Spins of Supermassive Black Holes M87* and SgrA* Revealed from the Size of Dark Spots in Event Horizon Telescope Images

Round 1

Reviewer 1 Report

please see the attached file

Comments for author File: Comments.pdf

Author Response

Question1 of referee1: The author has based this work on the dark spots, however a rigorous mathematical definition is missing. The reader is left to wonder what is the BH shadow and what is the dark spot and how they differ. Without this the reader cannot fully anticipate the results. 

Reply1: See Eq. (A8) for the definition of the BH shadow in revised paper.

Question2 of referee1: Moreover, the definition of r_eh is missing but most importantly how this is calculated for Fig. 9. 

Reply2: Definition of r_eh is added in the section Introduction of the revised paper.

Question3 of referee1: Which is the main result of the paper? 

Reply3: The  mager result is a crude estimation of the spin values for SgrA* an d M87*

Question4 of referee1: The author discusses in several places about photons emmited near the event horizon, but what how close is near, this needs to be put in firmer grounds.

Reply4: In the revised paper is added the additional pragraphfor explanation of about photons emmited near the event horizon:

In particular, the photon trajectories at Figs.~\ref{fig1}--\ref{fig6} are calculated by using solutions of equations  \ref{A9}--\ref{A11} with a starting radial coordinate $r_s=1.01r_{\rm h}$, with a finishing radial coordinate $r_0=10^5r_{\rm h}$ and with the different starting polar coordinates $\theta_s$ in the range $0\leq\theta_s\leq\pi$, and, respectively, with the different azimuth coordinates $\varphi_s$ in the range $0\leq\varphi_s\leq2\pi$ (see details in [48–53])).

Question5 of referee1: Another point to make is about M87 and what mass is used in the calculations, the EHT collaboration fitted the data also for this quantity, this of course has a great impact on the size of the shadow, shilouette or dark spot of the BH.

Reply5: The form of the lensed dark spot of the black hole image does not depend on  the black hole mass. The form of this gravittionally lensed dark spotis completely governed by the Kerr metric.

Question6 of referee1: Less important points, the caption of all figures should state in detail what is shown, which curves are plotted and what they represent. For example for Fig. 2, what is the face profile? there are so many curves in the plot, what do they represent? blue lines, black lines, dashed magenta lines, and grey regions?

Reply6: In the revised paper there are definituions for all these numerous curves.

Question7 of referee1:  The author discussed that black hole shadows are invisible for SgrA* and M87,
What does it mean that?

Reply7: This means that the form ofdark spots a t hte corresponding black hole images are defined by the emission of the hot accreting plasma in the vicinity of the black hole horizn in accordance with the Blandford-Znajec mechanism.

Question8 of referee1: Minor point, reference in figs. 7 & 8 come after reference to fig. 5.

Reply8: Thanks for finding of this misprint. It is proofed in therevised version.

Reviewer 2 Report

The author presents an approach (new method called by the author), to constrain the spin of the SMBH SgrA* and M87 through the EHT images.

I will firstly address some major points that considerably affect the whole of the paper.

The author has based this work on the dark spots, however a rigorous mathematical definition is missing. The reader is left to wonder what is the BH shadow and what is the dark spot and how they differ. Without this the reader cannot fully anticipate the results. Moreover, the definition of r_eh is missing but most importantly how this is calculated for Fig. 9. Which is the main result of the paper?

The author discusses in several places about photons emmited near the event horizon, but what how close is near, this needs to be put in firmer grounds.

 

Another point to make is about M87 and what mass is used in the calculations, the EHT collaboration fitted the data also for this quantity, this of course has a great impact on the size of the shadow, shilouette or dark spot of the BH.

Less important points, the caption of all figures should state in detail what is shown, which curves are plotted and what they represent. For example for Fig. 2, what is the face profile? there are so many curves in the plot, what do they represent? blue lines, black lines, dashed magenta lines, and grey regions?

The author discussed that black hole shadows are invisible for SgrA* and M87,

What does it mean that?

 

Minor point, reference in figs. 7 & 8 come after reference to fig. 5.

Author Response

Question1 of referee2: One of my main issues is that it is not clear to me what is the motivation of the study in the introduction. Authors are invited to add more info about the value of the study. What is the
motivation for this work?

Reply1: The main motivation is in aa crude evaluation of the spin values of the supoermassive black holes SgrA* and M87* from the first observations by EHT collaboration. This motivation is especially stressed in the revised version of the paper.

Reply2: There is a special paragraph, explaining this point:

``There are known at least 5 different methods to contain spins of SMBHs: thermal continuum fitting, disk reflection modelling, high frequency quasi-periodic oscillations, X-ray polarimetry, and using the EHT images (see, e.\.g., the review by Laura Brenneman 2013 \cite{Brenneman} or Christopher S. Reynolds \cite{Reynolds}). Measuring spins with the EHT images is only suitable for SMBHs Sgr A* and M87*.''
  
Question3 of referee2: How do you calculate the dependence of the dark spot size?

Reply3: In the revised paper is added the special paragraph fot this explanatiion:

In particular, the photon trajectories at Figs.~\ref{fig1}--\ref{fig6} are calculated by using solutions of equations  \ref{A9}--\ref{A11} with a starting radial coordinate $r_s=1.01r_{\rm h}$, with a finishing radial coordinate $r_0=10^5r_{\rm h}$ and with the different starting polar coordinates $\theta_s$ in the range $0\leq\theta_s\leq\pi$, and, respectively, with the different azimuth coordinates $\varphi_s$ in the range $0\leq\varphi_s\leq2\pi$ (see details in \cite{Dokuch22,doknaz21,doknaz20,Dokuch19,doknaz19,doknazsm19}).

Question4 of referee2: Please add a sentence to present your main finding in the abstract.

Reply4: The abstract of the revised paper is a little bit modified

Question4 of referee2: Please conduct a full revision to improve the language of the manuscript to correct few typos such as: ORCID must be corrected in the author list L..5 coma after ``in the used model''

Reply4: The language is corrected in the revised paper. ORCID presntation is govrned by th MDPI rules

Round 2

Reviewer 2 Report

I suggest that this paper can be accepted in its present form.