Small Optical Depth Research Articles

Measurement of aerosol optical depth and sub-visual cloud detection using the optical depth sensor (ODS)

Abstract. A small and sophisticated optical depth sensor (ODS) has been designed to work in the atmosphere of Mars. The instrument measures alternatively the diffuse radiation from the sky and the attenuated direct radiation from the Sun on the surface. The principal goals of ODS are to retrieve the daily mean aerosol optical depth (AOD) and to detect very high and optically thin clouds, crucial parameters in understanding the Martian meteorology and climatology. The detection of clouds is undertaken at twilight, allowing the detection and characterization of clouds with opacities below 0.03 (sub-visual clouds). In addition, ODS is capable to retrieve the aerosol optical depth during nighttime from moonlight measurements. Recently, ODS has been selected at the METEO meteorological station on board the ExoMars 2018 Lander. In order to study the performance of ODS under Mars-like conditions as well as to evaluate the retrieval algorithms for terrestrial measurements, ODS was deployed in Ouagadougou (Africa) between November 2004 and October 2005, a Sahelian region characterized by its high dust aerosol load and the frequent occurrence of Saharan dust storms. The daily average AOD values retrieved by ODS were compared with those provided by a CIMEL sunphotometer of the AERONET (Aerosol Robotic NETwork) network localized at the same location. Results represent a good agreement between both ground-based instruments, with a correlation coefficient of 0.77 for the whole data set and 0.94 considering only the cloud-free days. From the whole data set, a total of 71 sub-visual cirrus (SVC) were detected at twilight with opacities as thin as 1.10−3 and with a maximum of occurrence at altitudes between 14 and 20 km. Although further optimizations and comparisons of ODS terrestrial measurements are required, results indicate the potential of these measurements to retrieve the AOD and detect sub-visual clouds.

ALMA observations of the submillimetre hydrogen recombination line from the type 2 active nucleus of NGC 1068

Hydrogen recombination lines at the submillimetre band (submm-RLs) can serve as probes of ionized gas without dust extinction. One therefore expects to probe the broad line region (BLR) of an obscured (type 2) active galactic nucleus (AGN) with those lines. However, admitting the large uncertainty in the continuum level, here we report on the non-detection of both broad and narrow H26$\alpha$ emission line (rest frequency = 353.62 GHz) towards the prototypical type 2 AGN of NGC 1068 with the Atacama Large Millimeter/submillimeter Array (ALMA). We also investigate the nature of BLR clouds that can emit submm-RLs with model calculations. As a result, we suggest that clouds with an electron density ($N_e$) of $\sim$ 10$^9$ cm$^{-3}$ can mainly contribute to broad submm-RLs in terms of the line flux. On the other hand, line flux from other density clouds would be insignificant considering their too large or too small line optical depths. However, even for the case of $N_e \sim 10^9$ cm$^{-3}$ clouds, we also suggest that the expected line flux is extremely low, which is impractical to detect even with ALMA.

On the aerosol and cloud phase function expansion moments for radiative transfer simulations

AbstractThe accuracy of the Henyey‐Greenstein (HG) approximation in radiative transfer process is systematically investigated for aerosol and cloud particles. For small‐size aerosols, the phase function moment by the HG approximation is close to that of the true phase function; therefore, the differences in four‐stream radiative transfer calculations are very small whether using the HG approximation or the true phase function moments. However, for large‐size aerosols, liquid cloud droplets and ice crystals, the HG approximation produces very different phase function moment result compared to the true phase function. In case of small optical depth, the single layer four‐stream radiative transfer calculations show that the HG approximation can produce relative errors larger than 20% while the errors are generally less than 5% by using true phase function moments. By applying the four‐stream radiative transfer scheme to a multilayer atmosphere with aerosol and cloud, the differences in flux error are generally less than 2 Wm−2 by using either the HG approximation or true phase function moments, but can be over 10 Wm−2 for ice cloud case. The aerosol/cloud instantaneous radiative forcing has been analyzed in climate simulation. Compared to the four‐stream radiative transfer scheme, the two‐stream radiative transfer scheme overestimates the aerosol radiative flux in low‐latitude regions and underestimates the upward flux for the high‐latitude regions. By using the HG approximation, the difference in aerosol radiative forcing between the four‐stream and the two‐stream radiative transfer schemes is enhanced; the enhancement can be up to 0.5 Wm−2. The parameterization schemes for aerosol and cloud optical properties must be extended to contain higher‐order phase function moments, if higher‐order stream radiative transfer algorithms are to be used.

Underestimation of oceanic warm cloud occurrences by the Cloud Profiling Radar aboard CloudSat

The Cloud Profiling Radar (CPR) onboard CloudSat is an active sensor specifically dedicated to cloud detection. Compared to passive remote sensors, CPR plays a unique role in investigating the occurrence of multi-layer clouds and depicting the internal vertical structure of clouds. However, owing to contamination from ground clutter, CPR reflectivity signals are invalid in the lowest 1 km above the surface, leading to numerous missed detections of warm clouds. In this study, by using 1-yr CPR and MODIS (Moderate Resolution Imaging Spectroradiometer) synchronous data, those CPR-missed oceanic warm clouds that are identified as cloudy by MODIS are examined. It is demonstrated that CPR severely underestimates the occurrence of oceanic warm clouds, with a global-average miss rate of about 0.43. Over the tropical and subtropical oceans, the CPR-missed clouds tend to occur in regions with relatively low sea surface temperature. CPR misses almost all warm clouds with cloud tops lower than 1 km, and the miss rate reduces with increasing cloud top. As for clouds with cloud tops higher than 2 km, the negative bias of CPR-captured warm cloud occurrence falls below 3%. The cloud top height of CPR-missed warm clouds ranges from 0.6 to 1.2 km, and these clouds mostly have evidently small optical depths and droplet effective radii. The vertically integrated cloud liquid water content of CPR-missed warm clouds is smaller than 50 g m−2. It is also revealed that CPR misses some warm clouds that have small optical depths or small droplet sizes, besides those limited in the boundary layer below about 1 km due to ground clutter.

SDSS J083253.18+064316.7: one strange object with double-peaked narrow Hα but single-peaked narrow Hβ

Abstract In this Letter, we first report one unique object SDSS J0832+0643 with particular features of narrow Balmer emission lines: double-peaked narrow Hα but single-peaked narrow Hβ. The particular features cannot be expected by currently proposed kinematic models for double-peaked narrow emission lines, because the proposed kinematic models lead to similar line profiles of narrow Balmer emission lines. However, due to radiative transfer effects, the non-kinematic model can be naturally applied to well explain the particular features of narrow Balmer emission lines: larger optical depth in Hα than 10 leads to observed double-peaked narrow Hα, but smaller optical depth in Hβ around 2 leads to observed single-peaked narrow Hβ. Therefore, SDSS J0832+0643 can be used as strong evidence to support the non-kinematic model for double-peaked narrow emission lines.

Implications of a PeV neutrino spectral cut-off in gamma-ray burst models

The recent discovery of extragalactic PeV neutrinos opens a new window to the exploration of cosmic-ray accelerators. The observed PeV neutrino flux is close to the Waxman-Bahcall upper bound implying that gamma-ray bursts (GRBs) may be the source of ultra-high energy cosmic rays (UHECRs). Starting with the assumption of the GRB-UHECR connection, we show using both analytical estimates and numerical simulations that the observed neutrinos can originate at the jet as a result of photopion interactions with the following implications: the neutrino spectra are predicted to have a cutoff at energy $\le 10$ PeV; the dissipation responsible for the GRB emission and cosmic-ray acceleration takes place at distances $r_{\rm diss} \approx 3 \times 10^{11}-3 \times 10^{13} \ {\rm cm}$ from the central engine; the Thomson optical depth at the dissipation region is $\tau_{\rm T} \sim 1$; the jet carries a substantial fraction of its energy in the form of Poynting flux at the dissipation region, and has a Lorentz factor $\Gamma \simeq 100-500$. The non-detection of $\sim$PeV neutrinos coincident with GRBs will indicate that GRBs are either poor cosmic accelerators or the dissipation takes place at small optical depths in the jet.

Comparison of marine boundary layer cloud properties from CERES‐MODIS Edition 4 and DOE ARM AMF measurements at the Azores

AbstractMarine boundary layer (MBL) cloud properties derived from the NASA Clouds and the Earth's Radiant Energy System (CERES) project using Terra and Aqua Moderate Resolution Imaging Spectroradiometer (MODIS) data are compared with observations taken at the Department of Energy Atmospheric Radiation Measurement (ARM) Mobile Facility at the Azores (AMF‐Azores) site from June 2009 through December 2010. Cloud properties derived from ARM ground‐based observations were averaged over a 1 h interval centered at the satellite overpass time, while the CERES‐MODIS (CM) results were averaged within a 30 km × 30 km grid box centered over the Azores site. A total of 63 daytime and 92 nighttime single‐layered overcast MBL cloud cases were selected from 19 months of ARM radar‐lidar and satellite observations. The CM cloud top/base heights (Htop/Hbase) were determined from cloud top/base temperatures (Ttop/Tbase) using a regional boundary layer lapse rate method. For daytime comparisons, the CM‐derived Htop (Hbase), on average, is 0.063 km (0.068 km) higher (lower) than its ARM radar‐lidar‐observed counterpart, and the CM‐derived Ttop and Tbase are 0.9 K less and 2.5 K greater than the surface values with high correlations (R2 = 0.82 and 0.84, respectively). In general, the cloud top comparisons agree better than the cloud base comparisons, because the CM cloud base temperatures and heights are secondary products determined from cloud top temperatures and heights. No significant day‐night difference was found in the analyses. The comparisons of MBL cloud microphysical properties reveal that when averaged over a 30 km × 30 km area, the CM‐retrieved cloud droplet effective radius (re) at 3.7 µm is 1.3 µm larger than that from the ARM retrievals (12.8 µm), while the CM‐retrieved cloud liquid water path (LWP) is 13.5 gm−2 less than its ARM counterpart (114.2 gm−2) due to its small optical depth (9.6 versus 13.7). The differences are reduced by 50% when the CM averages are computed only using the MODIS pixel nearest the AMF site. Using the effective radius retrieved using 2.1 µm channel to calculate LWP can reduce the difference between the CM and ARM microwave radiometer retrievals from −13.7 to 2.1 gm−2. The 10% differences between the ARM and CERES‐MODIS LWP and re retrievals are within the uncertainties of the ARM LWP (~20 gm−2) and re (~10%) retrievals; however, the 30% difference in optical depth is significant. Possible reasons contributing to this discrepancy are increased sensitivities in optical depth from both surface retrievals when τ~10 and topography. The τ differences vary with wind direction and are consistent with the island orography. Much better agreement in τ is obtained when using only those data taken when the wind is from the northeast, where topographical effects on the sampled clouds are minimal.

Water vapor retrieval from OMI visible spectra

Abstract. There are distinct spectral features of water vapor in the wavelength range covered by the Ozone Monitoring Instrument (OMI) visible channel. Although these features are much weaker than those at longer wavelengths, they can be exploited to retrieve useful information about water vapor. They have an advantage in that their small optical depth leads to fairly simple interpretation as measurements of the total water vapor column density. We have used the Smithsonian Astrophysical Observatory (SAO) OMI operational retrieval algorithm to derive the slant column density (SCD) of water vapor using the 430–480 nm spectral region after extensive optimization. We convert from SCD to vertical column density (VCD) using the air mass factor (AMF), which is calculated using look-up tables of scattering weights and assimilated water vapor profiles. Our Level 2 product includes not only water vapor VCD but also the associated scattering weights and AMF. In the tropics, our standard water vapor product has a median SCD of 1.3 × 1023 molecules cm−2 and a median relative uncertainty of about 11%, about a factor of 2 better than that from a similar OMI algorithm that uses a narrower retrieval window. The corresponding median VCD is about 1.2 × 1023 molecules cm−2. We have examined the sensitivities of SCD and AMF to various parameters and compared our results with those from the GlobVapour product, the Moderate Resolution Imaging Spectroradiometer (MODIS) and the Aerosol Robotic NETwork (AERONET).

The optical properties and longwave radiative forcing in the lateral boundary of cirrus cloud

AbstractThrough observations from the Cloud‐Aerosol Lidar with Orthogonal Polarization onboard the Cloud‐Aerosol Lidar and Infrared Pathfinder Satellite Observation, we detected a common feature of narrow and subvisible lateral boundary layer in cirrus cloud. In this layer the lidar backscatter, the depolarization ratio, the ice water content, the effective radius of ice particles, and the cloud optical depth all decrease sharply toward the cloud edge. In general, the width of this layer (6.4 ± 3.1 km over land) decreases with increasing ambient temperature. The estimated longwave radiative forcing associated with the layer is about 10 W/m2. Due to its extremely small optical depth (less than 0.3), such lateral boundary layer may be missed by conventional satellite passive optical sensors. As a consequence, the mentioned radiative forcing has not been credited with its deserved share in the Earth's radiative energy budget.

Cavity-assisted revival of silenced echo quantum memory

We present a detailed analysis of a quantum memory for photons based on cavity-assisted revival of silenced echo. Our protocol, without the need of any external field or spectral hole burning technique, takes the full advantage of the available optical thickness, can provide a broader bandwidth for the practical material systems. Under the impedance matching condition to an optical cavity, we show that the efficiency of the memory can approach unity even for a small optical depth. It overcomes a formidable drawback in existing photon echo quantum memory and can help to realize a high-efficiency quantum memory within the present experimental conditions.

Turbid media optical properties derived from the characteristics of propagating laser radiation beams

The possibility is studied to develop a straightforward analytical approach to the determination of the optical properties of liquid turbid media having forward-peaked scattering indicatrices. The approach is based on investigating the in-depth behavior of the radius and the axial intensity of a laser radiation beam propagating through the turbid medium. Based on the small-angle approximation, the detected forward-propagating light power spatial distribution, at relatively small or large optical depths along the beam axis, is obtained asymptotically in analytical form allowing one to derive relatively simple expressions of the extinction, reduced-scattering and absorption coefficients and the anisotropy factor of the medium through the characteristics of the propagating light beam. Preliminary experiments have also been performed, using Intralipid dilutions of different relatively low concentrations and measuring the cross-sectional radial distribution of the detected light power at different depths along the beam axis. The corresponding on-axis detected light power profiles have been measured independently as well. The experimental results are consistent with the analytical expressions obtained that allow one to estimate the optical coefficients and the anisotropy factor of the investigated media on the basis of the measured beam characteristics. The values obtained are near those predicted by other researchers.

Insights into the content and spatial distribution of dust from the integrated spectral properties of galaxies

[Abridged] We present a new approach to investigate the content and spatial distribution of dust in structurally unresolved star-forming galaxies from the observed dependence of integrated spectral properties on galaxy inclination. We develop an innovative combination of generic models of radiative transfer (RT) in dusty media with a prescription for the spectral evolution of galaxies, via the association of different geometric components of galaxies with stars in different age ranges. We show that a wide range of RT models all predict a quasi-universal relation between slope of the attenuation curve at any wavelength and V-band attenuation optical depth in the diffuse interstellar medium (ISM), at all galaxy inclinations. This relation predicts steeper (shallower) dust attenuation curves than both the Calzetti and MW curves at small (large) attenuation optical depths, which implies that geometry and orientation effects have a stronger influence on the shape of the attenuation curve than changes in the optical properties of dust grains. We use our combined RT and spectral evolution model to interpret the observed dependence of the H\alpha/H\beta\ ratio and ugrizYJH attenuation curve on inclination in a sample of ~23 000 nearby star-forming galaxies. From a Bayesian MCMC fit, we measure the central face-on B-band optical depth of this sample to be tau_B\perp~1.8\pm0.2. We also quantify the enhanced optical depth towards newly formed stars in their birth clouds, finding this to be significantly larger in galaxies with bulges than in disc-dominated galaxies, while tau_B\perp is roughly similar in both cases. Finally, we show that neglecting the effect of geometry and orientation on attenuation can severely bias the interpretation of galaxy spectral energy distributions, as the impact on broadband colours can reach up to 0.3-0.4 mag at optical wavelengths and 0.1 mag at near-infrared ones.

An examination of Mars’ north seasonal polar cap using MGS: Composition and infrared radiation balance

A detailed analysis of data from one revolution of the Mars Global Surveyor (MGS) is presented. Approximately 80% of this revolution observes the mid-winter northern seasonal polar cap, which covers the surface to <60°N, and which is predominantly within polar night. The surface composition and temperature are determined through analysis of 6–50μm infrared spectra from the Thermal Emission Spectrometer (TES). The infrared radiative balance, which is the entire heat balance in the polar night except for small subsurface and atmospheric advection terms, is calculated for the surface and atmospheric column. The primary constituent, CO2 ice, also dominates the infrared spectral properties by variations in its grain size and by admixtures of dust and water ice, which cause large variations in the 20–50μm emissivity. This is modified by incomplete areal coverage, and clouds or hazes. This quantitative analysis reveals CO2 grain radii ranging from ∼100μm in isolated areas, to 1–5mm in more widespread regions. The water ice content varies from none to about one part per thousand by mass, with a clear increase towards the periphery of the polar cap. The dust content is typically a few parts per thousand by mass, but is as much as an order of magnitude less abundant in “cold spot” regions, where the low emissivity of pure CO2 ice is revealed. This is the first quantitative analysis of thermal spectra of the seasonal polar cap and the first to estimate water ice content. Our models show that the cold spots represent cleaner, dust-free ice rather than finer grained ice than the background. Our guess is that the dust in cold spots is hidden in the center of the CO2 frost particles rather than not present. The fringes of the cap have more dust and water ice, and become patchy, with warmer water snow filling the gaps on the night side, and warmer bare soil on the day side. A low optical depth (<1 in the visible) water ice atmospheric haze is apparent on the night side, and appears with smaller optical depth on the day side. The infrared radiative balance at the surface is typically 20–25Wm−2 in the central polar cap, with ∼25% dips in the regions of dust-free CO2. The atmospheric radiative terms are typically 1–3Wm−2.

Quantum Memory with Natural Inhomogeneous Broadening in an Optical Cavity

We propose an efficient quantum memory scheme with natural inhomogeneous broadening in an asymmetric optical cavity. The scheme uses the strong rephasing pulses like traditional photon echo techniques rather than spectral holeburning, which enables us to have the potential implementation in a much broader range of material systems. In the condition of impedance matching to an optical cavity, we find that the input light pulse can be completely absorbed by an atomic ensemble. We also show that the quantum memory efficiency can be equal to unity even for a small optical depth of the atomic system.

ON THERMALIZATION IN GAMMA-RAY BURST JETS AND THE PEAK ENERGIES OF PHOTOSPHERIC SPECTRA

The low energy spectral slopes of the prompt emission of most gamma-ray bursts (GRBs) are difficult to reconcile with radiatively efficient optically thin emission models irrespective of the radiation mechanism. An alternative is to ascribe the radiation around the spectral peak to a thermalization process occurring well inside the Thomson photosphere. This quasi-thermal spectrum can evolve into the observed non-thermal shape by additional energy release at moderate to small Thomson optical depths, which can readily give rise to the hard spectral tail. The position of the spectral peak is determined by the temperature and Lorentz factor of the flow in the termalization zone, where the total number of photons carried by the jet is established. To reach thermalization, dissipation alone is not sufficient and photon generation requires an efficient emission/absorption process in addition to scattering. We perform a systematic study of all relevant photon production mechanisms searching for possible conditions in which thermalization can take place. We find that a significant fraction of the available energy should be dissipated at intermediate radii, $\sim 10^{10}$-- a few$\times 10^{11}$ cm and the flow there should be relatively slow: the bulk Lorentz factor could not exceed a few tens for all but the most luminous bursts with the highest $\Epk$-s. The least restrictive constraint for successful thermalization, $\Gamma\lesssim 20$, is obtained if synchrotron emission acts as the photon source. This requires, however, a non-thermal acceleration deep below the Thomson photosphere transferring a significant fraction of the flow energy to relativistic electrons with Lorentz factors between 10 and 100. Other processes require bulk flow Lorentz factors of order of a few for typical bursts. We examine the implications of these results to different GRB photospheric emission models.

Ultrafast transverse thermoelectric response in c‐axis inclined epitaxial La0.5Sr0.5CoO3 thin films

AbstractUltrafast transverse thermoelectric voltage response has been observed in c‐axis inclined epitaxial La0.5Sr0.5CoO3thin films. Voltage signals with the rise time of 7 ns have been detected under the irradiation of pulse laser with duration of 28 ns. A concept, named response rate ratio, has been proposed to evaluate the intrinsic response rate, and this ratio in La0.5Sr0.5CoO3is smaller than that in other reported materials. The low resistivity is thought to be responsible for the ultrafast response, as low resistivity induces small optical penetration depth, and response time has a monotonous increasing relationship with this depth.magnified imageTransverse thermoelectric voltage with the rise time of 7 ns in La0.5Sr0.5CoO3thin film.(© 2013 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

在非对称光腔中利用CRIB技术进行光量子态存储

We propose an efficient quantum memory scheme with controlled reversible inhomogeneous broadening in an optical cavity.When the absorption per cavity round trip exactly matches the transmission of the coupling mirror(impedance matching condition),we show that the input light field can be completely absorbed by an inhomogeneous broadening atomic ensemble.From calculation we obtain that either a forward or a backward retrieval process the quantum memory efficiency can reach unity even for a small optical depth of the atomic system.If there is no cavity,100% efficiency is obtained only for extremely large optical depth.A high efficiency for the quantum memory at a small optical depth is easily achieved in practice.Therefore our proposal offers promising possibilities for the practical realization of high quantum memory efficiency.

Aerosols in central California: Unexpectedly large contribution of coarse mode to aerosol radiative forcing

The majority of previous studies dealing with effect of coarse mode aerosols (supermicron) on the radiation budget have focused primarily on regions where total aerosol loadings are substantial. We reexamine this effect for a relatively clean area using a unique 1‐month dataset collected during the recent Carbonaceous Aerosol and Radiative Effects Study (CARES, June 2010) in the central California region near Sacramento. Here we define “clean” as aerosol optical depths less than 0.1 at 0.5μm. We demonstrate that coarse mode particles contributed substantially (more than 50%) and frequently (up to 85% of time) to the total aerosol volume during this study. In contrast to conventional expectations that the radiative impact of coarse mode aerosols should be small for clean regions, we find that neglecting large particles may lead to significant overestimation, up to 45%, of direct aerosol radiative forcing despite very small aerosol optical depths. Our findings highlight the potential for substantial impacts of coarse mode aerosols on radiative properties over clean areas and the need for more explicit inclusion of coarse mode aerosols in climate‐related observational studies.

The water-ice librational band: radiative transfer model for AFGL 961

On the basis of radiative transfer models we have investigated the librational band of amorphous water-ice (H2O-ice), which might be expected to occur in the spectra of young stellar objects with dust shells, producing an absorption feature at about 13 μm. Models have been run with only bare silicate grains present, together with models including water-ice mantles. We find that the 13-μm feature may be substantially explained without the use of water-ice if laboratory silicates are employed and the 10-μm silicate feature is in self-absorption, a phenomenon which may arise for shells of intermediate optical depth (). The self-absorption process may lead to a variation in the shape of the local minimum between the 10- and 20-μm silicate features, the local minimum occurring at about 13 μm, close to the position of the water-ice librational band. This local minimum could easily be mistaken for the librational water-ice band itself. Nevertheless, the librational band undoubtedly does contribute to the finer details of the shape of the 13-μm absorption feature. We find that for shells of small optical depth the water-ice feature is not distinguishable from the local minimum between the 10- and 20-μm silicate features, while for shells of large optical depth it is washed out completely, so in neither of these two extreme cases should the feature be readily observable. The fact that relatively few objects exist in the intermediate state where the silicate feature is in self-absorption may account for the extremely small number of cases in which the librational band apparently appears as a distinct feature. Radiative transfer models have been produced for the massive pre-main-sequence object AFGL 961E, with the aim of investigating whether the 13-μm feature in its spectrum is due to the librational band of water-ice. Clearly there is water-ice somewhere in the line of sight to AFGL 961, as evidenced by the strong 3.1-μm water-ice band, and the case for the 13-μm feature being due to the librational band is uniquely strong amongst all astronomical objects. However, we present evidence that the grains may have properties similar to laboratory silicates rather than astronomical silicates and that the feature may largely be a manifestation of the 10-μm silicate feature being in self-absorption, with amorphous water-ice being a minor contributor. Neither bare astronomical silicates nor water-ice mantled astronomical silicates will reproduce the 13-μm feature.

Photon absorption in a hybrid slurry photocatalytic reactor: Assessment of differential approximations

AbstractRadiative transfer inside a slurry photocatalytic reactor with hybrid illumination from both solar radiation and lamps is examined. The local volumetric rate of photon absorption is evaluated. For this purpose, the P1 and the modified differential approximations (MDAs) are used, and results compared to a solution by the Monte Carlo method. It is found that significant differences may arise between the predictions of the above approximations and the exact results provided by the Monte Carlo simulations. The P1 approximation is very inaccurate near to the radiation entrance for the partially collimated solar radiation, although it improves, as optical depth increases. As expected, the MDA improves the results near to the boundary. Surprisingly, it turns out to be much worse than the P1 approximation at medium and large optical depths. In the case of lamp irradiation, the behavior of the MDA is the opposite; it works better at small optical depths. © 2011 American Institute of Chemical Engineers AIChE J, 58: 3256–3265, 2012