Derivation of the physical parameters of the jet in S5 0836+710 from stability analysis

A number of extragalactic jets show periodic structures at different scales that can be associated with growing instabilities. The wavelengths of the developing instability modes and their ratios depend on the flow parameters, so the study of those structures can shed light on jet physics at the scales involved. In this work, we use the fits to the jet ridgeline obtained from different observations of S5 B0836$+$710 and apply stability analysis of relativistic, sheared flows to derive an estimate of the physical parameters of the jet. Based on the assumption that the observed structures are generated by growing Kelvin-Helmholtz (KH) instability modes, we have run numerical calculations of stability of a relativistic, sheared jet over a range of different jet parameters. We have spanned several orders of magnitude in jet-to-ambient medium density ratio, and jet internal energy, and checked different values of the Lorentz factor and shear layer width. This represents an independent method to obtain estimates of the physical parameters of a jet. By comparing the fastest growing wavelengths of each relevant mode given by the calculations with the observed wavelengths reported in the literature, we have derived independent estimates of the jet Lorentz factor, specific internal energy, jet-to-ambient medium density ratio and Mach number. We obtain a jet Lorentz factor $\gamma \simeq 12$, specific internal energy of $\varepsilon \simeq 10^{-2}\,c^2$, jet-to-ambient medium density ratio of $\eta\approx 10^{-3}$, and an internal (classical) jet Mach number of $M_\mathrm{j}\approx 12$. We also find that the wavelength ratios are better recovered by a transversal structure with a width of $\simeq 10\,\%$ of the jet radius. This method represents a powerful tool to derive the jet parameters in all jets showing helical patterns with different wavelengths.


Introduction
Extragalactic jets of relativistic plasma are formed in the immediate vicinity of supermassive black holes (SMBH) residing in the nuclei of active galaxies (AGN, Blandford & Znajek 1977). The jets not only carry energy from this region to hundreds of kiloparsecs, but also condition galaxy evolution since the moment they are initially triggered (see, e.g., McNamara & Nulsen 2012). Therefore, determination of their physical parameters can allow us to understand the precise processes governing them and understand their impact on the ambient medium in more detail.
Radio images of many extragalactic jets from AGN show helical patterns on parsec and kiloparsec scales (see, e.g., Lister et al. 2013). In different works, these patterns have been associated either with Alfvén (Cohen et al. 2015) or with pressure waves (Perucho et al. 2012a). A periodicity in the direction of ejection of the flow due to jet precession is probably the origin of the oscillations. Precession can arise from the gravitational effect of the accretion disk on the compact object in the center of the AGN or in a binary-black-hole (see, e.g., Lister et al. 2003;Stirling et al. 2003;Lobanov & Roland 2005;Bach et al. 2006;Savolainen et al. 2006). Once triggered, the induced oscillation can couple to instability modes of the Kelvin-Helmholtz instability (KH, Perucho et al. 2005;Mizuno et al. 2007;Perucho et al. 2010) or current driven instability (CD, Mizuno et al. 2009;McKinney & Blandford 2009;Mignone et al. 2010;Mizuno et al. 2011) and grow in amplitude with time/distance as the wave is advected downstream.
The radio source S5 0836+710 (4C +71.07; J0841+7053) is a powerful low-polarization quasar (LPQ) located at a redshift of 2.17 (Osmer et al. 1994), which corresponds to the luminosity distance of 16.9 Gpc and a linear scale of 8.4 pc/mas, assuming the standard ΛCDM cosmology (H 0 = 73 km s −1 Mpc −1 , Ω m = 0.27, Ω Λ = 0.73) (Planck Collaboration et al. 2016). It has a long and one-sided jet at parsec and kiloparsec scales. Largescale radio emission was revealed by the VLA 1 and MERLIN 2 at distances larger than 1 arcsecond (Hummel et al. 1992;Perucho et al. 2012b). The jet shows apparent speeds of up to 21 c at 15 GHz, with a mean of 17 c (γ j 17), as reported by Lister et al. (2013). The estimated jet viewing angle is θ ≈ 3.2 • (Otterbein et al. 1998), who also gave a Lorentz factor of γ j = 12 from jet kinematics at 8 GHz.
The source morphology suggests the existence of plasma instability. Krichbaum et al. (1990) observed the source with VLBI 3 at 326 MHz and 5 GHz and recognized several kinks in the flow that can be associated with the growth of instability.
From high-resolution images obtained with VLBI Space Observatory Program (VSOP; Hirabayashi et al. 2000) at 1.6 GHz and 5 GHz, Lobanov et al. (1998) suggested the presence of Kelvin-Helmholtz (KH) instability in the flow, based on the observed morphology of the jet brightness ridgeline. The oscillations in the ridgeline were identified with helical and elliptical surface modes of the instability. Perucho et al. (2012a) provided further evidence for the interpretation of these oscillations as caused by KH instability modes. Furthermore, observations with MERLIN show an emission gap between 0.2 arcsec and 1.0 arcsec and a large-scale, apparently decollimated structure beyond 1 arcsec. This was explained as the expansion and disruption of the jet due to a helical instability mode (Perucho et al. 2012b).
In this paper, we use the results obtained from different observational campaigns on this source (see Lobanov et al. 2006;Perucho et al. 2012a), complemented by the Global VLBI array observations performed on October 24th, 2013, (L-Band), as part of RadioAstron space VLBI observations of S5 0836+710 (Vega García 2018, Vega-García et al., submitted). We apply a numerical solver of the stability equation for sheared, relativistic flows Perucho & Lobanov 2007) to derive an estimate of the physical parameters in the jet. In contrast with Perucho & Lobanov (2007), where the authors fixed the physical parameters of the jet to those obtained via previous estimates (Lobanov et al. 1998(Lobanov et al. , 2006 and studied the possibility of the presence of a shear-layer separating the jet and its environment, here we provide an independent estimate of all the relevant parameters without any initial restriction. The wavelengths corresponding to the fastest growing modes derived from the calculations are compared to the observed wavelengths in the jet, which allows us to provide a set of jet parameters.
The paper is structured as follows. In Section 2 we summarize the observational results that have provided fits to the intrinsic wavelengths of the jet ridgeline. Section 3 includes an explanation of the method that we have used. The results obtained are presented in Section 4. Finally, the discussion of those results and conclusions of this work are given in Section 5.

Ridgeline oscillations
The ridgeline of the jet is defined as a line connecting the maxima of Gaussian profiles fitted to the jet brightness profiles measured transversally to the jet direction. Figure 1 shows an example of the ridgeline in the jet in 0836+710 from a VLBI image at 1.6 GHz (Vega García 2018, Vega-García et al., submitted). At this frequency, the jet in 0836+710 can be traced up to ≈ 130 mas (deprojected linear distance of ≈ 10 kpc), which enables making detailed studies of a number of instability wavelengths.
A comparison of the ridgelines obtained from the VLBI images of 0836+710 made at 1.6 GHz at different epochs shows a consistent picture of a slowly evolving instability pattern in the jet (Perucho et al. 2012a). These ridgelines are shown in Fig. 2, tracing the jet up to a distance of 65 mas. At distances larger than about 60 mas, the jet is substantially resolved in the transverse direction. At these scales the formal ridgeline derived from the brightness profiles is not as reliable for reflecting the plasma instability development, and a more complex description comprising multiple threads inside the flow may be needed (e.g., the one similar to the approach employed for strongly resolved jets in 3C273; Lobanov & Zensus 2001). In addition to this, it is also likely that the plasma instability modes developing in the inner jet would saturate or dissipate at these distances (see Perucho et al. 2012a). Because of these considerations, we limit our analysis of the jet instability in 0836+710 to the structures   Ridgelines at 1.6 GHz from the observations described in Perucho et al. (2012a) together with those derived from the 2013/2014 Global VLBI observations. The main described structure is similar. The differences are due to the time difference, different data quality and different uv-coverages.
traced by the jet ridgeline within the inner 65 mas of the flow. The ridgelines offsets are plotted with respect to the overall jet axis assumed to be oriented at a position angle of −162 • . The small differences observed among the different epochs can be attributed to image noise and small changes in time (Perucho et al. 2012a;Vega García 2018).
Since the oscillations displace the jet axis from a straight line (as expected for the helical modes of KH instability) and the transverse profiles show hints of double-peaked structure (expected for the elliptical mode of the instability) only in the close vicinity of the core Perucho et al. (2012a), we consider that helical modes are largely determining the appearance of the ridgelines. The oscillations can thus be modelled as waves developing along the jet. From now on, we assume that these waves correspond to KH instability modes developing in the jet. Therefore, one can take the offset of the ridge from the straight line defining the jet direction as the projected amplitude of the wave, ∆ r(z). Taking into account that the jet opening angle is small (< 1 • , Perucho et al. 2012a), the expected effect of jet expansion on the mode wavelength (λ ∝ R, Hardee 1982) can be considered to be small. Then, assuming a constant wave amplitude, 4 the offsets can be modelled as: where a i is the amplitude, ψ i the phase, and λ i the wavelength corresponding to the i-th mode. The offsets of the ridge on the plane of the sky are not affected by projection effects. However, taking into account that we model a three-dimensional helical structure, there is a distortion on the observed ridgeline produced by travel time effects across the jet cross section. In this case, the distortion is caused by the oscillation in the direction orthogonal to both the jet axis and the direction defined by the observed ridgeline ripples on the plane of the sky. Our approach neglects the role of that distortion because we focus on the longest wavelengths (λ >> R j ), where these distortions are less relevant than for shorter ones (λ ∼ R j ). The χ 2 fits to the ridges can be done in two different ways: either by fitting one mode at a time and substracting it from the original ridgeline before fitting the next mode, or by fitting all modes simultaneously, starting from an initial set of mode parameters determined in earlier works (e.g., Lobanov et al. 2006;Perucho et al. 2012a). Both approaches give the same results (see, e.g., Vega García 2018 for the fits to the ridgline of Global VLBI 1.6 GHz image of the source). For the purposes of this work, we only need the result of this fit, which is reproduced in Table 1. It is important to stress that the wavelengths obtained by this analysis are similar to those derived in previous works (Lobanov et al. 2006;Perucho et al. 2012a), with improved resolution in our case since the observations included the global VLBI array. It is interesting to stress that the shorter wavelengths (∼ 35, 16.5, 6.6 mas) listed in Table 1 are around integer fractions of the longest excited mode (∼ 1/3, 1/6 and 1/15 of ∼ 100 mas). 4 The instability wave amplitude is expected to grow with distance, but this leaves the wavelengths unaffected so, assuming that the growth lengths are long enough, this represents an acceptable assumption. Otherwise, the growth should be included to improve the quality of the fits. This indicates that they could be triggered in the jet as harmonics of the fundamental frequency and suggests that a single triggering mechanism (e.g., jet base precession) could be responsible for the whole set of observed oscillation patterns.

Stability analysis
The simplest way to explain the observed combination of the oscillations is to assume that all the observed modes grow around the frequencies that correspond to the maximum growth rates for each mode (minimum growth lengths). An initial derivation of the jet parameters can be done using the approximation to the solution of the stability problem given by Hardee (2000). This approximation assumes that the jet is separated from the ambient medium by a contact discontinuity and that the jet Mach number is >> 1. The parameters derived using this approach also depend on an a priori identification of the modes and on the wave speed taken of the mode propagation. The basic jet parameters are then derived from the following equations. The jet Mach number, where λ * = λ i (n i + 2m b i + 1 2 ) is the characteristic wavelength (Lobanov & Zensus 2001), λ i are the observed (projected) wavelengths and n i is the azimuthal wavenumber, and m b i indicates the order of the mode (0 for the surface mode and m b i for the m b i -th body mode). The jet-to-ambient medium density ratio, where the external jet Mach number, M x , and the intrinsic jet pattern speeds are calculated with: and where R j is the jet radius at the jet base, θ j is the viewing angle β app is the apparent speed and β w,app is the apparent pattern speed. In these expressions, β w is also difficult to determine, and it has to be taken into account that each developing mode may have a different wave speed. With all these caveats, using the values given in Otterbein et al. (1998), i.e., θ j = 3 • and a bulk Lorentz factor γ j = 12, Lobanov et al. (1998) derived a jet classical Mach number of M j = 6 and a jet-to-ambient medium density ratio of η = 0.04. The three longest wavelengths have been indentified as corresponding to the surface, first body and second body modes, respectively (Vega García 2018). This identification results in M j = 12 ± 3 and η = 0.33 ± 0.08. Differences may be attributed to the different aproaches or to the difficulties in the measure of β w .
In order to overcome these problems and provide a selfconsistent framework for jet modelling, we propose an independent method to derive estimates of the jet parameters from the direct solutions of the stability equation. Taking into account that the presence of a shear layer surrounding the jet in S5 0836+710 has been suggested in earlier works (Perucho & Lobanov 2007, we solve the differential equation of the linearized stability problem of two flows in pressure equilibrium and relative velocity, for relativistic, sheared jets in cylindrical coordinates (see, e.g., Birkinshaw 1984Birkinshaw , 1991: where r and z are the radial and axial coordinates, k and n are the axial and azimuthal components of the wavenumber, respectively, ω is the frequency, Γ is the adiabatic index, p is the pressure, and subscripts 0 and 1 refer to unperturbed and perturbed variables, respectively. The perturbations are assumed to be proportional to g(r) exp{(i(k z z − ωt))}, where g(r) describes the radial structure of the wave. Like in previous works using the same approach Perucho & Lobanov 2007), the shear layer that we consider is described by the variable a(r), corresponding to the velocity or rest-mass density: where a 0 is the value of the parameter at r = 0, a ∞ its value when r → ∞, and m defines the steepness of the layer. This has been shown to converge to the vortex-sheet solution for large enough values of m (Perucho et al. 2005).
We assume an ideal gas equation of state with adiabatic index Γ = 4/3 to describe both the jet and ambient medium. We take a relativistic hydrodynamics approach, i.e., we implicitly assume that the magnetic field is dynamically negligible at the studied scales. Finally, we impose the following boundary conditions: i non incoming waves from infinity (Sommerfeld condition), and ii symmetry or antisymmetry of the perturbation and its first derivative on the jet axis.
Then, we obtain the solutions to the differential equation using the shooting-method (see Perucho et al. 2005, which provides us with values for k and ω that satisfy Eq. 7. We used the spatial approach to do this, i.e., we assume that the unstable modes grow in distance, because we study how the perturbations grow downstream along the jet. This approach thus uses real frequencies and complex wave numbers. The real part of the wave number describes the wavelength of the perturbation, λ int , and the imaginary part describes the growth length, λ e : We have solved the equation using a range of shear layer widths, and we have used different values of the Lorentz factor and have swept several orders of magnitude in jet-to-ambient medium density ratios and specific internal energy for each shearing width (see Appendix C for the complete set of solutions obtained). In the calculations, we use units of ρ a = 1, c = 1 and R j = 1 (jet radius). Therefore, the jet density in the equations, ρ, represents the density ratio, η, between the jet and the ambient medium. Specifically, in our parameter space, η ranges from 10 −5 to 10 −1 , the jet specific internal energies, ε j take values from 0.001 c 2 to c 2 , i.e., the calculations include both cold and hot jets, the shear layer widths range from a narrow layer, as given by m = 16 or m = 12 (≤ 10 % of the jet radius), to broader ones with m = 8 and m = 4 ( 20 % of the jet radius). Finally, we considered Lorentz factors from from 5 to 17, including γ j = 12 as derived from VLBI observations at 8 GHz (Otterbein et al. 1998), γ j = 17 from VLBI observations at 15 GHz (Lister et al. 2013), and γ j = 5 to study the option of a slower layer of plasma surrounding a fast inner spine.
As a result of the calculations, we obtain a set of physical parameters that provide the best fit to the observed wavelengths, and we can use them to derive an approximation of the Mach number of the flow, using the classical Mach number definition, M = v j /c sj , where the jet sound speed, c sj , can be calculated as: where h = (1 + Γε) is the specific enthalpy.
In the lower panels of Fig. 4, we indicate the maxima of the imaginary part of the wavenumber (or minimum growth lengths) for each mode. This allows us to derive the intrinsic wavelength of the modes by looking for the maximum in values of the wavenumber in the upper plots. Since the curves do not have sharp maxima, but plateau-like maxima, we consider the points where the maximum value of the imaginary part for each mode of the wave-number is reduced by 5% as lower and upper bounds. In order to compare our results with the fitted structures from observations, we convert the distance unit used in the calculations (jet radius, R j ) into milliarcseconds, using the jet radius at the jet base derived from observations (1 R j = 2 mas, Vega García 2018). In this respect, we note that the stability problem is solved for an infinite, cylindrical jet, but jet expansion introduces an increase of the mode peak wavelengths (e.g., Hardee 1982Hardee , 1984Hardee , 1986) so our assumption represents a source of uncertainty. However, the jet opening angle for the jet in S5 0836+710 is very small (< 1 • , see Perucho et al. 2012a), and we can regard our solutions as a plausible approach to the jet parameters.
The corresponding observed wavelength can be then derived from the calculated intrinsic wavelength, using the viewing angle, θ, and the intrinsic wave speed, β ω : The intrinsic wave speed has been estimated using the values of the complex k and real ω from the calculations using: using that Re(k) Im(k).

Results: Jet parameters
The values of the peak wavelengths that result from the solution of Eq. 7 for each set of parameters and the Mach numbers derived from them can be found in the tables in Appendix A.
In order to illustrate how the predicted observed wavelengths change with the shear layer width and the jet parameters, we have plotted in Fig. 5 the wavelengths obtained for the relevant modes as a function of rest-mass density ratio for each value of the shear layer and jet internal energy. This figure only shows the best result (given by the most probable set of jet parameters) and other three interesting examples. Additional figures can be found in the Appendix A. In the plots, the horizontal lines and shades indicate the observed wavelengths and their formal errors. Each family of three points in the y direction represents the wavelengths at the maxima in growth rate of the fundamental, first and second body modes for each different model (defined by the given set of physical parameters). The most likely model is thus the one for which the points lie closer to the observed wavelengths (horizontal lines) within the error given by the shaded areas. If we first compare the effect of the shear layer on the simulated observed wavelengths, we can see that the broader the shear layer, the larger the separation between the peaks of the fundamental mode and the first body mode, while the narrower the shear layer, the smaller the separation between these modes. This ratio excludes wide shear-layers extending more than 1 R j (e.g., m ≥ 4). The ratio between the observed wavelengths indicates a narrower shear layer width m = 16 (with an extension of 0.1 − 0.2 R j ) as a most likely scenario. Regarding the effect of the physical parameters of the jet, an increase of the density ratio reduces the peak wavelengths, since it causes an increase of the specific internal energy, whereas increasing the Lorentz factor increases those wavelengths. These solid trends have allowed us to exclude wide regions of the parameter space.
The dispersion between the observed wavelengths and the calculated ones can be used to determine which of the parameter sets best recovers the observed values given in Section 2. The two models with lower dispersions are the ones with η = 2×10 −3 and ε j = 0.01c 2 , and η = 10 −3 and ε j = 0.01c 2 . Both cases would correspond to a Mach number of M ∼ 12. This value coincides with the value obtained independently from the approximations used in previous works (see Section 3). The specific internal energy of the jet is clearly lower than one (c 2 ), which places the jet in the cold regime, possibly dominated by the kinetic energy.
We have also computed the solutions for different jet Lorentz factors values, γ j = 5 and γ j = 17, for the best fitting values of ε j and η. We found that the case of γ j = 17 results in peak wavelengths that are compatible with the observed wavelengths for each of the helical modes proposed (surface, and first and second body modes), whereas the γ j = 5 models systematically result in lower peak wavelengths for each of the modes. Comparing the results given by Lorentz factors 12 and 17, we see that although both models could still be compatible, the dispersion between the calculated wavelengths and the observed ones is lower for the Lorentz factor of 12 (see the Tables in the Appendix). It is noteworthy that the parameter sets that give the lowest dispersion for Lorentz factor 17 include options with larger values of ε j (0.01 − 0.1 c 2 ) and smaller η (10 −5 − 10 −4 ). Finally, although we found a relatively small dispersion values for paremeter sets with γ = 12 and higher values of ε j (1 − 10 c 2 ), the density ratios are extremely small (10 −7 − 10 −6 ), and still the dispersion is larger than for the aforementioned parameter sets.
Our results thus allow us to confirm that the Lorentz factor of the jet at the scales revealed by the 1.6 GHz jet is probably in the range 12 − 17, i.e., in agreement with the estimated value by Otterbein et al. (1998) from jet kinematics at 8 GHz and that derived by Lister et al. (2013) from 15 GHz observations. For this values of the Lorentz factor, we find that the jet is remarkably underdense with respect to its environment (η ∼ 10 −5 − 10 −3 ) and relatively cold (ε j ∼ 0.01 − 0.1 c 2 ). This results in a relativistic jet Mach number, M j γ j M j , range of 60 − 144.

Discussion and conclusions
Our calculations include the following basic assumptions: 1) infinite cylindrical jet, and 2) dynamically negligible magnetic field. Because we model the observed jet structures over > tance, we have to discuss the validity of our results as estimates of mean jet parameters. On the one hand, taking into account that the analysis pictures a cold and fast jet, and that our estimate of the Lorentz factor coincides with that derived by Otterbein et al. (1998) from jet kinematics at 8 GHz, we can infer that any kind of Bernoulli acceleration (internal energy conversion into kinetic energy) must have taken place upstream and that no further acceleration takes place along the observed jet at 1.6 GHz. This argumentation consolidates the value of the Lorentz factor of γ 12 for the whole VLBI jet in S5 0836+710. The high degree of collimation of this jet also allows us to take the derived values of the specific internal energy and density ratio can be regarded as an order of magnitude estimate because the limited jet expansion will lead to a relatively small drop in both parameters. As far as the dynamical role of the magnetic field, magnetic acceleration models (e.g., Vlahakis & Königl 2004;Komissarov 2012) and observational (e.g., Homan et al. 2015;Boccardi et al. 2016;Mertens et al. 2016) results seem to point towards magnetic acceleration taking place at the inner tens of parsecs, at most, from injection, so a magnetically dominated jet is not expected at kiloparsec scales, as those involved in the VLBI jet of S5 0836+710. In addition, our results tend to favor a kinetically dominated jet, with relatively low internal energy and high Lorentz factor (as expected for particle dominated flows, Martí et al. 2016). This can be taken as a self-consistency test for our initial hypothesis of the observed structures being generated by KH instability modes. The similarity between the inferred Lorentz factor and that given by previous kinematic studies (Otterbein et al. 1998;Lister et al. 2013) validates our approach and encourages us to propose it as a possible way to derive jet parameters in other cases, for the range of distances in which the assumptions are fulfilled. Namely, small jet opening angles and linear-growth regime of the modes (implying relatively small changes in the wave amplitude).
Because the calculations assume that the jet parameters are constant in the axial direction, we have to consider the parameters derived as mean values along the observed VLBI jet. Further work and should include genetic/machine learning algorithms in the application of this method to the determination of jet parameters.
The values derived for the jet Lorentz factor and Mach number result in a relativistic Mach number of M j ∼ 60 − 144 (c.f. the classical Mach number M j = 5 − 12). The jet would thus be well within the stable region of the relativistic jet stability plane shown in (Perucho et al. 2005, Fig. 21 in that paper), with very small growth-rate unstable modes (which validates the modelling assumption of constant wave amplitudes in Eq. 1) and prone to the development of stabilising, short-wavelength resonant modes (Perucho et al. , 2010. This is probably in contradiction with the conclusion of Perucho et al. (2012b), where the authors claim that the jet may be disrupted by the growth of instabilities, giving rise to the decollimated structure observed with MERLIN at arcsecond scales. This structure was interpreted by the authors as a sign of jet disruption, but this work and recent observational results (Kappes et al., in preparation) may indicate that the arcsecond emission region related to this source may be revealing the interaction site of the jet with the intracluster medium.
The results obtained in this work can be summarized as follows: -We have used the results from previous studies of the S5 0836+710 jet structure that have modelled its ridgeline at 1.6 GHz, which covers up to 130 mas, by fitting transverse brightness profiles with a single Gaussian component. -The ridgeline shows the presence of several oscillatory terms, which have been reported in several different works (Lobanov et al. 1998(Lobanov et al. , 2006Perucho et al. 2012a). All those previous works give similar values for the wavelengths of the most prominent oscillations, using different arrays and epochs of observation. -We assume that these oscillations correspond to KH instability modes developing in the flow, and propose a method to derive the physical parameters of the jet. The method requires solving numerically the stability problem for a wide range of values of the relevant jet parameters and comparing the wavelengths at the maximum growth-rates for the identified modes plus their ratios with the observed structures. -This method does not need any a priori known parameters, but only solving the linear stability problem for a wide region of the parameter space. From this approach, we find a jet Lorentz factor in the range γ 12 − 17, in coincidence with kinematical studies (Otterbein et al. 1998;Lister et al. 2013), and, within that range, the jet-to-ambient medium density ratio is found to be η ∼ 10 −5 −10 −3 , and the jet specific internal energy is ε ∼ 10 −2 − 10 −1 c 2 , resulting in a jet classical Mach number of M ∼ 5−12. We also found that the observed radio jet is surrounded by a thin shear layer, with a width 10 % of the jet radius. The value derived for the specific internal energy and the Lorentz factor point towards a kinetically dominated jet. Vega-García, Perucho & Lobanov: Instability analysis of S5 0836+710