Kelvin-Helmholtz modes revealed by the transversal structure of the

jet in 0836+710

Manuel PeruchoAndrei P. Lobanov

Max-Planck-Institut für Radioastronomie - BonnGirdwood, Alaska – May 2007

0836+710: A short summary

• Quasar. z=2.16. • Kiloparsec scale:

– Classified as an FRII source from the luminosity of the secondary component (2’’ from core),

• Log P (21cm)=27 Watts/Hz (O’Dea et al.1988).

– Hummel et al. (1992) observed loss of collimation.

• Parsec scale:– Otterbein et al. (1998) showed the

link between the ejection of a superluminal component (B3) and a gamma-ray-optical flaring.

• Spectral evolution of the component is consistent with self-absorption and adiabatic expansion.

• Component Lorentz factor =11, jet viewing angle Φ=3º, Mach number M=6, jet to ambient density ratio ρj/ρa=0.04.

• They observed kinks in the jet: instabilities?

18 cm, Merlin and combined Merlin+VLBI. Hummel et al. (1992)

22 GHz, VLBI 1993.71 and 1995.65. Otterbein et al. (1998)

0836+710: A short summary

• Lobanov et al. (1998, 2006): – Observations with VLBA and VSOP at 1.6

and 5 GHz. • The peaks in the spectral index coincide with

enhanced emission regions, which appear at similar intervals, with average separation of 4.8 mas.

• Identify displacements in the ridge line at 5 GHz.

– Confirm the twisting of the jet (Kelvin-Helmholtz?).

=11, Φ=3º, M=6, ρj/ρa=0.04.– Using approximations to the linear

problem: Helical surface mode: =7.7 mas.

Similar to oscillations found in the ridge line.

Elliptical surface mode: =4.6 mas. Similar to 4.8 mas found for the separation between peaks in the spectral index.

Longer helical mode 100 mas is not explained.

– Are the shocks from injection of dense plasma or Kelvin-Helmholtz instabilities responsible for the 4.8 mas structure? Maybe both (see Perucho et al. 2006).

• We solve the stability equation for a cylindrical jet with different widths of the shear-layer (see Perucho et al. 2005, 2007 for details).

•Shear layer defined for density and velocity (r is the radius and a is any of both variables)

• m controls the thickness of the shear layer (the higher m the thinner the layer) • The perturbation has wave form a1 = g(r) exp(i (kz+nθ-ωt)).

•g(r) includes the radial dependence, k is the wave number, z the axial coordinate, n is the azimuthal number, θ is the azimuthal coordinate, ω is the frequency of the perturbation and t is time.

•Differential (Bessel) equation for the pressure perturbation to be solved:

• where P1 is the pressure perturbation, γ0 is the jet Lorentz factor, v0z is the axial velocity, ρe0 is the equilibrium relativistic density, P0 is the equilibrium pressure and cs0 is the sound speed. Primes indicate radial derivatives.

Linear Analysis

Linear Analysis• This equation is solved for real frequency and complex wave-

number (spatial view).• Appropriate boundary conditions are applied: Symmetry or

antisymmetry of the solution and its first derivatives in the axis, and Sommerfeld condition (the amplitude of the waves tends to zero at infinity).

• We use variable step size Runge Kutta method for the integration and Muller method for the root finding. – Solutions are searched for point by point.

• The equation has been solved for:– the given parameters of 0836+710 in Lobanov et al. (1998, 2006),– different values of m, i.e., different shear layer thicknesses, and– pinching, helical and elliptic modes.

Helical m=8 Helical m=200 Elliptic m=8 Elliptic m=200

Linear Analysis

HELICAL MODE SOLUTIONS

m=8 (shear layer width = 0.6 Rj) m=200 (shear layer width = 0.1 Rj)

frequency of maximum growth for the surface mode: ω0.025 c/Rj

wave number of the maximally unstable mode:160 Rj

In this description: =2π/Re(k)growth-length=1/Im(k)

14 Rj 25 Rj

100 Rj

• The key equation to compare the observed wavelengths with those in the solutions is the following:

int is the intrinsic wavelength, obs is the observed wavelength, z is the redshift, θj is the viewing angle, and vω=ω/k is the wave speed.

•The speeds of the waves are derived from the solutions shown in the previous slide:•For m=8

• =160 Rj turns into 6.6 Rj in the jet reference frame (jrf).• =14 Rj turns into 0.29 Rj in the jrf.

•For m=200=100 Rj turns into 8.2 Rj in the jrf. =25 Rj turns into 1.4 Rj in the jrf.

From theory to observations (i)

• Which is the radius of the jet?

– Di is the observed width of the jet and b is the beam width transversally to the jet axis.

– We measure Di at 1% of the peak emission at the base of the jet (Wehrle et al. 1992), assuming that the outer parts of the shear layer emit less than 1% of the radio power at these frequencies.

• Rj = 17 mas at 1.6 GHz

• Rj = 0.6 mas at 5 GHz

• The 100 mas helical mode:– 6 Rj at 1.6 GHz

– 170 Rj at 5 GHz

• The 7.7 mas helical mode:– 0.2 Rj at 1.6 GHz

– 12.8 Rj at 5 GHz

• Clearly, we have to consider the radius of the jet at the lowest frequency.• The results point towards the jet being sheared:

– Different radius of the jet at different frequencies and showing different structures.

– The larger radius has to be taken into account in order to explain all the wavelengths.

– The wavelengths fit better to the case of a thicker shear layer (m=8).

COMPARED TO 0.29 Rj for m=8, or 1.4 Rj for m=200

COMPARED TO 6.6 Rj for m=8, or 8.2 Rj for m=200

From theory to observations (ii)

Implications for the active nucleus

• The helical surface mode can be driven by an external periodic process (Hardee 1994).

• If the 100 mas wavelength is associated to the helical surface mode at its maximum growth, the driving frequency is 0.025 c/Rj.

– This implies a driving period of Tdr5.6 107 yrs. – Hardee et al. found a similar period (2 107 yrs) for 3C 449.

• This long periodicity can be produced by a number of processes (Appl et al. 1996):

– Misaligned torus.– Binary black hole.

• We show –basing on results by Lu & Zhou (2005)- that misalignment can only produce precession periods of the order 103 – 105 yrs (depending on the properties of the system), quite below that found here.

• The derived period can be explained by a binary black hole: – we have used the mass of the primary black hole in 0836+710 (2 108 – 109 M○,

Tavecchio et al. 2000), – We derive a possible companion mass of 104 – 107 M○, depending on the

separation between the two black holes (1017-1018 cm), following Appl et al. (1996).

Conclusions

• We solve the differential equation of pressure perturbation for sheared relativistic jets in cylindrical coordinates.

• With the parameters derived for the jet on the basis of spectral evolution (Otterbein et al. 1998), we are able to explain the structures observed in the jet at several frequencies assuming that a shear layer exists between the jet and the ambient medium.

• Confirm the parameters found for the jet by computing the instabilities for different changes in them.

– We don’t expect big differences with respect to the numbers given.

• A driving precession period of the order of 5 107 yrs is derived.– It can be explained by a binary black hole system with masses 2 108 – 109 M○ for

the primary and 104 – 107 M○ for the companion, depending on the separation between the two black holes (1017-1018 cm).

• We postulate that the loss of collimation in the kiloparsec scales may be due to the 100 mas structure observed in the jet possibly causing its disruption.