Appendix E: Frequency-integration of the line intensity distribution function

In this appendix, we give a brief derivation of the frequency integrated line intensity distribution function, Eq. (73). Similar to the frequency dependent one, Eq. (61), which is the starting point of our considerations, we have to account for different regimes, since the maximum possible (logarithmic) oscillator strength $\tilde r_{\rm max}$ depends both on line intensity and frequency. Furthermore, we assume that under certain circumstances only levels below a cutoff $x_{\rm max}$ shall contribute. In this case then, all lines with lower levels energetically higher than $x_{\rm max}$ are neglected. This generalization will turn out to be important if NLTE-effects are included into our approach (cf. Sect. 4.2.3).

Finally, we allow for an integration between $0 < \nu < \nu_{\rm max}\le i_{\rm e}$, since for high ionization energies our line list may be incomplete (and useless, if one accounts for the vanishing flux) beyond a certain maximum frequency.

At first note, that the maximum possible oscillator strength is given by

$$\tilde r_{\rm max}= {\rm Min}\left(\frac{i_{\rm e} - \nu}{t} - l,\, -l_{\rm min}\right),$$

in accordance with Eq. (64). Accounting additionally for a possible incompleteness of the level list, $x_{1}\le x_{\rm max}$, and noting that the minimum value for r is given by $r_{\rm min}$, we have to extend this restriction further:

$$\tilde r_{\rm max}{=}{\rm Max}\Bigl({\rm Min}\left(\frac{i_{... ..._{\rm max}}{t}{-l}, {-l_{\rm min}} \right),~r_{\rm min}\Bigr).$$ (E1)

From this expression, we can derive the maximum frequency until which the integration has to be performed,

$$\nu_{\rm up}= {\rm Min}\bigl(\nu_{\rm max},\,\nu_r= i_{\rm e} - t(l+r_{\rm min})\bigr),$$ (E2)

since for frequencies larger than $\nu_r$ we have $\tilde r_{\rm max}= r_{\rm min}$ and the according number of lines is zero (cf. Eq. 61). Furthermore, the allowed range of l is given by

$$l_{\rm min}\le l \le l_{\rm max}= \frac{x_{\rm max}}{t} - r_{\rm min},$$ (E3)

where two subranges have to be defined in order to satisfy Eq. (E1):

At low line intensities, $l < x_{\rm max}/t + l_{\rm min}$, we have to account for the minimum ${\rm Min}\bigl((i_{\rm e} - \nu)/t - l,\, -l_{\rm min}\bigr)$, which introduces a threshold frequency $\nu_l= i_{\rm e} - t(l-l_{\rm min})\,<\,\nu_r$:

$$l < \frac{x_{\rm max}}{t}+l_{\rm min}\!:$$

$$\tilde r_{\rm max}= \left\{ \begin{array}{ll} -l_{\rm min}& ... ...}& \,\,{\rm for} \;\; \nu_l> \nu > 0 \\ \end{array} \right. .$$ (E4)

In the second regime with $x_{\rm max}/t - l < -l_{\rm min}$, the maximum oscillator strength $-l_{\rm min}$ cannot be reached any longer, and the appropriate threshold frequency $\nu_x=i_{\rm e} -x_{\rm max}$ introduces the following possibilities:

$$\frac{x_{\rm max}}{t}+l_{\rm min}< l < l_{\rm max}\!:$$

$$\tilde r_{\rm max}= \left\{ \begin{array}{ll} \frac{\display... ... & \,\,{\rm for} \;\; \nu_x> \nu > 0 \\ \end{array} \right. ,$$ (E5)

since $\nu_r> \nu_x$ always. ( $\nu_r= \nu_x$ is reached only at $l = l_{\rm max}$.)

By integrating Eq. (61) over frequency and accounting for these different cases (four in total), we finally obtain the following expressions for the function F defined in Eq. (73).

 

$$(l,t,\sigma,i_{\rm e},x_{\rm max},\nu_{\rm max},\gamma,l_{\rm min},r_{\rm min}) =$$ F

$$\left(10^{\displaystyle{-A l_{\rm min}}} - 10^{\displaystyle{A r_{\rm min}}}\right) \left(10^{\frac{{\nu_{\rm max}}}{{\sigma}}} -1 \right),$$ =

$$\left(l< \frac{x_{\rm max}}{t} + l_{\rm min}, \,\, \nu_{\rm max}< \nu_l\right).$$ (E6)

$${=} \frac{10^{{\displaystyle A \left(\frac{i_{\rm e}}{t}-l \r... ...u_l}}{{\sigma}}{\left(1{-}\frac{A\sigma}{t} \right)}} \right) +$$

 

$$+ 10^{\displaystyle{-A l_{\rm min}}} \left( 10^{\frac{{\nu_l}}{{\... ...laystyle{A r_{\rm min}}} \left(10^{\frac{{\nu_{\rm up}}}{{\sigma}}} -1 \right),$$

$$\left( l< \frac{x_{\rm max}}{t} + l_{\rm min}, \,\, \nu_{\rm max}> \nu_l \right).$$ (E7)

 

$$\left( 10^{\displaystyle{A \left(\frac{x_{\rm max}}{t}-l \right)}... ...A r_{\rm min}}} \right) \left(10^{\frac{{\nu_{\rm max}}}{{\sigma}}} -1 \right),$$ =

$$(l< l_{\rm max}, \,\, \nu_{\rm max}< \nu_x).$$ (E8)

$${=}\frac{10^{\displaystyle{A\left(\frac{i_{\rm e}}{t}-l \righ... ...ma}} \displaystyle{\left(1{-}\frac{A\sigma}{t}\right)}}\right)$$

 

$${+} 10^{\displaystyle{A\left( \frac{x_{\rm max}}{t}-l \right)}}\l... ...playstyle{A r_{\rm min}}}\left(10^{\frac{{\nu_{\rm up}}}{{\sigma}}} -1 \right),$$

$$(l < l_{\rm max}, \nu_{\rm max}> \nu_x).$$ (E9)

For convenience, we have summarize the required threshold values below:

$$\nu_{\rm up} {\rm Min}\bigl(\nu_{\rm max},\,\nu_r\bigr)$$ =

$$\nu_r i_{\rm e} - t(l+r_{\rm min})$$ =

$$\nu_l i_{\rm e} - t(l-l_{\rm min})$$ =

$$\nu_x i_{\rm e} - x_{\rm max}$$ =

$$l_{\rm max} \frac{x_{\rm max}}{t}-r_{\rm min}.$$ =

Note that due to the specific combinations of energies, frequencies and temperature derived above, the function F depends not independently on any of the variables $(t,\sigma,i_{\rm e},x_{\rm max},\nu_{\rm max})$, however only on certain ratios: The line intensity distribution function remains completely unchanged if the above variable set preserves its ratio with respect to either $t, \sigma$ or $i_{\rm e}$. In the following, we will mostly consider the normalization with respect to $\sigma$, i.e., understand F as function of $(t/\sigma,i_{\rm e}/\sigma,x_{\rm max}/\sigma$ and $\nu_{\rm max}/\sigma)$.

The most important fact concerning the derived function is the following: Depending on line intensity and the specific value of A, the distribution can show three different slopes, namely $2t/\sigma, t/\sigma$ and $\gamma-1$, where the first one and the last have been found already for the frequency dependent distribution function (cf. Sect. 4.2.1), whereas the second one is a new feature arising from the frequency integration.

Let us briefly consider under which condition which slope will show up. At first, assume that $x_{\rm max}= \nu _{\rm max}= i_{\rm e}$, i.e., both the level list and the line list shall be complete. In this case, Eqs. (E7) and (E9) have to be applied ($\nu_x= 0$).

For (very) low line intensities, l is approximately $l_{\rm min}$ and $\nu_l \approx i_{\rm e} = \nu_{\rm up}$. Thus, the second term in (E7) dominates and the result is similar to (E6), with maximum frequency $i_{\rm e}$. In consequence, F is independent on l and the apparent slope of the distribution is $2t/\sigma$ (remember, that $\Delta N(l) \propto 10^{2lt/\sigma} F$). In this situation, for (almost) all possible line frequencies the contributing oscillator strengths stretch from $r_{\rm min}$ to $r_{\rm max}= -l_{\rm min}$.

For larger l then, $\nu_l$ decreases, whereas $\nu_{\rm up}= i_{\rm e}$ for l being smaller than $-r_{\rm min}$. We encounter the case that the maximum possible oscillator strength $-l_{\rm min}$ can no longer be reached for large frequencies (cf. Eq. (E4), middle panel), and the apparent slope is controlled by the sign of A. For negative and not too small A, i.e., $\gamma > \gamma_{\rm crit}$,

$$\gamma_{\rm crit}= 1 + 2t/\sigma,$$ (E10)

the lower end of the oscillator strength distribution dominates by far, and F depends solely on the last term of (E7). In consequence, $\Delta N$ retains its slope of $2t/\sigma$.

For positive A, i.e., $\gamma < \gamma_{\rm crit}$, the situation is different. Now the first bracket of (E7) dominates, giving rise (via the combination of exponents $2lt/\sigma +A(i_{\rm e}/t-l) +\nu_l/\sigma(1-A\sigma/t)$) to an exact slope of $t/\sigma$, since the dependence on $\gamma$ cancels completely. This behaviour is finally reached also for negative A and larger l: For $l > -r_{\rm min}$, the upper frequential boundary $\nu_{\rm up}$ is $\nu_r$, which has the same dependency on l as $\nu_l$. Additionally, the impact of small oscillator strengths becomes smaller, simply because the last term in (E6) decreases with $\nu_r$ as function of l.

Figure E1: Frequency integrated line intensity distribution function: variation with $\gamma$ and $t/\sigma$. Basic parameters similar to the case of N IV, $i_{\rm e}/\sigma = 3.0, \nu _{\rm max}=x_{\rm max}= i_{\rm e} = 620$ kK, $l_{\rm min}= -1, r_{\rm min}= -8$ and ${\rm log}N(l_{\rm max}) =3.6$. Fully drawn: $\gamma = 1.0, t/\sigma = 0.23$ (corresp. to $T = 30\,000$ K); dotted: $\gamma = 1.3,1.6,3.0, t/\sigma = 0.23$. Dashed-dotted: $\gamma = 1.0, t/\sigma = 0.93$ (corresponding to $T= 120\,000$ K); dashed: $\gamma = 1.3,1.6, 3.0, t/\sigma =0.93$

Nothing changes for negative A and even larger l, when Eq. (E9) applies. The slope remains at its value $t/\sigma$. For positive A, however, there is a dramatic change for $l > i_{\rm e}/t +l_{\rm min}$. Due to the interrelation of line intensity and $\tilde r_{\rm max}$ (cf. the discussion in Sect. 4.2.1), the dependence on $t/\sigma$ cancels and only the $\gamma-1$ slope survives (the 2nd term in (E9) being now the dominating one). Thus for $\gamma$ (well) below $\gamma_{\rm crit}$ the apparent slope at high line intensities is coupled to the oscillator strength statistics. Finally if $\gamma \approx \gamma_{\rm crit}$ (corresponding to the case A=0 which cannot be treated by the above formalism), it turns out that the slope at large l smoothly changes from $(\gamma -1)$ to $t/\sigma$.

In summary, we have the following behaviour of $\Delta N$ if $x_{\rm max}= \nu _{\rm max}= i_{\rm e}$:

  

$$\gamma > \gamma_{\rm crit}\!: {\rm log}\Delta N \propto \left\{ \begin{array}{rl} \frac{\displayst... ...playstyle\sigma}\,l & \,\, {\rm for} \;\; l > - r_{\rm min} \end{array} \right.$$ (E11)

$${\rm log}\Delta N \propto \left\{ \begin{array}{rl} \frac{\displayst... ...frac{\displaystyle i_{\rm e}}{\displaystyle t}+l_{\rm min}. \end{array} \right.$$ (E12)

Figure E1 illustrates the described behaviour by means of the line intensity distribution function of N IV, normalized to ${\rm log}N(l_{\rm max}) =3.6$. Note that the ordinate stretches to (unphysical) negative values, in order to display the changes of apparent slope as function of l.

We have considered two case, namely $t/\sigma = 0.23$ and 0.93, respectively, to demonstrate the dependence on $\gamma_{\rm crit}$. In the first case then, $\gamma_{\rm crit}$ = 1.46, and the fully drawn line ( $\gamma = 1.0$) and the dotted ones $\gamma =1.3,1.6,3.0$ display the resulting distribution functions. As predicted by Eq. (E12), the curve for $\gamma = 1.0$displays two effective slopes, namely $t/\sigma$ and $(\gamma -1)$, where the dividing line intensity is given by $i_{\rm e}/t +l_{\rm min}= 12$. For $\gamma=1.3$, only one slope ($t/\sigma$) is present due to the third condition in Eq. (E12). In contrast, the behaviour for $\gamma = 1.6$ and 3 ( $>\gamma_{\rm crit}$, Eq. (E11)) depends on $2t/\sigma$ and $t/\sigma$, with a corresponding boundary at $l=-r_{\rm min}= 8$in our example.

For the larger value of $t/\sigma$ with $\gamma_{\rm crit}= 2.86$, we see the transition of slope $t/\sigma$ to $\gamma-1$ at $l= i_{\rm e}/t +l_{\rm min}= 2$ for the curves with $\gamma = 1.0$ (dashed-dotted), $\gamma=1.3$ and 1.6 (dashed). Only the case with $\gamma = 3 > \gamma_{\rm crit}$ reaches its asymptotic value of $t/\sigma \approx 1.9$, where also the steeper slope of $2t/\sigma$ for $l<-r_{\rm min}$ is clearly visible.

Figure E2: Frequency integrated line intensity distribution function: variation with $\nu _{\rm max}$ and $x_{\rm max}$. Basic parameters as in Fig. E1 with $T = 30\,000$ K and $\gamma = 1.0$. Fully drawn: $x_{\rm max}= \nu _{\rm max}= i_{\rm e}$; dotted: $x_{\rm max}= i_{\rm e}, \nu _{\rm max}= 400, 200, 100$ kK; dashed: $\nu _{\rm max}= i_{\rm e}, x_{\rm max}= 400, 200, 100$ kK

In Fig. E2 we demonstrate the influence of either varying the highest energetic level ( $x_{\rm max}$) or the maximum line frequency ( $\nu _{\rm max}$), if all other parameters are kept constant. As long as $x_{\rm max}= i_{\rm e}$ (dotted curves), the only influence of decreasing $\nu _{\rm max}$ concerns the region with $l < i_{\rm e}/t +l_{\rm min}$, i.e., Eq. (E6) (partly) replaces Eq. (E7), where the largest influence is close to $l_{\rm min}$. Only for very small values of $\nu _{\rm max}$ the complete first interval is affected. In consequence, even for $\gamma < \gamma_{\rm crit}$ the apparent slope becomes $2t/\sigma$, since F depends no longer on l.

In contrast, by diminishing $x_{\rm max}$ while keeping $\nu_{\rm max}= i\rm _e$ (dashed curves), the slope of the first interval is barely affected, unless $x_{\rm max}$becomes (very) small compared to $i_{\rm e}$ so that the transition value is close to $l_{\rm min}$ (cf. E12, first panel). The major effect of decreasing $x_{\rm max}$, however, is by shifting the dividing line (which is now a linear function of $x_{\rm max}$) to smaller line intensities. Thus, a line-distribution with $x_{\rm max}$ different from $i_{\rm e}$ resembles the line-distribution of a similar ion, however with much smaller ionization energy. This behaviour turns out to be important if one considers the NLTE line-strength statistics. Finally, if $x_{\rm max}$approaches zero, the line distribution becomes independent on any excitation effects. This limiting case, which corresponds to accounting for resonance lines only, leads to a line-statistics influenced solely by the underlying gf-distribution.

Figure E3: Frequency integrated line intensity distribution function: saturation effect. Basic parameters as in Fig. E2. Fully drawn: $x_{\rm max}= \nu _{\rm max}= i_{\rm e}$; dashed: $x_{\rm max}= 400$ kK, $\nu _{\rm max}= i_{\rm e}$; dotted: $x_{\rm max}= 400$ kK, $\nu _{\rm max}= 500, 400, 300$ kK; dashed-dotted: "saturation'' limit for $\nu _{\rm max}\le 220$ kK

Figure E3 illustrates the effect of diminishing both $x_{\rm max}$and $\nu _{\rm max}$. In principle, the effects are similar to the cases studied above, namely the transition value is changed via $x_{\rm max}$, and the slope of the first region increases to $2t/\sigma$. However, there exists another interesting effect, displayed by the dashed dotted curve: If the maximum considered frequency $\nu _{\rm max}$ falls below the value of $\nu_x=i_{\rm e} -x_{\rm max}$, the distribution function becomes "saturated'', i.e., does no longer change in shape (of course, the absolute value of F and thus the total line number decreases with $\nu _{\rm max}$). The reason for this saturation is given by the fact that Eqs. (E6) and  (E8) now control the behaviour of F, and that for $\nu_{\rm max}< \nu_x$ this function depends on $\nu _{\rm max}$ solely by a constant factor $(10^{\nu_{\rm max}/\sigma}-1)$ for all l.

Note, that in those cases when $x_{\rm max}$ is small compared to $i_{\rm e}$ (as is typical under NLTE-conditions, see Sect. 4.2.3), this condition applies for fairly large $\nu _{\rm max}$. In other words: The shape of the function is the same for all cutoff frequencies smaller than $i_{\rm e} -x_{\rm max}$: At maximum two slopes are present, namely either $2t/\sigma$ and $\gamma-1$ for $\gamma < \gamma_{\rm crit}$ or $2t/\sigma$ for $\gamma > \gamma_{\rm crit}$. This fact is essential for flux-weighted line-strength distribution function (Sect. 4.2.8), since the maximum frequency which has to be considered for this function is fairly small due to the decreasing flux at high energies.

Acknowledgements

We like to thank Ken Gayley, Stan Owocki and Achim Feldmeier for useful comments and suggestions. This project has been supported in part by the Deutsche Forschungsgemeinschaft under DFG grants Pu 117-1/2 and Pa 477-1/2.