# Quantifying the Transit Light Source Effect: Measurements of Spot
Temperature and Coverage on the Photosphere of AU Microscopii with High-
Resolution Spectroscopy and Multi-Color Photometry

William C. Waalkes NSF Graduate Research Fellow  Department of Astrophysical &
Planetary Sciences, University of Colorado Boulder, 2000 Colorado Ave,
Boulder, CO 80309, USA  Zachory K. Berta-Thompson Department of Astrophysical
& Planetary Sciences, University of Colorado Boulder, 2000 Colorado Ave,
Boulder, CO 80309, USA  Elisabeth Newton Department of Physics and Astronomy,
Dartmouth College, Hanover NH 03755, USA  Andrew W. Mann Department of Physics
and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill,
NC 27599-3255, USA  Peter Gao Earth and Planets Laboratory, Carnegie Institute
of Washington, Washington DC, USA  Hannah R. Wakeford School of Physics,
University of Bristol, Bristol, UK  Lili Alderson School of Physics,
University of Bristol, Bristol, UK  Peter Plavchan Department of Physics and
Astronomy, George Mason University, Fairfax, VA 22030, USA

###### Abstract

AU Mic is an active 24$\pm$3 Myr pre-main sequence M dwarf in the stellar
neighborhood (d=9.7 pc) with a rotation period of 4.86 days. The two
transiting planets orbiting AU Mic, AU Mic b and c, are warm sub-Neptunes on
8.5 and 18.9-day periods and are targets of interest for atmospheric
observations of young planets. Here we study AU Mic’s unocculted starspots
using ground-based photometry and spectra in order to complement current and
future transmission spectroscopy of its planets. We gathered multi-color LCO
0.4m SBIG photometry to study the star’s rotational modulations and LCO NRES
high-resolution spectra to measure the different spectral components within
the integrated spectrum of the star, parameterized by 3 spectral components
and their coverage fractions. We find AU Mic’s surface has at least 2 spectral
components, a $\rm{T_{amb}}$$=$$4003^{+15}_{-14}$ K ambient photosphere with
cool spots that have a temperature of $\rm{T_{spot}}$$=$$3003^{+63}_{-71}$ K
covering a globally-averaged area of $39\pm 4\%$ which increases and decreases
by $5.1\pm 0.3\%$ from the average throughout a rotation. We also detect a
third flux component with a filling factor less than $0.5\%$ and a largely
uncertain temperature between 8500-10000K that we attribute to flare flux not
entirely omitted when time-averaging the spectra. We include measurements of
spot characteristics using a 2-temperature model, which we find agree strongly
with the 3-temperature results. Our expanded use of various techniques to
study starspots will help us better understand this system and may have
applications for interpreting the transmission spectra for exoplanets
transiting stars of a wide range of activity levels.

##  1 Introduction

Observations from JWST are now revealing exoplanet atmospheres in more detail
than ever before (Ahrer et al., 2023; Alderson et al., 2023; Feinstein et al.,
2023; Rustamkulov et al., 2023; Fu et al., 2022) using an observational
technique called transmission spectroscopy (Seager & Sasselov, 2000; Brown et
al., 2001; Pont et al., 2007; Berta et al., 2012; Sing et al., 2011).
Transmission spectroscopy is done by measuring the transit depth (which is a
proxy for the planet’s radius) of an exoplanet as a function of wavelength and
inferring atmospheric absorption (e.g., Seager & Sasselov, 2000) and/or
scattering (e.g., Robinson et al., 2014; Sing et al., 2016) at wavelengths
where the planet’s transit is deeper. The stellar photons that are absorbed by
the planet and its atmosphere originate specifically from the transit chord,
the swathe of stellar surface occulted by the planet which in general is
indistinguishable from the surrounding photosphere except in compact systems
which exhibit transits at multiple latitudes. Stellar surfaces can be
homogeneous (i.e., spatially “smooth” aside from granulation and limb-
darkening effects), in which case the chord spectrum is the same as the disk-
integrated spectrum, or they can be heterogeneous (containing active regions),
in which case the transit chord is not necessarily representative of the disk-
integrated stellar spectrum.

A homogeneous stellar background surface has typically been adopted in
transmission studies. While this assumption holds true in some cases, most
stars do not have smooth, single-temperature surfaces but are instead spotted
with activity-induced heterogeneities. Spots are created where magnetic field
lines pass through the photosphere and the magnetic pressure overwhelms the
local gas pressure, suspending convection and causing the region within the
intersecting field to cool. Faculae arise from weaker concentrations of field
lines where this pressure isn’t enough to suspend convection but is enough to
reduce the local opacity and increase the flux emanating from deeper in the
photosphere, creating a brightening effect (Basri, 2021).

A non-axisymmetric distribution of spots and faculae creates time- and
wavelength-dependent changes in the stellar surface flux, which has been
observed in high-resolution stellar spectra (Wing et al., 1967; Afram &
Berdyugina, 2015), color-magnitude relations (Vogt, 1979; Olah et al., 1997),
stellar rotational modulation (Vogt, 1979; Pass et al., 2023), and more
recently in exoplanet transits (Brown et al., 2001; Pont et al., 2008; Sing et
al., 2011; Sanchis-Ojeda & Winn, 2011). When spots or faculae lie on the
transit chord and are occulted by transiting planets, they create bumps or
dips in the transit light curve that can bias the exoplanet radius
measurement. Occulted active regions, provided they occupy discrete regions of
the transit chord and their flux contrast appears above the noise, show up
directly in transit light curves and can be identified and removed from the
measured transmission spectrum. Unocculted active regions, however, alter the
disk-averaged spectrum such that it is no longer representative of the true
source spectrum of photons entering the planetary atmosphere. This in turn
creates spurious $\lambda$-dependent changes to the exoplanet transit depth in
what is now known as the transit light source effect (TLSE, Rackham et al.,
2018, 2019).

On cool stellar surfaces (below about 4000 K), molecular absorption lines
(like H2O, VO, and TiO, Jones et al., 1995; Allard et al., 2012) begin to
appear in the stellar spectrum and become entangled with molecular absorption
signals in planetary atmospheres. Cool unocculted spots with different or
deeper molecular absorption lines than the surrounding photosphere will appear
to add molecular absorption at those wavelengths in planetary transmission
spectra and lead to mischaracterization of exoplanets and their atmospheres.
Additionally, unocculted spots give rise to an increasing transit depth toward
bluer wavelengths as their contrast against the surrounding photosphere
increases, which can be mistakenly identified as Rayleigh scattering in a
transiting exoplanet’s atmosphere (e.g., Robinson et al., 2014). Until we can
precisely and reliably determine spot characteristics on our host stars, the
signature of exoplanet atmospheres will be very challenging or impossible to
disentangle from spot contamination for nearly all transmission observations
of exoplanets M and K dwarfs. This degeneracy, exemplified in recent
transmission observations of sub-Neptune exoplanets TOI 270d (Mikal-Evans et
al., 2023), Gl 486b (Moran et al., 2023), K2-33b (Thao et al., 2023), L 98-59
c (Barclay et al., 2023), and temperate terrestrial exoplanet TRAPPIST-1b (Lim
et al., 2023) is what we aim to mitigate for transmission observations of AU
Mic b by precisely measuring spot characteristics for its host star in this
work.

Quantity | Value
---|---
AU Mic |
D [pc] | $9.714\pm 0.002^{\rm a}$
Teff [K] | $3665\pm 31^{\rm b}$
M∗ [M☉] | $0.60\pm 0.03^{\rm c}$
R∗ [R☉] | $0.82\pm 0.05^{\rm c}$
Prot [days] | $4.86\pm 0.005^{\rm c,d}$
Metallicity [dex] | $0.12\pm 0.10^{\rm c}$
log$g$ [log10(cm/s2)] | $4.52\pm 0.05^{\rm c}$
AU Mic b |
P [days] | 8.4631427e
Rp/R∗ | 0.0433$\pm$0.0017e
Table 1: System parameters relevant to this study. a: Gaia Collaboration et
al. (2023), b: Cristofari et al. (2023), c: Donati et al. (2023), d: Martioli
et al. (2021), e: Szabó et al. (2022).

##### AU Microscopii

AU Mic (Torres & Ferraz Mello, 1973) is a nearby (9.7 pc; Gaia Collaboration
et al., 2023), young ($24\pm 3$ Myr; Mamajek & Bell, 2014), rapidly rotating
(P${}_{\rm{rot}}=4.86$ d; Plavchan et al., 2020; Donati et al., 2023) pre-main
sequence M dwarf with a debris disk (Kalas et al., 2004; Chen et al., 2005;
MacGregor et al., 2013). This star has an inflated radius as it settles onto
the main sequence, with M$=0.60$ M⊙ and R$=0.82$ R⊙ (Donati et al., 2023) and
an effective temperature of 3600-3700K (e.g., Afram & Berdyugina, 2019;
Plavchan et al., 2020; Cristofari et al., 2023) There are two transiting warm
Neptunes on 8.46 and 18.86-day periods (Hirano et al., 2020; Plavchan et al.,
2020; Martioli et al., 2021; Gilbert et al., 2022; Zicher et al., 2022) and 2
unconfirmed candidate non-transiting planets on 12.74 and 33.39-day periods
recently discovered through transit timing variations and radial velocity
analysis (Wittrock et al., 2022, 2023; Donati et al., 2023). The existence of
an observable debris disk with interior transiting planets is a rare and
exciting architecture that holds vast scientific potential.

Furthermore, this system is one of the best cases we have for studying star-
planet interactions and the effects of young M dwarf activity on planetary
atmospheres, an issue of great interest and concern in the search for
terrestrial atmospheres and potentially habitable planets orbiting M dwarf
stars (e.g., Shields et al., 2016; Louca et al., 2023). This star’s frequent
high-energy flaring that may eventually lead to photo-evaporation of the
atmospheres of AU Mic b and c (Feinstein et al., 2022) and even in the case
where atmospheres are preserved, the long term implications of young M dwarf
activity on planetary habitability are ominous. By continuing to study AU Mic
and its planets, we can build an internally consistent understanding of a
nearby multi-planet system in the early stages of formation with a stellar
surface evolving on months-long timescales (e.g., Donati et al., 2023). System
parameters are summarized in Table 1 and a thorough review of AU Mic’s stellar
and planetary characteristics can be found in Donati et al. (2023).

##### Transmission Observations of AU Mic b

The observations we present in this work are part of a companion study
alongside HST/WFC3 transmission spectra of AU Mic b; the first on 2021 Aug 30
(BJD 2459455.98) and the second on 2022 Apr 14 (BJD 2459684.40). Because AU
Mic’s spots evolve noticeably over time (e.g., Szabó et al., 2021, 2022;
Gilbert et al., 2022), we have collected photometric and spectroscopic
observations contemporaneous with the WFC3 observations to provide constraints
on spot contamination at the time of both transmission visits. This
transmission spectrum will be analyzed in the context of our results and
presented in a subsequent paper.

##### Objectives and Layout

In order to characterize AU Mic’s spots and forward-model the spot
contamination level in AU Mic b’s transmission spectrum, we assembled a self-
consistent statistical framework that combines multi-color time-series
photometry with contemporaneous high-resolution spectroscopy. This method
allows us to break observational degeneracies between spot coverage and spot
contrast and better understand the bulk characteristics of AU Mic’s spots. We
use bulk to mean the spatially-unresolved flux-weighted characteristics based
on the treatment of spots as discrete regions without complex temperature
profiles, ignoring for example the distinction between umbra and penumbra.
These are heavy assumptions, but we argue that our models are appropriate for
the quality of our data and the information we hope to obtain.

This paper is laid out as follows: in Section 2 we describe the types of
observations used in spot analysis and the specific observations we acquired,
along with the data reduction and processing. In Section 3 we describe our
methods of analyzing AU Mic’s rotational modulations (3.1), assembling the
self-consistent statistical framework for modeling spot filling factor and
temperature (3.2), and forward-modeling the spot contamination in AU Mic b’s
transmission spectrum (3.3). In Section 4 we report the results of our spot
model and in Section 5 we discuss the physical implications of our results,
how they compare to previous studies, and the limitations of our approach.

Figure 1: Processed and baseline-corrected photometry data for all filters in
visits F21 (top) and S22 (bottom) with vertical markers showing the temporal
distribution of WFC3 (red) and NRES (green) observations. The color of each
photometric data point is based on the LCO site where that data was observed.
The time between transit observations is 229 days.

##  2 Observations and Data Reduction

##### Observing Spots

Starspot characteristics are difficult to disentangle in practice as there is
a degeneracy between spot coverage and temperature contrast that creates
similar observational effects within a single waveband. In addition, the
physics and structure of stellar surfaces is poorly understood for all but the
most heavily studied stars. Stars of different type, rotation rate, and
magnetic field strength exhibit differing forms of surface phenomena which are
or will eventually be relevant to understand for the future of exoplanet
discovery and characterization.

Breaking spot degeneracies can be done by combining observations across
optical and infrared wavelengths. Short-wavelength broadband photometry allows
us to probe starspots where they stand out the most against the stellar
background (higher flux contrast toward the Wien limit), while long-wavelength
observations are useful for identifying molecular characteristics of starspots
where they overlap with planetary atmospheric absorption. Broadband
photometric variability measurements help us probe different temperature
components on rotating stars, but generally only provides a lower limit on
total spot coverage due to unknown axisymmetries in spot distribution (Apai et
al., 2018). Photometric variability amplitudes decrease with wavelength as the
two flux components approach the Rayleigh-Jeans limit, so measuring rotational
variability across the optical to the infrared provides strong constraints on
the spot-to-photosphere temperature contrast (e.g., Strassmeier & Olah, 1992).

High-resolution, time-series spectra have been used to study the relationships
between stellar activity tracers and spot coverage (e.g., Schöfer et al.,
2019; Medina et al., 2022). For spectroscopic studies, stellar spectra are
modeled as a combination of two or more temperature components (often referred
to as spectral decomposition, e.g., Gully-Santiago et al., 2017; Zhang et al.,
2018; Wakeford et al., 2019). Specific molecular lines are often used as spot
tracers including TiO (Wing et al., 1967; Vogt, 1979), CaH, MgH, FeH, and CrH
(e.g., Neff et al., 1995; Afram & Berdyugina, 2019). We recommend Berdyugina
(2005); Apai et al. (2018); Rackham et al. (2018) for a more thorough review
of starspots and the techniques used to study them.

###  2.1 Data

We acquired Las Cumbres Observatory (LCO) 0.4m $g^{\prime}$-, $r^{\prime}$-,
and $i^{\prime}$-band111https://lco.global/observatory/instruments/filters/
SBIG photometry and Network of Robotic Echelle Spectrographs (NRES) high-
resolution echelle spectra on two separate visits spanning 2-3 weeks around
their respective transmission observations (occurring in Fall 2021 and Spring
2022, hereafter F21 and S22). These data were acquired contemporaneously with
observations of AU Mic b’s HST/WFC3 transmission spectrum (see Figure 1) with
the hope of precisely constraining the magnitude of spot contamination at the
appropriate stellar epoch and phase.

Order |  $\lambda$ ($\mu$m) | Note
---|---|---
53∗ | 0.873-0.888 | TiO line [8860 Å]
54 | 0.858-0.872 |
55 | 0.842-0.857 | Excluded - Telluric Contamination
56 | 0.826-0.841 | Excluded - Telluric Contamination
57 | 0.810-0.826 | Excluded - Telluric Contamination
58 | 0.798-0.812 | Excluded - Telluric Contamination
59∗ | 0.784-0.798 |
60 | 0.771-0.784 | Excluded - Poor Fit
61∗ | 0.759-0.772 | TiO line [7600 Å]
62 | 0.746-0.760 | Excluded - Poor Fit
63 | 0.734-0.748 | Excluded - Telluric Contamination
64 | 0.723-0.736 | Excluded - Telluric Contamination
65 | 0.712-0.724 | Excluded - Telluric Contamination — TiO line [7150 Å]
66 | 0.701-0.713 | Excluded - Telluric Contamination — TiO line [7050 Å]
67 | 0.690-0.702 | Excluded - Telluric Contamination
68 | 0.680-0.692 | Excluded - Telluric Contamination
69∗ | 0.671-0.682 |
70 | 0.661-0.672 | Excluded - Poor Fit
71∗ | 0.652-0.663 | H-$\alpha$ band
72∗ | 0.642-0.653 |
73∗ | 0.634-0.645 | Excluded - Poor Fit
74 | 0.625-0.636 | Excluded - Telluric Contamination
75∗ | 0.617-0.627 |
76∗ | 0.609-0.619 |
77 | 0.601-0.611 | Excluded - Poor Fit
78 | 0.593-0.603 | Excluded - Telluric Contamination
79 | 0.586-0.596 | Excluded - Telluric Contamination
80 | 0.578-0.588 | Excluded - Poor Fit
81∗ | 0.571-0.581 |
82∗ | 0.564-0.574 |
83∗ | 0.558-0.567 |
Table 2: Details on the NRES Echelle spectra acquired for this study. The full
spectrum spans 0.39-0.91 $\mu$m (orders 119-52) but we truncate the table and
the analysis at orders 53 and 83 to focus on orders which are not dominated by
noise. Orders that we omit from the final analysis are noted with a brief
explanation, and further discussion of modeling specific orders is provided in
the Appendix. Most omitted orders were heavily contaminated by telluric
absorption, whereas the orders labeled “Poor Fit” typically exhibit extremely
cold spots, at the limit of the spectral library. Note that wavelength
decreases with order.

####  2.1.1 SBIG Imaging

Photometry was acquired between 2021 Aug 12-2021 Sept 03 (F21) and 2022 Apr
01-2022 Apr 27 (S22) using 5 separate telescopes automatically scheduled
depending on weather, telescope availability, and target observability.
Typical exposure times were 20s in $g^{\prime}$, 10s in $r^{\prime}$, and 6s
in $i^{\prime}$. Photometric data were automatically reduced into calibrated
images by the BANZAI pipeline (McCully et al., 2022) and downloaded from the
LCO Science Archive222https://archive.lco.global/. We performed aperture
photometry with AstroImageJ’s multi-aperture photometry tool (Collins et al.,
2017). For each filter, the same 3 comparison stars are used to account for
local atmospheric effects throughout the night and measure the target star’s
relative flux. The fact that AU Mic is far brighter than its nearby comparison
stars (g’, r’, and i’ magnitudes of 9.579$\pm$0.05, 8.636$\pm$0.09, and
7.355$\pm$0.14, respectively Zacharias et al., 2012) means that the photon
noise in the photometry is set by the comparison star brightness, rather than
by AU Mic itself. Scintillation also contributes significantly to the
photometric noise budget, particularly in the $i^{\prime}$ photometry where
exposure times are very short to avoid AU Mic saturating. After performing
aperture photometry, S/N per exposure was 300-600 in F21 $g^{\prime}$ and
$r^{\prime}$, 200-300 in F21 $i^{\prime}$, and 500-1000 in S22 $g^{\prime}$
and $r^{\prime}$. For the first visit (F21), we have 450 exposures in SDSS
$g^{\prime}$, 1036 exposures in $r^{\prime}$, and 1175 in $i^{\prime}$. For
the second visit (S22) we have 328 in $g^{\prime}$ and 330 in $r^{\prime}$.

For each night of observations, we stitch together all data in a single filter
and take the median in order to minimize the effect of flares. We use
scipy.optimize to fit an initial sinusoid model for sigma-clipping. We clip
$10-\sigma$ outliers from the initial model to account for flux variations
outside of the rotational modulation (i.e., flares), generate an optimized fit
with scipy.optimize.minimize, clip $5-\sigma$ outliers from that model, re-
optimize a final time, and normalize uncertainties based on the reduced
$\chi^{2}$ statistic from this final optimized fit. The first cutoff is set at
$10-\sigma$ because after median binning, the uncertainties were
underestimated and a slightly wrong initial model could easily exclude
otherwise useful data. The second cutoff is set at $5-\sigma$ to account for
any extreme outliers still remaining without being too restrictive, accounting
for known uncertainties in the chosen model. After processing, the typical
per-night S/N was 70-100 for $g^{\prime}$ and $r^{\prime}$ and 30-50 in
$i^{\prime}$, with a total of 19 data points for F21 $g^{\prime}$, 18 for
$r^{\prime}$, 17 for F21 $i^{\prime}$, 28 data points for S22 $g^{\prime}$,
and 30 for S22 $r^{\prime}$. The per-night S/N only reaches to 100 because we
set a minimum uncertainty of 1% on the post-processing photometry based on the
per-night spread in flux.

Figure 2: Cartoon of a spotted star showing the parameters used in this study.

####  2.1.2 NRES Spectra

AU Mic’s spectrum was observed in the $0.39-0.91\mu$m NRES bandpass with 600s
exposure times resulting in a total of 38 observations from 2021 August
24-2021 September 09 (F21) and 48 observations from 2022 April 02-2022 April
18 (S22). The R=53,000 NRES spectra were reduced by the BANZAI-NRES pipeline
(McCully et al., 2022) and downloaded from the LCO Science Archive. Spectra
from individual observations show typical peak S/N of 35 where the star is
brightest (around order 60) and drops off to below 10 in order 52 and beyond
order 83, so we omit orders outside of this range from our analysis. We
processed the spectra with the chromatic333https://github.com/zkbt/chromatic
tool, first correcting for the velocity shift in each order’s spectra from the
movement of the Earth in different positions of its orbit. This was -9.5km/s
in F21 and 33.0 km/s in S22 derived from a $\chi^{2}$ grid search using a
single-temperature optimized model spectrum and scipy.optimize.minimize. We
median-combine spectra in time to one averaged spectrum per night resulting in
7 spectra in F21 and 17 spectra in S22. We also bin each spectrum to 0.05 nm
(roughly R=11000-18000); this is greater than the Doppler broadening width of
0.015-0.024 nm that we estimate for the NRES bandpass based on AU Mic’s
v$sin$i of $8-9$ km/s (e.g., Donati et al., 2023). Zeeman broadening is an
additional effect which alters line profiles in magnetically active stars
(e.g., Gray, 1984), but we calculate the Zeeman broadening (Reiners et al.,
2013) to be less than rotational broadening at these wavelengths and accounted
for within our chosen bin size. Many of our spectral orders overlap with
absorption bands in Earth’s atmosphere so we trim out any wavelengths where
the molecular line transmission fraction of the atmosphere is $<0.995$ (i.e.,
any wavelength at which $\geq$0.5% of the photons are absorbed) based on time-
averaged telluric data from Skycalc (Noll et al., 2012; Jones et al., 2013).
We run a $\sigma$-clipping routine that first calculates an optimized single-
temperature PHOENIX model (Husser et al., 2013) for each spectrum, and second
clips emission lines, defined as points $>3-\sigma$ above the optimized model.
Uncertainties on the time-averaged spectra are inflated to give a reduced
$\chi^{2}$ of 1 when fit against a 3650K template. This is an increase in
uncertainty of 7-55x depending on the order, resulting in typical per-order
S/N of 10-40.

After processing, we omitted two-thirds of the spectral orders from the final
analysis based on their level of telluric overlap or in some cases because the
ambient or spotted component was very poorly constrained, possibly due to the
spectral model fidelity problem (Iyer & Line, 2020; Rackham & de Wit, 2023).
The orders we include in the final analysis are 53, 54, 59, 61, 69, 71, 72,
75, 76, 81, 82, and 83, with details in Table 2. In F21, there are total of
1831 spectral data points, with 1828 in S22, for a total of 3659 spectral data
points. Discussion on the choice of orders to include and spectral model
results for each individual order are in the Appendix.

##  3 Methods

Our analysis is ordered in three steps:

  1. 1.

Measuring the stellar rotation signal, where photometric data is modeled as a
sine wave to infer the semi-amplitude of stellar variability.

  2. 2.

Modeling spot characteristics, where we infer spot characteristics based on AU
Mic’s $\rm{T_{eff}}$, measured photometric variabilities, and the time-
averaged spectra. This is the primary focus of our analysis.

  3. 3.

Forward modeling the TLSE, where we take the posterior samples from our
modeling to calculate the range of spectral contamination we can expect in the
HST/WFC3 transmission data for AU Mic b.

Figure 2 shows the cartoon stellar surface we model as a combination of
ambient photosphere with a characteristic spectrum S($\lambda$,$\rm{T_{amb}}$)
and spots with characteristic spectra S($\lambda$,$\rm{T_{spot}}$) covering a
globe-averaged $f_{\rm spot}$ which deviates from the average coverage by
$\pm$$\Delta f_{\rm spot}$ throughout the stellar rotation. We include a third
flux component (not shown on the figure) and label this component “hot” (with
attributes $f_{\rm hot}$ and $\rm{T_{hot}}$) as we are uncertain about the
physical source of the measured hot component, if it exists. The primary
results we report in section 4 come from this 3-temperature modeling, but we
also test a 2-temperature model and discuss its results and implications in
section 5. The two visits are modeled with the same set of parameters
(implying no change to the surface components between visits), an assumption
we test and discuss later in the paper.

###  3.1 Measuring Photometric Variability

Once the photometry data were processed as described in Section 2, we measure
the rotational variability with a sinusoidal model with the following form:

$F(t)={A~{}\sin}(2\pi t/P)+{B\cos}(2\pi t/P)+{C}_{k},$ (1)

where $t$ is the time of an individual data point, $P$ is the stellar rotation
period which we keep fixed at 4.86 days, $A$ and $B$ are amplitude parameters,
and $C_{\rm k}$ is the offset parameter unique to each camera in each visit.
We fit each camera’s data separately as we expect different cameras to have
slightly different responses and should be normalized to their separate
average fluxes. While AU Mic has a notably asymmetric light curve (e.g., the
TESS light curves shown in Martioli et al., 2021) and Angus et al. (2018)
caution against using simple sinusoidal models to fit stellar rotation curves,
our goal here is not to infer a precise rotation curve morphology and spot
distribution but to measure the relative amplitude of flux variability between
separate bandpasses.

Using emcee (Foreman-Mackey et al., 2013), we ran a Markov Chain Monte Carlo
(MCMC) with 100 walkers, 1000 steps, and 25% burn-in. We used the auto-
correlation time to judge when the sampler had converged for each parameter
(e.g., Hogg et al., 2010). Median-value parameters and their 1-$\sigma$
uncertainties calculated from the sample distributions are propagated through
to the following reformulation of Equation 1:

$F(t)=X\sin(2\pi t/P+\theta)+{C}_{k},$ (2)

where $X=\sqrt{A^{2}+B^{2}}$ and $\theta=\arccos(A/X)$. $X$ is the photometric
semi-amplitude of variability (or $\rm\frac{\Delta S}{S_{avg}}$) for a given
photometric bandpass, which we use as inputs for the spot characteristics
model described below. We inflate the uncertainty on each variability
measurement by 25$\%$ to account for the assumptions of a simple rotation
curve and negligible facular contribution.

Quantity | Prior
---|---
$\rm{T_{eff}}$ [K] | $3650\pm 100$
$\rm{T_{hot}}$[K] |  $\mathcal{U}$[$\rm{T_{amb}}$, 12000]
$\rm{T_{spot}}$ [K] |  $\mathcal{U}$[2300, $\rm{T_{amb}}$]
$\rm{T_{amb}}$ [K] |  $\mathcal{U}$[$\rm{T_{spot}}$, $\rm{T_{hot}}$]
$f_{\rm hot}$ |  $\mathcal{U}$[0,0.5]
$f_{\rm spot}$ |  $\mathcal{U}$[0,(1.0-$f_{\rm hot}$)]
$\Delta f_{\rm spot}$ |  $\mathcal{U}$[0, $f_{\rm spot}$]
Table 3: Priors placed on our model parameters in the spot characteristics
Monte Carlo simulation.

###  3.2 Spot Characteristics Model

To draw inferences about AU Mic’s spot and facula characteristics, we
assembled a model for 3 data components: The effective temperature,
$\rm{T_{eff}}$, the photometric semi-amplitude of variability, $\frac{\Delta
S}{S_{\rm{avg}}}$, and the time-averaged stellar spectrum, Savg. Each of these
components is modeled as a function of some combination of $f_{\rm hot}$,
$f_{\rm spot}$, $\rm{T_{hot}}$, $\rm{T_{spot}}$, and $\rm{T_{amb}}$, with
model priors described in Table 3.

Modeling photometric variability requires one additional parameter, the
$|$peak-average$|$ amplitude of the change in spot coverage throughout a
rotation, $\Delta f_{\rm spot}$. This parameter represents a change in spot
coverage relative to the average coverage in a way that is not relevant to our
models of the time-averaged spectral data or $\rm{T_{eff}}$. It can range from
0, where the surface is homogeneous or the surface features are distributed
symmetrically around the rotation axis, to $\Delta f_{\rm spot}$$=$$f_{\rm
spot}$, where the total spot coverage is clustered on the surface such that it
rotates entirely in and out of view. While photometric variabilities only
provide a lower limit on the average spot coverage fraction, the magnitude of
variability depends strongly on this change in spot coverage throughout a
rotation and can be precisely constrained with sufficient evidence of the
spectral contrast between the ambient and spotted photosphere.

##### Effective Temperature

Similar to Libby-Roberts et al. (2022), we treat $\rm{T_{eff}}$ in the
following form:

$\mathrm{T_{eff}}^{4}=f\mathrm{{}_{spot}T_{spot}}^{4}+f\mathrm{{}_{hot}T_{hot}}^{4}+f\mathrm{{}_{amb}T_{amb}}^{4},$
(3)

where $f_{\rm spot}$ is the globally-averaged spot coverage fraction with
temperature $\rm{T_{spot}}$, $f_{\rm hot}$is the average coverage of any
potential third component (which may be faculae, flares, or something else)
with temperature $\rm{T_{hot}}$, and $f_{\rm amb}$ is the coverage of the
ambient photosphere which has temperature $\rm{T_{amb}}$. The ambient coverage
is not a unique parameter in the model but is calculated as $f_{\rm
amb}=1-(f_{\rm spot}+f_{\rm hot})$. This constraint effectively ensures that
whatever combination of spectral components is being modeled accurately
reproduces the known surface-averaged bolometric flux emitted from the stellar
surface.

##### Photometric Variability

Following the formalism in Libby-Roberts et al. (2022), we can calculate the
semi-amplitude of variability due to spots as the following:

$\frac{\Delta S(\lambda)}{S_{\rm avg}(\lambda)}=-\Delta f_{\rm
spot}\left(\frac{1-\frac{S(\lambda,\rm T_{spot})}{S(\lambda,\rm
T_{amb})}}{1-f_{\rm spot}[1-\frac{S(\lambda,\rm T_{spot})}{S(\lambda,\rm
T_{amb})}]}\right).$ (4)

The expression above, the only calculation in our model which depends on
$\Delta f_{\rm spot}$, is integrated across the filter bandpasses to generate
a single variability datum for each filter. We account for the filter response
curves by normalizing our variability integral by the filter response
function:

$\frac{\Delta S}{S_{\rm
avg}}=\frac{\int^{\lambda_{2}}_{\lambda_{1}}\frac{\Delta
S(\lambda)}{S_{\rm{avg}(\lambda)}}W(\lambda)S(\lambda,\rm
T_{eff})d\lambda}{\int^{\lambda_{2}}_{\lambda_{1}}W(\lambda)S(\lambda,\rm
T_{eff})d\lambda},$ (5)

where the SDSS filter response functions (Wλ) are acquired through
Speclite444https://speclite.readthedocs.io/en/latest/filters.html. The stellar
spectrum term ($S(\lambda,\rm T_{eff})$, calculated at T${}_{\rm{eff}}=3650K$)
accounts for the non-uniform distribution of stellar flux emitted across
bandpasses. These bandpass-integrated model variabilities are then fit to the
broadband variability measurements extracted from the stellar rotation curve
models (section 3.1).

We ignore a facular contribution to the rotational variability because
magnetically active stars are expected have photometric variabilities
dominated by spots (Shapiro et al., 2016), an assumption we argue is valid
considering the low filling factor of this component we measure, discussed in
the results. It does factor into the calculation of $\rm{T_{eff}}$ however,
and therefore indirectly affects the model results when we examine the
variability separate from the spectra. The variability light curve is also
highly under-sampled, so adding another component for this stage would be
over-fitting the very few (5) photometric variability data points we have.

##### Average Spectrum

From the NRES Echelle spectra, we calculate a time-averaged spectrum which we
model as a combination of a spotted spectrum and an ambient spectrum weighted
by their globally-averaged coverage:

$\mathrm{S_{avg}}=f_{\rm spot}S(\lambda,\mathrm{T_{spot}})+f_{\rm
hot}S(\lambda,\mathrm{T_{hot}})+f_{\rm amb}S(\lambda,\mathrm{T_{amb}}).$ (6)

Older studies of starspots have been limited in this approach due to the
computation time required to model thousands of spectral lines and as a result
they typically probed specific regions and rotational or vibrational
temperatures which may not be indicative of the bulk spot properties. Here we
modeled as many possible regions of the spectrum as possible, including orders
which have weak or non-existent spot signatures as well as those with strong
signatures indicative of very cool regions, to understand the most complete
picture of the star provided by the spectral data.

###  3.3 Spot Contamination Model

Atmospheric absorption will induce a wavelength-dependent change in the
transit depth ($\Delta\rm{D(\lambda)}$) of the planet, expressed as

$\Delta\rm{D(\lambda)}=\left(\frac{R_{p}}{R_{*}}\right)^{2}+\Delta\rm{D(\lambda)_{atm}}+\Delta\rm{D(\lambda)_{spot}},$
(7)

with $\Delta\rm{D(\lambda)_{atm}}$ is defined as:

$\Delta\rm{D(\lambda)_{atm}}=\frac{2R_{p}}{R^{2}_{*}}H\times n(\lambda),$ (8)

where H is the scale height and $n(\lambda)$ is the number of opaque scale
heights at each wavelength, which typically varies between 0-5 for cloud-free
atmospheres (Seager et al., 2000).

Following the derivations in Rackham et al. (2018); Zhang et al. (2018);
Libby-Roberts et al. (2022), we can express $\Delta\rm{D(\lambda)_{spot}}$ as:

$\mathrm{\Delta
D(\lambda)_{spot}=\left(\frac{R_{p}}{R_{*}}\right)^{2}}\left[\frac{(1-f_{\rm
spot,tra})+f_{\rm
spot,tra}\mathrm{\frac{S(\lambda,T_{spot})}{S(\lambda,T_{amb})}}}{(1-f_{\rm
spot})+f_{\rm
spot}\mathrm{\frac{S(\lambda,T_{spot})}{S(\lambda,T_{amb})}}}-1\right].$ (9)

This expression can similarly be used to calculate facular depth contribution
but we assume this contribution is negligible on AU Mic. Equation 9 does not
explicitly rely on $\Delta f_{\rm spot}$, but the value we derive for $\Delta
f_{\rm spot}$ can be used to project the spot coverage at a given time or
phase, which is needed to account for spot contamination at the time of
transit. In this work we assume $f_{tra}$ to be zero for both spots and
faculae, which implies that the contamination calculated for a given set of
parameters represents an upper limit relative to a spotted transit chord.
Samples for $f_{\rm spot}$, $f_{\rm hot}$, $\rm{T_{spot}}$, $\rm{T_{hot}}$,
and $\rm{T_{amb}}$ are injected into this model to generate a posterior
distribution of $\Delta\rm{D(\lambda)_{spot}}$.

###  3.4 Experimental Design

We run several different iterations of the spot characteristics model in order
to examine the information and constraints provided by each component of the
data: we model the photometric measurements separate from the spectra, the
spectra without photometric models, and the ensemble model which includes both
data types. The model is set up as a Monte-Carlo simulation using the emcee
sampler (Foreman-Mackey et al., 2013), run with 100 walkers and 2000 steps
with a 25$\%$ burn-in. Models ran past convergence in accordance with the
auto-correlation time of the sampler chains (for a discussion of convergence
and autocorrelation, see Foreman-Mackey et al., 2013). This framework
maintains a distinction between the temperatures of the different spectral
surface components ($\rm{T_{spot}}$$\leq$$\rm{T_{amb}}$$\leq$$\rm{T_{hot}}$)
while allowing $f_{\rm spot}$ and $f_{\rm hot}$ to vary, enabling models to
arrive at solutions where the the surface is $\geq$50% covered in spots.
Technically $f_{\rm hot}$ is allowed to vary as high as 50% but in practice
the models almost never preferred values of $f_{\rm hot}$ greater than a few
percent. Time-domain analysis of the 7 NRES spectra from F21 and 17 spectra
from S22 should contain information about the change in spot coverage with
stellar phase but no periodic signal could be found so we do not include a
time-domain spectral model in the analysis. Independent modeling of the
separate visits returned strongly consistent measurements for $f_{\rm spot}$,
$f_{\rm hot}$, and $\Delta f_{\rm spot}$ which could be be a robust finding,
considering the time between visits is $2\times$ the 120-150 day activity
evolution timescale (which we can approximate to be a spot decay timescale)
measured by Donati et al. (2023).

##  4 Results

###  4.1 Variability Amplitude Results

For the F21 visit, we measure variability semi-amplitudes, $\Delta S/S$, in
$g^{\prime}$, $r^{\prime}$, and $i^{\prime}$ of 0.075$\pm$0.006,
0.071$\pm$0.006, and 0.041$\pm$0.007, and for the S22 visit we measure $\Delta
S/S$ of 0.075$\pm$0.003 in $g$’ and 0.075$\pm$0.003 in $r$’, shown in Figure 3
and summarized in Table 4.

Filter | Amplitude | phase
---|---|---
First Visit (F21) |  |
$g$’ | 0.075$\pm$0.006 | $1.89\pm 0.05$
$r$’ | 0.071$\pm$0.006 | $1.93\pm 0.05$
$i$ ’ | 0.041$\pm$0.007 | $2.00\pm 0.04$
Second Visit (S22) |  |
$g$’ | 0.075$\pm$0.003 | $2.02\pm 0.04$
$r$’ | 0.075$\pm$0.003 | $2.03\pm 0.04$
Table 4: Parameters from the MCMC fits of the photometry. Period was kept
fixed at the literature period of 4.86d while the phase and amplitude were
modeled as a combination of sine and cosine terms.

(a)

(b)

(c)

(d)

(e)

Figure 3: LCO 0.4m photometry of AU Mic in the $g^{\prime}$, $r^{\prime}$, and
$i^{\prime}$ filters with the rotation model fits described in section 2.
Variability decreases with wavelength, exhibiting a significant decrease
between $r^{\prime}$ and $i^{\prime}$ measurements. The shaded regions are
randomly sampled models showing the distribution we quote as uncertainty on
the amplitude, and the color of each data set corresponds to the filter
response curves in Figure 12. Model results are provided in Table 4.

Measuring the signal for $i$’ was slightly challenging because, as the reddest
bandpass, this filter showed the weakest variability signal and therefore
smallest signal-to-noise. Moreover, AU Mic is much brighter than the nearby
comparison stars and exposure times are short, scintillation noise is
prevalent. To account for poorly constrained measurements in $i^{\prime}$ and
improve the internal consistency of our multi-color variability measurements,
we impose a prior on the phase ($\theta$) for $i$’. The independently-modeled
$g^{\prime}$ and $r^{\prime}$ rotation curves agree within their 1-$\sigma$
uncertainties in phase, so we use the average of their phases (1.91+/-0.04) as
a prior when modeling $i$’, resulting in a modeled phase in agreement with
$g$’ and $r$’. This is a reasonable approach to improving our $i^{\prime}$
results because we would expect the phase of 3 separate but contemporaneous
data sets to be equal, so forcing the $i^{\prime}$ phase to be consistent with
$g^{\prime}$ and $r^{\prime}$ increases the consistency between the three
measurements and lends confidence to the relative amplitudes our signal
measurements. The phase is not used further in this work but will be useful
for the analysis of AU Mic b’s atmospheric transmission spectrum when we need
to estimate the spot coverage at the time of transit.

###  4.2 Spot Characteristics Model

We examine the spot characteristics results when we model only the stellar
effective temperature and photometric variability data (excluding NRES
spectral models) and similarly when we model only the NRES spectra with
stellar effective temperature (excluding the multi-color photometric
variability data). Finally, we examine the results of modeling the stellar
$\rm{T_{eff}}$, photometric variabilities, and NRES spectra together in an
“ensemble” model which models the 12 spectral orders and multi-color
variabilities for both visits, from which we report the final results.

####  4.2.1 Photometric Variability

Figure 4 shows our measured photometric variability semi-amplitudes with
random samples drawn from the photometry-only posterior distributions, with
statistical results in Figure 6. The photometric variabilities, along with AU
Mic’ $\rm{T_{eff}}$, show evidence for essentially any spot coverage between
10-90$\%$ with temperature $3141^{+266}_{-389}$ K, changing throughout an
orbit by $6\pm 3\%$. Solutions for ambient temperature are
$3719^{+302}_{-502}$ K, with an upper limit of 40$\%$ coverage of a hot
component with temperature $4873^{+1922}_{-727}$K. The hot component is only
relevant to the effective temperature calculation in the variability-only
modeling, so it is acting more as an extra free parameter to improve the fit
than it is related to anything physical. The measurement of $\rm{T_{spot}}$ is
surprisingly accurate compared to the other measurements in this paper and
others.

Figure 4: All 5 measured rotational variabilities with random models (blue)
drawn from the variability model samples. Shaded regions in the background are
the filter response curves for our observations. We fit both visits together
because the photometric and spectroscopic data are consistent despite the 6
months between visits (e.g., Robertson et al., 2020). Variability decreases
with wavelength as the spot-to-photosphere flux contrast decreases, and when
modeling the visits separately we find that our solutions were very sensitive
to the magnitude and uncertainty of the $i$’ measurement. We have photometry
for all 3 filters in F21 but only $g^{\prime}$ and $r^{\prime}$ in S22.

2-Temperature Models of the variabilities return tighter constraints on the
ratio of $\rm{T_{spot}}$ to $\rm{T_{amb}}$ and their individual measurements,
but the measurement of $\rm{T_{spot}}$ is significantly warmer
($\rm{T_{spot}}$=$3454^{+155}_{-208}$ K) than virtually every other
measurement of $\rm{T_{spot}}$ we present in this paper. Measurements of
$f_{\rm spot}$ and $\Delta f_{\rm spot}$ are consistent with the 3-temperature
fits. Results for two-temperature modeling are in Table 6.

####  4.2.2 Spectral Decomposition Results

When modeling the 12 spectral orders (Figure 5) simultaneously and without
photometric variability, we find more precisely constrained temperatures and
coverages of all three components. The change in spot coverage, $\Delta f_{\rm
spot}$ is unconstrained by these models because we are modeling time-averaged
spectra. The stellar spectra indicate a large fraction ($f_{\rm
spot}$=$0.39\pm 0.04$) of cool spots with temperature $2974^{+72}_{-71}$ K, a
dominant ”ambient” (warmer) photosphere with temperature $4002^{+14}_{-15}$ K,
and a very tenuous detection of a hot component with temperature
$8681^{+868}_{-629}$ K covering less than 0.5$\%$ of the surface.

(a)  Figure 5: Spectra used in this analysis. The red and blue data points
(F21 and S22, respectively) are median-averaged in time and have had their
uncertainties normalized so a 3650 K template model has a reduced $\chi^{2}$
of 1. 100 randomly sampled models are plotted in black, which in most cases is
a tight spread and difficult to notice in these plots. Despite some poorly fit
line depths, there is a tight constraint on spot temperatures and filling
factor from the spectra with or without photometric variabilities.

2-Temperature Models of the spectra result in fully consistent measurements of
each parameter. The spot component measured to cover $41\pm 3\%$ of AU Mic
with temperature $3083^{+31}_{-45}$ K, and the ambient photosphere temperature
is measured to be $3998^{9}_{11}$ K.

####  4.2.3 Ensemble Model

Results from our ensemble model, where all 5 photometric variability
measurements are jointly modeled with the spectra of both visits, are shown in
Figures 6, 7, and 8, along with Table 5. This model finds well-constrained
spot characteristics of $f_{\rm spot}$=$0.39\pm 0.04$, $\Delta f_{\rm
spot}$=$0.05^{+0.004}_{-0.003}$, and $\rm{T_{spot}}$=$3003^{+63}_{-71}$ K,
with $\rm{T_{amb}}$=$4003^{+15}_{-14}$ K.

Some degenerate solutions can be seen which prefer what looks like a fourth
flux component between 3600-3900 K with a coverage fraction of 6-10$\%$. This
could be very weak evidence of either the spot penumbra or faculae, but
attempts to extract that component were unsuccessful and can be pursued more
in future work.

(a)  Figure 6: Posterior samples for the variability model (left), spectral
model (middle) and the ensemble model (right), which retains characteristics
of the separate model results. The spectra constrain how hot spot temperatures
can be while the photometry constrains how cool they can be. Limits on spot
coverage fraction are mostly provided by the spectral modeling, with the
variability providing constraints on $\Delta f_{\rm spot}$. The color of each
point corresponds to the log of the temperature ratio
($\rm{T_{spot}}$/$\rm{T_{amb}}$), with redder points being spots that are
further from the ambient temperature.

We can see that this ensemble model exhibits characteristics of both the
photometry-only and spectra-only models. The photometric variabilities
strongly constrain $\Delta f_{\rm spot}$ and the temperature ratio but provide
poor constraints on the quantity of spots. Variability measurements in
multiple wavebands across the optical-NIR constrain temperature contrast
because the relative change in variability with wavelength is set by the ratio
of spectral temperature components. Further into the red, the spectral
components are more similar and variability decreases. At bluer wavelengths,
variability reaches a maximum as the difference in spectra is greatest.

The spectral decomposition returns a precise estimate of $f_{\rm spot}$, which
is further constrained with the inclusion of photometric modeling. The cold
spot preference of the spectral models is balanced by the photometric limits
on how cool the spots can be, given the contrast at longer wavelengths. The
spot coverage fraction is unconstrained when modeling the photometry due to
degenerate observational effects between $f_{\rm spot}$ and $\rm{T_{spot}}$
but is precisely constrained when spectral modeling is included. The ensemble
model results are primarily driven by the spectra, with the photometry being
most important for the measurement of $\Delta f_{\rm spot}$.

(a)

(b)

(c)

Figure 7: Posterior histograms for the variability (top left) and spectral
models (top right), with the ensemble results (bottom) exhibiting what looks
like the separate-model posteriors multiplied together.

We argue that the consistency between measurements of the separate components
based on different data-model combinations indicates that the results we
report from the ensemble model are physically realistic.

2-Temperature Models agree with the 3-temperature results remarkably well,
showing spot coverage of $41\pm 3\%$ that changes throughout a rotation by
$5.1\pm 0.3\%$, with $\rm{T_{spot}}$=$3093^{+29}_{-41}$ and
$\rm{T_{amb}}$=$3998^{+10}_{-12}$.

Parameter | Photometry Model | Spectral Model | Ensemble Model
---|---|---|---
$\rm{T_{spot}}$ (K) | $3141^{+266}_{-389}$ | $2974^{+72}_{-71}$ | $3003^{+63}_{-71}$
$f_{\rm spot}$ | $0.57^{+0.22}_{-0.28}$ | $0.39\pm 0.04$ | $0.39\pm 0.04$
$\Delta f_{\rm spot}$ | $0.06\pm 0.03$ | $<0.40$ | $0.05^{+0.004}_{-0.003}$
$f_{\rm hot}$ | $<0.40$ | $<0.01$ | $<0.005$
$\rm{T_{hot}}$ (K) | $4873^{+1922}_{-727}$ | $8681^{+868}_{-629}$ | $8671^{+890}_{-629}$
$\rm{T_{amb}}$ (K) | $3719^{+302}_{-502}$ | $4002^{+14}_{-15}$ | $4003^{+15}_{-14}$
$\rm{T_{eff}}$ (K) | $3664\pm 101$ | $3783\pm 37$ | $3789\pm 35$
$\chi^{2}$ | $2.59~{}(2.59)$ | $2124~{}(0.581)$ | $2136~{}(0.584)$
Table 5: Parameters from fitting the different data combinations with a
3-temperature model. The photometry model uses the 5 photometric variability
measurements, the spectral model uses 12 spectral orders from both visits. The
ensemble model is applied to all the photometric and spectroscopic data.
Results are broadly consistent with the 2-T model, most importantly recovering
an approximately 3000K spot in either case. The results we recommend citing
for AU Mic’s surface components are the Ensemble Model results in the
rightmost column of this table.

###  4.3 Spot Contamination

We forward-modeled spot contamination (Equation 9) under the assumption of
$0\%$ spot coverage on the transit chord using ensemble model posteriors
generated in the previous step, shown in Figure 9. At short wavelengths like
the TESS bandpass (0.6-1$\mu$m), contamination ranges from 550-1400 ppm. In
the WFC3 bandpass, contamination is between 500-750ppm, and decreases to
250-500ppm at longer wavelengths. The transit depth of AU Mic b reported in
Szabó et al. (2022) calculated from TESS and CHEOPS (0.33-1.1$\mu$m) transits
translates to about 1875 ppm, so the spot contamination is 25-75$\%$ of that
signal if $f_{\rm tra}=0$. This means that without accounting for spot
contamination, AU Mic b’s true radius (3.46 RE if the Szabó et al. (2022)
value is uncontaminated) may be overestimated by as much as 0.5-1.7 Earth
radii. For AU Mic c, the Szabó et al. (2022) depth is about 980 ppm, with
contamination calculated to be between 400-750 ppm. This translates to an
over-estimated planetary radius of between 0.6-1.3 Earth radii.

The magnitude of this shorter-wavelength contamination may be much different
at different points in AU Mic’s activity cycle and long term magnetic
evolution, so these are rough estimates and contemporaneous measurements of
transit depths across the optical to infrared would shed more light on the
true radius of AU Mic b.

##  5 Discussion

In this work, we present an analysis of broadband photometry and high-
resolution spectroscopy to constrain the spot coverage fraction and
temperature contrast on AU Mic. This is crucial for addressing the spot
contamination in the transmission spectrum of AU Mic b and therefore
understanding its atmosphere. With observational data of stellar effective
temperature, multi-color time-series photometry, and high resolution stellar
spectra, we parameterized the spectroscopic and photometric effects of
starspots into spot coverage and temperature, with a well-constrained ambient
temperature and tentative measurement of a hot component.

Parameter | Photometry Model | Spectral Model | Ensemble Model
---|---|---|---
$\rm{T_{spot}}$ (K) | $3454^{+155}_{-208}$ | $3083^{+31}_{-45}$ | $3093^{+29}_{-41}$
$f_{\rm spot}$ | $0.61^{+0.22}_{-0.27}$ | $0.41\pm 0.03$ | $0.41\pm 0.03$
$\Delta f_{\rm spot}$ | $0.08^{+0.06}_{-0.04}$ | $<0.4$ | $0.05\pm 0.003$
$\rm{T_{amb}}$ (K) | $3876^{+234}_{-157}$ | $3998^{9}_{11}$ | $3998^{+10}_{-12}$
$\rm{T_{eff}}$ (K) | $3649\pm 98$ | $3704\pm 24$ | $3707\pm 24$
$\chi^{2}$ | $1.9~{}(1.9)$ | $3060~{}(0.837)$ | $3058~{}(0.836)$
Table 6: Parameters from fitting the different data combinations with a
2-temperature model. The photometry model uses the 5 photometric variability
measurements, the spectral model uses 12 spectral orders from both visits. The
ensemble model is applied to all the photometric and spectroscopic data. The
primary difference compared to 3-T results can be seen in the temperatures
measured for $\rm{T_{spot}}$ and $\rm{T_{amb}}$.

###  5.1 The Photometric Variability Spectrum

Photometric variability has wavelength-dependent characteristic which can be
used to constrain the primary flux components. Amplitudes increase to some
maximum at blue wavelengths as the flux contrast increases to 1, and decreases
at redder wavelengths toward the Rayleigh-Jeans limit. The shape and extent of
the amplitude of rotational variability as a function of wavelength provides
an important constraint on spot temperatures. Because our measured variability
amplitude is significantly less in $i^{\prime}$ than $g^{\prime}$ or
$r^{\prime}$, the spot temperature contrast is very sensitive to the magnitude
and uncertainty of $i^{\prime}$. The $i^{\prime}$ variability measurement thus
carries more weight than any other single data point in this study, as it
strongly constrains spot contrast and limits how cool the spots can be. If the
$i^{\prime}$ measurement showed greater variability, this would allow spot
temperature solutions to be cooler, as the contrast would be greater into the
red. The variability must instead decrease in this bandpass, which forces the
range of solutions to be narrower and the spot contrast to be within a certain
range. The importance of this $i^{\prime}$ measurement indicates that further
photometric studies of spotted stars must be sure to include multiple
bandpasses in the optical-to-infrared in order to precisely determine the
wavelength regime where variability decrease or “turn-off” happens. The
wavelengths where this turn-off is observed are highly descriptive of the
stars spot spectra.

###  5.2 Two Flux Components or Three?

The primary results in this work are reported from the 3-temperature modeling,
but we examine how those results changed when modeled with only two
temperatures, finding it returns tightly constrained and consistent
measurements for $f_{\rm spot}$, $\Delta f_{\rm spot}$, $\rm{T_{spot}}$, and
$\rm{T_{amb}}$. When we allow a third component, it finds a poorly constrained
temperature between 7000-10000 K, with a small number of solutions closer to
the ambient photosphere, around 4000-5000 K. If the 7000-10000 K component is
physical, it may be due to flares which are persistent enough to leave their
flux in the median-averaged spectra.

Faculae on M dwarfs may be up to a few hundred K above the ambient photosphere
(e.g., Norris et al., 2023) and a small cluster of solutions in this
temperature range stand apart from the primary results (seen in Figures 6 and
7). This could be indicative of faculae but we did not recover that component
in concerted attempts and its filling factor must be even less than the
filling factor measured for the hotter temperature, detected at less than
0.5$\%$.

There is one key difference in outcome between 2- and 3-temperature models.
When modeling only $\rm{T_{eff}}$ and photometric variabilities, the
2-temperature model returns a significantly warmer $\rm{T_{spot}}$ and a
tighter ratio of $\rm{T_{spot}}$ to $\rm{T_{amb}}$ compared with the
3-temperature model. A third temperature, which only factors into the
calculation of $\rm{T_{eff}}$, brings spot temperature solutions into
agreement with the other measurements we report, with larger uncertainties and
a weaker (though still noticeable) relationship between spotted and unspotted
temperatures.

Further investigation into multi-component rotation models and independent
measurements of spot temperatures will be needed to help clarify when 2- or
3-temperature rotation models are most appropriate, and what cautions to
impose on interpreting spot characteristics from photometric variabilities
alone.

###  5.3 Physical Interpretation of AU Mic’s Spot Characteristics

Much work has been done to measure and theoretically determine spot
temperatures, distributions, and filling factors as a function of stellar
type. Still, observational evidence of consistent spot temperatures and
precise filling factors for AU Mic is tenuous. Spot characteristics are
notoriously difficult to determine and different approaches can lead to
inconsistent results. Here we will discuss the results, limitations of this
approach, and physical interpretation of AU Mic’s starspots.

##### Spot Temperature

We report a characteristic spot temperature for AU Mic of
$\rm{T_{spot}}$$=$$3003^{+63}_{-71}$ K. A study of diatomic molecular lines in
AU Mic’s spot spectra found in Berdyugina et al. (2011) implies a $\Delta$T
($\rm{T_{eff}}$-$\rm{T_{spot}}$) of 500-700 K, which would translate to spot
temperatures of 3000-3200 K for a star with $\rm{T_{eff}}$ of 3700 K. More
recently, Ikuta et al. (2023) calculate $\rm{T_{spot}}$$=3140\pm 64$K for AU
Mic based on Equation 4 of Herbst et al. (2021), which is based on the work
done by Berdyugina (2005). Rackham et al. (2018) suggest that for cool stars,
$\rm{T_{spot}}$ can be estimated as 0.86$\times$Tphot (which we have labeled
$\rm{T_{amb}}$ in this work). Given our modeled $\rm{T_{amb}}$ of
$4003^{+15}_{-14}$, the estimated spot temperature would be roughly
3300-3400K, which is over 200K warmer than our spectral and ensemble results
but in agreement with our photometry-only model results. Recent work by Flagg
et al. (2022) found evidence of a cold H2 layer in AU Mic’s photosphere with
$1000\rm K<$$\rm{T_{spot}}$$<2400$K. Afram & Berdyugina (2019) use molecular
lines to measure spot temperatures and find that M0 stars may have spot
temperatures $\sim$2200K less than the stellar effective temperature, implying
spots on AU Mic may have T${}_{\rm{spot}}~{}\lesssim~{}1500$ K and Tspot-to-
Teff ratios $\sim\leq 0.4$.

Placed in context, the spot temperatures we measure in this work are much
cooler than the $\Delta T=400$K or $\rm{T_{spot}}$=0.86$\rm{T_{amb}}$
estimates for low mass stars and slightly cooler than (but consistent with)
the Ikuta et al. (2023) and Berdyugina et al. (2011) estimates for AU Mic,
warmer than some of the molecular and vibrational estimates for spots, but
consistent with older measurements of spot temperatures on young active stars
similar to AU Mic (e.g., Ramsey & Nations, 1980; Vogt, 1981). The range of
temperatures measured using different methods may be indicative that our
assumption of spots as a feature with a single descriptive bulk temperature is
breaking down, and we need to further develop our theoretical spot models to
account for a temperature gradient. If the spots on AU Mic have complex
temperature profiles and extensive penumbrae, then the $f_{\rm spot}$recovered
in these models may be an overestimate of the true spot umbra and an under-
estimate of the spot umbra + penumbra.

##### Spot Filling Factor

Our models indicate that the surface of AU Mic is $f_{\rm spot}$$=$ $0.39\pm
0.04$ covered in cool spots, with a change in spot coverage throughout a
stellar rotation of $\Delta f_{\rm spot}$$=$ $0.05^{+0.004}_{-0.003}$. This
estimate is within the very broad and uncertain range of possible spot
coverage fractions measured for sun-like and cooler stars and indicates a
heavily spotted stellar surface. Yamashita et al. (2022) estimate $f_{\rm
spot}$ of between 1-21% for Zero Age Main Sequence stars using a fixed
starspot temperature variability model, but such models are typically
underestimates on $f_{\rm spot}$ (Rackham et al., 2018; Apai et al., 2018).
Cao et al. (2022) use APOGEE H-band spectra to measure average spot filling
factors to be 0.248$\pm$0.005 for active stars in the Pleiades cluster and
0.03$\pm$0.008 for main sequence G and K stars in M67. Other estimates of
$f_{\rm spot}$ for low mass stars range from $<1\%$ to $50\%$ (see Table 3 in
Rackham et al., 2018). Our results are consistent with the range of expected
coverage fractions, and our derived $\Delta f_{\rm spot}$ is consistent with
values modeled by Libby-Roberts et al. (2022) of $\Delta f_{\rm spot}$$\leq
0.1$, though this is a much older and slowly rotating star.

###  5.4 Caveats

Here we will briefly describe some of the limitations of this work and
possible directions for future spot studies.

  * •

We use a simple sinusoidal rotation model, which is not necessarily the best
choice for complex rotation curves like AU Mic’s, but the photometry is not
densely sampled and individual measurements have large uncertainties, so for
the primary purpose of measuring the relative amplitude of variability between
filters we argue this is an appropriate model.

  * •

AU Mic is a bright star, so ground-based differential photometry is difficult
with LCO’s field of view. Without similarly bright field stars, our photometry
is very noisy, limiting the precision with which we can measure variability
signals, especially in redder bandpasses.

  * •

Our spot contamination forward-models assume that $f_{\rm{tra}}=0$, which
means our contamination results are upper-limit cases for different sets of
spot parameters. In reality, there may be some non-zero fraction of spot
coverage on the transit chord, which will lower the contamination level in the
transmission spectrum.

  * •

It is unclear what the source of the hot component is or whether it is truly a
real signal, and the choice to include a third component affects the
measurement of spot temperature from photometric variability models.

  * •

Our understanding of M star photosphere spectra is limited with the current
generation of high-resolution synthetic spectral libraries like the Husser et
al. (2013) PHOENIX models, described in detail by Iyer & Line (2020) and
Rackham & de Wit (2023). Choosing the best spectral template is a problem for
cool stars generally, but describing AU Mic with these models is further
complicated by its pre-main sequence age, a specific environment for which no
truly appropriate model spectra yet exist. Alternative spectral models for M
dwarfs exist, such as BT-SETTL (Allard et al., 2011; Allard, 2014) or SPHINX
(Iyer et al., 2023), but those are low- to mid- resolution libraries whereas
the Husser et al. (2013) library has a resolution of R=100000-500000 and is
more appropriate for the R=53000 echelle spectra we acquired.

Figure 8: Violin plot posteriors for the variability modeling (left
distribution on each panel), spectral modeling (middle distribution), and the
ensemble model (right distribution). Top left: temperature posteriors showing
$\rm{T_{spot}}$, $\rm{T_{amb}}$, and $\rm{T_{eff}}$. Top right: $f_{\rm
spot}$and $\Delta f_{\rm spot}$ posteriors. Bottom left: temperature ratio,
$\rm{T_{spot}}$/$\rm{T_{amb}}$, and bottom right: $\Delta$T, the difference
between the spotted and ambient temperatures. The ensemble results exhibit a
narrower parameter space for spot characteristics which is effectively the
product of the variability and spectral posteriors. Figure 9: Forward-modeled
spot contamination in the transmission spectrum of AU Mic b (left) and AU Mic
c (right) with models calculated using sampled parameters from the ensemble
model posterior. Red corresponds to models with $f=$$f_{\rm spot}$$+$$\Delta
f_{\rm spot}$ (The point of maximum spot coverage and minimum flux throughout
AU Mic’s rotation), black to $f=$$f_{\rm spot}$ and turquoise to models with
$f=$$f_{\rm spot}$$-$$\Delta f_{\rm spot}$. Atmospheric depth estimates come
from Equation 8 and are calculated to be 52 ppm for AU Mic b and 10ppm for AU
Mic c at one scale height. For an optimistic case where we can measure 5 scale
heights of a cloud-free atmosphere, AU Mic b’s atmospheric features will be
comparable to lower estimates of spot contamination, while AU Mic c’s
atmospheric features will still be a factor of a few below the lowest
contamination scenarios.

##  6 Conclusions

In this work, we have demonstrated the use in acquiring broadband photometry
and high-resolution spectroscopy of only the star (without data acquired
during transit) for the purpose of studying starspots and estimating their
effect on transmission spectra. The results of our modeling lead us to the
following conclusions:

  * •

At the time of these observations, AU Mic may have been $f_{\rm spot}$=39$\pm
4$% covered in spots with a change of $\Delta f_{\rm spot}$=6$\pm$1%
throughout a rotation. The spots have a bulk (flux-weighted, not
distinguishing umbra from penumbra) temperature of
$\rm{T_{spot}}$=$3003^{+63}_{-71}$ K surrounded by a
$\rm{T_{amb}}$=$4003^{+15}_{-14}$ K photosphere. We found very weak evidence
for facular coverage, and tentatively detect evidence of flux from flares with
characteristic temperature $8671^{+890}_{-629}$ K. The fractional
uncertainties measured on our final results for $\rm{T_{amb}}$ and
$\rm{T_{spot}}$ come out to $0.25\%$ and $2\%$, respectively. Berardo et al.
(2023) report theoretical precision limits on stellar spectral models with
current instruments and spectral libraries, finding we can constrain
photospheric spectra to $\geq 0.2\%$ and spot spectra down to $1-5\%$, which
translate to temperature uncertainties of $\leq 0.05\%$ on $\rm{T_{amb}}$ and
$0.25-1.25\%$ on $\rm{T_{spot}}$İf our results are physically accurate, we
have measured the temperature of AU Mic’s spots with a precision near but
slightly worse than the Berardo et al. (2023) theoretical precision limit, and
AU Mic’s ambient photosphere temperature down to near its limit.

  * •

Spot contamination in transmission spectra for AU Mic b will be significant,
adding between $250-1200$ppm contamination across the 0.5-5$\mu$m range,
overlapping with wavelengths where we measure planetary absorption features.
This contamination is based on zero spot coverage on the transit chord, which
could likely be an incorrect assumption if the large spot coverage fraction we
measure is accurate. Nevertheless, spot contamination may be causing us to
significantly overestimate the radii of AU Mic b and c. Determining the true
planetary radii may require further acquisition and study of contemporaneous
multi-color transit observations.

  * •

From these measurements we calculate $\rm{T_{amb}}$-$\rm{T_{spot}}$$=$ 1000 K
and $\rm{T_{spot}}$/$\rm{T_{amb}}$$=$ 0.75. Spots this cool should show up
noticeably if occulted in transit if our measurements and interpretation of AU
Mic’s photosphere are correct, but none have yet been confirmed. With AU Mic’s
short period and large radius, it is possible that spots exist primarily or
only at high latitudes, in which case spot crossings would be rarely or never
observed.

  * •

For stellar surfaces dominated by a single heterogeneity like spots (or
faculae, but the picture becomes more complicated with significant filling
factors of both), multi-color photometric variabilities provide significant
constraints on the ratio of $\rm{T_{spot}}$/$\rm{T_{amb}}$(or
Tfac/$\rm{T_{amb}}$) and potentially accurate measurements of $\rm{T_{spot}}$.
Sampling the sensitive regions of a star’s variability spectrum, even with
broad photometric bandpasses, is useful for roughly estimating the
temperatures of different spectral components.

  * •

Modeling the data with either 2-T or 3-T spectral decomposition did not
significantly change measurements of $\rm{T_{spot}}$, $\rm{T_{amb}}$, or
$\Delta f_{\rm spot}$, which indicates that these measurements are robust. As
we are uncertain of the nature and significance of the hot component we
detect, further investigation of the phase-resolved stellar spectrum may help
distinguish the nature of the different flux components.

  * •

Modeling only the variabilities provides a less accurate result for the
coverage fraction or when only 2 temperatures are included, even though this
only affects the calculation of $\rm{T_{eff}}$. The 3-T $\rm{T_{eff}}$ and
variability model returns an accurate (but highly uncertain) spot temperature
while the 2-T $\rm{T_{eff}}$ and variability model prefers spots of nearly
half the temperature. We recommend further investigation into the accuracy of
measuring spot and/or facula temperatures with multi-color rotation
modulations using both 2- and 3-temperature modeling.

  * •

Improving measurements of spot characteristics on exoplanet host stars and
understanding spot contamination is challenging but tractable with a multi-
modal approach that covers a broad range of the visible-IR electromagnetic
spectrum. As spot models and synthetic spectra improve, spot characteristics
on our host stars will become much clearer.

##### Software

Python code used in this paper is available on the author’s GitHub
(https://github.com/will-waalkes/AUMicTLSE). This project made use of many
publicly available tools and packages for which the authors are immensely
grateful. In addition to the software cited throughout the paper, we also used
Astropy (Astropy Collaboration et al., 2013, 2018, 2022), NumPy (van der Walt
et al., 2011), Matplotlib (Hunter, 2007), Pandas (McKinney, 2011), and
Anaconda’s JupyterLab.

##### Acknowledgements

The authors thank the anonymous referee for their timely, thorough, and
encouraging review which led to significant improvements to the manuscript.
This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship Program under Grant No. DGE-1650115. Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of
the National Science Foundation. This work makes use of observations from the
LCOGT network. WW thanks Jessica Libby-Roberts, Hannalore Gerling-Dunsmore,
Dennis Tilipman, Girish Duvuuri, Ward Howard, John Monnier, Steve Cranmer,
Gibor Basri, and Steve Vogt for conversations that helped aspects of this
project. WW and ZKBT were supported through STScI grant HST-GO-15788 and the
NSF CAREER program (AST-1945633). AWM was supported by grants from the NSF
CAREER program (AST-2143763) and NASA’s exoplanet research program (XRP
80NSSC21K0393). ERN was supported by HST-GO-15836.

## References

  * Afram & Berdyugina (2015) Afram, N., & Berdyugina, S. V. 2015, A&A, 576, A34, doi: 10.1051/0004-6361/201425314
  * Afram & Berdyugina (2019) —. 2019, A&A, 629, A83, doi: 10.1051/0004-6361/201935793
  * Ahrer et al. (2023) Ahrer, E.-M., Stevenson, K. B., Mansfield, M., et al. 2023, Nature, 614, 653, doi: 10.1038/s41586-022-05590-4
  * Alderson et al. (2023) Alderson, L., Wakeford, H. R., Alam, M. K., et al. 2023, Nature, 614, 664, doi: 10.1038/s41586-022-05591-3
  * Allard (2014) Allard, F. 2014, in Exploring the Formation and Evolution of Planetary Systems, ed. M. Booth, B. C. Matthews, & J. R. Graham, Vol. 299, 271–272, doi: 10.1017/S1743921313008545
  * Allard et al. (2011) Allard, F., Homeier, D., & Freytag, B. 2011, in Astronomical Society of the Pacific Conference Series, Vol. 448, 16th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, ed. C. Johns-Krull, M. K. Browning, & A. A. West, 91, doi: 10.48550/arXiv.1011.5405
  * Allard et al. (2012) Allard, F., Homeier, D., Freytag, B., & Sharp, C. M. 2012, in EAS Publications Series, Vol. 57, EAS Publications Series, ed. C. Reylé, C. Charbonnel, & M. Schultheis, 3–43, doi: 10.1051/eas/1257001
  * Angus et al. (2018) Angus, R., Morton, T., Aigrain, S., Foreman-Mackey, D., & Rajpaul, V. 2018, MNRAS, 474, 2094, doi: 10.1093/mnras/stx2109
  * Apai et al. (2018) Apai, D., Rackham, B. V., Giampapa, M. S., et al. 2018, arXiv e-prints, arXiv:1803.08708.  https://arxiv.org/abs/1803.08708
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Barclay et al. (2023) Barclay, T., Sheppard, K. B., Latouf, N., et al. 2023, arXiv e-prints, arXiv:2301.10866, doi: 10.48550/arXiv.2301.10866
  * Basri (2021) Basri, G. 2021, An Introduction to Stellar Magnetic Activity, 2514-3433 (IOP Publishing), doi: 10.1088/2514-3433/ac2956
  * Berardo et al. (2023) Berardo, D., de Wit, J., & Rackham, B. V. 2023, arXiv e-prints, arXiv:2307.04785, doi: 10.48550/arXiv.2307.04785
  * Berdyugina (2005) Berdyugina, S. V. 2005, Living Reviews in Solar Physics, 2, 8, doi: 10.12942/lrsp-2005-8
  * Berdyugina et al. (2011) Berdyugina, S. V., Berdyugin, A. V., Fluri, D. M., & Piirola, V. 2011, ApJ, 728, L6, doi: 10.1088/2041-8205/728/1/L6
  * Berta et al. (2012) Berta, Z. K., Charbonneau, D., Désert, J.-M., et al. 2012, ApJ, 747, 35, doi: 10.1088/0004-637X/747/1/35
  * Brown et al. (2001) Brown, T. M., Charbonneau, D., Gilliland, R. L., Noyes, R. W., & Burrows, A. 2001, ApJ, 552, 699, doi: 10.1086/320580
  * Cao et al. (2022) Cao, L., Pinsonneault, M. H., Hillenbrand, L. A., & Kuhn, M. A. 2022, ApJ, 924, 84, doi: 10.3847/1538-4357/ac307f
  * Chen et al. (2005) Chen, C. H., Patten, B. M., Werner, M. W., et al. 2005, ApJ, 634, 1372, doi: 10.1086/497124
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., Stassun, K. G., & Hessman, F. V. 2017, AJ, 153, 77, doi: 10.3847/1538-3881/153/2/77
  * Cristofari et al. (2023) Cristofari, P. I., Donati, J. F., Folsom, C. P., et al. 2023, MNRAS, 522, 1342, doi: 10.1093/mnras/stad865
  * Donati et al. (2023) Donati, J. F., Cristofari, P. I., Finociety, B., et al. 2023, MNRAS, 525, 455, doi: 10.1093/mnras/stad1193
  * Feinstein et al. (2022) Feinstein, A. D., France, K., Youngblood, A., et al. 2022, AJ, 164, 110, doi: 10.3847/1538-3881/ac8107
  * Feinstein et al. (2023) Feinstein, A. D., Radica, M., Welbanks, L., et al. 2023, Nature, 614, 670, doi: 10.1038/s41586-022-05674-1
  * Flagg et al. (2022) Flagg, L., Johns-Krull, C. M., France, K., et al. 2022, ApJ, 934, 8, doi: 10.3847/1538-4357/ac7643
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Fu et al. (2022) Fu, G., Espinoza, N., Sing, D. K., et al. 2022, ApJ, 940, L35, doi: 10.3847/2041-8213/ac9977
  * Gaia Collaboration et al. (2023) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1, doi: 10.1051/0004-6361/202243940
  * Gilbert et al. (2022) Gilbert, E. A., Barclay, T., Quintana, E. V., et al. 2022, AJ, 163, 147, doi: 10.3847/1538-3881/ac23ca
  * Gray (1984) Gray, D. F. 1984, ApJ, 277, 640, doi: 10.1086/161735
  * Gully-Santiago et al. (2017) Gully-Santiago, M. A., Herczeg, G. J., Czekala, I., et al. 2017, ApJ, 836, 200, doi: 10.3847/1538-4357/836/2/200
  * Herbst et al. (2021) Herbst, K., Papaioannou, A., Airapetian, V. S., & Atri, D. 2021, ApJ, 907, 89, doi: 10.3847/1538-4357/abcc04
  * Hirano et al. (2020) Hirano, T., Krishnamurthy, V., Gaidos, E., et al. 2020, ApJ, 899, L13, doi: 10.3847/2041-8213/aba6eb
  * Hogg et al. (2010) Hogg, D. W., Bovy, J., & Lang, D. 2010, arXiv e-prints, arXiv:1008.4686, doi: 10.48550/arXiv.1008.4686
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Husser et al. (2013) Husser, T.-O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6, doi: 10.1051/0004-6361/201219058
  * Ikuta et al. (2023) Ikuta, K., Namekata, K., Notsu, Y., et al. 2023, arXiv e-prints, arXiv:2302.09249, doi: 10.48550/arXiv.2302.09249
  * Iyer & Line (2020) Iyer, A. R., & Line, M. R. 2020, ApJ, 889, 78, doi: 10.3847/1538-4357/ab612e
  * Iyer et al. (2023) Iyer, A. R., Line, M. R., Muirhead, P. S., Fortney, J. J., & Gharib-Nezhad, E. 2023, ApJ, 944, 41, doi: 10.3847/1538-4357/acabc2
  * Jones et al. (2013) Jones, A., Noll, S., Kausch, W., Szyszka, C., & Kimeswenger, S. 2013, A&A, 560, A91, doi: 10.1051/0004-6361/201322433
  * Jones et al. (1995) Jones, H. R. A., Longmore, A. J., Allard, F., et al. 1995, MNRAS, 277, 767, doi: 10.1093/mnras/277.3.767
  * Kalas et al. (2004) Kalas, P., Liu, M. C., & Matthews, B. C. 2004, Science, 303, 1990, doi: 10.1126/science.1093420
  * Libby-Roberts et al. (2022) Libby-Roberts, J. E., Berta-Thompson, Z. K., Diamond-Lowe, H., et al. 2022, AJ, 164, 59, doi: 10.3847/1538-3881/ac75de
  * Lim et al. (2023) Lim, O., Benneke, B., Doyon, R., et al. 2023, arXiv e-prints, arXiv:2309.07047, doi: 10.48550/arXiv.2309.07047
  * Louca et al. (2023) Louca, A. J., Miguel, Y., Tsai, S.-M., et al. 2023, MNRAS, 521, 3333, doi: 10.1093/mnras/stac1220
  * MacGregor et al. (2013) MacGregor, M. A., Wilner, D. J., Rosenfeld, K. A., et al. 2013, ApJ, 762, L21, doi: 10.1088/2041-8205/762/2/L21
  * Mamajek & Bell (2014) Mamajek, E. E., & Bell, C. P. M. 2014, MNRAS, 445, 2169, doi: 10.1093/mnras/stu1894
  * Martioli et al. (2021) Martioli, E., Hébrard, G., Correia, A. C. M., Laskar, J., & Lecavelier des Etangs, A. 2021, A&A, 649, A177, doi: 10.1051/0004-6361/202040235
  * McCully et al. (2022) McCully, C., Turner, M., Collom, D., & Daily, M. 2022, BANZAI: Beautiful Algorithms to Normalize Zillions of Astronomical Images, Astrophysics Source Code Library, record ascl:2207.031.  http://ascl.net/2207.031
  * McKinney (2011) McKinney, W. 2011, Python for High Performance and Scientific Computing, 14
  * Medina et al. (2022) Medina, A. A., Charbonneau, D., Winters, J. G., Irwin, J., & Mink, J. 2022, ApJ, 928, 185, doi: 10.3847/1538-4357/ac5738
  * Mikal-Evans et al. (2023) Mikal-Evans, T., Madhusudhan, N., Dittmann, J., et al. 2023, AJ, 165, 84, doi: 10.3847/1538-3881/aca90b
  * Moran et al. (2023) Moran, S. E., Stevenson, K. B., Sing, D. K., et al. 2023, ApJ, 948, L11, doi: 10.3847/2041-8213/accb9c
  * Neff et al. (1995) Neff, J. E., O’Neal, D., & Saar, S. H. 1995, ApJ, 452, 879, doi: 10.1086/176356
  * Noll et al. (2012) Noll, S., Kausch, W., Barden, M., et al. 2012, A&A, 543, A92, doi: 10.1051/0004-6361/201219040
  * Norris et al. (2023) Norris, C. M., Unruh, Y. C., Witzke, V., et al. 2023, MNRAS, 524, 1139, doi: 10.1093/mnras/stad1738
  * Olah et al. (1997) Olah, K., Kővári, Z., Bartus, J., et al. 1997, A&A, 321, 811
  * Pass et al. (2023) Pass, E. K., Winters, J. G., Charbonneau, D., Irwin, J. M., & Medina, A. A. 2023, AJ, 166, 16, doi: 10.3847/1538-3881/acd6a2
  * Plavchan et al. (2020) Plavchan, P., Barclay, T., Gagné, J., et al. 2020, Nature, 582, 497, doi: 10.1038/s41586-020-2400-z
  * Pont et al. (2008) Pont, F., Knutson, H., Gilliland, R. L., Moutou, C., & Charbonneau, D. 2008, MNRAS, 385, 109, doi: 10.1111/j.1365-2966.2008.12852.x
  * Pont et al. (2007) Pont, F., Gilliland, R. L., Moutou, C., et al. 2007, A&A, 476, 1347, doi: 10.1051/0004-6361:20078269
  * Rackham et al. (2018) Rackham, B. V., Apai, D., & Giampapa, M. S. 2018, ApJ, 853, 122, doi: 10.3847/1538-4357/aaa08c
  * Rackham et al. (2019) —. 2019, AJ, 157, 96, doi: 10.3847/1538-3881/aaf892
  * Rackham & de Wit (2023) Rackham, B. V., & de Wit, J. 2023, arXiv e-prints, arXiv:2303.15418, doi: 10.48550/arXiv.2303.15418
  * Ramsey & Nations (1980) Ramsey, L. W., & Nations, H. L. 1980, ApJ, 239, L121, doi: 10.1086/183306
  * Reiners et al. (2013) Reiners, A., Shulyak, D., Anglada-Escudé, G., et al. 2013, A&A, 552, A103, doi: 10.1051/0004-6361/201220437
  * Robertson et al. (2020) Robertson, P., Stefansson, G., Mahadevan, S., et al. 2020, ApJ, 897, 125, doi: 10.3847/1538-4357/ab989f
  * Robinson et al. (2014) Robinson, T. D., Maltagliati, L., Marley, M. S., & Fortney, J. J. 2014, Proceedings of the National Academy of Science, 111, 9042, doi: 10.1073/pnas.1403473111
  * Rustamkulov et al. (2023) Rustamkulov, Z., Sing, D. K., Mukherjee, S., et al. 2023, Nature, 614, 659, doi: 10.1038/s41586-022-05677-y
  * Sanchis-Ojeda & Winn (2011) Sanchis-Ojeda, R., & Winn, J. N. 2011, ApJ, 743, 61, doi: 10.1088/0004-637X/743/1/61
  * Schöfer et al. (2019) Schöfer, P., Jeffers, S. V., Reiners, A., et al. 2019, A&A, 623, A44, doi: 10.1051/0004-6361/201834114
  * Seager & Sasselov (2000) Seager, S., & Sasselov, D. D. 2000, ApJ, 537, 916, doi: 10.1086/309088
  * Seager et al. (2000) Seager, S., Whitney, B. A., & Sasselov, D. D. 2000, ApJ, 540, 504, doi: 10.1086/309292
  * Shapiro et al. (2016) Shapiro, A. I., Solanki, S. K., Krivova, N. A., Yeo, K. L., & Schmutz, W. K. 2016, A&A, 589, A46, doi: 10.1051/0004-6361/201527527
  * Shields et al. (2016) Shields, A. L., Ballard, S., & Johnson, J. A. 2016, Physics Reports, 663, 1, doi: 10.1016/j.physrep.2016.10.003
  * Sing et al. (2011) Sing, D. K., Pont, F., Aigrain, S., et al. 2011, MNRAS, 416, 1443, doi: 10.1111/j.1365-2966.2011.19142.x
  * Sing et al. (2016) Sing, D. K., Fortney, J. J., Nikolov, N., et al. 2016, Nature, 529, 59, doi: 10.1038/nature16068
  * Strassmeier & Olah (1992) Strassmeier, K. G., & Olah, K. 1992, A&A, 259, 595
  * Szabó et al. (2021) Szabó, G. M., Gandolfi, D., Brandeker, A., et al. 2021, A&A, 654, A159, doi: 10.1051/0004-6361/202140345
  * Szabó et al. (2022) Szabó, G. M., Garai, Z., Brandeker, A., et al. 2022, A&A, 659, L7, doi: 10.1051/0004-6361/202243076
  * Thao et al. (2023) Thao, P. C., Mann, A. W., Gao, P., et al. 2023, AJ, 165, 23, doi: 10.3847/1538-3881/aca07a
  * Torres & Ferraz Mello (1973) Torres, C. A. O., & Ferraz Mello, S. 1973, A&A, 27, 231
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22, doi: 10.1109/MCSE.2011.37
  * Vogt (1979) Vogt, S. S. 1979, PASP, 91, 616, doi: 10.1086/130549
  * Vogt (1981) —. 1981, ApJ, 250, 327, doi: 10.1086/159379
  * Wakeford et al. (2019) Wakeford, H. R., Lewis, N. K., Fowler, J., et al. 2019, AJ, 157, 11, doi: 10.3847/1538-3881/aaf04d
  * Wing et al. (1967) Wing, R. F., Peimbert, M., & Spinrad, H. 1967, PASP, 79, 351, doi: 10.1086/128496
  * Wittrock et al. (2022) Wittrock, J. M., Dreizler, S., Reefe, M. A., et al. 2022, AJ, 164, 27, doi: 10.3847/1538-3881/ac68e5
  * Wittrock et al. (2023) Wittrock, J. M., Plavchan, P., Cale, B. L., et al. 2023, arXiv e-prints, arXiv:2302.04922, doi: 10.48550/arXiv.2302.04922
  * Yamashita et al. (2022) Yamashita, M., Itoh, Y., & Oasa, Y. 2022, PASJ, 74, 1295, doi: 10.1093/pasj/psac069
  * Zacharias et al. (2012) Zacharias, N., Finch, C. T., Girard, T. M., et al. 2012, VizieR Online Data Catalog, I/322A
  * Zhang et al. (2018) Zhang, Z., Zhou, Y., Rackham, B. V., & Apai, D. 2018, AJ, 156, 178, doi: 10.3847/1538-3881/aade4f
  * Zicher et al. (2022) Zicher, N., Barragán, O., Klein, B., et al. 2022, MNRAS, 512, 3060, doi: 10.1093/mnras/stac614

We include this appendix to show some of the spectral data (Figures 10, 11),
visit-specific models of the photometric variabilities (Figure 12), and order-
specific modeling of the NRES spectra (Figure 13). Comparing F21 and S22
variability models demonstrates how important the i’ measurement is and the
relative constraints provided with or without it. The order-specific spectral
models show agreement in many ways, as well as some poorly constrained orders
which are likely due to telluric contamination.

Figures 10 and 11 show spectral order 53 from both visits as examples of how
the data looked before and after being processed and the full figure sets can
be found in the online journal, including orders that were cut from final
analysis.

Fig. SetNRES Spectra - Visit F21

10.

Figure 10: Order 53 from visit F21 with spectral orders used in this analysis
extending from order 53 (0.881 $\mu$m) to order 83 (0.562 $\mu$m). Stacked in
gray are individual spectra collected from 2021 Aug 24 to 2021 Sep 9. In blue
is a single spectrum derived from the time series observations which has been
binned to 0.05 nm, telluric-corrected, median-averaged in time, and continuum
normalized. The complete figure set (31 images) is available in the online
journal.

Fig. SetNRES Spectra - Visit S22

Figure 11: Order 53 from visit S22 with spectral orders used in this analysis
extending from order 53 (0.881 $\mu$m) to order 83 (0.562 $\mu$m). Stacked in
gray are individual spectra collected from 2022 Apr 02 to 2022 Apr 17. In blue
is a single spectrum derived from the time series observations which has been
binned to 0.05 nm, telluric-corrected, median-averaged in time, and continuum
normalized. The complete figure set (31 images) is available in the online
journal.

##  Appendix A Visit-Specific Variability Measurements

While the magnitude of stellar variability in any given waveband is related to
degeneracies between spot temperature and spot coverage, the relative
variability between photometric bandpasses and the wavelengths where the
variability begins to decrease contain a lot of information about the spectral
temperatures on the surface of the star. If the bulk spot temperatures are
very cold relative to the photosphere, the variability will remain high at
redder wavelengths. Similarly, if spot temperatures are closer to the ambient
photosphere temperature, then the variability will decrease within optical and
bluer wavelengths. The right panel in Figure 12 shows the greater spread in
models at redder wavelengths resulting from the absence of an $i^{\prime}$
measurement, allowing the spot temperatures can be much cooler and much
further from the ambient photosphere. The different model solutions possible
are strongly constrained by this $i^{\prime}$ variability, and we extend that
observation to say that multi-color photometric variabilities covering the
optical-NIR provide very strong constraints on the temperature contrast for
spotted stars.

(a)

(b)

Figure 12: F21 (left) and S22 (right) sampled variability models (without
spectral fits). The tighter model constraints imposed by an $i^{\prime}$
measurement can be seen in the spread of model solutions in the red when
comparing visits. The presence or absence of $i^{\prime}$ is strongly
constraining and further work should emphasize multi-band measurements to
better constrain the spectral temperature contrast.

##  Appendix B Spectral Decomposition by Order

In the ensemble model described in the main text, we fit spotted spectral
models to 18 orders simultaneously, but here we examine the results from
modeling each order individually to check for consistency between orders and
which orders might be the most informative. In this part of the modeling, we
model the data from both F21 and S22 with the same set of parameters, under
the assumption that AU Mic’s spot characteristics have not changed measurably
between visits. Examining the results (Figure 13), we cut orders from the
analysis if their solutions are poorly constrained and they satisfy any of the
following criteria: there is telluric contamination, unconstrained ambient
characteristics (which should be the dominant signal), or extremely cold spot
temperatures (which is often coincident with heavy telluric contamination).
Spectral orders that satisfy these conditions are excluded if we suspect
useful information cannot be gained from the spectral order, most frequently
due to telluric contamination. Many orders also exhibit what appears to be a
hot component a few hundred Kelvin hotter than the ambient component, which
could be evidence of a facular flux component. Attempts to recover this
component were unsuccessful, but perhaps worth future investigation.

(a)  Figure 13: Combined-visit model violin plots showing the posterior
distributions when modeling only the spectra and $\rm{T_{eff}}$. Spectral
orders shown in this plot span 0.557-0.888 $\mu$m, with wavelength decreasing
to the right. Top: Temperature components for the hot (blue), middle (black)
and spot (red) components. In yellow is the corresponding $\rm{T_{eff}}$. Most
orders exhibit a hot component between 7000-10000 K, consistent with the
temperature of flares in this wavelength range. Spot temperatures are
generally poorly constrained, with the 3000K spot seemingly detected in a
handful of orders (53, 61, 72, 76, 79). Middle: the temperature ratio,
$\rm{T_{spot}}$/$\rm{T_{amb}}$. The 5 orders showing a spot solution tend to
show a temperature ratio of between 0.7-0.8. Bottom: coverage fractions for
the spotted (red), ambient (black), and hot (blue) components. The hot
component is very small, less than 3$\%$. There is no clear agreement with
spot coverage between orders, with the largest component being the ambient
photosphere in the early orders while the later orders show more spot-
dominated photospheres. Orders with very poor constraints tend to overlap with
orders that are contaminated by tellurics. Orders omitted from the analysis
are grayed out.