General relativistic model for the gravitational field of active galactic nuclei surrounded by a disk

An exact but simple general relativistic model for the gravitational field of active galactic nuclei is constructed, based on the superposition in Weyl coordinates of a black hole, a Chazy-Curzon disk and two rods, which represent matter jets. The influence of the rods on the matter properties of the disk and on its stability is examined. We find that in general they contribute to destabilize the disk. Also the oscillation frequencies for perturbed circular geodesics on the disk are computed, and some geodesic orbits for the superposed metric are numerically calculated.

Figure 1

Superposition of a black hole with mass $M$, two rods, and a Chazy-Curzon disk on the plane $z=0$.

Figure 2

Curves of $V^2=1$ (dotted curves) and of $hdh/dr=0$ (solid curves) for the Chazy-Curzon disk in presence of a black hole and two rods. We keep the mass of each rod constant $\beta =0.5$ and vary its mass density $\lambda$. In (a)-(b) we take, respectively, $\tilde{a}=1$ and $\tilde{a}=3$.

Figure 3

Curves of $V^2=1$ (dotted curves) and of $hdh/dr=0$ (solid curves) for the Chazy-Curzon disk in presence of a black hole and two rods. We keep the length of each rod constant ${\tilde{c}}_2=11$ and vary its mass density $\lambda$ (or equivalent, its mass). In (a)-(b) we take, respectively, $\tilde{a}=1$ and $\tilde{a}=3$.

Figure 4

(a) Energy density $\overline{\sigma }$, (b) azimuthal pressure $\overline{p}$, (c) counter-rotating velocity $V^2$ and (d) specific angular momentum $\overline{h}$ as functions of $\tilde{r}$ for $\alpha =1$, $\tilde{a}=3$, ${\tilde{c}}_2=11$ and three different values for the rod’s mass density $\lambda$.

Figure 5

Radial profiles of (a) the epicyclic frequency $m{\omega }_h$ and (b) the vertical oscillation frequency $m{\omega }_v$ for an isolated Chazy-Curzon disk.

Figure 6

Radial profiles of (a) horizontal and (b) vertical oscillation frequencies of Chazy-Curzon disk with a black hole without rods. Parameters: $\alpha =1$, $\tilde{a}=1,2,3\mathrm{a}\mathrm{n}\mathrm{d}4$.

Figure 7

Radial profiles of (a) horizontal and (b) vertical oscillation frequencies of Chazy-Curzon disk with black hole and rods. Parameters are ${\tilde{c}}_2=11$, $\alpha =1$, $\tilde{a}=3$, $\lambda =0,0.05\mathrm{a}\mathrm{n}\mathrm{d}0.1$.

Figure 8

Geodesic orbits for the superposition Curzon disk $+$ black hole without rods. Parameters: $\alpha =1$, $\tilde{a}=3$, $\mathcal{E}\approx 1.01$, ${\tilde{r}}_0=3.9$. (a) Projection on the $x-z$ plane of the curves in (b).

Figure 9

Geodesic orbits for the superposition Curzon disk $+$ black hole with rods. Parameters: $\alpha =1$, $\tilde{a}=3$, $\lambda =0.1$, ${\tilde{c}}_2=11$, $\mathcal{E}\approx 1.01$ ${\tilde{r}}_0=7.43$. The curve for $\psi ={89}^o$ is displayed in (c) and (d) for better visualization.