1388:(LTE). It is important to note that local equilibrium may apply only to a certain subset of particles in the system. For example, LTE is usually applied only to massive particles. In a radiating gas, the photons being emitted and absorbed by the gas do not need to be in a thermodynamic equilibrium with each other or with the massive particles of the gas in order for LTE to exist.
67:). For a very small area element in the radiation field, there can be electromagnetic radiation passing in both senses in every spatial direction through it. In radiometric terms, the passage can be completely characterized by the amount of energy radiated in each of the two senses in each spatial direction, per unit time, per unit area of surface of sourcing passage, per unit
683:
1750:
903:
Solutions to the equation of radiative transfer form an enormous body of work. The differences however, are essentially due to the various forms for the emission and absorption coefficients. If scattering is ignored, then a general steady state solution in terms of the emission and absorption
469:
42:
describes these interactions mathematically. Equations of radiative transfer have application in a wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more
1113:
1395:). The radiation field is not, however in equilibrium and is being entirely driven by the presence of the massive particles. For a medium in LTE, the emission coefficient and absorption coefficient are functions of temperature and density only, and are related by:
1374:
2928:
462:
The equation of radiative transfer simply says that as a beam of radiation travels, it loses energy to absorption, gains energy by emission processes, and redistributes energy by scattering. The differential form of the equation for radiative transfer is:
2563:
1767:. The two-stream approximation assumes that the intensity is constant with angle in the upward hemisphere, with a different constant value in the downward hemisphere. The Eddington approximation instead assumes that the intensity is a linear function of
397:
2321:. Higher order versions of the Eddington approximation also exist, and consist of more complicated linear relations of the intensity moments. This extra equation can be used as a closure relation for the truncated system of moments.
2265:
2967:
2167:
2770:
678:{\displaystyle {\frac {1}{c}}{\frac {\partial }{\partial t}}I_{\nu }+{\hat {\Omega }}\cdot \nabla I_{\nu }+(k_{\nu ,s}+k_{\nu ,a})\rho I_{\nu }=j_{\nu }\rho +{\frac {1}{4\pi }}k_{\nu ,s}\rho \int _{\Omega }I_{\nu }d\Omega }
1972:. The Eddington approximation can be used to obtain the spectral radiance in a "plane-parallel" medium (one in which properties only vary in the perpendicular direction) with isotropic frequency-independent scattering.
1514:
893:
2655:
2076:
453:
makes with a normal to the area element. The units of the spectral radiance are seen to be energy/time/area/solid angle/frequency. In MKS units this would be W·m·sr·Hz (watts per square-metre-steradian-hertz).
1459:
910:
2933:
This equation shows how the effective optical depth in scattering-dominated systems may be significantly different from that given by the scattering opacity if the absorptive opacity is small.
1875:
1238:
1962:
1172:
2801:
2434:
451:
227:
2319:
286:
1797:
2426:
1391:
In this situation, the absorbing/emitting medium consists of massive particles which are locally in equilibrium with each other, and therefore have a definable temperature (
278:
149:
1498:
799:
766:
2376:
2349:
733:
196:
103:
420:
248:
1230:
1203:
819:
2678:
1993:
1918:
173:
127:
2793:
2396:
1898:
706:
2173:
2082:
2972:
43:
realistic media, with complex multiple scattering effects, numerical methods are required. The present article is largely focused on the condition of
26:) is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by
2686:
1745:{\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau _{\nu }(s_{0},s)}+\int _{s_{0}}^{s}B_{\nu }(T(s'))\alpha _{\nu }(s')e^{-\tau _{\nu }(s',s)}\,ds'}
1755:
Knowing the temperature profile and the density profile of the medium is sufficient to calculate a solution to the equation of radiative transfer.
824:
2574:
2952:
2001:
1401:
2947:
1108:{\displaystyle I_{\nu }(s)=I_{\nu }(s_{0})e^{-\tau _{\nu }(s_{0},s)}+\int _{s_{0}}^{s}j_{\nu }(s')e^{-\tau _{\nu }(s',s)}\,ds'}
27:
3236:
3210:
3187:
3168:
3146:
3125:
3106:
3039:
3014:
31:
3286:
3136:
1369:{\displaystyle \tau _{\nu }(s_{1},s_{2})\ {\stackrel {\mathrm {def} }{=}}\ \int _{s_{1}}^{s_{2}}\alpha _{\nu }(s)\,ds}
3262:
3076:
1805:
3065:
Theory of
Stellar Atmospheres, An Introduction to Astrophysical Non-equilibrium Quantitative Spectroscopic Analysis
2923:{\displaystyle {\frac {d^{2}J_{\nu }}{dz^{2}}}=3\alpha _{\nu }(\alpha _{\nu }+\sigma _{\nu })(J_{\nu }-B_{\nu })}
1965:
1923:
1385:
2558:{\displaystyle \mu {\frac {dI_{\nu }}{dz}}=-\alpha _{\nu }(I_{\nu }-B_{\nu })+\sigma _{\nu }(J_{\nu }-I_{\nu })}
3291:
1121:
392:{\displaystyle dE_{\nu }=I_{\nu }(\mathbf {r} ,{\hat {\mathbf {n} }},t)\cos \theta \ d\nu \,da\,d\Omega \,dt}
1392:
425:
201:
63:
2273:
1384:
A particularly useful simplification of the equation of radiative transfer occurs under the conditions of
1969:
3254:
3202:
3068:
1770:
2977:
2404:
1900:
is the normal direction to the slab-like medium. Note that expressing angular integrals in terms of
1764:
253:
132:
1467:
3098:
3006:
771:
738:
2401:
The radiative transfer through an isotropically scattering medium with scattering coefficient
2354:
2327:
711:
178:
81:
3246:
3118:
Radiative
Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures
3032:
Radiative
Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures
405:
44:
232:
3228:
1208:
1181:
804:
72:
2968:
Radiative transfer equation and diffusion theory for photon transport in biological tissue
2663:
1978:
1903:
154:
108:
8:
3281:
2260:{\displaystyle K_{\nu }={\frac {1}{2}}\int _{-1}^{1}\mu ^{2}I_{\nu }d\mu ={\frac {a}{3}}}
3091:
2999:
2795:
allows the two above equations to be combined to form the radiative diffusion equation
2778:
2381:
1883:
691:
3258:
3232:
3206:
3183:
3164:
3142:
3121:
3102:
3072:
3035:
3010:
2162:{\displaystyle H_{\nu }={\frac {1}{2}}\int _{-1}^{1}\mu I_{\nu }d\mu ={\frac {b}{3}}}
57:
3086:
2942:
3156:
3060:
2962:
2765:{\displaystyle {\frac {dK_{\nu }}{dz}}=-(\alpha _{\nu }+\sigma _{\nu })H_{\nu }}
3275:
2957:
1175:
3220:
888:{\displaystyle {\frac {1}{4\pi }}k_{\nu ,s}\int _{\Omega }I_{\nu }d\Omega }
71:
of reception at a distance, per unit wavelength interval being considered (
2775:
Substituting in the closure relation, and differentiating with respect to
1975:
Extracting the first few moments of the spectral radiance with respect to
895:
term represents radiation scattered from other directions onto a surface.
2650:{\displaystyle {\frac {dH_{\nu }}{dz}}=\alpha _{\nu }(B_{\nu }-J_{\nu })}
68:
3225:
Fundamentals of
Atmospheric Radiation: an Introduction with 400 Problems
1501:
55:
The fundamental quantity that describes a field of radiation is called
35:
3155:
2071:{\displaystyle J_{\nu }={\frac {1}{2}}\int _{-1}^{1}I_{\nu }d\mu =a}
898:
3177:
2324:
Note that the first two moments have simple physical meanings.
1508:. The solution to the equation of radiative transfer is then:
1454:{\displaystyle {\frac {j_{\nu }}{\alpha _{\nu }}}=B_{\nu }(T)}
3085:
3058:
2270:
Thus the
Eddington approximation is equivalent to setting
61:
in radiometric terms (in other fields it is often called
16:
Energy transfer in the form of electromagnetic radiation
3196:
2804:
2781:
2689:
2666:
2577:
2437:
2407:
2384:
2357:
2330:
2276:
2176:
2085:
2004:
1981:
1926:
1906:
1886:
1808:
1773:
1517:
1470:
1404:
1241:
1211:
1184:
1124:
913:
827:
807:
774:
741:
714:
694:
472:
457:
428:
408:
289:
256:
235:
204:
181:
157:
135:
111:
84:
105:, the energy flowing across an area element of area
3245:
3115:
3090:
3029:
2998:
2922:
2787:
2764:
2672:
2649:
2557:
2420:
2390:
2370:
2343:
2313:
2259:
2161:
2070:
1987:
1956:
1912:
1892:
1869:
1791:
1744:
1492:
1453:
1379:
1368:
1224:
1197:
1166:
1107:
887:
813:
793:
760:
727:
700:
677:
445:
414:
391:
272:
242:
221:
190:
167:
143:
121:
97:
3138:A First Course in Atmospheric Radiation (2nd Ed.)
1763:The Eddington approximation is distinct from the
3273:
2996:
2973:Schwarzschild's equation for radiative transfer
2428:at local thermodynamic equilibrium is given by
899:Solutions to the equation of radiative transfer
3199:Radiative Transfer in the Atmosphere and Ocean
1870:{\displaystyle I_{\nu }(\mu ,z)=a(z)+\mu b(z)}
1758:
2680:, and then integrating over all angles gives
3178:George B. Rybicki; Alan P. Lightman (1985).
2990:
2953:List of atmospheric radiative transfer codes
422:is the angle that the unit direction vector
3134:
2351:is the isotropic intensity at a point, and
1957:{\displaystyle d\mu =-\sin \theta d\theta }
3141:. Sundog Publishing (Madison, Wisconsin).
3219:
1730:
1359:
1167:{\displaystyle \tau _{\nu }(s_{1},s_{2})}
1093:
382:
375:
368:
269:
239:
164:
118:
3274:
3197:G. E. Thomas & K. Stamnes (1999).
3161:Foundations of Radiation Hydrodynamics
3023:
2378:is the flux through that point in the
3120:. A. Deepak Publishing. p. 583.
3034:. A. Deepak Publishing. p. 583.
446:{\displaystyle {\hat {\mathbf {n} }}}
222:{\displaystyle {\hat {\mathbf {n} }}}
2948:Kirchhoff's law of thermal radiation
2314:{\displaystyle K_{\nu }=1/3J_{\nu }}
3180:Radiative Processes in Astrophysics
2568:Integrating over all angles yields
78:In terms of the spectral radiance,
13:
3097:. Dover Publications Inc. p.
3052:
3005:. Dover Publications Inc. p.
1300:
1297:
1294:
882:
864:
672:
654:
526:
514:
489:
485:
458:The equation of radiative transfer
379:
185:
14:
3303:
3159:; Barbara Weibel-Mihalas (1984).
1792:{\displaystyle \mu =\cos \theta }
1504:spectral radiance at temperature
75:will be ignored for the moment).
1178:of the medium between positions
433:
331:
320:
209:
137:
3251:Principles of Planetary Climate
1386:local thermodynamic equilibrium
1380:Local thermodynamic equilibrium
2917:
2891:
2888:
2862:
2749:
2723:
2644:
2618:
2552:
2526:
2510:
2484:
2421:{\displaystyle \sigma _{\nu }}
1864:
1858:
1846:
1840:
1831:
1819:
1725:
1708:
1687:
1676:
1663:
1660:
1649:
1643:
1603:
1584:
1563:
1550:
1534:
1528:
1487:
1481:
1448:
1442:
1356:
1350:
1278:
1252:
1161:
1135:
1088:
1071:
1050:
1039:
999:
980:
959:
946:
930:
924:
580:
542:
517:
437:
347:
335:
316:
213:
50:
40:equation of radiative transfer
1:
2983:
904:coefficients may be written:
735:is the emission coefficient,
1393:Zeroth Law of Thermodynamics
821:is the mass density and the
273:{\displaystyle \nu +d\nu \,}
144:{\displaystyle \mathbf {r} }
7:
3163:. Dover Publications, Inc.
3116:Jacqueline Lenoble (1985).
3030:Jacqueline Lenoble (1985).
2936:
1759:The Eddington approximation
1493:{\displaystyle B_{\nu }(T)}
801:is the absorption opacity,
768:is the scattering opacity,
10:
3308:
3255:Cambridge University Press
3203:Cambridge University Press
3087:Subrahmanyan Chandrasekhar
3069:Princeton University Press
1920:simplifies things because
794:{\displaystyle k_{\nu ,a}}
761:{\displaystyle k_{\nu ,s}}
229:in the frequency interval
3287:Electromagnetic radiation
2997:S. Chandrasekhar (1960).
2978:Vector radiative transfer
2371:{\displaystyle H_{\nu }}
2344:{\displaystyle J_{\nu }}
1765:two-stream approximation
728:{\displaystyle j_{\nu }}
191:{\displaystyle d\Omega }
98:{\displaystyle I_{\nu }}
708:is the speed of light,
415:{\displaystyle \theta }
3182:. Wiley-Interscience.
2924:
2789:
2766:
2674:
2651:
2559:
2422:
2392:
2372:
2345:
2315:
2261:
2163:
2072:
1989:
1958:
1914:
1894:
1871:
1793:
1746:
1494:
1455:
1370:
1226:
1199:
1168:
1109:
889:
815:
795:
762:
729:
702:
679:
447:
416:
393:
274:
244:
243:{\displaystyle \nu \,}
223:
192:
169:
145:
123:
99:
3292:Atmospheric radiation
3229:John Wiley & Sons
2925:
2790:
2767:
2675:
2652:
2560:
2423:
2393:
2373:
2346:
2316:
2262:
2164:
2073:
1990:
1970:spherical coordinates
1959:
1915:
1895:
1872:
1794:
1747:
1495:
1456:
1371:
1227:
1225:{\displaystyle s_{2}}
1200:
1198:{\displaystyle s_{1}}
1169:
1110:
890:
816:
814:{\displaystyle \rho }
796:
763:
730:
703:
680:
448:
417:
394:
275:
245:
224:
193:
170:
146:
124:
100:
45:radiative equilibrium
3135:Grant Petty (2006).
2802:
2779:
2687:
2673:{\displaystyle \mu }
2664:
2575:
2435:
2405:
2382:
2355:
2328:
2274:
2174:
2083:
2002:
1988:{\displaystyle \mu }
1979:
1924:
1913:{\displaystyle \mu }
1904:
1884:
1806:
1771:
1515:
1468:
1402:
1239:
1209:
1182:
1122:
911:
825:
805:
772:
739:
712:
692:
470:
426:
406:
287:
254:
233:
202:
198:about the direction
179:
168:{\displaystyle dt\,}
155:
133:
122:{\displaystyle da\,}
109:
82:
3247:R. T. Pierrehumbert
2217:
2126:
2045:
1632:
1339:
1028:
175:in the solid angle
24:radiation transport
3093:Radiative Transfer
3001:Radiative Transfer
2920:
2785:
2762:
2670:
2660:Premultiplying by
2647:
2555:
2418:
2388:
2368:
2341:
2311:
2257:
2200:
2159:
2109:
2068:
2028:
1985:
1954:
1910:
1890:
1867:
1789:
1742:
1611:
1490:
1451:
1366:
1311:
1222:
1195:
1164:
1105:
1007:
885:
811:
791:
758:
725:
698:
675:
443:
412:
389:
270:
240:
219:
188:
165:
141:
119:
95:
64:specific intensity
20:Radiative transfer
3238:978-3-527-40503-9
3212:978-0-521-40124-1
3189:978-0-471-82759-7
3170:978-0-486-40925-2
3148:978-0-9729033-1-8
3127:978-0-12-451451-5
3108:978-0-486-60590-6
3041:978-0-12-451451-5
3016:978-0-486-60590-6
2844:
2788:{\displaystyle z}
2715:
2603:
2466:
2391:{\displaystyle z}
2255:
2198:
2157:
2107:
2026:
1893:{\displaystyle z}
1427:
1310:
1305:
1283:
841:
701:{\displaystyle c}
628:
520:
496:
481:
440:
361:
338:
216:
58:spectral radiance
3299:
3268:
3242:
3216:
3193:
3174:
3152:
3131:
3112:
3096:
3082:
3046:
3045:
3027:
3021:
3020:
3004:
2994:
2943:Beer-Lambert law
2929:
2927:
2926:
2921:
2916:
2915:
2903:
2902:
2887:
2886:
2874:
2873:
2861:
2860:
2845:
2843:
2842:
2841:
2828:
2827:
2826:
2817:
2816:
2806:
2794:
2792:
2791:
2786:
2771:
2769:
2768:
2763:
2761:
2760:
2748:
2747:
2735:
2734:
2716:
2714:
2706:
2705:
2704:
2691:
2679:
2677:
2676:
2671:
2656:
2654:
2653:
2648:
2643:
2642:
2630:
2629:
2617:
2616:
2604:
2602:
2594:
2593:
2592:
2579:
2564:
2562:
2561:
2556:
2551:
2550:
2538:
2537:
2525:
2524:
2509:
2508:
2496:
2495:
2483:
2482:
2467:
2465:
2457:
2456:
2455:
2442:
2427:
2425:
2424:
2419:
2417:
2416:
2397:
2395:
2394:
2389:
2377:
2375:
2374:
2369:
2367:
2366:
2350:
2348:
2347:
2342:
2340:
2339:
2320:
2318:
2317:
2312:
2310:
2309:
2297:
2286:
2285:
2266:
2264:
2263:
2258:
2256:
2248:
2237:
2236:
2227:
2226:
2216:
2211:
2199:
2191:
2186:
2185:
2168:
2166:
2165:
2160:
2158:
2150:
2139:
2138:
2125:
2120:
2108:
2100:
2095:
2094:
2077:
2075:
2074:
2069:
2055:
2054:
2044:
2039:
2027:
2019:
2014:
2013:
1994:
1992:
1991:
1986:
1968:of integrals in
1963:
1961:
1960:
1955:
1919:
1917:
1916:
1911:
1899:
1897:
1896:
1891:
1876:
1874:
1873:
1868:
1818:
1817:
1798:
1796:
1795:
1790:
1751:
1749:
1748:
1743:
1741:
1729:
1728:
1718:
1707:
1706:
1686:
1675:
1674:
1659:
1642:
1641:
1631:
1626:
1625:
1624:
1607:
1606:
1596:
1595:
1583:
1582:
1562:
1561:
1549:
1548:
1527:
1526:
1499:
1497:
1496:
1491:
1480:
1479:
1460:
1458:
1457:
1452:
1441:
1440:
1428:
1426:
1425:
1416:
1415:
1406:
1375:
1373:
1372:
1367:
1349:
1348:
1338:
1337:
1336:
1326:
1325:
1324:
1308:
1307:
1306:
1304:
1303:
1291:
1286:
1281:
1277:
1276:
1264:
1263:
1251:
1250:
1231:
1229:
1228:
1223:
1221:
1220:
1204:
1202:
1201:
1196:
1194:
1193:
1173:
1171:
1170:
1165:
1160:
1159:
1147:
1146:
1134:
1133:
1114:
1112:
1111:
1106:
1104:
1092:
1091:
1081:
1070:
1069:
1049:
1038:
1037:
1027:
1022:
1021:
1020:
1003:
1002:
992:
991:
979:
978:
958:
957:
945:
944:
923:
922:
894:
892:
891:
886:
878:
877:
868:
867:
858:
857:
842:
840:
829:
820:
818:
817:
812:
800:
798:
797:
792:
790:
789:
767:
765:
764:
759:
757:
756:
734:
732:
731:
726:
724:
723:
707:
705:
704:
699:
684:
682:
681:
676:
668:
667:
658:
657:
645:
644:
629:
627:
616:
608:
607:
595:
594:
579:
578:
560:
559:
538:
537:
522:
521:
513:
507:
506:
497:
495:
484:
482:
474:
452:
450:
449:
444:
442:
441:
436:
431:
421:
419:
418:
413:
398:
396:
395:
390:
359:
340:
339:
334:
329:
323:
315:
314:
302:
301:
279:
277:
276:
271:
249:
247:
246:
241:
228:
226:
225:
220:
218:
217:
212:
207:
197:
195:
194:
189:
174:
172:
171:
166:
150:
148:
147:
142:
140:
128:
126:
125:
120:
104:
102:
101:
96:
94:
93:
3307:
3306:
3302:
3301:
3300:
3298:
3297:
3296:
3272:
3271:
3265:
3239:
3213:
3190:
3171:
3157:Dimitri Mihalas
3149:
3128:
3109:
3079:
3071:. p. 944.
3061:Dimitri Mihalas
3055:
3053:Further reading
3050:
3049:
3042:
3028:
3024:
3017:
2995:
2991:
2986:
2939:
2911:
2907:
2898:
2894:
2882:
2878:
2869:
2865:
2856:
2852:
2837:
2833:
2829:
2822:
2818:
2812:
2808:
2807:
2805:
2803:
2800:
2799:
2780:
2777:
2776:
2756:
2752:
2743:
2739:
2730:
2726:
2707:
2700:
2696:
2692:
2690:
2688:
2685:
2684:
2665:
2662:
2661:
2638:
2634:
2625:
2621:
2612:
2608:
2595:
2588:
2584:
2580:
2578:
2576:
2573:
2572:
2546:
2542:
2533:
2529:
2520:
2516:
2504:
2500:
2491:
2487:
2478:
2474:
2458:
2451:
2447:
2443:
2441:
2436:
2433:
2432:
2412:
2408:
2406:
2403:
2402:
2383:
2380:
2379:
2362:
2358:
2356:
2353:
2352:
2335:
2331:
2329:
2326:
2325:
2305:
2301:
2293:
2281:
2277:
2275:
2272:
2271:
2247:
2232:
2228:
2222:
2218:
2212:
2204:
2190:
2181:
2177:
2175:
2172:
2171:
2149:
2134:
2130:
2121:
2113:
2099:
2090:
2086:
2084:
2081:
2080:
2050:
2046:
2040:
2032:
2018:
2009:
2005:
2003:
2000:
1999:
1980:
1977:
1976:
1964:appears in the
1925:
1922:
1921:
1905:
1902:
1901:
1885:
1882:
1881:
1813:
1809:
1807:
1804:
1803:
1772:
1769:
1768:
1761:
1734:
1711:
1702:
1698:
1694:
1690:
1679:
1670:
1666:
1652:
1637:
1633:
1627:
1620:
1616:
1615:
1591:
1587:
1578:
1574:
1570:
1566:
1557:
1553:
1544:
1540:
1522:
1518:
1516:
1513:
1512:
1475:
1471:
1469:
1466:
1465:
1436:
1432:
1421:
1417:
1411:
1407:
1405:
1403:
1400:
1399:
1382:
1344:
1340:
1332:
1328:
1327:
1320:
1316:
1315:
1293:
1292:
1287:
1285:
1284:
1272:
1268:
1259:
1255:
1246:
1242:
1240:
1237:
1236:
1216:
1212:
1210:
1207:
1206:
1189:
1185:
1183:
1180:
1179:
1155:
1151:
1142:
1138:
1129:
1125:
1123:
1120:
1119:
1097:
1074:
1065:
1061:
1057:
1053:
1042:
1033:
1029:
1023:
1016:
1012:
1011:
987:
983:
974:
970:
966:
962:
953:
949:
940:
936:
918:
914:
912:
909:
908:
901:
873:
869:
863:
859:
847:
843:
833:
828:
826:
823:
822:
806:
803:
802:
779:
775:
773:
770:
769:
746:
742:
740:
737:
736:
719:
715:
713:
710:
709:
693:
690:
689:
663:
659:
653:
649:
634:
630:
620:
615:
603:
599:
590:
586:
568:
564:
549:
545:
533:
529:
512:
511:
502:
498:
488:
483:
473:
471:
468:
467:
460:
432:
430:
429:
427:
424:
423:
407:
404:
403:
330:
328:
327:
319:
310:
306:
297:
293:
288:
285:
284:
255:
252:
251:
234:
231:
230:
208:
206:
205:
203:
200:
199:
180:
177:
176:
156:
153:
152:
136:
134:
131:
130:
110:
107:
106:
89:
85:
83:
80:
79:
53:
38:processes. The
17:
12:
11:
5:
3305:
3295:
3294:
3289:
3284:
3270:
3269:
3263:
3243:
3237:
3217:
3211:
3194:
3188:
3175:
3169:
3153:
3147:
3132:
3126:
3113:
3107:
3083:
3077:
3054:
3051:
3048:
3047:
3040:
3022:
3015:
2988:
2987:
2985:
2982:
2981:
2980:
2975:
2970:
2965:
2960:
2955:
2950:
2945:
2938:
2935:
2931:
2930:
2919:
2914:
2910:
2906:
2901:
2897:
2893:
2890:
2885:
2881:
2877:
2872:
2868:
2864:
2859:
2855:
2851:
2848:
2840:
2836:
2832:
2825:
2821:
2815:
2811:
2784:
2773:
2772:
2759:
2755:
2751:
2746:
2742:
2738:
2733:
2729:
2725:
2722:
2719:
2713:
2710:
2703:
2699:
2695:
2669:
2658:
2657:
2646:
2641:
2637:
2633:
2628:
2624:
2620:
2615:
2611:
2607:
2601:
2598:
2591:
2587:
2583:
2566:
2565:
2554:
2549:
2545:
2541:
2536:
2532:
2528:
2523:
2519:
2515:
2512:
2507:
2503:
2499:
2494:
2490:
2486:
2481:
2477:
2473:
2470:
2464:
2461:
2454:
2450:
2446:
2440:
2415:
2411:
2387:
2365:
2361:
2338:
2334:
2308:
2304:
2300:
2296:
2292:
2289:
2284:
2280:
2268:
2267:
2254:
2251:
2246:
2243:
2240:
2235:
2231:
2225:
2221:
2215:
2210:
2207:
2203:
2197:
2194:
2189:
2184:
2180:
2169:
2156:
2153:
2148:
2145:
2142:
2137:
2133:
2129:
2124:
2119:
2116:
2112:
2106:
2103:
2098:
2093:
2089:
2078:
2067:
2064:
2061:
2058:
2053:
2049:
2043:
2038:
2035:
2031:
2025:
2022:
2017:
2012:
2008:
1984:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1909:
1889:
1878:
1877:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1816:
1812:
1788:
1785:
1782:
1779:
1776:
1760:
1757:
1753:
1752:
1740:
1737:
1733:
1727:
1724:
1721:
1717:
1714:
1710:
1705:
1701:
1697:
1693:
1689:
1685:
1682:
1678:
1673:
1669:
1665:
1662:
1658:
1655:
1651:
1648:
1645:
1640:
1636:
1630:
1623:
1619:
1614:
1610:
1605:
1602:
1599:
1594:
1590:
1586:
1581:
1577:
1573:
1569:
1565:
1560:
1556:
1552:
1547:
1543:
1539:
1536:
1533:
1530:
1525:
1521:
1489:
1486:
1483:
1478:
1474:
1462:
1461:
1450:
1447:
1444:
1439:
1435:
1431:
1424:
1420:
1414:
1410:
1381:
1378:
1377:
1376:
1365:
1362:
1358:
1355:
1352:
1347:
1343:
1335:
1331:
1323:
1319:
1314:
1302:
1299:
1296:
1290:
1280:
1275:
1271:
1267:
1262:
1258:
1254:
1249:
1245:
1219:
1215:
1192:
1188:
1163:
1158:
1154:
1150:
1145:
1141:
1137:
1132:
1128:
1116:
1115:
1103:
1100:
1096:
1090:
1087:
1084:
1080:
1077:
1073:
1068:
1064:
1060:
1056:
1052:
1048:
1045:
1041:
1036:
1032:
1026:
1019:
1015:
1010:
1006:
1001:
998:
995:
990:
986:
982:
977:
973:
969:
965:
961:
956:
952:
948:
943:
939:
935:
932:
929:
926:
921:
917:
900:
897:
884:
881:
876:
872:
866:
862:
856:
853:
850:
846:
839:
836:
832:
810:
788:
785:
782:
778:
755:
752:
749:
745:
722:
718:
697:
686:
685:
674:
671:
666:
662:
656:
652:
648:
643:
640:
637:
633:
626:
623:
619:
614:
611:
606:
602:
598:
593:
589:
585:
582:
577:
574:
571:
567:
563:
558:
555:
552:
548:
544:
541:
536:
532:
528:
525:
519:
516:
510:
505:
501:
494:
491:
487:
480:
477:
459:
456:
439:
435:
411:
400:
399:
388:
385:
381:
378:
374:
371:
367:
364:
358:
355:
352:
349:
346:
343:
337:
333:
326:
322:
318:
313:
309:
305:
300:
296:
292:
268:
265:
262:
259:
238:
215:
211:
187:
184:
163:
160:
139:
117:
114:
92:
88:
52:
49:
15:
9:
6:
4:
3:
2:
3304:
3293:
3290:
3288:
3285:
3283:
3280:
3279:
3277:
3266:
3264:9780521865562
3260:
3256:
3252:
3248:
3244:
3240:
3234:
3230:
3226:
3222:
3218:
3214:
3208:
3204:
3200:
3195:
3191:
3185:
3181:
3176:
3172:
3166:
3162:
3158:
3154:
3150:
3144:
3140:
3139:
3133:
3129:
3123:
3119:
3114:
3110:
3104:
3100:
3095:
3094:
3088:
3084:
3080:
3078:9780691163291
3074:
3070:
3066:
3062:
3059:Ivan Hubeny;
3057:
3056:
3043:
3037:
3033:
3026:
3018:
3012:
3008:
3003:
3002:
2993:
2989:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2959:
2958:Optical depth
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2940:
2934:
2912:
2908:
2904:
2899:
2895:
2883:
2879:
2875:
2870:
2866:
2857:
2853:
2849:
2846:
2838:
2834:
2830:
2823:
2819:
2813:
2809:
2798:
2797:
2796:
2782:
2757:
2753:
2744:
2740:
2736:
2731:
2727:
2720:
2717:
2711:
2708:
2701:
2697:
2693:
2683:
2682:
2681:
2667:
2639:
2635:
2631:
2626:
2622:
2613:
2609:
2605:
2599:
2596:
2589:
2585:
2581:
2571:
2570:
2569:
2547:
2543:
2539:
2534:
2530:
2521:
2517:
2513:
2505:
2501:
2497:
2492:
2488:
2479:
2475:
2471:
2468:
2462:
2459:
2452:
2448:
2444:
2438:
2431:
2430:
2429:
2413:
2409:
2399:
2385:
2363:
2359:
2336:
2332:
2322:
2306:
2302:
2298:
2294:
2290:
2287:
2282:
2278:
2252:
2249:
2244:
2241:
2238:
2233:
2229:
2223:
2219:
2213:
2208:
2205:
2201:
2195:
2192:
2187:
2182:
2178:
2170:
2154:
2151:
2146:
2143:
2140:
2135:
2131:
2127:
2122:
2117:
2114:
2110:
2104:
2101:
2096:
2091:
2087:
2079:
2065:
2062:
2059:
2056:
2051:
2047:
2041:
2036:
2033:
2029:
2023:
2020:
2015:
2010:
2006:
1998:
1997:
1996:
1982:
1973:
1971:
1967:
1951:
1948:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1907:
1887:
1861:
1855:
1852:
1849:
1843:
1837:
1834:
1828:
1825:
1822:
1814:
1810:
1802:
1801:
1800:
1786:
1783:
1780:
1777:
1774:
1766:
1756:
1738:
1735:
1731:
1722:
1719:
1715:
1712:
1703:
1699:
1695:
1691:
1683:
1680:
1671:
1667:
1656:
1653:
1646:
1638:
1634:
1628:
1621:
1617:
1612:
1608:
1600:
1597:
1592:
1588:
1579:
1575:
1571:
1567:
1558:
1554:
1545:
1541:
1537:
1531:
1523:
1519:
1511:
1510:
1509:
1507:
1503:
1484:
1476:
1472:
1445:
1437:
1433:
1429:
1422:
1418:
1412:
1408:
1398:
1397:
1396:
1394:
1389:
1387:
1363:
1360:
1353:
1345:
1341:
1333:
1329:
1321:
1317:
1312:
1288:
1273:
1269:
1265:
1260:
1256:
1247:
1243:
1235:
1234:
1233:
1217:
1213:
1190:
1186:
1177:
1176:optical depth
1156:
1152:
1148:
1143:
1139:
1130:
1126:
1101:
1098:
1094:
1085:
1082:
1078:
1075:
1066:
1062:
1058:
1054:
1046:
1043:
1034:
1030:
1024:
1017:
1013:
1008:
1004:
996:
993:
988:
984:
975:
971:
967:
963:
954:
950:
941:
937:
933:
927:
919:
915:
907:
906:
905:
896:
879:
874:
870:
860:
854:
851:
848:
844:
837:
834:
830:
808:
786:
783:
780:
776:
753:
750:
747:
743:
720:
716:
695:
669:
664:
660:
650:
646:
641:
638:
635:
631:
624:
621:
617:
612:
609:
604:
600:
596:
591:
587:
583:
575:
572:
569:
565:
561:
556:
553:
550:
546:
539:
534:
530:
523:
508:
503:
499:
492:
478:
475:
466:
465:
464:
455:
409:
386:
383:
376:
372:
369:
365:
362:
356:
353:
350:
344:
341:
324:
311:
307:
303:
298:
294:
290:
283:
282:
281:
266:
263:
260:
257:
236:
182:
161:
158:
115:
112:
90:
86:
76:
74:
70:
66:
65:
60:
59:
48:
46:
41:
37:
33:
29:
25:
22:(also called
21:
3250:
3224:
3198:
3179:
3160:
3137:
3117:
3092:
3064:
3031:
3025:
3000:
2992:
2963:Planck's law
2932:
2774:
2659:
2567:
2400:
2323:
2269:
1974:
1879:
1762:
1754:
1505:
1463:
1390:
1383:
1117:
902:
687:
461:
401:
77:
73:polarization
62:
56:
54:
39:
23:
19:
18:
2398:direction.
129:located at
69:solid angle
51:Definitions
3282:Radiometry
3276:Categories
2984:References
1502:black body
36:scattering
28:absorption
3221:C. Bohren
2913:ν
2905:−
2900:ν
2884:ν
2880:σ
2871:ν
2867:α
2858:ν
2854:α
2824:ν
2758:ν
2745:ν
2741:σ
2732:ν
2728:α
2721:−
2702:ν
2668:μ
2640:ν
2632:−
2627:ν
2614:ν
2610:α
2590:ν
2548:ν
2540:−
2535:ν
2522:ν
2518:σ
2506:ν
2498:−
2493:ν
2480:ν
2476:α
2472:−
2453:ν
2439:μ
2414:ν
2410:σ
2364:ν
2337:ν
2307:ν
2283:ν
2242:μ
2234:ν
2220:μ
2206:−
2202:∫
2183:ν
2144:μ
2136:ν
2128:μ
2115:−
2111:∫
2092:ν
2060:μ
2052:ν
2034:−
2030:∫
2011:ν
1983:μ
1952:θ
1946:θ
1943:
1937:−
1931:μ
1908:μ
1853:μ
1823:μ
1815:ν
1787:θ
1784:
1775:μ
1704:ν
1700:τ
1696:−
1672:ν
1668:α
1639:ν
1613:∫
1580:ν
1576:τ
1572:−
1546:ν
1524:ν
1477:ν
1438:ν
1423:ν
1419:α
1413:ν
1346:ν
1342:α
1313:∫
1248:ν
1244:τ
1131:ν
1127:τ
1067:ν
1063:τ
1059:−
1035:ν
1009:∫
976:ν
972:τ
968:−
942:ν
920:ν
883:Ω
875:ν
865:Ω
861:∫
849:ν
838:π
809:ρ
781:ν
748:ν
721:ν
673:Ω
665:ν
655:Ω
651:∫
647:ρ
636:ν
625:π
610:ρ
605:ν
592:ν
584:ρ
570:ν
551:ν
535:ν
527:∇
524:⋅
518:^
515:Ω
504:ν
490:∂
486:∂
438:^
410:θ
380:Ω
366:ν
357:θ
354:
336:^
312:ν
299:ν
267:ν
258:ν
237:ν
214:^
186:Ω
91:ν
3249:(2010).
3223:(2006).
3089:(1960).
3063:(2015).
2937:See also
1966:Jacobian
1739:′
1716:′
1684:′
1657:′
1102:′
1079:′
1047:′
151:in time
32:emission
1995:yields
1799:, i.e.
1500:is the
1174:is the
3261:
3235:
3209:
3186:
3167:
3145:
3124:
3105:
3075:
3038:
3013:
1880:where
1464:where
1309:
1282:
1118:where
688:where
402:where
360:
34:, and
3259:ISBN
3233:ISBN
3207:ISBN
3184:ISBN
3165:ISBN
3143:ISBN
3122:ISBN
3103:ISBN
3073:ISBN
3036:ISBN
3011:ISBN
1205:and
3099:393
3007:393
1940:sin
1781:cos
351:cos
280:is
250:to
3278::
3257:.
3253:.
3231:.
3227:.
3205:.
3201:.
3101:.
3067:.
3009:.
1232::
47:.
30:,
3267:.
3241:.
3215:.
3192:.
3173:.
3151:.
3130:.
3111:.
3081:.
3044:.
3019:.
2918:)
2909:B
2896:J
2892:(
2889:)
2876:+
2863:(
2850:3
2847:=
2839:2
2835:z
2831:d
2820:J
2814:2
2810:d
2783:z
2754:H
2750:)
2737:+
2724:(
2718:=
2712:z
2709:d
2698:K
2694:d
2645:)
2636:J
2623:B
2619:(
2606:=
2600:z
2597:d
2586:H
2582:d
2553:)
2544:I
2531:J
2527:(
2514:+
2511:)
2502:B
2489:I
2485:(
2469:=
2463:z
2460:d
2449:I
2445:d
2386:z
2360:H
2333:J
2303:J
2299:3
2295:/
2291:1
2288:=
2279:K
2253:3
2250:a
2245:=
2239:d
2230:I
2224:2
2214:1
2209:1
2196:2
2193:1
2188:=
2179:K
2155:3
2152:b
2147:=
2141:d
2132:I
2123:1
2118:1
2105:2
2102:1
2097:=
2088:H
2066:a
2063:=
2057:d
2048:I
2042:1
2037:1
2024:2
2021:1
2016:=
2007:J
1949:d
1934:=
1928:d
1888:z
1865:)
1862:z
1859:(
1856:b
1850:+
1847:)
1844:z
1841:(
1838:a
1835:=
1832:)
1829:z
1826:,
1820:(
1811:I
1778:=
1736:s
1732:d
1726:)
1723:s
1720:,
1713:s
1709:(
1692:e
1688:)
1681:s
1677:(
1664:)
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1622:0
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1609:+
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1601:s
1598:,
1593:0
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1568:e
1564:)
1559:0
1555:s
1551:(
1542:I
1538:=
1535:)
1532:s
1529:(
1520:I
1506:T
1488:)
1485:T
1482:(
1473:B
1449:)
1446:T
1443:(
1434:B
1430:=
1409:j
1364:s
1361:d
1357:)
1354:s
1351:(
1334:2
1330:s
1322:1
1318:s
1301:f
1298:e
1295:d
1289:=
1279:)
1274:2
1270:s
1266:,
1261:1
1257:s
1253:(
1218:2
1214:s
1191:1
1187:s
1162:)
1157:2
1153:s
1149:,
1144:1
1140:s
1136:(
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1095:d
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1086:s
1083:,
1076:s
1072:(
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1044:s
1040:(
1031:j
1025:s
1018:0
1014:s
1005:+
1000:)
997:s
994:,
989:0
985:s
981:(
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960:)
955:0
951:s
947:(
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931:)
928:s
925:(
916:I
880:d
871:I
855:s
852:,
845:k
835:4
831:1
787:a
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754:s
751:,
744:k
717:j
696:c
670:d
661:I
642:s
639:,
632:k
622:4
618:1
613:+
601:j
597:=
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576:a
573:,
566:k
562:+
557:s
554:,
547:k
543:(
540:+
531:I
509:+
500:I
493:t
479:c
476:1
434:n
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384:d
377:d
373:a
370:d
363:d
348:)
345:t
342:,
332:n
325:,
321:r
317:(
308:I
304:=
295:E
291:d
264:d
261:+
210:n
183:d
162:t
159:d
138:r
116:a
113:d
87:I
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