Quantifying the Transit Light Source Effect: Measurements of Spot Temperature and Coverage on the Photosphere of AU Microscopii with High-resolution Spectroscopy and Multicolor Photometry

AU Mic (Torres & Ferraz Mello 1973) is a nearby (9.7 pc; Gaia Collaboration et al. 2023), young (24 ± 3 Myr; Mamajek & Bell 2014), rapidly rotating (P_rot = 4.86 days; 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 3600–3700 K (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 two unconfirmed candidate nontransiting planets on 12.74 and 33.39 day periods recently discovered through transit timing variations and radial velocity analysis (Wittrock et al. 2022; Donati et al. 2023; Wittrock 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 photoevaporation of the atmospheres of AU Mic b and c (Feinstein et al. 2022) and even in the case where the 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 multiplanet 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).

The observations we present in this work are part of a companion study alongside Hubble Space Telescope (HST)/WFC3 transmission spectra of AU Mic b; the first on 2021 August 30 (BJD 2459455.98) and the second on 2022 April 14 (BJD 2459684.40). Because AU Mic's spots evolve 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. These transmission spectra will be analyzed in the context of our results and presented in a subsequent paper.

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 multicolor 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 the 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 (Section 3.1), assembling the self-consistent statistical framework for modeling spot filling factor and temperature (Section 3.2), and forward modeling the spot contamination in AU Mic b's transmission spectrum (Section 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.

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 wave band. 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 the 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 the 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), and Rackham et al. (2018) for more thorough reviews of starspots and the techniques used to study them.

We acquired Las Cumbres Observatory (LCO) 0.4 m $g^{\prime}$-, $r^{\prime}$-, and $i^{\prime}$-band ⁸ 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 of 2021 and spring of 2022, hereafter F21 and S22, respectively). 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.

Zoom In Zoom Out Reset image size

2.2.1. SBIG Imaging

2.2.2. NRES Spectra

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

$$\begin{eqnarray}&&F(t)\,=\,A\,\sin (2\pi t/P)+B\cos (2\pi t/P)+{C}_{k},\end{eqnarray} \tag{ 1 }$$

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_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 Markov Chain Monte Carlo (MCMC) sampling with 100 walkers, 1000 steps, and 25% burn-in. We used the autocorrelation time to judge when the sampler had converged for each parameter (e.g., Goodman & Weare 2010). Median-value parameters and their 1σ uncertainties calculated from the sample distributions are propagated through to the following reformulation of Equation (1):

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

where $X=\sqrt{{A}^{2}+{B}^{2}}$ and $\theta =\arccos (A/X)$. X is the photometric semiamplitude of variability (or $\tfrac{{\rm{\Delta }}{\rm{S}}}{{{\rm{S}}}_{{avg}}}$) for a given photometric bandpass, which we use as input 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.

To draw inferences about AU Mic's spot and facula characteristics, we assembled a model for three data components: the effective temperature, T_eff, the photometric semiamplitude of variability, $\tfrac{{\rm{\Delta }}S}{{S}_{\mathrm{avg}}}$, and the time-averaged stellar spectrum, S_avg. Each of these components is modeled as a function of some combination of f_hot, f_spot, T_hot, T_spot, and T_amb, with the model priors described in Table 3.

Table 3. Priors Placed on Our Model Parameters in the Spot Characteristics Monte Carlo Simulation

Quantity	Prior
Teff [K]	3650 ± 100
Thot [K]	${ \mathcal U }$[Tamb, 12,000]
Tspot [K]	${ \mathcal U }$[2300, Tamb]
Tamb [K]	${ \mathcal U }$[Tspot, Thot]
fhot	${ \mathcal U }$[0,0.5]
fspot	${ \mathcal U }$[0, (1.0 – fhot)]
Δfspot	${ \mathcal U }$[0, fspot]

Download table as:  ASCIITypeset image

Modeling photometric variability requires one additional parameter, the ∣peak – average∣ amplitude of the change in spot coverage throughout a rotation, Δf_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 T_eff. It can range from 0, where the surface is homogeneous or the surface features are distributed symmetrically around the rotation axis, to Δf_spot = f_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.

3.2.1. Effective Temperature

Similar to Libby-Roberts et al. (2022), we treat T_eff in the following form:

$$\begin{eqnarray}&&{{\rm{T}}}_{\mathrm{eff}}^{4}={f}_{\mathrm{spot}}{{\rm{T}}}_{\mathrm{spot}}^{4}+{f}_{\mathrm{hot}}{{\rm{T}}}_{\mathrm{hot}}^{4}+{f}_{\mathrm{amb}}{{\rm{T}}}_{\mathrm{amb}}^{4},\end{eqnarray} \tag{ 3 }$$

where f_spot is the globally averaged spot coverage fraction with temperature T_spot, f_hot is the average coverage of any potential third component (which may be faculae, flares, or something else) with temperature T_hot, and f_amb is the coverage of the ambient photosphere which has temperature T_amb. The ambient coverage is not a unique parameter in the model but is calculated as f_amb = 1 − (f_spot + f_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.

3.2.2. Photometric Variability

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

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}S(\lambda )}{{S}_{\mathrm{avg}}(\lambda )}=-{\rm{\Delta }}{f}_{\mathrm{spot}}\left(\displaystyle \frac{1-\tfrac{S(\lambda ,{T}_{\mathrm{spot}})}{S(\lambda ,{T}_{\mathrm{amb}})}}{1-{f}_{\mathrm{spot}}[1-\tfrac{S(\lambda ,{T}_{\mathrm{spot}})}{S(\lambda ,{T}_{\mathrm{amb}})}]}\right).\end{eqnarray} \tag{ 4 }$$

The expression above, the only calculation in our model which depends on Δf_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:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}S}{{S}_{\mathrm{avg}}}=\displaystyle \frac{{\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\tfrac{{\rm{\Delta }}S(\lambda )}{{S}_{\mathrm{avg}(\lambda )}}W(\lambda )S(\lambda ,{T}_{\mathrm{eff}}){\rm{d}}\lambda }{{\int }_{{\lambda }_{1}}^{{\lambda }_{2}}W(\lambda )S(\lambda ,{T}_{\mathrm{eff}}){\rm{d}}\lambda },\end{eqnarray} \tag{ 5 }$$

where the SDSS filter response functions (W_λ ) are acquired through Speclite. ¹¹ The stellar spectrum term (S(λ, T_eff), calculated at T_eff = 3650 K), accounts for the nonuniform distribution of stellar flux emitted across the 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 to have photometric variabilities dominated by spots (Shapiro et al. 2016), but it factors into the calculation of T_eff and is still included as a flux component when we examine the photometric variabilities without the spectra. The variability light curve is also highly undersampled, so adding another component into the rotation model would be overfitting the very few (five) photometric variability data points we have.

3.2.3. 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:

$$\begin{eqnarray}&&{S}_{\mathrm{avg}}={f}_{\mathrm{spot}}S(\lambda ,{T}_{\mathrm{spot}})+{f}_{\mathrm{hot}}S(\lambda ,{T}_{\mathrm{hot}})+{f}_{\mathrm{amb}}S(\lambda ,{T}_{\mathrm{amb}}).\end{eqnarray} \tag{ 6 }$$

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 nonexistent 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.

Atmospheric absorption will induce a wavelength-dependent change in the transit depth (ΔD(λ)) of the planet, expressed as:

$$\begin{eqnarray}&&{\rm{\Delta }}D(\lambda )={\left(\displaystyle \frac{{R}_{p}}{{R}_{* }}\right)}^{2}+{\rm{\Delta }}D{\left(\lambda \right)}_{\mathrm{atm}}+{\rm{\Delta }}D{\left(\lambda \right)}_{\mathrm{spot}},\end{eqnarray} \tag{ 7 }$$

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

$$\begin{eqnarray}&&{\rm{\Delta }}D{\left(\lambda \right)}_{\mathrm{atm}}=\displaystyle \frac{2{R}_{p}}{{R}_{* }^{2}}H\times n(\lambda ),\end{eqnarray} \tag{ 8 }$$

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

Following the derivations in Rackham et al. (2018), Zhang et al. (2018), and Libby-Roberts et al. (2022), we can express ΔD(λ)_spot as:

$$\begin{eqnarray}&&{\rm{\Delta }}D{(\lambda )}_{\mathrm{spot}}={\left(\displaystyle \frac{{R}_{p}}{{R}_{* }}\right)}^{2}\left[\displaystyle \frac{(1-{f}_{\mathrm{spot},\mathrm{tra}})+{f}_{\mathrm{spot},\mathrm{tra}}\tfrac{S(\lambda ,{T}_{\mathrm{spot}})}{S(\lambda ,{T}_{\mathrm{amb}})}}{(1-{f}_{\mathrm{spot}})+{f}_{\mathrm{spot}}\tfrac{S(\lambda ,{T}_{\mathrm{spot}})}{S(\lambda ,{T}_{\mathrm{amb}})}}-1\right].\end{eqnarray} \tag{ 9 }$$

This expression can similarly be used to calculate the facular depth contribution but we assume this contribution is negligible on AU Mic. Equation (9) does not explicitly rely on Δf_spot, but the value we derive for Δf_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_spot, f_hot, T_spot, T_hot, and T_amb are injected into this model to generate a posterior distribution of ΔD(λ)_spot.

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 separately 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 autocorrelation 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 (T_spot ≤ T_amb ≤ T_hot) while allowing f_spot and f_hot to vary, enabling the models to arrive at solutions where the the surface is ≥50% covered in spots. Technically f_hot is allowed to vary as high as 50% but in practice the models almost never preferred values of f_hot greater than a few percent. Time-domain analysis of the seven 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_spot, f_hot, and Δf_spot which could be be a robust finding, considering the time between visits is 2× the 120–150 day activity evolution timescale (which we can approximate to be a spot decay timescale) measured by Donati et al. (2023).

For the F21 visit, we measure variability semiamplitudes, ΔS/S, in $g^{\prime}$, $r^{\prime}$, and $i^{\prime}$ of 0.075 ± 0.006, 0.071 ± 0.006, and 0.041 ± 0.007, respectively, and for the S22 visit we measure ΔS/S of 0.075 ± 0.003 in g' and 0.075 ± 0.003 in r', shown in Figure 3 and summarized in Table 4.

Zoom In Zoom Out Reset image size

Measuring the signal for i' was slightly challenging because, as the reddest bandpass, this filter showed the weakest variability signal and therefore the smallest S/N. Moreover, as AU Mic is much brighter than the nearby comparison stars and the exposure times are short, scintillation noise is prevalent. To account for poorly constrained measurements in $i^{\prime}$ and improve the internal consistency of our multicolor variability measurements, we impose a prior on the phase (θ) for i'. The independently modeled $g^{\prime}$ and $r^{\prime}$ rotation curves agree within their 1σ 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 three 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.

We examine the spot characteristics results when we model only the stellar effective temperature and photometric variability data (excluding the NRES spectral models) and similarly when we model only the NRES spectra with the stellar effective temperature (excluding the multicolor photometric variability data). Finally, we examine the results of modeling the stellar T_eff, photometric variabilities, and NRES spectra together in an "ensemble" model which models the 12 spectral orders and multicolor variabilities for both visits, from which we report the final results.

4.2.1. Photometric Variability

Figure 4 shows our measured photometric variability semiamplitudes with random samples drawn from the photometry-only posterior distributions, with results listed in Table 5. The photometric variabilities, along with AU Mic' T_eff, show evidence for essentially any spot coverage between 10% and 90% with temperature ${3141}_{-389}^{+266}$ K, changing throughout an orbit by 6% ± 3%. Solutions for ambient temperature are ${3719}_{-502}^{+302}$ K, with an upper limit of 40% coverage of a hot component with temperature ${4873}_{-727}^{+1922}$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 being related to anything physical. The measurement of T_spot is surprisingly consistent with results from the spectral models.

Zoom In Zoom Out Reset image size

Two-temperature models of the variabilities return tighter constraints on the ratio of T_spot to T_amb and their individual measurements, but the measurement of T_spot is significantly warmer (T_spot = ${3454}_{-208}^{+155}$ K) than virtually every other measurement of T_spot we present in this paper. Measurements of f_spot and Δf_spot are consistent with the three-temperature fits. Results for the 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 for all three components. The change in spot coverage, Δf_spot is unconstrained by these models because we are modeling time-averaged spectra. The stellar spectra indicate a large fraction (f_spot = 0.39 ± 0.04) of cool spots with temperature ${2974}_{-71}^{+72}$ K, a dominant "ambient" (warmer) photosphere with temperature ${4002}_{-15}^{+14}$ K, and a very tenuous detection of a hot component with temperature ${8681}_{-629}^{+868}$ K covering less than 0.5% of the surface.

Zoom In Zoom Out Reset image size

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

4.2.3. Ensemble Model

Results from our ensemble model, where all five 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_spot = 0.39 ± 0.04, Δf_spot = ${0.05}_{-0.003}^{+0.004}$, T_spot = ${3003}_{-71}^{+63}$ K, and T_amb = ${4003}_{-14}^{+15}$ K. Similar results for the two-temperature modeling are shown in Table 6.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 5. Parameters from Fitting the Different Data Combinations with a Three-temperature Model

Parameter	Photometry Model	Spectral Model	Ensemble Model
Tspot (K)	${3141}_{-389}^{+266}$	${2974}_{-71}^{+72}$	${3003}_{-71}^{+63}$
fspot	${0.57}_{-0.28}^{+0.22}$	0.39 ± 0.04	0.39 ± 0.04
Δfspot	0.06 ± 0.03	<0.40	${0.05}_{-0.003}^{+0.004}$
fhot	<0.40	<0.01	<0.005
Thot (K)	${4873}_{-727}^{+1922}$	${8681}_{-629}^{+868}$	${8671}_{-629}^{+890}$
Tamb (K)	${3719}_{-502}^{+302}$	${4002}_{-15}^{+14}$	${4003}_{-14}^{+15}$
Teff (K)	3664 ± 101	3783 ± 37	3789 ± 35
χ2	2.59 (2.59)	2124 (0.581)	2136 (0.584)

Note. The photometry model uses the 5 photometric variability measurements and 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 two-temperature model, most importantly recovering an approximately 3000 K 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.

Download table as:  ASCIITypeset image

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

Table 6. Parameters from Fitting the Different Data Combinations with a Two-temperature Model

Parameter	Photometry Model	Spectral Model	Ensemble Model
Tspot (K)	${3454}_{-208}^{+155}$	${3083}_{-45}^{+31}$	${3093}_{-41}^{+29}$
fspot	${0.61}_{-0.27}^{+0.22}$	0.41 ± 0.03	0.41 ± 0.03
Δfspot	${0.08}_{-0.04}^{+0.06}$	<0.4	0.05 ± 0.003
Tamb (K)	${3876}_{-157}^{+234}$	${3998}_{11}^{9}$	${3998}_{-12}^{+10}$
Teff (K)	3649 ± 98	3704 ± 24	3707 ± 24
χ2	1.9 (1.9)	3060 (0.837)	3058 (0.836)

Note. The photometry model uses the 5 photometric variability measurements and 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 the three-temperature results can be seen in the temperatures measured for T_spot and T_amb.

Download table as:  ASCIITypeset image

We can see that this ensemble model exhibits characteristics of both the photometry-only and spectra-only models. The photometric variabilities strongly constrain Δf_spot and limit how cool the spots can be but provide poor constraints on the coverage fraction of spots. Variability measurements in multiple wave bands across the optical–near-infrared (NIR) constrain the temperature contrast because the relative change in variability with wavelength is set by the ratio of spectral components. Further into the red, the spot and ambient spectra are more similar and the variability is less. At bluer wavelengths, there is an increasing contrast between cool spots and the surrounding photosphere, up to a maximum of 100% (where the spot flux is effectively 0 relative to the surrounding photosphere).

The spectral decomposition returns a precise estimate of f_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_spot and 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 Δf_spot.

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.

The two-temperature models agree with the three-temperature results remarkably well, showing a spot coverage of 41% ± 3% that changes throughout a rotation by 5.1% ± 0.3%, with T_spot = ${3093}_{-41}^{+29}$ and T_amb = ${3998}_{-12}^{+10}$.

We forward modeled spot contamination (Equation (9)) under the assumption of 0% spot coverage on the transit chord using the ensemble model posteriors generated in the previous step, as shown in Figure 9. At short wavelengths like the TESS bandpass (0.6–1 μm), contamination ranges from 550 to 1400 ppm. In the WFC3 bandpass, contamination is between 500 and 750 ppm, and decreases to 250–500 ppm 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 μm) transits translates to about 1875 ppm, so the spot contamination is 25%–75% of that signal if f_tra = 0. This means that without accounting for spot contamination, AU Mic b's true radius (3.46 R_E 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 and 750 ppm. This translates to an overestimated planetary radius of between 0.6 and 1.3 Earth radii. 

Zoom In Zoom Out Reset image size

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.

Photometric variability has a 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 one, 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 amplitudes are 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 the spot contrast and limits how cool the spots can be. If the $i^{\prime}$ measurements showed greater variability, this would allow the 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–infrared in order to determine precisely the wavelength regime where variability decreases or "turn-off" happens. The wavelengths where this turn-off is observed are highly descriptive of the stellar spot spectra.

The primary results in this work are reported from the three-temperature modeling, but we examine how those results changed when modeled with only two temperatures, finding it returns measurements for f_spot, Δf_spot, T_spot, and T_ambthat are strongly consistent with the three-temperature results. When we allow a third component, it finds a poorly constrained temperature between 7000 and 10,000 K, with a small number of solutions closer to the ambient photosphere, around 4000–5000 K. If the 7000–10,000 K component is physical, it is likely a flux contribution from flares, which occur so frequently they cannot entirely be removed by time averaging.

Faculae on M dwarfs may be up to a few hundred kelvin 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, measured to be less than 0.5%.

There is one key difference in the outcomes of the two- and three-temperature models. When modeling only T_eff and photometric variabilities, the two-temperature model returns a significantly warmer T_spot and a tighter ratio of T_spot to T_amb compared with the three-temperature model. A third temperature, which only factors into the calculation of T_eff, brings the spot temperature solutions into agreement with the other measurements we report, with larger uncertainties and a weaker (though still noticeable) relationship between the spotted and unspotted temperatures.

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

Much work has been done to measure and theoretically determine the 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.

5.3.1. Spot Temperature

5.3.2. Spot Filling Factor

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

• 1.  
  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.
• 2.  
  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.
• 3.  
  Our spot contamination forward models assume that f_tra = 0, which means our contamination results are upper-limit cases for different sets of spot parameters. In reality, there may be some nonzero fraction of spot coverage on the transit chord, which will lower the contamination level in the transmission spectrum.
• 4.  
  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 the spot temperature from photometric variability models.
• 5.  
  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 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 = 100,000–500,000 and is more appropriate for the R = 53,000 echelle spectra we acquired.