© R. Sharma et al. 2022

1. Introduction

Relativistic jets are produced when the inflowing, accreting plasma around a black hole gets expelled and collimated by magnetic forces (e.g., Meier et al. 2001). When a jet is mapped at high resolution with an interferometer at frequency ν, the surface of the jet where optical depth τ_ν ≈ 1 corresponds to the peak flux in the map, and is referred to as the core (Blandford & Königl 1979). The position of the core is a function of frequency, and this is known as the core-shift effect. This characteristic feature of the jet is supported by very long baseline interferometry (VLBI) observations of X-ray binaries (e.g., Paragi et al. 1999, 2013; Massi et al. 2012; Rushton et al. 2012) and active galactic nuclei (AGNs) (e.g., Lobanov 1998; O’Sullivan & Gabuzda 2009; Hada et al. 2011; Sokolovsky et al. 2011).

The core-shift effect is essential for both astrophysical and astrometric applications. It can help us to understand the physical conditions of the jet, such as the magnetic field and the distance of the core from the base of the jet (Lobanov 1998; Hirotani 2005). Moreover, it affects the precise astrometric measurements performed by VLBI (Rioja et al. 2010; Porcas 2009). The position of the core can depend on the activity state of the source (e.g., Kudryavtseva et al. 2011), with recent studies revealing its variability with time (Rushton et al. 2012; Paragi et al. 2013; Niinuma et al. 2015; Lisakov et al. 2017; Plavin et al. 2019), an effect that, if not taken into account, can further affect the accurate astrometric observations of sources.

From a theoretical point of view, the relationship between core-shift and spectral characteristics was derived by Konigl (1981) for synchrotron emission by electrons with a power-law distribution of energy. Using synchrotron emission equations and imposing optical depth equal to unity, the position, l, of the core at various frequencies results in a core-shift of ν^−1/k_r, with k_r depending on the spectral index, magnetic field, and particle density distribution (Konigl 1981).

In the present work, we aim to explore the core position in the jet parameter space, that is, as a function of frequency, magnetic field configuration, relativistic electron density, and jet inclination angle, l ≡ l(ν, B, κ, η), and to monitor the corresponding evolution of the related spectrum. The importance of adding the inclination angle, η, results from the shown anisotropy of jet emission with the inclination (Zdziarski et al. 2016) and the evidence from observations and simulations that the spectral index α can vary by changing the orientation of the jet (Fine et al. 2011; DiPompeo et al. 2012, see their Fig. 1). We want to test here if the core position shows similar anisotropies to those shown for jet emission and spectral index. In particular, in this paper we analyse jets in microquasars. Our code (Massi & Torricelli-Ciamponi 2014), based on the synchrotron-emitting conical jet by Kaiser (2006), provides us with the relevant information regarding the key parameters at different frequencies, that is, the position of the core (l_τ = 1) along the jet where τ = 1 and also the peak flux. In Sect. 2, we describe the geometry of our synchrotron-emitting jet model. In Sect. 3, we present our results. Sections 3.1 and 3.2 describe the trend of the core position and spectral index, respectively, in response to variation in inclination angle, the electron density of the jet, and different magnetic field configurations. In Sect. 3.3, the relationship between core position and spectral index is analysed. Finally, we present our conclusions in Sect. 4.

2. Jet model

Here, we describe the geometry that we adopt for the self-absorbed stellar radio jet. The jet makes an angle η with respect to the line of sight (LOS) and has a half-opening angle ξ (see Fig. 1). We choose η such that ξ < η  ≤  75° (Table 1). Miller-Jones et al. (2006) calculated the half-opening angles of different X-ray binaries. Using the values from Table 1 provided by these latter authors, that is, for GRS 1915+105, Cygnus X-3, GRO J1655-40, XTE J1550-564, H 1743-322 and 1RXS J001442.2+580201, along with SS 433 (Marshall et al. 2013) and an updated value for Cygnus X-1 (Tetarenko et al. 2019), we find the average half-opening angle of eight different X-ray binaries to be ξ ∼ 3.5°. We use this value for our model.

Fig. 1. Schematic representation of a jet with an inclination angle η and a constant half-opening angle ξ with tan ξ = r0/x0 (see text). The jet propagates along the x-axis with the jet base at x0. Line segment AO represents the path in a stratified jet as viewed by an observer along the LOS. The optical depths at A and O are τ = 0 and τ = τend, respectively.

2.1. Geometry of the jet

We follow the model for adiabatic jets provided by Kaiser (2006) and described in Massi & Torricelli-Ciamponi (2014). We consider a conical jet with the jet propagating along the x-axis. All quantities are considered to be radially constant and change only along the x-axis. Any position along the x-axis is defined as x = x₀l with the jet’s base at position x₀, as shown in Fig. 1. The parameter l is a dimensionless coordinate and is equal to unity at the base of the jet. Adopting a distance of 2 kpc and the other values from Table 1, any position along the x-axis when projected on the sky plane is about 0.2 lsinη in mas. The radius of the cone is defined as r = r₀l, where r₀ is a constant scaling factor. The cone has a constant half-opening angle ξ with tan ξ = r₀/x₀. For our model, the base of the conical jet is located at x₀ > 10⁶ R_sch (see Table 1), where R_sch is the Schwarzschild radius. Asada & Nakamura (2012) discovered that the jet maintains a parabolic streamline over a range in size scale equal to 10⁵ R_sch, while further downstream the jet takes on a conical shape. This result is consistent with the size of the acceleration/collimation region of a jet that is around 10⁴ R_sch following Marscher (2006). Thus our jet model starts after the acceleration zone.

We parameterize the evolution of the magnetic field of the jet as B = B₀l^−a₂, where B₀ is the magnetic field strength at the base of the jet and a₂ can take different values depending on the configuration of its magnetic field. From Table 1 of Kaiser (2006), for a purely parallel magnetic field configuration, a₂ = 2 and for a purely perpendicular magnetic field configuration, a₂ = 1. We consider both of these cases in this work.

The number density of the relativistic electrons producing synchrotron radiation has a power-law distribution in energy,

$$\begin{array}{c}\hfill N_{\mathrm{rel}}={\displaystyle {\int }_{E_{\min }}^{E_{\max }}\kappa E^{-p}\mathrm{dE},}\end{array}$$(1)

where E is the energy of the electrons. The power-law index p influences the flux density S ∝ ν^α, with spectral index α related to p as α = −(p − 1)/2 (Longair 1994). For our model, we use the parameters as given in Table 1. We assume the electron density distribution along the jet to evolve as κ = κ₀l^−a₃, where κ₀ is a parameter related to the relativistic electron density at the base of the jet and a₃ = 2(2 + p)/3 as derived by Kaiser (2006) from the mass conservation condition in the case of adiabatic losses. The spectral index range for optically thin emission is −1 < α < −0.2 (Fender 2001) which corresponds to 1.4 < p < 3. For p = 1.8, κ₀ is expressed in units of [erg^0.8 cm⁻³] throughout the paper. The integral in the above equation is performed from E_min to E_max, where E_min and E_max are the minimum and maximum energy of the electron distribution, respectively, and are related to the Lorentz factor, as discussed in Sect. 2.3.

2.2. Synchrotron emission and optical depth

In the framework of the jet model described above and developed by Massi & Torricelli-Ciamponi (2014) for varying inclination angles, here we derive the flux density and opacity of the jet model in terms of inclination angle η and length of the jet, l. The perpendicular direction to the surface of the approaching jet makes an angle 90° − (η − ξ) with respect to the LOS, whereas it makes an angle 90° − (η + ξ) for the receding jet. Thus, the flux emanating from the approaching jet is given by

$$\begin{array}{c}\hfill S_{\mathrm{a}}={\displaystyle {\int }_1^L{r}_0{x}_0{I}_{{\nu }_{\mathrm{a}}}(\eta ,l)\frac{sin(\eta -\xi )}{D^2}l\mathrm{d}l,}\end{array}$$(2)

where D is the distance of the source, and I_νa is the intensity of the approaching jet. The integral is performed over the length of the jet from l = 1 to L. We get the same expression for the receding jet flux (S_r), but the sign of the angle is reversed. In this work, we consider a mildly relativistic jet with Doppler factor ∼ 1 because we want to study the impact of opacity on jet properties due to the change in κ₀ and η. On the other hand, a greater Doppler factor increases the opacity (e.g., Eq. (10) in Finke 2019) and only enhances the studied effects, especially at small inclination angles.

For inclination angles ξ < η < 90°, the observer views a stratified jet with varying plasma conditions along the LOS. For such a case, the total optical depth will have a contribution from each layer along the LOS. Therefore, using the radiative transfer equation, the intensity I_ν emanating from the surface of the jet and in the direction of the observer at an angle η is given as

$$\begin{array}{c}\hfill I_{\nu }(\eta ,l)={\displaystyle {\int }_0^{{\tau }_{\mathrm{end}}(l)}\frac{J_{\nu }}{{\chi }_{\nu }}{\mathrm{e}}^{-{\tau }^{\prime }/cos\eta }\mathrm{d}\left[\frac{{\tau }^{\prime }}{cos\eta }\right].}\end{array}$$(3)

From Longair (1994) and using Kaiser’s notation, J_ν is the emissivity per unit volume, and χ_ν is the absorption coefficient given by

$$\begin{array}{c}\hfill [J_{\nu }=J_0{\nu }^{-(p-1)/2}{l}^{-a_3-(p+1)a_2/2}]\mathrm{W}{\mathrm{m}}^{-3}{\mathrm{Hz}}^{-1},\end{array}$$(4)

and

$$\begin{array}{c}\hfill [{\chi }_{\nu }={\chi }_0{\nu }^{-(p+4)/2}{l}^{-a_3-(p+2)a_2/2}]{\mathrm{m}}^{-1},\end{array}$$(5)

where,

$$\begin{array}{c}\hfill J_0=2.3{10}^{-25}{(1.3{10}^{37})}^{(p-1)/2}a(p)B_0^{(p+1)/2}{\kappa }_0,\end{array}$$(6)

and

$$\begin{array}{c}\hfill {\chi }_0=3.4{10}^{-9}{(3.5{10}^{18})}^{p}b(p)B_0^{(p+2)/2}{\kappa }_0,\end{array}$$(7)

with a(p) and b(p) being constants as given in Longair (1994). The optical depth is given by (as in Kaiser 2006)

$$\begin{array}{c}\hfill \stackrel{\sim }{\tau }(l)={\tau }_0{l}^C,\end{array}$$(8)

where C = 1 − a₃ − (p + 2)a₂/2. Using Table 1 from Kaiser (2006), all possible values of the exponent in Eq. (8) will be negative, that is, C = −5.3 and C = −3.4 for parallel and perpendicular magnetic field, respectively. Therefore, the optical depth is maximum at the base of the jet l = 1 and is given by

$$\begin{array}{c}\hfill {\tau }_0=\stackrel{\sim }{\tau }(l=1)=\frac{{\chi }_0{x}_0}{-[1-a_3-(p+2)a_2/2]}{\nu }^{-(p+4)/2}.\end{array}$$(9)

The integral in Eq. (3) is performed over the optical depth along the LOS (along line segment AO in Fig. 1) from τ = 0 ≡ τ_A (if it is assumed that no emission or absorption occurs between the jet and the observer) to τ_end(l), which represents the maximum optical depth of the jet, that is, until the position at O. For an approaching jet, the optical depth τ_end(l) is given as

$$\begin{array}{c}\hfill {\tau }_{\mathrm{end}}(l)={\tau }_0{l}^C[{\left(\frac{tan\eta -tan\xi }{tan\eta +tan\xi }\right)}^C-1].\end{array}$$(10)

See Eq. (A.5) in the Appendix of Massi & Torricelli-Ciamponi (2014) for a comprehensive description.

To find the position along the jet where the optical depth attains a specific value τ = τ′/cos η (see Eq. (3)), Eq. (10) becomes τ₀l^Cf(η, ξ) = τcos η. Re-arranging the equation, we have

$$\begin{array}{c}\hfill l(\tau )={\left(\frac{\tau cos\eta }{{\tau }_0[{\left(\frac{tan\eta -tan\xi }{tan\eta +tan\xi }\right)}^C-1]}\right)}^{1/C}.\end{array}$$(11)

As the core position is defined where the optical depth τ = 1, we can use Eq. (11) to compute it along the jet (l_τ = 1) for any given frequency ν, inclination angle η, relativistic electron density (dependent on the parameter κ; see Sect. 2.3), and magnetic field configuration (dependent on the parameter a₂).

We check the consistency of our expression for τ with that of Eq. (1) in Lobanov (1998), which was derived in the limit of a small jet opening angle. This latter equation, with N = N₁(r₁/r)ⁿ, B = B₁(r₁/r)^m, r₁ = 1 pc, and α = −0.5, i.e., s = 1−2α = 2, in the limit of mildly relativistic flows, reads as

$$\begin{array}{c}\hfill \tau {(r)}_{\mathrm{Lobanov}}=6.1\times {10}^{44}{N}_1{B}_1^2{r}^{-(2m+n-1)}{\nu }^{-3}(\varphi /sin(\theta )).\end{array}$$(12)

From Eq. (11), our expression for τ is

$$\begin{array}{c}\hfill \tau =4.1\times {10}^{26}\frac{b(p){\kappa }_0{x}_0}{-Ccos\eta }{B}_0^2{\nu }^{-3}{l}^C[{\left(\frac{tan\eta -tan\xi }{tan\eta +tan\xi }\right)}^C-1].\end{array}$$(13)

This equation can be compared with Eq. (1) from Lobanov (1998) (here Eq. (12)), for small ξ and η > ξ (Konigl 1981). In this limit, the angular part can be approximated as

$$\begin{array}{c}\hfill \frac{[{(1-2\frac{\xi }{tan\eta })}^C-1]}{cos\eta }\simeq \frac{[(1-2C\frac{\xi }{tan\eta }-1]}{cos\eta }=-2C\frac{\xi }{sin\eta }·\end{array}$$(14)

Setting the normalization distance x₀ = 1 pc and the exponent p = 2, as in Lobanov, with b(p = 2) = 0.269 (Longair 1994),

$$\begin{array}{c}\hfill \tau =6.8\times {10}^{44}{\kappa }_0{B}_0^2{l}^{-(2a_2+a_3-1)}{\nu }^{-3}(\xi /sin\eta ).\end{array}$$(15)

This expression is the same as Eq. (12), taking into account the correspondence between our parameters and those of Lobanov (1998), a₂ = m, a₃ = n, ξ = Φ, and η = θ and the different approximations used for synchrotron absorption; in fact we used the formalization of Longair (1994) instead of that of Blumenthal & Gould (1970) and Rybicki & Lightman (1979), as in Lobanov (1998).

2.3. Model parameter κ and relativistic electron density

To understand the physical implications of the parameter κ, in this section, we calculate the relativistic electron density for Cygnus X-1, as it is the best example of a system with a flat radio spectrum over the entire radio band until millimetre range (Fender 2001). The range in energy for which these relativistic electrons exist is required to compute the relativistic electron density, and therefore,

$$\begin{array}{c}\hfill E_{\min }={\gamma }_{\min }m{c}^2\le E\le {\gamma }_{\max }m{c}^2=E_{\max },\end{array}$$(16)

where E_min and E_max is the minimum and maximum energy of the electron distribution, respectively and γ is the Lorentz factor. As for a synchrotron process, γ_min ≫ 1 (Dulk 1985), we set γ_min = 10. Integrating Eq. (1) gives

$$\begin{array}{c}\hfill N_{\mathrm{rel}}=\frac{\kappa }{1-p}(E_{\max }^{1-p}-E_{\min }^{1-p}).\end{array}$$(17)

For γ_max ≫ γ_min, the number of relativistic electrons per cubic centimetre at the base of the jet (l = 1) is given by

$$\begin{array}{c}\hfill N_{\mathrm{rel}}=\frac{{\kappa }_0}{p-1}{({\gamma }_{\min }m{c}^2)}^{1-p}.\end{array}$$(18)

For Cygnus X-1, the model of Kaiser (2006) with parameters from his Table 2 yield a relativistic electron density N_rel ≈ 7 × 10⁴ cm⁻³. We therefore chose κ₀ = 6 erg^0.8 cm⁻³ for our jet model, which corresponds to the relativistic electron density N_rel ≈ 7 × 10⁴ cm⁻³ as for Cygnus X-1. We also study an intermediate case with κ₀ = 0.6 erg^0.8 cm⁻³ with relativistic electron density N_rel ≈ 7 × 10³ cm⁻³. In the latter part of Sect. 3, we generalise our study for κ₀ = 0.06−60 erg^0.8 cm⁻³.

3. Results

In this section, we use the parameters of Table 1 to analyse the characteristics of the above-described model of a self-absorbed jet as a function of the angle the jet axis makes with respect to the LOS and the injected relativistic electrons. We study the core-shift and spectral properties of the jet for both parallel and perpendicular magnetic field configurations.

3.1. Core-shift

Figures 2 and 3 show the core position (l_τ = 1) for different frequencies (2.2, 5, 7, 8.3, and 15 GHz) for parallel and perpendicular magnetic field configurations. The core positions are plotted for different inclination angles, η = 5.2° and 7.6°. Comparing Figs. 2a with 2c, we see that for a jet with the parallel magnetic field, the core is farther away from the base of the jet than that for a perpendicular magnetic field. For instance, the 5 GHz core position is at l_τ = 1 ≈ 2.8 for the parallel magnetic field (Fig. 2a) and at l_τ = 1 ≈ 2.0 for the perpendicular magnetic field (Fig. 2c). Previous studies of a compact jet reveal that the core position is a function of the observing frequency, l_τ = 1 ∝ ν^−1/k_r, where k_r = ((3−2α)a₂ + 2a₃ − 2)/(5−2α). The power index k_r depends on the magnetic field, particle density, and electron energy spectrum (Konigl 1981). Therefore, first, we fit a power-law function of the form $l_{\tau =1}=a{\nu }^{-1/k_r^{\prime }}$ (blue lines in Figs. 2, 3) where a is a constant depending on the fit. For the performed fits, for the parallel magnetic field, $k_r^{\prime }=1.8$ and for the perpendicular magnetic field, $k_r^{\prime }=1.2$, irrespective of the angle η. To calculate the theoretical values of k_r, we then use the spectral index values derived from the jet emission calculation for the given inclination angle. For instance, for κ₀ = 0.6 and B_||, we have α = −0.13 for η = 5.2° (top panel of Fig. 7). Using a₂ and a₃ from Sect. 2.1, we get k_r = 1.8. Similarly, for B_⊥, we have α = −0.34 for η = 5.2° (top panel of Fig. 8) and we get k_r = 1.2. Thus, we see that the values of $k_r^{\prime }$ found by the fit of the derived core positions are equal to the theoretically calculated values of k_r when using the derived spectral index for the same inclination angle.

Fig. 2. Core positions (with reference to the base of the jet, l = 1) for two different inclination angles. Top panels: parallel magnetic field configuration. Bottom panels: perpendicular magnetic field configuration, with κ0 = 0.6. The best-fit power-law function $l\propto {\nu }^{-1/k_r^{\prime }}$ is represented with blue lines. To determine the angular distance projected on the sky in milliarcseconds (mas) the y-axis values must be multiplied by 0.2 sin η or 0.16 sin η for parallel and perpendicular B, respectively (for a distance of 2 kpc and the x0 values of Table 1, see Sect. 2.1).

Fig. 3. Same as Fig. 2 but for κ0 = 6.

We also compare our results with Eq. (11) in Lobanov (1998) where we set the bulk Lorentz factor equal to unity and redshift z = 0. Assuming equipartition between jet particle and magnetic field energies, the equation predicts the core shift Δr at two frequencies in a jet with synchrotron luminosity calculated using the model of Blandford & Königl (1979). To compare our work with Lobanov’s equation we chose the set of parameters of the perpendicular magnetic field (i.e., a₂ = 1, a₃ = 2.5, p = 1.8) which are more compatible with Lobanov’s set of parameters (i.e., m = 1, n = 2, s = 2). For η = 5.2°, ξ = 3.5°, using the values from Table 1 for perpendicular magnetic field, namely x₀ = 4.7 × 10¹²cm and B₀ = 9.6 Gauss, and a synchrotron luminosity L_sync = 1.7 × 10³⁴ erg s⁻¹ (following Eq. (23) in Blandford & Königl 1979), Lobanov’s Eq. (11) results in a shift between 2 GHz and 8 GHz of Δr = 0.015 mas. In addition, one has to consider that our results compare with Lobanov’s results for small ξ and η > ξ, as shown in Sect. 2.2. Figure 4 shows the ratio between our general equation (Eq. (11)) on which our plots of Figs. 2 and 3 are based, and its approximation for small ξ and η > ξ. Whereas the ratio tends to 1 for larger angles, at η = 5.2° the ratio is equal to 3.3. Taking into account this correction factor of 3.3, the predicted Δr = 0.015 mas rises to 0.05 mas, well within the range of 0.04 mas–0.07 mas obtained from Figs. 2c and 3c. Namely, the two values correspond to κ₀ of 0.6 (Fig. 2c) and 6.0 (Fig. 3c), and from the equipartition the value for κ₀ from B₀ = 9.6 Gauss is κ₀ = 3.6.

Fig. 4. Ratio of our Eq. (11) with its approximation for small ξ and η > ξ, using ξ = 3.5° (see Sect. 2.2). The symbol star indicates the ratio of 3.3 resulting for η = 5.2°.

In addition to verifying the known core-shift relationship with frequency and agreement of our findings with theoretical predictions, we obtain a new result. Figure 2 shows that for the same magnetic field configuration, increasing the inclination angle η shifts the core position of a given frequency closer to the base of the jet. For example, for the parallel magnetic field in Figs. 2a,b, the core position for 5 GHz emission is at l_τ = 1 ≈ 2.8 for η = 5.2° and decreases to l_τ = 1 ≈ 1.4 for η = 7.6°. Similarly, for perpendicular magnetic field (see Figs. 2c,d), let us consider again the core position for 5 GHz emission, which is at l_τ = 1 ≈ 2.0 for η = 5.2° but for η ≥ 7.6° the optical depth becomes τ < 1 all along the conical part of the jet. This shows that the core-shift depends not only on the frequency but also on the jet’s inclination angle.

Another important result from our analysis is shown in Fig. 3, which shows the changes in core position for a higher relativistic electron density, that is, for κ₀ = 6. We see that for higher relativistic electrons, the core moves downstream, that is, away from the base of the jet. Let us revisit the core position for 5 GHz. In Fig. 2a, the core position for 5 GHz is at l_τ = 1 ≈ 2.8 for κ₀ = 0.6, whereas it increases to l_τ = 1 ≈ 4.2 for κ₀ = 6 as shown in Fig. 3a.

The change in the core position for different inclination angles η is shown in Fig. 5 for 5 GHz and 7 GHz for the parallel magnetic field. The position of the core can be seen to decay with increasing inclination angle. The last angle for 5 GHz where the core is present in the conical jet is η ≈ 9.1° and is shown by a blue circle. For larger angles, the emission of 5 GHz becomes optically thin all along the conical jet. This holds true for the particular case of κ₀ = 0.6. As shown in Fig. 6, at larger electron densities, the core persists at much larger angles. In Fig. 5, for 7 GHz, the core is present in the conical jet until η ≈ 7.6°. We performed a fit with an exponential function of the form, l ∝ e^q/η, as shown by dashed lines where q is a constant depending on the fit.

Fig. 5. Core position as a function of the inclination angle for 5 GHz and 7 GHz with parallel magnetic field configuration and κ0 = 0.6. The dashed lines represent an exponential decay in the positions as the angle increases, given by l ∝ eq/η. The dotted line represents the jet base. The optical depth becomes τ < 1 all along the conical part of the jet for angles greater than 9.1° (blue circle) and 7.6° (red circle) for 5 GHz and 7 GHz, respectively. This last inclination angle where the core is present in the conical jet is plotted for all frequencies and different κ0 in Fig. 6.

In Fig. 6, we study the presence of the core in the conical jet for each frequency and different relativistic electron densities (ranging from κ₀ = 0.06−60.0). For a given frequency, each position marked in the (η, κ₀)–plane refers to the last inclination angle for which the core is present for that frequency, before moving below the base of the jet. This way, the blue circle at (η, κ₀) = (9.1°, 0.6) for 5 GHz represents the same condition as the blue circle in Fig. 5. The dependence of κ₀ on η is fitted with a power-law function for all frequencies. The curves divide the (η, κ₀)–plane into two distinct regions: any position on and above the curve represents the presence of the core in the conical jet for a given frequency; for example, for 2 GHz, it is the region above the blue shaded area. For instance, as shown in the top panel of Fig. 6, for the parallel magnetic field configuration and κ₀ = 0.06, for a frequency of 2 GHz, the core is present until an inclination angle of η ≈ 9.1°. On the other hand, for 15 GHz, the core is present until 5.2° at this relativistic electron density. To observe the core at a greater inclination angle, we would need to increase the number of relativistic electrons in the jet. Therefore, for the core to be present in the conical jet for 2 GHz at ≈27°, we need an increased electron density with κ₀ = 0.6. The same applies to the perpendicular magnetic field configuration (bottom panel of Fig. 6).

Fig. 6. Last inclination angle in the (η, κ0)–plane where the core is still present in the conical jet for the given frequencies (2, 5, 7, 8, and 15 GHz), before moving below the base. The curve divides the (η, κ0)–plane into two distinct regions: presence of core along the conical jet, i.e. lτ = 1 ⩾ 1 (points on and area above the curve) and core moved below the base of the jet lτ = 1 < 1 (area under the curve) for a given frequency. As an example, here we trace the area under the curve for 2 GHz (blue shaded region). Y-axis is in log scale. Top and bottom panels show results for parallel and perpendicular magnetic field configurations, respectively.

Very long baseline array (VLBA) astrometry of the microquasar LS I +61°303 (Wu et al. 2018) can be better understood in light of our results. A sequence of ten observations was spaced 3–4 days apart and the core of the precessing jet was expected to follow a regular path as well, that is, with regularly spaced astrometric points. Instead, VLBA astrometry (Wu et al. 2018) shows that the positions of the cores are displaced from the jet base, with the largest offsets for those observations corresponding to the smallest inclination angles. A future quantitative comparison of our determined angular distances projected on the sky with observed astrometry will make it possible to constrain important parameters such as κ₀ and B₀ for LS I +61°303 and other microquasars.

3.2. Spectral index analysis

Figures 7 and 8 show the spectra of the jet for different inclination angles for parallel and perpendicular magnetic field cases, respectively. We show the power-law fit required to get the spectral index α only for higher frequencies, i.e., excluding 2 GHz. This is because the 2 GHz core survives the largest inclination angle, and including it would bias the fit, resulting in large uncertainties for the spectral index. For instance, as shown by the blue dashed line, the spectra between 2 GHz and 5 GHz are inverted, contrary to the spectra between other frequencies (top panel of Fig. 7).

Fig. 7. Flux as a function of frequency for different inclination angles for a parallel magnetic field configuration. Both axes are in log scale. The blue dashed lines connect the model data and the red lines show the spectral index fit. Top panel: with κ0 = 0.6.. Bottom panel: with κ0 = 6.

Fig. 8. Same as Fig. 7, but for a perpendicular magnetic field configuration.

Considering the spectra from 5 to 15 GHz in the top panel of Fig. 7 for κ₀ = 0.6, the spectral index indicates optically thin emission (α < 0) at all inclination angles while becoming more negative for increasing angles. For the same electron density, but for the perpendicular magnetic field (top panel of Fig. 8), the emission is again optically thin. However, the spectral index is more negative than for the corresponding angles of the parallel magnetic field. When we increase the relativistic electron density to κ₀ = 6 (bottom panel of Fig. 7), the spectra become inverted (α > 0) and only become flat around 73°. For the perpendicular case (bottom panel of Fig. 8), the spectra are initially flat at 5.2° and later become optically thin. Our analysis shows that if we increase the relativistic electron density, the spectra become flat or even inverted. Thus we see that the spectral index changes significantly depending on the inclination angle, magnetic field configuration, and relativistic electron density, similar to core position changes as shown in the previous section.

We compare our results with 6 yr of radio observations of LS I +61°303. Spectral index data for LS I +61°303 are shown in Fig. 5 of Massi & Kaufman Bernadó (2009). The spectral index versus orbital phase presents a recurrent pattern: α ≥ 0 twice along the orbit, during the maximum of the long-term flux density modulation where Θ = 0.0−0.1, and Θ = 0.9−1.0, i.e., for ejections at small inclination angles (Massi & Torricelli-Ciamponi 2014). The trend changes and evolves to α < 0 along the orbit during the minimum of the long-term flux density modulation where Θ = 0.4−0.5, i.e., for ejections at the largest inclination angles (Massi & Torricelli-Ciamponi 2014).

3.3. Relationship between core position and spectral index

We analysed the core positions and spectral indices in previous sections by changing the inclination angles and relativistic electron density. In this section, we study their mutual relationship.

In Fig. 9a, we show the change in the spectral index between 5 GHz and 7 GHz for different inclination angles. For κ₀ = 0.6, we see that the spectrum is inverted for angles below η = 19° and becomes optically thin from η = 19.1° (marked by the green square). As noted above, the core position at the jet base, l_τ = 1 = 1, is present until η = 9.1° for 5 GHz (blue circle in Figs. 5 and 9a) and until η = 7.6° for 7 GHz (red circle in Fig. 5). This is different from the expected angle of η = 19.1°. This means that even though the core is not present, α still remains positive near the base of the jet. For the spectrum to evolve from inverted (i.e., α ⩾ 0) to optically thin (i.e., α < 0), the optical depth must fulfil the condition τ ≪ 1 all along the jet, including at the base of the jet. We quantified this value τ ≪ 1 with our code. For that purpose, we performed several iterations of the code assuming different values for τ = τ′/cos η (see Eq. (3)). At each iteration, we checked for the spectral index transition from α ⩾ 0 to α < 0. We find that for our jet model, the transition occurs when τ′/cos η = 0.08 at the base of the jet (l = 1), confirming that the condition for an optically thin jet is τ = τ′/cos η ≪ 1. This implies that the transition of α to negative values on the jet surface does not happen at l_τ = 1 = 1, that is, at the core at the base of the jet, but at l_{τ = 0.08} = l(τ′/cos η = 0.08) = 1, that is, when the jet becomes completely optically thin until the base. Applying the condition τ = 0.08 in Eq. (11), for κ₀ = 0.6, the optically thin emission begins at η = 19.1° (marked by the green square) as shown in Figs. 9b,c.

Fig. 9. (a) Spectral index between 5 GHz and 7 GHz as a function of the inclination angle for κ0 = 0.6. The green square represents the angle where the emission becomes optically thin. Red and blue circles are the same as in Figs. 5 and 6. (b),(c) Positions of the jet surface where optical depth τ = 0.08 for 5 GHz and 7 GHz, respectively, with κ0 = 0.6. lτ = 0.08 = 1 represents the position on the jet after which the emission becomes optically thin all along the jet, i.e., when τ ≪ 1.

4. Conclusions

We studied the synchrotron-emitting conical, self-absorbed jet initially proposed by Blandford & Königl (1979), later developed by Kaiser (2006), and finally modified to vary the inclination angle by Massi & Torricelli-Ciamponi (2014). The aim of this study is to explore the core position, l, as a function of frequency, magnetic field alignment, relativistic electron density, and jet inclination angle.

First, we checked that our derived expression for the optical depth, that is, the fundamental equation in our analysis, is equal to the commonly used Eq. (1) in Lobanov (1998). From our jet model results, we find that without changing the inclination angle of the jet or without changing the density of the relativistic electrons, the position of the core for a higher frequency is closer to the base of the jet than for lower frequencies, as expected in light of the synchrotron self-absorption effects (Konigl 1981). We then checked the consistency of our results, specifically paying attention to the obtained position of the cores at various frequencies with the derived core-frequency relationship of the jets, l ∝ ν^−1/k_r by Konigl (1981) and also with the derived shift of the core position between two frequencies calculated by Eq. (11) in Lobanov (1998).

Finally, we proceeded to study the core position in the jet at different inclination angles and electron densities. Our results in Figs. 2 and 3 show that for a given frequency, the greater the inclination angle, the closer the core is to the base of the jet. This implies that the core position can vary in an irregular way if the jet is precessing. Therefore, the deviations from a regular path are expected, as indeed traced by astrometric observations by Wu et al. (2018) which are discussed here in Sect. 3.1. As shown in Fig. 3, the core is more displaced from the jet base for higher electron densities. This is consistent with the observations where increased relativistic electron density due to flares shows a displacement of the core position downstream from the jet (Niinuma et al. 2015; Lisakov et al. 2017; Plavin et al. 2019). Figure 6 summarise the results of variations in inclination angle, electron density, and magnetic field configuration. The curves in the (η, κ₀)–plane show where for a given frequency the conical jet becomes entirely optically thin. The corresponding spectra, functions of the same parameters, are shown in Figs. 7 and 8. Comparing the spectral index for different relativistic electron densities, we see that an increase in electron density causes the spectra to become less steep or even inverted. Such results have been confirmed by observations of radio flares (e.g., Lisakov et al. 2017). Moreover, we see that the spectral index varies by changing the jet orientation, with a steep spectrum at large inclination angles getting flatter as the angle decreases (top panel of Fig. 7 and bottom panel of Fig. 8), or a flat spectrum at large inclination angles becoming inverted as the angle decreases (bottom panel of Fig. 7). These results for the relationship between spectral index and inclination angle are consistent with observations, as discussed in Sect. 3.2. Moreover, simulations (Fig. 1 in DiPompeo et al. 2012) show the same trend between α and η as in Fig. 9. In Fig. 9a, the spectral index at low inclination begins with a high positive value (inverted spectrum). Increasing the value of η decreases α until it crosses the zero level, becomes negative (steep spectrum) and steeper at larger inclination angles. It is worth noting that in our work, we can follow the change of the spectral index with corresponding displacements of the cores along the jet (Figs. 9b,c).

Acknowledgments

We would like to express our gratitude to the referee for his valuable and profound comments. GTC thanks INAF-Osservatorio di Arcetri for support.

References

All Tables

All Figures

	Fig. 1. Schematic representation of a jet with an inclination angle η and a constant half-opening angle ξ with tan ξ = r0/x0 (see text). The jet propagates along the x-axis with the jet base at x0. Line segment AO represents the path in a stratified jet as viewed by an observer along the LOS. The optical depths at A and O are τ = 0 and τ = τend, respectively.
In the text

Fig. 2. Core positions (with reference to the base of the jet, l = 1) for two different inclination angles. Top panels: parallel magnetic field configuration. Bottom panels: perpendicular magnetic field configuration, with κ0 = 0.6. The best-fit power-law function $l\propto {\nu }^{-1/k_r^{\prime }}$ is represented with blue lines. To determine the angular distance projected on the sky in milliarcseconds (mas) the y-axis values must be multiplied by 0.2 sin η or 0.16 sin η for parallel and perpendicular B, respectively (for a distance of 2 kpc and the x0 values of Table 1, see Sect. 2.1).

In the text

	Fig. 3. Same as Fig. 2 but for κ0 = 6.
In the text

	Fig. 4. Ratio of our Eq. (11) with its approximation for small ξ and η > ξ, using ξ = 3.5° (see Sect. 2.2). The symbol star indicates the ratio of 3.3 resulting for η = 5.2°.
In the text

Fig. 5. Core position as a function of the inclination angle for 5 GHz and 7 GHz with parallel magnetic field configuration and κ0 = 0.6. The dashed lines represent an exponential decay in the positions as the angle increases, given by l ∝ eq/η. The dotted line represents the jet base. The optical depth becomes τ < 1 all along the conical part of the jet for angles greater than 9.1° (blue circle) and 7.6° (red circle) for 5 GHz and 7 GHz, respectively. This last inclination angle where the core is present in the conical jet is plotted for all frequencies and different κ0 in Fig. 6.

In the text

Fig. 6. Last inclination angle in the (η, κ0)–plane where the core is still present in the conical jet for the given frequencies (2, 5, 7, 8, and 15 GHz), before moving below the base. The curve divides the (η, κ0)–plane into two distinct regions: presence of core along the conical jet, i.e. lτ = 1 ⩾ 1 (points on and area above the curve) and core moved below the base of the jet lτ = 1 < 1 (area under the curve) for a given frequency. As an example, here we trace the area under the curve for 2 GHz (blue shaded region). Y-axis is in log scale. Top and bottom panels show results for parallel and perpendicular magnetic field configurations, respectively.

In the text

	Fig. 7. Flux as a function of frequency for different inclination angles for a parallel magnetic field configuration. Both axes are in log scale. The blue dashed lines connect the model data and the red lines show the spectral index fit. Top panel: with κ0 = 0.6.. Bottom panel: with κ0 = 6.
In the text

	Fig. 8. Same as Fig. 7, but for a perpendicular magnetic field configuration.
In the text

Fig. 9. (a) Spectral index between 5 GHz and 7 GHz as a function of the inclination angle for κ0 = 0.6. The green square represents the angle where the emission becomes optically thin. Red and blue circles are the same as in Figs. 5 and 6. (b),(c) Positions of the jet surface where optical depth τ = 0.08 for 5 GHz and 7 GHz, respectively, with κ0 = 0.6. lτ = 0.08 = 1 represents the position on the jet after which the emission becomes optically thin all along the jet, i.e., when τ ≪ 1.

In the text