218:
is known. These can generally be calculated from other equations if a fair amount of information is known about the chemical makeup of the star. From the definition, it is also clear that large optical depths correspond to higher rate of obscuration. Optical depth can therefore be thought of as the
769:
This illustrates not only that the observable temperature and actual temperature at a certain physical depth of a star vary, but that the optical depth plays a crucial role in understanding the stellar structure. It also serves to demonstrate that the depth of the photosphere of a star is highly
922:
246:. In most astrophysical problems, this is exceptionally difficult to solve since solving the corresponding equations requires the incident radiation as well as the radiation leaving the star. These values are usually theoretical.
710:
173:
1034:
331:
625:
produces a "gray" absorption in the atmosphere of a star, that is, it is independent of any specific wavelength and absorbs along the entire electromagnetic spectrum. In that case,
391:
of the incident light before being absorbed or scattered. It is important to note that the Beer–Lambert law is only appropriate when the absorption occurs at a specific wavelength,
416:
385:
611:
561:
819:
965:
811:
504:
484:
354:
271:
240:
196:
945:
788:
764:
740:
459:
439:
92:
52:
530:
216:
72:
631:
790:
is about 2/3, which corresponds to a state where a photon would experience, in general, less than 1 scattering before leaving the star.
998:
119:
279:
31:
106:
94:
is able to show the relationship between these two quantities and can lead to a greater understanding of the
1029:
743:
250:
74:
respectively, can vary widely depending on the absorptivity of the astrophysical environment. Indeed,
616:
770:
dependent upon the absorptivity of its environment. The photosphere extends down to a point where
418:. For a gray atmosphere, for instance, it is most appropriate to use the Eddington Approximation.
394:
363:
917:{\displaystyle T^{4}={\frac {3}{4}}T_{e}^{4}\left(\int _{0}^{z}(\alpha )dz+{\frac {2}{3}}\right)}
576:
243:
582:
441:
is simply a constant that depends on the physical distance from the outside of a star. To find
95:
950:
796:
489:
339:
256:
225:
181:
984:
357:
930:
773:
749:
718:
444:
424:
77:
37:
1002:
535:
509:
464:
620:
201:
57:
16:
This article is about optical depth in astrophysics. For optical depth in general, see
1023:
17:
27:
572:
705:{\displaystyle T^{4}={\frac {3}{4}}T_{e}^{4}\left(\tau +{\frac {2}{3}}\right),}
388:
110:
571:
Since it is difficult to define where the interior of a star ends and the
99:
178:
The assumption here is that either the extinction coefficient
567:
The
Eddington approximation and the depth of the photosphere
168:{\displaystyle \tau \equiv \int _{0}^{z}\alpha dz=\sigma N.}
985:"Optical Depth -- from Eric Weisstein's World of Physics"
1035:
Scattering, absorption and radiative transfer (optics)
326:{\displaystyle \alpha =e^{4\pi \kappa /\lambda _{0}},}
953:
933:
822:
799:
776:
752:
721:
634:
585:
538:
512:
492:
467:
447:
427:
397:
366:
342:
282:
259:
228:
204:
184:
122:
80:
60:
40:
619:the approximation takes into account the fact that
959:
939:
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734:
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555:
524:
498:
478:
453:
433:
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379:
348:
325:
265:
234:
210:
190:
167:
86:
66:
46:
793:The above equation can be rewritten in terms of
113:up to a specific 'depth' of a star's makeup.
8:
575:begins, astrophysicists usually rely on the
952:
932:
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870:
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827:
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227:
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183:
138:
133:
121:
79:
59:
39:
976:
486:, the above equation may be used with
7:
927:Which is useful, for example, when
579:to derive the formal definition of
105:Optical depth is a measure of the
34:. Optical depth and actual depth,
14:
887:
881:
30:refers to a specific level of
1:
198:or the column number density
411:{\displaystyle \lambda _{0}}
380:{\displaystyle \lambda _{0}}
242:can be calculated using the
222:The extinction coefficient
1051:
15:
606:{\displaystyle \tau =2/3}
253:can be useful in finding
999:"CHP - Beer-Lambert Law"
960:{\displaystyle \alpha }
806:{\displaystyle \alpha }
577:Eddington Approximation
499:{\displaystyle \alpha }
349:{\displaystyle \kappa }
266:{\displaystyle \alpha }
235:{\displaystyle \alpha }
191:{\displaystyle \alpha }
961:
941:
918:
813:in the following way:
807:
784:
766:is the optical depth.
760:
736:
706:
607:
557:
526:
500:
480:
461:at a particular depth
455:
435:
412:
381:
350:
327:
267:
236:
219:opacity of a medium.
212:
192:
169:
107:extinction coefficient
88:
68:
48:
962:
942:
940:{\displaystyle \tau }
919:
808:
785:
783:{\displaystyle \tau }
761:
759:{\displaystyle \tau }
744:effective temperature
737:
735:{\displaystyle T_{e}}
707:
608:
558:
527:
506:and integration from
501:
481:
456:
454:{\displaystyle \tau }
436:
434:{\displaystyle \tau }
413:
382:
351:
328:
268:
237:
213:
193:
170:
89:
87:{\displaystyle \tau }
69:
49:
47:{\displaystyle \tau }
951:
931:
820:
797:
774:
750:
719:
632:
617:Sir Arthur Eddington
583:
556:{\displaystyle z=z'}
536:
510:
490:
465:
445:
425:
395:
364:
340:
280:
257:
226:
202:
182:
120:
78:
58:
38:
880:
860:
672:
525:{\displaystyle z=0}
143:
957:
937:
914:
866:
846:
803:
780:
756:
746:at that depth and
732:
702:
658:
603:
553:
522:
496:
479:{\displaystyle z'}
476:
451:
431:
408:
377:
346:
323:
263:
249:In some cases the
232:
208:
188:
165:
129:
84:
64:
44:
947:is not known but
907:
844:
692:
656:
244:transfer equation
211:{\displaystyle N}
67:{\displaystyle z}
1042:
1014:
1013:
1011:
1010:
1001:. Archived from
995:
989:
988:
981:
966:
964:
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944:
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923:
921:
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913:
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879:
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859:
854:
845:
837:
832:
831:
812:
810:
809:
804:
789:
787:
786:
781:
765:
763:
762:
757:
741:
739:
738:
733:
731:
730:
711:
709:
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703:
698:
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693:
685:
671:
666:
657:
649:
644:
643:
623:
612:
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552:
531:
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528:
523:
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437:
432:
417:
415:
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409:
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386:
384:
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378:
376:
375:
358:refractive index
355:
353:
352:
347:
332:
330:
329:
324:
319:
318:
317:
316:
307:
272:
270:
269:
264:
251:Beer–Lambert law
241:
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217:
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209:
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142:
137:
93:
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90:
85:
73:
71:
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65:
53:
51:
50:
45:
1050:
1049:
1045:
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1041:
1040:
1039:
1020:
1019:
1018:
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1008:
1006:
997:
996:
992:
983:
982:
978:
973:
949:
948:
929:
928:
865:
861:
823:
818:
817:
795:
794:
772:
771:
748:
747:
722:
717:
716:
677:
673:
635:
630:
629:
621:
581:
580:
569:
545:
534:
533:
508:
507:
488:
487:
468:
463:
462:
443:
442:
423:
422:
398:
393:
392:
367:
362:
361:
338:
337:
308:
289:
278:
277:
255:
254:
224:
223:
200:
199:
180:
179:
118:
117:
76:
75:
56:
55:
36:
35:
21:
12:
11:
5:
1048:
1046:
1038:
1037:
1032:
1022:
1021:
1016:
1015:
990:
975:
974:
972:
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956:
936:
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906:
903:
898:
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835:
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826:
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602:
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568:
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541:
521:
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471:
450:
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405:
401:
374:
370:
345:
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136:
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125:
83:
63:
43:
13:
10:
9:
6:
4:
3:
2:
1047:
1036:
1033:
1031:
1028:
1027:
1025:
1005:on 2014-02-24
1004:
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723:
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108:
103:
101:
97:
81:
61:
41:
33:
29:
25:
24:Optical depth
19:
18:optical depth
1030:Astrophysics
1007:. Retrieved
1003:the original
993:
979:
926:
792:
768:
714:
614:
570:
420:
335:
248:
221:
177:
111:absorptivity
104:
32:transparency
28:astrophysics
23:
22:
615:Devised by
573:photosphere
421:Therefore,
1024:Categories
1009:2011-04-09
971:References
389:wavelength
955:α
935:τ
885:α
868:∫
801:α
778:τ
754:τ
679:τ
587:τ
494:α
449:τ
429:τ
400:λ
369:λ
344:κ
310:λ
301:κ
298:π
284:α
261:α
230:α
186:α
157:σ
145:α
131:∫
127:≡
124:τ
98:inside a
96:structure
82:τ
42:τ
550:′
473:′
742:is the
387:is the
356:is the
715:where
360:, and
336:where
967:is.
100:star
54:and
532:to
109:or
26:in
1026::
563:.
273:.
102:.
1012:.
987:.
911:)
905:3
902:2
897:+
894:z
891:d
888:)
882:(
877:z
872:0
863:(
857:4
852:e
848:T
842:4
839:3
834:=
829:4
825:T
728:e
724:T
700:,
696:)
690:3
687:2
682:+
675:(
669:4
664:e
660:T
654:4
651:3
646:=
641:4
637:T
622:H
601:3
597:/
593:2
590:=
547:z
543:=
540:z
520:0
517:=
514:z
470:z
404:0
373:0
321:,
314:0
305:/
295:4
291:e
287:=
206:N
163:.
160:N
154:=
151:z
148:d
140:z
135:0
62:z
20:.
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