T = 1010M; X = 10- T.1/2

Proceedings of the National Academy of Sciences Vol. 66, No. 2, pp. 282-288, June 1970

Excitation of Metastable Levels in Low Density Nebular Plasmas: [SIT] and [ArIV] S. J. Czyzak,* T. K. Krueger,t and L. H. Allert OHIO STATE AND OHIO WESLEYAN UNIVERSITIES; WRIGHT-PATTERSON AIR FORCE BASkE, OHIO; AND UNIVERSITY OF CALIFORNIA (LOS ANGELES) Communicated March 23, 1970 Abstract. Diagnostics of low density nebular plasma by means of its forbidden line spectrum require not only a knowledge of certain intensity ratios as a function of its density and temperature but also a knowledge of the occupation numbers of relevant ionic levels. We present the necessary data for important levels of ionized sulfur and triply ionized argon.

As reliable atomic parameters become available for an increasing number of ionic species, such as [Sil], [C1iii], [ArIV], their relevant forbidden lines become correspondingly more useful for the diagnostics of nebular plasmas, i.e., for deducing the densities and temperatures of individual condensations or filaments in which these lines originate. The first step is the calculation of the occupation numbers of each level as a function of density NE (electrons per cm3) and electron temperature T,, ('K) or the convenient variables introduced by Seaton.'

t = 1010m; x = 10- T.1/2- 1 We then express the population Nj of each level, j, in terms of the ground level population Nl,-by means of a Boltzmann-type equation: N1 Wj N,= - bj exp (xjlkT,) (2)

where co and co, are the statistical weights of the jth level and ground level respectively, Xj is the height (in energy units) of the jth level above the ground level, k is Boltzmann constant, and bj(x,t) is the factor introduced by Menzel2 to express the deviation of the population of the jth level from that appropriate to thermodynamic equilibrium at temperature Te. Let the number of collisional excitations from a lower level i to a higher level j per cm3 per sec be designated by NiNeqlj and the corresponding number of de- excitations be denoted by NjNeqj. The analogous number of radiative de-excitations will be NjAji; where A i is the Einstein coefficient of spontaneous emission. Of course there are no radiative excitations. 282 Downloaded by guest on September 25, 2021 VOL. 66, 1970 ASTRONOMY: CZYZAK, KRUEGER, AND ALLER 283

Consider a ground p3 configuration as is appropriate for [SH1], [ArIV]. There are five levels. We express all populations in terms of level 1, 4SS12. We set up the equations of statistical equilibrium for the five levels in the following form :3 4 5 t 5 Neqij + N. Niqij + a NjAij = Nj N i qji + EAj1f (3) i=20j i>j iij=1 i

a = N(2Ds) C = N(2P1) b2 = b(2Ds/2), b4 = b(2P1/2)

b = d = (2P31) b3 = = (2D.')N,' N, b(2D5/2), b5 b(2P312). (4) Tables 1 and 2 list the values of the bj's for [SIH] and [ArIV], respectively. The equations of statistical equilibrium for [SIH] are: 2.222 X 10-(06074/t)a + 2.923 X 10-(0.6054/t) b + 1.612c

- -[342.5 + 8.412]d + 10-(1.535/t) 0,

3.497 X 10-(°6074/i) a + 2.080 X 10(- 6024/t) b - . + 18.461 c

+ 1.611d + 10-(1.532/t) - 0

1.388a + [1 + 1.592 + 2.123 X 10 Ol6046/t) b + L1 + 1.6731c LX XJ

+ [538.3 + 2.348]d + 10-(0.9299/t) - 0,

-[6917 + 3.081 + 3.190 X 10 (0 6o/t)]a + 1.387b + [744.9 + 2.811}c

+ [658.2 + 1.785]d + 10-(09280/t) = 0. (5) Downloaded by guest on September 25, 2021 284 ASTRONOMY: CZYZAK, KRUEGER, AND ALLER PROC. N. A. S.

TABLE 1. bn values of [SII].

Te Ne x b2 N N b5 5000 71 0.01 1.492 X 10- 5.698 X 1O-3 5.203 X 10-6 2.996 X 10-6 223 0.0316 4.785 X 10-3 1.775 X 10-2 1.722 X 10- 1.001 X 10-5 707 0.10 1.569 X 10-2 5.153 X 10-2 6.181 X 10-5 3.673 X 10-5 2234 0.316 5.283 X 10-2 1.344 X 10-1 2.573 X 1O-4 1.510 X 1O-4 7071 1.0 1.632 X 10-1 2.904 X 10-1 1.254 X 10-3 7.664 X 10-4 22345 3.16 3.942 X 10-1 5.098 X 10-1 6.297 X 1O-3 3.761 X 1O-3 70711 10.0 6.698 X 10-1 7.240 X 10-1 2.726 X 10-2 1.614 X 10-2 7500 87 0.01 1.545 X 10-3 5.804 X 10-1 5.206 X 10-6 2.998 X 10-6 274 0.0316 4.955 X 10-2 1.784 X 10-3 1.724 X 10-5 1.003 X 10-6 866 0.10 1.626 X 10-1 5.203 X 10-2 6.199 X 10-5 3.683 X 10-5 2737 0.316 5.410 X 10-2 1.339 X 10-1 2.579 X 10-4 1.571 X 10-4 8660 1.0 1.659 X 10-1 2.835 X 10-1 1.248 X 10-3 7.594 X 10-4 27366 3.16 3.896 X 10-1 4.863 X 10-3 6.154 X 10-3 3.657 X 10-3 86603 10.0 6.387 X 10-1 6.760 X 10-1 2.608 X 10-2 1.537 X 10-2 10000 100 0.01 1.597 X 10-3 5.908 X 10-3 5.208 X 10-6 3.000 X 10-6 316 0.0316 5.124 X 10-3 1.811 X 10-2 1.727 X 10-5 1.004 X 10-5 1000 0.10 1.682 X 10-2 5.253 X 10-2 6.217 X 10-5 3.693 X 10-5 3160 0.316 5.573 X 10-2 1.335 X 10-1 2.587 X 10-4 1.572 X 10-4 10000 1.0 1.684 X 10-1 2.773 X 10-1 1.243 X 10-3 7.531 X 10-4 31600 3.16 3.854 X 10-1 4.661 X 10-1 6.031 X 10-3 3.568 X 10-3 100000 10.0 6.166 X 10-1 6.363 X 10-1 2.510 X 10-2 1.473 X 10-1 12500 112 0.01 1.642 X 10-3 5.998 X 10-3 5.211 X 10-6 3.001 X 10-6 353 0.316 5.269 X 10-1 1.835 X 10-2 1.729 X 10-5 1.005 X 10-5 1118 0.10 1.729 X 10-2 5.295 X 10-2 6.232 X 10-5 3.702 X 10-5 3533 0.316 5.713 X 10-2 1.331 X 10-1 2.593 X 10-4 1.574 X 10-4 11180 1.0 1.704 X 10-1 2.725 X 10-1 1.239 X 10-3 7.483 X 10-4 35330 3.16 3.820 X 10-1 4.506 X 10-1 5.936 X 10-3 3.499 X 10-8 111803 10.0 5.997 X 10-1 6.066 X 10-1 2.436 X 10-2 1.425 X 10-2 15000 122 0.01 1.679 X 10-3 6.073 X 10-3 5.213 X 10-6 3.002 X 106 387 0.0316 5.391 X 10-3 1.855 X 10-2 1.730 X 10-5 1.007 X 10-5 1225 0.10 1.769 X 10-2 5.333 X 10-2 6.244 X 10-5 3.708 X 1o-5 3870 0.316 5.828 X 10-2 1.328 X 10-1 2.597 X 10-4 1.575 X 10-4 12247 1.0 1.721 X 10-1 2.687 X 10-1 1.236 X 10-3 7.445 X 10-4 38702 3.16 3.794 X 10-1 4.388 X 10-1 5.863 X 10-2 3.446 X 10-3 122474 10.0 5.870 X 10-1 5.843 X 10-1 2.380 X 10-2 1.390 X 10-2 17500 132 0.01 1.711 X 10-3 6.135 X 10-3 5.214 X 10-6 3.003 X 10-6 418 0.0316 5.494 X 10-3 1.872 X 10-2 1.732 X 10-5 1.007 X 10-5 1323 0.10 1.803 X 10-2 5.359 X 10-2 6.255 X 10-5 3.715 X 10-5 4180 0.316 5.924 X 10-2 1.326 X 10-1 2.602 X 10-4 1.576 X 10-4 13229 1.0 1.734 X 10-1 2.657 X 10-1 1.234 X 10-3 7.416 X 10-4 41803 3.16 3.774 X 10-1 4.296 X 10-1 5.806 X 10-3 3.405 X 10-3 132288 10.0 5.770 X 10-1 5.674 X 10-1 2.337 X 10-2 1.362 X 10-2 20000 141 0.01 1.737 X 10-3 6.188 X 10-3 5.215 X 10-6 3.004 X 10-1 447 0.0316 5.580 X 10-3 1.886 X 10-2 1.733 X 10-5 1.008 X 10-5 1414 0.1 1.831 X 10-2 5.383 X 10-2 6.264 X 10-5 3.719 X 10-5 4469 0.316 6.004 X 10-2 1.324 X 10-1 2.606 X 10-4 1.577 X 10-4 14142 1.0 1.746 X 10-1 2.633 X 10-1 1.231 X 10-3 7.393 X 10-4 44689 3.16 3.757 X 10-1 4.223 X 10-1 5.761 X 10-8 3.373 X 10- 141421 10.0 5.692 X 10-1 5.542 X 10-2 2.304 X 10-2 1.341 X 10- Downloaded by guest on September 25, 2021 VOL. 66, 1970 ASTRONOMY: CZYZAK, KRUEGER, AND ALLER 285

TABLE 2. bn values of [ArIV].

Tf Nf X b2 b6 b b6 5000 71 0.01 4.366 X 10-5 5.507 X 10-4 5.169 X 10-7 2.215 X 10-7 223 0.0316 1.383 X 10-4 1.735 X 10-8 1.642 X 10-6 7.049 X 10-7 707 0.10 4.409 X 10-4 5.439 X 10-3 5.281 X 10- 2.280 X 10-6 2234 0.316 1.424 X 10-3 1.700 X 10-2 1.751 X 10- 7.691 X 10-6 7071 1.0 4.777 X 10-3 4.851 X 10-2 6.290 X 10-i 2.869 X 10-5 22345 3.16 1.688 X 10-2 1.226 X 10-1 2.571 X 10-4 1.236 X 10-4 70711 10.0 5.960 X 10-2 2.460 X 10-1 1.173 X 10-i 5.786 X 10-4 7500 87 0.01 4.435 X 10-6 5.534 X 10-4 5.169 X 10-7 2.215 X 10-7 274 0.0316 1.408 X 10-4 1.743 X 10-3 1.642 X 10-6 7.048 X 10-7 866 0.10 4.480 X 10-4 5.461 X 10-3 5.282 X 10-6 2.281 X 10-6 2737 0.316 1.448 X 10-3 1.673 X 10-2 1.752 X 10-5 7.692 X 10-6 8660 1.0 4.862 X 10-3 4.835 X 10-2 6.289 X 10-5 2.868 X 10-6 27366 3.16 1.717 X 10-2 1.208 X 10-1 2.561 X 10-4 1.229 X 10-4 86603 10.0 6.009 X 10-2 2.381 X 10-1 1.157 X 10-3 5.686 X 10-4 10000 100 0.01 4.523 X 10-6 5.569 X 10-4 5.169 X 10-7 2.215 X 10-7 316 0.0316 1.433 X 10-4 1.754 X 10-3 1.642 X 10- 7.050 X 10-7 1000 0.10 4.570 X 10-4 5.490 X 10-3 5.283 X 10- 2.282 X 10-6 3160 0.316 1.478 X 10-3 1.678 X 10-2 1.752 X 10- 7.695 X 10- 10000 1.0 4.966 X 10-3 4.816 X 10-2 6.288 X 10- 2.866 X 10-5 31600 3.16 1.750 X 10-2 1.185 X 10-1 2.548 X 10-4 1.220 X 10-4 100000 10.0 6.055 X 10-2 2.288 X 10-1 1.139 X 10-3 5.568 X 10-4 12500 112 0.01 4.612 X 10-5 5.604 X 10-4 5.169 X 10-7 2.215 X 10-7 353 0.0316 1.461 X 10-4 1.764 X 10-3 1.642 X 10-6 7.050 X 10-7 1118 0.10 4.662 X 10-4 5.519 X 10-3 5.284 X 10-6 2.282 X 10-6 30533 0.316 1.508 X 10-3 1.683 X 10-2 1.753 X 10-5 7.698 X 10-6 11180 1.0 5.070 X 10-3 4.797 X 10-2 6.286 X 10-6 2.864 X 10-5 35330 3.16 1.782 X 10-2 1.163 X 10-1 2.536 X 10-4 1.212 X 10-4 111803 10.0 6.010 X 10-2 2.202 X 10-1 1.122 X 10-3 5.460 X 10-3 15000 122 0.01 4.694 X 10-5 5.637 X 10-4 5.169 X 10-7 2.215 X 10-7 387 0.0316 1.487 X 10-4 1.774 X 10-3 1.642 X 10- 7.051 X 10-7 1225 0.10 4.746 X 10-4 5.545 X 10-3 5.284 X 10-6 2.282 X 10-6 3870 0.316 1.536 X 10-3 1.687 X 10-2 1.753 X 10-5 7.700 X 10-6 12247 1.0 5.163 X 10-3 4.779 X 10-2 6.285 X 10-5 2.862 X 10-6 38702 3.16 1.811 X 10-2 1.144 X 10-1 2.526 X 10-4 1.205 X 10-4 122474 10.0 6.139 X 10-2 2.130 X 10-1 1.108 X 10-3 5.369 X 10-4 17500 132 0.01 4.767 X 10-6 5.666 X 10-4 5.169 X 10-7 2.215 X 10-7 418 0.0316 1.511 X 10-4 1.783 X 10-3 1.642 X 10- 7.051 X 10-7 1323 0.10 4.821 X 10-4 5.569 X 10-3 5.285 X 10- 2.283 X 10-6 4180 0.316 1.561 X 10-3 1.691 X 10-2 1.753 X 10-6 7.702 X 10-' 13229 1.0 5.251 X 10-3 4.764 X 10-2 6.284 X 10-5 2.861 X 10-6 41803 3.16 1.836 X 10-2 1.128 X 10-1 2.517 X 10-4 1.199 X 10-4 132288 10.0 6.173 X 10-2 2.070 X 10-1 1.096 X 10-3 5.293 X 10-4 20000 141 0.01 4.833 X 10-5 5.692 X 10-4 5.170 X 10-7 2.215 X 10-7 447 0.0316 1.532 X 10-4 1.791 X 10-3 1.642 X 10-6 7.051 X 10-7 1414 0.10 4.888 X 10-4 5.590 X 10-3 5.286 X 10-6 2.283 X 10-6 4469 0.316 1.583 X 10-3 1.694 X 10-2 1.754 X 10-6 7.703 X 10- 14142 1.0 5.326 X 10-3 4.750 X 10-2 6.283 X 10-4 2.860 X 10-6 44689 3.16 1.859 X 10-2 1.114 X 10-1 2.510 X 10-4 1.193 X 10-4 141421 10.0 6.230 X 10-2 2.021 X 10-1 1.087 X 10-3 5.230 X 10-4 Downloaded by guest on September 25, 2021 286 ASTRONOMY: CZYZAK, KRUEGER, AND ALLER PROC. N. A. S.

Using again the notation of equation (4), similar expressions may be written down for the levels in [ArIV]. We now have:

2.927 X 10-(08618/t) a + 3.935 X 10-(086l8/') b + 2.838c - [4 + 11.247]d

+ lo-(2.185/t) = o

4.753 X 10 -./t) a + 2.720 X 1-(0.8618/t) b - [38742. + 25.286]c

+ 2.833d + 10 - (2.185/t) = o [11.94 1 F661.8 1.574a - - + 1.716 + 2.624 X 10-(08618/t) b + [ + 2.013]c

+ [ + 2.9232]d + 10 -(1.323/t) = 0,

- [227.9 + 3.362 + 3.944 X 10-(0.868/t)]a + 1 574b

= + [ * + 3.539]c + + 2.175 d + 10 (1.323/t) 0. (6) 6 As N' increases without bound, b 1, notice that b2 and ba approach 1 more rapidly than do b4 and b5. This situation develops because A21 and A31 are small compared with A41, A42, A61, and A52. Once the b factors are known, the intensity ratios of any two lines Xjj,/Xwi, be expressed as follows:

Xi 7ff' = bj (N,,j [exp( - Xi)] Aj, = rk(N ,T.). (7) Iii bi(N.,T.) wi k A ji Xj As an example, consider the ratio of the intensities of the nebular and auroral- type transition in [SII]. Using A values and collision cross sections previously obtained,6 we find Ineb I(2D - 4S) I(X6716) + I(X6730) laur I(2P - 4S) I(X10321) + I(X10287) + I(X10373) + I(X10339) 19.6b2(X,t) + t) (1.39\ 8 7.45b,3(x, X 8 b4(x,t) + 2.64bs(x,t) 10-1epetx ) The infrared lines are less frequently observed than are the transauroral lines X4076.5 (2P1/2- 4S/2) X4068.6 (2P,,2 - 4S3/2), and nebular lines X6716.5 (2D/2,- 4SI/2), X6730.8 (2Ds/, - 4S32). Tabulations of ri = I(2D,/2 - 4S32/)/I(2D./, - 4S=/2)-I(X4711)/I(X4740) for [ArIVI are given elsewhere.3 The b factors are also required if we want ratios of ionic concentrations. For example, suppose the [SII] emission and the H14 emission occurs in the same knot or condensation, Downloaded by guest on September 25, 2021 VOL. 66, 1970 ASTRONOMY: CZYZAK, KRUEGER, AND ALLER 287 then the emission per unit volume per second (ergs cm-3 sec-9) in X4068 is given by E(X4068) = N(2p3/2)Ahv = bsN(S+) [exp(- 0.)]o.34 X 4.87 X 10-12 E(Hp) = N(H+)NE1O2E4,20 (9) where E1 20 = 2.22, 1.241, 0.863, 0.660 for t = 0.5, 1.0, 1.5, and 2.0, respectively.6 Then

I(X406) 1.66 N(S+)exp(-3.53/t) I(Hp) )= X 1013b5(xN'+N.42t) (10) The left-hand side of the equation is identified with the observed intensity ratio I(X4068)/(H1) and one solves for the ionic concentration ratio, N(S+)/N(H+). Similarly, for [ArIV], X4740 (4S.1, - 2D3/2) we have

I(X4740))= 1.17 X 1012b2(x)t) exp( - 30) (11) while for the infrared lines of [ArIV] we have the (2P,/, - 2D) transitions: I(X7237) + I(X7171) 13 N(Ar+++) exp(-5.03/t) (12) I(H1) -= 4.36 X 1 b(xt)E42 and (2P1/2- 2D) transitions: I(-k7262) + I(X7332) < 34 N(Ar+++) exp(-5.03/t) (3 -1I(H4) = 1.09 X 1013N (X)0 ° (13)

Thus, ionic concentrations may be calculated from the observed line intensities as soon as x and t can be estimated for the filaments or condensations in question. Alternately, one could use the infrared [ArIV] lines and X4740 to derive a rela- tion between x and t, i.e., I(X7237) + I(X7171) - 72b5(,&t) ~1 .99\(4 (I(X4740)I()X+740() ) 37.2 b(2b(')exp( t) (14) At lower densities, N. <103, and b5/b2 depends almost entirely on t and slowly on x. This program was supported in part by grants 6077 and 6559 from the National Science Foundation to the Ohio State University and by grant 83-67 from the Air Force Office of Scientific Research to the University of California (Los Angeles). * Present address: Perkins Observatory of the Ohio State and Wesleyan Universities. t Present address: General Physics Laboratory, Wright-Patterson Air Force Base, Fair- born, Ohio. t Requests for reprints may be addressed to Dr. Aller, Department of Astronomy, Univer- sity of California, Los Angeles, Calif. 90024. 1Seaton, M. J., Monthly Notices Roy. Astron. Soc., 114, 154 (1954); Rept. Prog. Phys., 23, 313 (1960). Downloaded by guest on September 25, 2021 288 ASTRONOMY: CZYZAK, KRUEGER, AND ALLER PROC. N. A. S.

2Menzel, D. H., Astrophys. J., 85, 330 (1937). ' Krueger, T. K., L. H. Aller, and S. J. Czyzak, Astrophys. J., in press. 4Aller, L. H., these PROCEEDINGS, 65, 775 (1970). 5 Czyzak, S. J., and T. K. Krueger, Phil. Trans. (1970); Monthly Notices Roy. Astron. Soc., 126, 177 (1963); 129, 103 (1965). 6 Clarke, Wm., Ph.D. thesis, University of California (Los Angeles), 1965, quoted in Stars and Stellar Systems (Chicago: University of Chicago Press, 1968), vol. 7, p. 521. Downloaded by guest on September 25, 2021