A Limiting Approach for the Evaluation of Geometric Mean Transmittance in a Multidimensional Absorbing and Isotropically Scattering Medium

1984 ◽  
Vol 106 (2) ◽  
pp. 441-447 ◽  
Author(s):  
W. W. Yuen
Keyword(s):  
Geometric Mean ◽  
Accurate Estimate ◽  
Exact Expression ◽  
Integral Function ◽  
Scattering Medium ◽  
Exponential Integral ◽  
Physical Consideration ◽  
Upper Limits ◽  
Lower Limits ◽  
Transmittance Factor

The calculation of the geometric-mean transmittance factor between areas with an intervening absorbing and isotropically scattering medium is considered. While an exact expression for the factor is shown to be quite complicated, the upper and lower limits of the factor can be readily generated from physical consideration. Integral expressions for successively increasing (decreasing) values of the lower (upper) limits are obtained. For two-dimensional systems, these expressions are reduced to integrals involving Sn (x), a class of exponential integral function that has been tabulated in a previous work. Utilizing the kernel substitution technique, these integrals are evaluated analytically in closed form for some selected geometries. For cases with small optical thickness and large scattering albedo, both limits are shown to converge relatively slowly to the actual transmittance factor. But the decreasing difference between the two limits provides accurate estimate of the geometric-mean transmittance factor. Based on these results, some interesting conclusions concerning the effect of scattering on multidimensional radiative transmission are established.

scholarly journals Upper limits on the water vapour content of the β Pictoris debris disk

2019 ◽  
Vol 628 ◽  
pp. A127 ◽  
Author(s):  
M. Cavallius ◽  
G. Cataldi ◽  
A. Brandeker ◽  
G. Olofsson ◽  
B. Larsson ◽  
...  
Keyword(s):  
Solar System ◽  
Far Infrared ◽  
Kinetic Temperature ◽  
Line Flux ◽  
Water Vapour Content ◽  
Water Ice ◽  
Upper Limit ◽  
Upper Limits ◽  
Debris Disk ◽  
Lower Limits

Context. The debris disk surrounding β Pictoris has been observed with ALMA to contain a belt of CO gas with a distinct peak at ~85 au. This CO clump is thought to be the result of a region of enhanced density of solids that collide and release CO through vaporisation. The parent bodies are thought to be comparable to solar system comets, in which CO is trapped inside a water ice matrix. Aims. Since H2O should be released along with CO, we aim to put an upper limit on the H2O gas mass in the disk of β Pictoris. Methods. We used archival data from the Heterodyne Instrument for the Far-Infrared (HIFI) aboard the Herschel Space Observatory to study the ortho-H2O 110–101 emission line. The line is undetected. Using a python implementation of the radiative transfer code RADEX, we converted upper limits on the line flux to H2O gas masses. The resulting lower limits on the CO/H2O mass ratio are compared to the composition of solar system comets. Results. Depending on the assumed gas spatial distribution, we find a 95% upper limit on the ortho-H2O line flux of 7.5 × 10−20 W m−2 or 1.2 × 10−19 W m−2. These translate into an upper limit on the H2O mass of 7.4 × 1016–1.1 × 1018 kg depending on both the electron density and gas kinetic temperature. The range of derived gas-phase CO/H2O ratios is marginally consistent with low-ratio solar system comets.

Vertical zonation of the hydroid Dynamena crisioides (Hydrozoa, Sertulariidae) in a mangrove ecosystem at Twin Cays, Belize

1991 ◽  
Vol 69 (12) ◽  
pp. 2993-2999 ◽  
Author(s):  
Dale R. Calder
Keyword(s):  
Intertidal Zone ◽  
Prime Factor ◽  
Good Condition ◽  
Mangrove Ecosystem ◽  
Lower Half ◽  
Upper Limits ◽  
Shallow Subtidal ◽  
Lower Limits ◽  
Transplantation Experiments ◽  
Number Of Branches

Dynamena crisioides is generally restricted to lower intertidal and shallow subtidal habitats along tropical and subtropical coasts. The hydroid was abundant during this study on prop roots of red mangroves (Rhizophora mangle) at Twin Cays, Belize, where it ranged vertically from 1.1 m below mean tide level (MTL) to 2–3 cm above MTL. The estimated percentage cover of the hydroid was maximal in the lower half of the intertidal zone (MTL and 0.1 m below MTL) and declined rapidly above and below that level. Colony height and number of branches per colony were also consistently highest in hydroids from the lower half of the intertidal zone. The percentage of colonies with gonophores was lowest at the lower and upper vertical limits of the species. Field observations and transplantation experiments suggest that the lower limits of D. crisioides are established by competition for space (with algae, sponges, and ascidians), smothering (by algae, sponges, ascidians, and other hydroids), and predation (by fishes). Desiccation was considered the prime factor determining the upper limits of the hydroid. Colonies transplanted to the supratidal zone were in good condition after 24 h, alive but in poor condition after 48 h, and dead after 72 and 96 h of exposure.

scholarly journals On a family of logarithmic and exponential integrals occurring in probability and reliability theory

1994 ◽  
Vol 35 (4) ◽  
pp. 469-478 ◽  
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Closed Form ◽  
Recurrence Relation ◽  
Reliability Theory ◽  
Integral Function ◽  
Integral Representations ◽  
Exponential Integral ◽  
Error Functions ◽  
Special Cases ◽  
Image Position ◽  
Form Evaluation

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

scholarly journals Faint Photometry of Edge-on Spiral Galaxies: A Search for Massive Halos

1987 ◽  
Vol 117 ◽  
pp. 425-425
Author(s):  
Michael F. Skrutskie ◽  
Mark A. Shure ◽  
Steven Beckwith
Keyword(s):  
Spiral Galaxies ◽  
Hydrogen Burning ◽  
Dwarf Stars ◽  
Color Changes ◽  
Upper Limits ◽  
Massive Halos ◽  
Halo Masses ◽  
Stellar Masses ◽  
M Dwarf Stars ◽  
Lower Limits

Upper limits have been set to the luminosity from the massive halos of three late-type edge-on spiral galaxies: NGC 2683 (Sb), NGC 4244 (Scd), and NGC 5907 (Sc). The limits resulted from simultaneous photometry in the visual (V) and 2.2μm (K) photometric bands which is sensitive to both luminosity and color changes along the minor axes of the three galaxies. The 3σ lower limits to the mass-to-light ratios for the halo of NGC 5907 are the largest ever recorded: M/Lv > 2000 and M/Lk > 64 in solar units. Since K band M/L for M-dwarf stars lying just above the hydrogen-burning limit is about 35, the results virtually eliminate the possibility that hydrogen-burning stars comprise more than a fraction of the halo masses. If the halos contain a more realistic spectrum of stellar masses, for example Population II, the visual band measurements imply that these stars account for less than one percent of the halo mass. Similar limits were obtained for NGC 4244 and NGC 2683. Variations of the V-K colors along and perpendicular to the disks show no sign of population changes toward redder objects at large galactocentric radii.

scholarly journals VIII. On Abel’s Theorem and Abelian functions

1883 ◽  
Vol 174 ◽  
pp. 323-368
Keyword(s):  
Integral Function ◽  
Upper Limits ◽  
Abelian Functions ◽  
And Function

The present paper is divided into two sections. The object of Section I. is to obtain an expression for an integral more general than, but intimately connected with, that occurring in Abel’s theorem. The latter, as enunciated by Professor Rowe in his memoir in the Phil. Trans., 1881, is as follows:—If χ ( x, y ) = 0 be a rational algebraical equation between x and y , then an expression can always be found for Σ∫U dx / f ( x ) ∂χ / ∂y where f ( x ) is a function of x only, U a rational algebraical integral function of x and y , and the upper limits of the series of integrals are the roots of the eliminant with regard to y of χ ( x, y ) = 0 and function θ ( x, y ).

Lower limits and upper limits for tails of random sums supported on

2010 ◽  
Vol 80 (13-14) ◽  
pp. 1111-1120 ◽  
Author(s):  
Changjun Yu ◽  
Yuebao Wang ◽  
Zhaolei Cui
Keyword(s):  
Random Sums ◽  
Upper Limits ◽  
Lower Limits
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