# Magnetic Effects and 3D Structure in Theoretical High-Resolution
Transmission Spectra of Ultrahot Jupiters: the Case of WASP-76b

Hayley Beltz Department of Astronomy, University of Michigan, Ann Arbor, MI
48109, USA  Emily Rauscher Department of Astronomy, University of Michigan,
Ann Arbor, MI 48109, USA  Eliza M.-R. Kempton Department of Astronomy,
University of Maryland, College Park, MD 20742, USA  Isaac Malsky Department
of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA  Arjun B. Savel
Department of Astronomy, University of Maryland, College Park, MD 20742, USA

###### Abstract

High resolution spectroscopy has allowed for unprecedented levels of
atmospheric characterization, especially for the hottest gas giant exoplanets
known as ultrahot Jupiters (UHJs). High-resolution spectra are sensitive to 3D
effects, making complex 3D atmospheric models important for interpreting data.
Moreover, these planets are expected to host magnetic fields that will shape
their resulting atmospheric circulation patterns, but little modeling work has
been done to investigate these effects. In this paper, we generate high-
resolution transmission spectra from General Circulation Models for the
canonical UHJ WASP-76b with three different magnetic treatments in order to
understand the influence of magnetic forces on the circulation. In general,
spectra from all models have increasingly blueshifted net Doppler shifts as
transit progresses, but we find that the differing temperature and wind fields
in the upper atmospheres of these models result in measurable differences. We
find that magnetic effects may be contributing to the unusual trends
previously seen in transmission for this planet. Our $B=3$ Gauss active drag
model in particular shows unique trends not found in the models with simpler
or no magnetic effects. The net Doppler shifts are additionally influenced by
the dominant opacity sources in each wavelength range considered, as each
species probes different regions of the atmosphere and are sensitive to
spatial differences in the circulation. This work highlights the ongoing need
for models of planets in this temperature regime to consider both 3D and
magnetic effects when interpreting high resolution transmission spectra.

††journal: AJ

##  1 Introduction

High resolution spectroscopy (HRS, typically R $\gtrsim$ 30,000) has opened
windows into exoplanet atmospheres at an unprecedented level of precision. HRS
has allowed for detections of new atmospheric species as well as measurements
of net Doppler shifts and broadening due to atmospheric winds and rotation
(Snellen et al., 2010; Brogi et al., 2016; Schwarz et al., 2016). Brogi &
Birkby (2021) offers a recent review of the techniques and major results of
HRS.

Due to the level of precision offered by HRS (and the fact that planets are
multi-dimensional objects), three-dimensional (3D) atmospheric models are
ideal for interpreting these spectra. Previous work has found that not only do
3D effects show up in high resolution transmission (Louden & Wheatley, 2015;
Flowers et al., 2019) and emission (Herman et al., 2022; van Sluijs et al.,
2022; Pino et al., 2022) spectra, but detection strengths can increase when
using spectra generated from a 3D model compared to 1D models (Beltz et al.,
2021). Typically the 3D structure of exoplanet atmospheres are simulated with
General Circulation Models (GCMs). This type of numerical model solves the set
of fluid dynamical equations known as the “primitive equations of meteorology”
to simulate a planet’s atmospheric structure, its including temperature and
wind fields throughout its orbit.

Ultrahot Jupiters (UHJs) are ideal for testing differing treatments of
magnetic effects. Due to the thermal ionization of dayside species (Parmentier
et al., 2018; Helling et al., 2021), charged particles will be blown around
the planet and interact with magnetic field lines generated from the planet’s
interior dynamo (Perna et al., 2010). Partially due to their already
significant computational time, most GCMs do not have an explicit treatment
for magnetic effects in their simulated atmospheres. One commonly used
treatment is applying a global uniform Rayleigh drag timescale to the
atmosphere such as in GCMs from Tan & Komacek (2019); Carone et al. (2020);
Deitrick et al. (2020); Lee et al. (2022). Notably this timescale is also
sometimes used to encompass a variety of effects, also including
hydrodynamical ones. Although easy to numerically implement, this prescription
of magnetic drag includes assumptions that become problematic when applied to
planets with strong day-night temperature differences. Since the strength of
magnetic effects is a strong function of local ionization levels (and so
temperature), order of magnitude estimates of the global field strength
corresponding to a particular uniform drag timescale (such as those carried
out in Kreidberg et al., 2018) effectively imply that the global magnetic
field is nearly two orders of magnitude stronger on the nightside than the
dayside, for the case of the UHJ WASP-76b and assuming a $10^{4}$s uniform
timescale (Beltz et al., 2022b). Instead, for a uniform global magnetic field,
we should expect magnetic effects to be much stronger on the dayside, compared
to the negligibly ionized nightside (Perna et al., 2010; Beltz et al., 2022b).
The most physically consistent treatment of magnetic effects are found in
specialized non-ideal magnetohydrodynamic (MHD) models (such as those
presented in Rogers & Showman, 2014; Rogers & Komacek, 2014; Rogers, 2017),
but at the cost of simplifying other aspects of the modeling such as the
treatment of radiative transfer(see Beltz et al., 2022b, for a more detailed
discussion) , and greatly increased computational time required, resulting in
less than a handful of these types of models having been published for UHJs.
In this work, our models use a medium complexity “kinematic MHD” approach,
allowing the strength of the drag timescale to vary as a function of
temperature, pressure, and latitude. (For a more detailed description of this
approach, see Rauscher & Menou, 2013; Beltz et al., 2022b).

UHJs are the ideal laboratory for exploration with high-resolution
spectroscopy due to their extremely favorable signal to noise ratio caused by
their size and temperature. Here, we specifically focus on the UHJ WASP-76b,
an inflated gas giant orbiting an F-type star with a period of 1.81 days (West
et al., 2016). High resolution transmission spectra for this planet have been
observed and studied extensively (Casasayas-Barris et al., 2021; Deibert et
al., 2021; Landman et al., 2021; Tabernero et al., 2021) with a recent work
(Kesseli et al., 2022) exploring the wide range of species detected in the
atmosphere of the planet. An influential transmission result by Ehrenreich et
al. (2020) found a spatially asymmetric and extremely large blueshift ($-$11
km/s) of neutral iron, arguing this blueshift is a result of nightside
condensation of the species. Alternate physical processes have been suggested
to explain this large blueshift, including clouds, non-zero eccentricity
(Savel et al., 2022), or large temperature differences between limbs
(Wardenier et al., 2021), but so far, it has been difficult for GCMs to match
this magnitude of shift. A recent work by Gandhi et al. (2022) performs a deep
analysis on this dataset, providing constraints on both temperature and Fe
abundances for four different regions of the planet and confirming spatial
differences across the terminator.

In this work, we explore modeled high-resolution transmission spectra for
three different models of the UHJ WASP-76b: one with a Uniform drag timescale,
one with our kinematic MHD approach, and one without any treatments for
magnetic effects. The difference in temperature and wind structures of these
three different models result in spectra that vary throughout transit, opening
the door to the exploration of how magnetic effects can alter high resolution
transmission spectroscopy. This work represents the first time the impact of
magnetic drag assumptions on high resolution transmission spectra has been
studied (we similarly explored the impact of magnetic effects on high
resolution emission spectra in Beltz et al., 2022a). By identifying measurable
differences between high resolution transmission spectra simulated using
different prescriptions for magnetic effects, we can hope to predict how we
might empirically constrain the role of magnetism in UHJ atmospheres.

In Section 2, we briefly describe the models used in this analysis and the
different treatments for magnetic effects we tested. We also discuss our
radiative transfer post-processing and choice of wavelength ranges to generate
our predictive spectra. In Section 3, we explore the features of our predicted
spectra and examine the impact of magnetic model and wavelength choices. We
then put this work in context of the model’s assumptions and other
capabilities in Section 4. Finally, we summarize our main conclusions in
Section 5.

##  2 Methods

###  2.1 GCM

For this work, we post-process previously generated 3D GCMs of the ultrahot
Jupiter WASP-76b (first published in Beltz et al., 2022b, where more details
and specific numerical parameters can be found) using a ray-striking radiative
transfer code to generate high-resolution transmission spectra at multiple
wavelengths and resolutions. These models used the RM-GCM (Rauscher & Menou,
2012; Roman & Rauscher, 2017) with parameters appropriate for WASP-76b, with
65 vertical layers evenly spaced in log pressure, from 100 to $10^{-5}$ bars,
and a horizontal spectral resolution of T31, corresponding to roughly $\sim 4$
degree spacing at the equator. The simulations ran for a total of 2000
planetary days. Our GCM assumes hydrostatic equilibrium, which is a valid
assumption for the opacity sources included in the high-resolution spectra we
calculate from these models. This is relevant to note as recent work from
Zhang et al. (2022) finds that absorption strength of particular species often
detected in transmission of UHJs, such as FeII and H$\alpha$ can’t be
explained from hydrostatic equilibrium assumptions. This is not an issue for
our work due to our choices of opacity sources.

The models from Beltz et al. (2022b) were calculated for several different
magnetic drag prescriptions at a variety of field strengths; we choose to
analyze the same subset of models as in we did in Beltz et al. (2022a). These
models differ in the way they treat magnetic effects, as follows:

  * •

Drag Free/0 G: This is the baseline model that contains no additional forms of
drag to represent magnetic effects. The GCM does contain numerical
hyperdissipation and three sponge layers (see Beltz et al., 2022b, for a
discussion on sponge layers in GCMs), both of which are used for numerical
stability and are also present in the models listed below.

  * •

Uniform/$10^{4}$ s: This method of applying drag is often found in GCMs due to
its numerical simplicity. A single Rayleigh drag timescale—in this case
$10^{4}$ seconds—is applied throughout the simulation to the horizontal and
vertical momentum equation. This value was chosen to match the strong drag
case from (Tan & Komacek, 2019) and provide the same comparisons as the
analysis work presented in Beltz et al. (2022a).

  * •

Active drag/3 G: This method for treating magnetic effects, first used in
Rauscher & Menou (2013) and first applied to UHJs in Beltz et al. (2022b) is
the most physically complex treatment of magnetic effects that we test. Our
active drag prescription, also sometimes referred to as a “Kinematic MHD”
treatment, also applies a drag on the winds, but only in the east-west
direction (as geometrically appropriate for a dipole global field Perna et
al., 2010) and with a timescale calculated based on local conditions, using
the following expression from Perna et al. (2010):

$\tau_{mag}(B,\rho,T,\phi)=\frac{4\pi\rho\ \eta(\rho,T)}{B^{2}|sin(\phi)|}$
(1)

where $B$ is the chosen global magnetic field strength (in this case 3 G),
$\phi$ is the latitude, $\rho$ is the density, and the magnetic resistivity
($\eta$) is calculated in the same way as Menou (2012):

$\eta=230\sqrt{T}/x_{e}\textnormal{ cm${}^{2}$ s${}^{-1}$}$ (2)

where the ionization fraction, $x_{e}$, is calculated from the Saha equation,
taking into account the first ionization potential of all elements from
hydrogen to nickel (as in Rauscher & Menou, 2013).

There are currently no direct observational constraints on the magnetic
strength of this planet, or any exoplanet for that matter. Although in Beltz
et al. (2022b) we present a variety of active drag field strengths (0.3 G, 3
G, and 30 G), we are primarily focusing on the 3 G model, as it is the best
match for previously published Spitzer phase curves from May et al. (2021) (as
shown in Beltz et al., 2022b). We previously found that varying the magnetic
field strength changed how deep the magnetic circulation regime—characterized
by dayside flow up and over the poles—persisted. All of these models exhibit
this magnetic circulation at the high pressures probed by high-resolution
transmission spectroscopy. Thus, we chose the 3 G model as a representative
for the active drag models. This field strength is also in line with estimates
from interior modeling by Yadav & Thorngren (2017).

It is important to acknowledge that our GCM currently does not consider H2
dissociation and recombination. This process is expected to reduce the day-
night temperature contrast (Bell & Cowan, 2018; Pluriel et al., 2020) of UHJs.
Another important result of dissociation is the change in scale heights
throughout the atmosphere. On the dayside, the mean molecular weight is
decreased due to the dissociation, thus increasing the scale height. However,
at the same time, the temperature of this region is decreased, meaning a
potential reduction in scale height. The nightside wouuld show the opposite
trend (increasing in temperature and mean molecular weight). Recent work from
Savel et al. (2023) explores the effect of scale height differences in limb
asymmetry during transit. Future work should explore the interaction between
this process and magnetic effects.

The temperature distributions of the models, for the near-terminator regions
probed by transmission spectroscopy, are shown in Figure 1, which plots the
temperature structure at a slice of the planet as it would be oriented during
ingress, mid-transit, and egress. Note that only the upper atmosphere (maximum
pressure of $\sim$0.1 bars) is shown and the relative size between the
atmosphere and planet core is not to scale. From this plot, we can see that
the spatial vertical extent of each limb varies throughout transit, with the
hotter regions being much more extended. The 3 G active drag model shows the
most variation across the limbs at ingress and egress but at transit center
the 0 G model shows the strongest temperature variation. This is directly
related to the fact that the 3 G model has the largest day-night temperature
contrast of the models considered and that this planet rotates $>30^{o}$
throughout the entire transit.

Figure 1: Temperature projections for the three models used in this analysis,
for pressures less than $\sim$0.1 bars. Note that the core and atmosphere are
not to scale, but the relative altitudes at different locations are accurately
plotted. The east and west limb asymmetries in spatial extent is a result of
the difference in scale heights of each region due to non-uniform temperatures
between the east and west terminators. Because of the planet’s short orbital
period, it rotates considerably ($>30^{o}$) between ingress to egress, which
is reflected above.  Figure 2:  Line of sight velocities for the three models
considered in this work at ingress, mid-transit, and egress. Throughout
transit, the blueshifts dominate the net Doppler shift for all models, though
the magnitudes of the net Doppler shifts are both model and wavelength
dependent.

It is also important to consider the line of sight velocities due to strong
winds of each model, as shown in Figure 2. As the transit proceeds, the planet
rotates, allowing different parts of the atmosphere and their associated winds
to come into view. These winds will directly influence the net Doppler shifts
associated with each model. We delve deeper into these calculations in section
3, but by eye one can notice that the 3 G model has the strongest redshifted
regions of the three and the Uniform drag has the strongest blueshifted
regions. One can additionally see that the 0 G model displays some high-
altitude and high-latitude winds that are blowing in the substellar-to-
antistellar direction. But, since that direction includes an east-west
component, this flow structure is disrupted in the 3 G active magnetic drag
model and so that blue-shifted contribution to the net Doppler shift is
removed.

###  2.2 Radiative Transfer

We use the same method of calculating high resolution transmission spectra
that accounts for 3D effects as that described in detail in Miller-Ricci
Kempton & Rauscher (2012); Savel et al. (2022). In short, the output from our
GCM (containing temperature values, east-west wind speeds, and north-south
wind speeds at every grid point) is interpolated onto a constant altitude grid
so that the post-processing radiative transfer can consistently implement
line-of-sight ray striking that calculates intensity and then transit depths
at each wavelength. During this process, winds from the GCM and the planet’s
bulk rotation are incorporated via Doppler shifts in the local opacities.
Stellar limb darkening effects are accounted for, meaning that as the planet
progresses through transit, the projected stellar flux illuminating each
region of the planet’s atmosphere is adjusted based on the limb-darkening
coefficients found in Ehrenreich et al. (2020).

####  2.2.1 Calculated Transmission Spectra

We calculate high resolution transmission spectra from our three models for
three different wavelength ranges, each with a different opacity source of
interest:

  * •

Wavelength 1: 0.379-0.789 $\mu$m; R=400,000; Opacity source: Fe

  * •

Wavelength 2: 1.135-1.355 $\mu$m; R=125,000; Opacity source: H2O

  * •

Wavelength 3: 2.3-2.35 $\mu$m; R=125,000; Opacity source: CO

The latter two wavelength ranges match the work done in Beltz et al. (2022a).
The opacity sources of particular interest are noted above, but both sets
contain opacity from the following six species: CO, H2O, TiO, VO, K, and Na.
Relative abundances of these species were calculated assuming solar-abundance
(Lodders, 2003) equilibrium models with FastChem (Stock et al., 2018, 2022).
Wavelength range 2 covers a range accessible by multiple high-resolution
spectrographs including WINERED (Ikeda et al., 2016) and CARMENES (Quirrenbach
et al., 2014). Wavelength range 3 overlaps with the IGRINS instrument (Park et
al., 2014). Both of these wavelength ranges are probed by the CRIRES+
instrument (Follert et al., 2014). The first wavelength range matches the
observations of this planet taken by the ESPRESSO spectrograph, first
published in Ehrenreich et al. (2020). This set is also unique in that the
only included source of opacity is Fe.

The choice of opacity sources is motivated by theory and observational
results. To start, each of these three species are expected to absorb strongly
in their corresponding wavelength range (Kurucz, 1995; Rothman et al., 2010;
Polyansky et al., 2018; Stock et al., 2018). Fe was chosen to allow a direct
comparison to the data presented in Ehrenreich et al. (2020). We chose CO due
to its expected near uniform abundance (as shown in Figure 1 of Beltz et al.,
2022a). Additionally, recent work from Savel et al. (2023) suggests CO
represents an ideal tracer molecule for UHJ atmospheres, given this expected
uniformity in abundance in these atmospheres. Finally, we chose to examine H2O
due to its lack of uniformity in abundance (Parmentier et al., 2018). The
daysides of UHJ are hot enough to disassociate water, reducing its abundance
on the hotter limb. The net Doppler shifts resulting from this dissociation
provides an interesting comparison to those from the CO spectra. All spectra
were calculated assuming local thermochemical equilibrium and solar
abundances. Recent work from Gandhi et al. (2022) suggests a metallicity for
this planet slightly higher than solar, but consistent with solar within error
bars presented.

####  2.2.2 Spatial Distribution of Opacity Sources

The wavelength regimes that we produced spectra for were chosen partly because
of the differing main opacity source. Given the extreme temperature contrasts
of the planet, our opacity sources are not necessarily uniformly distributed
around the planet. We will briefly touch on the spatial distribution of the
main absorbers for each wavelength here.

  * •

Wavelength 1, Fe: Fe is expected to have a non-uniform abundance distribution
in the atmosphere. For cooler regions of the planet, Fe is expected to
condense, potentially into optically thick clouds. 111Notably, these models
were ran without active clouds, so Fe condensation is applied in the radiative
transfer post-processing. Work from Savel et al. (2022) suggests there would
be more Fe on the eastern limb of the planet. Additionally, Wardenier et al.
(2021) found a lack of gaseous iron on the western limb allows for the signal
from Ehrenreich et al. (2020) to be reproduced.

  * •

Wavelength 2, H2O: Given that the dayside is hot enough to thermally
disassociate water, which is accounted for in the radiative transfer, the
abundance of water between the morning and evening terminators differs by
roughly 3 orders of magnitude for the 0 G model, but less than one order of
magnitude for the other models.

  * •

Wavelength 3, CO: Given the extremely strong triple bond of this molecule,
even UHJ atmospheres will not dissociate this species (Parmentier et al.,
2018; Savel et al., 2023). Additionally, the night side is warm enough such
that CO is not expected to convert into methane. Thus, its global distribution
is essentially uniform the planet.

##  3 Results

We begin our analysis by first searching for differences in the spectra by
eye. In Figure 3 we show the calculated transmission spectra at mid-transit
(phase=0) for all three models from a subset of wavelength range 3 (2.3-2.35
$\mu$m) for versions of the spectra calculated with and without Doppler
effects from winds and rotation. The spectra without Doppler effects are
difficult to differentiate by eye as the differences between the spectra are
on order of $0.5\%$, but these small differences are a result of differing
temperature structures between the models. We more clearly see the differences
between the models in the broadened spectra, as these models have unique upper
atmosphere wind structures due to the different types of drag applied to each
model.

Figure 3: Mid transit (phase=0) spectra from our three models. The solid lines
show spectra that have been shaped by Doppler shifts due to winds and rotation
while the dotted curves do not have this influence. Vertical offsets have been
added for clarity. Differences between the spectral features are due to the
different temperature and wind patterns of the models. All three models have
similar vertical temperature structures, and so the spectra without Doppler
effects are only very subtly different, while the different wind patterns
between the models result in noticeable differences in the resulting spectra.

Because WASP-76b rotates significantly during transit, its spectra will vary
as different parts of the atmosphere come into view (Gandhi et al., 2022;
Wardenier et al., 2022). In Figure 4, we show transmission spectra produced
from the 3 G model, where Doppler shifts and stellar limb darkening have been
applied. Since the spectra are evenly spaced in phase from mid-transit
(phase=0), the spectra appear in “pairs” where spectra sharing the same
absolute offset from transit have similar continuum levels. These pairs are
not identical though; differences in line center (due to differing wind
patterns) and absorption strength (due to differing temperature structure)
exist. It is also noticeable that during the second half of transit, lines
become more blueshifted as the more spatially extended side of the planet
increasingly dominates the back-illuminated part of the planet’s atmosphere.
Similar trends were found in Savel et al. (2022) and Wardenier et al. (2021),
in line with the trend presented in Ehrenreich et al. (2020). Thus, the 3D
geometry of the model is making a noticeable difference in the resulting high
resolution transmission spectra.

Figure 4: Simulated high resolution transmission spectra from our 3 G model,
with Doppler effects from winds and rotation, shown at equally spaced times
throughout transit. As expected, mid transit (phase=0) has the strongest
absorption since the maximum amount of light is obscured by the planet’s
atmosphere at this phase. The first and last phases shown here are partial
transits, which is why their continuum values are lower compared to the other
spectra shown. We can compare spectra that are equally spaced in time before
and after mid-transit to identify differences due to east-west asymmetries
around the terminator. While the phases nearest to mid-transit are very
similar, those phases further away show larger differences, with the spectra
near the end of transit (where only a portion of the planet is transiting the
host star) showing very sharply blue-shifted lines.  Figure 5:  Cross
correlation curves for each model throughout transit (with the first and last
points being partial ingress and egress respectively) for wavelength range 1
(1.135-1.355 $\mu$m). The peak of the cross correlation curve, corresponding
to the net Doppler shift of the spectrum are shown with colored points. These
net Doppler shifts vary with time and differ between models due to the
differing circulation patterns. Notably, for this wavelength range, the 3 G
spectrum become less blueshifted for a time near the end of transit. This is a
unique feature of the 3 G model and is a result of the differing wind
structure caused by the active magnetic drag prescription.

A tool often used in high-resolution spectral analysis is that of cross-
correlation between the data and a template spectrum (in velocity space),
which we can use to combine the information from all of the lines in a
spectrum. If the Doppler-on version of a spectrum is cross correlated with the
corresponding Doppler-off version of the same spectrum, one can determine the
net Doppler shift at that phase by finding the corresponding velocity of the
peak of the cross correlation function, as shown by the points on the curves
in Figure 5, calculated for wavelength range 2 (1.135-1.355 $\mu$m). Broadly,
our models show a changing net Doppler shift becoming more blueshifted with
time, due to the increasing contribution from the more extended, hotter
eastern limb, whose motions from winds and rotation are oriented toward the
observer during transit. Interestingly, the 3 G models are an exception to
this for a short time after mid-transit where the net Doppler shift becomes
less blueshifted for a brief time before becoming more blueshifted by the end
of transit. It is also relevant to note that of all the cross-correlation
curves presented in this Figure, the 3 G curves are the most broadened and
least peaked, particularly near mid transit. The broadness of these curves can
be attributed to the dual existence of strong blueshifted and redshifted winds
in the upper atmosphere.

Figure 6: Net Doppler shifts for the simulated spectra from each model, over
the three wavelength regimes considered, as well as the data from Ehrenreich
et al. (2020). Overall, the net behavior shows the spectra becoming more
blueshifted as transit proceeds. However, the 3 G active drag model shows some
deviations from this shortly after mid-transit, but only when water is the
dominant absorber within the wavelength range considered. When iron is the
dominant absorber, we see instead that the starting net redshift persists
longer into transit, before switching to a net blueshift. Both of these
behaviors are unique to the 3 G model, distinguishing it from the 0 G or
Uniform drag models.

Figure 6 shows the net Doppler shifts for all the models considered in this
work at each wavelength range. Each model exhibits unique trends but overall,
the spectra become more blueshifted throughout transit. The Uniform drag model
consistently shows the strongest blueshifts at each phase examined. Similarly,
the 0 G model wavelength becomes more blueshifted throughout transit for each
wavelength range. The 3 G model shows interesting structure in each wavelength
regime, with the spectra covering the near-IR becoming slightly less
blueshifted right after mid-transit.

The differences in these net Doppler shifts between models can be attributed
to a variety of physical effects. First, the underlying velocity structure of
the upper atmosphere differs between each model. For example, in Beltz et al.
(2022b), we saw significantly different dayside wind structures for the active
drag model, with the dayside winds traveling up and over the poles in the
North-South direction. The influence of these differing flow patterns can be
seen in Figure 2. Additionally, differences in relative scale heights due to
atmospheric temperature differences will affect the net Doppler shifts
(Wardenier et al., 2022; Savel et al., 2023). Figure 1 shows this, with the
hottest atmospheric regions having the largest vertical extent.

The spatial extent of the dominant opacity source (determined by the
wavelength) will influence the net Doppler shift (Savel et al., 2022). We can
start by examining the wavelength range containing CO, as this species is
fairly uniform in abundance around the planet. The 0 G and Uniform drag models
show similar behavior for this wavelength range, but with the Uniform drag
having the strongest net blueshift. We can attribute this to the weakness of
the redshifted region on the western limb, which is stronger for both of the
other models. For the 3 G case, we see that the net Doppler shift is roughly
constant, aside from the first and last phases calculated. This means that
although more blueshifted regions are coming into view during transit, this
effect is roughly equaled out by the redshifted winds on the western limb and
the different scale heights associated with each limb.

Water on the other hand is certainly not uniformly distributed around the
planet. This is most easily seen in the case of the wavelength range 2 for the
3 G model, which actually becomes less blueshifted with time for part of
egress. Water will dissociate in hot temperatures, so it is less abundant in
blueshifted limb. However, as transit progresses, we see a water-depleted blue
limb and a relatively water-rich red limb, resulting in a brief period during
transit where the spectra becomes more redshifted. Since this effect is not
seen to the same degree in the other two wavelength ranges tested, one can
infer the feature is influenced by the dominant absorber, water. Although
temperature inhomogeneities between the limbs exist for all models presented,
the 3 G model has the strongest day-night temperature contrast and most
dominant red-shifted atmospheric winds. This particular combination of
atmospheric structures results in the behavior seen in Figure 6. Neither the
Uniform or 0 G models show this behavior, indicating that these net Doppler
shifts may be a way of testing approximations of active drag.

Fe abundance is slightly more temperature dependent than CO, but not nearly to
the same level as water. While water abundances can very by over 6 orders of
magnitude from the dayside to the nightside of this planet, Fe abundances only
change by less than a single order of magnitude, and is slightly more
abundance in the cooler regions of the planet.

For wavelength range 1 where Fe is the dominant opacity source, , we can also
make a direct comparison to the Doppler shifts measured in Ehrenreich et al.
(2020). Although the data displays stronger magnitudes of blueshifted values
than our models predict, the magnitude of this shift can be altered strongly
by small changes in orbital parameters (Savel et al., 2022), leading us to
instead focus on comparing the velocity trends with orbital phase between the
models and the data. The 3 G model does the best job of reproducing the trend
found in the data. Both experience a strong negative slope in Doppler shift
shortly before mid transit and roughly constant Doppler shifts throughout the
rest of transit. The Uniform and 0 G model have a roughly constant slope which
does not match the data as well. Thus, out of the different drag prescriptions
tested, our active drag model best matched the trend presented in Ehrenreich
et al. (2020). Notably, other GCM work has struggled to reproduce this
trend—particularly the“bottoming out” or “kink” behavior that occurs after mid
transit. Wardenier et al. (2021) removed iron from the leading limb of their
atmosphere to reproduce this “kink” while Savel et al. (2022) used optically
thick clouds and a slight non-zero eccentricity to best fit the data. However,
the models from these two works also incorporated uniform drag in their
atmospheres. Thus, this interesting behavior in the Ehrenreich et al. (2020)
dataset may be a result of some combination of magnetic effects, clouds, or Fe
condensation, however we refrain from making more detailed predictions until a
model with both of these effects in concert is presented, which we leave for
future work.

##  4 Discussion

While we have presented particular features in transmission spectra of UHJs
that could be used to assess drag mechanisms within the planet’s atmosphere,
it is important to recognize the necessary limitations of our modeling and any
potential impact this could have on our results.

One caveat to this work is that due to numerical stability purposes, the top
boundary of our model is $\sim 10^{-5}$ bars. Compared to emission
observations, transmission spectra probe a higher region of the atmosphere,
potentially at lower pressure values than what is contained in our model.
These regions are less dense with potentially stronger wind speeds. This upper
boundary could be contributing to why our net Doppler shifts are not as large
in value as those reported in Ehrenreich et al. (2020), although other GCMs
similarly struggle to produce such large shifts (Wardenier et al., 2021; Savel
et al., 2022, despite these GCMs covering nearly 2 orders of magnitude more in
pressure space).

Additionally, the GCMs studied in this work use a double-gray radiative
transfer scheme instead of a more complex picket-fence or correlated-k method.
A downside of the double-gray method is that it results in more isothermal
upper atmospheres than the other radiative transfer schemes mentioned (Lee et
al., 2021). This effect is minimized on transmission spectra, which is less
sensitive to temperature structure than emission spectra. We also note we
chose only one set of infrared and optical coefficients (Fu et al., 2021,
chosen to most closely match observations presented in ). Different choices in
these coefficients would lead to slightly different temperature profiles, but
this exploration is beyond the scope of this work. Future work will compare
the impact of using spectra generated from double gray and picket fence GCMs
to determine how robust the patterns identified here are along different
radiative transfer schemes and planet parameters.

A physical process absent from these models are clouds. Clouds should reduce
the depth of spectral features and flatten the resulting spectra.
Additionally, clouds could potentially sculpt the Doppler fields calculated by
blocking out particular regions and create the ”bottoming out” trend seen
after mid transit in Ehrenreich et al. (2020), as discussed in Savel et al.
(2022). For a planet of this temperature, one could potentially find some
clouds in the nightside upper latitudes (Roman et al., 2021), so we may
initially not expect them to have any strong signatures in the transmission
spectra. However, those cloudy models were run in the absence of magnetic
effects; when the advection of hot gas to the nightside is reduced, we may
expect a colder nightside and therefore more cloud formation. Additionally,
work from Helling et al. (2021) suggests that cloud opacity at the morning
terminator and ionic or atomic opacity sources at the evening terminator may
influence the resulting transmission spectra for UHJs. However, we leave the
interplay between magnetic drag and cloud physics for later work.

Our active magnetic drag 3 G model also makes simplifying assumptions
regarding the magnetic field of the planet. (For a detailed explanation of
model assumptions, see Beltz et al., 2022b). The most relevant of these
assumptions to this work is that any magnetic field induced in the atmosphere
is smaller than the global magnetic field. Mathematically, this results in our
prescription being most effective when the magnetic Reynolds number, $R_{m}$,
is $<1$. This holds true for the vast majority of the planet’s atmosphere, but
there is a small region in the dayside upper atmosphere where the values of
$R_{m}$ reach unity or slightly above. However, given that the dayside is
never fully in view during transit, this small region of the atmosphere is
likely not very influential in the transmission spectra presented here, but
would only influence them secondarily through any change in the day-night
circulation.

###  4.1 Combining Observations for Detecting Magnetic Effects

In this work, we have identified trends in high resolution transmission
spectroscopy of planets in the magnetic circulation regime. We perhaps see
this trend in the Ehrenreich et al. (2020) data, but to more reliably convince
ourselves that this planet (or any other planet) is operating within the
magnetic circulation regime, we can combine this trend with others described
in Beltz et al. (2022b) and Beltz et al. (2022a), therefore allowing our
conclusion to become more robust. Combining three independent observations
(phase curves, high resolution emission, and high resolution transmission
spectroscopy) offers a chance to more conclusively identify planets that are
strongly influenced by magnetic effects. We summarize these trends below:

  * •

High Resolution Transmission Spectra: In this work, we found that for
magnetically active models, the net Doppler shift showed less overall
blueshifting throughout transit and, depending on the wavelength, could become
more redshifted during parts of transit. Neither behavior was shown by the
drag free or uniform drag models.

  * •

High Resolution Emission Spectra: The magnetic circulation regime influences
the net Doppler shift as a function of phase, especially around secondary
eclipse for high resolution emission spectra. Our work in Beltz et al. (2022a)
found that our active drag shows a unique trend in net Doppler shift compared
to the ones found in the drag free and uniform models near secondary eclipse
(see Figure 7 in that paper).

  * •

Phasecurves:  Our work in Beltz et al. (2022b) found that increasing our
magnetic drag strength resulted in a decrease in hotspot offset and an
increase in day-night temperature contrast.

This set of three papers and the trends discussed within can act as roadmap
for finding exoplanet atmospheres influenced by magnetic effects.

##  5 Conclusion

In this work, we post-processed three different models of the UHJ WASP-76b
with varying forms of magnetic drag treatment to generate high resolution
transmission spectra for three different wavelength regimes. The main results
of this work are as follows:

  * •

3D effects of both varying temperature and wind structure are present in this
high-resolution transmission spectra and alter the line shape and depth of
various features, offering an avenue for assessing sources of drag or magnetic
effects within the atmosphere.

  * •

While transmission spectra from all models generally show increasingly
blueshifted net Doppler shifts as transit progresses, the specific patterns
and magnitudes depend on the model and wavelength range (and the dominant
source of opacity) considered.

  * •

The 3 G model shows the largest differences in Doppler shifts from the other
models, beginning with the strongest net redshift of any model as well as
actually becoming less blueshifted from phase 0-0.02 for the spectra generated
at 1.135-1.355 $\mu$m. This is due to the model possessing the strongest
redshifted line of sight velocities during transit, as seen in Figure 2, and
may provide a unique way to constrain the role of magnetism within UHJ
atmospheres.

  * •

Our 3 G model was best able to match the Doppler shift trends in the data
presented by Ehrenreich et al. (2020) including the “bottoming out” behavior
during the second half of transit, which only appeared in the kinematic MHD
models. Thus, magnetic effects may help explain this particular dataset.

High resolution spectroscopy has opened the door to planetary atmospheric
characterization at an unprecedented level, uniquely probing physical
processes which were previously unobservable. In order to extract the most
meaningful, unbiased conclusions from this data, high complexity atmospheric
models and sophisticated post-processing routines are needed in order to
account for 3D gradients in temperature, winds, and chemical composition.
UHJs, due to their favorable signal to noise ratio, remain the best planetary
target for investigating analysis techniques for this type of data. However,
these planets have the largest spatial gradients and, due to their high
temperatures, must have partially ionized atmospheres. It is therefore
necessary to consider how magnetic effects may shape the spectra of these
planets and, in turn, how those spectra can give us insight into the physical
states of the atmospheres.

##  6 Acknowledgments

This work was generously supported by a grant from the Heising-Simons
Foundation. Many of the calculations in this paper made use of the Great Lakes
High Performance Computing Cluster maintained by the University of Michigan.
We thank the reviewer for their comments, which improved the quality of this
manuscript.

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