arXiv subjects

Michal Zajaček

Publications and source records attributed to Michal Zajaček.

At least 73 records · Page 4Linked to original sources

arXiv2020

Missing bright red giants in the Galactic center: A fingerprint of its once active state?

In the Galactic center nuclear star cluster, bright late-type stars exhibit a flat or even a decreasing surface-density profile, while fainter late-type stars maintain a cusp-like profile. Historically, the lack of red giants in the Galactic center was discovered via the drop in the strength of the CO absorption bandhead by Kris Sellgren et al. (1990), later followed by the stellar number counts based on the high angular resolution near-infrared observations. Several mechanisms were put forward that could have led to the preferential depletion of bright red giants: star-star collisions, tidal stripping, star-accretion disc collisions, or an infall of a massive cluster or a secondary black hole. Here we propose a novel scenario for the bright red-giant depletion based on the collisions between red giants and the nuclear jet, which was likely active in the Galactic center a few million years ago and could have led to the formation of the large-scale $γ$-ray Fermi bubbles. The process of the jet-induced ablation of red giants appears to be most efficient within $\sim 0.04\,{\rm pc}$ (S-cluster), while at larger distances it was complemented by star-accretion disc collisions and at smaller scales, tidal stripping operated. These three mechanisms likely operated simultaneously and created an apparent core of late-type stars within $\sim 0.5\,{\rm pc}$.

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arXiv2020

Constraining the accretion flow density profile near Sgr A* using the $L'$-band emission of the S2 star

The density profile of the ambient medium around a supermassive black hole plays an important role in understanding the inflow-outflow mechanisms in the Galactic Centre. We constrain the spherical density profile using the stellar bow shock of the star S2 which orbits the supermassive black hole at the Galactic Centre with the pericentre distance of 14.4 mas ($\sim$ 1500 R$_\text{s}$). Assuming an elliptical orbit, we apply celestial mechanics and the theory of bow shocks that are at ram pressure equilibrium. We analyse the measured infrared flux density and magnitudes of S2 in the L'-band (3.8 micron) obtained over seven epochs in the years between 2004-2018. We detect no significant change in S2 flux density until the recent periapse in May 2018. The intrinsic flux variability of S2 is at the level of 2 - 3%. Based on the dust-extinction model, the upper limit on the number density at the S2 periapse is $\sim$1.87$\times$10$^9$ cm$^{-3}$, which yields a density slope of at most 3.20. Using the synchrotron bow-shock emission, we obtain an ambient density of $\leq$ 1.01$\times$10$^5$ cm$^{-3}$ and a slope of $\leq$ 1.47. These values are consistent with a wide variety of media from hot accretion flows to potentially colder and denser media comparable in properties to broad-line region clouds. A standard thin disc can be, however, excluded at the distance of the S2 pericentre. Based on our sensitivity of 0.01 mag, we can distinguish between hot accretion flows and thin, cold discs, where the latter can be excluded at the scale of the S2 periapse. Future observations of stars with smaller pericentre distances in the S cluster using instruments such as METIS@ELT with a photometric sensitivity of as much as 10$^{-3}$ mag will allow to probe the Galactic Centre medium at intermediate scales at densities as low as $\sim$ 700 cm$^{-3}$ in case of non-thermal bow-shock emission.

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arXiv2020

Depletion of bright red giants in the Galactic center during its active phases

Observations in the near-infrared domain showed the presence of the flat core of bright late-type stars inside $\sim 0.5\,{\rm pc}$ from the Galactic center supermassive black hole (Sgr A*), while young massive OB/Wolf-Rayet stars form a cusp. Several dynamical processes were proposed to explain this apparent paradox of the distribution of the Galactic center stellar populations. Given the mounting evidence about a significantly increased activity of Sgr A* during the past million years, we propose a scenario based on the interaction between the late-type giants and a nuclear jet, whose past existence and energetics can be inferred from the presence of $γ$-ray Fermi bubbles and bipolar radio bubbles. Extended, loose envelopes of red giant stars can be ablated by the jet with kinetic luminosity in the range of $L_{\rm j}\approx 10^{41}$-$10^{44}\,{\rm erg\,s^{-1}}$ within the inner $\sim 0.04\,{\rm pc}$ of Sgr A* (S cluster region), which would lead to their infrared luminosity decrease after several thousand jet-star interactions. The ablation of the atmospheres of red giants is complemented by the process of tidal stripping that operates at distances of $\lesssim 1\,{\rm mpc}$, and by the direct mechanical interaction of stars with a clumpy disc at $\gtrsim 0.04\,{\rm pc}$, which can explain the flat density profile of bright late-type stars inside the inner half parsec from Sgr A*.

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arXiv2020

Scatter Analysis Along the Multidimensional Radius-Luminosity Relations for Reverberation-Mapped Mg II Sources

The usage of the radius-luminosity (R-L) relation for the determination of black hole masses across the cosmic history as well as its application for cosmological studies motivates us to analyze its scatter, which has recently increased significantly both for the optical (H$β$) and UV (Mg II) lines. To this purpose, we determined the scatter along the R-L relation for an up-to-date reverberation-mapped Mg II sample. Studying linear combinations of the luminosity at 3000 A with independent parameters such as the Full Width at Half Maximum (FWHM), Fe II strength (R$_\mathrm{FeII}$), and the fractional variability ($F\mathrm{_{var}}$) for the whole sample, we get only a small decrease in the scatter ($σ{_{\rm rms}}=0.29-0.30$ dex). Linear combinations with the dimensionless accretion rate ($\dot{\mathcal{M}}$) and the Eddington ratio ($α_\mathrm{Edd}$) lead to significant reductions of the scatter ($σ_{\rm rms}\sim 0.1$ dex), albeit both suffering from the inter-dependency on the observed time-delay. After the division into two sub-samples considering the median value of the $\dot{\mathcal{M}}$ in the full sample, we find that the scatter decreases significantly for the highly accreting sub-sample. In particular, the smallest scatter of $σ{_{\rm rms}}=0.17$ dex is associated with the independent parameter R$_\mathrm{FeII}$, followed by the combination with $F\mathrm{_{var}}$, with $σ{_{\rm rms}}=0.19$ dex. Both of these independent observationally-inferred parameters are in turn correlated with $\dot{\mathcal{M}}$ and $α_\mathrm{Edd}$. These results suggest that the large scatter along the R-L relation is driven $\textit{mainly}$ by the accretion rate intensity.

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arXiv2020

S62 and S4711: Indications of a population of faint fast moving stars inside the S2 orbit -- S4711 on a 7.6 year orbit around Sgr~A*

We present high-pass filtered NACO and SINFONI images of the newly discovered stars S4711-S4715 between 2004 and 2016. Our deep H+K-band (SINFONI) and K-band (NACO) data show the S-cluster star S4711 on a highly eccentric trajectory around Sgr~A* with an orbital period of 7.6 years and a periapse distance of 144 AU to the super massive black hole (SMBH). S4711 is hereby the star with the shortest orbital period and the smallest mean distance to the SMBH during its orbit to date. The used high-pass filtered images are based on co-added data sets to improve the signal to noise. The spectroscopic SINFONI data let us determine detailed stellar properties of S4711 like the mass and the rotational velocity. The faint S-cluster star candidates, S4712-S4715, can be observed in a projected distance to Sgr~A* of at least temporarily $\leq$ 120 mas. From these stars, S4714 is the most prominent one with an orbital period of 12 years and an eccentricity of 0.985. The stars S4712-S4715 show similar properties with comparable magnitudes and stellar masses to S4711. The MCMC simulations determine confidently precise uncertainties for the orbital elements of S62 and S4711-S4715. The presence of S4711 in addition to S55, S62, and the also newly found star S4714 implies a population of faint stars that can be found at distances to Sgr~A* that are comparable to the size of our solar system. These short orbital time period stars in the dense cluster around the SMBH in the center of our Galaxy are perfect candidates to observe gravitational effects such as the periapse shift.

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arXiv2020

Distribution of Radio Spectral Slopes of Galaxies in Optical Diagnostic Diagrams

For about 500 intermediate-redshift sources ($0.04<z<0.4$), whose radio flux densities at $1.4\,{\rm GHz}$ are larger than 10 mJy, we performed additional observations at 4.85 GHz and 10.45 GHz using 100-m Effelsberg telescope. Our radio-optical galaxies are located preferentially in the composite and AGN spectral classes in the narrow line optical diagnostic diagrams (ODD). In the analysis, we focused on the distribution of radio spectral indices of radio synchrotron power-law profiles, $S_ν\propto ν^{+α}$, in the ODDs. Using different analysis techniques, both observationally motivated and machine-learning based, we found three distinct groups--clusters in the radio loudness, [OIII]/H$β$ ratio, and spectral index volume: (1) sources with steep radio spectral index, large radio loudness and large [OIII]/H$β$ ratio; (2) sources with flat radio spectral index, intermediate radio loudness and lower [OIII]/H$β$ ratio; (3) sources with inverted radio spectral index, low radio loudness and low [OIII]/H$β$. The groups (1), (2), and (3) are located along the Seyfert-LINER spectral classes towards lower ionization ratios in the ODDs and hence can represent different activity cycles/accretion modes of AGNs, which could be in some cases associated with different merger stages.

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arXiv2020

Effect of Electromagnetic Interaction on Galactic Center Flare Components

Recently, near-infrared GRAVITY@ESO observations at $2.2\,μ{\rm m}$ have announced the detection of three bright "flares" in the vicinity of the Galactic center supermassive black hole (SMBH) that exhibited orbital motion at a distance of about $6 - 11$ gravitational radii from an $\sim 4\times 10^6\, M_{\odot}$ black hole. There are indications of the presence of a large-scale, organized component of the magnetic field at the Galactic center. Electromagnetic effects on the flare dynamics were previously not taken into account despite the relativistic motion of a plasma in magnetic field leading to the charge separation and nonnegligible net charge density in the plasma. Applying various approaches, we find the net charge number density of the flare components of the order of $10^{-3} - 10^{-4}$ cm$^{-3}$, while the particles' total number density is of the order of $10^{6} - 10^{8}$ cm$^{-3}$. However, even such a tiny excess of charged particles in the quasi-neutral plasma can significantly affect the dynamics of flare components, which can then lead to the degeneracy in the measurements of spin of the SMBH. Analyzing the dynamics of recent flares in the case of the rapidly rotating black hole, we also constrain the inclination angle between the magnetic field and spin axis to $α< 50^{\circ}$, as for larger angles, the motion of the hot spot is strongly chaotic.

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arXiv2020

Near- and mid-infrared observations in the inner tenth of a parsec of the Galactic center -- Detection of proper motion of a filament very close to Sgr~A*

We analyze the gas and dust emission in the immediate vicinity of the supermassive black hole Sgr~A* at the Galactic center (GC) with the ESO VLT (Paranal/Chile) instruments SINFONI and VISIR. The SINFONI H+K data cubes show several emission lines with related line map counterparts. From these lines, the Br$γ$ emission is the most prominent one and appears to be shaped as a bar extending along the North-South direction. With VISIR, we find a dusty counterpart to this filamentary emission. In this work, we present evidence that this feature can be most likely connected to the mini-spiral and potentially influenced by the winds of the massive stars in the central cluster or an accretion wind from Sgr~A*. To this end, we co-add the SINFONI data between 2005 and 2015. The spectroscopic analysis reveals a range of Doppler-shifted emission lines. We also detect substructures in the shape of clumps that can be investigated in the channel maps of the Br$γ$-bar. In addition, we compare the detection of the near-infrared (NIR) Br$γ$ feature to PAH1 mid-infrared (MIR) observations and published 226 GHz radio data. These clumps show a proper motion of about $320$km/s that are consistent with other infrared continuum detected filaments in the Galactic center. Deriving a mass of $2.5\,\times\,10^{-5}\,M_{\odot}$ for the investigated Br$γ$-feature shows an agreement with former derived masses for similar objects. Besides the North-South Br$γ$-bar, we find a comparable additional East-West feature. Also, we identify several gas reservoirs that are located west of Sgr~A* that may harbor dusty objects.

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arXiv2020

Time-delay measurement of MgII broad line response for the highly-accreting quasar HE 0413-4031: Implications for the MgII-based radius-luminosity relation

We present the monitoring of the AGN continuum and MgII broad line emission for the quasar HE 0413-4031 ($z=1.38$) based on the six-year monitoring by the South African Large Telescope (SALT). We managed to estimate a time-delay of $302.6^{+28.7}_{-33.1}$ days in the rest frame of the source using seven different methods: interpolated cross-correlation function (ICCF), discrete correlation function (DCF), $z$-transformed DCF, JAVELIN, two estimators of data regularity (Von Neumann, Bartels), and $χ^2$ method. This time-delay is below the value expected from the standard radius-luminosity relation. However, based on the monochromatic luminosity of the source and the SED modelling, we interpret this departure as the shortening of the time-delay due to the higher accretion rate of the source, with the inferred Eddington ratio of $\sim 0.4$. The MgII line luminosity of HE 0413-4031 responds to the continuum variability as $L_{\rm line}\propto L_{\rm cont}^{0.43\pm 0.10}$, which is consistent with the light-travel distance of the location of MgII emission at $R_{\rm out} \sim 10^{18}\,{\rm cm}$. Using the data of 10 other quasars, we confirm the radius-luminosity relation for broad MgII line, which was previously determined for broad H$β$ line for lower-redshift sources. In addition, we detect a general departure of higher-accreting quasars from this relation in analogy to H$β$ sample. After the accretion-rate correction of the light-travel distance, the MgII-based radius-luminosity relation has a small scatter of only $0.10$ dex.

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arXiv2019

Current and future applications of Reverberation-mapped quasars in Cosmology

Reverberation mapping technique is an important milestone in AGN demographics, kinematics and the structure of the Broad Line Region (BLR) based on the time-delay response between the continuum and emission line. The time delay is directly related to the size of BLR which is related to the continuum luminosity of the source, producing the well-known Radius-Luminosity (RL) relation. The majority of sources have been monitored for their H$β$ emission line in low redshift sources ($z<0.1$), while there are some attempts using the MgII line for higher redshifts. We present a recent MgII monitoring for the quasar CTS C30.10 ($z=0.90$) observed with the 10-meter SALT, for which RL scaling based on MgII holds within measurement and time-delay uncertainties. One of the most important advantages of reverberation mapping technique is the independent determination to the distant source, and considering the large range of redshifts and luminosities found in AGNs, their application to cosmology is promising. However, recently it has been found that highly accreting sources show the time delays shorter than expected from RL relation. We propose correction for this effect using a sample of 117 H$β$ reverberating-mapped AGNs with $0.02<z<0.9$, recovering the low scatter along with the relation. We are able to determine the cosmological constants, $Ω_m$ and $Ω_Λ$ within 2$σ$ confidence level. We present the first steps in modelling of light curves for H$β$ and MgII and discuss the quasar selection in the context of photometric reverberation mapping with Large Synoptic Survey Telescope (LSST). With the onset of the LSST era, we expect a huge rise in the overall quasar counts and redshift range covered ($z\lesssim 7.0$), which will provide a better constraint of AGN properties with cosmological purposes.

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arXiv2019

Electric charge of black holes: Is it really always negligible?

We discuss the problem of the third black hole parameter, an electric charge. While the mass and the spin of black holes are frequently considered in the majority of publications, the charge is often neglected and implicitly set identically to zero. However, both classical and relativistic processes can lead to a small non-zero charge of black holes. When dealing with neutral particles and photons, zero charge is a good approximation. On the other hand, even a small charge can significantly influence the motion of charged particles, in particular cosmic rays, in the vicinity of black holes. Therefore, we stress that more attention should be paid to the problem of a black-hole charge and hence, it should not be neglected a priori, as it is done in most astrophysical studies nowadays. The paper looks at the problem of the black-hole charge mainly from the astrophysical point of view, which is complemented by a few historical as well as philosophical notes when relevant. In particular, we show that a cosmic ray or in general elementary charged particles passing a non-neutral black hole can experience an electromagnetic force as much as sixteen times the gravitational force for the mass of the Galactic centre black hole and its charge being seventeen orders of magnitude less than the extremal value (calculated for a proton). Furthermore, a Kerr-Newman rotating black hole with the maximum likely charge of 1 Coulomb per solar mass can have the position of its innermost stable circular orbit (ISCO) moved by both rotation and charge in ways that can enhance or partly cancel each other, putting the ISCO not far from the gravitational radius or out at more than 6 gravitational radii. An interpretation of X-ray radiation from near the ISCO of a black hole in X-ray binaries is then no longer unique.

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arXiv2019

Quasars from LSST as dark energy tracers: first steps

In the near future, new surveys promise a significant increase in the number of quasars (QSO) at large redshifts. This will help to constrain the dark energy models using quasars. The Large Synoptic Survey Telescope (LSST) will cover over 10 million QSO in six photometric bands during its 10-year run. QSO will be monitored and subsequently analyzed using the photometric reverberation mapping (RM) technique. In low-redshift quasars, the combination of reverberation-mapped and spectroscopic results have provided important progress. However, there are still some facts which have to be taken into account for future results. It has been found that super-Eddington sources show time delays shorter than the expected from the well-known Radius-Luminosity ($R\mathrm{_{Hβ}}-L_{5100}$) relation. Using a sample of 117 H$β$ reverberation-mapped AGN with $0.02<z<0.9$, we propose a correction by the accretion rate effect recovering the classical $R\mathrm{_{Hβ}}-L_{5100}$ relation. We determined the cosmological constants, which are in agreement with $Λ$-Cold Dark Matter model within 2$σ$ confidence level, which is still not suitable for testing possible departures from the standard model. Upcoming LSST data will decrease the uncertainties in the dark energy determination using reverberation-mapped sources, particularly at high redshifts. We show the first steps in the modeling of the expected light curves for H$β$ and MgII.

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arXiv2019

Can reverberation-measured quasars be used for cosmology?

Quasars have been proposed as a new class of standard candles analogous to Supernovae, as their large redshift range and high luminosities make them excellent candidates. Reverberation mapping (RM) method enables to estimate the distance to the source from the time delay measurement of the emission lines with respect to the continuum, since the time delay depends on the absolute luminosity of the source. This radius-luminosity (RL) relation exhibits a low scatter and it was thus proposed for cosmological purposes. However, in recent years the increase in the studied sample, and in particular the inclusion of highly accreting QSO has increased the dispersion in the RL relation, with many objects showing time delays shorter than the expected. Using H$β$ RM measurements for 117 sources with $0.002<z<0.9$ and $41.5<\mathrm{log}~L_{5100}<45.9$, we find a correction for the time delay based on the dimensionless accretion rate considering the virial factor anti-correlated with the FWHM of H$β$. This correction decreases the scattering in the accretion parameters compared with the typical value used, which is directly reflected by suppressing the RL relation dispersion. We also confirm the anti-correlation between the excess of variability and the accretion parameters. With this correction we are able to build the Hubble diagram and estimate the cosmological constants $Ω_m$ and $Ω_Λ$, which are consistent with the standard cosmological model at 2$σ$ confidence level. Therefore, RM results can be used to constrain cosmological models in the future.

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arXiv2019

Radio spectral index distribution of SDSS-FIRST sources across optical diagnostic diagrams

A detailed understanding of how the activity of a galactic nucleus regulates the growth of its host is still missing. To understand the activity and the types of accretion of supermassive black holes in different hosts, it is essential to study radio-optical properties of a large sample of extragalactic sources. In particular, we aim at studying the radio spectral index trends across the optical emission line diagnostic diagrams to search for potential (anti)correlations. To this goal, we combine flux densities from the radio FIRST survey at $1.4\,{\rm GHz}$ (with the flux density range $10\,{\rm mJy} \leq F_{1.4} \leq 1000\,{\rm mJy}$) for 209 SDSS sources at intermediate redshift $(0.04\leq z \leq 0.4)$ with the Effelsberg radiotelescope measurements at $4.85\,{\rm GHz}$ and $10.45\,{\rm GHz}$. The information about the optical emission-line ratios is obtained from the SDSS-DR7 catalogue. Using the Effelsberg data, we were able to infer the two-point radio spectral index distributions for star-forming galaxies, composite galaxies (with a combined contribution to the line emission from the star-formation and AGN activity), Seyferts, and low ionization narrow emission region (LINER) galaxies. While studying the distribution of steep, flat, and inverted sources across optical diagnostic diagrams, we found three distinct classes of radio emitters for our sample: (i) sources with steep radio index, high ionization ratio and high radio loudness, (ii) sources with flat radio index, lower ionization ratio and intermediate radio loudness, (iii) sources with inverted radio index, low ionization ratio and low radio loudness. The classes (i), (ii), (iii) cluster mainly along the transition from Seyfert to LINER sources in the BPT diagram. We interpret these groups as a result of the recurrent nuclear-jet activity.

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arXiv2019

Reverberation-mapping of distant quasars: time-lag determination using different methods

By applying different statistically robust methods, we analyze the time-lag between the continuum and ionized line-emission (Mg II line) light curves for the distant bright quasar CTS C$30.10$ (redshift $z \sim 0.9$). The data were obtained by the 10-meter SALT telescope in South Africa. In detail, we demonstrate the application of several methods using interpolated cross-correlation function (ICCF), discrete correlation function (DCF), z-transformed discrete correlation function (zDCF), von Neumann estimator, and the JAVELIN code package. In particular, we discuss the uncertainties of these methods. In conclusion, we find that the quasar lies on the broad-line region (BLR) size -- monochromatic luminosity power-law scaling, $R_{\rm BLR}\propto L_{5100}^{1/2}$, which was already confirmed for low-redshift sources. In case the BLR size-luminosity relation holds for other distant sources, quasars could be used for probing cosmological models as ``standard candles'' complementary to supernovae Ia.

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arXiv2019

Constraining the charge of the Galactic centre black hole

In this contribution, we summarize our results concerning the observational constraints on the electric charge associated with the Galactic centre black hole - Sgr A*. According to the no-hair theorem, every astrophysical black hole, including supermassive black holes, is characterized by at most three classical, externally observable parameters - mass, spin, and the electric charge. While the mass and the spin have routinely been measured by several methods, the electric charge has usually been neglected, based on the arguments of efficient discharge in astrophysical plasmas. From a theoretical point of view, the black hole can attain charge due to the mass imbalance between protons and electrons in fully ionized plasmas, which yields about $\sim 10^8\,{\rm C}$ for Sgr A*. The second, induction mechanism concerns rotating Kerr black holes embedded in an external magnetic field, which leads to electric field generation due to the twisting of magnetic field lines. This electric field can be associated with the induced Wald charge, for which we calculate the upper limit of $\sim 10^{15}\,{\rm C}$ for Sgr A*. Although the maximum theoretical limit of $\sim 10^{15}\,{\rm C}$ is still 12 orders of magnitude smaller than the extremal charge of Sgr A*, we analyse a few astrophysical consequences of having a black hole with a small charge in the Galactic centre. Two most prominent ones are the effect on the X-ray bremsstrahlung profile and the effect on the position of the innermost stable circular orbit.

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arXiv2018

On the charge of the Galactic centre black hole

The Galactic centre supermassive black hole (SMBH), in sharp contrast with its complex environment, is characterized by only three classical parameters -- mass, spin, and electric charge. Its charge is poorly constrained. It is, however, usually assumed to be zero because of neutralization due to the presence of plasma. We revisit the question of the SMBH charge and put realistic limits on its value, timescales of charging and discharging, and observable consequences of the potential, small charge associated with the Galactic centre black hole. The electric charge due to classical arguments based on the mass difference between protons and electrons is $\lesssim 10^9\,{\rm C}$ and is of a transient nature on the viscous time-scale. However, the rotation of a black hole in magnetic field generates electric field due to the twisting of magnetic field lines. This electric field can be associated with induced charge, for which we estimate an upper limit of $\lesssim 10^{15}\,{\rm C}$. Moreover, this charge is most likely positive due to an expected alignment between the magnetic field and the black-hole spin. Even a small charge of this order significantly shifts the position of the innermost stable circular orbit (ISCO) of charged particles. In addition, we propose a novel observational test based on the presence of the bremsstrahlung surface brightness decrease, which is more sensitive for smaller unshielded electric charges than the black-hole shadow size. Based on this test, the current upper observational limit on the charge of Sgr A* is $\lesssim 3\times 10^{8}\,{\rm C}$.

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arXiv2018

A stellar fly-by close to the Galactic centre: Can we detect stars on highly-relativistic orbits?

The Galactic centre Nuclear Star Cluster is one of the densest stellar clusters in the Galaxy. The stars in its inner portions orbit the supermassive black hole associated with compact radio source Sgr~A* at the orbital speeds of several thousand km/s. The B-type star S2 is currently the best case to test the general relativity as well as other theories of gravity based on its stellar orbit. Yet its orbital period of $\sim 16\,{\rm yr}$ and the eccentricity of $\sim 0.88$ yields the relativistic pericentre shift of $\sim 11'$, which is observationally still difficult to reliably measure due to possible Newtonian perturbations as well as reference-frame uncertainties. A naive way to solve this problem is to find stars with smaller pericentre distances, $r_{\rm p}\lesssim 1529$ Schwarzschild radii ($120\,{\rm AU}$), and thus more prominent relativistic effects. In this contribution, we show that to detect stars on relativistic orbits is progressively less likely given the volume shrinkage and the expected stellar density distributions. Finally, one arrives to a sparse region, where the total number of bright stars is expected to fall below one. One can, however, still potentially detect stars crossing this region. In this contribution, we provide a simple formula for the detection probability of a star crossing a sparse region. We also examine an approximate time-scale on which the star reappears in the sparse region, i.e. a "waiting" time-scale for observers.

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