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Na pervuyu stranicu Dvizhushiesya obolochki zvezd << 1.3 Otnositel'nye intensivnosti linii | Oglavlenie | 1.5 Sravnenie s nablyudeniyami >>

1.4 O prozrachnosti sredy dlya izlucheniya v liniyah

Kak bylo vyyasneno, s uvelicheniem parametra h stepen' vozbuzhdeniya bystro ubyvaet, i pri dostatochno bol'shih znacheniyah h sreda stanovitsya prozrachnoi dlya izlucheniya v liniyah vseh serii, nachinaya c j+1-i. Usloviem neprozrachnosti sredy v liniyah j-i serii yavlyaetsya vypolnenie neravenstva βjk < 1. Odnako zaranee my ne mozhem skazat', nachinaya s kakoi serii sreda budet prozrachnoi, tak kak velichiny βjk zavisyat ot velichin nj yavlyayushihsya neizvestnymi. Zhelatel'no poetomu imet' prostoi sposob ocenki chisel nj.

Seichas my dadim priblizhennyi metod resheniya sistemy (9), osnovannyi na tom obstoyatel'stve, chto pri bol'shih znacheniyah h chisla nj bystro ubyvayut s vozrastaniem nomera j.

Skladyvaya pochlenno vse uravneniya (9), nachinaya s j-go, poluchaem

\begin{eqnarray} \sum_{i=j}^{\infty} n_i\left[x\sum_{k=1}^{j-1} \frac{\frac{g_2}{g_1}n_1-n_2}{\frac{g_i}{g_k}n_k-n_i}\left(\frac{\nu_{ki}}{\nu_{12}}\right)^3+\frac{B_{ic}\rho_{ic}^*}{A_{21}}\right]=\frac{n_e n^+}{WA_{21}}\sum_{i=j}^{\infty} C_i,\nonumber
\end{eqnarray} (21)

Prenebregaya einshteinovskim otricatel'nym poglosheniem i oboznachaya

\begin{eqnarray} 4A_{21}\frac{g_k}{g_i}\left(\frac{\nu_{ki}}{\nu_{12}}\right)^3=D_{ki}, \qquad \sum_{i=j}^{\infty} C_i=S_j,\nonumber
\end{eqnarray} (22)

vmesto (21) nahodim

\begin{eqnarray} \sum_{i=j}^{\infty} n_i\left(x\sum_{k=1}^{j-1} D_{ki}\frac{n_1}{n_k}+B_{ic}\rho_{ic}^*\right)=\frac{n_e n^+}{W}S_j.\nonumber
\end{eqnarray} (23)

Prinimaya vo vnimanie ubyvanie chisel ni i Dki s vozrastaniem i, ogranichimsya v summirovanii po i odnim pervym chlenom. Togda poluchaem

\begin{eqnarray} n_j\left(x\sum_{k=1}^{j-1} D_{kj}\frac{n_1}{n_k}+B_{jc}\rho_{jc}^*\right)=\frac{n_e n^+}{W}S_j.\nonumber
\end{eqnarray} (24)

My prishli k rekurentnoi formule, dayushei chislo nj, esli izvestny chisla n1, n2,... nj-1.

Dlya j=1 my mozhem vzyat' bolee tochnuyu formulu, chem ta, kotoraya sleduet iz (24), a imenno:

\begin{eqnarray} n_1B_{ic}\rho_{ic}^* + n_2B_{2c}\rho_{2c}^*=\frac{n_e n^+}{W}S_1.\nonumber
\end{eqnarray} (25)

Togda uravnenie (25) i vtoroe iz uravnenii (24) dayut

$$
\frac{n_2}{n_1}=\frac{(1-p)B_{ic}\rho_{ic}^*}{xA_{21}+pB_{2c}\rho_{2c}^*}
$$ (26)

gde $p=\frac{C_1}{S_1}$. Privlekaya sleduyushee iz uravnenii (24), poluchaem

\begin{eqnarray} \frac{n_3}{n_1} = \frac{S_3}{S_2}\frac{xA_{21}+B_{2c}\rho_{2c}^*}{xD_{13}+xD_{23}\frac{n_1}{n_2} + B_{3c}\rho_{3c}^*}\cdot\frac{n_2}{n_1}.\nonumber
\end{eqnarray} (27)

I tak dalee.

Esli pri vychislenii po formulam (24) my poluchim takoe chislo nj, chto okazhetsya βjk > 1, to eto budet oznachat', chto sreda prozrachna dlya izlucheniya v j-i serii. Dopustim, chto my poluchili β3k > 1, no β2k < 1 . Eto znachit, chto sreda neprozrachna dlya izlucheniya v bal'merovskoi serii, no prozrachna dlya izlucheniya v posleduyushih seriyah. Vspominaya sootnoshenie (8), my nahodim, chto v etom sluchae dolzhno byt'

\begin{eqnarray} \frac{A_{43}}{D_{34}}\cdot\frac{n_3}{n_1}<\beta_{12}<\frac{A_{32}}{D_{23}}\cdot\frac{n_2}{n_1},\nonumber
\end{eqnarray} (28)

ili

\begin{eqnarray} 30\frac{n_3}{n_1}<\beta_{12}<8\frac{n_2}{n_1}.\nonumber
\end{eqnarray} (29)

Vychislyaya dlya primera otnosheniya $\frac{n_2}{n_1}$ i $\frac{n_3}{n_1}$ po formulam (26) i (27) pri T=20000o, poluchaem, chto pri x=1,0 dolzhno byt'

\begin{eqnarray}
2\cdot 10^{-4} < \beta_{12}<3\cdot 10^{-3},\nonumber
\end{eqnarray} (30)

a pri x=10

\begin{eqnarray}
2\cdot 10^{-6} < \beta_{12}<3\cdot 10^{-4}.\nonumber
\end{eqnarray} (31)

Takovy usloviya, kotorye dolzhny vypolnyat'sya, esli sreda neprozrachna dlya izlucheniya v laimanovskoi i bal'merovskoi seriyah i prozrachna dlya izlucheniya v drugih seriyah. Predstavlyaet interes nahozhdenie bal'merovskogo dekrementa dlya tol'ko chto rassmotrennogo sluchaya. Pri vypolnenii poslednih uslovii my dolzhny polozhit' βjk=1 (j=3, 4, 5,...). Togda sistema uravnenii (9) privoditsya k vidu:

\begin{eqnarray}
\left.
\begin{array}{l} n_1B_{1c}W\rho_{1c}^* = \beta_{12}\sum\limits_{k=2}^{\infty} n_k D_{1k} + n_e n^+ C_1 \nonumber \\ \\ n_2(A_{21}\beta_{12} + B_{2c}W\rho_{2c}^*) = \frac{n_1}{n_2} \beta_{12}\sum\limits_{k=3}^{\infty} n_k D_{2k} + n_e n^+ C_2 \nonumber \\ \\ n_i\left[ \beta_{12}\left( D_{1i} + \frac{n_1}{n_2} D_{2i}\right) + \sum\limits_{k=3}^{i-1} A_{ik}+B_{ic}W\rho_{ic}^* \right]= \nonumber \\ \\
= \sum\limits_{k=i+1}^{\infty} n_k A_{ki} + n_e n^+ C_i \quad (i=3,4,5,...)\nonumber
\end{array}
\right\}
\end{eqnarray} (32)

Sistema uravnenii (32) byla reshena nami chislenno dlya sleduyushih dvuh sluchaev:

  1. T=20000o, β12=10-3, W=10-3, x=1.0,
  2. T=20000o, β12=10-5, W=10-6, x=10.

Bal'merovskii dekrement, poluchennyi v rezul'tate resheniya, priveden v tabl. VII.

Tablica VII
Bal'merovskii dekrement pri βjk=1 (j=3,4,5,..)
  I II
Hα 2.0 8.9
Hβ 1.00 1.00
Hγ 0.80 0.91
Hδ 0.61 0.84

Iz tablicy vidno, chto intensivnosti linii Hβ, Hγ i Hδ blizki drug k drugu, no otnoshenie intensivnostei $\frac{H_{\alpha}}{H_{\beta}}$ ves'ma veliko. Etot bal'merovskii dekrement rezko otlichaetsya ot poluchennogo ranee (cm. tablicy IV i V). Na samom dele iz-za neodnorodnosti obolochek vozmozhny kombinacii oboih bal'merovskih dekrementov.

V sleduyushem paragrafe my uvidim, chto bal'merovskii dekrement ukazannogo v tabl. VII tipa deistvitel'no osushestvlyaetsya v nekotoryh obolochkah.


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