed in our prior analysis shown in Fig. 3.
Because the mean radial velocity of the flow away from the planet is lower,
the density is higher and there is an increase in the number of cells with
higher optical depth. This is particularly evident in the greater presence of
light green cells in the top of Fig. 6 for models $A_{50\%}$ and $A_{10\%}$.
This increase leads, compared to the isotropic case, to a greater absorption
depth of the helium line.

We note that in Figure 6, the line of zero line-of-sight velocity shifts
toward the star in the asymmetric cases. More of the line-forming zones have
line-of-sight velocity directed toward the observer, though the typical
magnitude of the velocity is lower (as can be seen in the lighter colors on
the colorscale). Consequently, the resulting spectrum exhibits a net
blueshift. The blueshift is amplified when considering cells with smaller
optical depths. The majority of the blueshift arises from cells satisfying the
condition $10^{-3}<\Delta\tau<10^{-2}$, corresponding to a radial distance of
$1-2\leavevmode\nobreak\ R_{p}$. As anisotropy increases, the line profile
undergoes a shift towards shorter wavelengths and becomes narrower.
Importantly, this narrowing is not linked to a reduction in the planetary
mass-loss rate. Even when the mass-loss rate matches that of the isotropic
case, as shown in the left panel of Fig. 8, the line remains narrow. Instead,
this phenomenon can be attributed to the narrowing of the velocity
distribution of the line-forming area. The maximum of the velocity
distribution is reached in the blue-shifted regime. At the same time, a higher
absorption depth is achieved, which leads to a narrow line profile.

The discussion above highlights the importance of the radiative transfer and
optical depth in shaping a helium 1083 nm line profile. If we increase the
density of the planetary outflow, as discussed in Appendix A, we can model the
effect of a higher planetary mass-loss rate. For example, if the density is a
factor of 13 higher, the line forming region can extend up to
$7\leavevmode\nobreak\ R_{p}$, and the regions below $1\leavevmode\nobreak\
R_{p}$ close to the planetary surface can become opaque. This result aligns
with the findings of Linssen & Oklopčić (2023), who utilize the
photoionization code Cloudy. They predict that the helium line forms between
$2.5-6\leavevmode\nobreak\ R_{p}$ for a Neptune-like planet with a mass-loss
rate similar to $10^{10}$ g s-1, orbiting a K-type star.

To determine whether a portion of the lower atmosphere is optically thick, it
is useful to measure the line ratios of the helium triplet. As discussed by
Wang & Dai (2021a), the line ratio deviates from the quantum degeneracy ratio
if most of the absorption occurs in a region of higher density. According to
their self-consistent photochemistry models, the line forming region of
metastable helium in the atmosphere of the Saturn-mass gas giant WASP-69b is
situated above the base of the photoevaporative outflow at
$1.2\leavevmode\nobreak\ R_{p}$. Within the range of
$1.9-3.8\leavevmode\nobreak\ R_{p}$, they observed a blueshift of $2-3$ km
s-1, of similar magnitude to that found in our models. They attribute this
blueshift to the ongoing day-night advection of the tidally locked planet,
resulting from the temperature gradient between the day and night sides of the
region below.

According to our models, there is a direct correlation between the blue shift
and the altitude at which the line forms. Specifically, as the line forms at
higher altitudes in the escaping atmosphere, the magnitude of the blueshift
increases due to higher wind speeds. This dependence on altitude is, in turn,
dependent on the density of the outflow, which is directly related to the
mass-loss rate of the planet. Thus, the blueshift of the helium line is
influenced by two factors: day-night anisotropy and the altitude of line
formation.

Nevertheless, we consider the line velocity shift to be a valuable diagnostic
tool for understanding planetary wind kinematics, as it is less vulnerable to
the influence of other factors. While anisotropy does influence various
aspects of the line, including depth, width, and skewness, we argue that the
blueshift is a metric that is more easily determined in observed spectra.
Consequently, our emphasis in this paper is directed towards exploring this
feature.

###  3.2 Parametric Study

Figure 7: Mid-transit velocity shift of the dominant (red) component in the
helium 1083 nm triplet as a function of the night-to-day side pressure ratio.
The data points represent the average velocity shift over five snapshots in
the steady-state, with error bars indicating the corresponding standard
deviation. All models show the same trend: the line becomes more blue-shifted
with increasing day-night anisotropy. However, for ratios $<50\%$, this
behavior shifts, due to the occurrence of incident material on the night side.
$A$ represents the fiducial model group, including $A^{\prime}$, with the
distinction that the latter employs a constant $\dot{m}_{p}$ instead of a
constant $\rho_{p}$ as outlined in Sect 3.3 (see full model overview in Table
1).

In this section, we aim to examine the influence of the stellar wind, orbital
distance, and the planetary hydrodynamic escape parameter on the velocity
shift of the helium line. Additional line features, including absorption depth
and width, are detailed in Table 2. Across all model groups, a consistent
trend emerges: as day-night anisotropy intensifies, the line narrows, and
absorption depth increases.

Figure 7 shows the mid-transit velocity shift of the helium main component as
a function of the ratio of night-to-day side pressure. A notable trend of
increasing the net blueshift with increasing the level of anisotropy is
observed for most models. However, the blueshift of most of the model groups
reaches a plateau after an intermediate degree of anisotropy ($\lesssim
50\%$). A change in slope around this value in Fig. 7 is related to the onset
of the material fallback on the night side. As previously discussed in Fig. 4,
despite setting the initial ratio of night-to-day side pressure to 10%, it
effectively reaches $\sim 40\%$ during the simulation’s evolution. This
indicates that the anisotropy, and consequently the net blueshift, has reached
a saturated state.

An overview of the outflow structure of the most anisotropic cases for each
model group we consider is shown in Figure 14.

####  3.2.1 Stellar Wind

In this section, we examine models with increased stellar wind mass-loss rate
relative to a fixed planetary mass-loss rate. While our fiducial model
represents the least important stellar wind, cases $B$ and $C$ show
progressively strengthening stellar wind shaping of the planetary outflow.

In the scenario of an intermediate stellar wind, case $B$, the cavity
surrounding the planet experiences a higher compression as compared to case
$A$. Consequently, time variations in the blueshift of the helium line, shown
in Fig. 7 as error bars, become more prominent due to the presence of
turbulent, shocked gas within in the line forming regions. However, when
comparing it to case $A$ (which represents a weak stellar wind scenario),
there is a small difference in magnitude of the helium line velocity shift.

Moving on to the scenario of a strong stellar wind, denoted as case $C$, the
interaction between the intense stellar wind and the planetary wind results in
the suppression of the cavity surrounding the planet. This occurs due to the
compression of the wind-wind interaction region, bringing it into close
proximity with the planetary surface. Consequently, the outward flow
originating from the planet is redirected around the planet before it reaches
its sonic radius, giving rise to the formation of a tail-like structure.

In contrast to other cases, where boundary layers between the winds were
observed, in case $C$, the majority of the planetary outflow is found to be
blue-shifted and swept up within the compressed tail, giving rise to Rayleigh-
Taylor and Kelvin-Helmholtz instabilities. As a result, the magnitude of
blueshift and the variability over time are much larger compared to the other
models. With a significant increase in the day-night contrast, the pressure on
the night side is no longer sufficient to counteract the force of the stellar
wind. Consequently, the cavity surrounding the planet on the night side
dissipates, leaving only a bow shock on the day side and turbulent eddies on
the night side (see Fig. 16).

####  3.2.2 Orbital Distance

Model $D$ is characterized by a smaller orbital distance relative to model
$A$. The orbital shear velocity at this distance is higher (we emphasize that
this is sometimes called the Coriolis acceleration in the context of the
corotating reference frame). This shear leads to a stretched shape and
contraction of the cavity around the planet, as shown in Table 2.
Consequently, higher pre- and post-transit absorptions occur with material
compressed into denser streams. Notably, the helium line in model $D$ displays
a Doppler-broadened helium line profile compared to case $A$. Nonetheless, the
correlation between the helium blueshift and the degree of day-night
anisotropy follows a similar trend in both models.

####  3.2.3 Hydrodynamic Escape Parameter

We examined the impact of two different values of the parameter $\lambda_{p}$
on the dynamics of escaping gas in the planetary atmosphere. Specifically, we
analyzed models denoted by $L3$ (representative of a lower thermal energy
compared to the fiducial model)and models denoted by $L{1.5}$ (representative
of a higher thermal energy).

When the thermal energy is decreased, the pressure $P_{p}$ of the launching
wind is subsequently lowered as shown by Eq 1. As a result, the speed of sound
$c_{s}=\sqrt{\gamma P_{p}/\rho_{p}}$ is reduced, with
$c_{s}\propto\lambda_{p}^{-1/2}$. The wind accelerates more slowly and
consequently, the sonic surface is located at a greater distance from the
planet, scaling as $r_{s}\propto\lambda_{p}$. Slower outflow leads to smaller
cavities of planetary wind and narrower Doppler-shifted helium 1083 nm
spectral lines.

Our investigation further revealed that the slope of the blueshift in Fig. 7
is steeper for the $L1.5$ models compared to $L3$. At most degrees of
anisotropy, the cooler outflows of case $L$3 (with $\lambda_{p}=3$) yield
smaller magnitude of lower $v_{\rm shift}$ than model $A$. By contrast, the
hotter outflow of the $L$1.5 models leads to larger maximum blueshift before
it saturates around model $L1.5_{40\%}$. We also observe that the stronger
planetary outflows at smaller values of $\lambda_{p}$ require a higher degree
of anisotropy to produce a noticeable fallback of material.

###  3.3 Constant planetary atmospheric density vs. constant planetary mass-
loss rate

In our prior analyses, we maintained a constant initial density at the
planet’s surface to launch the wind, while modifying the degree of anisotropy.
This approach resulted in a gradual decrease in the planetary mass-loss rate
due to the reduced pressure on the night side, as outlined in Appendix B and
Fig. 11. We have opted for this approach to gain a deeper understanding of the
simulations without introducing additional parameters that might add
complexity to the analysis. Nevertheless, in order to untangle the impact of
the varying planetary mass-loss rate $\dot{m}_{p}$, we also explored the
behavior of the helium line properties for a constant $\dot{m}_{p}$ across all
degrees of day-night anisotropy. This adjustment was achieved through a
rescaling of the overall simulation density, as described in Sect A, to match
the planetary mass-loss rate of the isotropic case $A_{100\%}$.

In Fig. 8, we present a comparison of these two approaches: the left panel
depicts the scenario with a constant initial density, as discussed in our
previous analysis, while the right panel shows the situation with a constant
planetary mass-loss rate. As a consequence of the density rescaling in the
latter case, the density surrounding the planet increased, leading to a
greater number of cells contributing to line formation. Consequently, the
spectral line exhibited increased depth and equivalent width compared to the
case of constant initial density. Furthermore, we observed that the cells
responsible for line formation were located at higher altitudes in the right
panel. As a result, the blueshift of the helium line is higher with $\Delta
v=-2.4$ km s-1, in contrast to the $\Delta v=-1.7$ km s-1 observed in the
former case for model $A_{10\%}$.

To compare the mid-transit velocity shift behavior with increasing degree of
day-night anisotropy, we added model group $A^{\prime}$ to Fig. 7 (black
dotted line). This model group shows a trend similar to that of the fiducial
model group $A$. However, as day-night anisotropy increases, the blueshift of
the helium line in $A^{\prime}$ becomes slightly more prominent compared to
$A$. This is attributed to the helium line’s formation process occurring at
higher altitudes, as discussed previously. A spectral time series depicting
the excess absorption, akin to Fig. 5, is presented in Fig. 13.

Figure 8: Comparative analysis of constant initial density vs. constant
planetary mass-loss rate (same as Fig. 6). The left panel depicts the scenario
in which a uniform initial density ($\rho_{p}$) around the planet was
maintained across varying degrees of day-night anisotropy, as previously
discussed in our analysis. In contrast, the right panel shows the outcome of
maintaining a consistent planetary mass-loss rate ($\dot{m}_{p}$) by aligning
it with the values of the isotropic case $A_{100\%}$.

##  4 Discussion

###  4.1 Revealing day-night anisotropy

Figure 9: The mid-transit velocity shift of the strong (red) component of the
helium 1083 nm line shown as a function of the night-to-day side pressure
ratio, normalized by the escape velocity at the planet’s surface. Here, we
scale the density of the snapshots in a way that ensures the resulting spectra
attain the same equivalent width (EW) _(left)_. The observational measurements
of the helium velocity shift during mid-transit in units of the individual
escape velocity of the observed planet. The marker size corresponds to the
strength of the measured EW _(right)_. The dotted line shows the separation
between positive and negative velocity shifts. By adjusting our simulations to
match the properties of a specific planet, including the equivalent width
(EW), we could effectively constrain the day-night anisotropy of the planet’s
atmosphere. No velocity shift uncertainties were provided for HAT-P-11b. [1]
Alonso-Floriano et al. (2019), [2] Zhang et al. (2023a), [3] Nortmann et al.
(2018), [4] Allart et al. (2018), [5] Orell-Miquel et al. (2023), [6] Salz et
al. (2018), [7] Palle et al. (2020), [8] Kirk et al. (2022), [9] Kirk et al.
(2020).

The transport of heat between the day side and the night side in tidally
locked exoplanets, and the resulting day-night temperature gradients, have
been extensively studied over the years. However, most of the work has been
focused on investigating the energy circulation in the lower atmosphere, close
to the planet’s photosphere, while the uppermost atmospheric layers, such as
those probed by the helium 1083 nm line, still remain largely underexplored.

3D atmospheric models of hot Jupiters revealed the formation of an equatorial
jet at pressures around 1 bar. This jet leads to a displacement of the
temperature maximum away from the substellar point in the direction of the
planet’s rotation (e.g. Showman & Guillot 2002; Parmentier et al. 2013;
Showman & Polvani 2011). The existence of this hotspot offset has been
confirmed by phase curve observations in the infrared (e.g. Knutson et al.
2007; Stevenson et al. 2014; Bell et al. 2021; May et al. 2022).

At pressure levels lower than about 1 mbar, 3D atmospheric models find that
the wind structure changes to a flow directed from the day side to the night
side (Showman et al. 2009; Rauscher & Menou 2010; Heng et al. 2011; Miller-
Ricci Kempton & Rauscher 2012). The pressure level at which this qualitative
transition in the atmospheric flow occurs depends on the molecular weight of
the atmosphere, with heavier molecules causing the transition to occur deeper
within the atmosphere (Zhang & Showman 2017). The details of the day-to-night
wind pattern depend on the treatment of the magnetic effects in the model
atmosphere (e.g. Miller-Ricci Kempton & Rauscher 2012; Beltz et al. 2022), but
they are broadly consistent with spectroscopic observations, which find net
blueshifts of a few km s-1 (e.g. Snellen et al. 2010; Nortmann et al. 2018;
Salz et al. 2018).

Extending the predictions of 3D atmospheric models to even lower pressures is
not straightforward. Miller-Ricci Kempton & Rauscher (2012) found that at
pressures lower than 1 mbar, the winds can get stronger or weaker depending on
the treatment of the magnetic drag. Recently, Beltz et al. (2022) used the
active magnetic drag treatment (Rauscher & Menou 2013), which calculates the
local drag timescale throughout the atmosphere, and found that this approach
most strongly affects the upper atmosphere, where the winds mainly flow from
the day side to the night side over the poles. Observations of extended
atmospheres, such as those made using the helium 1083 nm line, could thus
provide valuable constraints on these models.

Understanding the efficiency of heat distribution, both from the day to the
night side and vertically from the photosphere to the upper low density regime
probed by He 1083 nm observations, would be valuable in gaining deeper
insights into the atmospheric dynamics. The question we aim to address in this
study is whether we can infer the day-night temperature gradient from high-
resolution helium 1083 nm observations, assuming the presence of day to night
side winds at high altitudes. Our findings reveal a correlation between the
velocity shift of the He 1083 nm triplet and the degree of day-night
anisotropy, offering the potential to constrain the latter through careful
analysis.

Therefore, for a particular planet, if we assume that the sole driver of the
helium triplet velocity shift is day-night temperature gradient, we could
simultaneously constrain three free parameters: the planetary mass loss rate,
characteristic temperature of the outflow, and degree of day-night anisotropy.
To construct these constraints requires a high-resolution transit spectrum
with a resolved He 1083 nm line profile. To apply this approach practically,
the spectral energy distribution (SED) of the star is needed to calculate the
population of metastable helium in the gas. Though these three parameters are
linked, we can make some generalizations about their impact on the
transmission spectrum. The temperature of the outflow informs the line’s
Doppler-broadened width (Linssen et al. 2022). At a given temperature, the
line equivalent width is determined by the density of the outflow, or the
planetary mass-loss rate. A 1D Parker wind approach can also provide a good
starting point for 3D modeling (e.g. Oklopčić & Hirata 2018; Linssen et al.
2022). Finally, as we have shown in this work, the velocity shift (blue or
red) is informed by any day-night anisotropy of the planetary upper
atmosphere. By combining these steps, we can constrain the day-night
anisotropy as a free parameter by assigning the velocity shift to the model
correlation.

We anticipate that one aspect of the constraints one could derive would be a
limiting value of the anisotropy parameter. Our study has provided insights
into the impact of a fallback of gas on the night side, as described in Sect
3.1.2. In such a scenario, the blueshift saturates at higher degrees of day-
night anisotropy. It becomes challenging to differentiate between a very
strong degree of anisotropy and an intermediate degree, as the blueshift
reaches a maximum value. This saturation effect poses a limitation on what can
be distinguished observationally based on the magnitude of the blueshift.

As a less planet-specific overview than the approach outlined above, in Fig.
9, we present a comparison between the helium line shifts observed in our
simulations from model group $A$ and those reported in helium transit
observations. We note that the simulations are not scaled to the properties of
the particular systems in question, which would require simulating each system
separately. However, the relative robustness of our results to model
variations (e.g., as seen in Fig. 7) motivates this comparison. To ensure
comparability, we express the observed velocity shift for each planet in terms
of the planet’s escape velocity at the surface. The right side of the figure
displays the observational results, ordered from left to right according to
increasing equivalent width. Many of them exhibit a blueshift, covering a
velocity range that aligns with the outcomes of our study. On the left side,
we present model group $A$ scaled to two different EW values. The solid line
corresponds to the smaller value of $13\leavevmode\nobreak\ \rm m\AA$, while
the dashed line represents the larger value of $340\leavevmode\nobreak\ \rm
m\AA$. To achieve a higher EW, we scaled the snapshots to a higher density,
effectively placing the line-forming region at higher altitudes with increased
wind speeds, leading to a higher velocity shift.

In Fig. 9 we note that many of the observed systems show a velocity blueshift
on the order of 10-40% of the planetary escape velocity. These sorts of
velocity shifts appear compatible with the magnitude of day-to-night flows
seen in our simulations. However, our simulations predict a correlation
between the magnitude of the velocity shift and the line equivalent width (all
else being equal), which is not seen when looking across the observed systems.
Future work with the planet-specific simulation approach described above will
probe how well the day-night anisotropy model applies to particular systems.

###  4.2 Limitations

When interpreting the helium line shift observed in transmission spectroscopy
of evaporating gas giants, we encounter two types of constraints. Those that
affect the blueshift when day to night side winds are present, which have been
discussed above in the context of jointly fitting outflow parameters, and
those that cause a blueshift by an entirely different physical process,
unrelated to the day-night anisotropy.

The latter includes a strong stellar wind relative to the planetary wind, as
shown by our model group C. Even using an isotropic boundary condition for the
planetary wind results in a blueshift of the helium line, as shown in Fig.
autoreffig:Blueshift. However, the work by MacLeod & Oklopčić (2022) has
demonstrated that, in such scenarios, a distinct post-transit signal emerges
due to the planet’s flow being considerably confined within a cometary
trailing tail. The resulting asymmetric light curve can help to distinguish
cases influenced by a strong stellar wind from those primarily driven by day-
night anisotropy.

Furthermore, we did not study the effects of a planetary magnetic field and
the extent to which they could impede the flow from the day to the night side.
Previous magnetohydrodynamic simulations have suggested that the outflow from
a dipole magnetic field would be confined to the polar regions, resulting in a
double-tailed structure, with the opening angle decreasing as the field
strength increases (Adams 2011; Trammell et al. 2011, 2014; Carolan et al.
2021). In a recent study by Schreyer et al. (2023), 2D simulations of
photoevaporation were performed for a configuration involving a strong
magnetic field. The results indicate that the day-to-night side winds can be
suppressed due to the presence of a planetary magnetic field, resulting in
either a lack of net blueshift or even a detectable red shift in the helium
line.

We note that the influence of 1083 nm radiation pressure on gas dynamics is
ignored in this study. There are different models and approaches in the
literature regarding the treatment of metastable helium in gas giant
atmospheres, with some models considering it as collisionless particles (e.g
Allart et al. 2019), while others treat it as a separate fluid species in a
multi-fluid approach (e.g Khodachenko et al. 2021). The latter model finds
substantial differential velocity for metastable helium as it is accelerated
by radiation between collisions, which differs from the results presented in
the study by Wang & Dai (2021b). The estimation of the short mean free path
near the planet, discussed in MacLeod & Oklopčić (2022), suggests that the
planetary outflow probed by the helium line is in a collisional regime for
observed mass-loss rates, making the use of hydrodynamic simulations
appropriate. However, it is important to note that radiation pressure could
still act as a radial force applied from the star – much like the action of a
stellar wind, collimating and confining planetary outflow in particularly
strong circumstances.

Finally, we emphasize the utility of having a long baseline for helium
observations in order to obtain precise measurements of velocity shift. As
revealed by recent studies of Zhang et al. (2023a) and Gully-Santiago et al.
(2023), the escaping atmosphere can extend a significant fraction of the
orbital phase – hundreds of planetary radii. If out-of-transit spectra used to
establish a baseline have the signature of planetary outflow kinematics, this
can introduce biases in determining the velocity shift measured at mid-
transit, as exemplified by Spake et al. (2021).

##  5 Summary and conclusions

The objective of this study was to investigate the gas dynamic effects of day-
to-night temperature contrasts on the escaping atmosphere of a tidally locked
planet, with emphasis on analyzing the effects on the helium 1083 nm triplet.

By combining hydrodynamic simulations and radiative transfer post-processing,
we modeled spectra of the metastable helium triplet for planets in transiting
configurations. Our simulations incorporate parameterized treatments of both
planetary and stellar winds, allowing us to investigate the influence of wind
properties on the observable characteristics of evaporating atmospheres in a
wide range of parameters. We conducted a parametric study, varying the stellar
mass-loss rate, orbital distance, and the ratio of thermal and gravitational
potential energies (i.e., the hydrodynamic escape parameter, $\lambda_{p}$)
associated with the planetary wind. The key findings of our study are:

  1. 1.

With increasing the day-night anisotropy, there is a general trend toward
narrowing of the helium line and an increase in the net blueshift of the line
centroid.

  2. 2.

The velocity shift of the line is contingent on the line forming altitude. A
higher planetary mass-loss rate causes the line to form at higher altitudes,
where the wind attains higher speeds. Consequently, this leads to a more
pronounced velocity shift in the observed spectra.

  3. 3.

We have identified a critical point of anisotropy, at which the blueshift
stagnates, indicating that a saturation of the day-night anisotropy is
reached. This saturation is attributed to turbulent flows generated by the
outflow material falling back onto the night side of the planet.

  4. 4.

The velocity shift of the helium line is affected the stellar wind strength
and the presence of turbulent flows, potentially leading to time variations in
the shift.

Assuming the day-night temperature gradient as the primary cause of the
observed blueshifts in the He-1083 nm line, we can make use of the correlation
found in Fig. 7 to constrain the temperature gradient. This requires a
simulation catered to the particular properties of an exoplanetary system.
Because of the sensitivity of the line shift to the formation altitude, it is
important to scale the density of the simulation appropriately to match the
observed equivalent width.

However, we emphasize that day-night temperature gradients are not the only
possible cause of blueshifts in He 1083 nm transit spectra. One other physical
process of potentially high importance is the impact of a strong stellar wind,
which could induce a blueshift in the helium line, regardless of the presence
of day-night anisotropy. Previous research by MacLeod & Oklopčić (2022) has
shown that examining the light curve for asymmetry can be helpful in
identifying this scenario. Additionally, processes not accounted for in our
model, such as magnetic drag, magnetically-controlled outflows, and radiation
pressure, could also impact the helium velocity shift.

The saturation of the blueshift in scenarios with falling back material on the
night side suggests that it may be challenging to distinguish between
intermediate and high day-night anisotropy cases. We find that consideration
should be given to the effects of time variability when interpreting the
observed line shift. This does not only concern the variability of the stellar
UV flux, which affect the population of metastable helium in the atmosphere.
Our models indicate that high variability implies that the line forming region
includes turbulent flow, either because of eddies caused by fallback, or wind-
wind collisions.

###### Acknowledgements.

We appreciate the helpful discussions with D. Linssen and C. Dominic. We thank
the referee for their insightful comments, and we thank SURFsara
(www.surfsara.nl) for their support in using the Lisa Compute Cluster.

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##  Appendix A Density rescaling

In hydrodynamic simulations conducted using codes such as Athena++, the
physical quantities in a single snapshot can be manipulated without disrupting
the flow patterns. This capability stems from the inherent scale invariance of
the hydrodynamic equations utilized in these simulations, as they are solved
numerically using discrete time steps. By uniformly increasing the pressure
and density across the entire simulation domain, the relative values and
ratios between pressure and density remain unaltered. Consequently, the flow
patterns, which are determined by the gradients and variations in pressure and
density, remain unchanged.

Taking advantage of this rescaling capability, we investigated the behavior of
the blueshift exhibited by the helium line within a higher density environment
surrounding the planet. Increasing the density is equivalent to increasing the
mass-loss rate of the planet. Fig. 10 illustrates the line properties of the
metastable helium triplet, specifically the mid-transit velocity shift as a
function of the equivalent width, for different density scalings. A clear
trend is observed, where the velocity shift increases with an increasing
equivalent width. The curves corresponding to individual models, ranging from
isotropic wind to anisotropic wind, do not intersect. This implies that there
is no degeneracy among different degrees of anisotropy. Consequently, the
results presented in Fig. 6 would undergo an offset shift along the velocity
axis in the case of different degrees of anisotropy.

An increase in gas density also leads to an increase in the number density of
metastable helium. Consequently, the extent of the region where line formation
occurs expands. In the context of day-night anisotropy, it is notable that the
line predominantly forms on the night side, where the gas moves in the
direction of the observer. At higher altitudes, the gas is more accelerated
and as a result, the line exhibits a more pronounced blueshift. Moreover, the
equivalent width of the line increases due to a greater contribution from the
additional gas involved in line formation, leading to an increase in
absorption depth.

Figure 10: The mid-transit velocity shift of the strong (red) component of the
helium 1083 nm line as a function of the equivalent width (EW), normalized by
the escape velocity at the planet’s surface. The points marked by squares
indicate the scaling value 1.0, which was used for the previous analysis in
this paper. When the density is increased, the mass-loss rate of the planet
also increases, leading to a larger equivalent width and a higher blue shift
in the helium line.

##  Appendix B Planetary mass-loss rate

The net mass-loss rate of the planet is calculated by determining the rate of
change of its mass through the gradient of a passive scalar with respect to
time. This gradient quantifies the variations in mass over small time
intervals. Dividing this gradient by the corresponding time intervals yields
the rate of change of mass per unit time. To account for the mass loss
attributed to the planet’s wind, which is already measured in the simulation,
we subtract this contribution from the rate of change of mass. Consequently,
the resulting value represents the net mass-loss rate of the planet,
encompassing all mass loss mechanisms. Finally, we take the time average
$\langle\dot{M}_{p}\rangle$ for $t>10^{6}$ s, when the quasi-steady state is
reached.

In Fig. 11, we present the time-averaged planetary mass-loss rates with
increasing day-night anisotropy. The mass-loss rates across all models exhibit
a twofold decrease, with the exception of model group $A^{\prime}$
intentionally scaled to maintain a constant $\dot{m}_{p}$. Firstly, the
introduction of an anisotropic surface lowers the pressure, thereby reducing
the driving force behind the wind, leading to a decrease in the mass-loss
rate. Since the pressure is uniformly decreased in each model, the slope of
the mass-loss rates as a function of the night-to-day side pressures should be
the same across all models. Secondly, the presence of gas falling back onto
the night side is expected to contribute to a decrease in the planetary mass-
loss rate. Figure 11 demonstrates that, in general, the mass-loss rates
exhibit a similar negative slope going from isotropic to anisotropic
conditions. However, model $B$ and $L3$ show a significant deviation at strong
anisotropic conditions, while models $A$ and $C$ show a negative increase in
slope. We observe that the extent of the deviations correlate with the
occurrence and strength and of the falling back material on the night side.
The remaining models, namely $D$, and $L1.5$, do not exhibit significant
variations.

Figure 11: Time-averaged planetary mass-loss rates. The error bars indicate
the standard deviation. The introduction of an anisotropic surface lowers the
pressure, leading to a decrease in the mass-loss rate for increasing day-night
anisotropy. Deviations for ratios $<50\%$ can be attributed to falling back
material on the night side.

##  Appendix C Numerical effects on radiative transfer results

Figure 12: Mid-transit velocity shift of the dominant (red) component in the
helium 1083 nm triplet as a function of the night-to-day side pressure; post-
processed with different spatial resolution levels of the simulation for model
group $A$. the overall numerical uncertainty of the velocity shift is in the
range of $\sim 0.5$ km s-1.

Figure 12 illustrates the impact of the spatial resolution during the
radiative transfer post-processing step on the mid-transit velocity shift of
the helium line. In the case of the isotropic model $A_{100\%}$, increasing
spatial resolution leads to a convergence of the velocity shift towards zero
(0.7, 0.5, 0.2 km s-1 for levels 3, 4, and 5, respectively). However, this
monotonic trend is not evident in all the anisotropic models due to their more
complex flow patterns. However, the curves corresponding to different
resolution levels follow the same trend and span a range of approximately
$\sim 0.5$ km s-1, which we estimate reflects the overall numerical
uncertainty associated with the velocity shifts measured in this study.

Furthermore, we investigated the effects of the convergence of the mid-transit
velocity shift with different total numbers of rays ($N_{\rm radial}$ = 30,
50, 100) and the step size coefficient along the rays ($f_{l}$ = 0.01, 0.5,
0.1). The differences in the blueshift for the tested parameters were not
significant. For the results shown in this work, we used for the radiative
transfer post-processing step the spatial resolution level 4, alongside
$N_{\rm radial}=100$, and $f_{l}$ = 0.1.

##  Appendix D Additional figures

Figure 13 shows Fig. 5 for model group $A^{\prime}$. To complement this visual
representation, Table 2 provides further details on line properties across all
models. Additionally, it lists simulation measurements of both stellar and
planetary mass-loss rates.

In Fig. 14, the density and line-of-sight velocity are displayed in the
vicinity of the planet for all anisotropic models with $\tilde{f}_{\rm pres}$=
0.1. This figure illustrates how the shape and size of the cavity surrounding
the planet change across the models studied in this work.

The online movies linked to Fig. 15 and Fig. 16 show snapshots of the line-of-
sight velocity and streamlines in the vicinity of the planet for increasing
day-night anisotropy.

Figure 13:  Analogous representation of metastable helium absorption in the
stellar frame as shown in Fig. 5, but for model group $A^{\prime}$, where the
planetary mass-loss rate is matching model $A_{100\%}$. Similarly to model
group $A$, where the density is the same across all models, the line becomes
more narrow and blue-shifted as day-night anisotropy increases. As discussed
in Sect 3.3, in the case of constant planetary mass-loss rate, the line forms
at higher altitudes which leads to a higher blueshift and absorption. Table 2:
Simulation results and helium line properties.

model | $\lambda_{p}$ | $a$ | $\langle\dot{M_{*}}\rangle$ | $\langle\dot{M_{p}}\rangle$ | $\delta$ | $\delta_{\rm model}/\delta_{A_{100\%}}$ | EW | FWHM | $\Delta v$
---|---|---|---|---|---|---|---|---|---
|  | [au] | [g s-1] | [g s-1] | [$\%$] |  | [m$\AA$] | [$\AA$] | [km s-1]
$A_{100\%}=A^{\prime}_{100\%}$ | 2 | 0.05 | $7.70\times 10^{10}$ | $3.80\times 10^{10}$ | 0.25 | 0.97 | 2.15 | 0.70 | 0.25
$A_{50\%}$ | 2 | 0.05 | $8.19\times 10^{10}$ | $2.79\times 10^{10}$ | 0.58 | 2.72 | 3.59 | 0.51 | -1.64
$A_{10\%}$ | 2 | 0.05 | $8.64\times 10^{10}$ | $1.63\times 10^{10}$ | 0.78 | 4.90 | 3.86 | 0.40 | -1.69
$A^{\prime}_{50\%}$ | 2 | 0.05 | $1.11\times 10^{11}$ | $3.80\times 10^{10}$ | 0.95 | 4.17 | 6.06 | 0.51 | -1.79
$A^{\prime}_{10\%}$ | 2 | 0.05 | $2.00\times 10^{11}$ | $3.80\times 10^{10}$ | 2.55 | 12.99 | 14.54 | 0.46 | -2.39
$B_{100\%}$ | 2 | 0.05 | $8.79\times 10^{11}$ | $3.11\times 10^{10}$ | 0.33 | 0.97 | 2.81 | 0.67 | 0.25
$B_{50\%}$ | 2 | 0.05 | $8.77\times 10^{11}$ | $2.46\times 10^{10}$ | 0.61 | 2.72 | 4.03 | 0.52 | -1.64
$B_{10\%}$ | 2 | 0.05 | $8.75\times 10^{11}$ | $1.65\times 10^{10}$ | 0.86 | 4.53 | 4.81 | 0.43 | -1.94
$C_{100\%}$ | 2 | 0.05 | $8.75\times 10^{12}$ | $1.07\times 10^{11}$ | 6.87 | 53.47 | 51.59 | 0.61 | -2.70
$C_{50\%}$ | 2 | 0.05 | $8.71\times 10^{12}$ | $8.47\times 10^{10}$ | 10.98 | 87.16 | 82.01 | 0.61 | -4.78
$C_{10\%}$ | 2 | 0.05 | $8.71\times 10^{12}$ | $5.11\times 10^{10}$ | 15.53 | 139.26 | 82.28 | 0.43 | -2.70
$D_{100\%}$ | 2 | 0.03 | $5.73\times 10^{10}$ | $5.44\times 10^{10}$ | 0.11 | 0.36 | 1.04 | 0.75 | -0.09
$D_{50\%}$ | 2 | 0.03 | $5.95\times 10^{10}$ | $4.43\times 10^{10}$ | 0.25 | 1.08 | 1.71 | 0.53 | -1.49
$D_{10\%}$ | 2 | 0.03 | $6.31\times 10^{10}$ | $3.00\times 10^{10}$ | 0.33 | 1.96 | 1.82 | 0.42 | -1.39
$L1.5_{100\%}$ | 1.5 | 0.05 | $7.40\times 10^{10}$ | $4.40\times 10^{10}$ | 0.12 | 0.37 | 1.30 | 0.84 | 0.20
$L1.5_{50\%}$ | 1.5 | 0.05 | $7.87\times 10^{10}$ | $3.54\times 10^{10}$ | 0.25 | 0.54 | 2.03 | 0.65 | -2.64
$L1.5_{10\%}$ | 1.5 | 0.05 | $8.35\times 10^{10}$ | $2.12\times 10^{10}$ | 0.45 | 1.95 | 2.62 | 0.47 | -2.64
$L3_{100\%}$ | 3 | 0.05 | $7.72\times 10^{10}$ | $4.15\times 10^{10}$ | 1.33 | 7.85 | 8.25 | 0.51 | 0.25
$L3_{50\%}$ | 3 | 0.05 | $8.55\times 10^{10}$ | $2.24\times 10^{10}$ | 2.08 | 15.14 | 10.02 | 0.40 | -0.74
$L3_{10\%}$ | 3 | 0.05 | $8.87\times 10^{10}$ | $1.31\times 10^{10}$ | 1.88 | 14.40 | 8.09 | 0.35 | -1.69

Notes. Time-averaged stellar $\langle\dot{M_{*}}\rangle$ and planetary
$\langle\dot{M_{p}}\rangle$ mass-loss rates measured from simulation results,
as well as line properties of the synthetic metastable helium spectrum at
phase 0, where $\delta$ = absorption depth, EW = equivalent width, FWHM = full
width at half maximum, $\Delta v$ = velocity shift of the line centroid. These
values are extracted from a single spectrum of a representative simulation
snapshot.

Figure 14: Anisotropic model overview of the density (left) and line-of-sight
velocity (right) showing the varying morphology of the planet’s vicinity. The
flow directions are indicated by gray streamlines. The observer is located in
the $-x$ and the star in the $+x$-direction. Figure 15: (Movie online) A
sequence of evolved snapshots displaying the line-of-sight velocity around the
planet, accompanied by over-plotted streamlines depicting the infalling
material on the night side at high degrees of day-night anisotropy; with model
$A_{100\%}$ representing the isotropic case and model $A_{10\%}$ exemplifying
the most anisotropic scenario. Notably, the occurrence of fallback can be
directly linked to the saturation of the blueshift shown in Figure 7. Figure
16: (Movie online) Same as Fig. 15, but showing model group $C$. It shows that
the cavity around the planet dissipates on the night side of the planet as the
planetary wind becomes too weak to form a shock front with increasing day-
night anisotropy. Additionally, a striking observation is that the shocked
regions now align with the helium line formation region. This alignment
results in a significant temporal variability, as depicted in Figure 7.

##  Appendix E Data and software availability

Simultaneously with the publication of this article, we are releasing the
necessary software and data to replicate the results and analysis. The
hydrodynamic model snapshots (in Athena++ hdf5 format), as well as 3D post-
processed species number densities (hdf5 format), and the synthetic spectra
(in ascii format) of models $A_{100\%}$, $A_{50\%}$, and $A_{10\%}$ can be
accessed on Zenodo at 10.5281/zenodo.10025849. Furthermore, we included the
radiative transfer post-processing software used in our study, the modified
Athena++ setup files, and the code necessary to replicate the presented
figures.