Local Scale Height Research Articles

Length and Velocity Scales in Protoplanetary Disk Turbulence

In the theory of protoplanetary disk turbulence, a widely adopted ansatz, or assumption, is that the turnover frequency of the largest turbulent eddy, Ω L , is the local Keplerian frequency Ω K . In terms of the standard dimensionless Shakura–Sunyaev α parameter that quantifies turbulent viscosity or diffusivity, this assumption leads to characteristic length and velocity scales given respectively by αH and αc , in which H and c are the local gas scale height and sound speed. However, this assumption is not applicable in cases when turbulence is forced numerically or driven by some natural processes such as vertical shear instability. Here, we explore the more general case where Ω L ≥ Ω K and show that, under these conditions, the characteristic length and velocity scales are respectively α/R′H and αR′c , where R′≡ΩL/ΩK is twice the Rossby number. It follows that α=α˜/R′ , where α˜c is the root-mean-square average of the turbulent velocities. Properly allowing for this effect naturally explains the reduced particle scale heights produced in shearing box simulations of particles in forced turbulence, and it may help with interpreting recent edge-on disk observations; more general implications for observations are also presented. For R′>1 , the effective particle Stokes numbers are increased, which has implications for particle collision dynamics and growth, as well as for planetesimal formation.

Trapping of H ii regions in Population III star formation

ABSTRACT Radiative feedback from massive Population III (Pop III) stars in the form of ionizing and photodissociating photons is widely believed to play a central role in shutting off accretion on to these stars. Understanding whether and how this occurs is vital for predicting the final masses reached by these stars and the form of the Pop III stellar initial mass function. To help us better understand the impact of UV radiation from massive Pop III stars on the gas surrounding them, we carry out high-resolution simulations of the formation and early evolution of these stars, using the arepo moving-mesh code coupled with the innovative radiative transfer module sprai. Contrary to most previous results, we find that the ionizing radiation from these stars is trapped in the dense accretion disc surrounding them. Consequently, the inclusion of radiative feedback has no significant impact on either the number or the total mass of protostars formed during the 20 kyr period that we simulate. We show that the reason that we obtain qualitatively different results from previous studies of Pop III stellar feedback lies in how the radiation is injected into the simulation. H ii region trapping only occurs if the photons are injected on scales smaller than the local scale height of the accretion disc, a criterion not fulfilled in previous 3D simulations of this process. Finally, we speculate as to whether outflows driven by the magnetic field or by Lyman-α radiation pressure may be able to clear enough gas away from the star to allow the H ii region to escape from the disc.

Probing the mid-layer structure of red giants

Context. The space-borne missions CoRoT and Kepler have already brought stringent constraints on the internal structure of low-mass evolved stars, a large part of which results from the detection of mixed modes. However, all the potential of these oscillation modes as a diagnosis of the stellar interior has not been fully exploited yet. In particular, the coupling factor or the gravity-offset of mixed modes, q and εg, are expected to provide additional constraints on the mid-layers of red giants, which are located between the hydrogen-burning shell and the neighborhood of the base of the convective zone. The link between these parameters and the properties of this region, nevertheless, still remains to be precisely established. Aims. In the present paper, we investigate the potential of the coupling factor in probing the mid-layer structure of evolved stars. Methods. Guided by typical stellar models and general physical considerations, we modeled the coupling region along with evolution. We subsequently obtained an analytical expression of q based on the asymptotic theory of mixed modes and compared it to observations. Results. We show that the value of q is degenerate with respect to the thickness of the coupling evanescent region and the local density scale height. On the subgiant branch and the beginning of the red giant branch (RGB), the model predicts that the peak in the observed value of q is necessarily associated with the important shrinking and the subsequent thickening of the coupling region, which is located in the radiative zone at these stages. The large spread in the measurement is interpreted as the result of the high sensitivity of q to the structure properties when the coupling region becomes very thin. Nevertheless, the important degeneracy of q in this regime prevents us from unambiguously concluding on the precise structural origin of the observed values. In later stages, the progressive migration of the coupling region toward the convective zone is expected to result in a slight and smooth decrease in q, which is in agreement with observations. At one point just before the end of the first-dredge up and the luminosity bump, the coupling region becomes entirely located in the convective region and its continuous thickening is shown to be responsible for the observed decrease in q. We demonstrate that q has the promising potential to probe the migration of the base of the convective region as well as convective extra-mixing during this stage. We also show that the frequency-dependence of q cannot be neglected in the oscillation spectra of such evolved RGB stars, which is in contrast with what is assumed in the current measurement methods. This fact can have an influence on the physical interpretation of the observed values. In red clump stars, in which the coupling regions are very thin and located in the radiative zone, the small variations and spread observed in q suggest that their mid-layer structure is very stable. Conclusions. A structural interpretation of the global observed variations in q was obtained and the potential of this parameter in probing the dynamics of the mid-layer properties of red giants is highlighted. This analytical study paves the way for a more quantitative exploration of the link of q with the internal properties of evolved stars using stellar models for a proper interpretation of the observations. This will be undertaken in the following papers of this series.

The Eccentric Cavity, Triple Rings, Two-armed Spirals, and Double Clumps of the MWC 758 Disk

Abstract Spatially resolved structures in protoplanetary disks hint at unseen planets. Previous imaging observations of the transitional disk around MWC 758 revealed an inner cavity, a ring-like outer disk, emission clumps, and spiral arms, all possibly generated by companions. We present ALMA dust continuum observations of MWC 758 at 0.87 mm wavelength with 43 × 39 mas angular resolution (6.9 × 6.2 au) and 20 μJy beam−1 rms. The central submillimeter emission cavity is revealed to be eccentric; once deprojected, its outer edge can be well fitted by an ellipse with an eccentricity of 0.1 and one focus on the star. The broad ring-like outer disk is resolved into three narrow rings with two gaps in between. The outer two rings tentatively show the same eccentricity and orientation as the innermost ring bounding the inner cavity. The two previously known dust emission clumps are resolved in both the radial and azimuthal directions, with radial widths equal to ∼4× the local scale height. Only one of the two spiral arms previously imaged in near-infrared (NIR) scattered light is revealed in ALMA dust emission, at a slightly larger stellocentric distance owing to projection effects. We also submit evidence of disk truncation at ∼100 au based on comparing NIR imaging observations with models. The spirals, the north clump, and the truncated disk edge are all broadly consistent with the presence of one companion exterior to the spirals at roughly 100 au.

A new catalogue of Galactic novae: investigation of the MMRD relation and spatial distribution

In this study, a new Galactic novae catalogue is introduced collecting important parameters of these sources such as their light curve parameters, classifications, full width half maximum (FWHM) of H$_\alpha$ line, distances and interstellar reddening estimates. The catalogue is also published on a website with a search option via a SQL query and an online tool to re-calculate the distance/reddening of a nova from the derived reddening-distance relations. Using the novae in the catalogue, the existence of a maximum magnitude-rate of decline (MMRD) relation in the Galaxy is investigated. Although an MMRD relation was obtained, a significant scattering in the resulting MMRD distribution still exists. We suggest that the MMRD relation likely depends on other parameters in addition to the decline time, as FWHM H$_\alpha$, the light curve shapes. Using two different samples depending on the distances in the catalogue and from the derived MMRD relation, the spatial distributions of Galactic novae as a function of their spectral and speed classes were studied. The investigation on the Galactic model parameters implies that best estimates for the local outburst density are 3.6 and 4.2 $\times 10^{-10}$ pc$^{-3}$ yr$^{-1}$ with a scale height of 148 and 175 pc, while the space density changes in the range of $0.4 - 16 \ \times 10^{-6}$ pc$^{-3}$. The local outburst density and scale height obtained in this study infer that the disk nova rate in the Galaxy is in the range of $\sim20$ to $\sim100$ yr$^{-1}$ with an average estimate $67^{+21}_{-17}$ yr$^{-1}$.

How supernovae launch galactic winds?

Abstract We use idealized three-dimensional hydrodynamic simulations of global galactic discs to study the launching of galactic winds by supernovae (SNe). The simulations resolve the cooling radii of the majority of supernova remnants (SNRs) and thus self-consistently capture how SNe drive galactic winds. We find that SNe launch highly supersonic winds with properties that agree reasonably well with expectations from analytic models. The energy loading ($\eta _{\rm E}= \dot{E}_{\rm wind}/ \dot{E}_{\rm SN}$) of the winds in our simulations are well converged with spatial resolution while the wind mass loading ($\eta _{\rm M}= \dot{M}_{\rm wind}/\dot{M}_\star$) decreases with resolution at the resolutions we achieve. We present a simple analytic model based on the concept that SNRs with cooling radii greater than the local scaleheight break out of the disc and power the wind. This model successfully explains the dependence (or lack thereof) of ηE (and by extension ηM) on the gas surface density, star formation efficiency, disc radius and the clustering of SNe. The winds our simulations are weaker than expected in reality, likely due to the fact that we seed SNe preferentially at density peaks. Clustering SNe in time and space substantially increases the wind power.

SUPERGRANULATION AS THE LARGEST BUOYANTLY DRIVEN CONVECTIVE SCALE OF THE SUN

ABSTRACT The origin of solar supergranulation remains a mystery. Unlike granulation, the size of which is comparable to both the thickness of the radiative boundary layer and local scale-height in the photosphere, supergranulation does not reflect any obvious length scale of the solar convection zone. Moreover, recent observations of flows in the photosphere using Doppler imaging or correlation or feature tracking show a monotonic decrease in horizontal flow power at scales larger than supergranulation. Both local area and global spherical shell simulations of solar convection by contrast show the opposite, an increase in horizontal flow amplitudes to a low wavenumber. We examine these disparities and investigate how the solar supergranulation may arise as a consequence of nonlocal heat transport by cool diving plumes. Using three-dimensional anelastic simulations with surface driving, we show that the kinetic energy of the largest convective scales in the upper layers of a stratified domain reflects the depth of transition from strong buoyant driving to adiabatic stratification below caused by the dilution of the granular downflows. This depth is quite shallow because of the rapid increase of the mean density below the photosphere. We interpret the observed monotonic decrease in solar convective power at scales larger than supergranulation to be a consequence of this rapid transition, with the supergranular scale the largest buoyantly driven mode of convection in the Sun.

Electron thermodynamics in GRMHD simulations of low-luminosity black hole accretion

Simple assumptions made regarding electron thermodynamics often limit the extent to which general relativistic magnetohydrodynamic (GRMHD) simulations can be applied to observations of low-luminosity accreting black holes. We present, implement, and test a model that self-consistently evolves an electron entropy equation and takes into account the effects of spatially varying electron heating and relativistic anisotropic thermal conduction along magnetic field lines. We neglect the back-reaction of electron pressure on the dynamics of the accretion flow. Our model is appropriate for systems accreting at $\ll 10^{-5}$ of the Eddington rate, so radiative cooling by electrons can be neglected. It can be extended to higher accretion rates in the future by including electron cooling and proton-electron Coulomb collisions. We present a suite of tests showing that our method recovers the correct solution for electron heating under a range of circumstances, including strong shocks and driven turbulence. Our initial applications to axisymmetric simulations of accreting black holes show that (1)~physically-motivated electron heating rates yield electron temperature distributions significantly different from the constant electron to proton temperature ratios assumed in previous work, with higher electron temperatures concentrated in the coronal region between the disc and the jet; (2)~electron thermal conduction significantly modifies the electron temperature in the inner regions of black hole accretion flows if the effective electron mean free path is larger than the local scale-height of the disc (at least for the initial conditions and magnetic field configurations we study). The methods developed in this work are important for producing more realistic predictions for the emission from accreting black holes such as Sagittarius A* and M87; these applications will be explored in future work.

ON THE FEEDING ZONE OF PLANETESIMAL FORMATION BY THE STREAMING INSTABILITY

The streaming instability is a promising mechanism to overcome the barriers in direct dust growth and lead to the formation of planetesimals. Most previous studies of the streaming instability, however, were focused on a local region of a protoplanetary disk with a limited simulation domain such that only one filamentary concentration of solids has been observed. The characteristic separation between filaments is therefore not known. To address this, we conduct the largest-scale simulations of the streaming instability to date, with computational domains up to 1.6 gas scale heights both horizontally and vertically. The large dynamical range allows the effect of vertical gas stratification to become prominent. We observe more frequent merging and splitting of filaments in simulation boxes of high vertical extent. We find multiple filamentary concentrations of solids with an average separation of about 0.2 local gas scale heights, much higher than the most unstable wavelength from linear stability analysis. This measures the characteristic separation of planetesimal forming events driven by the streaming instability and thus the initial feeding zone of planetesimals.

Treating gravity in thin-disk simulations

In 2D simulations of thin gaseous disks with embedded planets or self-gravity the gravitational potential needs to be smoothed to avoid singularities in the numerical evaluation of the gravitational potential or force. In order to correctly resemble the realistic case of vertically extended 3D disks the softening prescription used in 2D needs to be adjusted properly. In this paper we analyze the embedded planet and the self-gravity case and provide a method to evaluate the required smoothing in 2D simulations of thin disks. Starting from the averaged hydrodynamic equations and using a vertically isothermal disk model, we calculate the force. We compare our results to the often used Plummer form of the potential which runs as \propto 1/(r^2+\epsilon^2)^{1/2}. For that purpose we compute the required smoothing length \epsilon as a function of distance r to the planet or to a disk element within a self-gravitating disk. We find that for larger distances \epsilon is determined solely by the vertical disk thickness H. For the planet case we find that outside r\simH a value of \epsilon = 0.7H describes the averaged force very well, while in the self-gravitating disk the value needs to be larger, \epsilon = 1.2H. For smaller distances the smoothing needs to be reduced significantly. Comparing torque densities of 3D and 2D simulations we show that the modification to the vertical density stratification as induced by an embedded planet needs to be taken into account to obtain agreeing results. In disk fragmentation simulations the choice of \epsilon can determine whether a disk will fragment or not. Because a wrong smoothing length can change even the direction of migration, it is very important to include the effect of the planet on the local scale height in 2D planet-disk simulations. We provide an approximate and fast method for this purpose which gives very good agreement to full 3D simulations.

On the dynamics of planetesimals embedded in turbulent protoplanetary discs

(abridged) Angular momentum transport and accretion in protoplanetary discs are generally believed to be driven by MHD turbulence via the magneto-rotational instability (MRI). The dynamics of solid bodies embedded in such discs (dust grains, boulders, planetesimals and planets) may be strongly affected by the turbulence, such that the formation pathways for planetary systems are determined in part by the strength and spatial distribution of the turbulent flow. We examine the dynamics of planetesimals, with radii between 1m \^a 10 km, embedded in turbulent protoplanetary discs, using three dimensional MHD simulations. The planetesimals experience gas drag and stochastic gravitational forces due to the turbulent disc. We use, and compare the results from, local shearing box simulations and global models in this study. The main aims of this work are to examine: the growth, and possible saturation, of the velocity dispersion of embedded planetesimals as a function of their size and disc parameters; the rate of radial migration and diffusion of planetesimals; the conditions under which the results from shearing box and global simulations agree. We find good agreement between local and global simulations when shearing boxes of dimension 4H x 16H x 2H are used (H being the local scale height). The magnitude of the density fluctuations obtained is sensitive to the box size, due to the excitation and propagation of spiral density waves. This affects the stochastic forcing experienced by planetesimals. [...] Our models show that fully developed MHD turbulence in protoplanetary discs would have a destructive effect on embedded planetesimals. Relatively low levels of turbulence are required for traditional models of planetesimal accretion to operate, this being consistent with the existence of a dead zone in protoplanetary discs.

Recent results of zebra patterns in solar radio bursts

This review covers the most recent experimental results and theoretical research on zebra patterns (ZPs) in solar radio bursts. The basic attention is given to events with new peculiar elements of zebra patterns received over the last few years. All new properties are considered in light of both what was known earlier and new theoretical models. Large-scale ZPs consisting of small-scale fiber bursts could be explained by simultaneous inclusion of two mechanisms when whistler waves “highlight" the levels of double plasma resonance (DPR). A unique fine structure was observed in the event on 2006 December 13: spikes in absorption formed dark ZP stripes against the absorptive type III-like bursts. The spikes in absorption can appear in accordance with well known mechanisms of absorptive bursts. The additional injection of fast particles filled the loss-cone (breaking the loss-cone distribution), and the generation of the continuum was quenched at these moments. The maximum absorptive effect occurs at the DPR levels. The parameters of millisecond spikes are determined by small dimensions of the particle beams and local scale heights in the radio source. Thus, the DPR model helps to understand several aspects of unusual elements of ZPs. However, the simultaneous existence of several tens of the DPR levels in the corona is impossible for any realistic profile of the plasma density and magnetic field. Three new theories of ZPs are examined. The formation of eigenmodes of transparency and opacity during the propagation of radio waves through regular coronal inhomogeneities is the most natural and promising mechanism. Two other models (nonlinear periodic space – charge waves and scattering of fast protons on ion-sound harmonics) could happen in large radio bursts.

Type III migration in a low-viscosity disc

We study the type III migration of a Saturn mass planet in low viscosity discs. The planet is found to experience cyclic episodes of rapid decay in orbital radius, each amounting to a few Hill radii. We find this to be due to the scattering of large- scale vortices present in the disc. The origin and role of vortices in the context of type III migration is explored. It is shown through numerical simulations and semi- analytical modelling that spiral shocks induced by a sufficiently massive planet will extend close to the planet orbital radius. The production of vortensity across shock tips results in thin high vortensity rings with a characteristic width of the local scale height. For planets with masses equal to and above that of Saturn, the rings are co-orbital features extending the entire azimuth. Linear stability analysis show there exists unstable modes that are localised about local vortensity minima which coincide with gap edges. Simulations show that vortices are non-linear a outcome. We used hydrodynamic simulations to examine vortex-planet interactions. Their effect is present in discs with kinematic viscosity less than about an order of magnitude smaller than the typically adopted value of \nu = 10^{-5}\Omega_pr_p(0)^2, where r_p(0) and \Omega_p are the initial orbital radius and angular velocity of the planet respectively. We find that the magnitude of viscosity affects the nature of type III migration but not the extent of the orbital decay. The role of vortices as a function of initial disc mass is also explored and it is found that the amount of orbital decay during one episode of vortex-planet interaction is independent of initial disc mass. We incorporate the concept of the co-orbital mass deficit in the analysis of our results and link it to the presence of vortices at gap edges.

AN ANALYTIC MODEL FOR THE EVOLUTION OF A VISCOUS, IRRADIATED DISK

We describe an analytic model for the evolution of a protoplanetary disk heated by viscous accretion and radiation from the central star. The disk is assumed to be flared and viscosity is assumed to follow an alpha model, where viscosity is proportional to the local sound speed and scale height. In the inner disk, the midplane temperature is mainly determined by the energy released by the viscous accretion of material through the disk and onto the star. In the outer disk, stellar irradiation is the dominant heat source. A third regime is present in the innermost part of the disk, where viscous heating dominates but the opacity declines rapidly with increasing temperature due to dust grain sublimation. Changes in the protostellar radius and luminosity over time are readily incorporated into the model, although these have a relatively minor effect on the disk evolution. The model yields the surface density and midplane temperature at any point in space and time during the lifetime of the disk. It is especially suited to studies of planet formation that require a self-consistent model of disk evolution with minimal computational expense.

Stability and nonlinear adjustment of vortices in Keplerian flows

We investigate the stability, nonlinear development and equilibrium structure of vortices in a background shearing Keplerian flow. We make use of high-resolution global two-dimensional compressible hydrodynamic simulations. We introduce the concept of nonlinear adjustment to describe the transition of unbalanced vortical fields to a long-lived configuration. We discuss the conditions under which vortical perturbations evolve into long-lived persistent structures and we describe the properties of these equilibrium vortices. The properties of equilibrium vortices appear to be independent from the initial conditions and depend only on the local disk parameters. In particular we find that the ratio of the vortex size to the local disk scale height increases with the decrease of the sound speed, reaching values well above the unity. The process of spiral density wave generation by the vortex, discussed in our previous work, appear to maintain its efficiency also at nonlinear amplitudes and we observe the formation of spiral shocks attached to the vortex. The shocks may have important consequences on the long term vortex evolution and possibly on the global disk dynamics. Our study strengthens the arguments in favor of anticyclonic vortices as the candidates for the promotion of planetary formation. Hydrodynamic shocks that are an intrinsic property of persistent vortices in compressible Keplerian flows are an important contributor to the overall balance. These shocks support vortices against viscous dissipation by generating local potential vorticity and should be responsible for the eventual fate of the persistent anticyclonic vortices. Numerical codes have be able to resolve shock waves to describe the vortex dynamics correctly.

Vortices in Thin, Compressible, Unmagnetized Disks

We consider the formation and evolution of vortices in a hydrodynamic shearing-sheet model. The evolution is done numerically using a version of the ZEUS code. Consistent with earlier results, an injected vorticity field evolves into a set of long-lived vortices, each of which has a radial extent comparable to the local scale height. But we also find that the resulting velocity field has a positive shear stress, . This effect appears only at high resolution. The transport, which decays with time as t^-1/2, arises primarily because the vortices drive compressive motions. This result suggests a possible mechanism for angular momentum transport in low-ionization disks, with two important caveats: a mechanism must be found to inject vorticity into the disk, and the vortices must not decay rapidly due to three-dimensional instabilities.

Dark Matter in the Milky Way. I. The Isothermal Disk Approximation

We solve the combined Poisson-Boltzmann equation for the gravitational potential of the Milky Way for an isothermal distribution of stars and gas perpendicular to the Galactic disk. In contrast to the standard assumption that the dark matter is distributed in a spheroidal way, we focus our analysis on the question of whether a fraction of the Galactic dark matter might be distributed in a disklike way. Using gaseous tracers for the gravitational potential and taking all relevant observational constraints into account, we find that a disklike dark matter distribution with a local scale height of 4 kpc provides a self-consistent and stable solution of the Poisson-Boltzmann equation. This implies that both the mass distribution and the dynamics of the Milky Way for R 35 kpc are dominated by a massive dark matter disk. For larger galactocentric radii the gravitational potential approaches spherical symmetry, implying an approximately spherical dark matter distribution on such scales.

Three-dimensional MHD simulations of the emergence of twisted flux tubes

We carried out three-dimensional magnetohydrodynamic (MHD) simulations of the emergence of a twisted magnetic flux tube initially imbedded in the convection zone. When the magnetic twist exceeds a threshold value, the flux tube deforms itself into a helical structure through the kink instability. For sufficiently long wavelength perturbations ( λ p > 10 H, where H is the local scale height), the flux tube also subjects to the undular mode of the magnetic buoyancy instability (Parker instability). The emergent portions of such helically deformed, undulating flux tube show a series of S-shaped X-ray emitting regions as observed by the Yohkoh satellite.

Magnetic spiral arms in galaxies

Large-scale magnetic fields in spiral galaxies are strongest in spiral-shaped regions, the magnetic arms. It was recently discovered for the galaxy NGC 6946 that magnetic arms can be interlaced with the gaseous arms, rather than coinciding with them. There are indications that the magnetic and gaseous arms may cross in some other galaxies. We suggest that magnetic arms can be located away from regions of higher gas density (gaseous arms) because interstellar turbulence is stronger in the latter. We predict magnetic arms interlaced with gaseous arms in galaxies with weak dynamos, whereas the two spiral structures should overlap in galaxies with strong dynamo action; in an intermediate case, the magnetic and gaseous spiral structures overlap in the inner galaxy and are interlaced in the outer parts (as, possibly, in M51). Another plausible mechanism to produce displaced magnetic and gaseous spiral patterns results from a delay in the dynamo response to the enhancement of turbulence in the gaseous arms. This should lead to the magnetic and gaseous arms crossing at the corotation radius, as possibly observed in the galaxies IC 342 and M83. We also argue that spiral arms only weakly affect the local scaleheight of the galactic gas layer.

The two‐dimensional resolution kernel associated with retrieval of ionospheric and atmospheric refractivity profiles by Abelian inversion of radio occultation phase data

Errors in radio occultation experiments associated with departures from sphericity can be analyzed on the basis of straight‐ray tomography in which (1) only one parallel projection of the object is available, (2) diffraction is neglected, and (3) inversion is achieved by assuming the object to be circularly symmetric. The straight‐line tomographic case and the actual curved ray paths in a spherically symmetric atmosphere are related by an analytic transformation. A previously unknown kernel function transforms the two‐dimensional (2‐D) atmospheric refractivity map from the plane of propagation to a 1‐D radial profile along the locus of ray periapsides. This function consists of a discrete, positive singularity plus an extended negative branch; the negative branch rises smoothly from negative infinity in the neighborhood of the positive singularity toward zero. Knowledge of the resolving kernel allows study of horizontal and vertical resolution in a rigorous manner. Spherically symmetric structures are reproduced exactly with the kernel; processing of band‐limited projection data results in a modified kernel of finite vertical resolving power. In general, the horizontal extent of significant contributions, including levels of 10% of the peak, is of the order 2√4.6rH centered on the locus of periapsides, where r is the radius of the point in consideration and H is the refractivity scale height; the vertical extent of significant contributions extends from approximately one sampling distance below r to a few sampling distances above it. We confirm the expected result that a refractivity structure of radial thickness Δr will be retrieved properly only if its circumferential extent is approximately 2√2rΔr or greater centered on the periapsis point. In addition, a positive refractivity structure limited in circumferential extent introduces artifacts of negative refractivity in the retrieved profile. The artifacts first appear at the projected altitude onto the vertical of the structure's circumferential edges and continue downward. The error represented by such negative refractivity artifacts is diminished somewhat by the general exponential increase in refractivity with decreasing altitude of a real atmosphere. Thus the relative error depends on the local scale height and the circumferential extent of the refractivity structure.