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MEASURING PARAMETERS OF BLACK HOLES FROM SPACE
Alexander F. Zakharov National Astronomical Observatories of Chinese Academy of Sciences, Beijing 100012, China; Institute of Theoretical and Experimental Physics, 25, B.Cheremushkinskaya st., Moscow, 117259, Russia; Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia

ABSTRACT To describe black holes typically astronomers use Newtonian approaches for gravitational field because usually they consider systems in weak gravitational field and correspondingly a Newtonian approximation of General Relativity is good enough. Here we discuss phenomena where we have to use general relativistic approaches to explain present and future observational data. Different X-ray missions such as ASCA, XMM-Newton, Chandra etc. discovered features of Fe K lines and other X-ray lines as well. Attempts to fit spectral line shapes lead to conclusions that sometimes the profiles line shapes should correspond to radiating regions which are located in the innermost parts of accretion disks where contributions of general relativistic phenomena are extremely important. As an illustration we consider a radiating annulus ? model to clarify claims given recently by Muller & Camenzind (2004). We discuss properties of highly inclined disks and analyze a possibility to evaluate magnetic fields near black hole horizons. We mention also that shadows could give us another case when one could evaluate black hole parameters (namely, spins, charges and inclination angles) analyzing sizes and shapes shadows around black holes. Key words: Black hole physics; the Galactic Center; Tests of General Relativity.

1. INTRODUCTION Here we discuss samples where we really need general relativistic approaches in the strong gravitational field limit to explain observational data generating by radiation arising in black hole vicinities and typically one could get the data with space missions such as ASCA, RXTE, XMM-Newton, Chandra etc. Several years ago it was predicted that profiles of lines emitted by AGNs and X-ray binary systems could have

asymmetric double-peaked, double horned or triangular ? shape according to classification done by Muller and Camenzind [1]. Comprehensive review summarizes the detailed discussion of theoretical aspects of possible scenarios for generation of broad iron lines in AGNs [2] (an influence of microlensing on Fe K line shapes and spectra was discussed in [3] but optical depths for the phenomena were calculated in [4, 5, 6]). A formation of shadows (mirages) is another sample when general relativistic effects are extremely important and in principle they could be detected with forthcoming interferometrical facilities [7, 8, 9, 10, 11, 12, 13, 14, 15] (perspective studies of microlensing with Radioastron facilities were discussed recently [16]). Observations of shadows could give a real chance to observe "faces" of black holes and confirm general relativity predictions in the framework of strong gravitational field approach and obtain new constraints on alternative theories of gravity. The angular resolution of the space RADIOASTRON interferometer will be high enough to resolve radio images around black holes. In principle, measuring the mirage shapes one could evaluate the black hole mass, inclination angle (e.g. the angle between the black hole spin axis and line of sight) and spin if the black hole distance is known. For example, for the black hole at the Galactic Center the mirage size is 52Еas for the Schwarzschild case. In the case of a Kerr black hole [7], the mirage is deformed depending on the black hole spin a and on the angle of the line of sight, but its size is almost the same. In the case of ? a Reissner-Nordstrom black hole its charge changes the size of the shadows up to 30 % for the extreme charge case. Therefore, the charge of the black hole can measured by observing the shadow size, if the other black hole parameters are known with a sufficient precision. In general, one could say that a measure of the mirage shape (in size) allows to evaluate all the black hole "hairs". There is no doubt that the rapid growth of observational facilities will give a chance to measure the mirage shapes using not only RADIOASTRON facilities but also other instruments and spectral bands, like the X-ray interferometer MAXIM [17, 18], the RADIOASTRON mission or other space based interferometers in millimeter and submillimeter bands.


2. TOY MODEL LESSONS ? Recently Muller and Camenzind [1] presented results of their calculations and classified different types of spectral line shapes and described their origin. In particular, the authors claimed that usually "... triangular form follows from low inclination angles...", "...double peaked shape is a consequence of the space-time that is sufficiently flat. This is theoretically reproduced by shifting the inner edge to the disk outwards... A relatively flat space-time is already reached around 25 rg ..." We clarified their hypothesis about an origin of doubled peaked and double horned line shapes. Based on results of numerical simulations we showed that double peaked spectral lines arise for almost any locations of narrow emission rings (annuli) (except ? closest orbits as we could see below) although Muller and Camenzind [1] suggested that such profiles arise for relatively flat spacetimes and typical radii for emission region about 25 rg . Using a radiating annulus model we checked the statements and clarified them for the case. We could note here that in the framework of the model we do not use any assumptions about an emissivity law, but we only assume that radiating region is a narrow circular ring (annulus). Thus, below we do not use some specific model on surface emissivity of accretion (we only assume that the emitting region is narrow enough). But general statements (which will be described below) could be generalized on a wide disk case without any problem. We used an approach which was discussed in details in papers [19, 20, 21, 22, 23, 24, 27, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. The approach was used in particular to simulate spectral line shapes. This approach is based on results of qualitative analysis [40, 41]. Presenting their classification of different types of spectra ? line shapes Muller and Camenzind [1] noted that double peaked shapes arise usually for emission regions located far enough from black holes. Earlier, we calculated spectral line shapes for annuli for selected radii and distant observer position angles and found an essential fraction of spectral line gallery correspond to double peaked profiles ? [22]. To check the Muller and Camenzind [1] hypothesis about an origin of double peaked profiles we calculated a complete set of spectral line shapes for emitting annuli. Let us discuss results of our calculations for rapidly rotating black holes (for a = 0.998 one could find a detailed description of the calculations in [42, 43, 44]). We summarize results of the calculations. As it was shown in the framework of the simple model the double peaked spectral line shape arise almost for all parameters r and a except the case when radii are very small r (0.7, 2) and inclination angles are in the band [45 , 90 ] (for these parameters the spectral line shape has triangular structure). The phenomenon could be easy understood, since for this case the essential fraction of all photons emitted in the opposite direction in respect to emitting segment of annulus is captured by a black hole, therefore a red peak is strongly dumped. For other radii and angles spectral line profiles have double peaked structure. Therefore ? we clarify the statement Muller and Camenzind [1] that double peaked structure arises if radiation region is far

enough. If we assume that there is a weak dependence of emissivity function on radius, then a number of photons characterizes relative intensity in the line (roughly speaking for r = 0.7 an intensity (in counts) in 10 times lower then an intensity for r = 2) therefore in observations for small radii one could detect only a narrow blue peak but another part of spectra is non-distinguishable from a background. One could note also that for fixed radius there is a strong monotone dependence of intensity on inclination angle (maximal intensity corresponds to photon motion near equatorial plane and only a small fraction of photons reach a distant observer near the polar axis). That is a natural consequence of a photon boost due to a circular motion of emitting fragment of annulus in the equatorial plane and an influence of spin of a rotating black hole. In the framework of the simple model one could understand that sometimes the Fe K line has only one narrow peak like in observations of the Seyfert galaxy MCG-630-15 by the XMM-Newton satellite [45]. If radiating (or illuminating) region is a narrow annulus evolving along quasi-circular orbits, then initially two peak structure of the spectral line profile transforms in one peaked (triangular) form. Moreover, an absolute intensity in the line is increased for smaller radii since a significant fraction of emitted photons are captured by a black hole during the evolution of emitting region toward to black hole in observations we could detect only narrow blue peak and its height is essentially lower than its height was before for larger radii. Another part of the triangular spectral line shape could be non-distinguishable from a background. A relative low intensity for a triangular spectral line shape could give a narrow single peak structure in observations.

3. SIGNATURES OF ACCRETION DISCS WITH HIGH INCLINATION At inclination angles > 80 , new observational features of general relativity (GR) could arise. Matt et al. [46] discovered such phenomenon for a Schwarzschild black hole, moreover the authors predicted that their results could be applicable to a Kerr black hole over the range of parameters exploited. The authors mentioned that this problem was not analyzed in detail for a Kerr metric case and it would be necessary to investigate this case. In the detailed consideration [25] we did not use a specific model on surface emissivity of accretion (we only assume that the emitting region is narrow enough), therefore, we checked and confirmed their hypothesis for the Kerr metric case and for a Schwarzschild black hole using other assumptions about surface emissivity of accretion disks. In principle, such a phenomenon could be observed in microquasars and X-ray binary systems where there are neutron stars and black holes with stellar masses. We confirmed also the conclusion that extra peaks are generated by photons which are emitted by the far side of the disk, therefore we have a manifestation of gravitational lensing in the strong gravitational field approach for GR [25].


Some possibilities to observe considered features of spectral line profiles were considered [46]. The authors argued that there are non-negligible chances to observe such a phenomenon in some AGNs and X-ray binary systems. Thus, such properties of spectral line shapes are robust enough with respect to wide variations of rotational parameters of black holes and the surface emissivity of accretion disks as it was predicted [46]. So, their conjecture was confirmed not only for the Kerr black hole case but also for other dependencies of surface emissivity of the accretion disk. We use no astrophysical assumptions about the physical structure of the emission region except the assumption that the region should be narrow enough. Positions and heights of these extra peaks drastically depend on both the radial coordinate of the emitting region (annuli) and the inclination angle. It was found that these extra peaks arise due to gravitational lens effect in the strong gravitational field, namely they are formed by photons with some number of revolutions around black hole. This conclusion is based only on relativistic calculations without any assumption about physical parameters of the accretion disc like X-ray surface emissivity etc. A detailed description of the analysis was given in [25].

of magnetic field and simulate the spectral line shapes from observational data for these values, assuming that these observational data correspond to an object with no significant magnetic fields. We will try to find signatures of the triple blue peak analyzing the simulated data when magnetic fields are rather high. Assuming that there are no essential magnetic fields (compared to 1010 G) for some chosen object (for example, for MCG 6-30-15) we could simulate the spectral line shapes for the same objects but with essential magnetic fields. From results of simulations one can see that classical Zeeman splitting in three components, which can be revealed experimentally today, changes qualitatively the line profiles only for rather high magnetic field. Something like this structure can be detected, e.g. for H = 1.2 З 1011 G, but the reliable recognition of three peaks here is hardly possible [27]. It is known that neutron stars (pulsars) could have huge magnetic fields. So, it means that the effect discussed above could appear in binary neutron star systems. The quantitative description of such systems, however, needs more detailed computations. A detailed discussion of the magnetic field influence on spectral line shapes for flat accretion flows was discussed [27, 28] and for non-flat accretion flows in [31].

4. MAGNETIC FIELDS IN AGNS AND MICROQUASARS Magnetic fields play a key role in dynamics of accretion discs and jet formation. To obtain an estimation of the magnetic field we simulate the formation of the line profile for different values of magnetic field. As a result we find the minimal B value at which the distortion of the line profile becomes significant. Here we use an approach, which is based on numerical simulations of trajectories of the photons emitted by a hot ring moving along a circular geodesics near black hole, described in [20, 22, 21]. The influence of accretion disc model on the profile of spectral line was discussed [33]. We assume that the emitting region is located in the area of a strong quasi-static magnetic field. This field causes line splitting due to the standard Zeeman effect. There are three characteristic frequencies of the split line that arise in the emission. The energy of central component E0 remains unchanged, whereas two extra compoe = nents are shifted by БЕB H , where ЕB = 2me c 9.273 З 10-21 erg/G is the Bohr magneton. Therefore, in the presence of a magnetic field we have three energy levels: E0 - ЕB H, E0 and E0 + ЕB H . For the iron K H line they are as follows: E0 = 6.4 - 0.58 11 keV, 10 G H E0 = 6.4 keV and E0 = 6.4 + 0.58 11 keV [27]. 10 G Let us discuss possible influence of high magnetic fields on real observational data (see details in [27]). We will try to estimate magnetic fields when one could find the typical features of line splitting from the analysis of the spectral line shape. Further we will choose some values

ACKNOWLEDGMENTS I am grateful to the National Natural Science Foundation of China (NNSFC) (Grant # 10233050) and the National Basic Research Program of China (2006CB806300) for a partial financial support of the work.

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