Stability of Circular Orbits Around Kerr Black Holes Immersed in a Dehnen-Type Dark Matter Halo

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

My comments & questions are in a pdf Referee-report-Universe-Jan2026

Comments for author File: Comments.pdf

Author Response

Reviewer 1:

1.Comment: Question # 1 to the Authors: What if any are the implications of your paper about a plethora of data about (i) the orbits of stars rotating around at the Kerr BH present at the center of M87 & Sag A∗ ; (ii) the BH spin and (iii)

the bending of light around the black holes? Can the Authors put any limits on effects from DM or about the latter’s

density for the supermassive Kerr BH residing at the center of M87 or Sag A∗?

 

Response:Thank you very much for your careful review and insightful comments. Regarding your first question, our analysis mainly focuses on the stability of circular orbits—particularly the ISCO—around a rotating Kerr black hole when a Dehnen-type dark matter (DM) halo is introduced. While our model considers an idealized single BH + DM halo system and is simplified compared with the real environments of M87 or Sagittarius A∗ (Sgr A∗), the implications can be interpreted as follows:

Stellar Orbits: The presence of a DM halo increases the gravitational potential around the black hole, causing shifts or splitting of stable circular orbit regions. In our model, this is manifested as changes in the ISCO and marginally stable orbits with varying halo density $\rho_s$. For stars like S2, whose orbital radii are much larger than the ISCO, the direct dynamical effect of the DM halo is very small. However, at higher observational precision (e.g., measuring pericenter velocities or orbital perturbations), tiny corrections could in principle provide constraints on the DM distribution in the galactic center.

Black Hole Spin: The black hole spin strongly affects the ISCO and the distribution of stable circular orbits. Our calculations show that ISCOs of high-spin BHs are more sensitive to the surrounding DM potential. While precise measurements of BH shadow or close-in stellar orbits could potentially probe the coupling between spin and environmental effects, current observations cannot yet disentangle these to place direct limits on DM density.

 

Light Deflection: The DM halo slightly modifies the gravitational field near the black hole, causing small changes in light bending and shadow size. For the parameter ranges considered, this effect is minor, and the current EHT resolution is insufficient to detect it directly. Nevertheless, our model provides quantitative estimates of these small corrections that may be relevant for future high-resolution observations.

IMRI/EMRI Systems and Gravitational Waves: While the direct effects of the DM halo on stellar orbits and light bending are small, in extreme mass-ratio inspiral (EMRI) or intermediate mass-ratio inspiral (IMRI) systems, the DM halo can significantly influence orbital evolution and leave observable imprints on the gravitational wave signals. In particular, modifications of the ISCO and innermost orbits induced by the DM potential can alter the phase evolution of gravitational waves, providing a potentially detectable signal for future space-based gravitational wave detectors such as LISA, TianQin, and Taiji.

 

In summary, our study indicates that while the DM halo has negligible direct effects on stellar orbits, BH spin measurements, or light bending with current instruments, it can leave significant imprints in EMRI/IMRI gravitational wave signals, offering valuable opportunities for future space-based GW observations.

 

 

2.Comment:Going on further, there would be gravitational wave radiation generated by any star rotating around a BH [3 ]

and as shown in [4 ] it can be substantial enough to affect the stability of any star orbiting very close to the BH.

This brings me to my second question:

Question # 2 to the Authors: What if any are the implications of your paper on the subject of stability due to

gravitational wave radiation for an ISCO star around a supermassive Kerr BH?

 

Response:Gravitational wave radiation indeed introduces dissipative effects on stars orbiting a supermassive Kerr black hole, especially those near the innermost stable circular orbit (ISCO). In EMRI or IMRI systems, the ISCO corresponds to the cutoff radius for the gravitational wave phase integral. Prior to reaching this radius, the extreme mass-ratio inspiral can be well approximated as an adiabatic evolution, where the orbit gradually shrinks but the system remains well-described by the slow-variation approximation.

Our study shows that the presence of a dark matter halo modifies both the location and continuity of the ISCO, thereby directly affecting the cutoff of the gravitational wave phase integral. In other words, the dark matter halo not only shifts the ISCO radius but also indirectly alters the accumulated gravitational wave phase, producing potentially observable differences in the waveforms of EMRI/IMRI systems. This effect is particularly pronounced for high-spin black holes and dense dark matter halos and could be detectable by future space-based gravitational wave observatories such as LISA, TianQin, and Taiji.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Universe-4144193

Stability of Circular Orbits around Kerr Black Holes Immersed in a Dehnen-type Dark Matter Halo

In this manuscript, authors investigate the dynamical stability of equatorial circular orbits around a Kerr black hole surrounded by a Dehnen type dark matter halo. An effective metric for the combined system is constructed using the Newman-Janis algorithm, and the effective potential for test-particle motion is derived. Orbital stability is analyzed via the Hessian matrix of the potential. The presence of the dark matter halo is shown to modify the spatial distribution of stable circular orbits. It is shown that the location of the innermost stable circular orbit is shifted relative to the vacuum Kerr case due to the combined effects of black hole spin and halo parameters. Furthermore, a parameter-space scans and two and three dimensional visualizations are displayed to illustrate these effects. Finally, the influence of the halo on the black hole event horizon is also described.
 
The manuscript is rigorous and well presented, and I recommend publication after the issues listed below have been adequately resolved.

* Could the authors provide the form of the pressures in Eq. 8? this would benefit the readability and understanding of the spacetime.
* The derivations of the metric, in particular the choice of coordinates used in the Newman-Janis algorithm should appear in an Appendix.
* Could the authors provide a plot similar to Figure 4, for the purely Kerr case and for some other values of rho_s in order to better understand the impact of the dark matter halo?
* A similar plot for Figure 5, would help to compare with the pure Kerr-case.
* The authors may provide a further explanation on how the effect displayed in Fig 9 "produce observable  changes in EMRI dynamics if the BH is rapidly spinning".
lines 231-233. 

Author Response

Reviewer 2:

1.Comment: Could the authors provide the form of the pressures in Eq. 8? this would benefit the readability and understanding of the spacetime.

 

Response:We thank the referee for this helpful comment. In Eq. (8), the energy–momentum tensor is written in the general anisotropic fluid form, where P_r(r) and P_t(r) represent the radial and tangential pressures, respectively.

In the present work, we adopt the standard cold dark matter (CDM) assumption, according to which the dark matter halo is effectively pressureless. Therefore, both the radial and tangential pressures are taken to vanish. Under this assumption, the spacetime geometry is completely determined by the density profile of the Dehnen halo.

We have clarified this point in the revised manuscript immediately after Eq. (8) to improve the readability and understanding of the spacetime structure.

 

2.Comment: The derivations of the metric, in particular the choice of coordinates used in the Newman-Janis algorithm should appear in an Appendix.

 

 

Response: We thank the reviewer for the valuable suggestion. In the revised manuscript, we have moved the detailed derivation of the Kerr-like metric using the Newman–Janis algorithm, including the choice of coordinates, to a new Appendix (Appendix A). A reference to this Appendix has been added in the main text where the metric is first introduced, so that readers can consult it for the full derivation while keeping the main text concise.

 

3.Comment: Could the authors provide a plot similar to Figure 4, for the purely Kerr case and for some other values of rho_s in order to better understand the impact of the dark matter halo?

 

 

Response: We thank the referee for this helpful suggestion. To illustrate the impact of the dark matter halo, we have generated a new figure (Figure~6) showing the stability regions of circular orbits for different combinations of the black hole spin $a$ and the halo characteristic density $\rho_s$, including the purely Kerr case ($\rho_s = 0$). Each panel indicates the specific values of $a$ and $\rho_s$ in the upper-right corner, allowing a clear comparison of how the presence and variation of the dark matter halo affect the continuity and structure of the stable orbital regions.

 

4.Comment:A similar plot for Figure 5, would help to compare with the pure Kerr-case.

Response: We thank the reviewer for the suggestion. We have added a new panel showing the pure Kerr case ($\rho_s = 0$) alongside the original Figure~5. This allows for a direct comparison between the effects of the Dehnen-type dark matter halo and the baseline Kerr geometry. As shown, the presence of the dark matter halo modifies the continuity and shape of the stable circular orbit regions, whereas in the pure Kerr case the stable region remains continuous. This comparison clearly illustrates the influence of the surrounding dark matter environment on orbital stability.

Comment: The authors may provide a further explanation on how the effect displayed in Fig 9 "produce observable  changes in EMRI dynamics if the BH is rapidly spinning".lines 231-233. 

 

Response:We thank the referee for the helpful suggestion. In the revised manuscript, we have added a detailed explanation of Fig.~\ref{fig:9}. The figure now illustrates the ISCO as a function of the dark matter density $\rho_s$ for several representative black hole spins $a$. We clarify that denser halos shift the ISCO inward, and this effect is more pronounced for high-spin black holes due to relativistic frame-dragging and centrifugal balance near the innermost stable orbit. We also note that changes in the ISCO directly affect the cutoff of the gravitational wave phase integral in EMRI systems, which can produce potentially observable modifications in the waveform. These additions address the mechanism behind the effect displayed in Fig.~\ref{fig:9} and emphasize the importance of accounting for dark matter halos in EMRI modeling.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

If I may offer an advice, the authors should study in depth the subject of changes in the gravitational wave radiation loss due to strong dark matter on close by solar orbits to a Kerr black hole as they are likely to be important (and perhaps unexpected) for its stability.