# A Relativistic Orbit Model for Temporal Properties of AGN

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## Abstract

**:**

## 1. Introduction

## 2. Relativistic Circular and Spherical Orbits as Solutions to X-Ray QPOs

#### 2.1. Method for the Error Estimation

- 1.
- We assume that the frequencies, ${\nu}_{1}$ and ${\nu}_{2}$, of QPOs are Gaussian distributed with their mean values at the centroid of observed QPO frequencies, ${\nu}_{10}$ and ${\nu}_{20}$ (with ${\nu}_{10}>{\nu}_{20}$). The joint probability density distribution of these frequencies is given by$$P\left(\nu \right)=\prod _{i=1}^{2}{P}_{i}\left({\nu}_{i}\right),$$$${P}_{i}\left({\nu}_{i}\right)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma}_{i}^{2}}}}exp\left[-{\displaystyle \frac{{\left({\nu}_{i}-{\nu}_{i0}\right)}^{2}}{2{\sigma}_{i}^{2}}}\right],$$
- 2.
- We find the Jacobian, $\mathcal{J}$, of the transformation from frequency to orbital parameter space using the formulae of fundamental frequencies, which is given by$$\mathcal{J}=\left[\begin{array}{cc}\frac{\partial {\nu}_{1}}{\partial {x}_{1}}& \frac{\partial {\nu}_{1}}{\partial {x}_{2}}\\ \frac{\partial {\nu}_{2}}{\partial {x}_{1}}& \frac{\partial {\nu}_{2}}{\partial {x}_{2}}\end{array}\right].$$
- 3.
- Next, we write the probability density distribution in the parameter space given by$$P\left(\left[x\right]\right)=P\left(\nu \right)\left|\mathcal{J}\right|,$$
- 4.
- We calculate the exact solutions for parameters by solving ${\nu}_{\varphi}={\nu}_{10}$ and ${\nu}_{pp}={\nu}_{20}$ using Equation (7) for circular trajectories {${r}_{0}$, ${a}_{0}$}, and Equation (8) for spherical trajectories {${r}_{s0}$, ${a}_{0}$} for fixed Q. We fix ${M}_{\u2022}$ to the previously known values. We find 1$\sigma $ errors in the parameters by taking an appropriate parameter volume around the exact solution, and generate sets of parameter combinations with resolution $\Delta {x}_{j}$ in this volume. The chosen parameter range, exact solutions, and corresponding resolutions are summarized in Table 3 and Table 4. We then calculate the probability density using Equation (4), for all the generated parameter combinations and normalize the probability density by the normalization factor$$\mathcal{N}={\displaystyle \frac{{\sum}_{k}P\left({\left[x\right]}_{k}\right)\Delta {V}_{k}}{V}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\Delta {V}_{k}=\prod _{j=1}^{2}\Delta {x}_{j,k},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}V=\sum _{k}\Delta {V}_{k},$$$$\mathcal{P}\left(\left[x\right]\right)={\displaystyle \frac{P\left(\left[x\right]\right)}{\mathcal{N}}}.$$The normalization of the probability density in the parameter space, discussed above, is done because only a sub-volume in the parameter space is astrophysically allowed for bound orbits, which is discussed below.
- 5.
- The allowed parameter combinations for the bound orbits is governed by the condition given by [54]$$\left[{\mu}^{3}{a}^{2}Q{\left(1+e\right)}^{2}+{\mu}^{2}\left(\mu {a}^{2}Q-{x}^{2}-Q\right)\left(3-e\right)\left(1+e\right)+1\right]\ge 0,$$
- 6.
- For the circular orbit case, there are two parameters to estimate {r, a} using two QPO frequencies. For the case of spherical orbits, there are three unknown parameters {${r}_{s}$, a, Q}; hence, we first take $Q=${1, 4, 8, 12} for the spherical trajectory solutions, where the extrema of $\theta $ coordinate deviates away from the equatorial plane with an increase in Q. For each fixed value of Q, we find the normalized probability density distribution in the parameter space $\{{x}_{1},{x}_{2}\}=\{{r}_{s},a\}$ using Equation (5b). Later, using the calculated spin values and their errors for each fixed Q, we estimate the distribution of spin and the most probable spin. Using this distribution and the most probable value of the spin, we then determine the probability distribution in the $\{{x}_{1},{x}_{2}\}=\{{r}_{s},Q\}$ parameter space.
- 7.
- Next, we integrate the normalized probability density, $\mathcal{P}\left(\left[x\right]\right)$, Equation (5b), in one dimension to obtain the profile in the other dimension. Thus, we finally obtain the one dimensional distributions {${\mathcal{P}}_{1}\left(r\right)$, ${\mathcal{P}}_{1}\left(a\right)$} for circular orbits, and {${\mathcal{P}}_{1}\left({r}_{s}\right)$, ${\mathcal{P}}_{1}\left(a\right)$} for spherical orbits.
- 8.
- Finally, we fit the normalized probability density profiles in each of the parameter dimensions to find the corresponding mean values and quoted errors are obtained such that it contains a probability of 68.2% about the peak value of the probability density. The results of these fit are given in Table 3 and Table 4.

#### 2.2. Circular Orbits

- 1.
- We have computed the contours of ${\nu}_{\varphi}\left(r,a\right)$, using Equation (7a), for the QPO frequencies (given in Table 2) of RE J1034+396 (blue), MS 2254.9-3712 (red), and MCG-06-30-15 (magenta), shown in the $\left(r,a\right)$ plane in Figure 3a. The masses of these black holes were assumed from the previous estimations (see Table 2). We see that the QPO emission originates from a very narrow region of the accretion disk, where $r\sim $(9.4–9.9) for RE J1034+396, r∼(10.4–11.4) for MS 2254.9-3712, and $r\sim 14.2$ for MCG-06-30-15 even though a ranges from 0 to 1. This implies that the QPO emission region is very close to the black hole, and this emission region remains very narrow and nearly independent of the spin of the black hole.
- 2.
- For the case of Mrk 766, two QPO frequencies were detected (see Table 2), but at different epochs. We have shown ${\nu}_{\varphi}\left(r,a\right)$ contours for both these frequencies in Figure 3a, where ${\nu}_{1}=2.38\times {10}^{-4}$ Hz (orange) and ${\nu}_{2}=1.55\times {10}^{-4}$ Hz (green). The mass of the black hole was fixed to ${M}_{\u2022}=4.3\times {10}^{6}{M}_{\odot}$[58]. The QPO origin range is $r\sim 10$ for ${\nu}_{1}$ and $r\sim $ (12.6−14) for ${\nu}_{2}$, which is again found to be in a narrow range and very close to the black hole. Although these QPOs were not detected simultaneously, we tried to estimate a simultaneous solution for $\left(r,a\right)$ by equating ${\nu}_{\varphi}={\nu}_{1}$ and ${\nu}_{pp}={\nu}_{2}$ as per GRPM. We show them as curves in the $\left(r,a\right)$ plane in Figure 3b, and we see that these contours do not cross each other, implying that there is no simultaneous solution for $\left(r,a\right)$.
- 3.
- For the Type-2 AGN 2XMM J123103.2+110648, the detected QPO (see Table 2) was suggested as an LFQPO type because of its large rms value [9]. If this QPO frequency is equated to the high-frequency component, ${\nu}_{\varphi}\left(r,a\right)$, of the GRPM, we found that $r\sim 200$, which is far from the black hole to emit X-rays. Hence, the GRPM predicts that this should be an LFQPO. We show the contours of the LFQPO component of the GRPM, ${\nu}_{np}\left(r,a\right)$, in the $\left(r,a\right)$ plane for the QPO frequency of 2XMM J123103.2+110648 in Figure 3c, where we fixed ${M}_{\u2022}={10}^{5}{M}_{\odot}$[56]. We see that the emission region for this LFQPO is r∼(6–20), for the whole range of a. Hence, the detected QPO of 2XMM J123103.2+110648 is an LFQPO that originated very close to the black hole.
- 4.
- For the case having two simultaneous X-ray QPOs, 1H 0707-495 (see Table 2), we first solve the equations {${\nu}_{\varphi}\left(r,a\right)={\nu}_{10}$, ${\nu}_{pp}\left(r,a\right)={\nu}_{20}$} (using Equation (7a,b)), assuming ${M}_{\u2022}=5.2\times {10}^{6}{M}_{\odot}$[36], as per GRPM to estimate the exact solution for (r, a), which is found to be (${r}_{0}=8.214$, ${a}_{0}=0.0662$). We then apply the method, described in Section 2.1, to estimate the errors in the parameters (r, a) implied due to the errors of the QPO frequencies. The range of (r, a) and corresponding resolutions used for our simulations are summarized in Table 3. Finally, we generate the probability density profiles in each parameter dimension {${\mathcal{P}}_{1}\left(r\right)$, ${\mathcal{P}}_{1}\left(a\right)$}, shown in Figure 4, where we have also shown the probability contours in the parameter space. The results of the model fits to the probability density profiles are summarized in Table 3. The errors in the parameters are quoted with respect to the exact solution (${r}_{0}$, ${a}_{0}$), whereas the simulated {${\mathcal{P}}_{1}\left(r\right)$, ${\mathcal{P}}_{1}\left(a\right)$} profiles peak at ($r=8.092$, $a=0.038$), which slightly differs from the exact solution. Hence, our analysis assuming the circular orbit frequencies as the origin of QPOs, using the GRPM, in NLSy1 1H 0707-495, suggests that it harbors a slowly rotating black hole ($a\sim 0.0662$) at the center, and that the X-ray QPOs originate in the inner region of the accretion disk and very close to the black hole ($r\sim 8.214$).

#### 2.3. Spherical Orbits

- 1.
- We explore the parameter space (${r}_{s}$, a, Q) for the spherical orbits. Since there are two input QPO frequencies, we first vary the Q value to find various solutions of {${r}_{s}$, a} by solving equations {${\nu}_{\varphi}={\nu}_{1}$, ${\nu}_{pp}={\nu}_{2}$} as per GRPM. $Q=13$ is at the limit of astrophysically allowed bound orbits, Equation (6); $Q<13$ in the case of 1H 0707-495. The $Q=13$ orbit is an unstable orbit very close to the separation of bound and unbound (called a separatrix orbit), and such an unstable orbit is not relevant to our study; hence, we fix our parameter exploration between Q= 1 and 12. In Figure 5, we have shown these solutions in the (Q, a) and (Q, ${r}_{s}$) planes.
- 2.
- Next, we fix $Q=\{1,4,8,12\}$ and find the errors in the {${r}_{s}$, a} parameters using the method described in Section 2.1. The range of {${r}_{s}$, a}, resolution taken in the simulations, along with the exact solutions and their errors obtained by fitting ${\mathcal{P}}_{1}\left({r}_{s}\right)$ and ${\mathcal{P}}_{1}\left(a\right)$ are summarized in Table 4.
- 3.
- The ranges of {a, ${r}_{s}$, Q}, shown in Table 4 and Figure 5, span the complete parameter volume for QPO frequencies of 1H 0707-495. As the spin of the black hole does not change in the timescale of a few months or years, we need to find the most probable value of spin. We first find the variance of ${\mathcal{P}}_{1}\left(a\right)$ with respect to the exact solution of a for each Q, given in Table 4, which is given by$${{\sigma}_{ai}}^{2}={\int}_{0}^{0.9}{\left(a-{a}_{0i}\right)}^{2}{\mathcal{P}}_{1i}\left(a\right)da,$$$$L\left(a\right)=\sum _{i}^{4}\frac{{\left(a-{a}_{0i}\right)}^{2}}{{{\sigma}_{ai}}^{2}},$$$${a}_{p}=\frac{{\sum}_{i}^{4}\left({a}_{01}/{{\sigma}_{ai}}^{2}\right)}{{\sum}_{i}^{4}\left(1/{{\sigma}_{ai}}^{2}\right)}.$$We find the peak value to be ${a}_{p}=0.139$ for 1H 0707-495, and corresponding solution of {${r}_{s}$, Q} for the QPO frequencies is {${r}_{sp}=8.246$, ${Q}_{p}=9.814$}.
- 4.
- Next, we obtain the ${\chi}_{a}^{2}$ distribution function of a given by$${\chi}_{a}^{2}\left(a\right)=exp\left[-L\left(a\right)\right].$$A plot of ${\chi}_{a}^{2}/{\chi}_{p}^{2}$ is shown in Figure 6a, where ${\chi}_{p}^{2}={\chi}_{a}^{2}\left({a}_{p}\right)$. We obtain the $2\sigma $ errors with respect to ${a}_{p}$ by normalizing the ${\chi}_{a}^{2}\left(a\right)$ function and obtain $0.{139}_{-0.139}^{0.183}$, where the region of 95% probability is indicated by the vertical dashed line in Figure 6a. We also show the range of ${r}_{s}$ and Q in Figure 6b,c within the $2\sigma $ region of a, as seen in Figure 6a, where the parameter ranges are ${r}_{s}=$ (8.214–8.323) and $Q=$ (0.0001–12.264).
- 5.
- Hence, we conclude that the spherical orbits, close to the black hole in the region, ${r}_{s}=$ (8.214–8.323) with Q values between (0.0001–12.264), are possible sources of the QPO frequencies observed in 1H 0707-495, while the most probable spin value to be ${a}_{p}=0.{139}_{-0.139}^{0.183}$ with $2\sigma $ confidence.

## 3. Relativistic Jet Model for the Optical and $\gamma $ Ray QPOs

## 4. Relativistic Orbit Model (ROM) and PSD Shape

#### The ROM

- 1.
- We associate the temporal frequency, $\nu $, in the observed power spectral density with the fundamental azimuthal frequency of the particles orbiting in the circular orbits in the accretion disk outside ISCO, ${r}_{I}$, and both circular and spherical trajectories between ${r}_{I}$ and marginally bound spherical orbit (MBSO) radius, ${r}_{M}$. These frequencies are functions of the orbital radius, r or ${r}_{s}$, (Equations (7a) and (8a)), and hence they are also fundamentally related to the mechanical energy of the orbit through Equations (14) and (15).
- 2.
- We assume a prior distribution of the energy of particles (or electrons) given by a power-law$$\begin{array}{ccc}\hfill N\left(E\right)=& & A{\left(\frac{E}{{E}_{I}}\right)}^{-{\alpha}_{1}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{IC}:\phantom{\rule{4pt}{0ex}}\mathrm{radial}\phantom{\rule{4pt}{0ex}}\mathrm{range}\phantom{\rule{4pt}{0ex}}{r}_{M}<r<{r}_{I},\hfill \end{array}$$$$\begin{array}{cccccc}& & & \hfill =& & A{\left(\frac{E}{{E}_{I}}\right)}^{-{\alpha}_{2}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{OC}:\phantom{\rule{4pt}{0ex}}\mathrm{radial}\phantom{\rule{4pt}{0ex}}\mathrm{range}\phantom{\rule{4pt}{0ex}}{r}_{I}<r<{r}_{X}.\hfill \end{array}$$$$\int N\left(E\right)dE={N}_{0},$$$$A{\int}_{{E}_{I}}^{1}{\left(\frac{E}{{E}_{I}}\right)}^{-{\alpha}_{1}}dE+A{\int}_{{E}_{I}}^{{E}_{X}}{\left(\frac{E}{{E}_{I}}\right)}^{-{\alpha}_{2}}dE={N}_{0},$$$$A\left(a,{\alpha}_{1},{\alpha}_{2}\right)={N}_{0}{\left[\frac{\left({E}_{I}{\left(a\right)}^{{\alpha}_{1}}-{E}_{I}\left(a\right)\right)}{\left(1-{\alpha}_{1}\right)}+\frac{\left({E}_{X}{\left(a\right)}^{\left(1-{\alpha}_{2}\right)}{E}_{I}{\left(a\right)}^{{\alpha}_{2}}-{E}_{I}\left(a\right)\right)}{\left(1-{\alpha}_{2}\right)}\right]}^{-1}.$$We redefine $A\left(a\right)$ such that$$A\left(a,{\alpha}_{1},{\alpha}_{2}\right)={N}_{0}B\left(a,{\alpha}_{1},{\alpha}_{2}\right),$$$$B\left(a,{\alpha}_{1},{\alpha}_{2}\right)={\left[\frac{\left({E}_{I}{\left(a\right)}^{{\alpha}_{1}}-{E}_{I}\left(a\right)\right)}{\left(1-{\alpha}_{1}\right)}+\frac{\left({E}_{X}{\left(a\right)}^{\left(1-{\alpha}_{2}\right)}{E}_{I}{\left(a\right)}^{{\alpha}_{2}}-{E}_{I}\left(a\right)\right)}{\left(1-{\alpha}_{2}\right)}\right]}^{-1}.$$
- 3.
- We assume that the break frequency of the PSD corresponds to the temporal frequency at the ISCO radius.
- 4.
- We also assume that the particle distrbution in the temporal frequency space, $F\left(\overline{\nu}\right)$, directly translates to the observed intensity for a given temporal frequency, so that the power density is given by $P\left(\overline{\nu}\right)\propto F{\left(\overline{\nu}\right)}^{2}$.

- 1.
- If ${\beta}_{1}$ is the average slope of the observed PSD after the break frequency, $\overline{\nu}>{\overline{\nu}}_{b}$, given by$$\frac{\Delta log\left[P\left(\overline{\nu}\right)\right]}{\Delta log\left[\overline{\nu}\right]}={\beta}_{1}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\Rightarrow \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}2\frac{\Delta log\left[{F}_{1}\left(\overline{\nu}\right)\right]}{\Delta log\left[\overline{\nu}\right]}={\beta}_{1},$$$$2log\left[\frac{{f}_{1}\left({\alpha}_{1},{\alpha}_{2},{\overline{\nu}}_{M}\left(a\right),a,Q\right)}{{f}_{1}\left({\alpha}_{1},{\alpha}_{2},{\overline{\nu}}_{I}\left(a\right),a,Q\right)}\right]={\beta}_{1}log\left[\frac{{\overline{\nu}}_{M}\left(a\right)}{{\overline{\nu}}_{I}\left(a\right)}\right],$$$${f}_{1}\left({\alpha}_{1},{\alpha}_{2},{\overline{\nu}}_{M}\left(a\right),a,Q\right)={u}_{1}\left(a,{\beta}_{1}\right),$$$$\Rightarrow \left[1-\frac{{C}_{1IM}\left({\alpha}_{1},a,Q\right)V\left(a,{\alpha}_{1},{\alpha}_{2}\right)}{{N}_{I}\left({\alpha}_{1},{\alpha}_{2},a,Q\right)}\right]={u}_{1}\left(a,{\beta}_{1}\right),$$$${u}_{1}\left(a,{\beta}_{1}\right)={\left(\frac{{\overline{\nu}}_{M}\left(a\right)}{{\overline{\nu}}_{I}\left(a\right)}\right)}^{{\beta}_{1}/2},$$$$\frac{1-{u}_{1}\left(a,{\beta}_{1}\right)}{1+{u}_{1}\left(a,{\beta}_{1}\right)}=V\left(a,{\alpha}_{1},{\alpha}_{2}\right){C}_{1IM}\left({\alpha}_{1},a,Q\right).$$Hence, for a given combination of {a, Q}, we obtain a relation, given by Equation (25f), where {${\alpha}_{1}$, ${\alpha}_{2}$} are unknowns.
- 2.
- Similarly, if ${\beta}_{2}$ is the average slope of the observed PSD before the break frequency, $\overline{\nu}<{\overline{\nu}}_{b}$, we have$$2\frac{\Delta log\left[{F}_{2}\left(\overline{\nu}\right)\right]}{\Delta log\left[\overline{\nu}\right]}={\beta}_{2}.$$The lower extreme of the PSD at $r={r}_{X}$, for $\overline{\nu}<{\overline{\nu}}_{b}$, is given by ${\overline{\nu}}_{X}\left(a\right)$, so that$$2log\left[\frac{{f}_{2}\left({\alpha}_{1},{\alpha}_{2},{\overline{\nu}}_{I}\left(a\right),a,Q\right)}{{f}_{2}\left({\alpha}_{1},{\alpha}_{2},{\overline{\nu}}_{X}\left(a\right),a,Q\right)}\right]={\beta}_{2}log\left[\frac{{\overline{\nu}}_{I}\left(a\right)}{{\overline{\nu}}_{X}\left(a\right)}\right],$$$${n}_{X}\left({\alpha}_{1},{\alpha}_{2},a,Q\right)={u}_{2}\left(a,{\beta}_{2}\right),$$$${u}_{2}\left(a,{\beta}_{2}\right)={\left(\frac{{\overline{\nu}}_{I}\left(a\right)}{{\overline{\nu}}_{X}\left(a\right)}\right)}^{-{\beta}_{2}/2}.$$The substitution of ${N}_{I}\left({\alpha}_{1},{\alpha}_{2},a,Q\right)$ and ${N}_{X}\left({\alpha}_{1},{\alpha}_{2},a,Q\right)$ using Equations (24) and (23c) gives$$\frac{{u}_{2}\left(a,{\beta}_{2}\right)-1}{1+{u}_{1}\left(a,{\beta}_{1}\right)}=W\left(a,{\alpha}_{1},{\alpha}_{2}\right){C}_{2XI}\left({\alpha}_{2},a\right).$$
- 3.
- We compute the slopes {${\alpha}_{1}$, ${\alpha}_{2}$} by the above mentioned criteria for different combinations of (a, Q), which are shown in Table 6. We find that ${\alpha}_{1}$ ranges between ∼[2.3–4] and ${\alpha}_{2}$ is in the range ∼[3.7–8.9], indicating that a power-law model for the intrinsic mechanical energy of the orbiting matter describes the shape of the observed PSD reasonably well. Additionally, if we reverse the analysis to estimate {${\beta}_{1}$, ${\beta}_{2}$} by fixing {${\alpha}_{1}=2.5$, ${\alpha}_{2}=3.5$} for ($a=0.5$, $Q=2$), we find {${\beta}_{1}=-1.97$, ${\beta}_{2}=-0.77$} which are in good agreement with observations. We also show contours of ${\alpha}_{1}$ and ${\alpha}_{2}$ in the (Q, a) plane in Figure 10, where the values of ${\alpha}_{1}$ and ${\alpha}_{2}$ increase with a. We also see that contours are independent of Q for small a, which is expected because the non-equatorial orbits do not exist in Schwarzschild spacetime.
- 4.
- The examples of PSD profile obtained in the scaled frequency space, $\overline{\nu}$, are shown in Figure 11. We see that the PSD profiles for given parameter combinations in Table 6 show good fits to the expected bending power-law model, Equation (13). The PSD represents a general power spectrum obtained independent of the mass of the black hole; hence, it applies to the stellar-mass black holes also. This validates the ROM as a plausible model for PSD observed in black holes.

## 5. Summary

- In Section 2, we motivated the creation of (G)RPM models for X-ray QPOs and extracted the spins and radii for the sources, listed in Table 2, based on the model given in [48,49,50]. The GRPM model confirms that the detected QPO in Type-2 AGN 2XMM J123103.2+110648 is an LFQPO, as it was also suggested by [9]. In a statistical analysis, we were able to determine these parameters and their errors for 1H 0707-495, the case of two simultaneous QPOs, based on the observed QPO frequencies and their errors. The results are presented in Table 3 for circular orbits and in Table 4 for spherical orbits. We found non-planar orbits, with $Q\sim $ (1–12), which are very close to a Kerr black hole, that (${r}_{s}\phantom{\rule{3.33333pt}{0ex}}\sim $ (8.2–8.3); $a\sim 0.14$) are the possible solutions for QPO frequencies of 1H 0707-495.
- Next, in Section 3, we applied the relativistic kinematic jet model to check its validity by comparing the basic frequency with the observed QPO periods in BL Lac objects, given in Table 5. The ratio ${T}_{0}/{T}_{F}$ is typically in the range $1-20$, which is reasonable, given the range of footpoint radii of the field lines and typical location of the Alfvén point up to which the field line is rigid [52]. It motivates detailed relativistic MHD models along with polarization profile predictions (as given in [51]) to compare with observations.
- In Section 4, we built a relativistic orbit model consisting of circular and spherical orbits that have a power-law distribution, and its mechanical energy is split into two parts (above and below the energy at ISCO). This formulation leads to unique results relating to the PSD slopes (before ($\overline{\nu}<{\overline{\nu}}_{b}$) and after ($\overline{\nu}>{\overline{\nu}}_{b}$) the break) with those of the energy spectrum for the given spin and mass of the black hole (Figure 10 and Figure 11). We plan to test this model against observations to extract {a, ${M}_{\u2022}$}.

## 6. Discussion and Conclusions

- 1.
- The periastron and nodal precession of the particle orbits is an intrinsic phenomenon in Kerr geometry, which is a consequence of strong gravity and axisymmetry of the spacetime. We propose in the GRPM [50] that the precession frequencies of matter blobs orbiting in these trajectories, very close to the Kerr black hole, modulate the X-ray flux, from the thin accretion disk where the flow is hot. The origin of these non-equatorial orbits of blobs in a slim torus region having a single radius is motivated in [50], where a model of fluid flow in the general relativistic thin accretion disk [73] is studied. In this study, we suggest that the edge region, defined in [73], is a launchpad for plasma instabilities, where blobs orbit with fundamental frequencies of the geodesics near the edge and in the geodesic region (defined in [73]), in which Hamiltonian dynamics is applicable. We also show in the GRPM that these geodesics span a torus region, which overlaps with the edge and geodesic region of [73].
- 2.
- The QPOs in NLSy1s are usually observed when $L/{L}_{Edd}$ is very high; for example, $L/{L}_{Edd}\sim 10$ in the case of RE J1034+396 [8] implies a high accretion rate, but the association of $L/{L}_{Edd}$ with the QPO frequencies is not clear. Moreover, even if one assumes that the accretion process in AGN and BHXRB is the same and that both show similar characteristic $\mathcal{Q}$ shape in the hardness-intensity diagram [6], over a timescale, T, this would be ${10}^{5}$−${10}^{6}$ times more than BHXRB timescales, as $T\propto {M}_{\u2022}$.
- 3.
- Our relativistic orbit model (ROM) is built on the formulation of the intrinsic mechanical energy distribution of the plasma in motion, where three frequencies ${\nu}_{X}<{\nu}_{I}<{\nu}_{M}$ correspond to the low-frequency end, break frequency, and the high-frequency end of the PSD. However, there is a noise component to be added at higher frequencies of the PSD to obtain a more realistic PSD shape to the intrinsic energy distribution related to the frequencies of the unstable orbits inside MBSO. A more generalized approach will be to incorporate frequencies of the more general eccentric and non-planar orbits ($e\ne 0$, $Q\ne 0$) contributing to the PSD shape. This is planned as future work.
- 4.
- The fundamental frequencies of the spherical geodesics in the Kerr geometry seem to explain the PSD in the Inner Corona (IC) region, where $P\left({\nu}_{I}<\nu <{\nu}_{M}\right)$; whereas the frequencies of the Outer Corona (OC) region are associated with the circular orbits, where $P\left({\nu}_{X}<\nu <{\nu}_{I}\right)$. The results of this toy statistical model, ROM, seem promising. A detailed physical model is required to predict the power law indices in the energy spectrum. Furthermore, including a more ellaborate transfer function taking into account the GR effects like light bending and Doppler boosting, is in order for further study.
- 5.
- The paradigm of the ROM can be tested against observations by extracting {${M}_{\u2022}$, a} from observed {${\nu}_{X},{\nu}_{I},{\nu}_{M},{\beta}_{1},{\beta}_{2}$}, and by exploring the parameter space {${\alpha}_{1}$, ${\alpha}_{2}$} which is the basis of the PSD for the ROM model. In the future, we plan to apply and test this model against several observed PSD of various AGN sources.
- 6.
- The total power of a PSD having a power-law profile is given by$$\begin{array}{c}\hfill {\mathcal{P}}_{T}\propto {\int}_{0}^{{\nu}_{c}}{\left(\frac{\nu}{{\nu}_{c}}\right)}^{\tau}\mathrm{d}\nu \phantom{\rule{4pt}{0ex}}\propto {\nu}_{c}{\int}_{0}^{1}{X}^{\tau}\mathrm{d}X\phantom{\rule{4pt}{0ex}}\propto {\nu}_{c},\end{array}$$

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AGN | Active Galactic Nuclei |

BHXRB | Black Hole X-ray Binaries |

ULX | Ultra-Luminous X-ray source |

QPO | Quasi-Periodic Oscillation |

IC | Inner Corona |

OC | Outer Corona |

ISCO | Innermost Stable Circular Orbit |

MBCO | Marginally Bound Circular Orbit |

MBSO | Marginally Bound Spherical Orbit |

NLSy1 | Narrow-Line Seyfert 1 |

GRPM | General Relativistic Precession Model |

ROM | Relativistic Orbit Model |

PSD | Power Spectral Density |

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1 | $pp$ stands for the periastron precession. |

2 | $np$ stands for the nodal precession. |

**Figure 1.**The figure shows a unified picture of the models for X-ray, optical, and $\gamma $ ray QPOs and the origin of X-ray power spectral density (PSD) shape in AGN. The X-ray QPOs observed in NLSy1 galaxies are associated with the fundamental frequencies of the equatorial orbits in the accretion disk sandwiched by a corona region, which we call the outer corona (OC) region, ${r}_{I}<r<{r}_{X}$; the inner corona (IC) region, ${r}_{M}<r<{r}_{I}$, is associated with the fundamental frequencies of the spherical orbits around a Kerr black hole. The optical and $\gamma $ ray QPOs in Blazars are shown as the harmonics of the timescale of a blob of matter moving along the jet. The shape of the PSD is studied using the fundamental frequency of matter which is governed by the radial effective potential, ${V}_{eff}(E,L,a,Q)$, providing the gravitational background responsible for the geodesic motion, in IC and OC regions to derive the energy distribution of the orbiting matter, $N\left(E\right)$, which is directly related to the observed intensity, $I\left(\nu \right)$, where $\nu $ is the temporal frequency.

**Figure 2.**The figure represents the generalized relativistic precession phenomenon, near a Kerr black hole (BH) at the center rotating anti-clockwise, of the non-equatorial orbits ($Q\ne 0$). ${\Omega}_{pp}$ represents the periastron precession and ${\Omega}_{np}$ represents the nodal precession frequency. The particle starts from the initial point A, and follows an eccentric and non-equatorial trajectory before completing one (

**a**) radial, or (

**b**) vertical oscillation to reach point B, where it sweeps an extra $\Delta \varphi $ azimuthal angle during one (

**a**) radial, or (

**b**) vertical oscillation because the azimuthal motion is faster than the radial or vertical motion. Image courtesy: [50].

**Figure 3.**The figure shows the circular orbit frequency contours of (

**a**) ${\nu}_{\varphi}$, Equation (7a), for the QPO frequencies of RE J1034+396, MS 2254.9-3712, Mrk 766, and MCG-06-30-15, given in Table 2; (

**b**) ${\nu}_{\varphi}$ and ${\nu}_{pp}$ contours, Equation (7b), for two QPO frequencies of Mrk 766; and (

**c**) ${\nu}_{np}$ contour, Equation (7c), for the QPO frequency of 2XMM J123103.2+110648.

**Figure 4.**The integrated probability density profiles for 1H 0707-495 are shown in (

**a**) ${\mathcal{P}}_{1}\left(r\right)$ and (

**d**) ${\mathcal{P}}_{1}\left(a\right)$, where the dashed vertical lines enclose a region with 68.2% probability, and the solid vertical line corresponds to the peak of the profiles. The inner probability contours of the parameter solution are shown: (

**b**) in the (a, r) plane, and (

**c**) the outer contours in the (r, a) plane, where the + sign marks the exact solution.

**Figure 5.**The figure shows the solutions of spherical orbit parameters {${r}_{s0}$, ${a}_{0}$, ${Q}_{0}$} for QPO frequencies of 1H 0707-495 in (

**a**) (Q, a), and in (

**b**) (Q, ${r}_{s}$) plane.

**Figure 6.**The figure shows (

**a**) ${\chi}_{a}^{2}/{\chi}_{p}^{2}$ function for a, where the vertical solid black curve depicts ${a}_{p}$ and the vertical dashed black curve encloses the 95% probability region, (

**b**) the range of Q and (

**c**) ${r}_{s}$ corresponding to the $2\sigma $ region of a, where the vertical dashed black curves mark {${a}_{p}$, ${r}_{sp}$, ${Q}_{p}$}.

**Figure 7.**The figure shows $E\left(r\right)$ as a function r for (

**a**) the equatorial circular orbits ($Q=0$) and for (

**b**) the spherical orbits with $Q=8$, where $a=0.5$. The vertical black curves correspond to the innermost stable circular orbit (ISCO) and to the innermost stable spherical orbit (ISSO) for $Q=8$.

**Figure 8.**The figure shows a comparison of the ISCO and ISSO radii in the (r, a) plane. The ISSO radius moves outwards as Q increases.

**Figure 9.**The figure shows E as a function of ${\nu}_{\varphi}$ for the circular orbits (

**a**) outside ${r}_{I}$, and (

**b**) inside ${r}_{I}$ for $a=0.25$ and ${M}_{\u2022}={10}^{7}{M}_{\odot}$. The minima of E is at ${r}_{I}$ in both diagrams.

**Figure 10.**The figure shows contours of (

**a**) ${\alpha}_{1}$, and (

**b**) ${\alpha}_{2}$ in the (Q, a) plane.

**Figure 11.**The figure shows examples of PSD, $P\left(\overline{\nu}\right)\propto F{\left(\overline{\nu}\right)}^{2}$, profile obtained using the ROM for the parameter combinations (

**a**) #1, and (

**b**) #2 given in Table 6. The red curve shows the bending power-law model fit, given by Equation (13), where the fitting parameters are shown in Table 6. The vertical black dashed curve corresponds to the ISCO (break) frequency.

Symbol | Explanation | Symbol | Explanation |
---|---|---|---|

c | speed of light | ${\mathcal{P}}_{1}$ | one-dimensional and normalized probability |

G | gravitational constant | density in parameter space | |

${M}_{\u2022}$ | mass of the black hole | $L\left(a\right)$ | liklihood function for spin |

${M}_{\odot}$ | mass of the sun | ${a}_{p}$ | most probable value of spin |

a | spin of the black hole | ${\chi}_{a}^{2}$ | distribution function of spin |

Q | Carter’s constant | ${\sigma}_{ai}$ | variance of spin |

e | eccentricity of the orbit | ${T}_{0}$ | QPO period |

$\mu $ | inverse-latus rectum of the orbit | ${T}_{F}$ | theoretical timescale for jet-based QPOs |

$\nu $ | frequency in Hz | ${r}_{F}$ | radial footpoint of the magnetic field |

$\overline{\nu}$ | frequency scaled by (${c}^{3}/G{M}_{\u2022}$) | ${r}_{L}$ | light cylinder radius |

${\overline{\nu}}_{\varphi}$ | scaled azimuthal frequency | ${\mathcal{P}}_{s}\left(\nu \right)$ | bending power-law profile for PSD |

${\overline{\nu}}_{r}$ | scaled radial frequency | ${\nu}_{b}$ | break-frequency of PSD |

${\overline{\nu}}_{\theta}$ | scaled vertical oscillation frequency | ${\alpha}_{l}$ & ${\alpha}_{h}$ | PSD slopes for $\nu <{\nu}_{b}$ & $\nu >{\nu}_{b}$ |

r | radius of a circular orbit | $N\left(E\right)$ | distribution function for energy |

${r}_{s}$ | radius of a spherical orbit | $F\left(\nu \right)$ | distribution function for frequency |

${p}_{\theta}$ | conjugate momentum of $\theta $ coordinate | ${\alpha}_{1}$ | power-law index of $N\left(E\right)$ inside ISCO |

E | energy per unit rest mass of a test particle | ${\alpha}_{2}$ | power-law index of $N\left(E\right)$ outside ISCO |

${L}_{z}$ | z-component of the angular momentum | ${r}_{I}$ | ISCO radius |

per unit rest mass of a test particle | ${r}_{M}$ | MBSO radius | |

$\tau $ | proper time | ${r}_{X}$ | outer edge of the accretion disk |

${V}_{eff}$ | radial effective potential in Kerr geometry | ${\overline{\nu}}_{I}$ | scaled azimuthal frequency at ISCO |

$P\left(\nu \right)$ | probability density in frequency space | ${\overline{\nu}}_{M}$ | scaled azimuthal frequency at MBSO |

$\mathcal{J}$ | jacobian of transformation from frequency | ${\overline{\nu}}_{X}$ | scaled azimuthal frequency at outer edge |

to parameter space | of the accretion disk | ||

${\nu}_{i0}$ | observed centroid frequency of the ith QPO | ${\beta}_{1}$ | average slope of PSD for $\nu >{\nu}_{b}$ |

${\sigma}_{i}$ | observed standard dispersion of the ith QPO | ${\beta}_{2}$ | average slope of PSD for $\nu <{\nu}_{b}$ |

$\mathcal{P}\left(\left[x\right]\right)$ | normalized probability density in parameter | ${\nu}_{c}$ | upper cut off frequency of PSD |

space | ${\mathcal{P}}_{T}$ | total integrated power of PSD |

**Table 2.**A list of statistically significant QPOs detected in the X-ray band (0.3–10 keV) by the XMM-Newton in AGN along with their black masses. The lower-case letters (a to m) provide links to references given in the last column.

# | Source | Class of AGN | ${\mathit{M}}_{\u2022}/{\mathit{M}}_{\odot}$ ($\times {10}^{6}$) | QPO Period ks | QPO Frequency $\left(\times {10}^{-4}\right)$ Hz | References |
---|---|---|---|---|---|---|

1. | RE J1034+396 | NLSy1 | 4 ${}^{a}$ | $3.73\pm 0.13$ | 2.681 ± 0.093 ${}^{\mathrm{b}}$ | [55] ${}^{a}$, [8] ${}^{\mathrm{b}}$ |

2. | 2XMM J123103.2+110648 | Type-2 AGN | 0.1 ${}^{\mathrm{c}}$ | $13.71$ | 0.729 ${}^{\mathrm{d}}$ | [56] ${}^{\mathrm{c}}$, [9] ${}^{\mathrm{d}}$ |

3. | MS 2254.9-3712 | NLSy1 | 4 ^{e} | $6.667$ | $1.5$${}^{\mathrm{f}}$ | [57] ^{e}, [10] ${}^{\mathrm{f}}$ |

4. | 1H 0707-495 | NLSy1 | 5.2 ${}^{\mathrm{g}}$ | $3.8\pm 0.17$ | $2.632\pm 0.118$${}^{(\mathrm{g},\mathrm{h})}$ | [36] ${}^{\mathrm{g}}$, [37] ${}^{\mathrm{h}}$ |

$8.265\pm 1.366$ | $1.21\pm 0.2$${}^{\mathrm{h}}$ | |||||

5. | Mrk 766 | NLSy1 | 4.3 ${}^{\mathrm{i}}$ | $6.452\pm 0.458$ | $1.55\pm 0.11$${}^{\mathrm{j}}$ | [58] ${}^{\mathrm{i}}$, [38] ${}^{\mathrm{j}}$ |

$4.2$ | 2.38 ${}^{\mathrm{k}}$ | [39] ${}^{\mathrm{k}}$ | ||||

6. | MCG-06-30-15 | NLSy1 | 3.26 ${}^{\mathrm{l}}$ | $3.6\pm 0.229$ | 2.778 ± 0.177 ${}^{\mathrm{m}}$ | [59] ${}^{\mathrm{l}}$, [40] ${}^{\mathrm{m}}$ |

**Table 3.**The table summarizes results of {r, a} parameter solution, and corresponding errors for X-ray QPOs in NLSy1 1H 0707-495. The columns provide the range of parameter volume taken for {r, a}, the chosen resolution to calculate the normalized probability density at each point inside the parameter volume, the exact solutions, and the results of the model fit to the integrated profiles. The mass of the black hole is fixed to ${M}_{\u2022}=5.2\times {10}^{6}{M}_{\odot}$ [36].

Source | r Range | Resolution $\mathbf{\Delta}\mathit{r}$ | Exact Solution ${\mathit{r}}_{0}$ | Model Fit | a Range | Resolution $\mathbf{\Delta}\mathit{a}$ | Exact Solution ${\mathit{a}}_{0}$ | Model Fit |
---|---|---|---|---|---|---|---|---|

1H 0707-495 | 7–9.5 | 0.01 | 8.214 | 8.214${}_{-0.359}^{+0.116}$ | 0–0.9 | 0.001 | 0.0662 | 0.0662${}_{-0.0662}^{+0.2695}$ |

**Table 4.**The table summarizes results of spherical orbit parameter solution, {${r}_{s}$, a}, and corresponding errors for X-ray QPOs in NLSy1 1H 0707-495. The columns provide the range of parameter volume taken for {${r}_{s}$, a} by fixing $\{Q=1,4,8,12\}$, the chosen resolution to calculate the normalized probability density at each point inside the parameter volume, the exact solutions, the results of the model fit to the integrated profiles, and variance ${\sigma}_{a}$. The mass of the black hole is fixed to ${M}_{\u2022}=5.2\times {10}^{6}{M}_{\odot}$ [36].

Q | ${\mathit{r}}_{\mathit{s}}$ Range | Resolution $\mathbf{\Delta}{\mathit{r}}_{\mathit{s}}$ | Exact Solution ${\mathit{r}}_{\mathit{s}0}$ | Model Fit | a Range | Resolution $\mathbf{\Delta}\mathit{a}$ | Exact Solution ${\mathit{a}}_{0}$ | Model Fit | ${\mathit{\sigma}}_{\mathit{a}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 6.5–9.5 | 0.01 | 8.215 | $8.{215}_{-0.354}^{+0.118}$ | 0–0.9 | 0.001 | 0.069 | $0.{069}_{-0.069}^{+0.28}$ | 0.290 |

4 | 6.5–9.5 | 0.01 | 8.219 | $8.{219}_{-0.331}^{+0.127}$ | 0–0.9 | 0.001 | 0.080 | $0.{080}_{-0.080}^{+0.316}$ | 0.317 |

8 | 6.5–10 | 0.01 | 8.233 | $8.{233}_{-0.278}^{+0.157}$ | 0–0.9 | 0.001 | 0.109 | $0.{109}_{-0.109}^{+0.366}$ | 0.348 |

12 | 6.5–10 | 0.01 | 8.301 | $8.{301}_{-0.196}^{+0.231}$ | 0–0.9 | 0.001 | 0.269 | $0.{269}_{-0.269}^{+0.127}$ | 0.277 |

**Table 5.**A list of statistically significant QPOs detected in the $\gamma $ ray and optical energy bands in BL Lacertae type of AGN, along with their redshifts and black hole masses. The theoretical timescales are calculated, using Equation (12), such that the lower and upper limit correspond to ${r}_{F}=30$ and ${r}_{F}=80$ respectively. The lower-case letters (a to m) are links to references given in the last column.

# | Source | z | Log$\left({\mathit{M}}_{\u2022}/{\mathit{M}}_{\odot}\right)$ | Energy Band | QPO Period ${\mathit{T}}_{0}$ (Days) | ${\mathit{T}}_{\mathit{F}}$ (Days) | References |
---|---|---|---|---|---|---|---|

1. | PKS 2155-304 | 0.116 ${}^{\mathrm{a}}$ | 8.7 ${}^{\mathrm{b}}$ | 100 MeV–300 GeV | 620 ± 41 ${}^{\mathrm{c}}$ | 33–143 | [70] ${}^{\mathrm{a}}$,[71] ${}^{\mathrm{b}}$,[11,12,14] ${}^{\mathrm{c}}$ |

100 MeV–300 GeV | 612 ± 42 ${}^{\mathrm{d}}$ | [47] ${}^{\mathrm{d}}$ | |||||

R (optical) | 315 ± 25 ${}^{\mathrm{c}}$ | ||||||

2. | PG 1553+113 | 0.36 ^{e} | ∼8 ${}^{\mathrm{f}}$ | 100 MeV–300 GeV | 780 ± 63 ${}^{\mathrm{g}}$ | 8–35 | [71] ^{e},[72] ${}^{\mathrm{f}}$,[13,14] ${}^{\mathrm{g}}$ |

R (optical) | 810 ± 52 ${}^{\mathrm{g}}$ | ||||||

3. | PKS 0537-441 | 0.892 ${}^{\mathrm{h}}$ | 8.56 ${}^{\mathrm{i}}$ | 100 MeV–300 GeV | 280 ± 39 ${}^{\mathrm{j}}$ | 40–176 | [70] ${}^{\mathrm{h}}$,[71] ${}^{\mathrm{i}}$,[15] ${}^{\mathrm{j}}$ |

R (optical) | 148 ± 17 ${}^{\mathrm{j}}$ | ||||||

4. | BL Lac | 0.0686 ${}^{\mathrm{k}}$ | 8.21 ${}^{\mathrm{l}}$ | 100 MeV–300 GeV | 680 ± 35 ${}^{\mathrm{m}}$ | 10–44 | [70] ${}^{\mathrm{k}}$,[71] ${}^{\mathrm{l}}$,[14,16] ${}^{\mathrm{m}}$ |

R (optical) | 670 ± 40 ${}^{\mathrm{m}}$ |

**Table 6.**The table summarizes the computed values of (${\alpha}_{1}$, ${\alpha}_{2}$) and the parameter fits to the bending power-law, Equation (13), for various combinations of (a, Q), where we fixed ${r}_{X}=10$ and {${\beta}_{1}=-2$, ${\beta}_{2}=-1$}, and the frequencies were scaled by $({c}^{3}/G{M}_{\u2022})$.

# | (a, Q) | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{\mathit{l}}$ | ${\mathit{\alpha}}_{\mathit{h}}$ | ${\mathit{P}}_{0}$ |
---|---|---|---|---|---|---|

1 | ($0.1,0$) | $2.286$ | $3.753$ | 0.282 | 2.74 | 0.866 |

2 | ($0.5,4$) | $2.615$ | $4.864$ | 0.413 | 3.453 | 0.818 |

3 | ($0.9,0$) | $2.462$ | $8.873$ | 0.488 | 5.561 | 1.112 |

4 | ($0.9,4$) | $3.944$ | $7.407$ | 0.497 | 5.328 | 0.925 |

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**MDPI and ACS Style**

Rana, P.; Mangalam, A. A Relativistic Orbit Model for Temporal Properties of AGN. *Galaxies* **2020**, *8*, 67.
https://doi.org/10.3390/galaxies8030067

**AMA Style**

Rana P, Mangalam A. A Relativistic Orbit Model for Temporal Properties of AGN. *Galaxies*. 2020; 8(3):67.
https://doi.org/10.3390/galaxies8030067

**Chicago/Turabian Style**

Rana, Prerna, and A. Mangalam. 2020. "A Relativistic Orbit Model for Temporal Properties of AGN" *Galaxies* 8, no. 3: 67.
https://doi.org/10.3390/galaxies8030067