Issue 
A&A
Volume 641, September 2020



Article Number  A175  
Number of page(s)  16  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/202038414  
Published online  28 September 2020 
Determining massaccretion and jet massloss rates in postasymptotic giant branch binary systems^{⋆}
^{1}
Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia
email: dylan.bollen@kuleuven.be
^{2}
Astronomy, Astrophysics and Astrophotonics Research Centre, Macquarie University, Sydney, NSW 2109, Australia
^{3}
Instituut voor Sterrenkunde (IvS), Celestijnenlaan 200D, KU Leuven 3001, Belgium
Received:
13
May
2020
Accepted:
27
July
2020
Aims. In this study we determine the morphology and massloss rate of jets emanating from the companion in postasymptotic giant branch (postAGB) binary stars with a circumbinary disc. In doing so we also determine the massaccretion rates onto the companion, and investigate the source feeding the circumcompanion accretion disc.
Methods. We perform a spatiokinematic modelling of the jet of two wellsampled postAGB binaries, BD+46°442 and IRAS 19135+3937, by fitting the orbital phased time series of Hα spectra. Once the jet geometry, velocity, and scaled density structure are computed, we carry out radiative transfer modelling of the jet for the first four Balmer lines to determine the jet densities, thus allowing us to compute the jet massloss rates and massaccretion rates. We distinguish the origin of the accretion by comparing the computed massaccretion rates with theoretically estimated massloss rates, both from the postAGB star and from the circumbinary disc.
Results. The spatiokinematic model of the jet reproduces the observed absorption feature in the Hα lines. The jets have an inner region with extremely low density in both objects. The jet model for BD+46°442 is tilted by 15° with respect to the orbital axis of the binary system. IRAS 19135+3937 has a smaller tilt of 6°. Using our radiative transfer model, we find the full 3D density structure of both jets. By combining these results, we can compute the massloss rates of the jets, which are of the order of 10^{−7} − 10^{−5} M_{⊙} yr^{−1}. From this we estimate massaccretion rates onto the companion of 10^{−6} − 10^{−4} M_{⊙} yr^{−1}.
Conclusions. Based on the massaccretion rates found for these two objects, we conclude that the circumbinary disc is most likely the source feeding the circumcompanion accretion disc. This is in agreement with the observed depletion patterns in postAGB binaries, which is caused by reaccretion of gas from the circumbinary disc that is underabundant in refractory elements. The high accretion rates from the circumbinary disc imply that the lifetime of the disc will be short. Mass transfer from the postAGB star cannot be excluded in these systems, but it is unlikely to provide a sufficient masstransfer rate to sustain the observed jet massloss rates.
Key words: stars: AGB and postAGB / binaries: spectroscopic / circumstellar matter / stars: massloss / ISM: jets and outflows / accretion, accretion disks
© ESO 2020
1. Introduction
Binarity can have a significant impact on the evolution of stars from low to intermediate mass. The binary interactions in these systems can alter their massloss history, orbital parameters, and lifetimes and can lead to other phenomena such as excretion and accretion discs, jets, and bipolar nebulae (Hilditch 2001). PostAGB stars in binary systems are no exception. They are stars of low to intermediate mass in a final transition phase after the AGB (Van Winckel 2003). The luminous postAGB star in these binary systems is in orbit with a mainsequence (MS) companion of low mass (0.1−2.5 M_{⊙}, Oomen et al. 2018). Due to their binary interaction history, postAGB binary systems end up with periods and eccentricities that are currently unexplained by theory (Van Winckel 2018).
During the previous AGB phase, the star endures a period of mass loss as high as 10^{−4} − 10^{−3} M_{⊙} yr^{−1} (Ramstedt et al. 2008). When in a binary system, the mass loss of the AGB star can be concentrated on the orbital plane of the system, with the bulk of the mass being ejected via the L2 Lagrangian point (Hubová & Pejcha 2019; BermúdezBustamante et al. 2020). The focused mass loss of the star can then become a circumbinary disc (Shu et al. 1979; Pejcha et al. 2016; MacLeod et al. 2018). Observational studies have confirmed the presence of such discs in postAGB binary systems. Many postAGB stars have a nearIR dust excess in their spectral energy distribution (SED) that can be explained by dust in the proximity of the central binary system. The observed dust excess is a clear signature of dust residing in a circumbinary disc, close to the system (De Ruyter et al. 2006; Deroo et al. 2006, 2007; Kamath et al. 2014, 2015). The compact nature of the infrared dust excess has also been confirmed through interferometric studies (Bujarrabal et al. 2013; Hillen et al. 2013, 2016; Kluska et al. 2018). Additionally, Hillen et al. (2016) and Kluska et al. (2018) identified a flux excess at the location of the companion in the reconstructed interferometric image of postAGB binary IRAS 08544−4431. This flux excess is too large to originate from the companion, and most likely stems from an accretion disc around the companion.
Another commonly observed phenomenon in postAGB binaries is a highvelocity outflow or jet (Gorlova et al. 2012). Optical spectra of these objects show a blueshifted absorption feature in the Balmer lines during superior conjunction when the companion star is located between the postAGB star and the observer (Gorlova et al. 2012, 2015), as can be seen in Fig. A.1. The absorption feature in the Balmer lines is interpreted in terms of a jet launched from the vicinity of the companion that scatters the continuum light from the postAGB star travelling towards the observer during this phase in the binary orbit (Gorlova et al. 2012). Due to the orbital motion of the binary, the photospheric light of the postAGB star shines through various parts of the jet, providing a tomography of the jet. Hence, the orbitalphase dependent variations in the Balmer lines of these jetcreating postAGB binaries contain an abundance of information about the jet and the binary system (Bollen et al. 2017, 2019).
The jets are likely launched by an accretion disc around the companion. An unknown component in these jetcreating postAGB binaries is the source feeding the circumcompanion accretion disc that launches the jet. Direct observations of the masstransfer to the circumcompanion disc do not exist. The two plausible sources are the postAGB star, which could transfer mass via the first Lagrangian point (L1) to the companion, and the reaccretion from the circumbinary disc around the system. While this mass transfer has not yet been observed directly, we observe refractory element depletion in the atmosphere of postAGB stars in binary systems (Waters et al. 1992; Van Winckel et al. 1995; Gezer et al. 2015; Kamath & Van Winckel 2019). It has been suggested that this depletion pattern is caused by reaccretion of circumbinary gas, which is depleted of refractory elements by the formation of dust in the disc. Oomen et al. (2019) modelled this depletion pattern by implementing the reaccretion of metalpoor gas in their evolutionary models obtained using the Modules for Experiments in Stellar Astrophysics (MESA) code. They compared these models with 58 observed postAGB stars and found that initial massaccretion rates must be greater than 3 × 10^{−7} M_{⊙} yr^{−1} in order to obtain the observed depletion patterns.
In our previous study (Bollen et al. 2017) we used the time series of Hα line profiles to show that jets in postAGB binaries are wide and can reach velocities of ∼700 km s^{−1}. These velocities are of the order of the escape velocity of a MS star, pointing to the nature of the companion. In our recent study (Bollen et al. 2019) we created a more sophisticated spatiokinematic model for the jets, from which we determined their geometry, velocity, and scaled density structure.
In this paper, we fully exploit the potential of the tomography of the jet from the first four Balmer lines: Hα, Hβ, Hγ, and Hδ. We compute a radiative transfer model of the jet, with the aim of estimating the massloss rate of the jet. We do this in two main parts: (1) the spatiokinematic modelling, as described by Bollen et al. (2019), and (2) the radiative transfer modelling of the jet. Here, we focus on part 2 and the massejection and accretion rates. We use two wellsampled, jetlaunching postAGB binaries for our analysis: BD+46°442 (Gorlova et al. 2012; Bollen et al. 2017) and IRAS 19135+3937 (Gorlova et al. 2015; Bollen et al. 2019). Both objects have been observed for the past ten years with the HERMES spectrograph mounted on the Mercator telescope, La Palma, Spain (Raskin et al. 2011), providing a good amount of data covering the orbital phase of the binary.
The paper is organised as follows. We describe the methods of our spatiokinematic modelling and radiative transfer modelling in Sect. 2. We present the results for BD+46°442 and IRAS 19135+3937 in Sects. 3 an 4, respectively. We discuss these results in Sect. 5 and give a conclusion and summary in Sect. 6.
2. Methods
In this study, we expand on the spatiokinematic model carried out in our previous work (Bollen et al. 2019) by adding new components in the jet structure and we include a new radiative transfer model. Splitting the calculations into two parts, the spatiokinematic modelling and the radiative transfer modelling, allows us to fit the jet structure and obtain the jet massloss rates. In the following subsections, we give a short description of the spatiokinematic modelling part of the fitting, including improvements of the technique pioneered by Bollen et al. (2019), followed by a description of the new radiative transfer modelling.
2.1. Spatiokinematic modelling of the jet
To obtain the geometry and kinematics of the jet, we follow the modelfitting routine used by Bollen et al. (2019). In brief, we create a spatiokinematic model of the jet, from which we reproduce the absorption features in the Hα line. The modelled lines are then fitted to the observations. To fit our model to the data, we use the emceepackage, which applies an MCMC algorithm (ForemanMackey et al. 2013). This gives us the bestfitting parameters for the jet.
The model consists of three main components: the postAGB star, the MS companion, and the jet. The location of the postAGB star and the companion are determined for each orbital phase by the orbital parameters listed in Table B.1. The jet in the model is a double cone, centred on the companion. The postAGB star is approximated as a uniform flat disc facing the observer. We trace the light travelling from the postAGB star, along the line of sight towards the observer. When a ray from the postAGB star goes through the jet, the amount of absorption by the jet is calculated. The absorption is determined by the optical depth
$$\begin{array}{c}\hfill \mathrm{\Delta}{\tau}_{\nu}(s)={c}_{\tau}\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathrm{sc}}(s)\phantom{\rule{0.166667em}{0ex}}\mathrm{\Delta}s,\end{array}$$(1)
with c_{τ} the scaling parameter and Δs the length of the line element at position s(θ, z). The scaled density ρ_{sc} in this model is dimensionless and is a function of the polar angle θ and height z in the jet:
$$\begin{array}{c}\hfill {\rho}_{\mathrm{sc}}(\theta ,z)={\left(\frac{\theta}{{\theta}_{\mathrm{out}}}\right)}^{p}{\left(\frac{z}{1\phantom{\rule{0.166667em}{0ex}}\mathrm{AU}}\right)}^{2}.\end{array}$$(2)
Here p is the exponent, which is a free parameter in the model; θ_{out} is the outer jet angle; and z is the height of the jet above the centre of the jet cone. Hence, we calculate the relative density structure, which can then be scaled by the scaling parameter c_{τ}, in order to fit the synthetic spectra to the observations. By doing so the computations of optical depth are fast. The absolute density of the jet is estimated in Sect. 2.2.
In this model we implement the same three jet configurations as in Bollen et al. (2019): a stellar jet, an Xwind, and a disc wind. The velocity profile used for the stellar jet and Xwind models is defined as
$$\begin{array}{c}\hfill v(\theta )={v}_{\mathrm{out}}+({v}_{\mathrm{in}}{v}_{\mathrm{out}})\xb7{f}_{1}(\theta ),\end{array}$$(3)
where v_{out} and v_{in} are the outer and inner velocities, and with
$$\begin{array}{c}\hfill {f}_{1}(\theta )=\frac{{e}^{{p}_{v}\xb7{f}_{2}(\theta )}{e}^{{p}_{v}}}{1{e}^{{p}_{v}}}.\end{array}$$(4)
Here p_{v} is a free parameter, and f_{2} is defined as
$$\begin{array}{c}\hfill {f}_{2}(\theta )=\left\frac{\theta {\theta}_{\mathrm{cav}}}{{\theta}_{\mathrm{out}}{\theta}_{\mathrm{cav}}}\right,\end{array}$$(5)
where θ_{out} is the outer jet angle and θ_{cav} the cavity angle of the jet.
The velocity profile of the disc wind is dependent on the Keplerian velocity at the location in the disc from where the material is ejected. For the inner jet region between the jet cavity and the inner jet angle (θ_{cav} < θ < θ_{in}), we have
$$\begin{array}{c}\hfill v(\theta )={v}_{\mathrm{in},\mathrm{cav}}+({v}_{\mathrm{in},\mathrm{sc}}{v}_{\mathrm{in},\mathrm{cav}})\xb7{\left(\frac{\theta {\theta}_{\mathrm{cav}}}{{\theta}_{\mathrm{in}}{\theta}_{\mathrm{cav}}}\right)}^{{p}_{v}},\end{array}$$(6)
with v_{in, cav} the jet velocity at the cavity angle (θ_{cav}) and v_{in, sc} the jet velocity at the inner boundary angle (v_{in}). For the outer jet region (θ_{in} < θ < θ_{out}) the velocity is defined as
$$\begin{array}{c}\hfill v(\theta )=\frac{{v}_{M}}{\sqrt{tan\theta}},\end{array}$$(7)
with
$$\begin{array}{c}\hfill {v}_{M}={v}_{\mathrm{out},\mathrm{sc}}\sqrt{tan{\theta}_{\mathrm{out}}}.\end{array}$$(8)
We define the scaled inner velocity as v_{in, sc} = v_{M} ⋅ (tanθ_{in})^{−1/2} and the scaled outer velocity as v_{out, sc} = c_{v} ⋅ v_{out}. The parameter v_{out} is the outer jet velocity, which is equal to the Keplerian velocity at the launching point in the disc. The scaling factor c_{v} can have values between 0 and 1. Hence, the disc wind velocity is lower than or equal to the Keplerian velocity from its launching point in the disc.
In the three jet configurations, we included two important updates. The first update is the ability to model a jet whose axis is tilted with respect to the direction perpendicular to the orbital plane. As can be seen in Fig. 1, the absorption feature is not completely centred on the phase of superior conjunction, i.e. when the MS companion is between the postAGB primary and the observer. This can be explained by a tilt in the jet, causing the absorption feature to be observed later in the orbital phase. A tilted jet in the binary system would lead to a precessing motion of the jet. This is not uncommon and has been previously observed in preplanetary nebulae (Sahai et al. 2017; Yung et al. 2011). We implemented this jet tilt as an extra free parameter in our fitting routine.
Fig. 1. Dynamic spectra for the Balmer lines of BD+46°442. Upper left: Hα, upper right: Hβ, lower left: Hγ, lower right: Hδ. The black dashed line indicates the phase of superior conjunction. The white line indicates the radial velocity of the postAGB star. The colour gradient represents the strength of the line at each phase. 
The second update to our model presented in Bollen et al. (2019) is the introduction of a jet cavity for the Xwind and the disc wind configurations. In Bollen et al. (2019) we showed that the density in the innermost region of the outflow is extremely low, thus barely contributing to the absorption. This is in agreement with the disc wind theory of Blandford & Payne (1982) and the Xwind theory of Shu et al. (1994). According to these theories the disc material is launched at angles of 30° with respect to the jet axis, although farther from the launch point the angle can decrease substantially due to magnetic collimation. In our model, we allow some flexibility for the cavity angle parameter by giving it a lower limit of 20°. We compare the new version of the spatiokinematic model with the older version, which does not include the cavity and tilt, during our analysis in Sects. 3.1 and 4.1.
2.2. Radiative transfer model of the jet
The spatiokinematic model is used as input for the radiative transfer model. Hence, the geometry, velocity, and scaled density structure are fixed with the values estimated from the spatiokinematic model. By calculating the radiative transfer through the jet, the absolute jet densities can be determined, from which we can then calculate the jets’ massejection rate. Here, we use the equivalent width (EW) of the Balmer lines to fit the model to the observations. The fitting parameters are the absolute jet densities and temperatures instead of relative density differences throughout the jet. Hence, the optical depth calculations become more CPU intensive.
2.2.1. Radiative transfer
In our radiative transfer code we assume thermodynamic equilibrium, and that the jet medium is isothermal. Hence, each line of sight through the jet to the disc of the star will have the same temperature. Here we use the formal solution of the 1D radiative transfer equation, where the source function S_{ν} is described by the Planck function B_{ν} (see chapter 1 Rybicki & Lightman 1979). For the incident intensity of the postAGB star in the model we use a synthetic stellar spectrum from Coelho (2014), which is chosen based on the parameters of the postAGB star.
Using the Boltzmann equation, and by expressing the Einstein coefficients in terms of the oscillator strength f_{12}, the absorption coefficient α_{ν} can be written as
$$\begin{array}{c}\hfill {\alpha}_{\nu}(s)=\frac{\pi {e}^{2}}{{m}_{e}c}{\varphi}_{\nu}{n}_{l}(s){f}_{\mathit{lu}}[1{e}^{\mathrm{\Delta}E/kT}],\end{array}$$(9)
with n_{l} and n_{u} the densities in the lower and upper energy level, f_{lu} the oscillator strength, and ΔE the energy difference between the upper and lower energy levels. Hence, the computation of the intensity is dependent on the number density n_{l}, the temperature T, and the normalised line profile ϕ_{ν}. This normalised line profile ϕ_{ν} is described as a Doppler profile for Hβ, Hγ, and Hδ. For Hα, we follow the description in Muzerolle et al. (2001), Kurosawa et al. (2006), and Kurosawa et al. (2011) instead. As Muzerolle et al. (2001) showed, the Stark broadening effect can become significant in the optically thick Hα line. Hence, we describe the line profile of Hα with the Voigt profile:
$$\begin{array}{c}\hfill \varphi (\nu )=\frac{1}{{\pi}^{1/2}\mathrm{\Delta}{\nu}_{\mathrm{D}}}\frac{a}{\pi}{\displaystyle {\int}_{\infty}^{\infty}\frac{{e}^{{y}^{2}}}{{(\frac{\mathrm{\Delta}\nu}{\mathrm{\Delta}{\nu}_{\mathrm{D}}}y)}^{2}+{a}^{2}}\mathrm{d}y.}\end{array}$$(10)
Here Δν_{D} is the Doppler width of the line, a = Γ/4πΔν_{D}, and y = (ν − ν_{0})/Δν_{D} with ν_{0} the line centre. The Doppler line width Δν_{D} is a function of the thermal velocity v_{D}:
$$\begin{array}{c}\hfill \mathrm{\Delta}{\nu}_{\mathrm{D}}={\nu}_{0}\frac{{v}_{\mathrm{D}}}{c}=\frac{{\nu}_{0}}{c}\sqrt{\frac{2kT}{{m}_{p}}}.\end{array}$$(11)
We use the damping constant Γ described by Vernazza et al. (1973), which is given by the sum of the natural broadening, Van der Waals broadening, and the linear Stark broadening effects:
$$\begin{array}{c}\hfill \mathrm{\Gamma}={C}_{\mathrm{Rad}}+{C}_{\mathrm{VdW}}(\frac{{n}_{\mathit{HI}}}{{10}^{22}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}})(\frac{T}{5000\phantom{\rule{0.166667em}{0ex}}\mathrm{K}}{)}^{0.3}+{C}_{\mathrm{Stark}}(\frac{{n}_{e}}{{10}^{18}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}}{)}^{2/3}.\end{array}$$(12)
Here C_{Rad}, C_{VdW}, and C_{Stark} are the broadening constants of the natural broadening, Van der Waals, and Stark broadening effects, respectively; n_{HI} is the neutral hydrogen number density; and n_{e} is the electron number density. For the broadening constants, we use the values from Luttermoser & Johnson (1992): C_{rad} = 8.2 × 10^{−3} Å, C_{VdW} = 5.5 × 10^{−3} Å, and C_{Stark} = 1.47 × 10^{−2} Å.
2.2.2. Numerical integration of the radiative transfer equation
In our model we divide the light from the postAGB star into N_{r} rays. To compute the radiative transfer through the jet, we solve the 1D radiative transfer equation numerically. Hence, we iterate over each grid point along each ray, as shown in Fig. 2. This ray is split into N_{j} grid points between the point of entry and exit in the jet. The intensity at a grid point i along the ray is computed as follows:
$$\begin{array}{c}\hfill {I}_{\nu}({\tau}_{i})={I}_{\nu}({\tau}_{i1})\phantom{\rule{0.166667em}{0ex}}{e}^{({\tau}_{i1}{\tau}_{i})}+{B}_{\nu}({\tau}_{i})\phantom{\rule{0.166667em}{0ex}}[1{e}^{({\tau}_{i1}{\tau}_{i})}].\end{array}$$(13)
Fig. 2. Schematic representation of the radiative transfer calculations in the jet. The ray travelling from the star to the observer is split into N_{j} grid points where it passes through the jet. Each grid point has a density ρ and a velocity v. We iterate over each grid point in order to determine the resulting intensity along this line for each wavelength. 
Hence, if we want to calculate the intensity through the whole line of sight through the jet, the observed intensity I_{ν}(τ_{n}) will be
$$\begin{array}{cc}& {I}_{\nu}({\tau}_{n})=\phantom{\rule{0.166667em}{0ex}}{I}_{\nu}({\tau}_{0})\phantom{\rule{0.166667em}{0ex}}{e}^{({\tau}_{0}{\tau}_{n})}+{B}_{\nu}({\tau}_{n})\phantom{\rule{0.166667em}{0ex}}[1{e}^{({\tau}_{n1}{\tau}_{n})}]\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+{\displaystyle \sum _{i=1}^{n1}{B}_{\nu}({\tau}_{i})\phantom{\rule{0.166667em}{0ex}}[1{e}^{({\tau}_{i1}{\tau}_{i})}]\phantom{\rule{0.166667em}{0ex}}{e}^{({\tau}_{i}{\tau}_{n})}.}\hfill \end{array}$$(14)
Since the ray has been divided into discrete intervals, the optical depth τ_{i} will be calculated as
$$\begin{array}{c}\hfill {\tau}_{i+1}{\tau}_{i}={\rho}_{i}{\kappa}_{\nu ,i}\phantom{\rule{0.166667em}{0ex}}\mathrm{\Delta}s={\alpha}_{\nu ,i}\phantom{\rule{0.166667em}{0ex}}\mathrm{\Delta}s,\end{array}$$(15)
with κ_{ν, i} the opacity. This procedure is iterated for each ray from the postAGB star and each frequency ν. Hence, in general, the model consists of N_{r} rays leaving the postAGB surface, which are divided in N_{j} grid points and for which the intensity is computed for a total of N_{ν} frequencies. As we are assuming that the jet is isothermal and the jet velocities significantly smaller than the speed of light (v/c ≪ 1), we do not need to compute the Planck function B_{ν} for each grid point. However, this more general formulation will allow nonisothermal jet models to be computed in the future.
2.2.3. Equivalent width as tracer of absorption
The photospheric light from the postAGB star that travels through the jet will be scattered by the hydrogen atoms in the jet, causing the absorption features in the Balmer lines. To quantify this scattering in our model and the observations we use the EW of the Balmer lines as fitting parameter. We do this for two main reasons. The first is that the EW of a line will be higher for stronger extinction. Hence, the EW quantifies the amount of scattering by the jet. The EW is highly dependent on the level populations of hydrogen at the location where the line of sight passes through the jet. In our model these level populations are determined by the local density and by the temperature of the jet at those locations.
Second, the ratio of EW between the four Balmer lines (Hα, Hβ, Hγ, and Hδ) is also dependent on the chosen jet temperature and density. This ratio can change dramatically when these two parameters are changed. This makes the EW an ideal quantity in our fitting to find the absolute jet densities and temperatures.
3. Jet model for BD+46°442.
BD+46°442 is a jetlaunching postAGB binary system for which we obtained 36 spectra during oneandahalf orbital cycles of the binary orbit of 140.82 days (Van Winckel et al. 2009; Oomen et al. 2018). In this study we adopt the orbital parameters listed in Oomen et al. (2018, see Appendix B). The scattering by the jet is observable in the first four Balmer lines (Hα, Hβ, Hγ, and Hδ), hence we focus on these line for our analysis. The Balmer lines are shown in Appendix C. The signaltonoise ratio of the spectra lies between S/N = 22 and S/N = 60 in the Hα line, and drops to values between S/N = 12 and S/N = 37 in the Hδ line. In Fig. 1 we present the dynamic Balmer line spectra for BD+46°442. In the dynamic spectra we plot the continuumnormalised spectra as a function of orbital phase and interpolate between each of the spectra. In this way the orbital phasedependent variations in the line become apparent.
3.1. Spatiokinematic model of BD+46°442
We compare the quality of the fit for the three jet configurations through their reduced chisquare and Bayesian Information Criteria (BIC) values^{1}. The bestfitting model is the Xwind with a reduced chisquare of ${\chi}_{\nu}^{2}=0.23$. A chisquare lower than unity indicates that the model is overfitting the data. In our case, this is caused by overestimating the uncertainty on the data, which is determined from the signaltonoise ratio of the spectra (σ = (S/N)^{−1}) and the uncertainty in the emission feature of the synthetic spectra that is provided as input for the modelling. We impose a χ^{2} of unity for the bestfitting model and scale the χ^{2} of the other models appropriately, in order to compare their relative difference. The scaled chisquare values are ${\chi}_{\text{stellar}}^{2}=1.12$, ${\chi}_{\text{X0wind}}^{2}=1$, and ${\chi}_{\text{discwind}}^{2}=1.17$. Hence, the Xwind configuration gives a slightly better fitting result compared to the other two configurations. This is also confirmed by the BIC values of the three models. The Xwind has the lowest BIC and therefore fits the data best: BIC_{stellar} − BIC_{Xwind} = 1007 and BIC_{discwind} − BIC_{Xwind} = 1452. For this reason we use the bestfitting parameters from the spatiokinematic modelling of the Xwind for further calculations. We note, however, that the relative difference in χ^{2} between the three model configurations is not significant, and thus we conclude that the three model configurations fit the data equally well.
The bestfitting parameters of the model are tabulated in Table 1, and its model spectra are shown in the upper right panel of Fig. 3. The binary inclination for this model is about 50°. The jet has a halfopening angle of 35°. The inner boundary angle θ_{in} = 29° is the polar angle in the jet along which the bulk of the mass will be ejected. The geometry of the binary system and the jet are represented in Fig. 4. The material that is ejected in the inner regions of the jet reaches velocities up to 490 km s^{−1}. These velocities are of the order of the escape velocity from the surface of a MS star, confirming the nature of the companion. The velocities at the outer edges are lower at 41 km s^{−1}. The radius of the postAGB star in the bestfitting model is 27.2 R_{⊙} (0.127 AU).
Fig. 3. Interpolated observed and modelled dynamic spectra of the Hα line. The upper spectra are the observations (left) and model spectra (right) of BD+46°442. The lower spectra are the observations and model spectra of IRAS 19135+3937. The colours represent the normalised flux. 
Fig. 4. Geometry of the binary system and the jet of BD+46°442 at superior conjunction, when the postAGB star is directly behind the jet, as viewed by the observer. In all three plots the full orange circle denotes the postAGB star. The orange star indicates the location of the companion. The red cross is the location of the centre of mass of the binary system. The radius of the postAGB star is to scale. The jet is represented in blue, and the colour indicates the relative density of the jet (see colour scale). The dashed black line is the jet axis and the dotted white lines are the inner jet edges. The jet cavity is the inner region of the jet. Upper left panel: system viewed along the orbital plane from a direction perpendicular to the line of sight to the observer. Right panel: jet viewed from an angle perpendicular to the Xaxis. The postAGB star is located behind the companion and its jet in this image. The jet tilt is noticeable from this angle. Lower left panel: binary system viewed from above, perpendicular to the orbital plane. The grey dashed lines represent the Roche radii of the two binary components and the full black line shows the Roche lobes. 
Bestfitting jet configuration and parameters for the spatiokinematic model of BD+46°442 and IRAS 19135+3937.
Additionally, we implemented a jet tilt and jet cavity in this model (see Sect. 2.1). The resulting model has a cavity angle of 20°. The jet tilt for BD+46°442 is relatively large with ϕ_{tilt} = 15°. The effect of this tilt is noticeable in the resulting model spectra. The jet absorption feature is not centred at orbital phase 0.5, but at a later phase between 0.55 and 0.6. To evaluate the performance of the new modified spatiokinematic model that includes a jet cavity and tilt, we do an additional model fitting with the old version that does not include these features and compare the two model fitting results. The χ^{2} of the new model (${\chi}_{\text{new}}^{2}=1$) is lower than the old version (${\chi}_{\text{old}}^{2}=1.35$). If we account for the two extra parameters in the new model by comparing the BIC instead, we get a difference in BIC between the two results of Δ BIC = 2980, with the lower BIC for the new model, implying a better fit for this model. This demonstrates that the implemented jet cavity and jet tilt improve the spatiokinematic model for this object.
3.2. Radiative transfer model of BD+46°442
We apply the radiative transfer model for BD+46°442 to compute the amount of absorption caused by the jet that blocks the light from the postAGB star. The setup for the radiative transfer model is similar to that described in Sect. 2.1. For each ray of light the background intensity I_{ν, 0} is the background spectrum given in Sect. 3.1. For each orbital phase we calculate the amount of absorption by the jet for each ray. Additionally, the output from the MCMC fitting routine in Sect. 3.1 (i.e. the spatiokinematic model of BD+46°442) is used as input in our the radiative transfer model. Hence, there are only two fitting parameters: jet number density n_{j} and jet temperature T_{j}. We assume the jet temperature to be uniform for the segment of the jet through which the rays travel. The jet number density n_{j} is defined as the number density at the inner edge of the jet θ_{in} at a height of 1 AU. In the case of BD+46°442 the bestfitting jet configuration is an Xwind. Hence, the density in the jet at each grid point can be determined from the density profile of the jet that was used for the Xwind in the spatiokinematic model. This density profile is defined as
$$\begin{array}{c}\hfill n(\theta ,z)={n}_{j}\phantom{\rule{0.166667em}{0ex}}{\left(\frac{\theta}{{\theta}_{\mathrm{in}}}\right)}^{p}\phantom{\rule{0.166667em}{0ex}}{z}^{2},\end{array}$$(16)
with p either the exponent for the innerjet region p_{in} or the outerjet region p_{out}, which was determined in Sect. 3.1.
We use a grid of jet temperatures between 4400 K and 6000 K in steps of 100 K and the logarithm of the jet densities ${log}_{10}\left(\frac{{n}_{j}}{{\mathrm{m}}^{3}}\right)$ between 14 and 18 in logarithmic steps of 0.1. This makes a total of 697 grid calculations.
As described in Sect. 2.2.3, the EW of the Balmer lines represents the amount of absorption by the jet. In order to find the bestfitting model for our grid of temperatures and densities, we fit the EW of the Balmer lines in the model to those of the observed Balmer lines for each spectra.
We fit the model to the data with a χ^{2}–goodnessoffit test. Hence, the reduced ${\chi}_{\nu}^{2}$ value for a model will be
$$\begin{array}{cc}& {\chi}_{\nu}^{2}=\frac{1}{\nu}({\displaystyle \sum _{i}^{{N}_{o}}\left[\frac{({\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}o,\phantom{\rule{0.166667em}{0ex}}\mathrm{H}\alpha}{\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}m,\mathrm{H}\alpha}{)}^{2}}{({\sigma}_{i}^{\mathrm{H}\alpha}{)}^{2}}\right]+\sum _{i}^{{N}_{o}}\left[\frac{({\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}o,\phantom{\rule{0.166667em}{0ex}}\mathrm{H}\beta}{\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}m,\mathrm{H}\beta}{)}^{2}}{({\sigma}_{i}^{\mathrm{H}\beta}{)}^{2}}\right]}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}+{\displaystyle \sum _{i}^{{N}_{o}}\left[\frac{({\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}o,\phantom{\rule{0.166667em}{0ex}}\mathrm{H}\gamma}{\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}m,\mathrm{H}\gamma}{)}^{2}}{({\sigma}_{i}^{\mathrm{H}\gamma}{)}^{2}}\right]+\sum _{i}^{{N}_{o}}\left[\frac{({\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}o,\phantom{\rule{0.166667em}{0ex}}\mathrm{H}\delta}{\mathrm{EW}}_{i}^{\phantom{\rule{0.166667em}{0ex}}m,\mathrm{H}\delta}{)}^{2}}{({\sigma}_{i}^{\mathrm{H}\delta}{)}^{2}}\right]),}\hfill \end{array}$$(17)
with ν the degrees of freedom, N_{o} the number of spectra (36 for BD+46°442, and 22 for IRAS 19135+3937), EW^{o} and EW^{m} the equivalent width of the observed and modelled line, and σ the standard deviation determined by the signaltonoise ratio of the spectra.
The resulting 2D ${\chi}_{\nu}^{2}$ distribution for jet densities and temperatures is shown in Fig. 5 and the corresponding EW of the model in Fig. 7. The bestfitting model has a jet density of n_{j} = 2.5 × 10^{16} m^{−3} and jet temperature of T_{j} = 5600 K. In order to determine the uncertainties on the fitting parameters, we convert the 2D chisquared distribution into a probability distribution:
$$\begin{array}{c}\hfill {P}_{2\mathrm{D}}({n}_{j},{T}_{j})\propto exp({\chi}^{2}/2).\end{array}$$(18)
Fig. 5. Twodimensional reduced chisquared distribution for the grid of jet densities n_{j} and temperatures T_{j} for the fitting of BD+46°442. The white dot gives the location of the minimum reduced chisquare value ${\chi}_{\nu ,\text{min}}^{2}=43.9$. The contours represent the 1σ, 2σ, and 3σ intervals. 
Fig. 7. Equivalent width of the absorption by the jet for BD+46°442 as a function of orbital phase. The four panels show the EW in Hα (top left), Hβ (top right), Hγ (bottom left), and Hδ (bottom right). The circles are the measured EWs of the absorption feature by the jet in the observations with their respective errors. The full line is the EW of the absorption feature for the bestfitting model. 
The marginalised probability distribution can be found for each parameter via
$$\begin{array}{c}\hfill {P}_{1\mathrm{D}}({n}_{j})={\displaystyle \sum _{{T}_{j}}{P}_{2\mathrm{D}}({n}_{j},{T}_{j}),}\end{array}$$(19)
$$\begin{array}{c}\hfill {P}_{1\mathrm{D}}({T}_{j})={\displaystyle \sum _{{n}_{j}}{P}_{2\mathrm{D}}({n}_{j},{T}_{j}).}\end{array}$$(20)
From these distributions we can determine a mean and standard deviation. This gives us a jet density and temperature of ${n}_{j}=2.{5}_{0.7}^{+0.9}\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}$ and T_{j} = 5600 ± 80 K.^{2}
4. Jet model for IRAS 19135+3937
The second postAGB binary system that we model is IRAS 19135+3937. For this object, we obtained 22 spectra during a full cycle with an orbital period of 126.97 days (Van Winckel et al. 2009; Oomen et al. 2018). As for BD+46°442, we adopt the orbital parameters of this system found by Oomen et al. (2018). The individual and dynamic spectra are shown in Appendix C and Fig. 6, respectively. The signaltonoise ratio for these spectra lie between S/N = 26 and S/N = 49 in Hα. In Hδ the signaltonoise ratio ranges between S/N = 5 and S/N = 20.
Fig. 6. Dynamic spectra for the Balmer lines of IRAS 19135+3937. Upper left: Hα, upper right: Hβ, lower left: Hγ, lower right: Hδ. The black dashed line indicated the phase of superior conjunction. The white line indicates the radial velocity of the postAGB star. The colour gradient represents the strength of the line at each phase. 
4.1. Spatiokinematic model of IRAS 19135+3937
The spatiokinematic structure of IRAS 19135+3937 was modelled by Bollen et al. (2019). Here, we update it with the addition of the jet tilt and the jet cavity. The bestfitting jet configuration is a disc wind, but all three models produce similar fits, as was the case in the fitting of Bollen et al. (2019) (BIC_{stellar} − BIC_{disc wind} = 87 and BIC_{Xwind} − BIC_{disc wind} = 116). The bestfitting model parameters are tabulated in Table 1. This model has an inclination angle of i = 72° for the binary system and a jet angle of θ_{out} = 67°. These angles are about 7° lower than those found in the model fitting of Bollen et al. (2019). The jet reaches velocities up to 640 km s^{−1}. At its edges the jet velocity is v_{out} ⋅ c_{v} = 3 km s^{−1}. The postAGB star in our model has a radius of 22.5 R_{⊙} (0.105 AU), which is about 30% smaller than that found by Bollen et al. (2019). The geometry of the binary system and the jet are shown in Fig. 8. We compare the quality of the fit for the bestfitting model of Bollen et al. (2019) with the bestfitting model in this work. The BIC for the model fitting in our work is significantly lower than the BIC found by Bollen et al. (2019) (ΔBIC_{old} − BIC_{new} = 4190). This shows that the jet tilt and jet cavity significantly improve the model fitting. The jet tilt for this object is relatively small (ϕ_{tilt} = 5.7°). This is expected since there is no noticeable lag in the absorption feature in the spectra. The jet for this object has a significant jet cavity of θ_{cav} = 24°.
4.2. Radiative transfer model of IRAS 19135+3937
We apply the radiative transfer model for IRAS 19135+3937. The bestfitting spatiokinematic model found in Sect. 4.1 for IRAS 19135+3937 is a disc wind. We use this spatiokinematic model and its model parameters as input to calculate the radiative transfer in the jet for a grid of jet densities and temperatures. The density profile for the disc wind is similar to the Xwind (see Eq. (16)). The grid of temperatures and densities is the same as for BD+46°442.
The 2D χ^{2} distribution for the fitting is shown in Fig. 9 and the associated EW of the model is shown in Fig. 10. We calculate the marginalised probability distributions for n_{j} and T_{j}, given by Eqs. (19) and (20), from which we can determine the mean and standard deviations. This gives a jet density of ${n}_{j}=1.{0}_{0.4}^{+0.5}\phantom{\rule{0.166667em}{0ex}}\times \phantom{\rule{0.166667em}{0ex}}{10}^{16}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}$ and jet temperature of T_{j} = 5330 ± 180 K.
Fig. 9. Twodimensional reduced chisquare distribution for the grid of jet densities n_{j} and temperatures T_{j} for the fitting of IRAS 19135+3937. The white dot gives the location of the minimum reduced chisquare value ${\chi}_{\nu ,\text{min}}^{2}=60.8$. The contours represent the 1σ, 2σ, and 3σ intervals. 
5. Discussion
By fitting the spatiokinematic structure of the jet and estimating its density structure, we obtained crucial information about the jet. We can now estimate how much mass is being ejected by the jet, which is essential for understanding the mass accretion onto the companion and determining the source feeding this accretion.
5.1. Jet massloss rate
The velocity and density structure of the jets, calculated by fitting the models, is used to estimate the massejection rate. The massejection rate of the jet for both systems is estimated by calculating how much mass passes through the jet at a height of 1 AU from the launch point.
In the case of BD+46°442, the velocity and density profiles are determined by the Xwind configuration (see Sect. 4.1). We calculate the density at a height of z = 1 AU using Eq. (3). The massejection rate can be found by the integral
$$\begin{array}{c}\hfill {\dot{M}}_{\mathrm{jet}}={\displaystyle {\int}_{0}^{R}\phantom{\rule{0.166667em}{0ex}}\rho (r)\xb7v(r)\xb72\pi r\phantom{\rule{0.166667em}{0ex}}\mathrm{d}r,}\end{array}$$(21)
with r = 1 AU ⋅ tanθ. The velocity at each location in the jet is defined by Eq. (3). In this way we find a massejection rate of ${\dot{M}}_{\mathrm{jet}}={7}_{2}^{+3}\times {10}^{7}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}\phantom{\rule{0.166667em}{0ex}}{\mathrm{yr}}^{1}$ for BD+46°442.
For IRAS 19135+3937, the data is best fit by a disc wind model (see Sect. 3.1), whose velocity profile is described by Eqs. (6) and (7) for the inner and outer jet regions, respectively. From Eq. (21), we find a massejection rate of ${\dot{M}}_{\mathrm{jet}}=2.{0}_{0.7}^{+2}\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}\phantom{\rule{0.166667em}{0ex}}{\mathrm{yr}}^{1}$. We list these values as lower and upper limits in Table 2.
Derived massaccretion and massloss rates in the two binary systems.
5.2. Ejection efficiency
By assuming an ejection efficiency Ṁ_{jet}/Ṁ_{acc}, we can link the jet massloss rate (Ṁ_{jet}) to the massaccretion rate (Ṁ_{acc}), and hence obtain a range of possible accretion rates onto the circumcompanion disc. By doing so, we can assess if the mass transfer from either the postAGB or the circumbinary disc, or both can contribute enough mass to the circumcompanion disc in order to sustain the observed jet massloss rates.
Ejection efficiency has not been determined for postAGB binary systems, but the same theory (i.e., magneto centrifugal driving) applies to YSOs, which have been studied extensively (Ferreira et al. 2007, and references therein). Moreover, discs in YSOs are comparable in size to those in our postAGB binary systems (Hillen et al. 2017) and their massejection rates are similar to the rates estimated for jets in postAGB systems (10^{−8} − 10^{−4} M_{⊙} yr^{−1}; Calvet et al. 1998; Ferreira et al. 2007). Current estimates of ejectioEWn efficiencies for T Tauri stars are in the range Ṁ_{jet}/Ṁ_{acc} ∼ 0.01 − 0.1 (Cabrit et al. 2007; Cabrit 2009; Nisini et al. 2018). This said, the spread in these values is large and some studies have even found ratios higher than 0.3 (Calvet et al. 1998; Ferreira et al. 2006; Nisini et al. 2018).
In this work we adopt a wide range of ejection efficiencies for both of our postAGB binary objects, according to the typical ranges found through observations of YSOs: 0.01 < Ṁ_{jet}/Ṁ_{accr} < 0.3. Under these assumptions, and by using the jet massloss rates from Sect. 5.1, the accretion rates onto the two companions are (see Table 2)
$$\begin{array}{c}\hfill \{\begin{array}{cc}1.7\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}\phantom{\rule{0.166667em}{0ex}}{\mathrm{yr}}^{1}<\phantom{\rule{0.166667em}{0ex}}{\dot{M}}_{\mathrm{acc},\mathrm{BD}}\hfill & <1\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}\phantom{\rule{0.166667em}{0ex}}{\mathrm{yr}}^{1}\hfill \\ 5\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}\phantom{\rule{0.166667em}{0ex}}{\mathrm{yr}}^{1}<\phantom{\rule{0.166667em}{0ex}}{\dot{M}}_{\mathrm{acc},\mathrm{IRAS}}\hfill & <4\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}{M}_{\odot}\phantom{\rule{0.166667em}{0ex}}{\mathrm{yr}}^{1}.\hfill \end{array}\end{array}$$(22)
Next, we use these results to look at possible sources feeding the accretion.
5.3. Sources of accretion onto the companion
There are two possible sources of mass transfer onto the companion. The first is the postAGB primary itself, moving mass via the first Lagrange point L1 and creating a circumcompanion accretion disc. The second possibility is reaccretion of gas from the circumbinary discs (Van Winckel 2003; De Ruyter et al. 2006; Dermine et al. 2013).
5.3.1. Scenario 1: Mass transfer from the postAGB star to the companion
We first assume that the accretion onto the companion is due to mass transfer from the primary via L1. To estimate the masstransfer rate by the postAGB star, we follow the prescription in Ritter (1988):
$$\begin{array}{c}\hfill {\dot{M}}_{1}=\frac{2\phantom{\rule{0.166667em}{0ex}}\pi}{\sqrt{e}}{\left(\frac{{k}_{\mathrm{B}}}{{m}_{\mathrm{H}}\phantom{\rule{0.166667em}{0ex}}{\mu}_{1,\mathrm{ph}}}{T}_{1}\right)}^{3/2}\frac{{R}_{1}^{3}}{G{M}_{1}}\phantom{\rule{0.166667em}{0ex}}{\rho}_{1,\mathrm{ph}}\phantom{\rule{0.166667em}{0ex}}F(q).\end{array}$$(23)
Here e is Euler’s number; k_{B} is the Boltzmann constant; G is the gravitational constant; m_{H} is the hydrogen mass; and μ_{1, ph}, T_{1}, R_{1}, M_{1}, and ρ_{1, ph} are the mean molecular weight, the temperature, the radius, and the mass of the primary star, respectively. The parameter F(q) is defined as
$$\begin{array}{c}\hfill F(q)=[(g(q)(1+q))\phantom{\rule{0.166667em}{0ex}}g(q){]}^{1/2}{\left(\frac{{R}_{1,\mathrm{RL}}}{a}\right)}^{3},\end{array}$$(24)
with q = M_{2}/M_{1} the mass ratio, R_{1, RL} the Roche lobe radius of the postAGB star, and a the binary separation. In Eq. (24) g(q) is defined as
$$\begin{array}{c}\hfill g(q)=\frac{q}{{x}^{3}}+{(1x)}^{3},\end{array}$$(25)
with x the distance between the mass centre of the postAGB star and L1 in terms of the binary separation (a).
We assume a neutral cosmic mixture which implies a mean molecular weight of μ_{1, ph} = 0.8. We note that this prescription is based on Roche lobe overflow for a star filling its Roche lobe, transferring mass to the companion. In this case the radius of the star R_{1} is equal to the Roche radius R_{1, RL}. However, from our results in in the spatiokinematic modelling, the postAGB stars in these two systems do not fill their Roche lobes, as is shown in the geometrical representation of the systems in Figs. 4 and 8. The observations also support this result since a star that could fill at least 80% of its Roche lobe would show ellipsoidal variations in its light curve (Wilson & Sofia 1976). The light curves of BD+46°442 and IRAS 19135+3937 do not show these variations (Bollen et al. 2019). Hence, we extrapolate Eq. (23) by using the radius of the primary R_{1} instead of the Roche radius R_{1, RL}.
Additionally, since we find that the postAGB star does not fill its Roche lobe, the mass transfer would occur via a mechanism that is less efficient and weaker than RLOF. A few other possibilities being windRLOF and BondiHoyleLyttleton (BHL) accretion. In the case of windRLOF, the stellar wind will be focused to the orbital plane and most of the mass will be lost through the L1 point, towards the secondary (Mohamed et al. 2007). The masstransfer efficiency of windRLOF would vary between a few percent and 50% (de ValBorro et al. 2009; Abate et al. 2013). In the case of BHL accretion, the accretion efficiency would be significantly lower at about 1−10% (Abate et al. 2013; Mohamed & Podsiadlowski 2012). Hence, the upper limit for mass transfer through windRLOF would be lower than that of RLOF. The upper limit for BHL accretion would be several orders of magnitude lower. Hence, by equating the masstransfer from the postAGB star using Eq. (23), we can get a good estimation for an upper limit of masstransfer from the postAGB star to the companion.
To determine the photospheric density, we use the MESA stellar evolution code (MESA Paxton et al. 2011, 2013, 2015, 2018, 2019) to calculate the evolution of a postAGB star with the correct mass. We subsequently use the photospheric density from the MESA output at the timestep when the postAGB star is the same size as our star. This gives us a value of 10^{−10} g cm^{−2} for the photospheric density of the star (ρ_{1, ph}). The mass of the postAGB star M_{1} is set to 0.6 M_{⊙}, which is a typical value for these objects. The mass of the companion M_{2} is determined from the mass function f(M_{1}) and the inclination of the binary system found in the spatiokinematic model fitting:
$$\begin{array}{c}\hfill f({M}_{1})=\frac{{M}_{2}^{3}\phantom{\rule{0.166667em}{0ex}}{sin}^{3}i}{{({M}_{1}+{M}_{2})}^{2}}.\end{array}$$(26)
This gives a mass of 1.07 M_{⊙} for the companion star of BD+46°442, resulting in a mass ratio of q = 1.79. We find the Roche radius of the postAGB star using the formula by Eggleton (1983):
$$\begin{array}{c}\hfill {R}_{\mathrm{RL},1}=a\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{0.49\phantom{\rule{0.166667em}{0ex}}{q}^{2/3}}{0.6\phantom{\rule{0.166667em}{0ex}}{q}^{2/3}+ln\phantom{\rule{0.166667em}{0ex}}(1+{q}^{1/3})}.\end{array}$$(27)
Using these values for Eq. (23), we find the masstransfer rate from the postAGB star to the companion in BD+46°442 to be Ṁ_{1} = 3.5 × 10^{−7} M_{⊙} yr^{−1}. This value is less than the lower limit of the jet massloss rate in BD+46°442 (see Table 2). Moreover, in Sect. 5.1, we found the massaccretion rate to be in the range of 1.7 × 10^{−6} M_{⊙} yr^{−1} < Ṁ_{accr} < 1 × 10^{−4} M_{⊙} yr^{−1}, which is about five times the theoretical value for the upperlimit of masstransfer from the postAGB star to the companion. Hence, it is unlikely that the postAGB star can contribute enough mass to the circumcompanion disc to sustain the observed jet outflow.
We conduct a similar analysis for IRAS 19135+3937 and we come to a similar conclusion. The mass of the companion is M_{2} = 0.46 M_{⊙} and the mass ratio would be q = 0.77. We use the same values for μ_{1, ph} and ρ_{1, ph}, giving us a masstransfer rate of Ṁ_{RLOF} = 1.7 × 10^{−7} M_{⊙} yr^{−1}. Hence, this upper limit for masstransfer from the postAGB star for IRAS 19135+3937 is also lower than the lower limit for the jet massloss rate of IRAS 19135+3937 and thus too low to match a measured massaccretion rate in the range 4 × 10^{−6} M_{⊙} yr^{−1} < Ṁ_{accr} < 4 × 10^{−4} M_{⊙} yr^{−1}.
We conclude that it is unlikely that the mass transfer from the postAGB star alone is responsible for feeding the accretion disc around the companion.
5.3.2. Scenario 2: Reaccretion from the circumbinary disc
Here, we consider the possibility of mass accretion from the circumbinary disc onto the central binary system. In order to give an estimate of reaccretion by the circumbinary disc of BD+46°442 and IRAS 19135+3937, we use the massloss equation by Rafikov (2016), that defines the mass loss from the disc to the central binary as a function of time,
$$\begin{array}{c}\hfill {\dot{M}}_{\mathrm{disc}}(t)=\frac{{M}_{0,\mathrm{disc}}}{{t}_{0}}{(1+\frac{t}{2{t}_{0}})}^{3/2},\end{array}$$(28)
where M_{0, disc} is the initial disc mass. These circumbinary discs have average disc masses of M_{0, disc} = 10^{−2} M_{⊙} (Gielen et al. 2007; Bujarrabal et al. 2013, 2018; Hillen et al. 2017; Kluska et al. 2018). Bujarrabal et al. (2013, 2018) derived disc masses of circumbinary discs of postAGB binary systems ranging from 6 × 10^{−4} to 5 × 10^{−2} M_{⊙}. We use this range to estimate the massloss rate from the disc.
The initial viscous time of the disc t_{0} is defined by Rafikov (2016) as
$$\begin{array}{c}\hfill {t}_{0}=\frac{4}{3}\frac{\mu}{{k}_{\mathrm{B}}}\frac{{a}_{b}}{\alpha}{\left[\frac{4\pi \sigma {(G{M}_{b})}^{2}}{\zeta {L}_{1}}\right]}^{1/4}{(\frac{\eta}{{I}_{L}},)}^{2},\end{array}$$(29)
where μ is the mean molecular weight, a_{b} is the binary separation, α is the viscosity parameter, σ is StefanBoltzmann constant, L_{1} is the luminosity of the postAGB star, ζ is a constant factor that accounts for the starlight that is intercepted by the disc surface at a grazing incidence angle, η is the ratio of angular momentum of the disc compared to that of the central binary, and I_{L} characterises the spatial distribution of the angular momentum in the disc. We fix several values at the same values as Rafikov (2016) and Oomen et al. (2019): μ = 2m_{p}, α = 0.01, ζ = 0.1, and I_{L} = 1, with m_{p} the mass of a proton. The luminosity L_{1} of BD+46°442 and IRAS 19135+3937 are ${2100}_{800}^{+1500}\phantom{\rule{0.166667em}{0ex}}{L}_{\odot}$ and ${2100}_{400}^{+500}\phantom{\rule{0.166667em}{0ex}}{L}_{\odot}$, respectively (Oomen et al. 2019). The angular momentum of the circumbinary disc is typically of the order of the angular momentum of the central binary system (Bujarrabal et al. 2018; Izzard & Jermyn 2018). We set a range of η between 1.4 and 2, where a value of 1.4 is appropriate for a disc with the bulk of its mass located at the inner disc rim.
Using Eq. (28) and assuming t = 0, we can calculate a range of possible massloss rates by the disc. We find a range of 3 × 10^{−8} M_{⊙} yr^{−1} < Ṁ_{disc} < 6 × 10^{−6} M_{⊙} yr^{−1} for BD+46°442, while in the case of IRAS 19135+3937, we find that the reaccretion rate from the circumbinary disc is in the range of 5 × 10^{−8} M_{⊙} yr^{−1} < Ṁ_{disc} < 9 × 10^{−6} M_{⊙} yr^{−1} (Table 2). When the disc matter falls onto the central binary, it will be accreted by both the postAGB star and the companion. Hence, the mass lost by the circumbinary disc should be twice the mass accreted by the circumcompanion accretion disc. If we compare the massaccretion rate for BD+46°442 and IRAS 19135+3937 with the estimated massloss rate by the circumbinary disc, it shows that reaccretion from the circumbinary disc is a plausible mechanism for the formation of the jet.
We note that only the higher estimates for massaccretion rates from the circumbinary disc can explain our observationally derived rates. Hence, this would imply that for these two systems the disc masses are at the high end of the range (M_{0, disc} > 10^{−2} M_{⊙}) and that we are observing the early stages of the reaccretion by the circumbinary disc. Nevertheless, our findings are in good agreement with Oomen et al. (2019), who estimated that accretion rates should be higher than 3 × 10^{−7} M_{⊙} yr^{−1} and that disc masses should be higher than ∼10^{−2} M_{⊙}.
6. Summary and conclusion
In this paper our aim was to determine masstransfer rates of jetcreating postAGB binaries. We fully exploited the time series of highresolution optical spectra from these binary systems. We presented a new radiative transfer model for these jets and applied this model to reproduce the Balmer lines of two wellsampled postAGB binary systems: BD+46°442 and IRAS 19135+3937. With this model we were able to study the massloss rate of the jet and massaccretion rate onto the companion, and to constrain the source of the accretion in these systems: the postAGB star or the circumbinary disc. Additionally, we expanded the spatiokinematic model from Bollen et al. (2019). Our main conclusions can be summarised as follows:

We successfully reproduced the observed absorption feature in the Hα line profiles of our test sources with our improved spatiokinematic model of the jet. By doing so, we obtained the kinematics and 3D morphology of the jet. The implementation of the jet tilt in the model reproduced the observed lag of the absorption feature in the Balmer lines. This tilt is significant for both objects, with values of 15° and 6° for BD+46°442 and IRAS 19135+3937, respectively. Likewise, the new jet cavity in the model improves the jet representation, as was suggested by Bollen et al. (2019).

We showed that we can acquire a 3D jet morphology by modelling the amount of absorption in the Hα lines from our spatiokinematic model of the jet. By combining the results of the spatiokinematic and radiative transfer modelling, we found the crucial parameters to calculate jet massloss rates: jet velocity and geometry from the spatiokinematic model and jet density structure from the radiative transfer model.

We computed the massloss rate of the jet by combining the results of our spatiokinematic model and radiative transfer model. The computed massloss rates for the jets in BD+46°442 and IRAS 19135+3937 are in the ranges (5−10) × 10^{−7} M_{⊙} yr^{−1} and (1.3−4) × 10^{−6} M_{⊙} yr^{−1}, respectively, as shown in Table 2. These massejection rates are comparable to the massejection rates for the jets in planetary nebulae and preplanetary nebulae (Tocknell et al. 2014; Tafoya et al. 2019). Tocknell et al. (2014) found the massejection rates to be 1−3 × 10^{−7} M_{⊙} yr^{−1} and 8.8 × 10^{−7} M_{⊙} yr^{−1} for the Necklace and NGC 6778, respectively. These massejection rates imply correspondingly high massaccretion rates onto the companion that range between 1.7 × 10^{−6} M_{⊙} yr^{−1} and 1 × 10^{−4} M_{⊙} yr^{−1} for BD+46°442 and 4 × 10^{−6} M_{⊙} yr^{−1} and 4 × 10^{−4} M_{⊙} yr^{−1} for IRAS 19135+3937.

By determining the jet massloss rate we added an additional constraint on the nature of the accretion onto these systems. While the uncertainties are high, the circumbinary disc is the preferred source of accretion feeding the jet rather than the postAGB star: the accretion rates from the postAGB stars are too low to justify the observed jet massloss rates. We note, however, that the simultaneous accretion from the circumbinary disc and from the postAGB star cannot be ruled out. Reaccretion from the circumbinary disc also naturally explains the abundance pattern of the postAGB star and is in agreement with the study by Oomen et al. (2019), who showed that high reaccretion rates (> 3 × 10^{−7} M_{⊙} yr^{−1}) are needed in order to reproduce the observed depletion patterns of postAGB stars. These high reaccretion rates from the circumbinary disc can prolong the lifetime of the postAGB star, and can thus have an important impact on the evolution of these objects, provided that the disc can sustain the mass loss.
In our future studies, we will perform a comprehensive analysis of the whole diverse sample of jetcreating postAGB binary systems by using both the spatiokinematic and radiative transfer models. The observational properties of these binaries and their jets are nonhomogeneous. Hence, by analysing the whole sample, we aim to obtain strong constraints on the source of the accretion and identify correlations between mass accretion, depletion patterns, and the orbital properties of postAGB binaries.
Acknowledgments
This work was performed on the OzSTAR national facility at Swinburne University of Technology. OzSTAR is funded by Swinburne University of Technology and the National Collaborative Research Infrastructure Strategy (NCRIS). DK acknowledges the support of the Australian Research Council (ARC) Discovery Early Career Research Award (DECRA) grant (95213534). HVW acknowledges support from the Research Council of the KU Leuven under grant number C14/17/082. The observations presented in this study are obtained with the HERMES spectrograph on the Mercator Telescope, which is supported by the Research Foundation  Flanders (FWO), Belgium, the Research Council of KU Leuven, Belgium, the Fonds National de la Recherche Scientifique (F.R.S.FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Genève, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany.
References
 Abate, C., Pols, O. R., Izzard, R. G., Mohamed, S. S., & de Mink, S. E. 2013, A&A, 552, A26 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 BermúdezBustamante, L. C., GarcíaSegura, G., Steffen, W., & Sabin, L. 2020, MNRAS, 493, 2606 [NASA ADS] [CrossRef] [Google Scholar]
 Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883 [NASA ADS] [CrossRef] [Google Scholar]
 Bollen, D., Van Winckel, H., & Kamath, D. 2017, A&A, 607, A60 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bollen, D., Kamath, D., Van Winckel, H., & De Marco, O. 2019, A&A, 631, A53 [CrossRef] [EDP Sciences] [Google Scholar]
 Bujarrabal, V., CastroCarrizo, A., Alcolea, J., et al. 2013, A&A, 557, L11 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bujarrabal, V., CastroCarrizo, A., Van Winckel, H., et al. 2018, A&A, 614, A58 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Cabrit, S. 2007, in StarDisk Interaction in Young Stars, eds. J. Bouvier, & I. Appenzeller, IAU Symp., 243, 203 [Google Scholar]
 Cabrit, S. 2009, Astrophys. Space Sci. Proc., 13, 247 [NASA ADS] [CrossRef] [Google Scholar]
 Calvet, N. 1998, in AIP Conf. Ser., eds. S. S. Holt, & T. R. Kallman, 431, 495 [CrossRef] [Google Scholar]
 Coelho, P. R. T. 2014, MNRAS, 440, 1027 [NASA ADS] [CrossRef] [Google Scholar]
 De Ruyter, S., Van Winckel, H., Maas, T., et al. 2006, A&A, 448, 641 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 de ValBorro, M., Karovska, M., & Sasselov, D. 2009, ApJ, 700, 1148 [NASA ADS] [CrossRef] [Google Scholar]
 Dermine, T., Izzard, R. G., Jorissen, A., & Van Winckel, H. 2013, A&A, 551, A50 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deroo, P., van Winckel, H., Min, M., et al. 2006, A&A, 450, 181 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Deroo, P., Acke, B., Verhoelst, T., et al. 2007, A&A, 474, L45 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Eggleton, P. P. 1983, ApJ, 268, 368 [NASA ADS] [CrossRef] [Google Scholar]
 Ferreira, J., Dougados, C., & Cabrit, S. 2006, A&A, 453, 785 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ferreira, J., Dougados, C., & Whelan, E. 2007, Jets from Young Stars I: Models and Constraints, 723 [CrossRef] [Google Scholar]
 ForemanMackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 [CrossRef] [Google Scholar]
 Gezer, I., Van Winckel, H., Bozkurt, Z., et al. 2015, MNRAS, 453, 133 [NASA ADS] [CrossRef] [Google Scholar]
 Gielen, C., Van Winckel, H., Waters, L. B. F. M., Min, M., & Dominik, C. 2007, A&A, 475, 629 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Gorlova, N., Van Winckel, H., Gielen, C., et al. 2012, A&A, 542, A27 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Gorlova, N., Van Winckel, H., Ikonnikova, N. P., et al. 2015, MNRAS, 451, 2462 [NASA ADS] [CrossRef] [Google Scholar]
 Hilditch, R. W. 2001, An Introduction to Close Binary Stars (Cambridge: Cambridge University Press) [CrossRef] [Google Scholar]
 Hillen, M., Verhoelst, T., Van Winckel, H., et al. 2013, A&A, 559, A111 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hillen, M., Kluska, J., Le Bouquin, J.B., et al. 2016, A&A, 588, L1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hillen, M., Van Winckel, H., Menu, J., et al. 2017, A&A, 599, A41 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hubová, D., & Pejcha, O. 2019, MNRAS, 489, 891 [CrossRef] [Google Scholar]
 Izzard, R., & Jermyn, A. 2018, Galaxies, 6, 97 [NASA ADS] [CrossRef] [Google Scholar]
 Kamath, D., & Van Winckel, H. 2019, MNRAS, 486, 3524 [NASA ADS] [CrossRef] [Google Scholar]
 Kamath, D., Wood, P. R., & Van Winckel, H. 2014, MNRAS, 439, 2211 [NASA ADS] [CrossRef] [Google Scholar]
 Kamath, D., Wood, P. R., & Van Winckel, H. 2015, MNRAS, [Google Scholar]
 Kluska, J., Hillen, M., Van Winckel, H., et al. 2018, A&A, 616, A153 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Kurosawa, R., Harries, T. J., & Symington, N. H. 2006, MNRAS, 370, 580 [NASA ADS] [CrossRef] [Google Scholar]
 Kurosawa, R., Romanova, M. M., & Harries, T. J. 2011, MNRAS, 416, 2623 [NASA ADS] [CrossRef] [Google Scholar]
 Luttermoser, D. G., & Johnson, H. R. 1992, ApJ, 388, 579 [NASA ADS] [CrossRef] [Google Scholar]
 MacLeod, M., Ostriker, E. C., & Stone, J. M. 2018, ApJ, 868, 136 [NASA ADS] [CrossRef] [Google Scholar]
 Mohamed, S., & Podsiadlowski, P. 2007, in 15th European Workshop on White Dwarfs, eds. R. Napiwotzki, & M. R. Burleigh, ASP Conf. Ser., 372, 397 [Google Scholar]
 Mohamed, S., & Podsiadlowski, P. 2012, Balt. Astron., 21, 88 [Google Scholar]
 Muzerolle, J., Calvet, N., & Hartmann, L. 2001, ApJ, 550, 944 [NASA ADS] [CrossRef] [Google Scholar]
 Nisini, B., Antoniucci, S., Alcalá, J. M., et al. 2018, A&A, 609, A87 [EDP Sciences] [Google Scholar]
 Oomen, G.M., Van Winckel, H., Pols, O., et al. 2018, A&A, 620, A85 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Oomen, G.M., Van Winckel, H., Pols, O., & Nelemans, G. 2019, A&A, 629, A49 [CrossRef] [EDP Sciences] [Google Scholar]
 Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3 [Google Scholar]
 Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4 [NASA ADS] [CrossRef] [Google Scholar]
 Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15 [Google Scholar]
 Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Pejcha, O., Metzger, B. D., & Tomida, K. 2016, MNRAS, 461, 2527 [NASA ADS] [CrossRef] [Google Scholar]
 Rafikov, R. R. 2016, ApJ, 830, 8 [NASA ADS] [CrossRef] [Google Scholar]
 Ramstedt, S., Schöier, F. L., Olofsson, H., & Lundgren, A. A. 2008, A&A, 487, 645 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Raskin, G., van Winckel, H., Hensberge, H., et al. 2011, A&A, 526, A69 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ritter, H. 1988, A&A, 202, 93 [NASA ADS] [Google Scholar]
 Rybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics (New York: Wiley) [Google Scholar]
 Sahai, R., Vlemmings, W. H. T., Gledhill, T., et al. 2017, ApJ, 835, L13 [NASA ADS] [CrossRef] [Google Scholar]
 Shu, F. H., Lubow, S. H., & Anderson, L. 1979, ApJ, 229, 223 [NASA ADS] [CrossRef] [Google Scholar]
 Shu, F., Najita, J., Ostriker, E., et al. 1994, ApJ, 429, 781 [NASA ADS] [CrossRef] [Google Scholar]
 Tafoya, D., Orosz, G., Vlemmings, W. H. T., Sahai, R., & PérezSánchez, A. F. 2019, A&A, 629, A8 [CrossRef] [EDP Sciences] [Google Scholar]
 Tocknell, J., De Marco, O., & Wardle, M. 2014, MNRAS, 439, 2014 [NASA ADS] [CrossRef] [Google Scholar]
 Van Winckel, H. 2003, ARA&A, 41, 391 [NASA ADS] [CrossRef] [Google Scholar]
 Van Winckel, H. 2018, ArXiv eprints [arXiv:1809.00871] [Google Scholar]
 Van Winckel, H., Waelkens, C., & Waters, L. B. F. M. 1995, A&A, 293, L25 [NASA ADS] [Google Scholar]
 Van Winckel, H., Lloyd Evans, T., Briquet, M., et al. 2009, A&A, 505, 1221 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Vernazza, J. E., Avrett, E. H., & Loeser, R. 1973, ApJ, 184, 605 [NASA ADS] [CrossRef] [Google Scholar]
 Waters, L. B. F. M., Trams, N. R., & Waelkens, C. 1992, A&A, 262, L37 [NASA ADS] [Google Scholar]
 Wilson, R. E., & Sofia, S. 1976, ApJ, 203, 182 [NASA ADS] [CrossRef] [Google Scholar]
 Yung, B. H. K., Nakashima, J.I., Imai, H., et al. 2011, ApJ, 741, 94 [NASA ADS] [CrossRef] [Google Scholar]
Appendix A: The absorption feature in the Hα line
Fig. A.1. Hα line of BD+46°442 at two different phases in the orbital period. The Hα line displays a doublepeaked emission feature with a central absorption feature during inferior conjunction (black solid line), when the postAGB star is between the jet and the observer. During superior conjunction, when the jet is between the postAGB star and the observer, we observe a blueshifted absorption feature in the Hα line (blue dotted line). 
Appendix B: Orbital parameters of BD+46°442 and IRAS 19135+3937
Spectroscopic orbital solutions of the primary component of BD+46°442 and IRAS 19135+3937 (Oomen et al. 2018).
Appendix C: Balmer lines of BD+46°442 and IRAS 19135+3937
Fig. C.1. Balmer lines of BD+46°442 as a function of wavelength. The spectra are given in arbitrary units and offset according to their orbital phase. Numbers on the right vertical axis indicate the orbital phase of the spectra (from 0 to 100). The dashed vertical lines represent the centre of each Balmer line. 
All Tables
Bestfitting jet configuration and parameters for the spatiokinematic model of BD+46°442 and IRAS 19135+3937.
Spectroscopic orbital solutions of the primary component of BD+46°442 and IRAS 19135+3937 (Oomen et al. 2018).
All Figures
Fig. 1. Dynamic spectra for the Balmer lines of BD+46°442. Upper left: Hα, upper right: Hβ, lower left: Hγ, lower right: Hδ. The black dashed line indicates the phase of superior conjunction. The white line indicates the radial velocity of the postAGB star. The colour gradient represents the strength of the line at each phase. 

In the text 
Fig. 2. Schematic representation of the radiative transfer calculations in the jet. The ray travelling from the star to the observer is split into N_{j} grid points where it passes through the jet. Each grid point has a density ρ and a velocity v. We iterate over each grid point in order to determine the resulting intensity along this line for each wavelength. 

In the text 
Fig. 3. Interpolated observed and modelled dynamic spectra of the Hα line. The upper spectra are the observations (left) and model spectra (right) of BD+46°442. The lower spectra are the observations and model spectra of IRAS 19135+3937. The colours represent the normalised flux. 

In the text 
Fig. 4. Geometry of the binary system and the jet of BD+46°442 at superior conjunction, when the postAGB star is directly behind the jet, as viewed by the observer. In all three plots the full orange circle denotes the postAGB star. The orange star indicates the location of the companion. The red cross is the location of the centre of mass of the binary system. The radius of the postAGB star is to scale. The jet is represented in blue, and the colour indicates the relative density of the jet (see colour scale). The dashed black line is the jet axis and the dotted white lines are the inner jet edges. The jet cavity is the inner region of the jet. Upper left panel: system viewed along the orbital plane from a direction perpendicular to the line of sight to the observer. Right panel: jet viewed from an angle perpendicular to the Xaxis. The postAGB star is located behind the companion and its jet in this image. The jet tilt is noticeable from this angle. Lower left panel: binary system viewed from above, perpendicular to the orbital plane. The grey dashed lines represent the Roche radii of the two binary components and the full black line shows the Roche lobes. 

In the text 
Fig. 5. Twodimensional reduced chisquared distribution for the grid of jet densities n_{j} and temperatures T_{j} for the fitting of BD+46°442. The white dot gives the location of the minimum reduced chisquare value ${\chi}_{\nu ,\text{min}}^{2}=43.9$. The contours represent the 1σ, 2σ, and 3σ intervals. 

In the text 
Fig. 7. Equivalent width of the absorption by the jet for BD+46°442 as a function of orbital phase. The four panels show the EW in Hα (top left), Hβ (top right), Hγ (bottom left), and Hδ (bottom right). The circles are the measured EWs of the absorption feature by the jet in the observations with their respective errors. The full line is the EW of the absorption feature for the bestfitting model. 

In the text 
Fig. 6. Dynamic spectra for the Balmer lines of IRAS 19135+3937. Upper left: Hα, upper right: Hβ, lower left: Hγ, lower right: Hδ. The black dashed line indicated the phase of superior conjunction. The white line indicates the radial velocity of the postAGB star. The colour gradient represents the strength of the line at each phase. 

In the text 
Fig. 8. Similar to Fig. 4, but for IRAS 19135+3937. 

In the text 
Fig. 9. Twodimensional reduced chisquare distribution for the grid of jet densities n_{j} and temperatures T_{j} for the fitting of IRAS 19135+3937. The white dot gives the location of the minimum reduced chisquare value ${\chi}_{\nu ,\text{min}}^{2}=60.8$. The contours represent the 1σ, 2σ, and 3σ intervals. 

In the text 
Fig. 10. As for Fig. 7, but for IRAS 19135+3937. 

In the text 
Fig. A.1. Hα line of BD+46°442 at two different phases in the orbital period. The Hα line displays a doublepeaked emission feature with a central absorption feature during inferior conjunction (black solid line), when the postAGB star is between the jet and the observer. During superior conjunction, when the jet is between the postAGB star and the observer, we observe a blueshifted absorption feature in the Hα line (blue dotted line). 

In the text 
Fig. C.1. Balmer lines of BD+46°442 as a function of wavelength. The spectra are given in arbitrary units and offset according to their orbital phase. Numbers on the right vertical axis indicate the orbital phase of the spectra (from 0 to 100). The dashed vertical lines represent the centre of each Balmer line. 

In the text 
Fig. C.2. Similar to Fig. C.1, but for IRAS 19135+3937. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.