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# Infrared variability due to magnetic pressure-driven jets, dust ejection and quasi-puffed-up inner rims

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## Abstract

The interaction between a YSO stellar magnetic field and its protostellar disc can result in stellar accretional flows and outflows from the inner disc rim. Gas flows with a velocity component perpendicular to disc mid-plane subject particles to centrifugal acceleration away from the protostar, resulting in particles being catapulted across the face of the disc. The ejected material can produce a ‘dust fan’, which may be dense enough to mimic the appearance of a ‘puffed-up’ inner disc rim. We derive analytical equations for the time-dependent disc toroidal field, the disc magnetic twist, the size of the stable toroidal disc region, the jet speed, and the disc region of maximal jet flow speed. We show how the observed infrared variability of the pre-transition disc system LRLL 31 can be modelled by a dust ejecta fan from the inner-most regions of the disc whose height is partially dependent on the jet flow speed. The greater the jet flow speed, the higher is the potential dust fan scale height. An increase in mass accretion on to the star tends to increase the height and optical depth of the dust ejection fan, increasing the amount of 1–8 µm radiation. The subsequent shadow reduces the amount of light falling on the outer disc and decreases the 8–40 µm radiation. A decrease in the accretion rate reverses this scenario, thereby producing the observed ‘see-saw’ infrared variability.

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... Then, these grains could have been lifted up from the disk upper layers via the drag forces caused by the outflowing gas. Recent studies have investigated the possibility of dust ejection from the inner disk (Liffman et al. 2020;Vinkovic & Čemeljić 2021;Hutchison & Clarke 2021;Booth & Clarke 2021), but one must verify they don't get shattered and can survive in the outflow (Wong et al. 2016). Indeed, it would be interesting to assess the grain fragmentation timescale versus the grain aggregation timescale in outflow cavities. ...
Thesis
With the aim of characterizing the role played by magnetic fields in the formation of young protostars, several recent studies have revealed unprecedented features toward high angular resolution ALMA dust polarization observations of Class 0 protostellar cores. Observations of polarized dust emission allow us to investigate the physical processes involved in the Radiative Alignment Torques (RATs) acting on dust grains from the core to disk scales, that align the angular momentum of grains with magnetic field. We find that the dust polarization is enhanced along the cavity walls of bipolar outflows, which are subject to high irradiation from the reprocessed radiation field emanating from the center of the protostar. In addition, highly polarized dust thermal emission has been detected in region most likely linked with the infalling envelope, in the form of filamentary structure being potential magnetized accretion streamer. Notably, we propose that the polarized emission we see at millimeter wavelengths along the irradiated cavity walls can be reconciled with the expectations of RAT theory if the aligned grains present in these cavities have grown larger than what is typically expected in young protostellar cores. To approach an estimation of the efficiency of dust alignment in protostars, we gathered a large sample of ALMA dust polarization observations of Class 0 protostars in order to perform a statistical analysis examining the trend between the dispersion of polarization position angles and the fractional polarization. We report a significant correlation between these two quantities, whose power-law index differs significantly from the one observed by Planck in star-forming clouds, confirming the different nature for the disorganized component of magnetic fields at the scales of protostellar envelopes. The grain alignment efficiency, is surprisingly constant across three orders of magnitude in envelope column density. Synthetic observations of non-ideal magneto-hydrodynamic simulations of protostellar cores implementing RATs, show that the ALMA values of grain alignment efficiency lie among those predicted by a perfect alignment of grains, and are significantly higher than the ones obtained with RATs. Ultimately, our results suggest dust alignment mechanism(s) are efficient at producing polarized dust emission in the local conditions typical of Class 0 protostars. The grain alignment efficiency found in these objects seems to be higher than the efficiency produced by the standard RAT alignment of paramagnetic grains. We performed further detailed modelling of the protostellar inner envelope physical conditions, alongside tentative comparisons between ALMA molecular line observations of UV-sensitive chemical tracers and dust polarization observations. We found that indeed, grains with super-paramagnetic inclusions, significant irradiation conditions (qualitatively comforted by the chemical observations), and large grains (10 micron) of compact structure are necessary to reproduce the observed grain alignment efficiency. However, further studies leading to a better characterization of dust grain characteristics, and additional grain alignment mechanisms, will be required to investigate deeper the cause of strong polarized dust emission located in regions of the envelope where alignment conditions are not favorable.
Article
In order to improve the physical interpretation about innermost dusty regions in protoplanetary discs around brown dwarf (BD), and even very low mass star (VLMS), we present a grid of models taking into account two different sets: (i) The set called standard model, that simulates an axisymmetric dusty disc with an inner curved wall. (ii) The perturbed one called non-standard where the axisymmetry of the inner edge has been broken. We have achieved a fitting for the disc structure able to explain the spectral energy distribution (SED). As the main condition, we assume that the changes of the inner wall geometry in the tongue-like shape depend on the Rayleigh-Taylor instability (R-TIns) generated in the inner disc edge. For each object, we parametrize the shape of the inner wall to find a time-dependent model that enables us to explain the photometric near-Infrared variability and connect the changes on the inner disc structure with the amplitude of such variability. We re-analyzed photometric measurements from near to mid-infrared wavelengths of a sample of 6 BDs and one VLMS in different cloud associations which were previously studied by other authors. We also show that the flux change calculated between the non-standard and the standard configurations models the observed variability in LRLL 1679. The magnitude changes due to these fluctuations slightly depend on the wavelength and they can present differences of up to 0.9 mag. We suggest that if the R-TIns persist enough time, the features in the protoplanetary inner disc, e.g. inner holes or gaps evolve.
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We explore dust flow in the hottest parts of protoplanetary discs using the forces of gravity, gas drag, and radiation pressure. Our main focus is on the optically thin regions of dusty disc, where the dust is exposed to the most extreme heating conditions and dynamical perturbations: the surface of optically thick disc and the inner dust sublimation zone. We utilize results from two numerically strenuous fields of research. The first is the quasi-stationary solutions on gas velocity and density distributions from mangetohydrodynamical (MHD) simulations of accretion discs. This is critical for implementing a more realistic gas drag impact on dust movements. The second is the optical depth structure from a high-resolution dust radiation transfer. This step is critical for a better understanding of dust distribution within the disc. We describe a numerical method that incorporates these solutions into the dust dynamics equations. We use this to integrate dust trajectories under different disc wind models and show how grains end up trapped in flows that range from simple accretion on to the star to outflows into outer disc regions. We demonstrate how the radiation pressure force plays one of the key roles in this process and cannot be ignored. It erodes the dusty disc surface, reduces its height, resists dust accretion on to the star, and helps the disc wind in pushing grains outwards. The changes in grain size and porosity significantly affect the results, with smaller and porous grains being influenced more strongly by the disc wind and radiation pressure.
Preprint
A small subset of young stellar objects (YSOs) exhibit "see-saw" temporal variations in their mid-infrared SED; as the flux short-ward of a fulcrum wavelength ($\lambda_{f}$) increases the flux long-wards of this wavelength decreases (and vice-versa) over timescales of weeks to years. While previous studies have shown that an opaque, axisymmetric occulter of variable height can cause this behaviour in the SED of these objects, the conditions under which a single $\lambda_{f}$ occurs have not previously been determined, nor the factors determining its value. Using radiative transfer modelling, we conduct a parametric study of the exemplar of this class, LRLL 31 to explore this phenomenon, and confirm that the cause of this flux variation is likely due to the change in height of the optically thick inner rim of the accretion disc at the dust sublimation radius, or some other phenomenon which results in a similar appearance. We also determine that a fulcrum wavelength only occurs for high inclinations, where the line of sight intersects the accretion disc. Accepting that the disc of LRLL 31 is highly inclined, the inner rim radius, radial and vertical density profiles are independently varied to gauge what effect this had on $\lambda_{f}$ and its position relative to the silicate feature near $10 \mu$m. While $\lambda_{f}$ is a function of each of these parameters, it is found to be most strongly dependent on the vertical density exponent $\beta$. All other factors being held constant, only for flatter discs ($\beta < 1.2$) did we find a $\lambda_{f}$ beyond the silicate feature.
Preprint
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Classical T Tauri stars with ages of less than 10 Myr possess accretion discs. Magnetohydrodynamic processes at the boundary between the disc and the stellar magnetosphere control the accretion and ejections gas flows. We carried out a long series of simultaneous spectroscopic and photometric observations of the classical T Tauri stars RY Tau and SU Aur with the aim to quantify the accretion and outflow dynamics at time scales from days to years. It is shown that dust in the disc wind is the main source of photometric variability of these stars. In RY Tau we observed a new effect: during events of enhanced outflow the circumstellar extinction gets lower. The characteristic time of changes in outflow velocity and stellar brightness indicates that the obscuring dust is near the star. The outflow activity in both stars is changing on a time scale of years. Periods of quiescence in H$\alpha$ profile variability were observed during 2015-2016 season in RY Tau and during 2016-2017 season in SU Aur. We interpret these findings in the framework of the magnetospheric accretion model, and discuss how the global stellar magnetic field may influence the long-term variations of the outflow activity.
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Protostellar jets are one of the most intriguing signposts in star formation. Recent detection of a jet rotation indicates that they can carry away angular momenta from the innermost edges of the disks, allowing the disks to feed the central protostars. In current jet-launching models, magnetic fields are required to launch and collimate the jets, however, observationally, it is still uncertain if magnetic fields are really present in the jets. Here we report a clear detection of SiO line polarization in the HH 211 protostellar jet. Since this line polarization has been attributed to the Goldreich-Kylafis effect in the presence of magnetic field, our observations show convincingly the presence of magnetic field in a jet from a low-mass protostar. The implied magnetic field could be mainly toroidal, as suggested in current jet-launching models, in order to collimate the jet at large distances.
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We investigate the properties of magnetized stars in the propeller regime using axisymmetric numerical simulations. We modelled the propeller regime for stars with realistically large magnetospheres (5-7 stellar radii), so that our results could be applied to different types of magnetized stars, including Classical T Tauri stars, cataclysmic variables, and accreting millisecond pulsars. A wide range of propeller strengths has been studied, from very strong propellers (where the magnetosphere rotates much more rapidly than the inner disk) to very weak propellers (where the magnetosphere rotates only slightly faster than the inner disc. In both propellers, matter is accumulated at the inner disc for the majority of the time, while episodes of accretion onto the star and ejection into the wind are relatively brief. The efficiency of the propeller, which characterizes the part of inner disk matter flowing into the wind, strongly depends on the fastness parameter omega_s: propeller efficiency increases with omega_s. The properties of the winds are different in strong and weak propellers. In the strong propellers, matter is accelerated rapidly above the escape velocity and flows at a relatively small opening angle of 40-45 degrees. In the weak propellers, matter may flow faster or slower than the escape velocity and at a large opening angle of 60-70 degrees. A star-disk system loses energy and angular momentum. A part of the rotational energy of the star is ejected to the magnetically-dominated (Poynting flux) jet, which is only present in the strong propellers. The other part of the energy flows from the inner disk into a propeller-driven wind. A star spins down partly due to the flow of angular momentum from the star to the corona or to the Poynting flux jet along the open field lines, and partly due to the flow of angular momentum to the inner disk along the closed field lines.
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A simple but general theory for the formation and propagation of nonrelativistic jets and winds from magnetized accretion disks is derived from the equations of ideal magnetohydrodynamics (MHD). The theory characterizes a jet by its axial velocity, radius, angular rotation rate, and temperature as functions of axial distance z from the central object. Differential equations for these four variables are derived from the conservation laws of ideal MHD for the flow of mass, momentum, angular momentum, and energy in the jet. The jets are assumed to carry no net current because this is energetically favored. The magnetic field is however essential in that the zz-component of the magnetic stress acts, in opposition to gravity, to "drive" matter through the slow magnetosonic critical point. The centrifugal force has no direct role at this critical point. Close to the disk, the gravitational field of the central object provides weak collimation of the flow. At much larger distances, good collimation can result from the focusing effect of the external medium. Further, the pressure of the external medium can act, in opposition to gravity, to "drive" the flow through the fast magnetosonic critical point. Conservation of mass, angular-momentum, and energy of the disk/jet system significantly constrains the jet solutions. For a representative self-consistent disk/jet solution relevant to a protostellar system, the fraction of the accreted mass expelled in the jets is 0.1, the ratio of the power carried by the jets (kinetic and Poynting fluxes) to the disk luminosity is 0.66, and the ratio of the boundary layer to disk luminosities is less-than-or-similar-to 0.13. The star's rotation rate is found to decrease with time even for rotation rates much less than the breakup rate.
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Context. Transition disks are protoplanetary disks with inner depleted dust cavities that are excellent candidates for investigating the dust evolution when there is a pressure bump. A pressure bump at the outer edge of the cavity allows dust grains from the outer regions to stop their rapid inward migration towards the star and to efficiently grow to millimetre sizes. Dynamical interactions with planet(s) have been one of the most exciting theories to explain the clearing of the inner disk. Aims. We look for evidence of millimetre dust particles in transition disks by measuring their spectral index α_mm with new and available photometric data. We investigate the influence of the size of the dust depleted cavity on the disk integrated millimetre spectral index. Methods. We present the 3-mm (100 GHz) photometric observations carried out with the Plateau de Bure Interferometer of four transition disks: LkHα 330, UX Tau A, LRLL 31, and LRLL 67. We used the available values of their fluxes at 345 GHz to calculate their spectral index, as well as the spectral index for a sample of twenty transition disks. We compared the observations with two kinds of models. In the first set of models, we considered coagulation and fragmentation of dust in a disk in which a cavity is formed by a massive planet located at different positions. The second set of models assumes disks with truncated inner parts at different radii and with power-law dust-size distributions, where the maximum size of grains is calculated considering turbulence as the source of destructive collisions. Results. We show that the integrated spectral index is higher for transition disks (TD) than for regular protoplanetary disks (PD) with mean values of α^(TD)_(mm) = 2.70 ± 0.13 and α^(PD)_(mm) = 2.20 ± 0.07 respectively. For transition disks, the probability that the measured spectral index is positively correlated with the cavity radius is 95%. High angular resolution imaging of transition disks is needed to distinguish between the dust trapping scenario and the truncated disk case.
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A general theory is developed for relativistic, steady, axisymmetric, ideal magnetohydrodynamic flows around a black hole or a rotating magnetized star. The theory leads to an autonomous second-order partial differential equation - a Grad-Shafranov equation - for the magnetic flux function ψ(r,z). One limit of this equation gives the familiar Grad-Shafranov equation which describes the equilibrium of axisymmetric fusion plasmas. Another limit gives the equation describing general nonmagnetic flows of matter with angular momentum. A further limit gives the "pulsar equation" of Scharlemann, Wagoner, and Michel for relativistic plasma flows around an aligned, rotating, magnetized neutron star. Applications of the theory are made to thin, magnetized disks around a Schwarzschild black hole and around an aligned, rotating, magnetized star.
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We present a nozzle equation for isentropic, magnetohydrodynamic (MHD) flow in a channel, where the critical flow speed is the magnetosonic speed. A similar equation for a jet from a young stellar object (YSO) is obtained, where the jet flow is produced from a toroidal field, embedded in an accretion disc, which surrounds a YSO. For at least one case, the critical flow point for a YSO jet occurs on or near the surface of the accretion disc. We use this result to model how the stability of the toroidal fields, in the jet region, may be enhanced by the ram pressure of the jet.
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We apply ionization balance and MHD calculations to investigate whether magnetic activity moderated by recombination on dust can account for the mass accretion rates and the mid-infrared spectra and variability of protostellar disks. The MHD calculations use the stratified shearing-box approach and include grain settling and the feedback from the changing dust abundance on the resistivity of the gas. The two-decade spread in accretion rates among T Tauri stars is too large to result solely from variety in the grain size and stellar X-ray luminosity, but can be produced by varying these together with the disk magnetic flux. The diversity in the silicate bands can come from the coupling of grain settling to the distribution of the magneto-rotational turbulence, through three effects: (1) Recombination on grains yields a magnetically inactive dead zone extending above two scale heights, while turbulence in the magnetically active disk atmosphere overshoots the dead zone boundary by only about one scale height. (2) Grains deep in the dead zone oscillate vertically in waves driven by the turbulent layer above, but on average settle at the laminar rates, so the interior of the dead zone is a particle sink and the disk atmosphere becomes dust-depleted. (3) With sufficient depletion, the dead zone is thinner and mixing dredges grains off the midplane. The MHD results also show that the magnetic activity intermittently lifts clouds of dust into the atmosphere. The photosphere height changes by up to one-third over a few orbits, while the extinction along lines of sight grazing the disk surface varies by factors of two over times down to 0.1 orbit. We suggest that the changing shadows cast by the dust clouds on the outer disk are a cause of the daily to monthly mid-infrared variability in some young stars. (Abridged.) Comment: ApJ in press
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We present multi-epoch Spitzer Space Telescope observations of the transitional disk LRLL 31 in the 2-3 Myr-old star forming region IC 348. Our measurements show remarkable mid-infrared variability on timescales as short as one week. The infrared continuum emission exhibits systematic wavelength-dependent changes that suggest corresponding dynamical changes in the inner disk structure and variable shadowing of outer disk material. We propose several possible sources for the structural changes, including a variable accretion rate or a stellar or planetary companion embedded in the disk. Our results indicate that variability studies in the infrared can provide important new constraints on protoplanetary disk behavior.
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A model is developed for the time-dependent twisting of an initial poloidal magnetic field threading a conducting protostellar disk. The region outside the disk is assumed to be filled, at least initially, with a low-density, force-free 'coronal' plasma. The differential rotation of the disk acts to twist the B field in the space outside the disk thus generating a toroidal magnetic field. In turn, the toroidal field acts to pinch the plasma and the poloidal field toward the system axis producing a collimated channel. This channel could facilitate the formation of bipolar flows. The magnitude and duration of the field twisting is expected to be limited by magnetohydrodynamic instability.
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A model for disk accretion by a rotating magnetic neutron star is proposed which includes a detailed description of matter flow in the transition region between the disk and the magnetosphere. It is shown that the disk plasma cannot be completely screened from the stellar magnetic field and that the resulting magnetic coupling between the star and the disk exerts a significant torque on the star. On the assumption that the distortion of the residual stellar field lines threading the disk is limited by reconnection, the total accretion torque on the star is calculated. The calculated torque gives period changes in agreement with those observed in the pulsating X-ray sources and provides a natural explanation of why a fast rotator like Her X-1 has a spin-up rate much below the conventional estimate for slow rotators. It is shown that for such fast rotators, fluctuations in the mass-accretion rate can produce fluctuations in the accretion torque about 100 times larger. For sufficiently fast rotators or, equivalently, for sufficiently low accretion rates, the star experiences a braking torque even while accretion continues and without any mass ejection from its vicinity.
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An investigation is made of disk accretion of matter onto a rotating star with an aligned dipole magnetic field. A new aspect of this work is that when the angular velocity of the star and disk differ substantially we argue that the $\bf B$ field linking the star and disk rapidly inflates to give regions of open field lines extending from the polar caps of the star and from the disk. The open field line region of the disk leads to the possibility of magnetically driven outflows. An analysis is made of the outflows and their back affect on the disk structure assuming an $\ap$" turbulent viscosity model for the disk and a magnetic diffusivity comparable to this viscosity. The outflows are found to extend over a range of radial distances inward to a distance close to $r_{to}$, which is the distance of the maximum of the angular rotation rate of the disk. We find that $r_{to}$ depends on the star's magnetic moment, the accretion rate, and the disk's magnetic diffusivity. The outflow regime is accompanied in general by a spin-up of the rotation rate of the star. When $r_{to}$ exceeds the star's corotation radius $r_{cr} = (GM/\om_*^2)^{1\ov 3}$, we argue that outflow solutions do not occur, but instead that magnetic braking" of the star by the disk due to field-line twisting occurs in the vicinity of $r_{cr}$. The magnetic braking solutions can give spin-up or spin-down (or no spin change) of the star depending mainly on the star's magnetic moment and the mass accretion rate. For a system with $r_{to}$ comparable to $r_{cr}$, bimodal behavior is possible where extraneous perturbations cause the system to flip between spin-up and spin-down. Comment: 24 pages, Plain TeX, 9 figures available upon request from R. Lovelace (rvl1@cornell.edu), Submitted to MRNAS
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The accretion disk around a rotating magnetic star (neutron star, white dwarf or T Tauri star) is subjected to periodic vertical magnetic forces from the star, with the forcing frequency equal to the stellar spin frequency or twice the spin frequency. This gives rise bending waves in the disk that may influence the variabilities of the system. We study the excitation, propagation and dissipation of these waves using a hydrodynamical model coupled with a generic model description of the magnetic forces. The $m=1$ bending waves are excited at the Lindblad/vertical resonance, and propagate either to larger radii or inward toward the corotation resonance where dissipation takes place. While the resonant torque is negligible compared to the accretion torque, the wave nevertheless may reach appreciable amplitude and can cause or modulate flux variabilities from the system. We discuss applications of our result to the observed quasi-periodic oscillations from various systems, in particular neutron star low-mass X-ray binaries. Comment: Small changes/clarifications. To be published in ApJ, Aug.20,2008 issue
Article
Comet samples returned to Earth by the NASA Stardust mission have provided a surprising glimpse into the nature of early Solar System materials and an epiphany on the origin of the initial rocky materials that once filled the cold regions of the solar nebula. The findings show that the cold regions of the early Solar System were not isolated and were not a refuge where interstellar materials could commonly survive. Wild 2, the sampled comet, appears to be a typical active Jupiter family comet, and yet most of its sampled micron and larger grains are familiar high-temperature meteoritic materials, such as chondrule fragments, that were transported to cold nebular regions. The rocky components in primitive asteroids and comets may differ because asteroid formation was dominated by local materials, whereas comets formed from mixed materials, many of which were transported from very distant locations.
Article
The structure of inner region of protoplanetary disks around young pre-main-sequence stars is still poorly understood. This part of the disk is shaped by various forces influencing dust and gas dynamics and by dust sublimation, which creates abrupt drops in the dust density. This region also emits a strong near-infrared excess that cannot be explained by classical accretion disk models, which suggests the existence of some unusual dust distribution or disk shape. The most prevalent explanation to date is the puffed-up inner disk rim model, where the disk exhibits an optically thin cavity around the star up to the distance of dust sublimation. The critical parameter in this model is the inner disk rim height $z_{\rm max}$ relative to the rim's distance from the star $R_{\rm in}$. Observations often require $z_{\rm max}/R_{\rm in}\gtrsim0.2$ to reproduce the near-infrared excess in the spectra. In this paper we put together a comprehensive list of processes that can shape the inner disk rim and combined them together into a self-consistent model. Two of them, radiation pressure force and the gas velocity profile, have never been applied in this context before. The aim was to find the most plausible theoretical values of $z_{\rm max}/R_{\rm in}$. The results show that this value is $\lesssim$0.13 for Herbig Ae stars, $\lesssim$0.11 for T Tau stars and $\lesssim$0.10 for young brown dwarfs. This is below the observational requirements for Herbig Ae stars. We argue that the same problem exists also in T Tau stars. We conclude that the puffed-up inner rim model cannot be the sole explanation for the near-infrared excess in young pre-main-sequence stars.
Article
Transitional disks are objects whose inner disk regions have undergone substantial clearing. The Spitzer Space Telescope produced detailed spectral energy distributions (SEDs) of transitional disks that allowed us to infer their radial dust disk structure in some detail, revealing the diversity of this class of disks. The growing sample of transitional disks also opened up the possibility of demographic studies, which provided unique insights. There now exist (sub)millimeter and infrared images that confirm the presence of large clearings of dust in transitional disks. In addition, protoplanet candidates have been detected within some of these clearings. Transitional disks are thought to be a strong link to planet formation around young stars and are a key area to study if further progress is to be made on understanding the initial stages of planet formation. Here we provide a review and synthesis of transitional disk observations to date with the aim of providing timely direction to the field, which is about to undergo its next burst of growth as ALMA reaches its full potential. We discuss what we have learned about transitional disks from SEDs, color-color diagrams, and imaging in the (sub)mm and infrared. We then distill the observations into constraints for the main disk clearing mechanisms proposed to date (i.e., photoevaporation, grain growth, and companions) and explore how the expected observational signatures from these mechanisms, particularly planet-induced disk clearing, compare to actual observations. Lastly, we discuss future avenues of inquiry to be pursued with ALMA, JWST, and next generation of ground-based telescopes.
Article
Article
We have updated our publicly available dust radiative transfer code (HOCHUNK3D) to include new emission processes and various 3-D geometries appropriate for forming stars. The 3-D geometries include warps and spirals in disks, accretion hotspots on the central star, fractal clumping density enhancements, and misaligned inner disks. Additional axisymmetric (2-D) features include gaps in disks and envelopes, "puffed-up inner rims" in disks, multiple bipolar cavity walls, and iteration of disk vertical structure assuming hydrostatic equilibrium. We include the option for simple power-law envelope geometry, which combined with fractal clumping, and bipolar cavities, can be used to model evolved stars as well as protostars. We include non-thermal emission from PAHs and very small grains, and external illumination from the interstellar radiation field. The grid structure was modified to allow multiple dust species in each cell; based on this, a simple prescription is implemented to model dust stratification. We describe these features in detail, and show example calculations of each. Some of the more interesting results include the following: 1) Outflow cavities may be more clumpy than infalling envelopes. 2) PAH emission in high-mass stars may be a better indicator of evolutionary stage than the broadband SED slope; and related to this, 3) externally illuminated clumps and high-mass stars in optically thin clouds can masquerade as YSOs. 4) Our hydrostatic equilibrium models suggest that dust settling is likely ubiquitous in T Tauri disks, in agreement with previous observations.
Article
We consider accreting systems in which the central object interacts, via the agency of its magnetic field, with the disc that surrounds it. The disc is turbulent and, so, has a finite effective conductivity. The field sweeps across the face of the disc, thereby forming a current that is directed radially within the disc. In turn, this disc current creates a toroidal field, where the interaction between the disc current and the toroidal field produces a Lorentz force that compresses the disc. We investigate this compression, which creates a magnetic scaleheight of the disc that can be much smaller than the conventional scaleheight. We derive an analytic expression for the magnetic scaleheight and apply it to fully ionized discs.
Article
This paper examines the outflows associated with the interaction of a stellar magnetosphere with an accretion disk. In particular, we investigate the magnetospheric ejections (MEs) due to the expansion and reconnection of the field lines connecting the star with the disk. Our aim is to study the dynamical properties of the outflows and evaluate their impact on the angular momentum evolution of young protostars. Our models are based on axisymmetric time-dependent magneto-hydrodynamic simulations of the interaction of the dipolar magnetosphere of a rotating protostar with a viscous and resistive disk, using alpha prescriptions for the transport coefficients. Our simulations are designed in order to model: the accretion process and the formation of accretion funnels; the periodic inflation/reconnection of the magnetosphere and the associated MEs; the stellar wind. Similarly to a magnetic slingshot, MEs can be powered by the rotation of both the disk and the star so that they can efficiently remove angular momentum from both. Depending on the accretion rate, MEs can extract a relevant fraction of the accretion torque and, together with a weak but non-negligible stellar wind torque, can balance the spin-up due to accretion. When the disk truncation approaches the corotation radius, the system enters a "propeller" regime, where the torques exerted by the disk and the MEs can even balance the spin-up due to the stellar contraction. The MEs spin-down efficiency can be compared to other scenarios, such as the Ghosh & Lamb, X-wind or stellar wind models. Nevertheless, for all scenarios, an efficient spin-down torque requires a rather strong dipolar component, which has been seldom observed in classical T Tauri stars. A better analysis of the torques acting on the protostar must take into account non-axisymmetric and multipolar magnetic components consistent with observations.
Article
We present the Spitzer Space Telescope Infrared Spectrograph spectrum of the Orion A protostar HOPS-68. The mid-infrared spectrum reveals crystalline substructure at 11.1, 16.1, 18.8, 23.6, 27.9, and 33.6 μm superimposed on the broad 9.7 and 18 μm amorphous silicate features; the substructure is well matched by the presence of the olivine end-member forsterite (Mg2SiO4). Crystalline silicates are often observed as infrared emission features around the circumstellar disks of Herbig Ae/Be stars and T Tauri stars. However, this is the first unambiguous detection of crystalline silicate absorption in a cold, infalling, protostellar envelope. We estimate the crystalline mass fraction along the line of sight by first assuming that the crystalline silicates are located in a cold absorbing screen and secondly by utilizing radiative transfer models. The resulting crystalline mass fractions of 0.14 and 0.17, respectively, are significantly greater than the upper limit found in the interstellar medium (0.02-0.05). We propose that the amorphous silicates were annealed within the hot inner disk and/or envelope regions and subsequently transported outward into the envelope by entrainment in a protostellar outflow.
Article
We consider the interaction between a magnetic star and its circumstellar disk under the assumption that the stellar magnetic field permeates the disk and that the system's magnetosphere is force-free. Using simplified axisymmetric models (both semianalytic and numerical), we study the time evolution of the magnetic field configuration induced by the relative rotation between the disk and the star. We show that if both the star and the magnetosphere are perfectly conducting, then there is a maximum disk surface conductivity Σmax for which a steady state field configuration can be established. For larger values of conductivity, no steady state is possible, and the field lines inflate and effectively open up when a critical twist angle (which for an initially dipolar field is on the order of a few radian) is attained. We argue that for thin astrophysical disks, surface conductivities are likely to exceed the local Σmax except in the immediate vicinity of the corotation radius in a Keplerian disk. If the disk conductivity is high enough, then the radial magnetic field at the disk surface will become large and induce radial migration of the field lines across the disk. We find, however, that the radial diffusion in the disk is generally much slower than the field-line expansion in the magnetosphere, which suggests that the opening of the magnetosphere is achieved before the diffusive outward expulsion of the field lines from the disk can occur. The effects of magnetospheric inertial effects and of field-line reconnection are considered in the companion paper.
Article
We present results of 3D simulations of magnetohydrodynamics (MHD) instabilities at the accretion disc–magnetosphere boundary. The instability is Rayleigh–Taylor, and develops for a fairly broad range of accretion rates and stellar rotation rates and magnetic fields. It manifests itself in the form of tall, thin tongues of plasma that penetrate the magnetosphere in the equatorial plane. The shape and number of the tongues changes with time on the inner disc dynamical time-scale. In contrast with funnel flows, which deposit matter mainly in the polar region, the tongues deposit matter much closer to the stellar equator. The instability appears for relatively small misalignment angles, Θ≲ 30°, between the star's rotation and magnetic axes, and is associated with higher accretion rates. The hotspots and light curves during accretion through instability are generally much more chaotic than during stable accretion. The unstable state of accretion has possible implications for quasi-periodic oscillations and intermittent pulsations from accreting systems, as well as planet migration.
Article
The problem of the effect of a strongly magnetic star on a surrounding accretion disc is considered. For stellar rotation periods greater than a critical value, a numerical solution is found for a steady disc with turbulent magnetic diffusion, including electron scattering opacity and radiation pressure. Inside the corotation radius, the extraction of disc angular momentum by magnetic coupling to the star becomes strong and this leads to enhanced viscous stress and dissipation. The resulting elevated temperature causes electron scattering opacity and radiation pressure to become significant further from the star than in the absence of its magnetic field. The disc ends as its height increases rapidly due to the large central pressure, its density decreases and magnetically induced viscous instability occurs.
Article
We present model spectral energy distributions (SEDs), colors, polarization, and images for an evolutionary sequence of a low-mass protostar from the early collapse stage (Class 0) to the remnant disk stage (Class III). We find a substantial overlap in colors and SEDs between protostars embedded in envelopes (Class 0–I) and T Tauri disks (Class II), especially at mid-IR wavelengths. Edge-on Class I–II sources show double-peaked SEDs, with a short-wavelength hump due to scattered light and a long-wavelength hump due to thermal emission. These are the bluest sources in mid-IR color-color diagrams. Since Class 0 and I sources are diffuse, the size of the aperture over which fluxes are integrated has a substantial effect on the computed colors, with larger aperture results showing significantly bluer colors. Viewed through large apertures, the Class 0 colors fall in the same regions of mid-IR color-color diagrams as Class I sources and are even bluer than Class II–III sources in some colors. It is important to take this into account when comparing color-color diagrams of star formation regions at different distances or different sets of observations of the same region. However, the near-IR polarization of the Class 0 sources is much higher than the Class I–II sources, providing a means to separate these evolutionary states. We varied the grain properties in the circumstellar envelope, allowing for larger grains in the disk midplane and smaller grains in the envelope. In comparing with models with the same grain properties throughout, we find that the SED of the Class 0 source is sensitive to the grain properties of the envelope only—that is, grain growth in the disk in Class 0 sources cannot be detected from the SED. Grain growth in disks of Class I sources can be detected at wavelengths greater than 100 lm. Our image calculations predict that the diffuse emission from edge-on Class I and II sources should be detectable in the mid-IR with the Space Infrared Telescope Facility (SIRTF) in nearby star-forming regions (out to several hundred parsecs).
Article
Abstract Infrared molecular spectroscopy is a key tool for the observation of gas in the innermost region of disks around T Tauri stars. In this contribution, we examine how infrared spectroscopy of CO can be used to study the inner truncation region of disks around T Tauri stars. The inferred inner gas radii for T Tauri star disks are compared to the inner dust radii of disks, to the expectations of models for disk truncation, and to the orbital distribution of short-period extra-solar planets.
Article
Protostellar systems, ranging from low-luminosity T Tauri and Herbig Ae stars to high-luminosity Herbig Be stars, exhibit a near-infrared (NIR) excess in their spectra that is dominated by a bump in the monochromatic luminosity with a peak near 3 microns. The bump can be approximated by a thermal emission component of temperature 1500 K that is of the order of the sublimation temperature of interstellar dust grains. In the currently popular "puffed up rim" scenario, the bump represents stellar radiation that propagates through the optically thin inner region of the surrounding accretion disk and is absorbed and reemitted by the dust that resides just beyond the dust sublimation radius, Rsub. However, this model cannot account for the strongest bumps measured in these sources, and it predicts a large secondary bounce in the interferometric visibility curve that is not observed. In this paper we present an alternative interpretation, which attributes the bump to reemission of stellar radiation by dust that is uplifted from the disk by a centrifugally driven wind. Winds of this type are a leading candidate for the origin of the strong outflows associated with protostars, and there is observational evidence for disk winds originating on scales ~Rsub. Using a newly constructed Monte Carlo radiative transfer code, we show that this model can account for the NIR excess emission even in bright Herbig Ae stars such as AB Auriga and MWC 275, and that it successfully reproduces the basic features of the visibilities measured in these protostars. We argue that a robust dusty outflow in these sources could be self-limiting to a relatively narrow launching region between Rsub and 2Rsub. Finally, we suggest that our model could also naturally account for the NIR and scattered-light variability exhibited by a source like MWC 275, which may be triggered by the uplifting of dust clouds from the disk.
Article
We investigate the long-term optical–infrared variability of SV Cep and explain it in the context of an existing UX Ori (UXOR) model. A 25-month monitoring programme was completed with the Infrared Space Observatory in the 3.3–100 μm wavelength range. Following a careful data reduction, the infrared light curves were correlated with the variations of SV Cep in the V band. A remarkable correlation was found between the optical and the far-infrared light curves. In the mid-infrared regime, the amplitude of variations is lower, with a hint for a weak anti-correlation with the optical changes. In order to interpret the observations, we modelled the spectral energy distribution of SV Cep assuming a self-shadowed disc with a puffed-up inner rim, using a two-dimensional radiative transfer code. We found that modifying the height of the inner rim, the wavelength dependence of the long-term optical–infrared variations is well reproduced, except the mid-infrared domain. The origin of variation of the rim height might be fluctuation in the accretion rate in the outer disc. In order to model the mid-infrared behaviour, we tested adding an optically thin envelope to the system, but this model failed to explain the far-infrared variability. Infrared variability is a powerful tool to discriminate between models of the circumstellar environment. The proposed mechanism of variable rim height may not be restricted to UXOR stars, but might be a general characteristic of intermediate-mass young stars.
Article
The status of dust driven winds, constituting an important subclass of essentially radiation generated winds, is surveyed. Dust driven winds are conceived as a long lasting phenomenon of heavy mass loss concerning those luminous cool giants and supergiants, where dust condensation in the expanding flow determines both thestellar mass loss rate and thesubsonic-supersonic transition of the velocity field. Our contribution aims at a self-consistent description of the dynamical shell structure with particular emphasis to the theoretical aspects of this important phenomenon. Thus, not only the complex coupling of the various ingredients (hydrodynamics, chemistry, radiative transfer, dust nucleation, and growth) is outlined in detail, but also general arguments regarding the overall structure of such winds and the expected position of their central objects in the Hertzsprung-Russel diagram are conducted. A selected typical self-consistent model for a stationary C-star shell demonstrates the characteristic wind structure and gives insight into the close nonlinear interplay between dust formation and wind generation. During the late evolutionary stages of a star along the AGB dust driven mass loss provides a natural self-accelerating mechanism which easily can produce very high mass loss rates, an effect which possibly might play an important role for the Tip-AGB objects and the AGB-PN-transition.
Article
We present the first results of ionospheric conductivities at Titan based on measurements during 17 Titan flybys from the Cassini spacecraft. We identify an ionospheric region ranging from (approximately the location of the exobase) to approximately 1000 km where electrical currents perpendicular to the magnetic field may become important. In this region the ionosphere is highly conductive with peak Pedersen conductivities of 0.002–0.05 S/m and peak Hall conductivities of 0.01–0.3 S/m depending on Solar illumination and magnetospheric conditions. Ionospheric conductivities are found to be typically higher on the sunlit side of Titan. However, Hall and Pedersen conductivities depend strongly on the magnetic field magnitude which is highly variable, both in altitude and with respect to the draping configuration of Saturn's magnetic field around Titan. Furthermore, a consistent double peak nature is found in the altitude profile of the Pedersen conductivity. A high altitude peak is found to be located between 1300 and 1400 km. A second and typically more conductive region is observed below 1000 km, where the magnetic field strength drops sharply while the electron density still remains high. This nature of the Pedersen conductivity profile may give rise to complicated ionospheric–atmospheric dynamics and may be expected also at other unmagnetized objects with a substantial atmosphere, such as e.g. Mars and Venus. Estimates of the total Pedersen conductance are found to range between 1300 and 22,000 S. The Pedersen conductance is always higher than the local Alfvén conductance but the difference varies by two orders of magnitude (from a factor 4 to 100). Thus, the conditions for reflection or absorption of Alfvén waves in Titans ionosphere are highly variable and depends strongly on the magnetic field strength.
Article
We have performed smoothed particle magnetohydrodynamics (SPMHD) simulations demonstrating the production of collimated jets during collapse of 1 M⊙ molecular cloud cores to form the ‘first hydrostatic core’ in low‐mass star formation. Recently, a number of candidate first‐core objects have been observed, including L1448 IRS2E, L1451‐mm and Per‐Bolo 58, although it is not yet clear that these are first hydrostatic cores. Recent observations of Per‐Bolo 58 in particular appear to show collimated, bipolar outflows which are inconsistent with previous theoretical expectations. We show that low‐mass first cores can indeed produce tightly collimated jets (opening angles ≲10°) with speeds of ∼2–7 km s−1, consistent with some of the observed candidates. We have also demonstrated, for the first time, that such phenomena can be successfully captured in SPMHD simulations.
Article
(abridged) Accretion from disks onto young stars is thought to follow magnetic field lines from the inner disk edge to the stellar surface. The accretion flow thus depends on the geometry of the magnetic field. This paper extends previous work by constructing a collection of orthogonal coordinate systems, including the corresponding differential operators, where one coordinate traces the magnetic field lines. This formalism allows for an (essentially) analytic description of the geometry and the conditions required for the flow to pass through sonic points. Using this approach, we revisit the problem of magnetically controlled accretion flow in a dipole geometry, and then generalize the treatment to consider magnetic fields with multiple components, including dipole, octupole, and split monopole contributions. This approach can be generalized further to consider more complex magnetic field configurations. Observations indicate that accreting young stars have substantial dipole and octupole components, and that accretion flow is transonic. If the effective equation of state for the fluid is too stiff, the flow cannot pass smoothly through the sonic points in steady state. For a multipole field of order \ell, we derive a constraint on the polytropic index, n>\ell+3/2, required for steady transonic flow to reach free-fall velocities. For octupole fields, inferred on surfaces of T Tauri stars, n>9/2, so that the flow must be close to isothermal. The inclusion of octupole field components produces higher densities at the stellar surface and smaller hot spots, which occur at higher latitudes; the magnetic truncation radius is also modified. This contribution thus increases our understanding of magnetically controlled accretion for young stellar objects and can be applied to a variety of additional astrophysical problems.
Article
We formulate a general, steady-state model for the torque on a magnetized star from a surrounding accretion disc. For the first time, we include the opening of dipolar magnetic field lines due to the differential rotation between the star and disc, so the magnetic topology then depends on the strength of the magnetic coupling to the disc. This coupling is determined by the effective slip rate of magnetic field lines that penetrate the diffusive disc. Stronger coupling (i.e., lower slip rate) leads to a more open topology and thus to a weaker magnetic torque on the star from the disc. In the expected strong coupling regime, we find that the spin-down torque on the star is more than an order of magnitude smaller than calculated by previous models. We also use our general approach to examine the equilibrium (`disc-locked') state, in which the net torque on the star is zero. In this state, we show that the stellar spin rate is roughly an order of magnitude faster than predicted by previous models. This challenges the idea that slowly-rotating, accreting protostars are disc locked. Furthermore, when the field is sufficiently open (e.g., for mass accretion rates > 5 x 10^{-9} M_sun / yr, for typical accreting protostars), the star will receive no magnetic spin-down torque from the disc at all. We therefore conclude that protostars must experience a spin-down torque from a source that has not yet been considered in the star-disc torque models--possibly from a stellar wind along the open field lines. Comment: Accepted by MNRAS


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# Question: eqn output split long lines

January 04 2013 by
false
Maple

1

I use the eqn feature from Maple to generate troff output.Each equation is longer than one line.  How does one get the eqn/troffcombination to automatically break the equation into multiple lines.(I tried finding the division point myself and could not do it.  I am generating the output and I have a perl script that puts together the indivdualequations into the document to be printed.  Thus, I need something automatic)I tried Google and could not find anything relevant and searched mapleprimes.comI am using a much older version of Maple currently (Maple V Release Three)Dr. Laurence Leff  Western Illinois University, Macomb IL 61455 Stipes 447 Assoc. Prof. of Computer Sci. Pager: 309-367-0787

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# Hyperplane Sections of Determinantal Varieties over Finite Fields and Linear Codes

We determine the number of F_q-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number of F_q-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance and generalized Hamming weights. The previously known results about these are generalized and expanded significantly

## Authors

• 8 publications
• 5 publications
05/07/2019

### Two classes of linear codes and their generalized Hamming weights

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### Relative Generalized Hamming weights of affine Cartesian codes

We explicitly determine all the relative generalized Hamming weights of ...
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11/01/2017

### On the complete weight enumerators of some linear codes with a few weights

Linear codes with a few weights have important applications in authentic...
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### Generalized Hamming weights of projective Reed--Muller-type codes over graphs

Let G be a connected graph and let X be the set of projective points def...
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### Free Resolutions and Generalized Hamming Weights of binary linear codes

In this work, we explore the relationship between free resolution of som...
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### The Identification Problem for Linear Rational Expectations Models

We consider the problem of the identification of stationary solutions to...
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## 1 Introduction

The classical determinantal variety defined by the vanishing all minors of a fixed size in a generic matrix is an object of considerable importance and ubiquity in algebra, combinatorics, algebraic geometry, invariant theory and representation theory. The defining equations clearly have integer coefficients and as such the variety can be defined over any finite field. The number of -rational points of this variety is classically known. We are mainly interested in a more challenging question of determining the number of -rational points of such a variety when intersected with a hyperplane in the ambient projective space, or more generally, with a linear subvariety of a fixed codimension in the ambient projective space. In particular, we wish to know which of these sections have the maximum number of -rational points. These questions are directly related to determining the complete weight distribution and the generalized Hamming weights of the associated linear codes, which are caledl determinantal codes. In this setting, the problem was considered in [2] and a beginning was made by showing that the determination of the weight distribution is related to the problem of computing the number of generic matrices of a given rank with a nonzero “partial trace”. More definitive results were obtained in the special case of varieties defined by the vanishing of all minors of a generic matrix. Here we settle the question of determination of the weight distribution and the minimum distance of determinantal codes in complete generality. Further, we determine some initial and terminal generalized Hamming weights of determinantal codes. We also show that the determinantal codes have a very low dual minimum distance (viz., 3), which makes them rather interesting from the point of view of coding theory. Analogous problems have been considered for other classical projective varieties such as Grassmannians, Schubert varieties, etc., leading to interesting classes of linear codes which have been of some current interest; see, for example, [17], [13], [14], [20], [12], and the survey [16].

As was mentioned in [2] and further explained in the next section and Remark 3.7, the results on the weight distribution of determinantal codes are also related to the work of Delsarte [9]

on eigenvalues of association schemes of bilinear forms using the rank metric as distance. We remark also that a special case of these results has been looked at by Buckhiester

[6].

A more detailed description of the contents of this paper is given in the next section, while the main results are proved in the two sections that follow the next section. An appendix contains self-contained and alternative proofs of some results that were deduced from the work of Delsarte and this might be of an independent interest.

## 2 Preliminaries

Fix throughout this paper a prime power , positive integers , and an matrix whose entries are independent indeterminates over . We will denote by the polynomial ring in the variables (, ) with coefficients in . As usual, by a minor of size or a minor of we mean the determinant of a submatrix of , where is a nonnegative integer . As per standard conventions, the only minor of is . We will be mostly interested in the class of minors of a fixed size, and this class is unchanged if is replaced by its transpose. With this in view, we shall always assume, without loss of generality, that . Given a field , we denote by the set of all matrices with entries in . Often and in this case we may simply write for . Note that can be viewed as an affine space over of dimension . For , the corresponding classical determinantal variety (over ) is denoted by and defined as the affine algebraic variety in given by the vanishing of all minors of ; in other words

 Dt(ℓ,m)={M∈Mℓ×m(Fq):rank(M)≤t}.

Note that

only consists of the zero-matrix. For

, no minors of exist. This means that , which is in agreement with the above description of as the set of all matrices of rank at most .

It will also be convenient to define the sets

 Dt(ℓ,m):={M∈Mℓ×m(Fq):rank(M)=t},

for as well as their cardinalities . The map that sends to its row-space is a surjection of onto the space of -dimensional subspaces of . Moreover for a given , the number of with row-space is equal to the number of matrices over of rank or equivalently, the number of

-tuples of linearly independent vectors in

. Since is the Gaussian binomial coefficient , we find

 μt(ℓ,m)=|Dt(ℓ,m)|=[mt]qt−1∏i=0(qℓ−qi)=q(t2)j−1∏i=0(qℓ−i−1)(qm−i−1)qi+1−1. (1)

Using the Gaussian factorial , one can also give the following alternative expressions:

 μt(ℓ,m)=q(t2)[mt]q[ℓ]q![ℓ−t]q!=q(t2)[m]q![ℓ]q![m−t]q![t]q![ℓ−t]q!=q(t2)[ℓt]q[m]q![m−t]q!.

Note that . Next we define

 νt(ℓ,m):=t∑s=0μs(ℓ,m).

Since is the disjoint union of , we have

 νt(ℓ,m)=t∑s=0μs(ℓ,m)=|Dt(ℓ,m)|. (2)

The affine variety is, in fact, a cone; in other words, the vanishing ideal (which is precisely the ideal of generated by all minors of ) is a homogeneous ideal. Also it is a classical (and nontrivial) fact that is a prime ideal (see, e.g., [10]). Thus can also be viewed as a projective algebraic variety in , and viewed this way, we will denote it by . We remark that the dimension of is (cf. [10]). Moreover

 |ˆDt(ℓ,m)| = |Dt(ℓ,m)|−1q−1=1q−1t∑s=1μs(ℓ,m).

We are now ready to define the codes we wish to study. Briefly put, the determinantal code is the linear code corresponding to the projective system . An essentially equivalent way to obtain this code is as follows: Denote and choose an ordering of the elements of . Further choose representatives for . Then consider the evaluation map

 Ev:Fq[X]1→F^nqdefined byEv(f)=^cf:=(f(M1),…,f(M^n)),

where denotes the space of homogeneous polynomials in of degree together with the zero polynomial. The image of this evaluation map can directly be identified with . A different choice of representatives or a different ordering of these representatives gives in general a different code, but basic quantities like minimum distance, weight distribution, and generalized Hamming weights are independent on these choices.

In [2], also another code was introduced. It can be obtained by evaluating functions in in all elements of . The parameters of determine those of and vice-versa, see [2, Prop. 1]. It is therefore sufficient to study either one of these two codes. In the remainder of this article, we will focus on and determine some of its basic parameters. It is in a sense also a more natural code to study, since the code is degenerate, whereas is nondegenerate [2]. We quote this and some other useful facts from [2, Prop.1, Lem. 1, Cor. 1]:

###### Fact 2.1

The code is a nondegenerate code of dimension and length For , denote by the coefficient matrix of . Then the Hamming weight of the corresponding codeword of depends only on . Consequently, if , then

 wH(^cf)=wH(^cτr),% whereτr:=X11+⋯+Xrr.

As a result, the code has at most distinct weights, , given by for .

We call the function the partial trace. Note that , since . To determine the other weights , one would need to count the number of of rank at most with nonzero -th partial trace. Delsarte [9] used the theory of association schemes to solve an essentially equivalent problem of determining the number of with , and showed:

 wr(t;ℓ,m)=q−1q(μt(ℓ,m)−ℓ∑i=0(−1)t−iqim+(t−i2)[ℓ−iℓ−t]q[ℓ−ri]q).

The case was already dealt with by Buckhiester in [6]. In the appendix of this paper, we obtain using different methods an alternative formula for , which may be of independent interest. For future use, we define for and . From Delsarte’s result it follows that

 ^wr(t;ℓ,m)=1q(μt(ℓ,m)−ℓ∑i=0(−1)t−iqim+(t−i2)[ℓ−iℓ−t]q[ℓ−ri]q), (3)

In fact, Delsarte considered codes obtained by evaluating elements from in all matrices of rank . Therefore his code can be seen as the projection of on the coordinates corresponding to matrices of rank .

Using Equation (3), we see that the nonzero weights of are given by

 ^wr(t;ℓ,m) = t∑s=1^wr(s;ℓ,m) = t∑s=11q(μt(ℓ,m)−ℓ∑i=0(−1)t−iqim+(t−i2)[ℓ−iℓ−t]q[ℓ−ri]q),

for . However, for a fixed , it is not obvious how are ordered or even which among them is the least. We will formulate a conjecture based (among others) on the following examples.

###### Example 2.2

If the code is trivial (containing only the zero word), while the code is not defined. Therefore the easiest nontrivial case occurs for . This case was considered in [2], where is was shown that

 ^wr(1;ℓ,m)=qℓ+m−2+qℓ+m−3+⋯+qℓ+m−r−1=qℓ+m−r−1(qr−1)q−1.

These formulae also follow fairly directly from Equations (3) and (2). It follows directly that and that is the minimum distance of .

###### Example 2.3

In this example we consider the determinantal code in case and . Using the formulae in Equations (3) and (2), we find the following table:

 r1234^wr(1;ℓ,m)128192224240^wr(2;ℓ,m)13568162561657616480^wr(3;ℓ,m)201728212480211712211840^wr(4;ℓ,m)524288524288524288524288

One sees that it is not true in general that whenever . However, in this example it is true that for a given , the weight is the smallest among all nonzero weights .

###### Example 2.4

In case in the previous example, all weights were the same. This holds in general: If , then and is a first order projective Reed-Muller code (cf. [15]). All nonzero codewords in this code therefore have weight . Note that combining this with Equations (3) and (2) we obtain for the following identity

 qℓm−1=ℓ∑s=11q(μs(ℓ,m)−ℓ∑i=0(−1)s−iqim+(s−i2)[ℓ−iℓ−s]q[ℓ−ri]q).

Using Equation (2) with , we see that for apparently the following identity holds

 ℓ∑s=1ℓ∑i=0(−1)s−iqim+(s−i2)[ℓ−iℓ−s]q[ℓ−ri]q=−1.

This identity may readily be shown using for example [1, Thm 3.3] after interchanging the summation order. In any case, it is clear that Equations (3) and (2) may not always give the easiest possible expression for the weights.

While for and all weights are easy to compare with one another, the same cannot be said in case . We formulate the following conjecture.

###### Conjecture 2.5

Let be positive integers and an integer satisfying . The the following hold:

1. All weights are mutually distinct.

2. We have .

3. For all , the weight lies between and .

## 3 Minimum distance of determinantal codes

Recall that in general for a linear code of length , i.e., for a linear subspace of , the Hamming weight of a codeword , denoted is defined by

 wH(c):=|{i:ci≠0}|.

The minimum distance of , denoted , is defined by

 d(C) := min{wH(c):c∈C,c≠0}.

A consequence of Conjecture 2.5 would also be that is the minimum distance of . We will now show that this is indeed the case. We start by giving a rather compact expression for .

###### Proposition 3.1

Let , and be integers satisfying . Then

 ^w1(t;ℓ,m)=qℓ+m−2νt−1(ℓ−1,m−1).
###### Proof.

First suppose that . In this case Example 2.2 implies that . On the other hand, using Equations (1) and (2), we see that , so the proposition holds for .

From now on we assume that (implying that also ). We will show that

 ^w1(t;ℓ,m)=qℓ+m−2μt−1(ℓ−1,m−1). (5)

Once we have shown this, the proposition follows using Equations (2) and (2). Let and suppose that . In that case, we may find uniquely determined square matrices

 A=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝10a11⋮⋱aℓ−101⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ \makeboxand B=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝1b1⋯bm−110⋱01⎞⎟ ⎟ ⎟ ⎟ ⎟⎠,

such that

 AMB=⎛⎜ ⎜ ⎜ ⎜⎝m110⋯000˜M0⎞⎟ ⎟ ⎟ ⎟⎠. (6)

The matrices and are indeed uniquely determined, since for and we have

 0=(AMB)i1=ai−1(MB)11+(MB)i1=ai−1m11+mi1

and

 0=(AMB)1j=(AM)11bj−1+(AM)1j=m11bj−1+m1j.

These equations determine the values of given the matrix . The association of therefore is a well-defined map

 ϕ:{M∈Dt(ℓ,m)|m11≠0}→Dt−1(ℓ−1,m−1).

The map is clearly surjective (one can for example choose as in the right hand side of Equation (6)), while the preimage of any matrix consist of the matrices of the form , with and as above and again chosen as in the right-hand-side of Equation (6). Equation (5) (and hence the proposition) then follows, since

 w1(t;ℓ,m) = |{M∈Dt(ℓ,m)|m11≠0}| = ∑˜M∈Dt−1|ϕ−1(˜M)|=|Dt−1(ℓ−1,m−1)|(q−1)qℓ+m−2.

Equation (5), and hence the proposition, follows directly from this. ∎

Note that the expression for from Equation (3) is considerably more involved that the expression obtained in the proof of Proposition 3.1. We now turn our attention to proving that actually is the minimum distance of the code . The proof involves several identities concerning and . The key is the following theorem in which the following quantity occurs:

 A(r,t):=qt^wr−1(t;ℓ−1,m−1)+qt−1(μt(ℓ−1,m)−μt(ℓ−1,m−1)), \makeboxfor$0≤t<ℓ$.
###### Theorem 3.2

Let and , then

 ^wr(t;ℓ,m)=A(r,t)−A(r,t−1)+qm−1μt−1(ℓ−1,m).
###### Proof.

Given a matrix , we denote by the matrix obtained from by deleting its -th row. Since either or , this defines a map . It is not hard to see that is surjective. In fact:

 |ψ−1(N)|={qt\makeboxif$N∈Dt(ℓ−1,m)$qm−qt−1\makeboxif$N∈Dt−1(ℓ−1,m)$, (7)

since if we obtain all elements of by adding a row from the rowspace of , while if we obtain all elements of by adding any row not from the rowspace of .

We will now prove the theorem by carefully counting the number of matrices such that , thus computing . The theorem then follows easily, since . We distinguish four cases:

1. The -th column of is zero and has rank ,

2. The -th column of is zero and has rank ,

3. The -th column of is non zero and has rank ,

4. The -th column of is non zero and has rank .

Case 1: The -th column of is zero and has rank . In this case , since otherwise . Therefore if and only if . By Equation (7), we find the following contribution to :

 qtwr−1(t;ℓ−1,m−1). (8)

Case 2: The -th column of is zero and has rank . If , then by a similar reasoning as in case 1, we find a contribution to of magnitude

 (qm−1−qt−1)wr−1(t−1;ℓ−1,m−1). (9)

If , the situation is more complicated. If namely , then if and only if . Since and the -th column of is zero, all matrices with nonzero -th entry are in . This gives a contribution to of magnitude

 qm−1(q−1)(μt−1(ℓ−1,m−1)−wr−1(t−1;ℓ−1,m−1)). (10)

If on the other hand , then if and only if . Since we already assumed that , we find a contribution to of magnitude

 qm−1(q−2)wr−1(t−1;ℓ−1,m−1). (11)

Case 3: The -th column of is non zero and has rank . Since the -th column of is non zero, the -th coordinates of elements from the row space of are distributed evenly over the elements of . This implies that regardless of the value of , a -th fraction of the matrices in contribute to . In total we find the contribution:

 qt−1(q−1)(μt(ℓ−1,m)−μt(ℓ−1,m−1)). (12)

Case 4: The -th column of is non zero and has rank . Just as in case 3, since the -th column of is non zero, the -th coordinates of elements from the row space of are distributed evenly over the elements of . Therefore also the -th coordinates of elements not from the row space of are distributed evenly over the elements of . By a similar reasoning as in case 3, we find a contribution to of magnitude:

 (qm−1−qt−2)(q−1)(μt−1(ℓ−1,m)−μt−1(ℓ−1,m−1)). (13)

Adding all contributions to from Equations (8),(9),(10),(11),(12), and (13), the theorem follows. ∎

###### Corollary 3.3

Let and . Then

 ^wr(t;ℓ,m)=A(r,t)+qm−1νt−1(ℓ−1,m−1).
###### Proof.

By Equation (2) and Theorem 3.2 we have

 ^wr(t;ℓ,m) = t∑s=1^wr(s;ℓ,m) = t∑s=1(A(r,s)−A(r,s−1)+qm−1μs−1(ℓ−1,m)) = A(r,t)−A(r,0)+qm−1t∑s=1μs−1(ℓ−1,m).

The corollary now follows by the definition of and Equation (2). ∎

###### Corollary 3.4

Let and , then

 ^wr(t;ℓ,m)−^ws(t;ℓ,m)=qt(^wr−1(t;ℓ−1,m−1)−^ws−1(t;ℓ−1,m−1)).

In particular

 ^wr(t;ℓ,m)−^w1(t;ℓ,m)=qt^wr−1(t;ℓ−1,m−1).
###### Proof.

Using the previous corollary, we see that

 ^wr(t;ℓ,m)−^ws(t;ℓ,m) = A(r,t)−A(s,t) = qt(^wr−1(t;ℓ−1,m−1)−^ws−1(t;ℓ−1,m−1)).

This yields the first part of the corollary. The second part follows directly by choosing . ∎

We are now ready to prove our main theorem on the minimum distance.

###### Theorem 3.5

Let and . Then the minimum distance of the code is given by

 ^d=qℓ+m−2νt−1(ℓ−1,m−1).
###### Proof.

We already know that the only nonzero weights occurring in code are . Moreover, in case , we already know from Example 2.4 that the minimum distance is given by

 ^w1(ℓ;ℓ,m)=qℓm−1=qℓ+m−2νt−1(ℓ−1,m−1).

Therefore we may assume . However, in this case the second part of Corollary 3.4 implies that cannot be larger than any of the other weights, since

 ^wr(t;ℓ,m)−^w1(t;ℓ,m)=qt^wr−1(t;ℓ−1,m−1)≥0.

The theorem then follows from Proposition 3.1. ∎

The above theorem gives a start to proving Conjecture 2.5. Exploring the above methods, we can do a little more as well as gain some information about codewords of minimum weight in .

###### Proposition 3.6

Let , then The code has exactly codewords of minimum weight and these codewords generate the entire code. More precisely, any codeword in is the sum of at most minimum weight codewords.

###### Proof.

Choosing in Corollary 3.4 and , we obtain that

 ^wr(t;ℓ,m)−^w1(t;ℓ,m)=qt^wr−1(t;ℓ−1,m−1),

so the first part of the proposition follows once we have shown that . In order to this, it is sufficient to produce one matrix of rank such that . However, this is easy to do: Let be a permutation matrix corresponding to a permutation on elements that fixes , but does not have other fixed points. Then , while any other diagonal element is zero. Now take to be the matrix such that if and , while otherwise. Then for any , we have , which is exactly what we wanted to show.

Now that we know that is strictly smaller than all other nonzero weights, the minimum weight codewords are exactly those such that has a coefficient matrix of rank . This gives exactly possibilities for and hence for . Now let be given. Assume that has coefficient matrix of rank . Since any matrix of rank can be written as the sum of matrices of rank , we can write for certain all having a coefficient matrix of rank . This implies that


text:
+1 vote
73 views
A computer has a 256 KB, K-way set associative write-back data cache with block size of 32 B. The address sent to the cache controller by the processor is of 32 bits. In addition to the address tag, each cache tag directory contains 2 valid bits and 1 modified bit. If 16 bits are used to address tag. What is the minimum value of K?

A) 6

B) 5

C) 4

D) None of these

edited | 73 views
+1
It should be option C.

Total lines = $\frac{2^{18}}{2^5}= 2^{13}$

Let x is set number.

Now whole address is of 32 bits out of which 16 + x  + 5 (tagbits + set number + offset)

Hence x = 11 i.e $2^{11}$ sets will be there.

Hence k = $\frac{2^{13}}{2^{11}} = 4$
0
Yes 4 way set way associative is right..
0
Sir, my confusion is that should we always consider that these three bits (2 valid bits and 1 modified bit) are included in address tag ? Since the question is saying "In addition to the address tag, each cache tag directory contains 2 valid bits and 1 modified bit" i thought these three bits are additional to the address tag bits.

0
Actually you can't exceed the virtual address range...it's given 32 bits therefore you have to fit extra bits in that range only..
0

Got it..sir

This question (https://gateoverflow.in/2192/gate2012-54) is also helpful.


text:
Parmsys --- Introduction ---

A system of linear equations has 3 possibilities concerning its solution: it may have several solutions, a unique solution, or no solution at all. the type of solution of the system depends essentially on its coefficients. In particular, if the coefficients of a system depend on one or more parameters, the type of solution may vary according to the values of the parameters, which constitutes an important mathematical phenomene.

Parmsys is therefore an exercise designed to help you understand this phenomenon concerning linear systems with parameters. The server will propose such a system to you, and then ask you to determine which conditions on the parameters lead to which type of solution.

The principal tool used to solve the problem is the method of Gauss elimination. Therefore, the exercise incorporates a user-friendly interface allowing you to simplify the system by the method of Gauss, before replying to the questions.

Choose a level of difficulty suitable for you: 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 .

In order to access WIMS services, you need a browser supporting forms. In order to test the browser you are using, please type the word wims here: and press Enter''.


text:
# meas_base test_PluginLogs.py depends on global state

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## Details

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## Description

test_PluginLogs.py fails if you run all the tests for meas_base in a single process (such as would happen if you type pytest in the build directory. I think this is because the test is relying on the set of registered plugins but that set has been modified by earlier tests.

 ============================================== FAILURES ============================================== __________________________ RegisteredPluginsTestCase.testSingleFramePlugins __________________________   self =     def testSingleFramePlugins(self):  center = lsst.afw.geom.Point2D(50, 50)  bbox = lsst.afw.geom.Box2I(lsst.afw.geom.Point2I(0, 0),  lsst.afw.geom.Extent2I(100,100))  dataset = lsst.meas.base.tests.TestDataset(bbox)  dataset.addSource(1000000.0, center)  registry = SingleFramePlugin.registry  dependencies = registry.keys() > task = self.makeSingleFrameMeasurementTask("base_SdssCentroid", dependencies=dependencies)   tests/test_PluginLogs.py:132:  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  python/lsst/meas/base/tests.py:515: in makeSingleFrameMeasurementTask  return SingleFrameMeasurementTask(schema=schema, algMetadata=algMetadata, config=config) python/lsst/meas/base/sfm.py:267: in __init__  self.initializePlugins(schema=self.schema) python/lsst/meas/base/baseMeasurement.py:232: in initializePlugins  for executionOrder, name, config, PluginClass in sorted(self.config.plugins.apply()): ../../stack/DarwinX86/pex_config/13.0-3-g520d906/python/lsst/pex/config/registry.py:182: in apply  retvals.append(self._field.typemap.registry[c](*args, config=self[c], **kw)) python/lsst/meas/base/pluginRegistry.py:95: in __call__  return (self.PluginClass.getExecutionOrder(), self.name, config, self.PluginClass) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    cls = .NonApCorrPlugin'>    @classmethod  def getExecutionOrder(cls):  """Sets the relative order of plugins (smaller numbers run first).    In general, the following class constants should be used (other values  are also allowed, but should be avoided unless they are needed):  CENTROID_ORDER centroids and other algorithms that require only a Footprint  and its Peaks as input  SHAPE_ORDER shape measurements and other algorithms that require getCentroid() to return  a good centroid (in addition to a Footprint and its Peaks).  FLUX_ORDER flux algorithms that require both getShape() and getCentroid(),  in addition to a Footprint and its Peaks  DEFAULT_CATALOGCALCULATION plugins that only operate on the catalog    Must be reimplemented as a class method by concrete derived classes.    This approach was chosen instead of a full graph-based analysis of dependencies  because algorithm dependencies are usually both quite simple and entirely substitutable:  an algorithm that requires a centroid can typically make use of any centroid algorithms  outputs. That makes it relatively easy to figure out the correct value to use for any  particular algorithm.  """ > raise NotImplementedError("All plugins must implement getExecutionOrder()") E NotImplementedError: All plugins must implement getExecutionOrder()   python/lsst/meas/base/pluginsBase.py:78: NotImplementedError =============================== 1 failed, 197 passed in 72.78 seconds ================================

If I reverse the order of tests all tests pass (but "P" is late in the alphabet so there are only ten or so tests after it).

## People

• Assignee:
Tim Jenness
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John Swinbank
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text:
chapter  14
6 Pages

## Switching costs

Firms have no fixed costs and have constant marginal costs. The home firm’s marginal costs are normalized to zero, while c* represents the foreign firm’s marginal costs. Before moving to the trading equilibrium, let us briefly examine the equilibrium without the foreign firm’s entry. In this case, the home firm’s profit is represented by

Π = __

Π1 + __

Π2 = (a – bx1 – s)x1 + (a – bx2)x2,

where Πt represents profits in period t. We can obtain the equilibrium output as

_ x1 =

_ x2 =

2a – s_____4b , (14.1)

where an overbar indicates the equilibrium value without the foreign firm’s entry. Consumer surplus

follows:

CS2 = (2a – s)2_______16b , (14.2)

Π = __

Π1 + __

Π2 = (2a – s)2_______8b , (14.3)

Π = 3(2a – s) 2

________16b . (14.4)

Now, let us move to the case with the foreign firm’s entry. In this case, the analysis is simplified by considering the firm’s second-period reaction curves. We write R(y2) for the home firm’s reaction curve if consumers have no switching costs, and R´(y2) and R*(x2) when consumers have a switching cost s. The emboldened line in Figure 14.1 is the home firm’s reaction curve given x1 > 0.