arXiv subjects

Eve C. Ostriker

Publications and source records attributed to Eve C. Ostriker.

At least 91 records · Page 5Linked to original sources

arXiv2019

Evolution of supernovae-driven superbubbles with conduction and cooling

We use spherically symmetric hydrodynamic simulations to study the dynamical evolution and internal structure of superbubbles (SBs) driven by clustered supernovae (SNe), focusing on the effects of thermal conduction and cooling in the interface between the hot bubble interior and cooled shell. Our simulations employ an effective diffusivity to account for turbulent mixing from nonlinear instabilities that are not captured in 1D. The conductive heat flux into the shell is balanced by a combination of cooling in the interface and evaporation of shell gas into the bubble interior. This evaporation increases the density, and decreases the temperature, of the SB interior by more than an order of magnitude relative to simulations without conduction. However, most of the energy conducted into the interface is immediately lost to cooling, reducing the evaporative mass flux required to balance conduction. As a result, the evaporation rate is typically a factor of $\sim$3-30 lower than predicted by the classical similarity solution of Weaver et al. (1977), which neglects cooling. Blast waves from the first $\sim$30 SNe remain supersonic in the SB interior because reduced evaporation from the interface lowers the mass they sweep up in the hot interior. Updating the Weaver solution to include cooling, we construct a new analytic model to predict the cooling rate, evaporation rate, and temporal evolution of SBs. The cooling rate, and hence the hot gas mass, momentum, and energy delivered by SBs, is set by the ambient ISM density and the efficiency of nonlinear mixing at the bubble/shell interface.

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arXiv2019

Investigating the Complex Velocity Structures within Dense Molecular Cloud Cores with GBT-Argus

We present the first results of high-spectral resolution (0.023 km/s) N$_2$H$^+$ observations of dense gas dynamics at core scales (~0.01 pc) using the recently commissioned Argus instrument on the Green Bank Telescope (GBT). While the fitted linear velocity gradients across the cores measured in our targets nicely agree with the well-known power-law correlation between the specific angular momentum and core size, it is unclear if the observed gradients represent core-scale rotation. In addition, our Argus data reveal detailed and intriguing gas structures in position-velocity (PV) space for all 5 targets studied in this project, which could suggest that the velocity gradients previously observed in many dense cores actually originate from large-scale turbulence or convergent flow compression instead of rigid-body rotation. We also note that there are targets in this study with their star-forming disks nearly perpendicular to the local velocity gradients, which, assuming the velocity gradient represents the direction of rotation, is opposite to what is described by the classical theory of star formation. This provides important insight on the transport of angular momentum within star-forming cores, which is a critical topic on studying protostellar disk formation.

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arXiv2019

Modeling UV Radiation Feedback from Massive Stars: III. Escape of Radiation from Star-forming Giant Molecular Clouds

Using a suite of radiation hydrodynamic simulations of star cluster formation in turbulent clouds, we study the escape fraction of ionizing (Lyman continuum) and non-ionizing (FUV) radiation for a wide range of cloud masses and sizes. The escape fraction increases as H II regions evolve and reaches unity within a few dynamical times. The cumulative escape fraction before the onset of the first supernova explosion is in the range 0.05-0.58; this is lower for higher initial cloud surface density, and higher for less massive and more compact clouds due to rapid destruction. Once H II regions break out of their local environment, both ionizing and non-ionizing photons escape from clouds through fully ionized, low-density sightlines. Consequently, dust becomes the dominant absorber of ionizing radiation at late times and the escape fraction of non-ionizing radiation is only slightly larger than that of ionizing radiation. The escape fraction is determined primarily by the mean $\langle τ\rangle$ and width $σ$ of the optical-depth distribution in the large-scale cloud, increasing for smaller $\langle τ\rangle$ and/or larger $σ$. The escape fraction exceeds (sometimes by three orders of magnitude) the naive estimate $e^{-\langle τ\rangle}$ due to non-zero $σ$ induced by turbulence. We present two simple methods to estimate, within $\sim20\%$, the escape fraction of non-ionizing radiation using the observed dust optical depth in clouds projected on the plane of sky. We discuss implications of our results for observations, including inference of star formation rates in individual molecular clouds, and accounting for diffuse ionized gas on galactic scales.

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arXiv2019

The EDGE-CALIFA Survey: Evidence for Pervasive Extraplanar Diffuse Ionized Gas in Nearby Edge-On Galaxies

We investigate the prevalence, properties, and kinematics of extraplanar diffuse ionized gas (eDIG) in a sample of 25 edge-on galaxies selected from the CALIFA survey. We measure ionized gas scale heights from ${\rm Hα}$ and find that 90% have measurable scale heights with a median of $0.8^{+0.7}_{-0.4}$ kpc. From the ${\rm Hα}$ kinematics, we find that 60% of galaxies show a decrease in the rotation velocity as a function of height above the midplane. This lag is characteristic of eDIG, and we measure a median lag of 21 km s$^{-1}$ kpc$^{-1}$ which is comparable to lags measured in the literature. We also investigate variations in the lag with radius. $\rm H{\small I}$ lags have been reported to systematically decrease with galactocentric radius. We find both increasing and decreasing ionized gas lags with radius, as well as a large number of galaxies consistent with no radial lag variation, and investigate these results in the context of internal and external origins for the lagging ionized gas. We confirm that the ${\rm [S{\small II}]}$/${\rm Hα}$ and ${\rm [N{\small II}]}$/${\rm Hα}$ line ratios increase with height above the midplane as is characteristic of eDIG. The ionization of the eDIG is dominated by star-forming complexes (leaky ${\rm H{\small II}}$ regions). We conclude that the lagging ionized gas is turbulent ejected gas likely resulting from star formation activity in the disk as opposed to gas in the stellar thick disk or bulge. This is further evidence for the eDIG being a product of stellar feedback and for the pervasiveness of this WIM-like phase in many local star-forming galaxies.

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arXiv2019

A Simple and Accurate Network for Hydrogen and Carbon Chemistry in the ISM

Chemistry plays an important role in the interstellar medium (ISM), regulating heating and cooling of the gas, and determining abundances of molecular species that trace gas properties in observations. Although solving the time-dependent equations is necessary for accurate abundances and temperature in the dynamic ISM, a full chemical network is too computationally expensive to incorporate in numerical simulations. In this paper, we propose a new simplified chemical network for hydrogen and carbon chemistry in the atomic and molecular ISM. We compare results from our chemical network in detail with results from a full photo-dissociation region (PDR) code, and also with the Nelson & Langer (1999) (NL99) network previously adopted in the simulation literature. We show that our chemical network gives similar results to the PDR code in the equilibrium abundances of all species over a wide range of densities, temperature, and metallicities, whereas the NL99 network shows significant disagreement. Applying our network in 1D models, we find that the $\mathrm{CO}$-dominated regime delimits the coldest gas and that the corresponding temperature tracks the cosmic ray ionization rate in molecular clouds. We provide a simple fit for the locus of $\mathrm{CO}$ dominated regions as a function of gas density and column. We also compare with observations of diffuse and translucent clouds. We find that the $\mathrm{CO}$, $\mathrm{CHx}$ and $\mathrm{OHx}$ abundances are consistent with equilibrium predictions for densities $n=100-1000~\mathrm{cm^{-3}}$, but the predicted equilibrium $\mathrm{C}$ abundance is higher than observations, signaling the potential importance of non-equilibrium/dynamical effects.

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arXiv2019

Magnetohydrodynamic-Particle-in-Cell Simulations of the Cosmic-Ray Streaming Instability: Linear Growth and Quasi-linear Evolution

The gyro-resonant cosmic-ray (CR) streaming instability is believed to play a crucial role in CR transport, leading to growth of Alfvén waves at small scales that scatter CRs, and impacts the interaction of CRs with the ISM on large scales. However, extreme scale separation ($λ\ll \rm pc$), low cosmic ray number density ($n_{\rm CR}/n_{\rm ISM} \sim 10^{-9}$), and weak CR anisotropy ($\sim v_A/c$) pose strong challenges for proper numerical studies of this instability on a microphysical level. Employing the recently developed magnetohydrodynamic-particle-in-cell (MHD-PIC) method, which has unique advantages to alleviate these issues, we conduct one-dimensional simulations that quantitatively demonstrate the growth and saturation of the instability in the parameter regime consistent with realistic CR streaming in the large-scale ISM. Our implementation of the $δf$ method dramatically reduces Poisson noise and enables us to accurately capture wave growth over a broad spectrum, equally shared between left and right handed Alfvén modes. We are also able to accurately follow the quasi-linear diffusion of CRs subsequent to wave growth, which is achieved by employing phase randomization across periodic boundaries. Full isotropization of the CRs in the wave frame requires pitch angles of most CRs to efficiently cross $90^\circ$, and can be captured in simulations with relatively high wave amplitude and/or high spatial resolution. We attribute this crossing to non-linear wave-particle interaction (rather than mirror reflection) by investigating individual CR trajectories. We anticipate our methodology will open up opportunities for future investigations that incorporate additional physics.

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arXiv2019

A Fast Poisson Solver of Second-Order Accuracy for Isolated Systems in Three-Dimensional Cartesian and Cylindrical Coordinates

We present an accurate and efficient method to calculate the gravitational potential of an isolated system in three-dimensional Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two parts: an interior solver and a boundary solver. The interior solver adopts an eigenfunction expansion method together with a tridiagonal matrix solver to solve the Poisson equation subject to the zero boundary condition. The boundary solver employs James's method to calculate the boundary potential due to the screening charges required to keep the zero boundary condition for the interior solver. A full computation of gravitational potential requires running the interior solver twice and the boundary solver once. We develop a method to compute the discrete Green's function in cylindrical coordinates, which is an integral part of the James algorithm to maintain second-order accuracy. We implement our method in the {\tt Athena++} magnetohydrodynamics code, and perform various tests to check that our solver is second-order accurate and exhibits good parallel performance.

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arXiv2018

Bound Outflows, Unbound Ejecta, and the Shaping of Bipolar Remnants during Stellar Coalescence

Recent observations have revealed that the remnants of stellar-coalescence transients are bipolar. This raises the questions of how these bipolar morphologies arise and what they teach us about the mechanisms of mass ejection during stellar mergers and common-envelope phases. In this paper, we analyze hydrodynamic simulations of the lead-in to binary coalescence, a phase of unstable Roche lobe overflow that takes the binary from the Roche limit separation to the engulfment of the more compact accretor within the envelope of the extended donor. As mass transfer runs away at increasing rates, gas trails away from the binary. Contrary to previous expectations, early mass loss from the system remains bound to the binary and forms a circumbinary torus. Later ejecta, generated as the accretor grazes the surface of the donor, have very different morphologies and are unbound. These two components of mass loss from the binary interact as later, higher-velocity ejecta collide with the circumbinary torus formed by earlier mass loss. Unbound ejecta are redirected toward the poles, and escaping material creates a bipolar outflow. Our findings show that the transition from bound to unbound ejecta from coalescing binaries can explain the bipolar nature of their remnants, with implications for our understanding of the origin of bipolar remnants of stellar-coalescence transients and, perhaps, some preplanetary nebulae.

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arXiv2018

Forming Super Star Clusters in the Central Starburst of NGC 253

NGC 253 hosts the nearest nuclear starburst. Previous observations show a region rich in molecular gas, with dense clouds associated with recent star formation. We used ALMA to image the 350 GHz dust continuum and molecular line emission from this region at 2 pc resolution. Our observations reveal ~14 bright, compact (~2-3 pc FWHM) knots of dust emission. Most of these sources are likely to be forming super star clusters (SSCs) based on their inferred dynamical and gas masses, association with 36 GHz radio continuum emission, and coincidence with line emission tracing dense, excited gas. One source coincides with a known SSC, but the rest remain invisible in Hubble near-infrared (IR) imaging. Our observations imply that gas still constitutes a large fraction of the overall mass in these sources. Their high brightness temperature at 350 GHz also implies a large optical depth near the peak of the IR spectral energy distribution. As a result, these sources may have large IR photospheres and the IR radiation force likely exceeds L/c. Still, their moderate observed velocity dispersions suggest that feedback from radiation, winds, and supernovae are not yet disrupting most sources. This mode of star formation appears to produce a large fraction of stars in the burst. We argue for a scenario in which this phase lasts ~1 Myr, after which the clusters shed their natal cocoons but continue to produce ionizing photons. The strong feedback that drives the observed cold gas and X-ray outflows likely occurs after the clusters emerge from this early phase.

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arXiv2018

Geometry, Kinematics, and Magnetization of Simulated Prestellar Cores

We utilize the more than 100 gravitationally-bound dense cores formed in our three-dimensional, turbulent MHD simulations reported in Chen & Ostriker (2015) to analyze structural, kinematic, and magnetic properties of prestellar cores. Our statistical results disagree with the classical theory of star formation, in which cores evolve to be oblate with magnetic field parallel to the minor axes. Instead, we find that cores are generally triaxial, although the core-scale magnetic field is still preferentially most parallel to the core's minor axis and most perpendicular to the major axis. The internal and external magnetic field directions are correlated, but the direction of integrated core angular momentum is misaligned with the core's magnetic field, consistent with recent observations. The ratio of rotational/total kinetic and rotational/gravitational energies are independent of core size and consistent in magnitude with observations. The specific angular momentum also follows the observed relationship $L/M \propto R^{3/2}$, indicating rotation is acquired from ambient turbulence. With typical $E_\mathrm{rot}/E_K \sim 0.1$, rotation is not the dominant motion when cores collapse.

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arXiv2018

Runaway Coalescence at the Onset of Common Envelope Episodes

Luminous red nova transients, presumably from stellar coalescence, exhibit long-term precursor emission over hundreds of binary orbits, leading to impulsive outbursts with durations similar to a single orbital period. In an effort to understand these signatures, we present and analyze a hydrodynamic model of unstable mass transfer from a giant-star donor onto a more compact accretor in a binary system. Our simulation begins with mass transfer at the Roche limit separation and traces a phase of runaway decay leading up to the plunge of the accretor within the envelope of the donor. We characterize the fluxes of mass and angular momentum through the system and show that the orbital evolution can be reconstructed from measurements of these quantities. The morphology of outflow from the binary changes significantly as the binary orbit tightens. At wide separations, a thin stream of relatively high-entropy gas trails from the outer Lagrange points. As the orbit tightens, the orbital motion desynchronizes from the donor's rotation, and low-entropy ejecta trace a broad fan of largely ballistic trajectories. An order-of-magnitude increase in mass ejection rate accompanies the plunge of the accretor with the envelope of the donor. We argue that this transition marks the precursor-to-outburst transition observed in stellar coalescence transients.

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arXiv2018

The Star Formation Efficiency per Free Fall Time in Nearby Galaxies

We estimate the star formation efficiency per gravitational free fall time, $ε_{\rm ff}$, from observations of nearby galaxies with resolution matched to the typical size of a Giant Molecular Cloud. This quantity, $ε_{\rm ff}$, is theoretically important but so far has only been measured for Milky Way clouds or inferred indirectly in a few other galaxies. Using new, high resolution CO imaging from the PHANGS-ALMA survey, we estimate the gravitational free-fall time at 60 to 120 pc resolution, and contrast this with the local molecular gas depletion time to estimate $ε_{\rm ff}$. Assuming a constant thickness of the molecular gas layer ($H = 100$ pc) across the whole sample, the median value of $ε_{\rm ff}$ in our sample is $0.7\%$. We find a mild scale-dependence, with higher $ε_{\rm ff}$ measured at coarser resolution. Individual galaxies show different values of $ε_{\rm ff}$, with the median $ε_{\rm ff}$ ranging from $0.3\%$ to $2.6\%$. We find the highest $ε_{\rm ff}$ in our lowest mass targets, reflecting both long free-fall times and short depletion times, though we caution that both measurements are subject to biases in low mass galaxies. We estimate the key systematic uncertainties, and show the dominant uncertainty to be the estimated line-of-sight depth through the molecular gas layer and the choice of star formation tracers.

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arXiv2018

Modeling UV Radiation Feedback from Massive Stars: II. Dispersal of Star-Forming Giant Molecular Clouds by Photoionization and Radiation Pressure

UV radiation feedback from young massive stars plays a key role in the evolution of giant molecular clouds (GMCs) by photoevaporating and ejecting the surrounding gas. We conduct a suite of radiation hydrodynamic simulations of star cluster formation in marginally-bound, turbulent GMCs, focusing on the effects of photoionization and radiation pressure on regulating the net star formation efficiency (SFE) and cloud lifetime. We find that the net SFE depends primarily on the initial gas surface density, $Σ_0$, such that the SFE increases from 4% to 51% as $Σ_0$ increases from $13\,M_{\odot}\,{\rm pc}^{-2}$ to $1300\,M_{\odot}\,{\rm pc}^{-2}$. Cloud destruction occurs within $2$-$10\,{\rm Myr}$ after the onset of radiation feedback, or within $0.6$-$4.1$ freefall times (increasing with $Σ_0$). Photoevaporation dominates the mass loss in massive, low surface-density clouds, but because most photons are absorbed in an ionization-bounded Strömgren volume the photoevaporated gas fraction is proportional to the square root of the SFE. The measured momentum injection due to thermal and radiation pressure forces is proportional to $Σ_0^{-0.74}$, and the ejection of neutrals substantially contributes to the disruption of low-mass and/or high-surface density clouds. We present semi-analytic models for cloud dispersal mediated by photoevaporation and by dynamical mass ejection, and show that the predicted net SFE and mass loss efficiencies are consistent with the results of our numerical simulations.

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arXiv2018

The X_CO conversion factor from galactic multiphase ISM simulations

CO(J=1-0) line emission is a widely used observational tracer of molecular gas, rendering essential the X_CO factor, which is applied to convert CO luminosity to H_2 mass. We use numerical simulations to study how X_CO depends on numerical resolution, non-steady-state chemistry, physical environment, and observational beam size. Our study employs 3D magnetohydrodynamics (MHD) simulations of galactic disks with solar neighborhood conditions, where star formation and the three-phase interstellar medium (ISM) are self-consistently regulated by gravity and stellar feedback. Synthetic CO maps are obtained by post-processing the MHD simulations with chemistry and radiation transfer. We find that CO is only an approximate tracer of H_2. On parsec scales, W_CO is more fundamentally a measure of mass-weighted volume density, rather than H_2 column density. Nevertheless, $\langle X_\mathrm{CO} \rangle=0.7-1.0\times10^{20}~\mathrm{cm^{-2}K^{-1}km^{-1}s}$ consistent with observations, insensitive to the evolutionary ISM state or radiation field strength if steady-state chemistry is assumed. Due to non-steady-state chemistry, younger molecular clouds have slightly lower X_CO and flatter profiles of X_CO versus extinction than older ones. The CO-dark H_2 fraction is 26-79 %, anti-correlated with the average extinction. As the observational beam size increases from 1 pc to 100 pc, X_CO increases by a factor of ~ 2. Under solar neighborhood conditions, X_CO in molecular clouds is converged at a numerical resolution of 2 pc. However, the total CO abundance and luminosity are not converged even at the numerical resolution of 1 pc. Our simulations successfully reproduce the observed variations of X_CO on parsec scales, as well as the dependence of X_CO on extinction and the CO excitation temperature.

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arXiv2018

Galactic disk winds driven by cosmic ray pressure

Cosmic ray pressure gradients transfer energy and momentum to extraplanar gas in disk galaxies, potentially driving significant mass loss as galactic winds. This may be particularly important for launching high-velocity outflows of "cool" (T < 10^4 K) gas. We study cosmic-ray driven disk winds using a simplified semi-analytic model assuming streamlines follow the large-scale gravitational potential gradient. We consider scaled Milky Way-like potentials including a disk, bulge, and halo with a range of halo velocities V_H = 50-300 km/s, and streamline footpoints with radii in the disk R_0=1-16 kpc at height 1 kpc. Our solutions cover a wide range of footpoint gas velocity u_0, magnetic-to-cosmic-ray pressure ratio, gas-to-cosmic-ray pressure ratio, and angular momentum. Cosmic ray streaming at the Alfvén speed enables the effective sound speed C_eff to increase from the footpoint to a critical point where C_eff,c = u_c ~ V_H; this differs from thermal winds in which C_eff decreases outward. The critical point is typically at a height of 1-6 kpc from the disk, increasing with V_H, and the asymptotic wind velocity exceeds the escape speed of the halo. Mass loss rates are insensitive to the footpoint values of the magnetic field and angular momentum. In addition to numerical parameter space exploration, we develop and compare to analytic scaling relations. We show that winds have mass loss rates per unit area up to ~ Pi_0 V_H^-5/3 u_0^2/3 where Pi_0 is the footpoint cosmic ray pressure and u_0 is set by the upwelling of galactic fountains. The predicted wind mass-loss rate exceeds the star formation rate for V_H < 200 km/s and u_0 = 50 km/s, a typical fountain velocity.

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arXiv2018

Numerical Simulations of Multiphase Winds and Fountains from Star-Forming Galactic Disks: I. Solar Neighborhood TIGRESS Model

Gas blown away from galactic disks by supernova (SN) feedback plays a key role in galaxy evolution. We investigate outflows utilizing the solar neighborhood model of our high-resolution, local galactic disk simulation suite, TIGRESS. In our numerical implementation, star formation and SN feedback are self-consistently treated and well resolved in the multiphase, turbulent, magnetized interstellar medium. Bursts of star formation produce spatially and temporally correlated SNe that drive strong outflows, consisting of hot (T>5x10^5K) winds and warm (5050K < T < 2x10^4K) fountains. The hot gas at distance d>1kpc from the midplane has mass and energy fluxes nearly constant with d. The hot flow escapes our local Cartesian box barely affected by gravity and is expected to accelerate up to the terminal velocity of v_wind~350-500km/s. The mean mass and energy loading factors of the hot wind are 0.1 and 0.02, respectively. For warm gas, the mean outward mass flux through d=1kpc is comparable to the mean star formation rate, but only a small fraction of this gas is at velocity >50km/s. Thus, the warm outflows eventually fall back as inflows. The warm fountain flows are created by expanding hot superbubbles at d< 1kpc; at larger d neither ram pressure acceleration nor cooling transfers significant momentum or energy flux from the hot wind to the warm outflow. The velocity distribution at launching near d~1kpc better represents warm outflows than a single mass loading factor, potentially enabling development of subgrid models for warm galactic winds in arbitrary large-scale galactic potentials.

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arXiv2017

Numerical Simulations of Turbulent Molecular Clouds Regulated by Radiation Feedback Forces II: Radiation-Gas Interactions and Outflows

Momentum deposition by radiation pressure from young, massive stars may help to destroy molecular clouds and unbind stellar clusters by driving large-scale outflows. We extend our previous numerical radiation hydrodynamic study of turbulent, star-forming clouds to analyze the detailed interaction between non-ionizing UV radiation and the cloud material. Our simulations trace the evolution of gas and star particles through self-gravitating collapse, star formation, and cloud destruction via radiation-driven outflows. These models are idealized in that we include only radiation feedback and adopt an isothermal equation of state. Turbulence creates a structure of dense filaments and large holes through which radiation escapes, such that only ~50% of the radiation is (cumulatively) absorbed by the end of star formation. The surface density distribution of gas by mass as seen by the central cluster is roughly lognormal with sigma_ln(Sigma) = 1.3-1.7, similar to the externally-projected surface density distribution. This allows low surface density regions to be driven outwards to nearly 10 times their initial escape speed v_esc. Although the velocity distribution of outflows is broadened by the lognormal surface density distribution, the overall efficiency of momentum injection to the gas cloud is reduced because much of the radiation escapes. The mean outflow velocity is approximately twice the escape speed from the initial cloud radius. Our results are also informative for understanding galactic-scale wind driving by radiation, in particular the relationship between velocity and surface density for individual outflow structures, and the resulting velocity and mass distributions arising from turbulent sources.

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arXiv2017

Modeling UV Radiation Feedback from Massive Stars: I. Implementation of Adaptive Ray Tracing Method and Tests

We present an implementation of an adaptive ray tracing (ART) module in the Athena hydrodynamics code that accurately and efficiently handles the radiative transfer involving multiple point sources on a three-dimensional Cartesian grid. We adopt a recently proposed parallel algorithm that uses non-blocking, asynchronous MPI communications to accelerate transport of rays across the computational domain. We validate our implementation through several standard test problems including the propagation of radiation in vacuum and the expansions of various types of HII regions. Additionally, scaling tests show that the cost of a full ray trace per source remains comparable to that of the hydrodynamics update on up to $\sim 10^3$ processors. To demonstrate application of our ART implementation, we perform a simulation of star cluster formation in a marginally bound, turbulent cloud, finding that its star formation efficiency is $12\%$ when both radiation pressure forces and photoionization by UV radiation are treated. We directly compare the radiation forces computed from the ART scheme with that from the M1 closure relation. Although the ART and M1 schemes yield similar results on large scales, the latter is unable to resolve the radiation field accurately near individual point sources.

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