adsabs/FOCAL huggingface dataset:
To evaluate the turbulence heating term St, we need to describe the transport and dissipation of turbulence throughout the heliosphere, ensuring that the appropriate sources of turbulence are identified as well as that associated with pickup ion creation in the distant solar wind. The turbulence transport model we use is that of Zank et al. (2017). The current paradigm for fully developed turbulence in the solar wind is that it is a superposition of a majority quasi-2D component and a minority slab component, e.g., Matthaeus et al. (1995). The nearly incompressible (NI) reduction of MHD in the plasma beta 





 regime, appropriate to most of the solar wind, shows that such a decomposition (quasi-2D plus slab) arises naturally in the presence of a sufficiently strong guide magnetic field (Zank & Matthaeus 1992, 1993; Hunana & Zank 2010). The stationary form of the NI MHD transport equations describing the evolution of the majority quasi-2D component in a spherically symmetric solar wind with an azimuthal magnetic field can be expressed as (Adhikari et al. 2017; Zank et al. 2017)
22







23







24







25





Here, 





 are the forward and backward Elsässer variables, combining the quasi-2D fluctuating velocity field 





 and the fluctuating magnetic field 





 through the fluctuating (turbulent) Alfvén velocity 





. The variance of the fluctuating Elsässer variables is denoted by 





, where the operator 





 averages over small scales and high frequencies. The quasi-2D residual energy (or energy difference) is defined by 





. The total fluctuating quasi-2D energy is denoted by 





, and the quasi-2D correlation lengths 





 are related to the correlation functions 





 through 





. Finally, 





 is a correlation function related to the residual energy. The energy for the forward and backward slab Elsässer variables 





 and the slab correlation length 





 are discussed further below. However, we note that the quasi-2D and slab components are coupled, capturing the 3-mode coupling (Shebalin et al. 1983; Zank et al. 2017) that governs spectral transfer in the inertial range. This is expressed by the source-like terms in Equations (22) and (23) that contain the slab variables. Equations (22) and (23) contain genuine source terms associated with stream shear (Adhikari et al. 2017; Zank et al. 2017), where 





 denotes the strength of the stream-shear source of forward or backward quasi-2D fluctuations, 





 the characteristic velocity jump across the stream, and VA the large-scale Alfvén speed. A lengthy derivation and discussion of these equations can be found in Zank et al. (2017) and specific solutions appropriate to a prescribed large-scale solar wind can be found in Adhikari et al. (2017).
Citation: Hunana & Zank 2010
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To characterize the motions of the deeply embedded gas that could reveal signatures of rotation and infall and that would help us to understand the formation of massive stars, for many years we have been carrying out multiple high angular resolution studies of hot molecular cores (HMCs), first with the IRAM Plateau de Bure (PdB) and Submillimeter Array (SMA) interferometers (Beltrán et al. 2004, 2005, 2011a,b), and later on with the Atacama Large Millimeter/submillimeter Array (ALMA, Cesaroni et al. 2017; Beltrán et al. 2018, 2021; Moscadelli et al. 2018, 2021; Goddi et al. 2020) and the IRAM NOrthern Extended Millimeter Array (NOEMA, Gieser et al. 2019, 2021). Cesaroni et al. (2017) carried out an extensive and systematic search for circumstellar disks around O-type stars by observing a sample of HMCs at 1.4 mm and 0.′′2 resolution with ALMA. One of the HMCs of the sample is G31.41+0.31 (hereafter G31), which is located at 3.75 kpc Immer et al. (2019), has a luminosity of ~ 5 × 104 L⊙ Osorio et al. (2009), and displays a clear NE–SW velocity gradient suggestive of rotation (e.g., Beltrán et al. 2004). Among the HMCs studied, G31 stands out as the only source that displays accelerating infall and rotational spin-up (Beltrán et al. 2018). This characteristic suggests that the source is in an earlier evolutionary stage compared to all other targets in the Cesaroni et al. sample. At an angular resolution of 0.′′2 (~ 750 au), the dust continuum emission traces a well-resolved, monolithic, and featureless core, called Main by Beltrán et al. (2018), with no hint of fragmentation despite the high mass (~ 70 M⊙; Cesaroni et al. 2019) and the diameter of ~8000 au. However, new ALMA observations at 1.4 and 3.5 mm at a higher angular resolution of ~ 0.′′1 (~ 375 au), and Very Large Array (VLA) observations at 7 mm and ~ 0.′′05 (~ 190 au) resolution have resolved for the first time the HMC into a small protocluster composed of at least four massive sources, named A, B, C, and D and with masses ranging from ~15 to ~ 26 M⊙, within the central 1″ (~ 3750 au) region of the core (Beltrán et al. 2021). These observations have revealed that the homogeneous appearance previously observed at 1.4 mm and 0.′′2 is a consequence of both high dust opacity and insufficient angular resolution. Besides the four sources embedded in the Main core, there are six additional millimeter sources located very close to the core which appear to outline streams or filaments of matter pointing to the HMC (Beltrán et al. 2021).
Citation: Immer et al. (2019)
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Citation: Beltrán et al. 2018
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Citation: Beltrán et al. 2018
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Citation: Beltrán et al. (2018)
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The efficiency of pebble accretion is determined by the pebble scale height in the disc given by (5)\begin{equation} \label{eq:Hpebble}  H_{\rm peb} = H\sqrt{\alpha/\tau_{\rm f}} , \end{equation}Hpeb=Hα/τf,where τf is the Stokes number of the particles. If the planetary seed’s Hill radius is larger than the pebble scale height, the planetary seed can accrete efficiently in a 2D manner (6)\begin{equation} \label{eq:Mdotpebble}  \dot{M}_{\rm c, 2D} = 2 \left(\frac{\tau_{\rm f}}{0.1}\right)^{2/3} r_{\rm H} v_{\rm H} \Sigma_{\rm peb} , \end{equation}Ṁc,2D=2τf0.12/3rHvHΣpeb,where rH = r [Mc/ (3M⋆)] 1/3 is the Hill radius, vH = ΩKrH the Hill speed, and Σpeb the pebble surface density. For Stokes numbers larger than 0.1, the (τf/ 0.1)2/3 term vanishes in Eq. (6), because the planetary seed cannot accrete particles from outside its Hill radius (Lambrechts & Johansen 2012). However for small planetary seeds, the Hill sphere can be smaller than the pebble scale height, so that the seed accretes in a slow 3D fashion (7)\begin{equation} \dot{M}_{\rm c, 3D} = \dot{M}_{\rm c, 2D} \left( \frac{\pi (\tau_{\rm f}/0.1)^{1/3} r_{\rm H}}{2 \sqrt{2 \pi} H_{\rm peb}} \right) \cdot \end{equation}Ṁc,3D=Ṁc,2Dπ(τf/0.1)1/3rH22πHpeb·The transition from 3D to 2D pebble accretion is then reached (Morbidelli et al. 2015) when (8)\begin{equation} \label{eq:2D3D}  \frac{\pi (\tau_{\rm f}/0.1)^{1/3} r_{\rm H}}{2 \sqrt{2 \pi}} > H_{\rm peb} . \end{equation}π(τf/0.1)1/3rH22π>Hpeb.Clearly this transition depends on the scale height of the pebbles and thus on the scale height of the gas. Therefore, pebble accretion is slower in the initial planetary growth phase in the outer parts of the disc, because the aspect ratio is larger there. The core can then finally reach its pebble isolation mass (Lambrechts et al. 2014) (9)\begin{equation} \label{eq:Misolation}  M_{\rm iso} \sim 20 \left( \frac{H/r}{0.05}\right)^3~{M}_{\rm Earth} , \end{equation}Miso~20H/r0.053MEarth,where pebble accretion is terminated and the contraction of the gaseous envelope can start. We use for the contraction of the gaseous envelope the semi-analytical model described by Piso & Youdin (2014), where the contraction phase becomes shorter for more massive planetary cores. This contraction phase additionally depends on the opacity inside the of the planetary envelope, as this determines the cooling and with this the contraction rate of the envelope. Our nominal opacity in the envelope corresponds to similar values as in Movshovitz & Podolak (2008). We discuss the influence of a slower contraction rate in Appendix B. As soon as Menv = Mcore, rapid gas accretion can start.
Citation: Lambrechts & Johansen 2012
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Solar flares and coronal mass ejections (CMEs) are the most energetic phenomena in the solar coronae, which are considered as different manifestations of the same physical process (Zhang et al. 2001; Priest & Forbes 2002; Zhang et al. 2004; Shibata & Magara 2011). However, flares that are associated with no CMEs are known as “confined flares” or “failed eruptions” (e.g., Ji et al. 2003; Liu et al. 2014; Zhou et al. 2019; Mrozek et al. 2020). Whether an eruption is successful or not is heavily dependent on the strapping force exerted by the overlying magnetic field on the eruptive structure (e.g., Török & Kliem 2005; Guo et al. 2010; Cheng et al. 2011; Zuccarello et al. 2014; Myers et al. 2015; Sun et al. 2015; Amari et al. 2018), which is often a magnetic flux rope (MFR; Vourlidas et al. 2013; Liu 2020). A flux rope undergoes the torus instability (Bateman 1978; Kliem & Török 2006), when the magnitude of the overlying field decreases with height at a sufficiently steep rate, quantified by the decay index 



n=−dlnB/dlnh

 (Bateman 1978), exceeding about 1.5; the corresponding height is referred to as the critical height. Nevertheless, one should bear in mind that the theoretical critical value of n
cr = 1.5 adopted in many studies, as well as in this study, is derived from a thin, toroidal current ring (Bateman 1978; Kliem & Török 2006). In practice, n
cr typically scatters around in a range as large as [0.8, 2] for different theoretical considerations (e.g., Démoulin & Aulanier 2010; Olmedo & Zhang 2010), observational studies (e.g., Jing et al. 2018; Duan et al. 2019), numerical simulations (e.g., Fan & Gibson 2007; Török & Kliem 2007), and laboratory experiments (e.g., Myers et al. 2015; Alt et al. 2021). Obviously, n
cr is a function of the flux-rope parameters including the geometry and internal current profile, yet the discrepancy in different studies is also affected by where the decay index is computed (Zuccarello et al. 2016; Alt et al. 2021).
Citation: Myers et al. 2015; Myers et al. 2015
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A natural experiment for deciding between these possibilities is to search for a KS relation within individual molecular clouds using counts of the recently formed stars or protostars identified by their bright infrared emission from circumstellar dust. The rarity of massive stars and their disruptive effect on their host cloud means they are poor tracers on cloud scales. Protostars, by contrast, have the advantage that they sample a much shorter time interval and therefore provide a much better estimate of the “instantaneous” star formation rate (SFR), and they allow measurements of the SFR even in clouds that lack massive stars and have not been significantly affected by feedback. Studies based on this method generally do find a reasonable correlation between the number of young stellar objects (YSOs) in a cloud and its gas mass above a certain density, or its gas mass divided by its mean-density freefall time (Krumholz et al. 2012a, 2019; Lada et al. 2012; Heyer et al. 2016). Within molecular clouds, several studies have found a power-law correlation between the surface densities of YSOs and gas (Gutermuth et al. 2011; Lada et al. 2013; Willis et al. 2015). Most recently, Pokhrel et al. (2020) used high-accuracy YSO catalogs and high dynamic range gas column densities to show the presence of these laws in 12 nearby clouds. This correlation is consistent with a star formation surface density being proportional to the gas surface density squared. The scaling of this law, however, varies significantly between clouds. Moreover, their analysis technique examines the density of gas around stars on a star-by-star basis, and therefore cannot easily determine whether there is a KS relation based on the volume density of gas. Using the same data, we apply a different approach to determine the star formation law that includes a dependence on the volume density of the gas. We find that the star formation law can be recast as an effectively universal linear dependence of the surface densities between star formation rate and gas mass per freefall time, with much less scatter between clouds.
Citation: Heyer et al. 2016
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Citation: Gutermuth et al. 2011
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Citation: Pokhrel et al. (2020)
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NGC 6397 is the second-nearest globular cluster to the Sun and has been thoroughly observed by the Hubble Space Telescope. Thus, we have high-quality deep images for it that have allowed us to study not only the lower main sequence, but also the white dwarf cooling sequence. NGC 6397 is old and metal-poor, its metallicity is [ Fe/H ]  =  −1.8 (Hansen et al. 2013), although there exists some discrepancy about its precise value (Richer et al. 2008) in the recent literature. The same is true for the age of the cluster. For instance, Hansen et al. (2007) analyzed the position of the cutoff of the white dwarf luminosity function and, comparing this with theoretical cooling models, derived an age Tc = 11.47 ± 0.47 Gyr. Similarly, Winget et al. (2009) simultaneously fitted the main-sequence, the pre-white dwarf, and the white dwarf regions of the color–magnitude diagram, and obtained an age \hbox{$T_{\rm  c}=12.0^{+0.5}_{-1.0}$}Tc=12.0-1.0+0.5 Gyr. These ages, which are based on white dwarf evolutionary models, need to be compared with those obtained by fitting the luminosity of the main-sequence turn-off (MSTO) – see Richer et al. (2008) for a discussion of different methods and age estimates of NGC 6397. In particular, it is worth highlighting that using this method, Gratton et al. (2003) derived an age of 13.9 ± 1.1 Gyr. This prompted Strickler et al. (2009) to claim that this age determination was compatible with the possible existence of a putative population of very old helium white dwarfs. On the other hand, Anthony-Twarog & Twarog (2000) obtained 12.0 ± 0.8 Gyr and Chaboyer et al. (2001) derived 13.4  ±  0.8 Gyr. Thus, the precise age of NGC 6397 remains unclear and needs to be independently evaluated using the most recent white dwarf evolutionary tracks of the appropriate metallicity. To this we add that NGC 6397 shares some other interesting characteristics with other Galactic globular clusters, such as the unusual lithium enhancement of some of their stars – see Pasquini et al. (2014) and references therein – or the possible existence of multiple populations (di Criscienzo et al. 2010; Milone et al. 2012). Thus, deriving an independent white dwarf cooling age for this globular cluster is of crucial importance.
Citation: Winget et al. (2009)
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As reported in Section 2.4, the uncertainties introduced by chromatic aberration in our observations prevented an unambiguous determination of the detailed characteristics of the spectral slope at wavelengths in the Balmer continuum range (Î»  3646âÃ). However, we do note the apparent lack of a jump in excess intensity or flare contrast at Î»â  â3646âÃ relative to the intensity at redder wavelengths (Figure 9). A dominant optically thin 10,000Â K spectrum would have produced a large Balmer jump ratio (i.e., the ratio of intensity at blue wavelengths to that at red wavelengths of 3646âÃ) that is â¼14 (Kunkel 1970; Neidig et al. 1993); we think that this should have been noticeable even with the data quality degraded due to chromatic aberration at Î»â  â3646âÃ. It is possible that in our spectra the blending of Stark-broadened high-order Balmer lines77The spectral region from 3654â3674âÃ contains the rapidly converging hydrogen Balmer lines H23 through at least H40; the center wavelengths of H23 and H24 are separated by â¼2.5âÃ whereas H39 and H40 are separated by only 0.5âÃ. combined with the Stark broadening of the Balmer recombination edge could smear the Balmer jump making it non-detectable. Still, we note that other WL flares reported in the literature either did not display a Balmer jump, or had other properties not readily explained by a Hydrogen recombination spectrum. In particular, a strong blue continuum emission at Î»  4000âÃ has been often reported (Hiei 1982; Neidig & Wiborg 1984), and shown by Donati-Falchi et al. (1985) as the result of the blending of Stark-broadened high-order Balmer lines in a dense chromosphere. In the model of Donati-Falchi et al. (1985), this âbumpâ in the blue continuum peaks at Î»â â¼ â3675âÃÂ  and becomes more prominent and shifts to redder wavelengths as electron density in the flare region increases. Existing models of Stark broadening imply that such blue âcontinuumâ emission originates from a location with electron density in excess of 1013 cmâ3 and electron temperature between 7000 and 10,000Â K. Interestingly, in our excess spectra we observe a relatively featureless, broad bump peaking at Î» â¼Â 3675âÃ (Figure 9). Taken at face value, the electron densities inferred from interpreting this feature via Stark broadening appear at odd with those derived in Section 5.2, unless more optically thin features such as the higher order Balmer lines probe different portions of the flaring atmosphere. However, additional spectra that are not affected by chromatic aberration will be needed to confirm and understand this feature.
Citation: Neidig et al. 1993
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K2-138 is a remarkable dynamical system with five planets in a chain close to the 3:2 resonance. In addition, these planets form a chain of three-body Laplace resonances (Christiansen et al. 2018). This makes K2-138 an ideal target to study transit timing variations. We estimated the transit timing variations using the TTVFaster code (Agol & Deck 2016). We first assumed zero eccentricities for the six planets and took the median masses from Table A.2. We found amplitudes of the order of 2.0, 4.1, 7.3,4.5, 6.4, and 0.02 min for planets from b to g, respectively. Then, assuming median eccentricities from Table A.2, we obtained amplitudes of the order 34, 42, 66, 37, and 41 min for planets b to f (Fig. 9). These amplitudes are excluded by K2 observations which do not show any significant variations at the level of 8–10 min, asdemonstrated by Christiansen et al. (2018). Therefore, the planets are likely close to circular orbits, which is compatible with the posterior distributions we obtained. This is also in line with results showing that tightly packed multi-transiting planets have low eccentricities (Van Eylen et al. 2019). Finally, using the upper limits derived for the masses of planets f and g, we computed the corresponding TTVs of planets e and f. Planet g does not impact TTVs of planet f in a significant (detectable) way as it is far from resonance, being located after a double gap in the chain of resonance. Therefore, it is not possible to use TTVs of planet f to constrain the mass of planet g even if there are undetected and non-transiting planets filling the gap in the chain. Still assuming circular orbits, TTVs of planet e reach an amplitude of the order eight minutes only for masses of planet f above 12 M⊕, which is above the upper limit we derived. However, as we cannot exclude an absorption of the signal of planet f by the GP, it is more conservative to include masses for planet f up to around 12 M⊕. Using the upper limit mass for planet f, the predicted TTVs of planet e are of the order of 6.4 min, and using the median mass they are of 4.5 min. Both are well within reach of space mission like CHEOPS which has a cadence of up to 60 s. Observations of transits of planet e using CHEOPS would be beneficial to constrain further the mass of planet f through a photodynamical analysis (Barros et al. 2015). Additionally, more transit observations, with a higher cadence, of planets b, c and d would allow to constrain the masses of planets c, d and e from TTVs which would make K2-138 an interesting benchmark system for comparing RV and TTV masses. This would allow to better calibrate the two mass measurement techniques. Nevertheless, at this stage, and without more precise photometric observationsof planets e and f, an analysis including TTVs would not allow to measure the masses of planets f and g.
Citation: Christiansen et al. 2018
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Citation: Christiansen et al. (2018)
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Citation: Agol & Deck 2016
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Citation: Van Eylen et al. 2019
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Citation: Barros et al. 2015
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The de-reddened photometry of TYC 8252-533-1 B and its host star are presented in Table 5. The listed Gaia photometry is directly obtained from the EDR3 catalog and the infrared photometry originates from our high-contrast imaging data. We applied the previously determined extinction of AV = 0.13 mag to correct the presented fluxes. As the SPHERE H band data from 2017-04-02 were collected in poor atmospheric conditions and with a very unstable AO performance, we disregarded the photometry that was extracted from these observations in our further analysis. We fitted the full optical and infraredSED of the companion utilizing the MCMC approach described in Bohn et al. (2020b). We used a linearly interpolated grid of BT-Settl models (Allard et al. 2012; Baraffe et al. 2015) with effective temperatures, Teff, between 1500 K and 4000 K, surface gravity in the range of 

$0<\log\left(g\right)<6$0log(g)6
, and solar metallicity. We further allowed object radii R from 0.5 RJup to 5 RJup. The MCMC sampler was implemented in the emcee framework (Foreman-Mackey et al. 2013) and we used 100 walkers with 10 000 steps to sample the posterior distribution. In accordance with the computed autocorrelation time of approximately 200 steps, the first 1000 samples of each chain were discarded as burn-in phase and we further continued using each twentieth step of the remaining chains. This provided 45 000 samples for our posterior distribution in (Teff, 

$\log\left(g\right)$log(g)
, R), which are visualized in Fig. D.1. From these distributions we derived an effective temperature of 

$T_{\textrm{eff}}=3092^{+186}_{-91}\,K$Teff=3092−91+186 K
, a surface gravity of 

$\log\left(g\right)=3.41^{+1.07}_{-0.31}$log(g)=3.41−0.31+1.07
 dex, and a radius of 

$R=3.5^{+0.3}_{-0.4}\,R_{\textrm{Jup}}$R=3.5−0.4+0.3 RJup
 for TYC 8252-533-1 B. These values were obtained as the 95% confidence intervals around the medians of the posterior distributions. From these three parameters we derived the object luminosity as 

$\log\left(L_*/L_{\odot}\right)=-1.99^{+0.01}_{-0.02}$log(L*/L⊙)=−1.99−0.02+0.01
. The results of this SED fit are presented in Fig. 9. Interestingly, the derived object radius is markedly larger than usual radii of field brown dwarfs of this mass, which are on the order of 1 RJup (e.g., Chabrier et al. 2009). This inflated photometric radius, however, is nothing unusual for sub-stellar companions at this young age, and similarly large values have been reported for various planetary-mass objects (e.g., Schmidt et al. 2008; Bohn et al. 2019; Stolker et al. 2020).
Citation: Baraffe et al. 2015
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