Relativistic tidal acceleration of astrophysical jets

Within the framework of general relativity, we investigate the tidal acceleration of astrophysical jets relative to the central collapsed configuration (“Kerr source”). To simplify matters, we neglect electromagnetic forces throughout; however, these must be included in a complete analysis. The rest frame of the Kerr source is locally defined via the set of hypothetical static observers in the spacetime exterior to the source. Relative to such a fiducial observer fixed on the rotation axis of the Kerr source, jet particles are tidally accelerated to almost the speed of light if their outflow speed is above a certain threshold, given roughly by one-half of the Newtonian escape velocity at the location of the reference observer; otherwise, the particles reach a certain height, reverse direction and fall back toward the gravitational source.

Figure 1

The phase portrait of ODE (38) is depicted for the case $\mathrm{\Phi }=1/10$ and $\eta =1/20$. The vertical coordinate axis corresponds to $v≔dz/d\zeta$. Also shown are the ellipse (43) corresponding to $\mathrm{\Gamma }=\infty$, the position of the hyperbolic saddle rest point ($z=z_0$), the critical value of $v_0$ along the vertical axis ($v_0=v_{ta}$) and the line $v=1/\sqrt{\mathrm{\Phi }}$. The initial outflow velocity $v_0$ at $z=0$ is such that $0<v_0<1/\sqrt{\mathrm{\Phi }}$. The phase trajectory through the critical point is shown with a thick curve. It is part of the stable manifold of the saddle point. For $v_0>0$, all solutions starting below the critical value reach a maximum $z$ at a point where $v=0$. All solutions starting above the critical value and below the horizontal line $v=1/\sqrt{\mathrm{\Phi }}$ eventually meet the ellipse corresponding to $\mathrm{\Gamma }=\infty$. The critical value for tidal acceleration in the outflow, $v_{ta}\approx 0.79$, corresponds in this case to a Fermi speed of $V_{ta}\approx 0.25c$.

Figure 2

The behavior of the Lorentz factor $\mathrm{\Gamma }$ is shown versus $z$ by using the numerical integration of system (38) for $\mathrm{\Phi }=1/10$ and $\eta =1/20$. The initial conditions are $z=0$ and $v_0=1,\sqrt{5},3$.

Figure 3

The bifurcation diagram for the phase portrait of system (38) with respect to the parameters $\mathrm{\Phi }$ and $\eta$ is shown. In this $(\eta ,\mathrm{\Phi })$ plane, the thick curve, which corresponds to the locus of the event horizon $1-2\mathrm{\Phi }+{\eta }^2=0$, is the upper boundary of the relevant parameter region bounded also by the coordinate axes and the line $\eta =1$. The line $\mathrm{\Phi }=\eta$ is depicted as well as the bifurcation curve corresponding to $q-\mathrm{\Phi }{p}^2=0$. Below the lower branch of the latter curve, the phase portrait has a saddle-type rest point; above the curve it has a center. The upper branch is also depicted, but lies outside the domain of physical interest.

Figure 4

The phase portrait for system (38) where $q-\mathrm{\Phi }{p}^2<0$ is depicted for the rapidly rotating case of $\mathrm{\Phi }=1/10$ and $\eta =8/10$. The upper branch of the hyperbola corresponding to $\mathrm{\Gamma }=\infty$ is also shown. Also, the rest point, the critical point on the vertical axis, and the critical curve are depicted. The line $v=1/\sqrt{\mathrm{\Phi }}$ shows the limiting value of $v_0$, while the line $z=1/\sqrt{\mathrm{\Phi }}$ indicates the outer boundary of the admissibility of Fermi coordinates. The critical value for outflow tidal acceleration, $v_{ta}\approx 2.245$, corresponds in this case to a (Fermi) speed of $V_{ta}\approx 0.71c$. We note that in this case the outflow tidal acceleration is in fact insignificant in comparison to the case presented in Fig. 1.

Figure 5

Plots of ${\mathcal{V}}_{+}$ and ${\mathcal{V}}_0$ versus $z$. For ${\mathcal{V}}_{+}$, the parameters are $\mathrm{\Phi }=1/10$ and $\eta =1/20$, so that $z_0\approx 0.60$ and $v_{ta}\approx 0.79$, corresponding to the phase portrait in Fig. 1. The parameters for ${\mathcal{V}}_0$ are $\mathrm{\Phi }=1/10$ and $\eta \approx 0.5715$.

Figure 6

Plot of ${\mathcal{V}}_{-}$ versus $z$. The parameters are $\mathrm{\Phi }=1/10$ and $\eta =8/10$ so that $p\approx 0.14$, $q\approx -0.42$, $z_0\approx -0.34$ and $v_{ta}^2={\mathcal{V}}_{-}(\infty )\approx 5.04$. Note that ${\mathcal{V}}_{-}(-\infty )\approx 5.06>{\mathcal{V}}_{-}(\infty )$.