3 Emission regions of finite lateral width

In our radiation-hydrodynamical calculations, energy conservation is enforced and the radiative luminosity of the infinite layer per unit area equals the accretion energy $\dot m$ $\upsilon _{\rm o}$. Any real emision region, however, has a finite lateral width and looses energy not only from its top and bottom surfaces but also from its sides (Fig. 1). Two-dimensional radiation hydrodynamics would then be needed to calculate the temperature structure. In this section we discuss to what extent our one-dimensional results are still applicable to regions of finite extent.

  
3.1 Emitted spectra

We consider an emission region as depicted in Fig. 1, with finite width D, cross section D², and field strength B. In a first step, we adopt the temperature and velocity profiles along the flow lines, T(x) and $\upsilon(x)$, calculated for infinite D also for the case of finite D.

In the Rybicki code the radiation transport equation was solved with a mean cyclotron absorption coefficient and electron scattering was included. For the emission region of finite extent, we calculate the outgoing flux at angle $\vartheta$ by ray tracing using the temperature profiles along slanted paths as shown schematically in Fig. 1. For rays starting or ending on the side surfaces, the temperature and density profiles were truncated appropriately. We account separately for the cyclotron emissivities in the ordinary ray (index o) and the extraordinary ray (index e), and add 50% of the free-free emissivity with Gaunt factor to both. We neglect electron scattering in the emission region, but include the atmospheric albedo $A_{\nu}$ (van Teeseling et al. 1994). Each ray yields a contribution $\Delta {\cal I}_{\nu}$ to the integrated intensity ${\cal I}_{\nu}$(in ergs^-1Hz^-1sr^-1) in that direction and the summation is extended over n rays,

 

$${{\cal I}_{\nu}(\vartheta) = \sum_{i=1}^n \Delta {\cal I}_{\nu} (... ...theta, s)\,{\rm e}^{-\tau_{\nu,{\rm o}}^{(i)}(\vartheta,s)}\,{\rm d}s \right. } \hspace*{10mm}+~ \left.\,\int\limits_0^{s_{\max}^{(i)}}\epsilon_{... ...rtheta, s)\,{\rm e}^{-\tau_{\nu,{\rm e}}^{(i)}(\vartheta,s)}\,{\rm d}s \right),$$ (22)

where $\tau_{\nu,{\rm o}}^{(i)}(\vartheta,s)$ and $\tau_{\nu,{\rm e}}^{(i)}(\vartheta,s)$ are the optical depths of the ordinary and extraordinary rays along path i at angle $\vartheta$, $s_{\rm max}^{(i)}$ is the pathlength along that ray and $\Delta \sigma^{(i)}$ the effective projected area associated with it. The albedo contribution at $\vartheta$ is calculated as $A_{\nu}\,{\cal I}_{\nu} (\pi-\vartheta)$ and is not yet included in Eq. (22). Reprocessing of the flux absorbed in the white dwarf atmosphere is not considered in this paper and the corresponding flux is, therefore, missing from our spectra. The spectral luminosity $L_{\nu}$ is obtained by integrating Eq. (22) over $4\pi$ and the total luminosity L by integration over all frequencies.

Figure 8 shows the spectral flux at $\vartheta = 5^{\circ }$against the field direction emitted by an emission region with B = 30MG and an area of 10¹⁶cm² (D = 10⁸cm) on an 0.6 $M_{\odot }$ white dwarf at a distance of 10 pc. Cyclotron emission dominates for low $\dot m$ and bremsstrahlung for high $\dot m$. Free-free absorption becomes important near 10¹⁵Hz at the highest $\dot m$, but in reality this spectral region is dominated by the quasi-blackbody component produced by reprocessing of the incident flux in the white dwarf atmosphere. The results of WB96 on the ratio of the cyclotron vs. bremsstrahlung luminosities as a function of $\dot m$ and B remain basically valid, but will be modified if the shock is buried in the atmosphere and X-ray absorption is accounted for.

Figure 9 illustrates the optical depth dependence of the cyclotron spectra at $\vartheta = 80^{\circ }$. Cyclotron emission lines at low $\dot m$ change into absorption features at high $\dot m$. Since in real emission regions the fractional area of the high-$\dot m$ section is small (Rousseau et al. 1996) observed spectra show emission lines.

  
3.2 Specific luminosity

An isolated emission region of lateral width D, shock height $h_{\rm sh}$, and the temperature profile T(x) of the infinite layer appropriate for the mass flow density $\dot m$ will have ${\cal L} = \dot m$ $\upsilon _{\rm o}$ for optically thin and ${\cal L} \ge \dot m$ $\upsilon _{\rm o}$ for optically thick emission. The overestimate in the latter case results from radiation emerging from the sides of the region without a compensating influx. For the optical depths considered here, bremsstrahlung is practically free of such overestimate, cyclotron radiation is not.

Let us assume for the infinite layer that $\dot m \upsilon_{\rm o}$feeds two components of ${\cal L}$, namely ${\cal L}_{\rm thin}$ and ${\cal L}_{\rm thick} = \dot m \upsilon_{\rm o} - {\cal L}_{\rm thin}$. For finite D, we then have

$${\cal L} \simeq {\cal L}_{\rm thin} + \left(\dot m \upsilon_{... ...1+\frac{2\,h_{\rm sh}}{D}\right) \ge \dot m \upsilon_{\rm o} ,$$ (23)

where the second term in brackets is the ratio of the total surface area of the emission region over the sum of top and bottom areas. For simplicity, we have neglected the temperature variation over the surface of the emission region and taken the energy loss per unit area as constant. We rearrange the terms in Eq. (23) to form a quantity A which relates ${\cal L}/\dot m \upsilon_{\rm o}$ to the aspect ratio of the emission region, $h_{\rm sh}$/D,

$$A = \left(\frac{{\cal L}}{\dot m \upsilon_{\rm o}}-1\right)\f... ...q 1 - \frac{{\cal L}_{\rm thin}}{\dot m \upsilon_{\rm o}}\cdot$$ (24)

Figure 10 shows A as a function of ${\dot m}\,B_7^{-2.6}$, calculated for model columns with $h_{\rm sh}$/ D = 0.1, 1 and 10 in the way described in the previous section. To a first approximation, A is independent of $h_{\rm sh}$/D and the relative luminosity error $({\cal L} - \dot m )/\dot m \upsilon_{\rm o}$ increases proportional to $h_{\rm sh}$/D for a given ${\dot m}\,B_7^{-2.6}$. The quantity A is negligibly small for large ${\dot m}\,B_7^{-2.6}$ where bremsstrahlung dominates and reaches $A \simeq 0.4$ for low ${\dot m}\,B_7^{-2.6}$ where cyclotron radiation dominates. Note that A never reaches the optically thick limit of unity because bremsstrahlung and optically thin cyclotron emission always contribute. To give an example, $h_{\rm sh}$/D = 0.5 and $A \simeq 0.4$ imply ${\cal L}/\dot m \upsilon_{\rm o} \simeq 1.4$, i.e. an overestimate of the luminosity by 40%.

Figure 10: Quantity A from Eq. (24) measuring the excess luminosity ${\cal L}/\dot m$ $\upsilon _{\rm o}$ as function of the aspect ratio of the emission region, $h_{\rm sh}$/D. The symbols refer to narrow and pillbox-shaped emission regions with D/ $h_{\rm sh}$= 0.1 ( $\hbox{$^\circ$ }$), D/ $h_{\rm sh}$= 1 ( $\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil \penalty50\h... ...x{\rlap{$\sqcap$ }$\sqcup$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi$), D/ $h_{\rm sh}$= 10  ($\triangle$), the overplotted + and $\times$ refer to field strengths of 10 and 100 MG, respectively, the uncrossed symbols to B = 30 MG.

In order to assess the size of the possible error for AMHer stars, we estimate $h_{\rm sh}$/D for a typical accretion rate of $\dot M = 10^{16}$gs^-1 as a function of $\dot m$. The linear width of the emission region is $D \simeq (\dot M/\dot m)^{1/2} \simeq 10^8/\dot m^{1/2}$cm with $\dot m$ in gcm^-2s^-1. For a bremsstrahlung-dominated flow, Eq. (14) yields $h_{\rm sh,brems}/D \simeq 0.48/\dot m^{1/2}$ (for $M_{\rm wd}$ = 0.6 $M_{\odot }$) which is less than unity since in this case gcm^-2s^-1. For a cyclotron-dominated flow, Eq. (21) correspondingly yields $h_{\rm sh,cyc}/D \simeq 2\dot m^{1/2}/B_7^{2.6}$ which is again less than unity since now gcm^-2s^-1 and typically B₇ > 1. Hence, ${\cal L}$ is seriously overestimated only for isolated narrow subcolumns which are not radiatively shielded by neighboring fluxtubes.

  
3.3 Temperature correction

Let the application of the unmodified one-dimensional temperature profile T(x)yield a specific luminosity ${\cal L} = r\,\raisebox{0.4ex}{.}\,\dot m \upsilon_{\rm o}$ with r > 1. We can then take then either: (i) identify the parameters of this emission region with those appropriate for the increased mass flow rate $\dot m' = r\,\raisebox{0.4ex}{.}\,\dot m$; or (ii) recalculate the emission for a reduced mass flow rate $\dot m'' = \dot m/r$ and identify temperature and emission of that region as appropriate for the initial $\dot m$. In case (ii), $T_{\max}$ and $x_{\rm s}$ are reduced to $T_{\max}$($\dot m''$) and $x_{\rm s}$($\dot m''$). This approach demonstrates that the rising sections of the relations displayed in Fig. 5 (Eq. (19)) and Fig. 6 (Eq. (20)) are further depressed for narrow columns, while the horizontal parts, where optically thin bremsstrahlung dominates, are not affected. Both approaches secure energy conservation but cannot replace a proper treatment of the problem. They are not recommended for isolated tall columns.