Aerosols and hydrocarbons within the ambiance of a white dwarf planet

Data discount

Our PRISM remark used 33 teams throughout 240 integrations, with every integration lasting 29.8 s, for a complete observing time of 1.98 h. Although the transit length of WD 1856 b solely lasts 8 min, we chosen this observing window to make sure JWST captured a transit of WD 1856 b with enough out-of-transit baseline for detector settling. We used two unbiased codes to cut back the NIRSpec PRISM remark of WD 1856 b and extract transmission spectra: FIREFLy and Juniper. Here we element the discount and lightweight curve becoming method utilized by every code.

FIREFLy

We first decreased the WD 1856 b information utilizing the Fast InfraRed Exoplanet Fitting Lyghtcurve (FIREFLy)^13,29,30 discount suite. The discount began with the uncalibrated (uncal.suits) photographs and a custom-made jwst pipeline discount. During levels 1 and a pair of, the 1/f noise was eliminated on the group stage, utilizing the highest and backside six rows to measure the rely stage for every column and subtracting the median worth. The darkish present step was skipped and the bounce step was carried out with a rejection threshold of 20. During stage 3, we then used the custom-run pipeline 2D photographs after the jwst.assign_wcs step and carried out custom-made cleansing of unhealthy pixels, cosmic rays and scorching pixels. We used cross-correlation to measure the positional shift of the spectral hint throughout the detector and shift-stabilized the photographs with flux-conserving interpolation. This process has been discovered to cut back the amplitude of position-dependent systematic developments^13,29. An aperture measurement of 5.7 pixels was used to extract the spectra, with this product used to suit the transit mild curves and extract the exoplanet spectra.

During levels 4 and 5, we match the white mild curve utilizing a linear baseline and a limb-darkened transit mannequin³¹. The stellar limb darkening was modelled with the procedures from ref. ³² and a quadratic perform utilizing ExoTiC-LD³³. The values have been fastened to the best-fitting theoretical host white dwarf mannequin values (see the ‘White dwarf host spectrum’ part beneath). We measured the white mild curve utilizing the 0.55 to three μm area, such that the planetary emission wouldn’t bias the transit depth and ensuing system parameters. The NIRSpec PRISM spectral time sequence for FIREFLy is proven in Extended Data Fig. 1). Several pre-transit exposures confirmed abnormally low flux ranges, which we flagged as outliers and faraway from the remaining evaluation. These outliers appear to be due to clusters of vibrant/scorching pixels, so are most likely related to snowball occasions. We fastened the interval utilizing the outcomes from ref. ³⁴. The best-fit white mild curve system values are given in Extended Data Table 1. The transmission spectral mild curves used 3-pixel binning and have been match with the identical mannequin, setting the system parameters to be fastened to the white mild curve values.

Juniper

We additionally utilized Juniper, a brand new {custom} pipeline for JWST NIRSpec observations, to cut back our WD 1856 b information. Juniper accommodates a wrapper for levels 1 and a pair of of the jwst pipeline with {custom} steps. We begin our processing from Juniper stage 1 with the uncal.suits information from the Mikulski Archive for Space Telescopes (MAST). We choose to disable the jwst stage 1 bounce detection step and as a substitute deal with cosmic rays by {custom} procedures in later levels. Before ramp becoming, group-level background subtraction is carried out utilizing the highest and backside six rows because the background area to cut back scatter within the extracted mild curves. We spatially filter 3σ outliers from this area and common alongside columns to find out the background stage of counts per column, which is subtracted from the complete group. We then proceed with jwst stage 1 ramp becoming and achieve scaling. Juniper stage 2 is a pure wrapper for jwst stage 2; we supply out this stage with the flat discipline and photom steps disabled, as neither is required to measure transit depth and we noticed the previous to extend the noise within the spectral extraction.

Juniper stage 3 performs further cleansing on the integration stage. We first masks pixels flagged by the jwst pipeline for information high quality points. We then reject cosmic rays in time over two iterations, changing 6.5σ outliers with the median worth of the pixel in time. Finally, a second spherical of background subtraction, utilizing the identical technique because the group-level background subtraction in stage 1, is carried out utilizing the highest and backside three rows with outliers masked at 3σ.

Stage 4 of the Juniper pipeline extracts 1D spectra, that are subsequently binned to supply mild curves. Our aperture is centred on the brightest row of the hint and extends ±3 pixels above and beneath it. We carry out optimum extraction³⁵ to extract the 1D spectrum, taking as our extraction profile the median picture of the hint contained within the aperture, normalized alongside columns. We sum throughout all pixels from 0.552 to three μm to extract a broadband mild curve; we select to not embody mild from wavelengths longer than 3 μm as this mild is strongly affected by contamination from nightside thermal emission, which impacts the decided system parameters (for instance, semimajor axis, transit epoch, impression parameter). We then bin each 3 pixels to supply 137 median-normalized spectroscopic mild curves at almost native decision, spanning 0.552 to five.360 μm. Our extracted broadband and spectroscopic mild curves would sometimes be sigma-clipped to additional take away outliers; nevertheless, this system is vulnerable to clipping out the transit itself owing to the quick transit length and huge transit depth. We subsequently disable this process and use various outlier rejection procedures in stage 5.

Juniper stage 5 is the ultimate stage of the pipeline, which inserts transit fashions to every mild curve to extract transit depth and produce a transmission spectrum. Our becoming process is a two-step course of combining linear and nonlinear becoming strategies, which we use to wash outliers that sigma-clipping can not safely take away. We begin through the use of a linear least-squares estimator to suit a batman transit mannequin³¹, making use of a quadratic limb-darkening legislation with coefficients generated with ExoTiC-LD³³ utilizing a {custom} white dwarf mannequin produced by becoming a white dwarf spectrum to the out-of-transit flux of WD 1856 (described beneath). Our mannequin is additional multiplied by a linear-in-time pattern to account for visit-long ramp results. We then compute the residuals of the fitted transit and systematics mannequin and clip any factors within the mild curve that produce 3σ outliers within the residuals. We compute the usual deviation of the sigma-clipped residuals to estimate the photometric uncertainty, which we provide to the following step of our match process. We refit the sigma-clipped mild curve utilizing Markov chain Monte Carlo strategies³⁶ to extract our last planet–star radius ratio spectrum. We first match our 0.552 to three μm broadband mild curve utilizing this two-step becoming course of to find out the broadband depth, semimajor axis a/R_*, inclination i, mid-transit epoch t_C and linear-in-time systematics mannequin parameters, from which we derive the impression parameter b and its uncertainty. The orbit interval P was held fastened to 1.407939217 days based mostly on a follow-up paper finding out transit timing variations within the WD 1856+534 system³⁴. Our broadband mild curve evaluation yielded a/R_* = 339.25 ± 5.92 and b = 7.34 ± 0.20. We then repair these values in addition to t_C as decided by the broadband mild curve match for all subsequent spectroscopic mild curve suits. We match each spectroscopic mild curve with our two-step course of to find out R_p(λ)/R_* and the linear systematics pattern in each wavelength channel. Our broadband mild curve match achieves residuals of 522 ppm, whereas our spectroscopic suits obtain median residuals of 8,153 ppm. We current our fitted system parameters (a/R_*, b, R_p/R_*) and broadband transit depth in Extended Data Table 1.

Grazing transit spectroscopy

The distinctive transit geometry of the WD 1856 system required us to develop a brand new method to specific and mannequin transmission spectra. A transmission spectrum encodes the wavelength-dependent efficient space of a planet relative to its host star. Exoplanet analyses sometimes take the spectroscopic planet–star radius ratio from mild curve suits, R_p(λ)/R_*, after which specific the transmission spectrum as ({R}_{{rm{p}}}{(lambda )}^{2}/{R}_{ast }^{2}). This amount is equal to the transit depth for a planet with radius R_p(λ) totally occulting a non-limb-darkened star of radius R_*. However, as a result of WD 1856 b is seven occasions bigger than its white dwarf host with a grazing transit, the transmission spectrum can’t be written as ({R}_{{rm{p}}}{(lambda )}^{2}/{R}_{ast }^{2}). Indeed, the transit depth of WD 1856 b relies on time all through the transit (see ref. ¹¹), with a most transit depth equivalent to the time of best areal overlap between the planet and its host (Fig. 1). We subsequently specific the transmission spectrum of WD 1856 b because the wavelength-dependent most transit depth on the time of mid-transit, A_p/A_*.

We convert the spectroscopic planet–host radius ratio into the mid-transit transit depth by calculating the time-dependent space overlap of two discs. The overlapping space of two circles with radii R_p (representing the planet) and R_* (representing the white dwarf), separated by a distance d, is given by:

during which

Considering the time of mid-transit (when d = bR_*, during which b is the transit impression parameter), we are able to specific the utmost transit depth as:

during which

We use equations (4), (5) and (6) to map the spectroscopic radius ratio, R_p/R_*, and impression parameter from the spectroscopic mild curve suits of every information discount into the equal mid-transit transmission spectrum. We use the uncertainties Python bundle to propagate errors utilizing these formulae. This method routinely removes offsets between the completely different reductions for R_p/R_*, as every corresponding pair of R_p/R_* and b should yield constant A_p/A_* to have the identical transit form (that’s, to match Fig. 1).

We present our last transmission spectra of WD 1856 b, expressed because the mid-transit transmission spectrum (A_p/A_*), in Extended Data Fig. 2. Both reductions clearly detect the robust signature of nightside contamination (the slope to decrease transit depths at longer wavelengths) and result in constant atmospheric inferences from our retrieval evaluation (see the ‘Atmospheric retrieval analysis’ part). We observe that the 2 reductions partially deviate at wavelengths longer than 5 μm—primarily due to variations within the light-darkening remedies and uncertainties within the crimson edge detector behaviour—so we restricted our atmospheric evaluation for WD 1856 b to the NIRSpec PRISM information from 0.5–5 μm.

White dwarf host spectrum

We extracted a calibrated out-of-transit NIRSpec PRISM stellar spectrum for WD 1856 utilizing the FIREFLy information discount. Starting from the cleaned 2D photographs, we additional flat-fielded, flux-calibrated and extracted the ensuing host spectrum. The ensuing stellar spectra are proven in Extended Data Fig. 3.

We decided the atmospheric parameters of the host white dwarf by becoming the out-of-transit system flux. We minimized the χ² for a mannequin appropriate for cool white dwarfs¹⁴ outlined by three parameters: the efficient temperature of the white dwarf, T_eff, its photospheric hydrogen-to-helium abundance ratio and the strong angle πR_*²/D². Because the gap D is understood from the Gaia DR3 parallax, the strong angle straight constrains the radius of the white dwarf. The radius, in flip, determines the mass and floor gravity of the white dwarf given theoretical white dwarf construction fashions³⁷. The best-fit resolution (Extended Data Fig. 3) corresponds to T_eff = 4,920 Okay, logg = 8.05 and N_H/N_He = 4.1. This resolution yields an Hα line that extends 2% beneath the continuum, which is in step with beforehand obtained optical spectroscopy¹¹ (not thought-about right here in our match). We additionally tried to suit the PRISM spectrum utilizing pure-hydrogen fashions (that’s, with out contemplating N_H/N_He as a free parameter), however the best-fit resolution yields a a lot worse match to the PRISM information than the combined H and He ambiance resolution. We calculated limb-darkening coefficients for the best-fitting white dwarf mannequin (utilizing the method from ref. ³⁸), which we then fastened in the course of the WD 1856 b spectroscopic mild curve suits for our two information reductions.

Transmission spectrum modelling

The grazing transit geometry of WD 1856 b, coupled with the clear presence of planetary nightside thermal emission, requires a brand new modelling method. The transmission spectrum for a planet with a grazing transit and nightside thermal emission might be written as³⁹:

during which A_p(high) is the realm of the planet overlapping the star on the high of the modelled ambiance (given by equation (1) with R_p = R_p,high), ({{mathcal{T}}}_{lambda }) is the atmospheric transmissivity (({{rm{e}}}^{-{tau }_{lambda }}), during which τ_λ is the slant optical depth) within the space factor dA and F_{p(evening),λ} and F_*,λ are the noticed fluxes from the planetary nightside and white dwarf at Earth, respectively. We scale back the realm integral within the first time period to a single integral over the fractional annuli of the planet overlapping the star (that’s, dA = A_p(r_i,up) − A_p(r_i,low), during which we use the radii of the higher and decrease boundaries of every atmospheric layer rather than R_p in equation (1)). The first time period in equation (7) represents the wavelength-dependent efficient space of the fraction of the planet overlapping the white dwarf, relative to the projected disk space of the white dwarf. The second time period accounts for the ‘nightside pollution’/dilution of the transit depth¹⁷ owing to thermal emission from the planetary hemisphere dealing with the observer.

Transmission spectra of WD 1856 b can be expressed when it comes to emergent fluxes through the use of the strong angle relation between the noticed flux and the emergent (floor) flux, such that equation (7) turns into:

during which R_{p,(evening),λ} is the radius of the emitting thermal photosphere on the nightside (nominally the τ_v,λ = 2/3 strain stage, during which τ_v,λ is the vertical optical depth built-in downwards from the highest of the ambiance). The planet–star floor flux ratio featured in equation (8) is a regular output from radiative switch codes used to calculate exoplanet emission spectra. Similarly, the transmissivity, ({{mathcal{T}}}_{lambda }), is already calculated by radiative switch codes calculating commonplace transmission spectra. Therefore, to calculate the transmission spectra of WD 1856 b, we are able to assemble a mannequin ambiance after which calculate each the transmissivity from the slant optical depth and the emergent planet–host flux ratio. The noticed transmission spectrum then represents a product between a grazing transit transmission spectrum and an ‘upside-down’ emission spectrum.

Atmospheric retrieval evaluation

We infer the atmospheric properties of WD 1856 b utilizing the open-source Bayesian atmospheric retrieval code POSEIDON^15,16. We mannequin the ambiance of WD 1856 b utilizing 100 layers spaced uniformly in log-pressure from 10⁻⁷ to 100 bar. We assume that the ambiance is effectively combined, with constant atmospheric properties on the day–evening terminator and on the nightside, such that solely a single set of parameters describe the atmospheric state. We match for the planetary radius on the 10 bar strain stage and the planetary mass, whereas fixing the white dwarf radius to R_* = 0.0131 R_Sun and the transit impression parameter to 7.430234 (because the impression parameter uncertainty is already marginalized into the R_p/R_* uncertainties, it doesn’t must be an unbiased free parameter). We embody the log₁₀ mixing ratios of the next molecules as free parameters: CH₄, NH₃, H₂O, CO₂, CO, HCN, C₂H₂, C₂H₄, C₂H₆, H₂S and PH₃. The the rest of the ambiance consists of H₂ and He with an abundance ratio of He/H₂ = 0.17, in step with the large planets within the Solar System⁴⁰. We parameterize the temperature profile of WD 1856 b utilizing an adaptation of a prescription used for brown dwarfs⁴¹. This prescription retrieves the temperature at 9 strain nodes (spaced uniformly per decade in strain from 10⁻⁶ to 100 bar) and interpolates between them with a spline. The 9 free parameters defining this temperature profile are the 100 mbar temperature and eight ΔT_i parameters encoding the temperature distinction between every pair of nodes. Given the low exterior irradiation of WD 1856 b, we prohibit ΔT_i > 0 to contemplate bodily believable profiles with temperature monotonically growing with strain. Finally, we match for a three-parameter aerosol mannequin consisting of an influence legislation scattering slope (with exponent γ) and an optically thick cloud-top strain¹⁵. We don’t think about inhomogenous clouds across the terminator, as solely a small fraction of the terminator of WD 1856 b occults the floor of the white dwarf throughout transit.

Our retrieval mannequin is thus outlined by 25 free parameters, which we match utilizing MultiNest’s^42,43,44 Python wrapper PyMultiNest⁴⁵ with 1,000 stay factors. The priors for every parameter are summarized in Extended Data Table 2. We calculate Bayes elements (that’s, odds ratios; ({mathcal{B}})) utilizing Bayesian mannequin comparisons between nested retrieval fashions, with the retrieval mannequin statistics summarized in Extended Data Table 3. For consistency with the exoplanet literature, we additionally convert the Bayes issue between two nested fashions (for instance, our reference mannequin and a mannequin excluding CH₄) into an ‘equivalent detection significance’, Nσ, utilizing a regular relation⁴⁶. We observe, nevertheless, that there are a number of caveats related to the Bayes issue to detection significance mapping, so our most popular statistic for mannequin desire is the Bayes issue/odds ratio (see ref. ⁴⁷).

We calculate mannequin transmission spectra of WD 1856 b by fixing equation (8) on a wavelength grid starting from 0.5 to five.6 μm at R = 20,000. We pattern high-resolution pre-computed cross-sections³⁹ onto this wavelength grid, utilizing the next line record sources: CH₄ (ref. ⁴⁸), NH₃ (ref. ⁴⁹), H₂O (ref. ⁵⁰), CO₂ (ref. ⁵¹), CO (ref. ⁵²), HCN (ref. ⁵³), C₂H₂ (ref. ⁵⁴), C₂H₄ (ref. ⁵⁵), C₂H₆ (ref. ⁵⁵), H₂S (ref. ⁵⁶) and PH₃ (ref. ⁵⁷). We additionally embody continuum opacity from H₂ and He collision-induced absorption⁵⁸ and Rayleigh scattering. For the host flux, we use the best-fit white dwarf mannequin proven in Extended Data Fig. 3. Our mannequin transmission spectra are lastly convolved with the NIRSpec PRISM level unfold perform and binned right down to the decision of the observations to calculate the probability of every location within the retrieval mannequin parameter area.

Although Fig. 3 compares a number of retrieved atmospheric properties between the FIREFLy and Juniper information reductions, we offer the complete posterior distributions in Extended Data Fig. 4. We discover glorious settlement between FIREFLy and Juniper for all retrieved parameters.

To interpret the thermal historical past of WD 1856 b, we additionally calculate posterior distributions for the planetary efficient temperature from our retrieval outcomes. We calculated the emergent planetary floor flux of WD 1856 b for every ambiance within the full set of posterior samples from each the FIREFLy and Juniper reductions on a wavelength grid from 1 to 50 μm. For every set of atmospheric parameters, we calculate the corresponding efficient temperature utilizing the Stefan–Boltzmann legislation: ({T}_{{rm{eff}}}={left(frac{1}{{sigma }_{{rm{SB}}}}int {F}_{{rm{p}},{rm{surf}},lambda }{rm{d}}lambda proper)}^{1/4}). Extended Data Fig. 5 reveals our retrieved floor flux spectrum for WD 1856 b for each information reductions and the corresponding T_eff posterior distributions. Using the bottom 1σ credible interval (from FIREFLy) and the best 1σ credible interval (from Juniper), we discover a vary of 390–412 Okay for T_eff. We equally report the 1σ vary encompassing each information reductions for M_p in the principle textual content. The roughly 10 Okay 1σ uncertainty on T_eff is pushed by the quite a few CH₄ bands detected in our NIRSpec PRISM information setting the relative amplitude of different CH₄ options at longer wavelengths. However, the potential presence of different hydrocarbons, reminiscent of C₂H₆, permits a bigger floor flux uncertainty during which these species take up within the mid-infrared (for instance, 10–15 μm), which will increase the uncertainty within the built-in energy and therefore T_eff. Longer wavelength observations of WD 1856 b with MIRI LRS/MRS, reminiscent of these deliberate in JWST Cycle 4 (GO-9033 and GO-9157), will constrain T_eff even additional.

Mie scattering retrievals

We have established that fashions together with aerosol opacity are required to clarify the transmission spectrum of WD 1856 b. Specifically, our free retrieval evaluation above infers an opaque cloud deck close to 100 mbar and a haze to clarify the power-law scattering slope shortwards of 1 μm. The enhanced scattering slope signifies a group of small particles within the higher ambiance, however our parametric description is agnostic to the particular aerosol composition. Here we think about retrievals together with Mie scattering to research which particular aerosol species are in step with the transmission spectrum of WD 1856 b.

The composition of small, Mie scattering particles might be doubtlessly recognized by the use of aerosol absorption options at infrared wavelengths, whereas their particle measurement is encoded by the scattering slope. We assess right here which aerosol species and particle sizes can clarify the noticed scattering slope by the use of retrievals together with compositionally particular Mie scattering aerosols. We don’t take a look at straight for particular species inflicting the opaque cloud deck, as this deck might be composed of enormous particles with muted resonance options^59,60. Because such a condensate cloud deck has no spectroscopic options, it’s not doable to find out the composition until condensates are lofted above the deck and change into smaller in measurement.

We use the Mie scattering retrieval module and database launched in POSEIDON v1.2 (ref. ⁶¹). Our Mie scattering retrievals use aerosol extinction cross-sections pre-computed from refractive indices. We primarily think about a easy aerosol mannequin parameterized by the log₁₀ imply particle measurement (logr_m ({mathcal{U}},[-3,1])) and aerosol log₁₀ quantity mixing ratio (log aerosol ({mathcal{U}},[-30,-1]))—representing a well-mixed aerosol uniformly distributed inside the ambiance. We additionally examined extra complicated aerosol fashions that match for pressure-dependent aerosol mixing ratios however these all decreased to a pressure-independent mannequin. Our Mie retrievals use a six-parameter strain–temperature (P–T) profile⁶². We conduct these Mie retrievals on the FIREFLy information discount.

We ran retrievals with a collection of aerosol species representing three completely different aerosol formation regimes that could possibly be related in higher ambiance of WD 1856 b. The first aerosol regime represents disequilibrium hazes and soot species that may be produced by photochemistry: Titan tholins (tholins^63,64), carbon soot (C (ref. ⁶⁵)), water-rich natural haze at two temperatures (ExoHaze 300K, ExoHaze 400K (ref. ⁶⁶)) and hexene (C₆H₁₂ (ref. ⁶⁷)). The second aerosol regime represents the myriad of sulfide and chloride clouds that kind in brown dwarfs on the T–Y transition (400–1,300 Okay)²⁷ alongside Cr: chromium (Cr; Lynch and Hunter in ref. ⁶⁸), magnesium sulfide (MnS (ref. ⁶⁹)), sodium sulfide (Na₂S (ref. ²⁷)), zinc sulfide (ZnS (ref. ⁷⁰)) and potassium chloride (KCL; Palik and Addamiano in ref. ⁷¹) (ordered by condensation temperature). The third aerosol regime consists of condensed ices that kind deep cloud decks in Solar System planets and doubtlessly cooler Y dwarfs (≤400 Okay)^72,73. These ices may trigger the opaque cloud deck present in our retrievals above, that are then lofted to greater atmospheric pressures to trigger the noticed scattering slope, or they may condense in situ within the colder higher ambiance: water ice (H₂O (ref. ⁷⁴)), ammonia ice (NH₃ (ref. ⁷⁵)) and methane ice (CH₄ (ref. ⁷⁶)) (ordered by condensation temperature).

We discover that every one aerosol species, aside from MnS and hexene, present good suits to the scattering slope and solely imprint weak absorption options into the transmission spectrum. Using Bayesian mannequin comparisons, the best-fit haze and soot species is the water-rich natural ExoHaze (the 400K variant), the best-fit T–Y dwarf cloud species is KCl and the best-fit ice is NH₃. Of these three aerosols, KCl has the best Bayesian proof. The potential presence of KCl can be in step with expectations for chilly T–Y dwarf fashions²⁷, during which KCl varieties the best, low-density cloud. However, we observe {that a} easy gray cloud deck + haze mannequin (as utilized in the principle textual content) is most popular over KCl by about 2σ. Therefore, the current information for WD 1856 b shouldn’t be sufficiently exact to determine a transparent desire for which particular aerosols are current within the ambiance of WD 1856 b.

Our Mie scattering retrievals present insights into the vary of particle sizes and abundances appropriate with the quick wavelength scattering slope WD 1856 b (Extended Data Fig. 6). The ExoHaze and NH₃ ice fashions favour a group of small particles (about 0.03 μm) with low mixing ratios (about 10⁻¹⁴), whereas the KCl mannequin favours even smaller particles (roughly 0.01 μm) with the next abundance (about 10⁻⁸). Compared with our default gray cloud deck + haze retrieval mannequin, we discover constant outcomes for different mannequin parameters to inside 1σ. In explicit, we present that the retrieved planetary mass shouldn’t be delicate to the assumed aerosol mannequin. We do discover roughly 1 dex decrease median CH₄ abundances for the Mie scattering retrievals and therefore a decrease C/H ratio, however the CH₄ abundance distribution continues to be in step with our leads to the principle textual content. We observe that the marginal proof of C₂H₆ strengthens when together with Mie scattering in contrast with the deck + haze mannequin (Extended Data Fig. 6) however this molecule shouldn’t be strongly detected with the current information.

Our retrieved temperature construction from the Mie scattering retrievals additionally signifies an environment that’s a lot hotter than the equilibrium temperature of WD 1856 b (Extended Data Fig. 6). As with our gray cloud and haze retrieval, we additionally discover a temperature of about 400 Okay within the thermal photosphere close to 10–100 mbar. However, as a result of the Mie scattering retrievals can not produce an optically thick cloud deck on the pressures required to obscure thermal emission from the deep ambiance (roughly 10^−1.5 bar), the Mie retrievals compensate by making the P–T profile basically isothermal within the deep ambiance (that’s, the retrieved P–T profile proven in Fig. 2 is extra bodily). We observe that our uniform aerosol Mie scattering retrievals are incompatible with the P–T profile utilized in the principle textual content⁴¹, as a group of small aerosols aren’t capable of concurrently block the deep adiabatic thermal flux and match the scattering slope. The P–T profile parameterization chosen right here⁶² tends to favour a virtually isothermal higher ambiance, which suffices for the exploration of the aerosol properties in step with the scattering slope of WD 1856 b. Future explorations of the cloud construction and radiative properties of WD 1856 b, reminiscent of composite cloud fashions with a number of scattering, are a wealthy space to deepen our understanding of the ambiance of WD 1856 b.

Evolution of the WD 1856 system

Host progenitor and white dwarf

We examined the evolution of the progenitor star of WD 1856 by consulting the MIST evolutionary fashions⁷⁷ for non-rotating solar-metallicity stars within the acceptable mass vary (({M}_{{rm{progenitor}}}=1.3{6}_{-0.18}^{+0.29},{M}_{{rm{Sun}}})). From these fashions, we extracted fiducial estimates of the principle sequence lifetime (({4}_{-1.8}^{+2.4},{rm{Gyr}})) utilizing an preliminary–last mass relation^78,79, the length of the thermally pulsing AGB stage ((1.5{5}_{-0.10}^{+0.26},{rm{Myr}})) and the post-AGB/pre-white-dwarf stage ((0.03{4}_{-0.002}^{+0.053},{rm{Myr}})). The latter is outlined right here because the elapsed time between the ultimate thermal pulse and the cooling of the uncovered core to an efficient temperature of 100,000 Okay. This yields a complete system age of (9.{4}_{-1.9}^{+2.5},{rm{Gyr}}).

We calculated the cooling age of the white dwarf host by evolving MESA white dwarf fashions of the suitable mass right down to T_eff = 4,920 Okay. We used MESA r23.05.1 (ref. ⁸⁰). This MESA launch now contains carbon–oxygen fractionation⁸¹, which is necessary right here because the white dwarf is within the technique of crystallizing. A regular helium layer of (log {M}_{{rm{He}}}/{M}_{* }=-2) was assumed, whereas a comparatively skinny hydrogen layer of (log {M}_{{rm{H}}}/{M}_{* }=-6) was used. This is way thinner than the canonical worth of (log {M}_{{rm{H}}}/{M}_{* }=-4) (ref. ⁸²) however is motivated by the truth that the mannequin ambiance evaluation factors to an environment containing a mixture of hydrogen and helium. This presumably requires the superficial convection zone to increase slightly below the hydrogen layer, thereby diluting hydrogen with helium. From this constraint, we are able to estimate (log {M}_{{rm{H}}}/{M}_{* }=-6) (ref. ⁸³). Cooling calculations have been carried out for various carbon–oxygen core composition profiles to account for present mannequin uncertainties: a regular profile predicted by stellar evolution⁸⁰ and an asteroseismologically derived stratification⁸⁴ have been used. We additionally calculated cooling fashions utilizing completely different electron thermal conductivities^85,86 to account for present uncertainties on the transition between the regimes of average and robust degeneracy⁸⁷. From this evaluation, we discover a cooling age of 5.4 ± 0.7 Gyr, during which the uncertainty contains the systematic uncertainty sources listed above and a 2% uncertainty on the T_eff of the star and mass typical of white dwarfs on this temperature vary⁸⁸. This cooling age is in step with estimates produced by different stellar evolution codes^82,89.

Thermal historical past of WD 1856 b

We reconstructed the thermal evolution of WD 1856 b below the belief that the cooling of the planet after migration has been just like the cooling undergone by a substellar object after formation. We used cooling fashions from the ATMO2020 (ref. ⁹⁰) and Sonora Bobcat⁹¹ mannequin grids. Each grid tabulates international portions reminiscent of luminosity, efficient temperature, radius and floor gravity as a perform of age for substellar objects of a given mass and bulk chemical composition, ranging from an preliminary situation with excessive entropy. Both present self-consistent evolutionary–atmospheric modelling frameworks, during which the construction and evolution of the totally convective, adiabatic inside are computed with a cloudless, non-grey, rainout-chemical-equilibrium ambiance because the floor boundary situation. The most necessary distinction between ATMO2020 and Sonora Bobcat is that the previous neglects some related opacity sources at efficient temperatures above 2,000 Okay, resulting in quicker cooling at excessive temperatures within the ATMO2020 fashions. ATMO2020 gives fashions of solar-metallicity objects, whereas Sonora Bobcat gives fashions for each solar-metallicity and metal-enriched ([M/H] = +0.5) objects. We think about these three units of fashions in our evaluation beneath.

Reconstructing the thermal historical past of WD 1856 b requires us to decide on a mannequin grid and specify three parameters: planetary mass (M_p), present planetary efficient temperature (T_eff,p) and present white dwarf cooling age (t_wd). For a given M_p and mannequin grid, we obtained the efficient temperature as a perform of time by including a uniform offset (t₀) to the mannequin age (t_p) such that the mannequin temperature matches T_eff,p at t_p = t_wd − t₀ (utilizing linear interpolation between the tabulated mannequin ages and temperatures). The cooling fashions predict a excessive efficient temperature (about 1,500–3,000 Okay) at t₀; these values are believable for planets which were tidally heated throughout high-eccentricity migration^92,93 or have survived a common-envelope section^8,9,94. We subsequently interpret t₀ as an estimate of the time of the planet’s reheating throughout migration, expressed as a white dwarf cooling age. Because cooling is speedy at excessive T_eff,p, our estimate of t₀ is strong to theoretical uncertainties in what temperature the planet needs to be instantly after migration.

We thought-about cooling fashions with M_p between 0.5 and 20 M_J, overlaying the vary of samples from the atmospheric retrieval posterior distributions for the FIREFLy and Juniper JWST information reductions. Each grid samples a finite variety of mass values; when contemplating objects of arbitrary mass between grid factors, we used the cooling mannequin with the closest mass on the grid. Both the ATMO2020 and Sonora Bobcat fashions are spaced by about 0.5–1.0 M_J in mass over the vary we think about, so our method doesn’t introduce substantial error in a given reconstruction in contrast with interpolating between adjoining fashions.

We generated ensembles of doable thermal histories utilizing the mass and efficient temperature constraints derived from the NIRSpec PRISM transmission spectrum of WD 1856 b. Specifically, we thought-about almost 10,000 values of M_p from the atmospheric retrieval posterior distribution alongside the almost 10,000 corresponding values of T_eff,p obtained utilizing the process described above for every information discount (see the ‘Atmospheric retrieval analysis’ part). Our samples of M_p and T_eff,p aren’t statistically unbiased, as every pair of values is derived from a single pattern from the distribution of atmospheric fashions in step with our NIRSpec PRISM transmission spectrum. This has an necessary affect on the vary of thermal histories that we are able to infer from the info, as a result of the cooling fee is a delicate perform of mass.

On the opposite hand, our estimated t_wd is unbiased of our atmospheric retrieval evaluation. For every pair of M_p and T_eff,p values, we generated ten random values of t_wd drawn from a Gaussian distribution with imply 5.4 Gyr and commonplace deviation 0.7 Gyr. Each ensemble subsequently contains about 100,000 doable thermal histories in step with the transmission spectrum of WD 1856 b. We generated one ensemble for every mannequin grid. Extended Data Fig. 7 reveals the distribution of calculated t₀ values for the three cooling fashions and two information reductions, from which we derive a statistical constraint on t₀. The outcomes reported in the principle textual content have been obtained utilizing the solar-metallicity Sonora Bobcat fashions (strong orange and inexperienced histograms; see additionally Fig. 4). If we use the ATMO2020 fashions, we discover a comparable ({t}_{0}=4.{3}_{-1.1}^{+0.9},{rm{Gyr}}) for FIREFLy and (4.{6}_{-1.0}^{+0.8},{rm{Gyr}}) for Juniper. Using the metal-enriched Sonora Bobcat fashions yields ({t}_{0}=4.{2}_{-1.4}^{+1.0},{rm{Gyr}}) for FIREFLy and (4.{5}_{-1.1}^{+0.9},{rm{Gyr}}) for Juniper. The conclusions we draw from modelling the thermal evolution of WD 1856 b are subsequently sturdy to each the JWST information discount and the selection of cooling fashions, given the fashions out there at current.

In a small fraction of instances (<0.15%) for every ensemble, we calculate values t₀ < 0. These correspond to the best M_p values sampled from the atmospheric retrieval posterior. Negative values of t₀ come up in these instances as a result of we’ve calculated t₀ by extrapolating the cooling fashions again to the efficient temperatures anticipated amongst new child brown dwarfs of about 20 M_J (>2,000 Okay). However, these outcomes are unphysical in accordance with the interpretation of t₀ because the time elapsed between the top of the AGB section and the reheating/migration of the planet. If we stipulate that reconstructed thermal histories be truncated for t₀ < 0, then these few instances are in step with common-envelope evolution in that it’s doable for the planet to have achieved its present temperature by passively cooling for the reason that finish of the AGB section (albeit from a cooler, lower-entropy state than these implied in instances with t₀ > 0). The reality stays, nevertheless, that a lot of the instances (>99.85%) suggest t₀ values that can’t coincide with a common-envelope section in all three ensembles. Thus, we conclude that reheating in the course of the white dwarf section (in step with high-eccentricity migration) is most popular over reheating throughout common-envelope evolution at >2σ (for FIREFLy) and >3σ (for Juniper). Further theoretical examine is required to corroborate or qualify this conclusion, as we describe beneath.

Our technique of reconstructing thermal histories is predicated on backward extrapolation of the efficient temperature solely. However, the cooling fashions additionally predict the evolution of the radius of WD 1856 b; these predictions ought to agree in precept. In Extended Data Fig. 8, we present the radius evolution implied by our reconstruction technique for 100 samples from the Sonora Bobcat ensemble for each information reductions. For the noticed radius, we use the best-fitting worth R_p = 0.911 ± 0.020 R_J from the FIREFLy discount. We additionally embody a scientific error of ±0.050 R_J given by the vary of best-fitting radius values lined by the 2 information reductions, for a complete uncertainty of ±0.054 R_J. We see that lots of the temperature-based reconstructions overestimate the radius of WD 1856 b by about 2σ. Future efforts to know the thermal evolution of WD 1856 b ought to reproduce each the efficient temperature and radius. A clue as to the origin of this discrepancy comes from the heavy-element enrichment of the envelope of WD 1856 b, advised by our retrieved CH₄ abundance, as planetary radius decreases with growing metallicity at a hard and fast mass and inner entropy. Model grids of comparable high quality with ATMO2020 and Sonora Bobcat which are relevant to things as huge (about 7 M_J) and metal-rich (about 100 × photo voltaic) as WD 1856 b haven’t been developed or printed to our information.

We observe that we uncared for the irradiation of WD 1856 b by the host white dwarf in our reconstruction of the planet’s thermal historical past. Irradiation is a key ingredient in modelling the construction and evolution of short-period exoplanets round main-sequence stars, reminiscent of scorching Jupiters^95,96. The significance of irradiation within the case of WD 1856 b might be gauged by calculating the ratio of the facility emitted from the photosphere of the planet to the facility incident on the planet from the star:

during which T_eff,* and T_eff,p are measured efficient temperatures of the host star and planet, respectively, R_* is the host radius and a_orb is the orbital semimajor axis (assuming a near-circular orbit). Using the system parameters as decided on this work, we calculate ({mathcal{R}}approx 25), indicating that the self-luminosity of the planet overwhelms the facility acquired from the star. Our reconstructed histories usually discover that this ratio was bigger previously (besides maybe within the first a number of Myr after the white dwarf fashioned). Thus, we argue that irradiation has had a small impact on the earlier thermal evolution of WD 1856 b. It can be of curiosity to self-consistently mannequin the evolution of a substellar physique with time-dependent irradiation, as can be the case in proximity to a cooling white dwarf. We depart this for future work.

Alternatives to reheating throughout migration

We thought-about a number of various explanations for the elevated efficient temperature of WD 1856 b, all of which we deemed implausible or unlikely. We briefly describe every of them right here, together with our reasoning.

First, the noticed efficient temperature of WD 1856 b can’t be defined purely by passive cooling over the system’s whole age of about 10 Gyr (ref. ¹). This is quickly dominated out by consulting theoretical cooling fashions^90,91. To have an efficient temperature of about 400 Okay at an age of 10 Gyr, WD 1856 b would wish to have a mass of roughly 24 M_J. Our observations rule out such a excessive mass at >3σ confidence.

The mass of WD 1856 b could also be above the brink for deuterium fusion in its core (about 13 M_J) inside 2σ. However, though it’s doable that WD 1856 b was as soon as heated internally by nuclear reactions, this can not clarify its present-day properties. Models of deuterium-burning brown dwarfs predict a complete luminosity many orders of magnitude higher than that of WD 1856 b (ref. ⁹⁷). The length of deuterium burning (about 3–50 Myr relying on mass⁹⁷) is way shorter than the overall age of the system, so the primordial deuterium WD 1856 b would have been destroyed early within the main-sequence lifetime of the host.

Owing to the proximity of WD 1856 b to its host, tidal interactions are one other doable warmth supply contained in the planet; this may be analogous to the heating of the Galilean satellite tv for pc Io by its tidal interplay with Jupiter⁹⁸. For tidal heating to function, the orbit of WD 1856 b would must be barely eccentric reasonably than round as is usually assumed. Assuming that the facility dissipated by tidal friction is the same as the overall energy emitted by WD 1856 b, we calculate the efficient temperature of the planet, utilizing the usual ‘equilibrium tide’ concept⁹⁹, as:

Here G is the gravitational fixed, σ_SB is the Stefan–Boltzmann fixed, M_* is the mass of the host white dwarf, R_p is the radius of WD 1856 b and a_orb and e_orb are respectively the orbital semimajor axis and eccentricity (with e_orb ≪ 1). The portions ok_2p and τ_p, identified respectively because the tidal Love quantity and tidal lag time, characterize the dissipation inside WD 1856 b in the usual equilibrium tidal concept. The reference values of M_*, R_p and a_orb utilized in equation (10) match the noticed system parameters. For ok_2p and τ_p, we use values just like these inferred for Jupiter’s dissipation of the tide raised by Io^100,101. We see that tidal heating may, in precept, maintain the noticed efficient temperature of WD 1856 b for an orbital eccentricity of about 0.02 (the extremely unsure values of ok_2p and τ_p however). This can be in step with orbital circularization ultimately stage of high-eccentricity migration. However, the identical dissipation would damp the orbital eccentricity on a attribute timescale of roughly 0.075 Gyr (ref. ⁹⁹). In this image, we’re observing WD 1856 b within the final, short-lived stage of high-eccentricity migration. Although we can not rule it out based mostly on the out there information, we think about this clarification unlikely.