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A&A
Volume 690, October 2024
Article Number A357
Number of page(s) 11
Section Planets and planetary systems
DOI https://doi.org/10.1051/0004-6361/202451301
Published online 18 October 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1 Introduction

The study of variability in the atmospheres of planets and sub- stellar objects has a long and rich history, for instance, the observed variability of the atmosphere of Earth over a wide variety of timescales (Peixóto & Oort 1984). The variability of brown dwarfs at the ~1% level has been observationally established for over a decade (Artigau et al. 2009; Apai et al. 2013; Radigan et al. 2014; Metchev et al. 2015). It is believed to be caused by rotational modulated surface features that are the consequence of heterogeneous cloud cover (Showman & Kaspi 2013; Apai et al. 2013; Crossfield et al. 2014; Morley et al. 2014). Several time series observations support this idea (e.g., Radigan et al. 2012; Buenzli et al. 2014; Metchev et al. 2015). More recent studies have suggested that these clouds possess vertical structures (Apai et al. 2013; Yang et al. 2015; Zhou et al. 2018). Heterogeneities in thermal structures (Robinson & Marley 2014) and methane abundance (Tremblin et al. 2020) have also been proposed. Additionally, a recent study by Rowland et al. (2023) examined the impact of abundance profile assumptions by performing atmospheric retrievals on synthetic, cloud-free L dwarf spectra generated from the Sonora Bobcat models at SpeX resolution. They demonstrated that a non-uniform (step function) description for the iron hydride abundance profiles is needed to accurately retrieve bulk properties of early L dwarfs.

The study of brown dwarf variability is an important precursor to the study of variability in exoplanetary atmospheres, which is still in its infancy and its reports of variability are still a matter of debate (e.g., Agol et al. 2010; Apai et al. 2016; Armstrong et al. 2016; Cowan et al. 2017; Jackson et al. 2019; Hooton et al. 2019; Komacek & Showman 2020; Biller et al. 2021; Cho et al. 2021; Jones et al. 2022; Hardy et al. 2022; Lally & Vanderburg 2022).

VHS 1256 b (full name: VHS J125060.192-125723.9 b) was originally discovered by Gauza et al. (2015). It has a spectral type of L7, a distance of 21.14 ± 0.22 pc (Gaia Collaboration 2016, 2021), an age of 140 ± 20 Myr (Dupuy et al. 2023), and a low surface gravity (Petrus et al. 2023). Recently, Zhou et al. (2022) analyzed Wide Field Camera 3 (WFC3) spectra of VHS 1256 b from 1.12 to 1.65 µm using the Hubble Space Telescope (HST) at two different epochs (Bowler et al. 2020; Zhou et al. 2022). The object stands out as a prime example of a variable brown dwarf as it exhibits 24% color variability on the timescale of a rotation period (~22 h). Additionally, for long-term variability on a timescale of nearly ~900 rotation periods, a peak-to-valley flux difference of 33% ± 2% was found. An even higher amplitude was achieved, reaching 38% in the J band, the highest amplitude ever observed in a sub-stellar object (Bowler et al. 2020; Zhou et al. 2022).

The luminosity of VHS 1256 b at its current age is consistent with two possible evolutionary tracks: one where deuterium is inert and another where it fuses. As a result, there are two potential mass estimates for VHS 1256 b: 12.0 ± 0.1 MJup or 16 ± 1 MJup. These values are based on evolutionary models from Saumon & Marley (2008), which produces a bimodal probability distribution that places it on either side of the exoplanet-brown dwarf division. The corresponding surface gravities log 𝑔, with 𝑔 in cgs units, were recently constrained by Dupuy et al. (2023). They reported values of log 𝑔 = 4.268 ± 0.006 and log 𝑔 = 4.45 ± 0.03, respectively.

The near-IR (NIR) spectra of VHS 1256 b have revealed weak alkali absorption, suggestive of its youth (Gauza et al. 2015; Petrus et al. 2023; Dupuy et al. 2023). Fitting atmospheric forward models (Burrows et al. 2006) to spectrophotometric data suggests the presence of a cloudy atmosphere in VHS 1256 b. The observed variability suggests dynamic atmospheric processes, potentially influenced by a patchy cloud structure, as demonstrated by Vos et al. (2023) for two variable, planetary-mass early T dwarfs using HST and Spitzer data. Furthermore, weak methane absorption suggests a relative scarcity of methane, which may be attributed to disequilibrium chemistry (Miles et al. 2023). Large variability of NIR-spectra on the timescale of a rotation period motivates the question of whether the atmospheric properties of VHS 1256 b are changing on these timescales – and, if so, which the most variable properties would be.

The traditional approach of analyzing brown dwarf spectra is to extract its atmospheric and bulk properties. This process involves performing interpolation on a pre-computed model grid of spectra and evolutionary tracks and subsequently acquiring the best-fit spectra through the application of a goodness-of-fit metric (e.g., Marley et al. 1996; Burrows et al. 1997; Chabrier et al. 2000; Ackerman & Marley 2001; Allard et al. 2001; Baraffe et al. 2002; Burrows et al. 2003, 2011; Morley et al. 2014; Zhang et al. 2021a,b). This allows the practitioner to infer the surface gravity, mass, and effective temperature of the brown dwarf (e.g., Burgasser et al. 2006; Cushing et al. 2008; Kasper et al. 2009; King et al. 2010; Bowler et al. 2010; Hinkley et al. 2015; Franson et al. 2023). The caveat to such an approach is that the model parameters are usually sampled uniformly (either in a linear or logarithmic sense), which may lead to biased posterior distributions when sampling over more than two parameters; this is particularly true for parameters exhibiting nonlinear effects on the spectrum (Fisher & Heng 2022). It has also been shown that these self-consistent model grids, despite incorporating an array of sophisticated chemistry and physics, systematically under- or overestimate the inferred mass of brown dwarfs for objects where dynamical masses have been measured (Dupuy et al. 2009; Konopacky et al. 2010; Cheetham et al. 2018; Beatty et al. 2018; Rickman et al. 2020; Bowler et al. 2021; Brandt et al. 2021).

In more recent years, techniques have been developed applying machine learning methods to analyze the atmospheres of exoplanets and brown dwarfs (e.g., Waldmann 2016; Márquez- Neila et al. 2018; Zingales & Waldmann 2018; Cobb et al. 2019; Fisher et al. 2020; Yip et al. 2021; Matchev et al. 2022; Ardévol Martínez et al. 2022; Vasist et al. 2023; Gebhard et al. 2023). Instead of interpolating on a pre-computed grid of models, machine learning methods use the model grid as a training set, which is used to map a measured spectrum to the parameter values of the grid. Traditional Bayesian methods often sacrifice physical realism for computational feasibility and are typically not self-consistent, but are capable of exploring parameter space widely. Therefore, we have utilized self-consistent model grids in our interpretation of the HST WFC3, VLT/X-shooter, and ERS JWST spectra of VHS 1256 b, in tandem with a supervised machine learning approach (Márquez-Neila et al. 2018); however, we also interpreted the spectra using the standard Bayesian inference approach of nested sampling (Skilling 2006; Feroz et al. 2009; Trotta 2008; Kitzmann et al. 2020) for HST WFC3 and VLT/X-shooter.

In Section 2, we describe the archival data set that we are using for this study and the atmospheric retrieval techniques, as well as the legacy model grids used for the current study. Outcomes from a large number of atmospheric retrievals, both orbit-averaged and intra-orbit, as well as their time series analysis for HST data are presented in Section 3. This section also includes the BeAR retrieval analysis of the curated VLT/X- shooter spectrum and grid-based machine-learning retrievals for the ERS JWST spectrum using HELA. Section 4 presents implications from our study and opportunities for future work.

2 Methodology

2.1 Data

The two epochs of HST-WFC3 spectra of VHS 1256 b, taken in 2018 and 2020, were originally published in Bowler et al. (2020) and Zhou et al. (2022). The six-orbit continuous monitoring from UT 2018-03-05 16:02:30 to 2018-03-06 00:42:47 (Program ID: GO-15197; PI: Bowler) resulted in a light curve that spanned less than half of its rotation period (~9 h). In 2020, spectroscopic monitoring of VHS 1256 b was performed for fifteen orbits from UT 2020-05-26 09:27:17 to 2020-05-28 03:29:24 (Program ID: GO- 16036; PI: Zhou), covering approximately two rotation periods (~42 h). The spectral resolution is R~130 at 1.4 µm and the wavelength range is from 1.12 to 1.65 µm.

For illustration, the spectra of both epochs were combined separately and their minimum and maximum values are displayed in Fig. 1. It is apparent that the flux variation is wavelength-dependent. It was suggested that heterogeneous clouds are the main driver for the observed spectral variability (Bowler et al. 2020; Zhou et al. 2020, 2022). The variability was found to be greater in 2018 (red) than 2020 (blue), when considering the flux ratios between maximum and minimum values of each epoch (see the lower panel of Fig. 1).

As a complementary analysis, we considered the simultaneous 0.65-2.5µm wavelength-dependent medium-resolution (3300 ≤ Rλ ≤ 8100) VLT/X-shooter spectrum published by Petrus et al. (2023) (Fig. 8) and the latest Early Release Science (ERS) JWST spectrum collected by Miles et al. (2023), which observed VHS 1256 b with JWST’s NIRSpec IFU and MIRI MRS modes, generating a spectrum spanning 1-18 µm at a ~1000–3700 resolution (e.g., Fig. 9).

2.2 Bayesian atmospheric retrievals: BeAR

We performed Bayesian atmospheric retrievals using tools that have been widely adopted in past studies. For instance, it was previously applied to a curated sample of 19 L and T dwarfs (Lueber et al. 2022), as well as HIP 21152 B, the first T dwarf companion in the Hyades (Franson et al. 2023).

In this study, we used the open-source Bern Atmospheric Retrieval code (BeAR). BeAR is a new and renamed version of the former Helios-r2 code described by Kitzmann et al. (2020) with enhanced capabilities and additional forward models. For this work we employed the emission spectroscopy forward model of BeAR. The code, together with a full documentation, has been released under a GNU General Public License v3.0 and can be found in a public GitHub repository1. The retrieval code uses the MULTINEST version of the Bayesian nested sampling (Skilling 2006; Feroz et al. 2009) algorithm in order to explore the multi-dimensional parameter space of our models. The radiative transfer calculations were done via the method of short characteristics (Olson & Kunasz 1987).

The discretized temperature-pressure profile has 70 levels and is described by seven parameters via a finite element approach (Kitzmann et al. 2020). The Bayesian model comparison follows standard approaches for calculating the Bayesian evidence and Bayes factor of Bij (Trotta 2008), which are natural outputs of the nested sampling algorithm (Skilling 2006). A recognized association exists between the logarithm of the Bayes factor and the number of standard deviations by which one model is disfavored by the data (Trotta 2008). Values of In Bij = 1, 2.5, and >5 indicate “weak”, “moderate”, and “strong” evidence for a statistical preference between the two models (Trotta 2008). However, it is noteworthy that Bayesian model comparison may not always successfully rule out un-physical scenarios, as demonstrated by Fisher & Heng (2019).

The opacities (cross-sections per unit mass) of atoms and molecules were computed using the open-source HELIOS-K calculator (Grimm & Heng 2015; Grimm et al. 2021) and made publicly available via the DACE database (Grimm et al. 2021). We included the following molecules: H2O, CH4, NH3, CO2, CO, H2S, CrH, FeH, CaH, TiH, TiO, VO, HCN, as well as the alkali metals Na and K. The corresponding line lists are taken from the ExoMol database (Barber et al. 2006; Yurchenko et al. 2011; Yurchenko & Tennyson 2014; Azzam et al. 2016) and the HITEMP database (Rothman et al. 2010). Collision-induced absorption coefficients for H2–H2 and H2–He are based on Abel et al. (2011) and Abel et al. (2012), respectively. Following Kitzmann et al. (2020), we use the line profile descriptions of Allard et al. (2016) and Allard et al. (2019) for the resonance lines of Na and K.

Our cloud model follows basic principles of Mie theory and was calibrated using Mie calculations based on measured refractive indices (Kitzmann & Heng 2018). This description of a non-gray cloud layer is aimed at parameterizing the particles’ extinction coefficients. No attempt has actually been made to realistically model the formation of these clouds. The cross-section of monodisperse cloud particles is designed to continuously transition from small to large particles, which correspond to a spectral slope and a constant (gray) cross-section, respectively. The labels “small” and “large” refer to the “radiative” size of a particle; namely, its physical size compared to the wavelength of radiation being absorbed or scattered. This cloud model was implemented into Helios-r2 in Lueber et al. (2022) and is now also part of the newest version of the code, which was renamed BeAR.

All parameters and their prior distributions are shown in Table 1. In total, we have 22 free parameters for the cloud-free model, 25 free parameters for the gray cloud model, and 28 for the non-gray cloud description, respectively. Following Lueber et al. (2022), we adopted a uniform prior distributions for the cloud optical depth, because that results in it being constrained. We note that we allowed negative values in its prior range to ensure that the normal prior boundary of 0 was sampled correctly. Negative optical depths were then internally replaced by 0 to avoid unphysical results.

thumbnail Fig. 1

Minimum and maximum fluxes from the two epochs of HST observations of VHS 1256 b in 2018 and 2020. The data were first published by Bowler et al. (2020) and Zhou et al. (2022), displayed here for illustration.

2.3 Machine learning atmospheric retrievals: HELA

The use of supervised machine learning allows one to perform approximate Bayesian computation using pre-computed model grids. In our open-source HELA computer code, we implemented the random forest method for performing atmospheric retrievals (Márquez-Neila et al. 2018), which was subsequently used in several follow-up studies on analyzing high-resolution spectra of the ultra-hot Jupiter KELT-9b (Fisher et al. 2020), mediumresolution spectra of benchmark brown dwarfs (Oreshenko et al. 2020; Lueber et al. 2023), information content of James Webb Space Telescope spectra (JWST; Guzmán-Mesa et al. 2020), and the sampling strategy of model grids (Fisher & Heng 2022). Training and testing utilized 80 and 20% of the training set, respectively, as established in these earlier studies.

We selected three legacy model grids of brown dwarf atmospheres to use as training sets for HELA. The model grid of Hubeny & Burrows (2007) was computed using the COOLTLUSTY code (Sudarsky et al. 2003; Hubeny et al. 2003; Burrows et al. 2006). It includes both equilibrium and disequilibrium chemistry for cloud-free and cloudy atmospheres. For analyzing the spectra of VHS 1256 b, we utilized only the cloudy models with disequilibrium chemistry. This disequilibrium chemistry is represented by vertical mixing, characterized by a vertical eddy diffusion coefficient Kzz , which is set to 106 cm2s–1 in the current study. The model grid covers 4.5 ≤ log g ≤ 5.5 (cgs units) and 700 ≤ Teff/K ≤ 1800 in steps of ∆ log g = 0.5 and ∆Teff = 100 K, respectively. To improve the predictability of model parameters, we perform a one-dimensional (1D) linear interpolation of the measured flux for sufficiently small surface gravity steps (∆ log 𝑔 = 0.025) to increase the total grid size from 208 to 480 spectra (see discussion in Lueber et al. 2023).

The BT-Settl model grid (Allard et al. 2012) was computed using the PHOENIX computer code (Hauschildt 1992; Hauschildt et al. 1997). It accounts for vertical convective mixing and overshooting (Allard et al. 2011). The BT-Settl model grid covers 2.0 ≤ log 𝑔 ≤ 5.5 (cgs units) and 500 ≤ Teff /K ≤ 2400 in steps of ∆ log 𝑔 = 0.5 and ∆ Teff = 50 - 100 K, respectively. Again, we performed a 1D linear interpolation of the measured flux for surface gravity (∆ log 𝑔 = 0.05), resulting in an increase of the total grid size from 216 to 1237 spectra.

The most modern model grid among the three sets is the cloudy Sonora Diamondback model (Morley et al. 2024), which covers 3.5 ≤ log 𝑔 ≤ 5.5 (cgs units) and 900 ≤ Teff/K ≤ 2400 in steps of ∆ log 𝑔 = 0.50 and ∆Teff = 100 K, respectively. Based on the radiative-convective equilibrium model by Marley & McKay (1999), chemical equilibrium holds throughout the atmosphere. The model accounts for vertical mixing using mixing length theory within the Ackerman & Marley (2001) cloud parametrization. Three metallicities [M/H] are available (solar, +0.5, –0.5), as well as the cloud parameter fsed from 1.0 to 8.0. The 1D linear interpolation of the measured flux was performed to achieve surface gravity steps of ∆ log 𝑔 = 0.025 cm/s2 and a total grid size of 11 521 individual spectra, compared to 1440 in the original grid.

In addition to the mentioned varying parameters of our considered grids, we introduced an additional parameter, a so-called flux scaling factor, f, to our retrievals. Its purpose is to scale the radius-distance relationship for the outgoing flux, Fν+$F_\nu^ + $, of the brown dwarf to the one measured by the observer, (Fν): Fν=Fν+f(ℛd)2,${F_v} = F_v^ + f{\left( {{{\cal R} \over d}} \right)^2},$(1)

where d is the distance between the observer and the brown dwarf and ℛ the prior brown dwarf radius. To incorporate it into our training sets, we assume a uniform distribution of 0.5 ≤ f ≤ 2.0. We randomly drew an f value for each model spectrum and multiplied the entire set of fluxes by it.

Table 1

Summary of retrieval parameters and prior distributions for the free chemistry approach used in the cloud-free, gray cloud, and non-gray cloud models.

3 Results

3.1 Standard Bayesian retrieval analysis and sensitivity test of the retrieved radius using HST data

We perform a suite of HST retrievals using the BeAR code, applied on the minimum, average and maximum spectra (in terms of the flux) from each epoch using models without clouds, with gray clouds and with non-gray clouds. The outcome of this suite of retrievals is reported in Table A.1 in terms of the retrieved atmospheric properties. Generally, the most meaningful quantities that have been aptly constrained are the surface gravity, photometric radius, water abundance (by volume), cloud optical depth, and cloud-top pressure. Table A.2 compares the Bayesian evidences and natural logarithm of the Bayes factors, ln Bij , which demonstrates that cloudy models are strongly favored (Trotta 2008). Additionally, the Bayesian model comparison favors a gray cloud description over non-gray cloudy models. Therefore, Figs. 2 and A.1 show the best-fit model outcome of our HST atmospheric retrieval analysis for the HST 2018 maximum brightness spectrum with a gray cloud description, including the joint posterior distributions of the parameters, as well as the median spectra. The retrieved posterior distributions are for the surface gravity (log 𝑔), the radius (R; as explained in Eq. [10] of Kitzmann et al. 2020), the distance to VHS 1256 b (d), and the chemical abundances and the cloud properties. The cloud properties include the optical depth, τ, the cloud-top pressure, pt, the cloud-bottom pressure, pb = bcpt, the (spherical) cloud particle radius, a, the spectral slope associated with small particles, a0 , a proxy for the particle composition, Q0, and the reference optical depth, τref , at 1 µm. As we are displaying the exemplary outcome for the HST 2018 maximum brightness spectrum retrieval with gray clouds, we have only shown the posterior distributions of τ, pt, and pb. Except for water, the chemical compositions of molecules are poorly constrained or unconstrained.

The retrieved surface gravity values of log 𝑔 ~ 5.2–5.7, are reasonable for field and high-mass brown dwarfs, but certainly higher than expected by previous studies of Gauza et al. (2015) (log 𝑔 = 4.24 ± 0.35) or Miles et al. (2018) (log 𝑔 = 3.2), which are based on model grids. The retrieved effective temperatures of Teff ~ 1360-1390 K are approximately 100-150 K hotter than the value obtained by Miles et al. (2018) with 1240 K. The prediction by Gauza et al. (2015) with Teff=880−110+140${{\rm{T}}_{{\rm{eff}}}} = 880_{ - 110}^{ + 140}$ is about 400-700 K cooler than the temperature expected for late-L field dwarfs. The retrieved radii R ∼ 0.7–0.8 RJ are lower than expected given the age of VHS 1256 b. Similarly small radii have been inferred in numerous other retrieval studies (e.g., Zalesky et al. 2019; Gonzales et al. 2020; Lueber et al. 2022). Dupuy et al. (2023) derived estimates from using the hybrid cloudy evolutionary model by Saumon & Marley (2008), suggesting two approximate radii, R, of 1.30 RJup and 1.22 RJup, effective temperatures, Teff, of 1153 ± 5 K and 1194 ± 9 K, as well as surface gravities of log 𝑔 = 4.268 ± 0.006 and log 𝑔 = 4.45 ± 0.03, according to the two potential mass estimates for VHS 1256 b of 12.0 ± 0.1 MJup and 16± 1 MJup.

A closer look at the water abundances retrieved from the HST spectra with BeAR reveals somewhat high values and a strong positive correlation with surface gravity (see Fig. A.1 and Table A.1). These results are aligned with the findings of Phillips et al. (2024), who also reported somewhat high (∼1–10%) H2O abundances for two late-L objects, W0047 and BD+60 1417B, using SpeX data. This emphasizes the challenges associated with characterizing water in low-gravity L dwarfs. After performing several tests, which included fixing the gravity based on values from previous publications or adjusting the pressure range of the atmosphere, we conclude that water and surface gravity show a degeneracy in these atmospheric retrievals on HST spectra, since the H2O-content depends strongly on the assumed prior of log 𝑔. It should be noted that these adjustments did not affect the retrieved effective temperatures, radii, or cloud parameters. Thus, for our further runs, we decided to fix the prior of log 𝑔 to the two values taken from Dupuy et al. (2023): log 𝑔 = 4.268 ± 0.006 and log 𝑔 = 4.45 ± 0.03, respectively. Since these values are derived from evolutionary models, our approach is somewhat circular. We further note that the derived radius and effective temperatures are markedly different from the ones corresponding to the evolutionary models used by Dupuy et al. (2023).

As an additional test, we conduced retrieval test runs with a vertically non-uniform H2O abundance profile for the HST 2018 maximum brightness spectrum, for both fixed surface gravity values. For the vertically non-uniform description of the H2O abundance, BeAR implements a description of the abundance profile based on a finite-element approach, analogous to the temperature profile described in Eq. (12) of Kitzmann et al. (2020). Unless stated otherwise, we used two first-order elements for the abundance profile, which results in a total of three free additional parameters for the retrieval. Comparing the retrieval statistics of the vertically non-uniform water abundance models against the corresponding uniform models results in a strong preference for vertically varying H2O abundance profiles (ln Bij ≈ 13.4 forafixed log𝑔 = 4.268 ± 0.006 and ln Bij ≈ 13.1 for log 𝑔 = 4.45 ± 0.03, respectively). However, Fig. 3 also graphically highlights that both abundance profile descriptions agree in the range of greatest contribution (0.1–1 bars, see Fig. A.1) within the respective uncertainties. The non-constant vertical water profiles may arise from latitudinal or longitudinal inhomogeneities within the brown dwarf atmosphere. This is manifested in a non-constant profiles in a 1D atmospheric retrieval.

thumbnail Fig. 2

Posterior median spectra (F) and residuals (∆F) associated with the free-chemistry retrieval analyses of the HST 2018 maximum brightness spectrum with a gray-cloud model. The data are shown as red dots with associated uncertainties.
thumbnail Fig. 3

Retrieved volume mixing ratios for H2O of the HST 2018 maximum brightness spectrum (left). Four retrieval runs were conducted: a vertically uniform and a non-uniform H2O abundance profile one for each of the two fixed surface gravity values of log g = 4.268 ± 0.006 (low) and log g = 4.45 ± 0.03 (high). Right panel shows the corresponding posterior median spectra (F) for all four runs and differences between them (∆F). The HST 2018 maximum brightness spectrum data is shown as black dots with associated uncertainties.

3.2 Time series HST retrieval analysis

Next, we applied our retrieval framework towards analyzing an entire time series of spectra. Since data from the 2018 epoch displays the most variability, we chose to focus our efforts on the 66 spectra collected during the same HST visit. Following the outcome of the previous sub-section, we use only cloudy models with gray clouds, as well as both vertical uniform water and non-uniform H2O-abundance profiles. Figure 4 shows the retrieved atmospheric properties across the 66 different spectra, which are constructed from six different orbits of VHS 1256 b with 11 spectra in each orbit. We fixed the surface gravity to a value of log 𝑔 = 4.268 ± 0.006, due the degeneracy with the water abundance. The analogous plot for a surface gravity value of log 𝑔 = 4.45 ± 0.03 can be found in Fig. A.2.

We note that the retrieved radius (≈0.8 RJ) and effective temperatures (≈1350 K) are markedly different compared to the values associated with the evolutionary models used by Dupuy et al. (2023), who reported R ≈ 1.2–1.3 RJ and Teff ≈ 1200 K. In other words, despite using the surface gravities reported by Dupuy et al. (2023) as priors for our retrievals the retrieved radii and effective temperatures are discrepant. This suggests that different input physics and chemistry were used in the analyses and also highlights their model-dependent nature.

By fitting a flat line to the retrieved quantities across time, both for vertical uniform non-uniform description of H2O, we find that almost all of the quantities result in reduced chi-square values well below unity. This implies the existence of two potential scenarios: either the quantities remain invariant over time or our observations and retrievals lack the sensitivity required to detect any temporal variability. As expected from previous studies, the effective temperature does increase with time and is highly correlated with the variability amplitude of VHS 1256 b, resulting in Pearson correlation coefficients of ≈0.85. The only exception next to the effective temperature slightly increasing with time is the abundance of H2O, which exhibits a slight variation between the first, second and third orbits. The variation over time is limited to a pressure of around 0.1–1 bars in the vertically non-uniform description, corresponding to the pressure range in which the largest contribution comes from.

The temporal variability of VHS 1256 b cannot be attributed to a single varying factor, probably because the various atmospheric properties are degenerate with one another. It is tempting to conclude that the atmospheric properties adjust rapidly to seek an equilibrium and therefore a stable climate. However, without access to a set of spectra covering a wider wavelength range at different epochs, it is premature to draw this conclusion.

We next investigate whether performing the retrieval using pre-computed model grids will result in the same outcome. Using our three selected model grids (see Section 2.3), we report the posterior distributions of the effective temperature, Teff, surface gravity, log 𝑔, and radius scaling factor, f , in Fig. 5, as well as metallicity, [M/H], and cloud parameter, fsed, for Sonora Diamondback in Fig. 6. Consistent with our standard Bayesian analysis, we find that these retrieved quantities are invariant across the 231 spectra, for which we performed retrievals (in both epochs: 2018 and 2020). The only exception is the scaling factor f for the model grid by Morley et al. (2024), showing a reduced chi-square value of χr2=2.773$\chi _r^2 = 2.773$ when fitting a straight line trough all 231 posteriors. However, physically, we do not expect to see an ≈10% increase in the scaling factor, f . This might aptly correspond to the masked effect of a proportional increase in the effective temperature Teff (see Eq. (1)). Previous work has shown that retrievals performed with the supervised machine learning method of the random forest tend to produce more conservative (broader) posterior distributions (Fisher et al. 2020). In our case, the great error bars are likely influenced by the large grid step sizes and sparsely sampled grids, which can increase uncertainties (see, e.g., Lueber et al. 2023). While the broader posteriors from the random forest method reflect its inherent caution, the grid resolution and sampling density are additional factors contributing to the retrieved error bars.

Finally, Fig. 7 illustrates the best-fit spectra in a montage, combining a collection of model spectra with an effective temperature of 1100 K and varying surface gravity values for the three model grids (COOLTLUSTY in Hubeny & Burrows 2007, BT-Settl in Allard et al. 2012, and Sonora Diamondback in Morley et al. 2024), as well as the best-fit spectra of the nested sampling (BeAR). For illustration, we have opted to analyze the spectrum corresponding to the maximum brightness observed in the HST data from the year 2018. Substantial discrepancies exist between the actual best-fit spectra and the data sets. Such differences are not visible in retrievals with BeAR. Furthermore, the individual grids differ from each other. For a more thorough analysis, we refer to the discussion in Lueber et al. (2023). Overall, the poor fits of the model grids to the data (in Fig. 7) suggest that these models are unsuitable for quantifying the temporal variation of atmospheric properties (as shown in Figs. 5 and 6), unless they are being used to simulate patchy or non-uniform atmospheres (as shown in, e.g., Miles et al. 2023).

thumbnail Fig. 4

HST18 BeAR posteriors with a fixed surface gravity of log 𝑔 = 4.268 ± 0.006 and vertically uniform (left column) versus non-uniform (right column) H2O abundances as a sequence over all 6 orbits in 2018. Each orbit consist of 11 intra-orbit spectra. Orbit-averaged posteriors are shown in red. The reduced chi-square values correspond to a fit of a straight line through all posteriors of a given parameter. The lowest panel represents the brightness change in G141 broad band, normalized by 3.0 × 10−13 erg/s/cm2/µm (data taken from Bowler et al. 2020 and Zhou et al. 2022).
thumbnail Fig. 5

Machine learning retrieval outputs for legacy model grids. From left to right: COOLTLUSTY from Hubeny & Burrows (2007), BT-Settl by Allard et al. (2012), and Sonora Diamondback by Morley et al. (2024). The HST exposures cover data from 2018 (66) and 2020 (165), where the x-axes are broken to fit the two epochs of data. Each orbit consists of 11 intra-orbit spectra. Posterior median values are indicated as black dots, while the corresponding error bars are shown in green. The represented reduced chi-square values correspond to a fit of a straight line through all posteriors of a given parameter.
thumbnail Fig. 6

Additional machine learning retrieval outputs for legacy model grid Sonora Diamondback by Morley et al. (2024). The HST exposures cover data from 2018 (66) and 2020 (165), where the x-axes are broken to fit the two epochs of data. Each orbit consists of 11 intra-orbit spectra. Posterior median values are indicated as black dots, while corresponding error bars are shown in green. The represented reduced chi-square values correspond to a fit of a straight line trough all posteriors of a given parameter.
thumbnail Fig. 7

Collection of model spectra with an effective temperature of 1100 K and varying surface gravity values for the three legacy model grids (COOLTLUSTY from H+07c: Hubeny & Burrows 2007, BT-Settl by A+12: Allard et al. 2012, and Sonora Diamondback by M+23: Morley et al. 2024, with a fixed [M/H] of 0.0 and fsed = 2), as well as the best-fit spectra of the BeAR free-chemistry retrieval analyses with a gray-cloud model. Data is shown for the HST 2018 maximum brightness spectrum as dark blue dots with associated uncertainties.

3.3 Constraining C/O and [M/H] ratios from medium-resolution NIR VLT/X-shooter spectra

As a complementary analysis, we considered the 0.65–2.5 µm medium-resolution (3300 ≤ Rλ ≤ 8100) VLT/X-shooter spectrum taken by Petrus et al. (2023). In contrast to their study, which compared the data set to a spectral grid of cloudless ATMO models (Tremblin et al. 2015) with the Bayesian inference tool called ForMoSa (Petrus et al. 2023), we used the spectrum to calculate the C/O and [M/H] ratios from posteriors of a cloudy atmospheric retrieval with BeAR. Figs. 8 and A.3 show the posterior median spectra and joint posterior distributions from the free-chemistry retrieval analyses of the VLT/X-shooter spectrum with a gray-cloud model. Compared to HST retrievals of VHS 1256 b, no degeneracy between surface gravity and H2O was found, presumably due to the significantly greater spectral resolution compared to HST/WC3, as well as the extended wavelength range, which includes gravity-sensitive lines.

The retrieved surface gravity value is slightly higher than expected, namely, logg=4.9992−0.0008+0.0005$\log g = 4.9992_{ - 0.0008}^{ + 0.0005}$. The water abundance was constrained with a value of log H2O=−3.37−0.01+0.01$\log {{\rm{H}}_2}{\rm{O}} = - 3.37_{ - 0.01}^{ + 0.01}$, in good agreement with our previously constrained abundances of HST retrievals (log H2O ∼ 10−3, when fixing the surface gravity to log 𝑔 = 4.268 ± 0.006 and log 𝑔 = 4.45 ± 0.03, respectively). We note that the HST retrievals without fixed gravity values yield ∼1% water abundances (Fig. A.1 and Table A.1). The retrieved atmospheric properties of VHS 1256 b result in a solar to super-solar C/O-ratio of C/O = 0.77 ± 0.49 and a metallicity of [M/H] = 0.53 ± 0.27 based on the elements contained in the molecules constrained in the retrieval.

Following the enhanced data quality provided by the VLT/X- shooter spectrum, further investigation warrants a separate, indepth study into non-uniform abundance profiles. While the Bayesian model comparison strongly preferred the adoption of such non-uniform profiles of H2O (ln Bij ≈ 725), their detailed parametrization and analysis thereof exceed the scope of the current study. Future efforts will focus on exploring the implications of these findings and leveraging the enhanced data quality provided by X-shooter and JWST to deepen our understanding of the atmospheric composition of such objects as VHS 1256b.

thumbnail Fig. 8

Posterior median spectra (F) and residuals (∆F) associated with the free-chemistry retrieval analyses of the simultaneous 0.65–2.5 µm medium-resolution VLT/X-shooter spectrum taken by Petrus et al. (2023) with a gray-cloud model. The data are shown as red dots, with the associated uncertainties.
thumbnail Fig. 9

JWST spectra of VHS 1256 b (gray dots), overlaid by a collection of best fit model spectra over the full wavelength coverage of the three legacy model grids (COOLTLUSTY from H+07c: Hubeny & Burrows 2007, BT-Settl by A+12: Allard et al. 2012, and Sonora Diamondback by M+23: Morley et al. 2024).

3.4 HELA applied to JWST spectra

The latest publications on VHS 1256 b now focus on the JWST spectrum collected by Miles et al. (2023). They observed VHS 1256 b with JWST’s NIRSpec IFU and MIRI MRS modes, generating a spectrum spanning 1–18 microns at a ∼1000–3700 resolution. A forward modeling approach was used (Petrus et al. 2024), as well as the first atmospheric retrievals applied to the JWST data of a brown dwarf (Whiteford et al., in prep.). Both studies highlight the new challenges we are facing with JWST and indicate possible next steps towards a complete picture of VHS 1256 b. We have gone on to apply our random forest approach with HELA to analyze the JWST data.

Performing the random forest retrievals using pre-computed model grids result in best-fit spectra, shown in Fig. 9. None of the model grids considered fit the combined JWST spectrum, as indicated by the reduced chi-square values that greatly exceed unity. Despite the large deviations, the cloudy grid Sonora Diamondback (Morley et al. 2024) provides the best fit. Following the approach in Petrus et al. (2024), examining the spectra of the different instruments individually yields significantly better fits, which we show for Sonora Diamondback in Fig. 10 and BT-Settl in Fig. A.4. This goes along with the fact that the posteriors, effective temperatures, surface gravity, and scaling factors show a wavelength dependence.

4 Discussion and conclusions

Despite ∼30% variations in the emergent flux of VHS 1256 b, its retrieved atmospheric properties are consistent within errors when just considering HST/WFC3 data. This implies the existence of two potential scenarios: either the properties remain invariant over time or our observations and retrievals lack the sensitivity required to detect any temporal variability. Such substantial variations have never been definitively observed in exoplanetary atmospheres (e.g., Agol et al. 2010; Apai et al. 2016; Biller et al. 2021). A good agreement was found between retrieved water abundances for HST and VLT/X-shooter using BeAR (Figs. 4, A.2, and A.3) and retrieved parameters from the machine-learning retrievals using HELA for HST and JWST data (Figs. 5, 6, 10, and A.4). Additionally, our analysis of JWST spectra indicates consistency in retrieved properties across different wavelength channels, further strengthening the reliability of our findings. This bodes well for retrieval analyses that will be performed on next-generation spectra of exoplanetary atmospheres, suggesting that retrieval outcomes will be robust within the applied framework. JWST’s broader wavelength coverage will likely enable more robust, time-resolved retrievals, enhancing our understanding of atmospheric variability. Hitherto, a JWST Cycle 2 proposal (Program ID: GO #3375, PI: Whiteford) has been accepted to observe 1–14 µm spectra of VHS 1256b in time series mode to study its variability. Our study has already found that both HST and VLT/X-shooter data necessitate vertically varying abundance profiles of water to model the spectra accurately. Consequently, the choice of the retrieval method and its parametrization will become even more crucial in the future. Certainly, the final word will come following a detailed retrieval analysis on these spectra. Hence, it will be necessary to test future retrievals for non-uniform abundance profiles akin to those exemplified in this study or to employ self-consistent models, as executed by Rowland et al. (2023), to ensure capture of the full complexity imprinted in the data. With the broader wavelength range covered by the JWST, it will become essential not only for H2O but also for molecules that are expected to vary even more with pressure, as, for example, FeH, TiO or CH4 (Rowland et al. 2023).

A worrying outcome of Fig. 5 is that the retrieved surface gravities from machine learning grid-based retrievals are markedly different: log 𝑔 ≈ 5 (COOLTLUSTY) versus logg ≈ 3.2 (BT-Settl) versus log 𝑔 ≈ 4.2 (Sonora Diamondback). Consistently with previous works (e.g., Oreshenko et al. 2020 and Lueber et al. 2023), this suggests that the retrieved surface gravity is somewhat model-dependent, unlike for the effective temperature. For comparison, Gauza et al. (2015) derived log 𝑔 = 4.24 ± 0.35, and Miles et al. (2018) derived log 𝑔 = 3.2, based on the use of PHOENIX model grids.

An even more direct comparison can be made by looking at derived values by Petrus et al. (2024), where the authors compared parameters from fits across the full JWST wavelength range (0.97–18.02 µm) and from 15 distinct spectral windows. They combined NIRSpec and MIRI spectra in the full range and independently fitting smaller segments in the windowed approach. Values reported by Petrus et al. (2024) are comparable to our study, as they found log 𝑔 = 3.50 ± 0.01 (full range) and 3.62 ± 0.60 (windows) for the BT-Settl model, as well as log 𝑔 < 3.50 (full range) and 4.50 ± 0.60 (windows) for the Sonora Diamondback model. The COOLTLUSTY model was not used in their study.

Since it is difficult to decipher how each model grid is constructed thoroughly, we have not attempted to track down the source of this discrepancy and this will be the subject of future work. In general, model grids containing a large range of surface gravity values tend to result in predictions at the lower end of the prior range. Another opportunity for future work is to run general circulation models of brown dwarfs (Showman & Kaspi 2013), with a focus on the temporal variation of their atmospheric properties.

thumbnail Fig. 10

JWST spectra of VHS 1256 b (dark gray dots), overlaid by the Sonora Diamondback (Morley et al. 2024) best fit model spectra for each of the 15 independent spectral windows. Corresponding posteriors for effective temperature, Teff, surface gravity, log g, metallicity, [M/H], cloud parameter, fsed , and scaling factor, f , are presented in the five lower panels, where the parameter range of the grid itself is indicated by the colored area. The posterior values and corresponding errors when using the entire spectrum are shown by the gray line and shaded gray area, respectively. Dashed lines represent the corresponding averaged values from the machine learning retrievals with HST data (see Figs. 5 and 6).

Data availability

The appendix related to this paper is available on Zenodo: https://doi.org/10.5281/zenodo.13754670.

Acknowledgements

A.L. and K.H. acknowledge partial financial support from the Swiss National Science Foundation and the European Research Council (via a Consolidator Grant to K.H.; grant number 771620), as well as administrative support from the Center for Space and Habitability (CSH). Calculations were performed on UBELIX (https://www.id.unibe.ch/hpc), the HPC cluster at the University of Bern. J.M.V. acknowledges support from a Royal Society - Science Foundation Ireland University Research Fellowship (URF\R1\221932).

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All Tables

Table 1

Summary of retrieval parameters and prior distributions for the free chemistry approach used in the cloud-free, gray cloud, and non-gray cloud models.

In the text

All Figures

thumbnail Fig. 1

Minimum and maximum fluxes from the two epochs of HST observations of VHS 1256 b in 2018 and 2020. The data were first published by Bowler et al. (2020) and Zhou et al. (2022), displayed here for illustration.
In the text
thumbnail Fig. 2

Posterior median spectra (F) and residuals (∆F) associated with the free-chemistry retrieval analyses of the HST 2018 maximum brightness spectrum with a gray-cloud model. The data are shown as red dots with associated uncertainties.
In the text
thumbnail Fig. 3

Retrieved volume mixing ratios for H2O of the HST 2018 maximum brightness spectrum (left). Four retrieval runs were conducted: a vertically uniform and a non-uniform H2O abundance profile one for each of the two fixed surface gravity values of log g = 4.268 ± 0.006 (low) and log g = 4.45 ± 0.03 (high). Right panel shows the corresponding posterior median spectra (F) for all four runs and differences between them (∆F). The HST 2018 maximum brightness spectrum data is shown as black dots with associated uncertainties.
In the text
thumbnail Fig. 4

HST18 BeAR posteriors with a fixed surface gravity of log 𝑔 = 4.268 ± 0.006 and vertically uniform (left column) versus non-uniform (right column) H2O abundances as a sequence over all 6 orbits in 2018. Each orbit consist of 11 intra-orbit spectra. Orbit-averaged posteriors are shown in red. The reduced chi-square values correspond to a fit of a straight line through all posteriors of a given parameter. The lowest panel represents the brightness change in G141 broad band, normalized by 3.0 × 10−13 erg/s/cm2/µm (data taken from Bowler et al. 2020 and Zhou et al. 2022).
In the text
thumbnail Fig. 5

Machine learning retrieval outputs for legacy model grids. From left to right: COOLTLUSTY from Hubeny & Burrows (2007), BT-Settl by Allard et al. (2012), and Sonora Diamondback by Morley et al. (2024). The HST exposures cover data from 2018 (66) and 2020 (165), where the x-axes are broken to fit the two epochs of data. Each orbit consists of 11 intra-orbit spectra. Posterior median values are indicated as black dots, while the corresponding error bars are shown in green. The represented reduced chi-square values correspond to a fit of a straight line through all posteriors of a given parameter.
In the text
thumbnail Fig. 6

Additional machine learning retrieval outputs for legacy model grid Sonora Diamondback by Morley et al. (2024). The HST exposures cover data from 2018 (66) and 2020 (165), where the x-axes are broken to fit the two epochs of data. Each orbit consists of 11 intra-orbit spectra. Posterior median values are indicated as black dots, while corresponding error bars are shown in green. The represented reduced chi-square values correspond to a fit of a straight line trough all posteriors of a given parameter.
In the text
thumbnail Fig. 7

Collection of model spectra with an effective temperature of 1100 K and varying surface gravity values for the three legacy model grids (COOLTLUSTY from H+07c: Hubeny & Burrows 2007, BT-Settl by A+12: Allard et al. 2012, and Sonora Diamondback by M+23: Morley et al. 2024, with a fixed [M/H] of 0.0 and fsed = 2), as well as the best-fit spectra of the BeAR free-chemistry retrieval analyses with a gray-cloud model. Data is shown for the HST 2018 maximum brightness spectrum as dark blue dots with associated uncertainties.
In the text
thumbnail Fig. 8

Posterior median spectra (F) and residuals (∆F) associated with the free-chemistry retrieval analyses of the simultaneous 0.65–2.5 µm medium-resolution VLT/X-shooter spectrum taken by Petrus et al. (2023) with a gray-cloud model. The data are shown as red dots, with the associated uncertainties.
In the text
thumbnail Fig. 9

JWST spectra of VHS 1256 b (gray dots), overlaid by a collection of best fit model spectra over the full wavelength coverage of the three legacy model grids (COOLTLUSTY from H+07c: Hubeny & Burrows 2007, BT-Settl by A+12: Allard et al. 2012, and Sonora Diamondback by M+23: Morley et al. 2024).
In the text
thumbnail Fig. 10

JWST spectra of VHS 1256 b (dark gray dots), overlaid by the Sonora Diamondback (Morley et al. 2024) best fit model spectra for each of the 15 independent spectral windows. Corresponding posteriors for effective temperature, Teff, surface gravity, log g, metallicity, [M/H], cloud parameter, fsed , and scaling factor, f , are presented in the five lower panels, where the parameter range of the grid itself is indicated by the colored area. The posterior values and corresponding errors when using the entire spectrum are shown by the gray line and shaded gray area, respectively. Dashed lines represent the corresponding averaged values from the machine learning retrievals with HST data (see Figs. 5 and 6).
In the text

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