Wolf-Rayet Research Articles

Wind-fed GRMHD simulations of Sagittarius A*: tilt and alignment of jets and accretion discs, electron thermodynamics, and multiscale modelling of the rotation measure

ABSTRACTWind-fed models offer a unique way to form predictive models of the accretion flow surrounding Sagittarius A*. We present 3D wind-fed magnetohydrodynamic (MHD) and general relativistic magnetohydrodynamic (GRMHD) simulations spanning the entire dynamic range of accretion from parsec scales to the event horizon. We expand on previous work by including non-zero black hole spin and dynamically evolved electron thermodynamics. Initial conditions for these simulations are generated from simulations of the observed Wolf–Rayet stellar winds in the Galactic Centre. The resulting flow tends to be highly magnetized (β ≈ 2) with an ∼r−1 density profile independent of the strength of magnetic fields in the winds. Our simulations reach the magnetically arrested disc (MAD) state for some, but not all cases. In tilted flows, standard and normal evolution (SANE) jets tend to align with the angular momentum of the gas at large scales, even if that direction is perpendicular to the black hole spin axis. Conversely, MAD jets tend to align with the black hole spin axis. The gas angular momentum shows similar behaviour: SANE flows tend to only partially align while MAD flows tend to fully align. With a limited number of dynamical free parameters, our models can produce accretion rates, 230 GHz flux, and unresolved linear polarization fractions roughly consistent with observations for several choices of electron heating fraction. Absent another source of large-scale magnetic field, winds with a higher degree of magnetization (e.g. where the magnetic pressure is 1/100 of the ram pressure in the winds) may be required to get a sufficiently large rotation measure with consistent sign.

Stripped-envelope stars in different metallicity environments

Stripped-envelope stars can be observed as Wolf-Rayet (WR) stars or as less luminous hydrogen-poor stars with low mass-loss rates and transparent winds. Both types are potential progenitors of Type I core-collapse supernovae (SNe). We used grids of core-collapse models obtained from single helium stars at different metallicities to study the effects of metallicity on the transients and remnants these stars produce. We characterised the surface and core properties of our core-collapse models and investigated their ‘explodability’ using three criteria. In the cases where explosions are predicted, we estimated the ejecta mass, explosion energy, nickel mass, and neutron star (NS) mass. Otherwise, we predicted the mass of the resulting black hole (BH). We constructed a simplified population model and find that the properties of SNe and compact objects depend strongly on metallicity. The ejecta masses and explosion energies for Type Ic SNe are best reproduced by models with Z = 0.04 that exhibit strong winds during core helium burning. This implies that either their mass-loss rates are underestimated or that Type Ic SN progenitors experience mass loss through other mechanisms before exploding. The distributions of ejecta masses, explosion energies, and nickel mass for Type Ib SNe are not well reproduced by progenitor models with WR mass loss, but are better reproduced if we assume no mass loss in progenitors with luminosities below the minimum WR star luminosity. We find that Type Ic SNe become more common as metallicity increases, and that the vast majority of progenitors of Type Ib SNe must be transparent-wind stripped-envelope stars. We find that several models with pre-collapse CO masses of up to ∼30 M⊙ may form ∼3 M⊙ BHs in fallback SNe. This may have important consequences for our understanding of SNe, binary BH and NS systems, X-ray binary systems, and gravitational wave transients.

Exploring the influence of different velocity fields on Wolf–Rayet star spectra

ABSTRACT Given their strong stellar winds, Wolf–Rayet (WR) stars exhibit emission line spectra that are predominantly formed in expanding atmospheric layers. The description of the wind velocity field $\upsilon (r)$ is therefore a crucial ingredient in the spectral analysis of WR stars, possibly influencing the determination of stellar parameters. In view of this, we perform a systematic study by simulating a sequence of WR-star spectra for different temperatures and mass-loss rates using β-type laws with 0.5 ≤ β ≤ 20. We quantify the impact of varying $\upsilon (r)$ by analysing diagnostic lines and spectral classifications of emergent model spectra computed with the Potsdam Wolf–Rayet (PoWR) code. We additionally cross-check these models with hydrodynamically consistent – hydro – model atmospheres. Our analysis confirms that the choice of the β exponent has a strong impact on WR-star spectra, affecting line widths, line strengths, and line profiles. In some parameter regimes, the entire range of WR subtypes could be covered. Comparison with observed WR stars and hydro models revealed that values of β ≳ 8 are unlikely to be realized in nature, but a range of β values needs to be considered in spectral analysis. UV spectroscopy is crucial here to avoid an underestimation of the terminal velocity $\upsilon _\infty$. Neither single- nor double-β descriptions yield an acceptable approximation of the inner wind when compared to hydro models. Instead, we find temperature shifts to lower T2/3 when employing a hydro model. Additionally, there are further hints that round-lined profiles seen in several early WN stars are an effect from non-β velocity laws.

Line luminosities of Galactic and Magellanic Cloud Wolf–Rayet stars

ABSTRACT We provide line luminosities and spectroscopic templates of prominent optical emission lines of 133 Galactic Wolf–Rayet (WR) stars by exploiting Gaia DR3 parallaxes and optical spectrophotometry, and provide comparisons with 112 counterparts in the Magellanic Clouds. Average line luminosities of the broad blue (He ii λ4686, C iii λλ4647,51, N iii λλ4634,41, and N v λλ4603,20) and yellow (C iv λλ5801,12) emission features for WN, WN/C, WC, and WO stars have application in characterizing the WR populations of star-forming regions of distant, unresolved galaxies. Early-type WN stars reveal lower line luminosities in more metal-poor environments, but the situation is less clear for late-type WN stars. LMC WC4–5 line luminosities are higher than their Milky Way counterparts, with line luminosities of Magellanic Cloud WO stars higher than Galactic stars. We highlight other prominent optical emission lines, N iv λλ3478,85 for WN and WN/C stars, O iv λλ3403,13 for WC and WO stars, and O vi λλ3811,34 for WO stars. We apply our calibrations to representative metal-poor and metal-rich WR galaxies, IC 4870 and NGC 3049, respectively, with spectral templates also applied based on a realistic mix of subtypes. Finally, the global blue and C iv λλ5801,12 line luminosities of the Large Magellanic Clouds or LMCs (Small Magellanic Clouds) are 2.6 × 1038 erg s−1 (9 × 1036 erg s−1) and 8.8 × 1037 erg s−1 (4 × 1036 erg s−1), respectively, with the cumulative WR line luminosity of the Milky Way estimated to be an order of magnitude higher than the LMC.

Probing the Stellar Wind of the Wolf–Rayet Star in IC 10 X-1

IC 10 X-1 is an eclipsing high-mass X-ray binary containing a stellar-mass black hole (BH) and a Wolf–Rayet (WR) donor star with an orbital period of P = 34.9 hr. This binary belongs to a group of systems that can be the progenitors of gravitational-wave sources; hence understanding the dynamics of systems such as IC 10 X-1 is of paramount importance. The prominent He ii 4686 emission line (previously used in mass estimates of the BH) is out of phase with the X-ray eclipse, suggesting that this line originates somewhere in the ionized wind of the WR star or in the accretion disk. We obtained 52 spectra from the GEMINI/GMOS archive, observed between 2001 and 2019. We analyzed the spectra both individually, and after binning them by orbital phase to improve the signal-to-noise ratio. The radial-velocity curve from the stacked data is similar to historical results, indicating the overall parameters of the binary have remained constant. However, the He ii line profile shows a correlation with the X-ray hardness-ratio values; also, we report a pronounced skewness of the line profile, and the skewness varies with orbital phase. These results support a paradigm wherein the He ii line tracks structures in the stellar wind that are produced by interactions with the BH’s ionizing radiation and the accretion flow. We compare the observable signatures of two alternative hypotheses proposed in the literature: wind irradiation plus shadowing, and accretion disk hotspot; and we explore how the line-profile variations fit into each of these models.

A New Study on a Type Iax Stellar Remnant and its Probable Association with SN 1181

We report observations and modeling of the stellar remnant and presumed double-degenerate merger of Type Iax supernova Pa 30, which is the probable remnant of SN 1181 AD. It is the only known bound stellar SN remnant and the only star with Wolf–Rayet features that is neither a planetary nebula central star nor a massive Population I progenitor. We model the unique emission-line spectrum with broad, strong O vi and O viii lines as a fast stellar wind and shocked, hot gas. Non-LTE wind modeling indicates a mass-loss rate of ∼10−6 M ⊙ yr−1 and a terminal velocity of ∼15,000 km s−1, consistent with earlier results. O viii lines indicate shocked gas temperatures of T ≃ 4 MK. We derive a magnetic field upper limit of B < 2.5 MG, below earlier suggestions. The luminosity indicates a remnant mass of 1.0–1.65 M ⊙ with ejecta mass 0.15 ± 0.05 M ⊙. Archival photometry suggests the stellar remnant has dimmed by ∼0.5 mag over 100 yr. A low Ne/O < 0.15 argues against an O-Ne white dwarf in the merger. A cold dust shell is only the second detection of dust in an SN Iax and the first of cold dust. Our ejecta mass and kinetic energy estimates of the remnant are consistent with Type Iax extragalactic sources.

Limits on the non-thermal emission of the WR–WR system Apep

Colliding-wind binaries (CWBs) constitute an emerging class of γ-ray sources powered by strong, dense winds in massive stellar systems. The most powerful of them are those binaries hosting a Wolf-Rayet (WR) star. Following the recent discovery of Apep – the closest known Galactic WR–WR binary – we discuss the non-detection of its putative high-energy emission by the Fermi Large Area Telescope (Fermi-LAT) in this Letter. The limits reported in the GeV regime can be used to set a lower limit on the magnetic field pressure density within the shocked wind-collision region (WCR), and to exclude Apep as a bright γ-ray emitting binary. Given that this WR–WR system is the most luminous CWB identified until now at radio wavelengths, this result proves unambiguously that non-thermal synchrotron emission is not a suitable identifier for the subset of γ-ray emitters in this class of particle accelerators. Rather, Apep could be an interesting case of study for magnetic field amplification in shocked stellar winds.

A search for transient, monochromatic light from the Galactic plane

ABSTRACT The Galactic plane was searched for transient, monochromatic light at optical and near-IR wavelengths to detect pulses shorter than 1 s. An objective-prism Schmidt telescope of 0.28-m aperture and a CMOS camera were used to observe 973 square deg, with 8864 exposures of 1-s each, within a strip 2.1 deg wide along the Galactic plane, from Galactic longitude −4 deg to +248 deg. All exposures were analysed for transient, monochromatic sources using a ‘difference image’ algorithm that yielded 11 candidate sources. All 11 sources were found to be associated with either astrophysical emission-line objects or aircraft with subsecond blinking lights. Our survey ‘rediscovered’ many Wolf–Rayet stars, M dwarf flare stars, and planetary nebulae. It also identified an aircraft, of unknown type, that apparently had a nearly monochromatic lamp and a xenon lamp. This survey would have revealed optical and near-IR pulses having a power of ∼180 GW (wavelength dependent) if emitted by a 10-m aperture laser located 1 kiloparsec away. These non-detections of laser pulses from the Galactic plane, including a 10-deg region towards the Galactic centre, add to the non-detections from more than 5000 nearby stars. Indeed, all-sky surveys for emission-line objects (e.g. ionized gas, supernovae remnants, and active galactic nuclei) would have revealed lasers of a wide range of average brightness, wavelength, and cadence. The absence of beacons reveals more of a SETI desert, notably at the intensely surveyed optical and radio wavelengths.

Mapping progenitors of binary black holes and neutron stars with binary population synthesis

ABSTRACT The first directly observed gravitational wave event, GW150914, featuring the merger of two massive black holes, highlighted the need to determine how these systems of compact remnant binaries are formed. We use the binary population synthesis code Compact Object Synthesis and Monte Carlo Investigation Code (COSMIC) to predict the types of massive stars that will show significant radial velocity (RV) variations, indicative of a potential compact object (i.e. a black hole or neutron star) orbiting the star. We ‘observe’ the binaries generated in the populations with a similar number of epochs and RV accuracy as planned for the Milky Way Mapper (MWM) survey. In this analysis, we are especially interested in systems where a compact remnant is orbiting a massive O or B star as these systems survived the first supernova and neutron star kick. We test the ability of the Milky Way Mapper observing strategy to distinguish among different mass-loss and kick prescriptions. We find that Wolf–Rayet stars or hot subdwarfs in binaries could be detectable (i.e. luminous, high ΔRVmax), viable progenitors of such objects, while the different prescriptions primarily affect the number of sources.

Detection of He++ ion in the star-forming ring of the Cartwheel using MUSE data and ionizing mechanisms

ABSTRACT We here report the detection of the nebular He ii λ4686 line in 32 H ii regions in the metal-poor collisional ring galaxy Cartwheel using the Multi-Unit Spectroscopic Explorer (MUSE) data set. The measured I(He ii λ4686)/I(H β) ratio varies from 0.004 to 0.07, with a mean value of 0.010 ± 0.003. Ten of these 32 H ii regions are coincident with the location of an ultra luminous X-ray (ULX) source. We used the flux ratios of important diagnostic lines and results of photoionization by simple stellar populations (SSPs) to investigate the likely physical mechanisms responsible for the ionization of He+. We find that the majority of the regions (27) are consistent with photoionization by star clusters in their Wolf–Rayet (WR) phase with initial ionization parameter −3.5 &amp;lt; log 〈U〉 &amp;lt; −2.0. Blue bump (BB), the characteristic feature of the WR stars, however, is not detected in any of the spectra. We demonstrate that this non-detection is due to the relatively low equivalent width (EW) of the BB in metal-poor SSPs, in spite of containing sufficient number of WR stars to reproduce the observed I(He ii λ4686)/I(H β) ratio of ≤1.5 per cent at the Cartwheel metallicity of Z = 0.004. The H ii regions in the WR phase that are coincident with a ULX source do not show line ratios characteristic of ionization by X-ray sources. However, the ULX sources may have a role to play in the ionization of He+ in two (#99, 144) of the five regions that are not in the WR phase. Ionization by radiative shocks along with the presence of channels for the selective leakage of ionizing photons are the likely scenarios in #17 and #148, the two regions with the highest observed I(He ii λ4686)/I(H β) ratio.

The orbital kinematics of η Carinae over three periastra with a possible detection of the elusive secondary’s motion

ABSTRACT The binary η Carinae is the closest example of a very massive star, which may have formed through a merger during its Great Eruption in the mid-19th century. We aimed to confirm and improve the kinematics using a spectroscopic data set taken with the Cerro Tololo Inter-American Observatory 1.5-m telescope over the time period of 2008–2020, covering three periastron passages of the highly eccentric orbit. We measure line variability of H α and H β, where the radial velocity and orbital kinematics of the primary star were measured from the H β emission line using a bisector method. At phases away from periastron, we observed the He ii 4686 emission moving opposite the primary star, consistent with a possible Wolf–Rayet companion, although with a seemingly narrow emission line. This could represent the first detection of emission from the companion.

Discovery of GeV Gamma-Ray Emission from Pulsar Wind Nebula Kes 75 and PSR J1846–0258

We report the detection of gamma-ray emission from pulsar wind nebula (PWN) Kes 75 and PSR J1846−0258. Through modeling the spectral energy distribution incorporating the new Fermi-LAT data, we find that the observed gamma-ray emission is likely a combination of both the PWN and pulsar magnetosphere. The spectral shape of this magnetospheric emission is similar to the γ-ray spectrum of rotation-powered pulsars detected by Fermi-LAT, and the results from our best-fit model suggest that the pulsar’s magnetospheric emission accounts for 1% of the current spin-down luminosity. Prior works attempted to characterize the properties of this system and found a low supernova (SN) explosion energy and low SN ejecta mass. We reanalyze the broadband emission incorporating the new Fermi emission and compare the implications of our results to prior reports. The best-fit gamma-ray emission model suggests a second very hot photon field possibly generated by the stellar wind of a Wolf–Rayet star embedded within the nebula, which supports the low ejecta mass found for the progenitor in prior reports and here in the scenario of binary mass transfer.

A Survey for C ii Emission-line Stars in the Large Magellanic Cloud. II. Final Results and the Origin of C ii Emission in [WC] Spectra

We present the final results of an imaging and spectroscopic search for stars in the Large Magellanic Cloud (LMC) with C ii λλ7231, 7236 emission lines. The goal is to identify and study [WC11] stars, the coolest of the low-mass Wolf–Rayet sequence, and a subset of central stars of planetary nebulae where the C ii lines are known to be especially prominent. A recent serendipitous discovery of an LMC [WC11] raised the possibility that these objects, although difficult to identify, might in fact be more common than previously believed. Several new members of this rare class have been found in this survey. It now seems clear, however, that a significant number of these stars are not hiding among the general [WC] population. We point out that the C ii doublet intensity ratio observed in our spectra proves to neatly divide the objects into two distinct groups, with the C ii emission likely originating from either the stellar wind or a surrounding nebula. The physics of the C ii emission mechanism correctly explains this bifurcation. Spectral subtypes are suggested for most of the objects. The numerous spectroscopic clues now available for these objects should facilitate future detailed modeling.

Escape of cosmic rays from perpendicular shocks in the circumstellar magnetic field

We investigate the escape process of cosmic rays (CRs) from perpendicular shock regions of a spherical shock propagating to a circumstellar medium with the Parker-spiral magnetic field. The diffusive shock acceleration in perpendicular shocks of supernova remnants (SNRs) is expected to accelerate CRs up to PeV without upstream magnetic field amplification. Red supergiants (RSGs) and Wolf-Rayet (WR) stars are considered as progenitors in this work. We perform test particle simulations to investigate the escape process and escape-limited maximum energy without magnetic field amplification in the upstream region, where the magnetic field strength and rotation period expected from observations of RSGs and WR stars are used. We show that particles escape to the far upstream region while moving along the equator or poles and the maximum energy is about $10-100~{\rm TeV}$ when SNRs propagate to free wind regions of RSGs and WR stars. In most cases, the escape-limited maximum energy is given by the potential difference between the equator and pole. If progenitors are oblique rotators and SNRs are in the early phase just after the supernova explosion, the escape-limited maximum energy is limited by the half wavelength of the wavy current sheet. In addition, for RSGs, we show that the luminosity of CRs accelerated in the wind region is sufficient to supply the observed CR flux above $10~{\rm TeV}$ if a strong magnetic field strength is sustained in most RSGs. In terms of the CR luminosity, SNRs propagating to the free wind of WR stars can contribute to PeV CRs. As long as no magnetic field amplification works around SNR shocks, the maximum energy is decided by the magnetic field strength in the wind region, which depends on the rotation period, stellar wind, and surface magnetic field of RSGs and WR stars. Therefore, we need to observe these quantities to understand the origin of CRs.

Constraints on clumps in the representative wind of the WN8 Wolf–Rayet star HD 96548 = WR 40 with simultaneous broad-band light and linear-polarization variability

ABSTRACT We report precision ground-based broad-band optical intensity and linear-polarization light curves for the sky’s brightest WN8 star, WR 40. WN8 stars are notorious for their high level of variability, stemming from stochastic clumps in their strong winds that are slower and less hot than the winds of most other Wolf–Rayet (WR) stars. We confirm previous results that many WR stars display an amplitude of variability that is an order of magnitude higher in photometry than in polarimetry. For the first time, the unique nature of near simultaneity of our photometric and polarimetric observations of WR 40 allows us to check whether the two types of variability show correlated behaviour, of which we find none. Assuming simple temporal functions for the brightness and polarization of individual clumps, a model for simulated light curves is found to reproduce the properties of the observations, specifically the absence of correlation between photometric and polarimetric variations, the ratio of standard deviations for photometric and polarimetric variability, and the ratio of the average intrinsic polarization relative to its standard deviation. Mapping the solution for time variability to a spatial coordinate suggests that the wind clumps of WR 40 are free-free emitting in addition to light scattering.

Looking for nebular He ii emission south of the multiple-massive star system, HD 5980

ABSTRACT The nebular He ii λ1640 emission line is observed in star-forming galaxies out to large distances and can be used to constrain the properties of sources of He+-ionizing photons. For this purpose, it is crucial to understand which are the main stellar sources of these photons. In some nearby metal-poor starburst galaxies, nebular He ii λ4686 (optical equivalent) is accompanied by a broad underlying component, which is generally attributed to formation in the winds of classical (He-burning) Wolf–Rayet stars, primarily of the WN subtype. In such cases, the origin of the nebular component has been proposed to be the escape of He+-ionizing photons from the winds of the WN stars, at least partially. We use archival long-slit observations obtained with Focal Reducer Low Dispersion Spectrograph (FORS1) on the Very Large Telescope to look for nebular He ii λ4686 emission south of the WN6h + WN6-7 close binary in HD 5980. We only find broad He ii λ4686 emission, as far as ∼7.6 pc from the binary. A comparison with observations obtained with Space Telescope Imaging Spectrograph (STIS) on the Hubble Space Telescope, at a similar orbital phase, shows that the FORS1 broad He ii emission is likely contamination from the multiple-star system HD 5980. We use models to show that no significant He+-ionizing flux is expected from the WN stars in HD 5980 and that when similar stars are present in a coeval stellar population, the O stars can be far greater emitters of He+-ionizing radiation.

Supernovae and gamma-ray burst connection

I review the status of the Supernova/Gamma-Ray Burst connection. Two decades of observations have established that long-duration Gamma-ray Bursts are associated with very energetic SNe-Ib/c. These SN explosions are produced by the gravitational collapse of Wolf–Rayet progenitors, with masses on the main sequence, between 30–50 $M{\odot}$. Current measurements provide a GRB/SNe-Ibc ratio in the range $\sim$0.02–1.5$\%$, so GRBs are very rare events, even when compared with the SN frequency. There is a minority fraction of long-duration GRBs with no accompanying SN that are likely produced during the merging of neutron star or white dwarf neutron star merger.

DISCOVERY OF A NEW [WR] STAR

Wolf–Rayet (WR) stars are evolved massive objects, famous for their spectra rich with emission lines. For the first time WR stars were described more than 150 years ago, and now in our Galaxy we know around 667 objects of this type. However, the discovery of a new WR star, especially based on spectral observations, is still a rare event. The paper is devoted to J040901.83+323955.6 star, discovered a few years ago as WR star in LAMOST spectral survey by machine learning methods. We were interested that different researchers based on different methods classify this object in different ways – ranging from RR Lyr variable to nitrogen rich WR – and so we began our study. We combined spectral and archival photometric data, analysed the spatial location of the star in the Galaxy and concluded what J040901 is [WR] – central star of a planetary nebula.

Mechanisms for high spin in black-hole neutron-star binaries and kilonova emission: inheritance and accretion

ABSTRACT A black-hole neutron-star binary merger can lead to an electromagnetic counterpart called a kilonova if the neutron star is disrupted prior to merger. The observability of a kilonova depends on the amount of neutron star ejecta, which is sensitive to the aligned component of the black hole spin. We explore the dependence of the ejected mass on two main mechanisms that provide high black hole spin in isolated stellar binaries. When the black hole inherits a high spin from a Wolf–Rayet star that was born with least $\sim 10{{\ \rm per\ cent}}$ of its breakup spin under weak stellar core-envelope coupling, relevant for all formation pathways, the median of the ejected mass is ≳10−2 M⊙. Though only possible for certain formation pathways, similar ejected mass results when the black hole accretes $\gtrsim 20{{\ \rm per\ cent}}$ of its companion’s envelope to gain a high spin. Together, these signatures suggest that a population analysis of black-hole neutron-star binary mergers with observed kilonovae may help distinguish between mechanisms for spin and possible formation pathways. We show that these kilonovae will be difficult to detect with current capabilities, but that future facilities, such as the Vera Rubin Observatory, can do so even if the aligned dimensionless spin of the black hole is as low as ∼0.2. Our model predicts kilonovae as bright as Mi ∼ −14.5 for an aligned black hole spin of ∼0.9 and mass ratio Q = 3.6.

Using shell models to investigate clumping in the WN4 star HD 50896

ABSTRACT The spectra of Wolf–Rayet (WR) stars exhibit strong, broad emission lines that originate in the wind. These winds are radiatively driven and are susceptible to hydrodynamic instabilities that result in the formation of clumps. When modelling spectra of WR stars the volume-filling factor (VFF) approach is usually employed to treat clumpy winds. However, it is based on the assumption that the entire wind mass resides in optically thin clumps, which is not necessarily justifiable in dense winds. To test the validity of the VFF approach, we use a previously described method of treating clumping, the ‘shell’ approach, to study line and continuum formation in the dense wind of the WN4 star, HD 50896. Our models indicate that fully intact spherical shells are in tension with observed spectra; a persistent ‘dip’ in emission lines occurs at line centre. Removing this dip requires our models to use ‘broken’ shells – shells that are highly decoherent laterally. This insinuates that the wind of HD 50896, and by extension the winds of other WR stars, are comprised of small laterally confined and radially compressed clumps – clumps smaller than the Sobolev length. We discuss some of the conditions necessary for the VFF approach to be valid.