4 What mimics the reflection effect in symbiotic binaries?

To get a better idea of the origin of the orbitally related wave-like variation we discuss it within the basic model of symbiotic binaries - the cool giant losing material via the stellar wind and the hot luminous compact object ionizing a portion of the neutral wind. Such composition creates a strong source of nebular emission in the binary, which often dominates the optical region (e.g. BFCyg, AG Dra, He2-467, AS338). Therefore we discuss the apparent variation in the optical continuum within an ionization model.

First, we introduce the simplest model of the ionization structure in symbiotic binaries.

4.1 A zero level ionization model

 

 

Table 2: Observed and calculated EM for selected objects

Object	$F_{\lambda}^{\rm obs}/10^{-13}$	Ref.	$\int_{V}n^{2}{\rm d}V$ (Eq. 9)	$L_{\rm ph}/\alpha_{B}$ (Eq. 10)	$\int_{V}n^{2}{\rm d}V$ (Eq. 13)	$\int_{V}n^{2}{\rm d}V^{c}$
	[ergcm-2s-1Å-1]		[cm-3]	[cm-3]	[cm-3]	[cm-3]
ZAnd	7.5a	1	3.4 $\; 10^{59}$	2.9 $\; 10^{59}$	1.7 $\; 10^{59}$
ZAnd	2.4 - 9.6b	Fig. 1	$1.9 - 7.5\; 10^{59}$			$1.9 - 6.0\; 10^{59}$
BFCyg	5.0a	2	3.8 $\; 10^{60}$	2.9 $\; 10^{60}$	d
BFCyg	2.7 - 7.9b	Fig. 1	$3.6 - 10\; 10^{60}$			$2.6 - 5.3\; 10^{60}$
AGDra	2.4 a	3	1.5 $\; 10^{59}$	1.5 $\; 10^{59}$	1.1 $\; 10^{59}$
AGDra	0.9 - 2.2b	Fig. 1	$0.5 - 1.3\; 10^{59}$			$3.5 - 8.0\; 10^{58}$
AXPer	2.0a	4	2.1 $\; 10^{59}$	1.2 $\; 10^{59}$	d
AXPer	1.0 - 2.5b	Fig. 1	$1.8 - 4.6\; 10^{59}$			$1.6 - 5.5\; 10^{59}$
V443Her	2.1a	5	3.0 $\; 10^{59}$	4.2 $\; 10^{59}$	1.9 $\; 10^{59}$
V443Her	1.1 - 2.1b	Fig. 1	$2.7 - 5.2\; 10^{59}$			$1.0 - 2.0\; 10^{59}$

a - from the energy distribution in the spectrum at $\lambda$3646^-Å.
b - from the dereddened U-magnitude at minimum and maximum, respectively:
ZAnd: U = 12.3 - 10.8, E_B-V = 0.35, d = 1.12kpc, BFCyg:
U = 12.4 - 11.3, E_B-V = 0.4, d = 4.6kpc, AGDra: U = 12.0 - 11.0, E_B-V = 0.05,
d = 1.0kpc (Appendix A), AXPer: U = 12.9 - 11.9, E_B-V = 0.27, d = 1.73kpc, V443Her: U = 12.2 - 11.5, E_B-V = 0.15, d = 2.0kpc.

c - the range of the observed EM referred in the literature. d - not applicable.
Ref.: 1 - Fig. 1 of Fernández-Castro et al. (1988), 2 - Fig. 3b of Fernández-Castro et al. (1990), Skopal et al. (1997),
3 - Fig. 3 of Mikolajewska et al. (1995), Appendix A, 4 - Fig. 2 of Mikolajewska & Kenyon (1992), Skopal (2000), 5 - Fig. 1 of Dobrzycka et al. (1993).

Figure 3: The Hi/Hii boundary calculated for X = 0.3, 1, 10, the stellar wind model characterized by the parameters $\gamma$ = 2.5 and $R_{\rm g}/A$ = 0.28

The extent of the ionized zone can be obtained from a parametric equation

$$f(r,\vartheta) - X = 0,$$ (7)

the solution of which defines the boundary between neutral and ionized gas at the orbital plane determined by a system of polar coordinates, $r,\,\vartheta$, with the origin at the hot star. The function $f(r,\,\vartheta)$ was treated for the first time by Seaquist et al. (1984), hereafter STB, for a steady state situation and pure hydrogen gas. The parameter X is given mainly by the binary properties - separation of the components, number of hydrogen ionizing photons, terminal velocity of the wind, $v_{\infty}$, and the mass-loss rate (for details see STB, Nussbaumer & Vogel 1987). The particle density of the ionized material is given by the velocity distribution of the giant's wind, which is assumed to be of the form

$$v_{\rm wind} = v_{\infty}(1- R/r)^{\gamma},$$ (8)

where r is the distance from the centre of the cool star, and R is the origin of the stellar wind ($\approx$the radius of the giant). Figure 3 shows examples of the Hi/Hii boundary for X = 0.3, 1, 10 and the parameter $\gamma$ = 2.5 in the wind model (8). STB used $\gamma$ = 0, which represents a constant velocity of the wind (= $v_{\infty}$); no acceleration zone above the giant's photosphere exists. However, stellar wind characterized by the parameter $\gamma > 0$ will accelerate gradually above the photosphere to its terminal velocity; it is a more realistic model. Therefore, according to Schröder (1985), we adopted $\gamma$ = 2.5 in our calculations. In this model, the source of the nebular radiation is physically displaced from the cool giant photosphere.

We now test if the amount of emission produced by this model is consistent with observations.

4.2 Emission measure

We investigate the balance between the observed nebular emission in the continuum and that produced by the ionization model. The nebular flux largely depends on the number of hydrogen recombinations, and is proportional to $\int \!n_{+}n_{\rm e}\,{\rm d}V$, (the so called emission measure - EM); n₊ and $n_{\rm e}$ is the concentration of ions (protons) and electrons, respectively.

(i) Observations: The quantity of the EM can be estimated, for example, from the measured flux, $F^{\rm obs}_{\lambda}$ ( $\rm erg\,cm^{-2}\,s^{-1}\,\AA^{-1}$), of the nebular continuum at the wavelength $\lambda$, according to the equation

$$4\pi d^{2} F^{\rm obs}_{\lambda} = \varepsilon_{\lambda}\!\int_{V}\!\! n_{+}n_{\rm e}\,{\rm d}V,$$ (9)

in which d is the distance to the object, $\varepsilon_{\lambda}$ is the volume emission coefficient per electron and per ion ( $\rm erg\,cm^{3}\,s^{-1}\,\AA^{-1}$) and V is the volume of the ionized zone. The nebular flux can be obtained from the energy distribution in the spectrum. Its upper limit can also be estimated from the dereddened U-magnitude of systems, in which the nebular continuum dominates the optical.

(ii) Model: The source of the nebular radiation in the model is the ionized region, in which the rate of ionization/recombination processes is balanced by the rate of photons, $L_{\rm ph}$ (photonss^-1), capable of ionizing the element under consideration. In the case of pure hydrogen we can write the equilibrium condition as

$$L_{\rm ph} = \alpha_{B} \!\int_{V} \!\!n_{+}(r)n_{\rm e}(r)\,{\rm d}V,$$ (10)

where $\alpha_{B}$ ( $\rm cm^{3}\,s^{-1}$) is the total hydrogenic recombination coefficient. The number of hydrogen ionizing photons, $L_{\rm ph}$ is

$$L_{\rm ph}= \frac {L_{\rm h}}{\sigma T_{\star}^{4}}\pi (hc)^{... ...{912}_{0}\!\! \lambda B_{\lambda}(T_{\star})\,{\rm d}\lambda ,$$ (11)

where $L_{\rm h}$[ergs^-1] and $T_{\star}$ is the total luminosity and the temperature of the hot star, respectively. Having independently determined $L_{\rm ph}$ we compare the EM given by observations (Eq. 9) and that required by the ionization model (Eq. 10). In Table 2 we give results for the objects with available parameters. We obtained the observed fluxes from the energy distribution in the spectrum at $\lambda 3646^{-}$Å and used the hydrogen emission coefficient $\varepsilon_{3646^{-}} = 3.36\; 10^{-28}\,\,\rm erg\,cm^{3}\,s^{-1}\,\AA^{-1}$ (Gurzadyan 1997). For a comparison, we also estimated the nebular flux from the U-magnitude according to the calibration of Henden & Kaitchuck (1982). In this case we adopted $\varepsilon_{U} = 1.9\; 10^{-28}\,\,\rm erg\,cm^{3}\,s^{-1}\,\AA^{-1}$ as the average of $\varepsilon_{3646^{-}}$ and $\varepsilon_{3646^{+}}$.

We can calculate the EM directly by integrating emission contributions throughout the volume of the ionized zone defined by the model (Eq. 7). The calculation of the Hi/Hii and Hei/Heii boundary in 2-D representation can be found in STB and Nussbaumer & Vogel (1987), respectively. Using Nussbaumer & Vogel (1989), we derive an upper limit to the modeled EM assuming the sphere around the cool star to be fully ionized from r = Q to $r = \infty$. The parameter Q is the location of the Hi/Hii boundary on the line joining the cool and the hot star ( $Q > R \sim R_{\rm g}$). The particle density n(r) is given by the mass-loss rate of the wind

$$\dot M=4\pi r^{2}\mu m_{\rm H}n(r)v_{\rm wind},$$ (12)

where $\mu$ is the mean molecular weight, $m_{\rm H}$ is the mass of the hydrogen atom and the velocity of the wind is given by Eq. (8). Then the EM for a completely ionized medium ($n_{\rm e}$ = n₊) can be calculated from the following analytical expression

$$\int_{V} \!\!n(r)^{2}\,{\rm d}V = 4\pi\Big[\frac{\dot M}{4\... ...int_{Q}^{\infty}\!\! \frac{{\rm d}r}{r^{2}(1-R/r)^{2\gamma}}$$

$$=4\pi\Big[\frac{\dot M}{4\pi\mu {\rm m}_{\,\rm H} v_{\infty... ...ma)} \Big[1-\Big(1-\frac{R}{Q}\Big)^{1-2\gamma}\Big]\nonumber$$

for $\gamma \ne~ 0.5$
and

$$= - 4\pi\Big[\frac{\dot M}{4\pi\mu {\rm m} _{\rm H} v_{\inft... ...\frac{R}{Q}\right) \hspace*{0.5cm} {\rm for}~~\gamma~ =~ 0.5.$$ (13)

Nussbaumer & Vogel (1989) derived formula (13) for the special case of $\gamma$ = 2.5. This approximation can be used for cases when the major part of the circumbinary environment is ionized, i.e. the parameter X > 10 (Fig. 3). We applied the formula (13) to ZAnd, AGDra and V443Her (Col. 6 in Table 2). The calculated EM is somewhat lower than that indicated by observations. However, given the large uncertainties in fundamental parameters (mainly in d, $\dot M$ and $L_{\rm h}$, see Appendix A) we still consider the calculated values to be consistent with observations. However, if the calculated EM according to Eq. (13) is really lower than that measured, this could imply the presence of an additional source of particles in the system - for example, the hot star wind. Note that in the case of an open ionized zone a portion of the $L_{\rm ph}$ photons escapes the system, and thus an injection of new particles (emitters) into such a zone will produce an extra flux.

We find that the observed EM is consistent with that produced by the ionization model. This implies that all the $L_{\rm ph}$ photons consumed by the particles of the giant wind are needed to produce the observed nebular flux. However, in the model of the reflection effect only a small part of ionizing photons, $(R_{\rm g}/2A)^{2}\times L_{\rm ph}$ (Sect. 3), can be used to produce the nebular radiation. This causes the discrepancy between the ${\beta }$ parameter given by observations and much larger value, required by models of reflection effect to explain the amplitude of the LCs.

We now will demonstrate that the variation in the EM is responsible for the investigated wave-like variation in the LCs.

4.3 Variation in the emission measure

The quantity of the EM in the continuum also varies as a function of the orbital phase (e.g. Fernandez-Castro et al. 1988; Mikolajewska et al. 1989; Mikolajewska & Kenyon 1992; Dobrzycka et al. 1993). To investigate this variability we express Eq. (9) in the scale of magnitudes ( $m_{\lambda}=-2.5\log(F_{\lambda})+q_{\lambda}$) as

$$m_{\lambda} = -2.5\log(EM) + C_{\lambda},$$ (14)

where

$$C_{\lambda} = q_{\lambda} - 2.5\log\left(\frac{\varepsilon_{\lambda}} {4\pi d^{2}}\right),$$ (15)

in which the constant $q_{\lambda}$ defines magnitude zero. For the standard photometry and fluxes in units of $\rm erg\,cm^{-2}\,s^{-1}\AA^{-1}$, q_U = -20.9, q_B = -20.36 and q_V = -21.02 (Henden & Kaitchuck 1982). Using Eq. (14) we constructed the LCs from the measured values of the EM in BFCyg and ZAnd and compared them to those obtained photometrically. We calculated the B-magnitudes, because in these objects the nebular contribution still dominates the continuum and the emission coefficient is nearly constant in this region. We used $\varepsilon_{B} = 0.5~ 10^{-28}\,\rm erg\,cm^{3}\,s^{-1}\,\AA^{-1}$ (Gurzadyan 1997) and d = 4.6 and 1.12kpc for BFCyg and ZAnd, respectively. Figure 4 shows that the variation in the EM follows well that observed in the LCs. In other systems, where only a few measurements of the EM at different orbital phases are available (AXPer, AGDra, V443Her), we estimated the amplitude as

$$\Delta m_{EM} = -2.5\log\frac{EM_{\rm min}}{EM_{\rm max}}\cdot$$ (16)

We find that the observed range of the EM is comparable with the amplitude of LCs in the U band of these systems (the last column in Table 2). We conclude that the variation in the EM produces the observed phase-dependent variation in the LCs of symbiotic binaries.

Note that the decrease of the LC amplitude with wavelength is caused by an increase of the cool giant contribution, which does not vary with the orbital phase. The same effect takes place towards the short wavelengths due to an increase in the stellar contribution from the hot star (see Fig. 4 of Kenyon et al. 1993).

Figure 4: Top: variation in the EM of BFCyg as a function of the orbital phase. The data (Mikolajewska et al. 1989) were converted into the B-magnitudes according to Eq. (14). Compared is the LC in the B band obtained photometrically during the same period, between JD 2445700 and 2446718 (Hric et al. 1993). Bottom: the same as the top, but for ZAnd. Measurements of the EM were taken from Fernandez-Castro et al. (1988), and photometric B-magnitudes from Fig. 1, but omitting the active phase. These results show that the variation in the EM is fully responsible for the variation in the LCs

4.4 Variation in the hydrogen lines

Together with the periodic variation in the nebular continuum, the same type of variability is observed in fluxes of Balmer lines. The source of hydrogen emission in lines is also the Hii region. Therefore the variation in both the hydrogen continuum and the lines should be of the same nature. So the EM derived from the hydrogen continuum should be consistent with that given by the Balmer lines, assuming an optically thin regime and the case B of recombination. In Table 3 we summarize results for the line H${\beta }$.

 

 

Table 3: Emission measure in the line H${\beta }$ for selected objects

Object	$F_{\beta}/10^{-12}$a	Ref.	$EM_{\beta}$
	[ergcm-2s-1]		[cm-3]
ZAnd	37.3	1	4.5 $\; 10^{58}$
BFCyg	18 - 65	2	0.4 $- 1.3\; 10^{60}$
AGDra	6.4 - 9.4	3	6.2 $- 9.1\; 10^{57}$
AXPer	7.2 - 22	4	2.1 $- 6.3\; 10^{58}$
V443Her	6.5 - 15	5	2.5 $- 5.7\; 10^{58}$

^a - dereddened fluxes with E_B-V referred in Table 2.
Ref.: 1 - Mikolajewska & Kenyon (1996), 2 - Mikolajewska et al. (1989),
3 - Mikolajewska et al. (1995), 4 - Mikolajewska & Kenyon (1992), 5 - Dobrzycka et al. (1993).

We calculated the $EM_{\beta}$ according to Eq. (9) using the volume emission coefficient in the H${\beta }$ line, $\varepsilon_{\beta} = 1.23\; 10^{-25}$ergcm³s^-1. We can see that the $EM_{\beta}$ is by a factor of $\sim$5, on average, lower than that derived from the Balmer continum. This discrepancy can be caused by atenuation of the Hi emission due to (i) an absorption originating from the cool component wind and (ii) the self-absorption effect, when the nebula is partially opaque in the line H${\beta }$ and H$\gamma$, in addition to the case B. The first case is demonstrated well by sets of observations along the orbital cycle (e.g. Oliversen et al. 1985). The wind absorption component is more pronounced around the inferior conjunction of the giant (see Fig. 3), which makes the amplitude of the variation in line fluxes even larger. The second case is indicated by the observed very steep Balmer decrement (H${\alpha}$/H${\beta }$/H$\gamma$ $\approx$ 7/1/0.4, for objects in Table 3), which is not consistent with the theoretical prediction (2.85/1/0.47) that the excitation is due to photoionization. Moreover, attenuation of hydrogen fluxes could also be in part due to electron collisions in a dense Hii region, and/or partly caused by selective interstellar absorption. Therefore the $EM_{\beta}$ represents only the lower limit of the EM derived from the Balmer continuum.

As the variation in the emission of the Balmer lines is connected with the Hii region - the dominant source of the optical/near-UV continuum - the observed periodic wave-like variation in the continuum should always be followed by a similar variation in Balmer lines.

4.5 Why does the emission measure vary?

To produce the wave-like variation in the LCs along the orbital cycle, the nebula - the main source of the optical continuum in symbiotic binaries - has to be partially optically thick and of a non-spherical shape. In our simple ionization model the opacity, $\kappa$, of the ionized emission medium decreases with the distance from the cool star, since $\kappa \propto n \propto r^{-2}$ (i.e. its parts nearest to the giant's surface will be most opaque). It is probable that the observed emission will also depend on the extension of the ionized region. Below we give a qualitative description on how, or whether, it is possible to produce the observed profile of LCs within the ionization model mentioned above.

(i) In the case of an oval shape of the Hii zone (Fig. 3; a small parameter X), its total emission will be atenuated more at positions of the inferior and superior conjunction of the cool star (the orbital phase $\varphi$ = 0 and 0.5, respectively) than at positions of $\varphi$ = 0.25 and 0.75, respectively. Such apparent variation in the EM will produce both the primary and the secondary minimum in the LC and will thus mimic the ellipsoidal effect in binaries containing a giant star. This type of LC profile corresponds to the parameter $a \sim 1$ (right panels of Fig. 1).

(ii) A gradual opening of the Hii zone (approximately ) will make it optically thinner behind the hot star (outside the binary around $\varphi$ = 0.5). Thus, more of the nebular radiation, relative to the case (i), will be observed at the position of $\varphi \sim 0.5$. The secondary minimum in LCs will therefore become less pronounced or flat, and/or a maximum at $\varphi \sim 0.5$ can arise. The LC profile here should be characterized by the parameter a > 0.5 (mid panels of Fig. 1).

(iii) Given an extensive emission zone (X > 10) one can imagine the partially optically thick portion of the Hii region as a cap on the Hi/Hii boundary around the binary axis. This resembles the geometry of the reflection effect, but the emission region causing the light variation is physically displaced from the giant's surface (see also a sketch in Baratta & Viotti (1990), who drawn such regions for Feii and Ciii lines). In these cases the LC profile is similar to that of the reflection effect, i.e. the parameter $a \le 0.5$ (left panels of Fig. 1).

A relationship between the parameters a and X supports the connection between the shape of the LC and the extent of the symbiotic nebula mentioned above (Fig. 5, Appendix B).

However, the real structure of the ionized region in symbiotic binaries is probably much more complex. Currently it is being intensively investigated (e.g. Schmid 1998). For example, the observed systematic variation in the minima position (i.e. an apparent change of the orbital period) requires an asymmetrical shape of the Hii zone with respect to the binary axis (see Sect. 4.1 of Skopal 1998a in more detail). A modification of the STB model, which includes effects of the orbital motion, is outlined in Appendix C.