arXiv subjects

J. D. Landstreet

Publications and source records attributed to J. D. Landstreet.

At least 19 recordsLinked to original sources

arXiv2026

Magnetically guided accretion and extremely slow rotation in a metal-enriched white dwarf

White dwarfs that show atmospheric metal enrichment offer a direct window into the bulk composition of exoplanetary material and provide some of the most direct constraints on the building blocks of rocky exoplanetary systems. Recent work has suggested that magnetic fields may influence how this material is accreted, potentially channelling debris along field lines and retaining it near magnetic poles rather than allowing it to be redistributed uniformly across the stellar surface. Here we report the discovery of photometric, spectroscopic, and spectropolarimetric variability in the magnetic DZ white dwarf WD1532+129, revealing a rotation period of about 289 days, by far the longest yet directly measured for any degenerate star. The observed period may result from the combined effect of angular-momentum loss during the evolution leading to the white dwarf stage and an unusually slowly rotating progenitor, perhaps a magnetic Ap star. The white dwarf exhibits surface abundance inhomogeneities, with metals concentrated near both magnetic poles, and the observed modulation is consistent with localised metal-rich regions aligned with the magnetic geometry. WD1532+129 is the third of the four known magnetic DZ white dwarfs within 20 pc to show evidence for polar metal enhancement, indicating that magnetically guided inhomogeneities are a common property of magnetic metal-enriched white dwarfs.

astro-ph.SR↗

arXiv2024

Strong magnetic fields of old white dwarfs are symmetric about the stellar rotation axes

Many magnetic white dwarfs exhibit a polarised spectrum that periodically varies as the star rotates because the magnetic field is not symmetric about the rotation axis. In this work, we report the discovery that while weakly magnetic white dwarfs of all ages with M < 1Mo show polarimetric variability with a period between hours and several days, the large majority of magnetic white dwarfs in the same mass range with cooling ages older than 2 Gyr and field strengths > 10 MG show little or no polarimetric variability. This could be interpreted as extremely slow rotation, but a lack of known white dwarfs with measured periods longer than two weeks means that we do not see white dwarfs slowing their rotation. We therefore suggest a different interpretation: old strongly magnetic white dwarfs do not vary because their fields are roughly symmetric about the rotation axes. Symmetry may either be a consequence of field evolution or a physical characteristic intrinsic to the way strong fields are generated in older stars. Specifically, a strong magnetic field could distort the shape of a star, forcing the principal axis of maximum inertia away from the spin axis. Eventually, as a result of energy dissipation, the magnetic axis will align with the angular momentum axis. We also find that the higher-mass strongly magnetised white dwarfs, which are likely the products of the merging of two white dwarfs, may appear as either polarimetrically variable or constant. This may be the symptom of two different formation channels or the consequence of the fact that a dynamo operating during a merger may produce diverse magnetic configurations. Alternatively, the massive white dwarfs with constant polarisation may be rotating with periods much shorter than the typical exposure times of the observations.

astro-ph.SR↗

arXiv2024

Searching for magnetic fields in featureless white dwarfs with the DIPOL-UF polarimeter at the Nordic Optical Telescope

About 20% of the white dwarfs possess a magnetic field that may be detected by the splitting and/or polarization of their spectral lines. As they cool, the effective temperatures of the white dwarfs becomes so low that no spectral lines can be seen in the visible wavelength range. If their atmospheres are not polluted by the debris of a planetary system, these cool white dwarfs have featureless optical spectra. Until quite recently, very little was known about the incidence of magnetic fields in these objects. However, when observed with polarimetric techniques, a significant number of featureless white dwarfs reveal strong magnetic fields in their optical continuum spectra. Measuring the occurrence rate and strength of magnetic fields in old white dwarfs may help us to understand how these fields are generated and evolve. We report the results of an ongoing survey of cool white dwarfs with the high-precision broad-band polarimeter DIPOL-UF, which is deployed at the Nordic Optical Telescope on La Palma, Spain. This survey has led to the firm discovery of 13 cool magnetic white dwarfs in the solar neighborhood so far, including six new detections that we report in this paper.

astro-ph.SR↗

arXiv2024

Atmospheric heating and magnetism driven by $^{22}$Ne distillation in isolated white dwarfs

The origin of atmospheric heating in the cool, magnetic white dwarf GD 356 remains unsolved nearly 40 years after its discovery. This once idiosyncratic star with $T_{\rm eff}\approx7500$ K, yet Balmer lines in Zeeman-split emission is now part of a growing class of white dwarfs exhibiting similar features, and which are tightly clustered in the HR diagram suggesting an intrinsic power source. This paper proposes that convective motions associated with an internal dynamo can power electric currents along magnetic field lines that heat the atmosphere via Ohmic dissipation. Such currents would require a dynamo driven by core $^{22}$Ne distillation, and would further corroborate magnetic field generation in white dwarfs by this process. The model predicts that the heating will be highest near the magnetic poles, and virtually absent toward the equator, in agreement with observations. This picture is also consistent with the absence of X-ray or extreme ultraviolet emission, because the resistivity would decrease by several orders of magnitude at the typical coronal temperatures. The proposed model suggests that i) DAHe stars are mergers with enhanced $^{22}$Ne that enables distillation and may result in significant cooling delays; and ii) any mergers that distill neon will generate magnetism and chromospheres. The predicted chromospheric emission is consistent with the two known massive DQe white dwarfs.

astro-ph.SR↗

arXiv2024

Metal accretion scars may be common on magnetic, polluted white dwarfs

More than 30% of white dwarfs exhibit atmospheric metals, which are understood to be from recent or ongoing accretion of circumstellar debris. In cool white dwarfs, surface motions should rapidly homogenise photospheric abundances, and the accreted heavy elements should diffuse inward on a timescale much longer than that for surface mixing. The recent discovery of a metal scar on WD0816-310 implies its magnetic field has impeded surface mixing of metals near the visible magnetic pole. Here, we report the discovery of a second magnetic, metal-polluted white dwarf, WD2138-332, which exhibits periodic variability in longitudinal field, metal line strength, and broadband photometry. All three variable quantities have the same period, and show remarkable correlations: the published light curves have a brightness minimum exactly when the longitudinal field and line strength have a maximum, and a maximum when the longitudinal field and line strength have a minimum. The simplest interpretation of the line strength variability is that there is an enhanced metal concentration around one pole of the magnetic field; however, the variable line-blanketing cannot account for the observed multi-band light curves. More theoretical work is required to understand the efficiency of horizontal mixing of the accreted metal atoms, and the origin of photometric variability. Because both magnetic, metal-polluted white dwarfs that have been monitored to date show that metal line strengths vary in phase with the longitudinal field, we suggest that metal scars around magnetic poles may be a common feature of metal-polluted white dwarfs.

astro-ph.SR↗

arXiv2024

Discovery of magnetically guided metal accretion onto a polluted white dwarf

Dynamically active planetary systems orbit a significant fraction of white dwarf stars. These stars often exhibit surface metals accreted from debris disks, which are detected through infrared excess or transiting structures. However, the full journey of a planetesimal from star-grazing orbit to final dissolution in the host star is poorly understood. Here, we report the discovery that the cool metal polluted star WD0816-310 has cannibalized heavy elements from a planetary body similar in size to Vesta, and where accretion and horizontal mixing processes have clearly been controlled by the stellar magnetic field. Our observations unveil periodic and synchronized variations in metal line strength and magnetic field intensity, implying a correlation between the local surface density of metals and the magnetic field structure. Specifically, the data point to a likely persistent concentration of metals near a magnetic pole. These findings demonstrate that magnetic fields may play a fundamental role in the final stages of exoplanetary bodies that are recycled into their white dwarf hosts.

astro-ph.SR↗

arXiv2022

The magnetic field of the chemically peculiar star V352Peg

We present a spectropolarimetric analysis of the hot star V352Peg. We have acquired 18 spectropolarimetric observations of the star with ESPaDOnS at the CFHT between 2018 and 2019 and completed our dataset with one archival ESPaDOnS measurement obtained in 2011. Our analysis of the spectra shows that the star is on the main sequence and chemically peculiar, i.e. it is a Bp star, with overabundances of iron peak elements (Ti, Cr and Fe) and underabundance of He and O. Through a Least-Square Deconvolution of each spectrum, we extracted the mean Zeeman signature and mean line profile of thousands of spectral lines and detected a magnetic field in V352Peg. By modelling the Stokes I and V profiles and using the Oblique Rotator Model, we determined the geometrical configuration of V352Peg. We also performed Zeeman-Doppler Imaging (ZDI) to provide a more detailed characterization of the magnetic field of V352Peg and its surface chemical distributions. We find a magnetic field that is mainly dipolar, dominantly poloidal, and largely non-axisymmetric with a dipole field strength of $\sim$9 kG and a magnetic axis almost perpendicular to the rotation axis. The strong variability of Stokes I profiles also suggests the presence of chemical spots at the stellar surface.

astro-ph.SR↗

arXiv2022

MOBSTER -- VI. The crucial influence of rotation on the radio magnetospheres of hot stars

Numerous magnetic hot stars exhibit gyrosynchrotron radio emission. The source electrons were previously thought to be accelerated to relativistic velocities in the current sheet formed in the middle magnetosphere by the wind opening magnetic field lines. However, a lack of dependence of radio luminosity on the wind power, and a strong dependence on rotation, has recently challenged this paradigm. We have collected all radio measurements of magnetic early-type stars available in the literature. When constraints on the magnetic field and/or the rotational period are not available, we have determined these using previously unpublished spectropolarimetric and photometric data. The result is the largest sample of magnetic stars with radio observations that has yet been analyzed: 131 stars with rotational and magnetic constraints, of which 50 are radio-bright. We confirm an obvious dependence of gyrosynchrotron radiation on rotation, and furthermore find that accounting for rotation neatly separates stars with and without detected radio emission. There is a close correlation between H$α$ emission strength and radio luminosity. These factors suggest that radio emission may be explained by the same mechanism responsible for H$α$ emission from centrifugal magnetospheres, i.e. centrifugal breakout (CBO), however, whereas the H$α$-emitting magnetosphere probes the cool plasma before breakout, radio emission is a consequence of electrons accelerated in centrifugally-driven magnetic reconnection.

astro-ph.SR↗

arXiv2021

New insight into the magnetism of degenerate stars from the analysis of a volume limited sample of white dwarfs

Many stars evolve into magnetic white dwarfs, and observations may help to understand when the magnetic field appears at the stellar surface, if and how it evolves during the cooling phase, and what are the mechanisms that generate it. After obtaining new spectropolarimetric observations and combining them with previous literature data, we have checked the population of about 152 white dwarfs within 20 pc from the Sun for the presence of magnetic fields, with a sensitivity that ranges from better than 1 kG for most of the stars of spectral class DA, to 1 MG for some of the featureless white dwarfs. We find that 33 white dwarfs of the local 20 pc volume are magnetic. Statistically the data are consistent with the possibility that the frequency of the magnetic field occurrence is similar in stars of all spectral classes, except that in the local 20 pc volume, either DQ stars are more frequently magnetic, or host much stronger fields than average. The distribution of the observed field strength ranges from 40 kG to 300 MG and is uniform per decade, in striking contrast to the field frequency distribution resulting from spectroscopic surveys. No fields weaker than 40 kG are found. We confirm that magnetic fields are more frequent in white dwarfs with higher than average mass. We find a marked deficiency of magnetic white dwarfs younger than 0.5 Gyr, and that the frequency of the occurrence of the magnetic field is significantly higher in white dwarfs that have undergone the process of core crystallisation than in white dwarfs with fully liquid core. There is no obvious evidence of field strength decay with time. We discuss the implications of our findings in relation to some of the proposals that have been put forward to explain the origin and evolution of magnetic fields in degenerate stars, in particular those that predict the presence of a dynamo acting during the crystallisation phase.

astro-ph.SR↗

arXiv2021

A test of the planet-star unipolar inductor for magnetic white dwarfs

Despite thousands of spectroscopic detections, only four isolated white dwarfs exhibit Balmer emission lines. The temperature inversion mechanism is a puzzle over 30 years old that has defied conventional explanations. One hypothesis is a unipolar inductor that achieves surface heating via ohmic dissipation of a current loop between a conducting planet and a magnetic white dwarf. To investigate this model, new time-resolved spectroscopy, spectropolarimetry, and photometry of the prototype GD 356 are studied. The emission features vary in strength on the rotational period, but in anti-phase with the light curve, consistent with a cool surface spot beneath an optically thin chromosphere. Possible changes in the line profiles are observed at the same photometric phase, potentially suggesting modest evolution of the emission region, while the magnetic field varies by 10 per cent over a full rotation. These comprehensive data reveal neither changes to the photometric period, nor additional signals such as might be expected from an orbiting body. A closer examination of the unipolar inductor model finds points of potential failure: the observed rapid stellar rotation will inhibit current carriers due to the centrifugal force, there may be no supply of magnetospheric ions, and no anti-phase flux changes are expected from ohmic surface heating. Together with the highly similar properties of the four cool, emission-line white dwarfs, these facts indicate that the chromospheric emission is intrinsic. A tantalizing possibility is that intrinsic chromospheres may manifest in (magnetic) white dwarfs, and in distinct parts of the HR diagram based on structure and composition.

astro-ph.SR↗

arXiv2020

Discovery of six new strongly magnetic white dwarfs in the 20 pc local population

The sample of white dwarfs included in the local 20 pc volume documents, fairly accurately, the total production of white dwarfs over roughly 10 Gyr of stellar evolution in this part of the Milky Way Galaxy. In this sample, we have been systematically searching for magnetic white dwarfs. Here we report the discovery of six new magnetic white dwarfs, with a field strength from a few MG to about 200MG. Two of these stars show H lines that are split and polarised by the magnetic field. One star shows extremely weak spectral lines in intensity, to which highly polarised narrow features correspond. The three other stars have featureless flux spectra, but show continuum polarisation. These new discoveries support the view that at least 20% of all white dwarfs in the local 20 pc volume have magnetic fields, and they fully confirm the suspicion that magnetism is a common rather than a rare characteristic of white dwarfs. We discuss the level and the handedness of the continuum polarisation in the presence of a magnetic field in cool white dwarfs. We suggest that a magnetic field with a 15 MG longitudinal component produces 1% of continuum circular polarisation. We have also shown that the problem of cross-talk from linear to circular polarisation of the FORS2 instrument, used in our survey, represents an obstacle to accurate measurements of the circular polarisation of faint white dwarfs when the background is illuminated, and polarised, by the moon.

astro-ph.SR↗

arXiv2020

The Magnetic Early B-type Stars IV: Breakout or Leakage? H$α$ emission as a diagnostic of plasma transport in centrifugal magnetospheres

Rapidly rotating early-type stars with strong magnetic fields frequently show H$α$ emission originating in Centrifugal Magnetospheres (CMs), circumstellar structures in which centrifugal support due to magnetically enforced corotation of the magnetically confined plasma enables it to accumulate to high densities. It is not currently known whether the CM plasma escapes via Centrifugal Breakout (CB), or by an unidentified leakage mechanism. We have conducted the first comprehensive examination of the H$α$ emission properties of all stars currently known to display CM-pattern emission. We find that the onset of emission is dependent primarily on the area of the CM, which can be predicted simply by the value $B_{\rm K}$ of the magnetic field at the Kepler corotation radius $R_{\rm K}$. Emission strength is strongly sensitive to both CM area and $B_{\rm K}$. Emission onset and strength are {\em not} dependent on effective temperature, luminosity, or mass-loss rate. These results all favour a CB scenario, however the lack of intrinsic variability in any CM diagnostics indicates that CB must be an essentially continuous process, i.e.\ it effectively acts as a leakage mechanism. We also show that the emission profile shapes are approximately scale-invariant, i.e.\ they are broadly similar across a wide range of emission strengths and stellar parameters. While the radius of maximum emission correlates closely as expected to $R_{\rm K}$, it is always larger, contradicting models that predict that emission should peak at $R_{\rm K}$.

astro-ph.SR↗

arXiv2020

Discovery of a Sirius-like binary system with a very strongly magnetic white dwarf. Binarity among magnetic white dwarfs

Binary systems containing a magnetic white dwarf and a main-sequence star are considered extremely rare, perhaps non-existent. In the course of a search of magnetic fields in high-mass white dwarfs we have discovered a Sirius-like wide binary system composed of a main-sequence G0 star and a $M\sim 1.1\,M_\odot$ white dwarf with a huge (hundreds of MG) magnetic field. This star, WDS J03038+0608B, shows a circular polarisation amplitude of 5% in the continuum, with no evidence of variability in a 1d time-scale, little or no linear polarisation in the blue part of the spectrum, and about 2% linear polarisation in the red part of the optical spectrum. A search in the literature reveals the existence of at least four more binary systems that include a magnetic white dwarf and a non-degenerate companion - three such systems passed unremarked in previous studies. We estimate that up to a few percent of magnetic white dwarfs may be found to occur in wide binary pairs. However, at least four of the five known binary systems with a magnetic white dwarf are too widely separated to be expected to evolve into systems experiencing Roche lobe overflow, and cannot be considered as progenitors of magnetic cataclysmic variable (AM Her and DQ Her) systems.

astro-ph.SR↗

arXiv2019

Discovery of weak magnetic fields in four DZ white dwarfs in the local 20pc volume. Implications for the frequency of magnetic fields with cooling age

We report the discovery of weak magnetic fields in three white dwarfs within the local 20pc volume (WD 0816-310, WD 1009-184, and WD 1532+129), and we confirm the magnetic nature of a fourth star (WD 2138-332) in which we had previously detected a field at a 3 sigma level. The spectra of all these white dwarfs are characterised by the presence of metal lines and lack of H and He lines, that is, they belong to the spectral class DZ. The polarisation signal of the Ca II H+K lines of WD 1009-184 is particularly spectacular, with an amplitude of 20% that is due to the presence of a magnetic field with an average line-of-sight component of 40kG. We have thus established that at least 40% of the known DZ white dwarfs with an He-rich atmosphere contained in the 20pc volume have a magnetic field, while further observations are needed to establish whether the remaining DZ white dwarfs in the same volume are magnetic or not. Metal lines in the spectra of DZ white dwarfs are thought to have originated by accretion from rocky debris, and it might be argued that a link exists between metal accretion and higher occurrence of magnetism. However, we are not able to distinguish whether the magnetic field and the presence of a polluted atmosphere have a common origin, or if it is the presence of metal lines that allows us to detect a higher frequency of magnetic fields in cool white dwarfs, which would otherwise have featureless spectra. We argue that the new highly sensitive longitudinal field measurements that we have made in recent years are consistent with the idea that the magnetic field appears more frequently in older than in younger white dwarfs.

astro-ph.SR↗

arXiv2019

The variation of the magnetic field of the Ap star HD~50169 over its 29 year rotation period

Context. The Ap stars that rotate extremely slowly, with periods of decades to centuries, represent one of the keys to the understanding of the processes leading to the differentiation of stellar rotation. Aims. We characterise the variations of the magnetic field of the Ap star HD 50169 and derive constraints about its structure. Methods. We combine published measurements of the mean longitudinal field of HD 50169 with new determinations of this field moment from circular spectropolarimetry obtained at the 6-m telescope BTA of the Special Astrophysical Observatory of the Russian Academy of Sciences. For the mean magnetic field modulus , literature data are complemented by the analysis of ESO spectra, both newly acquired and from the archive. Radial velocities are also obtained from these spectra. Results. We present the first determination of the rotation period of HD 50169, Prot = (29.04+/-0.82) y. HD 50169 is currently the longest-period Ap star for which magnetic field measurements have been obtained over more than a full cycle. The variation curves of both and have a significant degree of anharmonicity, and there is a definite phase shift between their respective extrema. We confirm that HD 50169 is a wide spectroscopic binary, refine its orbital elements, and suggest that the secondary is probably a dwarf star of spectral type M. Conclusions. The shapes and mutual phase shifts of the derived magnetic variation curves unquestionably indicate that the magnetic field of HD 50169 is not symmetric about an axis passing through its centre. Overall, HD 50169 appears similar to the bulk of the long-period Ap stars.

astro-ph.SR↗

arXiv2018

Evidence for mass accretion driven by spiral shocks onto the white dwarf in SDSS J123813.73-033933.0

We present high-time-resolution photometry and phase-resolved spectroscopy of the short-period ($P_\mathrm{orb} = 80.52\,\mathrm{min}$) cataclysmic variable SDSS J123813.73-033933.0, observed with the $\mathit{Hubble}$ $\mathit{Space}$ $\mathit{Telescope}$ $\mathit{(HST)}$, the $\mathit{Kepler/K2}$ mission and the Very Large Telescope (VLT). We also report observations of the first detected super-outburst. SDSS J1238-0339 shows two types of variability: quasi-regular brightenings recurring every $\simeq 8.5$ h during which the system increases in brightness by $\simeq 0.5$mag, and a double hump quasi-sinusoidal modulation at the orbital period. The detailed $\mathit{K2}$ light curve reveals that the amplitude of the double-humps increases during the brightenings and that their phase undergoes a $\simeq 90^{\circ}$ phase shift with respect to the quiescent intervals. The $\mathit{HST}$ data unambiguously demonstrate that these phenomena both arise from the heating and cooling of two relatively large regions on the white dwarf. We suggest that the double-hump modulation is related to spiral shocks in the accretion disc resulting in an enhanced accretion rate heating two localised regions on the white dwarf, with the structure of the shocks fixed in the binary frame explaining the period of the double humps. The physical origin of the 8.5 h brightenings is less clear. However, the correlation between the observed variations of the amplitude and phase of the double-humps with the occurrence of the brightenings is supportive of an origin in thermal instabilities in the accretion disc.

astro-ph.SR↗

arXiv2017

Monitoring and modelling of white dwarfs with extremely weak magnetic fields

Magnetic fields are detected in a few percent of white dwarfs. The number of such magnetic white dwarfs known is now some hundreds. Fields range in strength from a few kG to several hundred MG. Almost all the known magnetic white dwarfs have a mean field modulus >= 1 MG. We are trying to fill a major gap in observational knowledge at the low field limit (<= 200 kG) using circular spectro-polarimetry. In this paper we report the discovery and monitoring of strong, periodic magnetic variability in two previously discovered "super-weak field" magnetic white dwarfs, WD2047+372 and WD2359-434. WD2047+372 has a mean longitudinal field that reverses between about -12 and +15 kG, with a period of 0.243 d, while its mean field modulus appears nearly constant at 60 kG. The observations can be intepreted in terms of a dipolar field tilted with respect to the stellar rotation axis. WD2359-434 always shows a weak positive longitudinal field with values between about 0 and +12 kG, varying only weakly with stellar rotation, while the mean field modulus varies between about 50 and 100 kG. The rotation period is found to be 0.112 d using the variable shape of the Halpha line core, consistent with available photometry. The field of this star appears to be much more complex than a dipole, and is probably not axisymmetric. Available photometry shows that WD2359-434 is a light variable with an amplitude of only 0.005 mag, our own photometry shows that if WD2047+372 is photometrically variable, the amplitude is below about 0.01 mag. These are the first models for magnetic white dwarfs with fields below about 100 kG based on magnetic measurements through the full stellar rotation. They reveal two very different magnetic surface configurations, and that, contrary to simple ohmic decay theory, WD2359-434 has a much more complex surface field than the much younger WD2047+372.

astro-ph.SR↗

arXiv2017

Magnetic field topology and chemical abundance distributions of the young, rapidly rotating chemically peculiar star HR5624

The young, rapidly rotating Bp star HR5624 (HD133880) shows an unusually strong non-sinusoidal variability of its longitudinal magnetic field. This behaviour was previously interpreted as the signature of an exceptionally strong, quadrupole-dominated surface magnetic field geometry. We studied the magnetic field structure and chemical abundance distributions of HR5624 with the aim to verify the unusual quadrupolar nature of its magnetic field and to investigate correlations between the field topology and chemical spots. We analysed high resolution, time series Stokes parameter spectra of HR5624 with the help of a magnetic Doppler imaging inversion code based on detailed polarised radiative transfer modelling of the line profiles. Our magnetic Doppler inversions reveal that the field structure of HR5624 is considerably simpler and the field strength is much lower than proposed by previous studies. We find a maximum local field strength of 12 kG and a mean field strength of 4 kG. Our model implies that overall large-scale field topology of HR5624 is better described as a distorted, asymmetric dipole. The chemical abundance maps of Mg, Si, Ti, Cr, Fe, and Nd obtained in our study are characterised by large-scale, high-contrast abundance patterns. These structures correlate weakly with the magnetic field geometry and show no distinct element concentrations in the horizontal field regions predicted by theoretical atomic diffusion calculations. We conclude that the surface magnetic field topology of HR5624 is not as unusual as previously proposed. Considering these results together with other recent magnetic mapping analyses of early-type stars suggests that predominantly quadrupolar magnetic field topologies, invoked to be present in a significant number of stars, probably do not exist in real stars.

astro-ph.SR↗