Depletion of Bright Red Giants in the Galactic Center during Its Active Phases

The Galactic center supermassive black hole (hereafter SMBH) with the mass of 4.1 × 10⁶ M_⊙ is located at the distance of 8.1 kpc (Boehle et al. 2016; Gillessen et al. 2017; Parsa et al. 2017; Gravity Collaboration et al. 2018) and provides a unique laboratory for studying detailed dynamical processes and the mutual interaction between a nuclear star cluster (NSC) and a central massive black hole (Alexander 2005; Genzel et al. 2010; Eckart et al. 2017; Ali et al. 2020) as well as with the multiphase gaseous-dusty circumnuclear medium (Morris & Serabyn 1996; Różańska et al. 2017). The compact radio source Sgr A* associated with the SMBH is embedded in the Milky Way NSC, which is one of the densest stellar systems in the Galaxy (Alexander 2017; Schödel et al. 2014), and in addition, it is surrounded by an ionized, neutral, and molecular gas and dust (see, e.g., Moser et al. 2017 and references therein).

The NSC consists of both late-type (red giants, supergiants and asymptotic giant branch stars) and early-type stars of O and B spectral classes (Krabbe et al. 1991; Buchholz et al. 2009; Do et al. 2009; Gallego-Cano et al. 2018), which imply star formation during the whole Galactic history, albeit most likely episodic (Pfuhl et al. 2011; Schödel et al. 2020) with the star formation peak at 10 Gyr, the minimum at 1–2 Gyr, and a recent increase in the last few hundred million years.

In the innermost parsec of the Galactic center, there is a surprising abundance of young massive OB/Wolf–Rayet stars that have formed in situ in the last 10 million years (Ghez et al. 2003). These young stars form an unrelaxed cusp-like distribution. On the other hand, previous studies of the distribution of late-type stars showed that they exhibit a core-like distribution inside the inner ∼0.5 pc, which has a projected flat or even decreasing profile toward the center (Buchholz et al. 2009; Do et al. 2009; Sellgren et al. 1990). More recent studies provided a precise analysis of the distribution of late-type stars because of increasing their sensitivity toward larger magnitudes, i.e., fainter giants (Gallego-Cano et al. 2018; Habibi et al. 2019). Using photometric number counts and diffuse light analysis, Gallego-Cano et al. (2018) found that fainter late-type stars with magnitudes of K ≈ 18 exhibit a cusp-like distribution within the sphere of influence of Sgr A* with a 3D power-law exponent of γ ≃ 1.43. In comparison, there is an apparent lack of bright red giants with K = 12.5–16 at the projected radii of ≲0.3 pc from Sgr A*. Gallego-Cano et al. (2018) estimate the number of missing giants to 100 for this distance range. The study of Habibi et al. (2019) also finds a cusp-like distribution for late-type stars with K < 17 within 0.02–0.4 pc. In agreement with Gallego-Cano et al. (2018), they found a core-like distribution for the brightest giants with K < 15.5, although the number of missing giants appears to be lower than 100 according to their analysis. Although the surface-brightness distribution of late-type stars brighter than 15.5 mag appears to be rather flat, already in Figure 11 of Buchholz et al. (2009) the inner point at 05 (where 1'' ∼ 0.04 pc at the Galactic center) of the distribution of late-type stars as well as of the distribution of all stars indicates the presence of a cusp.

In summary, there appears to be an internal mechanism within the NSC that preferentially depleted bright, large red giants on one hand, which has led to their apparent core-like distribution, and at the same time has been less efficient for early-type as well as fainter late-type stars, on the other hand. Such a mechanism has altered either the spatial, luminosity, or the temperature distribution of the bright red giant stars so that they effectively fall beyond the detection limit or they instead mimic younger, "bluer" stars. So far, it is mainly the following four mechanisms that have been discussed to explain the apparent lack of bright red giants:

• 1.  
  complete or partial tidal disruption of red giants by the SMBH (Hills 1975; Bogdanović et al. 2014; King 2020; envelope removal),
• 2.  
  envelope stripping by the collisions of red giants with the dense clumps within a self-gravitating accretion disk (Armitage et al. 1996; Amaro-Seoane & Chen 2014; Kieffer & Bogdanović 2016; Amaro-Seoane et al. 2020; envelope removal),
• 3.  
  collisions of red giants with field stars and compact remnants (Phinney 1989; Morris 1993; Genzel et al. 1996; Bailey & Davies 1999; Alexander 2005; Davies & King 2005; Dale et al. 2009; envelope removal),
• 4.  
  mass segregation effects: the dynamical effect of a secondary massive black hole (Baumgardt et al. 2006; Merritt & Szell 2006; Portegies Zwart et al. 2006; Matsubayashi et al. 2007; Löckmann & Baumgardt 2008; Gualandris & Merritt 2012) or of an infalling massive cluster (Kim & Morris 2003; Ernst et al. 2009; Antonini et al. 2012) or of stellar black holes (Morris 1993; altered spatial distribution),

where in parentheses we include the mechanism responsible for altering the population of late-type stars. The importance of star–star and star–disk interactions was also analyzed generally for active galactic nuclei (AGNs) in terms of the effects on the accretion disk and broad-line region structure as well as the NSC orbital distribution (Zurek et al. 1994; Armitage et al. 1996; Karas & Šubr 2001; Vilkoviskij & Czerny 2002; Kieffer & Bogdanović, 2016; MacLeod & Lin 2020).

We propose here another mechanism based on the jet–star interactions (Barkov et al. 2012a; Araudo et al. 2013; Araudo & Karas 2017), which most likely coexisted with the mechanisms proposed above. In particular, the star–accretion disk collisions are expected to be accompanied by star–jet crossings during previous active phases of Sgr A*. Because red giant stars have typically large, loosely bound tenuous envelopes, dense compact cores, and slow winds with the terminal velocity $\lesssim 100\,\mathrm{km}\,{{\rm{s}}}^{-1}$, they are in particular susceptible to mass removal in encounters with higher-pressure material (MacLeod et al. 2012; Amaro-Seoane & Chen 2014; Kieffer & Bogdanović 2016).⁶ Therefore, during the red giant–jet interactions, the jet ram pressure will remove the outer layers of the stellar envelope during the passage. We illustrate this idea in Figure 1.

Zoom In Zoom Out Reset image size

After several star–jet crossings, the atmosphere is removed similarly as for repetitive star–disk crossings (Amaro-Seoane et al. 2020), and the giant is modified in a way that it follows an evolutionary track in the Hertzsprung–Russell (HR) diagram approximately along the constant absolute magnitude toward higher effective temperatures. We show that this mechanism quite likely operated in the vicinity of Sgr A* during its active Seyfert-like phase in the past few million years (Bland-Hawthorn et al. 2019) when the jet kinetic luminosity could have reached $\sim {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. In principle, even in the quiescent state, a tidal disruption event (TDE) every ∼10⁴ yr, which can be estimated for the Galactic center (Syer & Ulmer 1999; Alexander 2005), can temporarily reactivate the jet of Sgr A*, and some of the bright red giants could be depleted during its existence. This makes the red giant–jet interaction in the Galactic center relevant and highly plausible in its recent history, and the dynamical consequences can be inferred based on the so-far detected traces of the past active phase of Sgr A* as well as the currently observed stellar density distribution.

Previous jet–star interaction studies were focused on the emergent nonthermal radiation, in particular in the gamma-ray domain, and mass-loading and chemical enrichment of jets by stellar winds (see e.g., Komissarov 1994; Barkov et al. 2012a; Bosch-Ramon et al. 2012; Araudo et al. 2013; Bednarek & Banasiński 2015; de la Cita et al. 2016). Here we focus on the effect of the jet on the stellar population. In most jetted AGNs, this is perhaps a secondary problem because stellar populations in the host bulge cannot be resolved out, i.e., one can only analyze the integrated starlight. In contrast, within the Galactic center NSC, one can not only disentangle late- and early-type stars, but it is also possible to study their distribution as well as the kinematics of individual stars. Although in the current low-luminosity state there is no firm evidence for the presence of a relativistic jet, there are nowadays several multiwavelength signatures of the past active Seyfert-like state of Sgr A* that occurred a few million years ago (Bland-Hawthorn et al. 2019; Heywood et al. 2019; Ponti et al. 2019). However, even in the current quiescent state of Sgr A*, studies by Yusef-Zadeh et al. (2012) and Li et al. (2013) indicate the existence of a low-surface-brightness parsec-scale jet. In addition, the presence of the cometary-shaped infrared-excess bow-shock sources X3, X7 (Mužić et al. 2010), and recently X8 (Peißker et al. 2019) indicates that the star–outflow interaction is ongoing even in a very low state of the Sgr A* activity. The morphology of these sources can be explained by the interaction with a strong accretion wind originating from Sgr A* or with the collective wind of the cluster of young stars.

This paper proposes a new mechanism that could have affected the current population of bright late-type stars, namely their appearance as well as number counts in specific magnitude bins, in the Galactic center. We apply analytical and semianalytical calculations to assess whether the potential past jet–star interactions could have had an effect on the stellar population in the sphere of influence of Sgr A*. Although the analytical calculations introduce several simplifications, we show that the mechanism could have operated and the estimated number of affected red giants is in accordance with the most sensitive, up-to-date studies (Gallego-Cano et al. 2018; Habibi et al. 2019). A more detailed computational treatment including magnetohydrodynamic numerical simulations as well as a stellar evolution will be presented in our future studies.

The paper is structured as follows. In Section 2, we derive the stagnation radius, basic timescales, and the envelope mass removed for red giant stars interacting with the jet of Sgr A* during its past active phase. In Section 3, we discuss the observational signatures in the near-infrared domain. Subsequently, in Section 4, we estimate the number of red giants that could be affected by the jet interaction and visually depleted from the immediate vicinity of Sgr A*. In Section 6, we discuss additional processes related to the red giant–jet interaction in the Galactic center. Finally, we summarize the main results and conclude with Section 7.

The evidence for the active phase of Sgr A* that is estimated to have occurred 4 ± 1 Myr ago is based on the X-ray/γ-ray bubbles with a total energy content of 10⁵⁶–10⁵⁷ erg (Bland-Hawthorn et al. 2019). The first evidence for the nuclear outburst was the kiloparsec-scale 1.5 keV ROSAT X-ray emission that originated in the Galactic center (Bland-Hawthorn & Cohen 2003). The X-ray structure coincides well with the more recently discovered Fermi γ-ray bubbles extending 50^◦ north and south of the Galactic plane at 1–100 GeV (Su et al. 2010). The X-ray/γ-ray bubbles are energetically consistent with the nuclear AGN-like activity associated with the jet and/or disk-wind outflows with jet power ${L}_{{\rm{j}}}={2.3}_{-0.9}^{+5.1}\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and age ${4.3}_{-1.4}^{+0.8}\,\mathrm{Myr}$ (Miller & Bregman 2016). In comparison, the starburst origin of the Fermi bubbles is inconsistent with the bubble energetics by a factor of ∼100 (Bland-Hawthorn & Cohen 2003). On intermediate scales of hundreds of parsecs, the base of the Fermi bubbles coincides with the bipolar radio bubbles (Heywood et al. 2019) as well as with the X-ray chimneys (Ponti et al. 2019).

Using hydrodynamic simulations, Guo & Mathews (2012) reproduce the basic radiative characteristics of the Fermi bubbles with the AGN jet duration of ∼0.1–0.5 Myr, which corresponds to the jet luminosity ${L}_{{\rm{j}}}\approx {10}^{56-57}\,\mathrm{erg}/(0.1-0.5\,\mathrm{Myr})\,=6.3\times {10}^{42}-3.2\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The jet is dominated by the kinetic luminosity ${L}_{{\rm{j}}}={\eta }_{{\rm{j}}}{L}_{\mathrm{acc}}$, where ${\eta }_{{\rm{j}}}$ is the conversion efficiency from the accretion luminosity L_acc to the jet kinetic luminosity. The accretion luminosity is ${L}_{\mathrm{acc}}\lesssim {L}_{\mathrm{Edd}}$, where the Eddington luminosity is

$$\begin{eqnarray}&&{L}_{\mathrm{Edd}}=5.03\times {10}^{44}\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\,{M}_{\odot }}\right)\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 1 }$$

and η_j < 0.7 for most radio sources (Ito et al. 2008). This yields the maximum jet kinetic luminosity for Sgr A* of ${L}_{{\rm{j}}}\approx 3.5\,\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. We will consider ${L}_{{\rm{j}}}\approx {10}^{41}-{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, where the lower limit is given by the putative jet present in the current quiescent state (Yusef-Zadeh et al. 2012), with the inferred kinetic luminosity of ${L}_{\min }\sim 1.2\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, and the upper limit is given by the Eddington luminosity.

We assume a conical jet with a half-opening angle θ and width ${R}_{{\rm{j}}}=z\tan \theta$, where z is the distance to Sgr A*. The jet footpoint for Sgr A* can be estimated to be located at ${z}_{0}\sim 2\,\times {10}^{-5}({M}_{\bullet }/4\times {10}^{6}\,{M}_{\odot })\mathrm{pc}=52{R}_{\mathrm{Schw}}$ (Junor et al. 1999), where ${R}_{\mathrm{Schw}}=2{{GM}}_{\bullet }/{c}^{2}=3.8\times {10}^{-7}({M}_{\bullet }/4\times {10}^{6}\,{M}_{\odot })\,\mathrm{pc}$ is the Schwarzschild radius. Any red giant or supergiant with radius⁷ R_⋆ and mass m_⋆ is not expected to plunge below z₀ as the tidal disruption radius ${r}_{{\rm{t}}}={R}_{\star }{\left(2{M}_{\bullet }/{m}_{\star }\right)}^{1/3}$ (Hills 1975; Rees 1988) is at least a factor of 2 larger,

$$\begin{eqnarray}&&\displaystyle \frac{{r}_{{\rm{t}}}}{{R}_{\mathrm{Schw}}}=1\,20\left(\displaystyle \frac{{R}_{\star }}{10\,{R}_{\odot }}\right){\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{\displaystyle \frac{1}{3}}{\left(\displaystyle \frac{{m}_{\star }}{{M}_{\odot }}\right)}^{-\tfrac{1}{3}}.\end{eqnarray} \tag{ 2 }$$

${R}_{\star }=10\,{R}_{\odot }$ and m_⋆ = 1 M_⊙ are typical intermediate values for the evolved late-type giants with extended envelopes (Merritt 2013). For numerical estimates, we consider the range of radii for red giants and supergiants, ${R}_{\star }\sim 4\mbox{--}1000\,{R}_{\odot }$, as indicated by the HR diagram. These late-type stars have a large range of bolometric luminosities, L_⋆ ∼ 10–72,000L_⊙, and the temperature range of T_⋆ ∼ 5000–3000 K, which corresponds to the spectral classes K and M, respectively. The K-band magnitude range is K ∼ 15.2 mag for R_⋆ = 4 R_⊙, T_⋆ = 5000 K, and K ∼ 4.4 mag for R_⋆ = 1000 R_⋆, T_⋆ = 3000 K. More specifically, we focus on late-type stars of K = 16 mag and brighter, which appear to form a core-like density distribution in the central 0.5 pc. Using the isochrones obtained with the Parsec code (Bressan et al. 2012), these stars have R_⋆ ∼ 4 R_⊙ and larger for an age of 5 Gyr. The late-type stars that are completely absent in the S cluster (inner ∼0.04 pc) were inferred to have R_⋆ = 30 R_⊙ and larger (Habibi et al. 2019). Therefore, numerical estimates are typically scaled to R_⋆ = 30 R_⊙ unless otherwise indicated.

We will further focus on the region between the tidal radius of red giants and the outer edge of the S cluster, which approximately corresponds to the Bondi radius of the hot bremsstrahlung plasma (Baganoff et al. 2003; Wang et al. 2013),

$$\begin{eqnarray}&&{R}_{{\rm{B}}}=\displaystyle \frac{{{GM}}_{\bullet }}{{c}_{{\rm{s}}}^{2}}=0.12\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\,{M}_{\odot }}\right){\left(\displaystyle \frac{{T}_{{\rm{p}}}}{{10}^{7}{\rm{K}}}\right)}^{-1}\,\mathrm{pc},\end{eqnarray} \tag{ 3 }$$

which is two to three orders of magnitude larger than the tidal radius of red giants, ${R}_{{\rm{B}}}\sim 3.1\times {10}^{5}\,{R}_{\mathrm{Schw}}$. More generally speaking, the population of predominantly B-type stars lies within the innermost arcsecond (∼0.04 pc, S cluster), while the population of young massive OB/Wolf–Rayet stars stretches from ∼0.04 pc up to ∼0.5 pc, a fraction of which forms a warped stellar disk (Genzel et al. 2010).

The stellar-wind ram pressure at a distance r from the center of the star can be estimated as ${P}_{\mathrm{sw}}\sim {\rho }_{{\rm{w}}}{v}_{{\rm{w}}}^{2}={\dot{m}}_{{\rm{w}}}{v}_{{\rm{w}}}/(4\pi {r}^{2})$, where ${\dot{m}}_{{\rm{w}}}$ is the mass-loss rate and v_w is the terminal wind velocity. Using typical values for red giants with ${\dot{m}}_{{\rm{w}}}\approx {10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ and ${v}_{{\rm{w}}}\approx 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (e.g., Reimers 1987; de la Cita et al. 2016),⁸

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{sw}} & = & 0.012\left(\displaystyle \frac{{\dot{m}}_{{\rm{w}}}}{{10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right)\left(\displaystyle \frac{{v}_{{\rm{w}}}}{10\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)\\ & & \times {\left(\displaystyle \frac{r}{30{R}_{\odot }}\right)}^{-2}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}.\end{array}\end{eqnarray} \tag{ 4 }$$

The ram pressure of a relativistic jet with bulk motion Lorentz factor Γ is ${P}_{{\rm{j}}}={\rm{\Gamma }}{\rho }_{{\rm{j}}}{v}_{{\rm{j}}}^{2}$, where the jet density is ${\rho }_{{\rm{j}}}={L}_{{\rm{j}}}/[({\rm{\Gamma }}-1){c}^{2}{\sigma }_{{\rm{j}}}{v}_{{\rm{j}}}]$ and ${\sigma }_{{\rm{j}}}=\pi {R}_{{\rm{j}}}^{2}$ is the jet cross-sectional area. By assuming v_j ∼ c and Γ ∼ 10, the jet kinetic pressure for Sgr A* can be written as

$$\begin{eqnarray}\begin{array}{rcl}{P}_{{\rm{j}}} & \approx & \displaystyle \frac{{L}_{{\rm{j}}}}{{\sigma }_{{\rm{j}}}c}=\displaystyle \frac{{L}_{{\rm{j}}}}{\pi {{cz}}^{2}{\tan }^{2}\theta }\\ & = & 0.014\left(\displaystyle \frac{{L}_{{\rm{j}}}}{{10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{z}{0.04\mathrm{pc}}\right)}^{-2}\,\mathrm{erg}\,{\mathrm{cm}}^{-3},\end{array}\end{eqnarray} \tag{ 5 }$$

where we have assumed θ ≃ 125, which corresponds to the jet sheath half-opening angle estimated for the current candidate jet of Sgr A* (Li et al. 2013). Note that the innermost arcsecond (z ∼ 0.04 pc) is also the outer radius of fast-moving stars in the S cluster (Genzel et al. 2010; Eckart et al. 2017).

By equating P_sw = P_j, we obtain the stagnation distance

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{R}_{\mathrm{stag}}}{{R}_{\odot }} & = & 27\left(\displaystyle \frac{z}{0.04\,\mathrm{pc}}\right){\left(\displaystyle \frac{{\dot{m}}_{{\rm{w}}}}{{10}^{-8}{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{\displaystyle \frac{1}{2}}\\ & & \times {\left(\displaystyle \frac{{v}_{{\rm{w}}}}{10\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{\displaystyle \frac{1}{2}}{\left(\displaystyle \frac{{L}_{{\rm{j}}}}{{10}^{42}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-\tfrac{1}{2}}\,,\end{array}\end{eqnarray} \tag{ 6 }$$

which characterizes by how much the red giant envelope can be ablated by the jet in one encounter. Note that ${R}_{\mathrm{stag}}\lt {R}_{\star }$ for late-type giants and supergiants with R_⋆ ∼ 100–1000 R_⊙ (see Figure 2). The very tenuous wind of red giants cannot balance the jet ram pressure, and therefore, the jet plasma impacts on the stellar surface. As a consequence, a fraction of the stellar envelope is removed as estimated by Equation (6).

Zoom In Zoom Out Reset image size

Interestingly, giant stars with ${R}_{\star }\gtrsim 30\,{R}_{\odot }$ appear to be missing in the S cluster. Only late-type stars with R_⋆ between 4 and 30 R_⊙ (with absolute bolometric magnitudes between −1.05 and 3.32, respectively, for the effective temperature of 4000 K) were detected by Habibi et al. (2019; see their Figure 2). Stars with 4 ≤ R_⋆/R_⊙ ≤ 30 within 0.02 pc have R_stag/R_⋆ ∼ 1 when $2\times {10}^{41}\leqslant {L}_{{\rm{j}}}/\mathrm{erg}\,{{\rm{s}}}^{-1}\leqslant 1.1\times {10}^{43}$, as indicated in Figure 2. This is in agreement with the estimated jet power ${L}_{{\rm{j}}}\sim 2.3\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ from X- and γ-ray bubbles (Miller & Bregman 2016). Therefore, an apparent lack of late-type giant stars with envelopes ${R}_{\star }\gtrsim 30\,{R}_{\odot }$ in the inner ∼0.04 pc of Galactic center could result from the jet-induced ablation of the stellar envelope during the last active phase of Sgr A*, a few million years ago.

2.1. Basic Timescales of the Jet–Star Interaction

The red giant will enter the jet and not mix with its sheath layers on the surface if v_orb ≳ v_sc, where ${v}_{\mathrm{orb}}\sim {\left({{GM}}_{\bullet }/z\right)}^{1/2}$ is the Keplerian orbital velocity of the star around Sgr A* and ${v}_{\mathrm{sc}}\sim c({\rm{\Gamma }}{\rho }_{{\rm{j}}}/{\rho }_{\star })$ is the sound speed inside the shocked obstacle. This condition can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{\star }}{{\rho }_{{\rm{j}}}}\gtrsim 5.2\times {10}^{5}\left(\displaystyle \frac{z}{0.01\,\mathrm{pc}}\right)\left(\displaystyle \frac{{\rm{\Gamma }}}{10}\right){\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{-1},\end{eqnarray} \tag{ 7 }$$

which means that stellar atmosphere layers of comparable density or greater than indicated by Equation (7) will enter the jet, and the less dense upper layers will mix with the jet surface layers.

Once inside, the bow shock is formed inside the jet on the very short timescale of ${t}_{\mathrm{bs}}\sim {R}_{\star }/c\sim 232({R}_{\star }/100\,{R}_{\odot }){\rm{s}}$. A shock also propagates through the red-giant atmosphere on the shock-crossing or dynamical timescale, t_d ∼ R_⋆/v_sc, whose lower limit is imposed by the condition of penetration, v_orb ≳ v_sc, which leads to ${t}_{{\rm{d}}}\gtrsim {R}_{\star }/{v}_{\mathrm{orb}}={R}_{\star }\sqrt{z/({{GM}}_{\bullet })}\,\sim 53063({R}_{\star }/100\,{R}_{\odot }){\left(z/0.01\mathrm{pc}\right)}^{1/2}{\left({M}_{\bullet }/4\times {10}^{6}{M}_{\odot }\right)}^{-1/2}\,{\rm{s}}$. The dynamical, shock-crossing time is at least ∼10 times longer than the bow-shock formation time close to the footpoint of the jet, but the ratio becomes larger with the distance from Sgr A* as ${t}_{{\rm{d}}}/{t}_{\mathrm{bs}}\gtrsim 229{\left(z/0.01\mathrm{pc}\right)}^{1/2}{\left({M}_{\bullet }/4\times {10}^{6}\right)}^{-1/2}$.

The star-crossing time through the jet can be estimated as ${t}_{\star }\sim 2{R}_{{\rm{j}}}/{v}_{\mathrm{orb}}$. Using ${R}_{{\rm{j}}}=z\tan \theta$ and the condition ${v}_{\mathrm{sc}}\lesssim {v}_{\mathrm{orb}}$, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{t}_{\star }}{{t}_{{\rm{d}}}}\lesssim \displaystyle \frac{2z\tan \theta }{{R}_{\star }}\sim 1966,\end{eqnarray} \tag{ 8 }$$

which implies that the shock propagates throughout the detached envelope, which is dragged by the jet and mixed with its material. Eventually, after several t_d, the envelope material will reach the velocity of v_j ∼ c. Note that t_⋆/t_d ∼ 1 when z ∼ 10⁻⁵ pc; hence, the removed envelope material should be dragged by the jet throughout the whole NSC.

The ablated red giant after the first crossing through the jet would first expand adiabatically to the original size on the thermal expansion timescale ${t}_{\exp }\sim {R}_{\star }/{c}_{{\rm{s}}}={R}_{\star }\sqrt{\mu {m}_{{\rm{H}}}/({k}_{{\rm{B}}}{T}_{\mathrm{atm}}})\,=0.2({R}_{\star }/100\,{R}_{\odot }){\left({T}_{\mathrm{atm}}/{10}^{4}{\rm{K}}\right)}^{-1/2}\,\mathrm{yr}$ because of the pressure of the warmer, underlying layers as the star adjusts its size to reach a hydrodynamic equilibrium. This expansion timescale is shorter than the orbital timescale

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{orb}} & = & 2\pi {\left(\displaystyle \frac{{z}^{3}}{{{GM}}_{\bullet }}\right)}^{1/2}\\ & = & 47{\left(\displaystyle \frac{z}{0.01\mathrm{pc}}\right)}^{3/2}{\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{-1/2}\,\mathrm{yr}.\end{array}\end{eqnarray} \tag{ 9 }$$

Kieffer & Bogdanović (2016) infer a similar timescale for the envelope expansion using the hydrodynamic simulations of red giant–accretion-clump collisions. According to their Figure 7, the envelope expands to a larger size than the original stellar radius in t_exp ∼ 1.5t_dyn after the star emerges from the accretion clump, where t_dyn is a dynamical timescale of the star,

$$\begin{eqnarray}&&{t}_{\mathrm{dyn}}\sim 0.32{\left(\displaystyle \frac{{R}_{\star }}{100{R}_{\odot }}\right)}^{3/2}{\left(\displaystyle \frac{{m}_{\star }}{1{M}_{\odot }}\right)}^{-1/2}\,\mathrm{yr},\end{eqnarray} \tag{ 10 }$$

which leads to ${t}_{\exp }\sim 1.5\times 0.32\,\mathrm{yr}\approx 0.5\,\mathrm{yr}$.

The timescale of the thermal evolution of a star after the jet–star interaction is expressed by the Kelvin–Helmholtz (KH) or thermal timescale,

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{KH}} & \approx & \displaystyle \frac{{{Gm}}_{\star }^{2}}{{R}_{\star }{L}_{\star }}\\ & = & 210{\left(\displaystyle \frac{{m}_{\star }}{1{M}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{{R}_{\star }}{100{R}_{\odot }}\right)}^{-1}{\left(\displaystyle \frac{{L}_{\star }}{1500{L}_{\odot }}\right)}^{-1}\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 11 }$$

where we estimated the stellar luminosity using ${L}_{\star }\,=4\pi {R}_{\star }^{2}\sigma {T}_{\star }^{4}$ for T_⋆ = 3500–3700 K typical of red giants. For the whole range of stellar radii R_⋆ ∼ 4–1000 R_⊙ and stellar luminosities L_⋆ ∼ 10–7.2 × 10³ L_⊙, t_KH differs considerably—from ∼10⁶ yr for the smallest giants to ∼0.43 yr for the largest ones.

Based on the comparison between the time between jet–star collisions t_c = P_orb/2 and the KH timescale, one can distinguish cool colliders when t_c ≳ t_KH, i.e., the star had enough time to radiate away the accumulated collisional heat, and it cools down and shrinks before the next collision. For the case when t_c < t_KH, there is not enough time to radiate away the excess collisional heat, and the star is warmer and larger at the time of the subsequent collision—these are the so-called warm colliders. In the nuclear star cluster when the jet was active, there were both types of colliders with the approximate division given by t_c ≈ t_KH, which leads to

$$\begin{eqnarray}\begin{array}{rcl}{z}_{{\rm{c}}} & \approx & \displaystyle \frac{{{Gm}}_{\star }^{4/3}{M}_{\bullet }^{1/3}}{{R}_{\star }^{2/3}{L}_{\star }^{2/3}{\pi }^{2/3}}\\ & = & 0.04{\left(\displaystyle \frac{{m}_{\star }}{1{M}_{\odot }}\right)}^{4/3}{\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{1/3}\\ & & \times {\left(\displaystyle \frac{{R}_{\star }}{100{R}_{\odot }}\right)}^{-2/3}{\left(\displaystyle \frac{{L}_{\star }}{1500{L}_{\odot }}\right)}^{-2/3}\,\mathrm{pc}.\end{array}\end{eqnarray} \tag{ 12 }$$

The length scale z_c implies that red giants located inside the inner S cluster were collisionally heated up and bloated, which increased their mass removal during repetitive encounters with the jet. Stars orbiting at larger distances managed to cool down and shrink in size before the next collision, which has subsequently diminished their overall mass loss. However, note that Equation (12) is a function of the stellar parameters m_⋆, R_⋆, and L_⋆, hence z_c differs depending on the red giant stage and its mass. For the smallest late-type stars with R_⋆ ∼ 4 R_⊙ and L_⊙ ∼ 8.9 L_⊙, z_c ∼ 10.5 pc; therefore, they can be classified as warm colliders throughout the nuclear star cluster. On the other hand, the late-type supergiants with R_⋆ ∼ 10³ R_⊙ and L_⊙ ∼ 7.2 × 10⁴ L_⊙ have z_c ∼ 0.7 mpc, hence they can be classified as cool colliders beyond milliparsec distances.

2.2. Jet-induced Envelope Removal

The stellar evolution after a jet–star encounter is generally complicated given that the envelopes of red giants become bloated after the first passage through the jet. This is because of the pressure of lower, hotter layers and their subsequent nearly adiabatic expansion, which can make the red giant even larger and brighter (Kieffer & Bogdanović 2016). Note that the number of encounters n_cross = 2t_jet/P_orb, where t_jet ∼ 0.5 Myr is the jet lifetime, is typically ≫1. In particular,

$$\begin{eqnarray}&&{n}_{\mathrm{cross}}\sim 2\times {10}^{4}\left(\displaystyle \frac{{t}_{\mathrm{jet}}}{0.5\,\mathrm{Myr}}\right){\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{\displaystyle \frac{1}{2}}{\left(\displaystyle \frac{z}{0.01\mathrm{pc}}\right)}^{-\tfrac{3}{2}}.\end{eqnarray} \tag{ 13 }$$

The number of encounters is also at least two orders of magnitude larger than the one expected from the star–accretion-clump interaction investigated by Kieffer & Bogdanović (2016) and Amaro-Seoane et al. (2020). After the first passage, the bloated red giant has an even larger cross section than before the encounter, which increases the mass removed during subsequent encounters (Armitage et al. 1996; Kieffer & Bogdanović 2016). The jet–star interaction phase proceeds during the red giant lifetime, which is t_rg ∼ 10⁸ yr (MacLeod et al. 2012). During the red giant phase, there are ${n}_{\mathrm{orb}}={t}_{\mathrm{rg}}/{P}_{\mathrm{orb}}\sim 2.1\,\times {10}^{6}{\left(z/0.01\mathrm{pc}\right)}^{-3/2}$ orbits around Sgr A*, out of which ${n}_{\mathrm{cross}}/{n}_{\mathrm{orb}}=2{t}_{\mathrm{jet}}/{t}_{\mathrm{rg}}\sim 1 \%$ involve the interaction with the jet, assuming there was only one period of increased activity of Sgr A* in the last 100 million years. This ensures that the repetitive jet–red giant interaction leads to substantial mass loss, and the upper layer of the envelope is eventually removed.

The mass removal in a single passage due to atmosphere ablation ΔM₁ can be estimated through the balance of the jet ram force and the gravitational force acting on the envelope, i.e., ${P}_{{\rm{j}}}\pi {R}_{\star }^{2}\simeq G{\rm{\Delta }}{M}_{1}{m}_{\star }/{R}_{\star }^{2}$, giving (Barkov et al. 2012a)

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{\Delta }}{M}_{1}^{\max }}{{M}_{\odot }} & \approx & 4\times {10}^{-10}\left(\displaystyle \frac{{L}_{{\rm{j}}}}{{10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{R}_{\star }}{100{R}_{\odot }}\right)}^{4}\\ & & \times {\left(\displaystyle \frac{z}{0.04\mathrm{pc}}\right)}^{-2}{\left(\displaystyle \frac{\theta }{0.22}\right)}^{-2}{\left(\displaystyle \frac{{m}_{\star }}{{M}_{\odot }}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 14 }$$

In Figure 3, we plot ΔM₁. Note that ΔM₁ ≪ m_⋆ and, therefore, the orbital dynamics is not significantly affected by a single passage through the jet. However, given that ${\rm{\Delta }}{M}_{1}\propto {R}_{\star }^{4}{z}^{-2}$, the mass loss can be about one-thousandth to one-hundredth of the mass of a star for the largest giants on the asymptotic giant branch with ${R}_{\star }\sim {10}^{3}\,{R}_{\odot }$ and distances an order of magnitude smaller than z ≃ 0.04 pc (see the yellowish region in Figure 3). In fact, the value of ${\rm{\Delta }}{M}_{1}=6\times {10}^{28}\,{\rm{g}}=3\times {10}^{-5}\,{M}_{\odot }$ discussed by Barkov et al. (2012a) for powerful blazars in connection to their very high-energy γ-ray emission can be reached in the Galactic center for red giants with radii R_⋆ > 200 R_⊙ at z < 0.01 pc. Such a large mass-loss with a certain momentum with respect to the star can already have an effect on the orbital dynamics, taking into account repetitive encounters of the red giant with the jet. In other words, the mass removal takes place at the expanse of the kinetic energy of the star, which has implications for the dynamics of the NSC; see also Kieffer & Bogdanović (2016) for discussion. In addition, already for jets with a lower power corresponding to the active phase of the Galactic center, jet–red giant interactions can affect the short-term TeV emission in these sources. These effects are beyond the scope of the current paper and will be investigated in our future studies.

Zoom In Zoom Out Reset image size

The mass removal during a single jet encounter given by Equation (14) can be considered as an upper limit as we assume that the cross section of the star is given by its radius during the entire passage of the star through the jet, hence ${\rm{\Delta }}{M}_{1}^{\max }\approx {P}_{{\rm{j}}}\pi {R}_{\star }^{4}/({{Gm}}_{\star })$. However, this is only an approximation as realistically, during a few shock-crossing or dynamical timescales t_d, the ram pressure of the jet will shape the red giant and its detached envelope into a comet-like structure (see Figure 1) for which the interaction cross section is given by R_stag rather than by R_⋆, which gives us a lower limit on the mass removal, ${\rm{\Delta }}{M}_{1}^{\min }\approx {P}_{{\rm{j}}}\pi {R}_{\mathrm{stag}}^{4}/({{Gm}}_{\star })\leqslant {\rm{\Delta }}{M}_{1}^{\max }$. Using Equation (6), ${\rm{\Delta }}{M}_{1}^{\min }$ can be expressed in terms of the basic parameters of the star and the jet, and it can numerically be expressed by the same units as in Equations (14) and (6),

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{\Delta }}{M}_{1}^{\min }}{{M}_{\odot }} & \approx & \displaystyle \frac{c{({\dot{m}}_{{\rm{w}}}{v}_{{\rm{w}}}z\tan \theta )}^{2}}{16{{Gm}}_{\star }{L}_{{\rm{j}}}}\\ & = & 2.1\times {10}^{-12}{\left(\displaystyle \frac{{L}_{{\rm{j}}}}{{10}^{42}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1}{\left(\displaystyle \frac{{m}_{\star }}{{M}_{\odot }}\right)}^{-1}\\ & & \times {\left(\displaystyle \frac{{\dot{m}}_{{\rm{w}}}}{{10}^{-8}{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{2}{\left(\displaystyle \frac{{v}_{{\rm{w}}}}{10\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}{\left(\displaystyle \frac{z}{0.04\mathrm{pc}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 15 }$$

where we adopted θ ∼ 125 as before. In comparison with ${\rm{\Delta }}{M}_{1}^{\max }$ in Equation (14), which is proportional to z⁻², ${\rm{\Delta }}{M}_{1}^{\min }$ in Equation (15) increases as z². This implies that ${\rm{\Delta }}{M}_{1}^{\min }\leqslant {\rm{\Delta }}{M}_{1}^{\max }$ holds for z ≤ z_stag, where at z_stag, R_⋆ = R_stag. In other words, only at z < z_stag is the mass removal due to the jet activity from the red giant atmosphere possible, while at distances larger than z_stag, the jet ablation is limited to the stellar-wind material, as is the case for the observed comet-shaped sources X3, X7, and X8 (Mužić et al. 2010; Peißker et al. 2019). From Equation (6), the relation for z_stag follows as

$$\begin{eqnarray}\begin{array}{rcl}{z}_{\mathrm{stag}} & \approx & 0.15\left(\displaystyle \frac{{R}_{\star }}{100\,{R}_{\odot }}\right){\left(\displaystyle \frac{\theta }{0.22}\right)}^{-1}{\left(\displaystyle \frac{{L}_{{\rm{j}}}}{{10}^{42}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{\tfrac{1}{2}}\\ & & \times {\left(\displaystyle \frac{{\dot{m}}_{{\rm{w}}}}{{10}^{-8}{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{{v}_{{\rm{w}}}}{10\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-\tfrac{1}{2}}\,\mathrm{pc}.\end{array}\end{eqnarray} \tag{ 16 }$$

The dependence of ${z}_{\mathrm{stag}}$ on the stellar radius and the mass-loss rate is shown in Figure 4 (left panel). It is apparent that the volume around the reactivated Sgr A*, where the jet ablation can occur for a particular red giant, depends considerably on ${\dot{m}}_{{\rm{w}}}$, which spans over three orders of magnitude depending on the evolutionary stage, ${\dot{m}}_{{\rm{w}}}\approx {10}^{-9}-{10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Reimers 1987). In particular, for red giants with R_⋆ = 10 R_⊙, z_stag shrinks from 0.047 pc to 0.0015 pc as the mass-loss rate increases from ${\dot{m}}_{{\rm{w}}}={10}^{-9}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ to ${10}^{-6}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$.

Zoom In Zoom Out Reset image size

In Figure 4 (right panel), we show an exemplary case for the mass removal range from the red giant atmosphere (red giant with the parameters of R_⋆ = 100 R_⊙, ${\dot{m}}_{{\rm{w}}}={10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, and ${v}_{{\rm{w}}}=10\,\mathrm{km}\,{{\rm{s}}}^{-1}$) due to the single crossing through the jet with the luminosity of ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and the opening angle of 25^◦. Toward z_stag, the mass removal due to a single encounter approaches the mean value of $\overline{{\rm{\Delta }}{M}_{1}}={\rm{\Delta }}{M}_{1}^{\max }(z={z}_{\mathrm{stag}})\,={\rm{\Delta }}{M}_{1}^{\min }(z={z}_{\mathrm{stag}})$,

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\overline{{\rm{\Delta }}{M}_{1}}}{{M}_{\odot }} & = & \displaystyle \frac{{\dot{m}}_{{\rm{w}}}{v}_{{\rm{w}}}{R}_{\star }^{2}}{4{{Gm}}_{\star }}\\ & = & 2.9\times {10}^{-11}\left(\displaystyle \frac{{\dot{m}}_{{\rm{w}}}}{{10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right)\left(\displaystyle \frac{{v}_{{\rm{w}}}}{10\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)\\ & & \times {\left(\displaystyle \frac{{R}_{\star }}{100{R}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{{m}_{\star }}{1{M}_{\odot }}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 17 }$$

Because of the estimated several thousands of red giant–jet encounters according to Equation (13), the cumulative mass loss from the red giant can be derived as ${\rm{\Delta }}M\sim {n}_{\mathrm{cross}}{\rm{\Delta }}{M}_{1}$, giving

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{\Delta }}M}{{M}_{\odot }} & \approx & {10}^{-4}\left(\displaystyle \frac{{L}_{{\rm{j}}}}{{10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{R}_{\star }}{100{R}_{\odot }}\right)}^{4}{\left(\displaystyle \frac{z}{0.01\mathrm{pc}}\right)}^{-\tfrac{7}{2}}\\ & & \times {\left(\displaystyle \frac{\theta }{0.22}\right)}^{-2}{\left(\displaystyle \frac{{m}_{\star }}{{M}_{\odot }}\right)}^{-1}\left(\displaystyle \frac{{t}_{\mathrm{jet}}}{0.5\,\mathrm{Myr}}\right){\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{\displaystyle \frac{1}{2}}.\end{array}\end{eqnarray} \tag{ 18 }$$

In Figure 5, we plot ΔM for L_j = 10⁴² and ${10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, and R_⋆ = 50 and 100 R_⊙. Additionally, we plot ΔM for the longer jet lifetime of t_jet = 1 Myr and ${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and R_⋆ = 100 R_⊙, which can be considered as an upper limit of ΔM for a red giant orbiting Sgr A*. We find that ΔM within the S cluster is comparable to the mass removal inferred from the red giant–clump collision simulations by Kieffer & Bogdanović (2016). In Figure 5, we plot the upper and lower limits of ΔM obtained by Kieffer & Bogdanović (2016). Beyond z = 0.03 pc, ΔM for the star–jet interaction is progressively smaller than 3.5 × 10⁻⁴ M_⊙, which implies that the jet impact on the stellar evolution is the most profound for S-cluster red giants. In this region, there also lies the division between the warm and the cool colliders as discussed in Section 2.1, with warm colliders present inside z_c ∼ 0.04 pc, which is also marked in Figure 5 with a vertical dotted–dashed line. Because warm colliders are warmer and bigger, this further enhances the mass removal inside the S cluster. In addition, the mass removal due to the jet interaction is of a comparable order of magnitude to the mass loss expected from cool winds during the time interval of the active jet ${\rm{\Delta }}{M}_{{\rm{w}}}\,\approx {\dot{m}}_{{\rm{w}}}{t}_{\mathrm{jet}}\sim 0.5\mbox{--}5\times {10}^{-4}\,{M}_{\odot }$ when ${\dot{m}}_{{\rm{w}}}\approx {10}^{-6}\mbox{--}{10}^{-9}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Reimers 1987), as indicated by the shaded rectangle in Figure 5. This implies that the jet–star interaction perturbs the stellar evolution of passing red giants, in particular in the innermost parts of the NSC.

Zoom In Zoom Out Reset image size

Note that, on one hand, ΔM is supposed to be a lower limit as after the first passage through the jet, the giant is expected to expand to an even larger radius before the next encounter, which increases the mass removal efficiency (Kieffer & Bogdanović 2016). On the other hand, the resonant relaxation of stellar orbits as well as a jet precession may change the frequency of the jet–star interactions (see Sections 4 and 6.1), and therefore, n_cross should be considered as an upper limit of the number of encounters. Overall, ΔM in Equation (18) can be applied as an approximation for the total mass removal due to the red giant–jet interactions. Hence, the truncation of stellar envelopes of late-type stars by the jet during active phases of Sgr A* appears to be efficient and complementary to other previously proposed processes, mainly tidal disruptions of giants and stellar collisions with other stars and/or the accretion disk.

Red giants are post-main-sequence evolutionary stages of stars with initial mass $0.5\,{M}_{\odot }\lesssim {m}_{\star }\lesssim 10{M}_{\odot }$. These stars exhausted hydrogen supplies in their cores, and the hydrogen fusion into helium continues in the shell. As a result, the mass of the helium core gradually increases, and this is linked to the increase in the atmosphere radius as well as luminosity. Stellar evolutionary models of red giants show that their atmosphere radius and bolometric luminosity depend primarily on the mass of the helium core m_c as (Refsdal & Weigert 1971; Joss et al. 1987)

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{L}_{\star }}{{L}_{\odot }} & \approx & \displaystyle \frac{{10}^{5.3}{\mu }_{{\rm{c}}}^{6}}{1+{10}^{0.4}{\mu }_{{\rm{c}}}^{4}+{10}^{0.5}{\mu }_{{\rm{c}}}^{5}},\\ \displaystyle \frac{{R}_{\star }}{{R}_{\odot }} & \approx & \displaystyle \frac{3.7\times {10}^{3}{\mu }_{{\rm{c}}}^{4}}{1+{\mu }_{{\rm{c}}}^{3}+1.75{\mu }_{{\rm{c}}}^{4}},\end{array}\end{eqnarray} \tag{ 19 }$$

where μ_c ≡ m_c/M_⊙. This also holds for red supergiants with carbon–oxygen cores and burning hydrogen and helium in their shells (Paczyński 1970). In the red giant stage, the dominant energy source is the p–p process, and hence, the luminosity is mainly determined by the growth rate of the helium core ${\dot{m}}_{{\rm{c}}}$,

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\star }}{{L}_{\odot }}\,\simeq \,1.02\left(\displaystyle \frac{{\dot{m}}_{{\rm{c}}}}{{10}^{-11}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right).\end{eqnarray} \tag{ 20 }$$

Equations (19) and (20) imply that the bolometric luminosity is not significantly affected by the jet–red giant interaction, as only the tenuous shell is ablated by the jet, and the dense core is left untouched. Then, the effective temperature T₁ of the ablated giant is

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{1}}{{T}_{0}}={\left(\displaystyle \frac{{R}_{0}}{{R}_{1}}\right)}^{\displaystyle \frac{1}{2}},\end{eqnarray} \tag{ 21 }$$

where T₀ is the original effective temperature, and R₀ and R₁ are the atmosphere radii before and after the truncation, respectively. Here we implicitly assume that the red giant underwent n_coll interactions with the jet during the active phase given by Equation (13), which eventually leads to the decreased radius of R₁ ≈ R_stag according to Equation (6). The luminosity in the infrared domain between frequencies ν₁ and ν₂ can be expressed using the Rayleigh–Jeans approximation⁹ as ${L}_{\mathrm{IR}}\lesssim 8/3{\left(\pi /c\right)}^{2}{R}_{\star }^{2}{k}_{{\rm{B}}}{T}_{\star }({\nu }_{2}^{3}-{\nu }_{1}^{3})\propto {R}_{\star }^{2}{T}_{\star }$ (see Alexander 2005 for a similar analysis), which using Equation (21) leads to

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{1}}{{L}_{0}}\sim {\left(\displaystyle \frac{{R}_{1}}{{R}_{0}}\right)}^{\displaystyle \frac{3}{2}}.\end{eqnarray} \tag{ 22 }$$

For instance, the ablation of a red giant atmosphere from 120 to 30 R_⊙ would result in the increase of effective temperature by a factor of 2 and a decrease by a factor of 8 in the IR luminosity or ∼2.26 mag. The ablation of the envelope from 120 to 4R_⊙ would result in the decrease by as much as ∼5.5 mag. The difference of 2–5 mag can already affect the count rate of late-type stars in the near-infrared domain in the central arcsecond of the Galactic center.

As an exemplary case, we set up a simplified temporal evolution of a red giant using Equations (19) and (20). We perform this calculation to estimate the potential difference in near-infrared magnitudes, and the color change for late-type stars before and after the active jet phase—it does not represent realistic stellar evolution tracks, but can provide insight into the basic trends in the near-infrared magnitude evolution and the effective temperature. We evolve the stellar luminosity and the radius for an increasing core mass ${\mu }_{{\rm{c}}}={\mu }_{{\rm{c}}0}+{\dot{\mu }}_{{\rm{c}}}{dt}$, where ${\mu }_{{\rm{c}}0}=0.104$ and the time step is ${dt}={10}^{4}\,\mathrm{yr}$. The overall evolution from ${\mu }_{{\rm{c}}0}$ to μ_c = 0.55 takes ∼8.41 × 10⁹ yr, when neither the effect of stellar winds nor that of rotation is taken into account. The initial and the final core masses were chosen according to the limiting values for lighter stars, m_⋆ < 2 M_⊙, in which case ${\mu }_{{\rm{c}}}^{\min }\sim 0.1$ and ${\mu }_{{\rm{c}}}^{\max }\sim 0.5$. These are stars with degenerate helium cores and hydrogen-burning shells (Refsdal & Weigert 1971). The lower core-mass value of 0.1 also approximately corresponds to the Schönberg–Chandrasekhar limit. The total duration is comparable to the time that stars of 1 M_⊙ spend on the giant and asymptotic giant branches, which is of the order of 10⁹ yr according to MacLeod et al. (2012).

To assess the observational effects of the star–jet collision in the near-infrared domain, we calculate the effective temperature at each step using ${T}_{\star }={T}_{\odot }{\left({L}_{\star }/{L}_{\odot }\right)}^{1/4}{\left({R}_{\star }/{R}_{\odot }\right)}^{-1/2}$. Subsequently, we calculate the monochromatic flux density in K band (2.2 μm) and L' band (3.8 μm) using ${F}_{\nu }={\left({R}_{\star }/{d}_{\mathrm{GC}}\right)}^{2}\pi {B}_{\nu }({T}_{\star })$, where B_ν(T_⋆) is the spectral brightness given by the Planck function at the given effective temperature. The corresponding magnitudes are calculated using ${m}_{K}=-2.5\mathrm{log}({F}_{K}/653\,\mathrm{Jy})$ and ${m}_{L^{\prime} }\,=-2.5\mathrm{log}({F}_{K}/253\,\mathrm{Jy})$, from which the color index follows as ${CI}={m}_{K}-{m}_{L^{\prime} }$.

For the analysis in this section as well as in Section 5, we calculate intrinsic stellar magnitudes m_K and ${m}_{L^{\prime} }$. The calculated magnitudes and colors can then be compared to extinction-corrected magnitudes and the derived surface-brightness profiles of the nuclear star cluster, i.e., those corrected for the foreground extinction. To compare our results to observed magnitudes and derived surface profiles that are just corrected for the differential extinction but not for the foreground extinction (Buchholz et al. 2009; Habibi et al. 2019; Schödel et al. 2020), it is necessary to increase the magnitudes using the corresponding mean extinction coefficients (see, e.g., Schödel et al. 2010), in particular A_K = 2.54 ± 0.12 mag and ${A}_{L^{\prime} }=1.27\pm 0.18$ mag.

The imprint of the ablation of the stellar atmosphere by a jet, whose kinetic luminosity is fixed to ${L}_{{\rm{j}}}=2.3\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, is modeled by assuming that the radius of an interacting red giant keeps evolving according to Equation (19) when ${R}_{\star }\lt {R}_{\mathrm{stag}}$ at a given distance z from Sgr A*. After the stellar radius reaches the scale of the stagnation radius at a given distance z, we set R_⋆ = R_stag for the rest of the evolution, which can by justified by the fact that the red giant propagates through the jet n_coll times and the envelope is removed after repetitive encounters. The expected number of encounters is 6 × 10⁵, 2 × 10⁴, and 632 for z = 0.001 pc, 0.01 pc, and 0.1 pc, respectively; see Equation (13). In Figure 6, we show three magnitude–effective temperature curves of the ablated red giants that underwent repetitive encounters with the jet at their orbital distances of 0.001, 0.01, and 0.1 pc from Sgr A*, which led to their truncation to a smaller radius close to the corresponding stagnation radius at a given distance. In addition, we compare the magnitude–temperature curves of ablated giants with an unaffected evolution. We list the stellar parameters at the time of the atmosphere truncation when R_⋆ = R_stag at the corresponding distance as well as the parameters for the final state of ablated giants in Table 1. This is compared to an unaffected final state with the core mass of 0.55 M_⊙; see the bottom row of Table 1. The basic signature of the jet–star interaction is that the star gets progressively warmer (with a bluer, more negative color index) and fainter in the near-infrared K_s band in comparison with the normal evolution without any atmosphere ablation. This trend is more apparent for red giants that are closer to Sgr A* because of the smaller jet–star stagnation radius and hence a larger fraction of the stellar atmosphere that is removed.

Zoom In Zoom Out Reset image size

Table 1.  Parameters at the Time of the Jet Ablation and Those of the Final State of a Red Giant, which is Evolved from the Core Mass of μ_c = 0.104 to 0.55

Ablation Distance (pc)	Ablation Time (yr)	${R}_{\star }^{\mathrm{abl}}$ (R⊙)	${T}_{\star }^{\mathrm{abl}}$ (K)	${T}_{\star }^{\mathrm{fin}}$ (K)	${m}_{K}^{\mathrm{abl}}$ (mag)	${m}_{K}^{\mathrm{fin}}$ (mag)	$\,{\mathrm{CI}}^{\mathrm{fin}}$
0.001	3.0 × 108	0.45	6205	68 759	19.6	16.4	−0.14
0.01	8.0 × 109	4.45	4662	21 744	15.2	12.8	−0.09
0.1	8.4 × 109	44.51	3526	6 876	10.7	9.4	−0.09
Normal Evolution	⋯	255.21 (final radius)	⋯	2 871	⋯	7.47	0.51

Note. The upper three lines contain parameters of the ablated red giants orbiting at distances 0.001, 0.01, and 0.1 pc (first column). The second column lists the time, at which the ablation occurred with respect to the initial state of μ_c = 0.104. The third column lists the truncation radius, when R_⋆ = R_stag. The fourth column lists the effective temperature at the time of truncation. The fifth column lists the final effective temperature at the final stage of their evolution (when μ_c = 0.55). The sixth and seventh columns contain dereddened magnitudes at the ablation and final times, respectively. The last column contains dereddened color indices, CI = m_K − m_L, at the final state. The bottom line corresponds to the final state of a normal, unaffected evolution.

Download table as:  ASCIITypeset image

Although we do not calculate stellar evolutionary tracks, only basic trends in terms of near-infrared magnitude and effective temperature, our results are consistent with those of Amaro-Seoane et al. (2020) who calculated evolutionary tracks specifically for late-type stars ablated due to the red giant–accretion-clump collisions. They show in their Figure 2 that the collision affects the stellar evolution of a red giant in a way that after repetitive encounters it follows a track along a nearly constant absolute bolometric magnitude toward higher effective temperatures. The constant absolute bolometric magnitude or bolometric luminosity in combination with an increasing effective temperature results in the drop in the near-infrared luminosity, because ${L}_{\mathrm{IR}}/{L}_{\mathrm{bol}}\propto {T}_{\star }^{-3}$. We used their evolutionary tracks calculated using the CESAM code (Morel & Lebreton 2008) for estimating K_s-band near-infrared magnitudes. These tracks are depicted in Figure 6 for the case of a normal evolution of a star with M_⋆ = 1.4 M_⊙ and R_⋆ = 60 R_⊙ (black solid line) and different collision cases for clumps with surface densities in the range ${\rm{\Sigma }}=2\times {10}^{4}\mbox{--}{10}^{6}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ (see the legend). Qualitatively, the perturbed stellar evolutionary tracks follow the temperature trends that we can also observe for giant–jet collisions: ablated giants move toward higher effective temperature. Also, they become fainter in the near-infrared domain in comparison with an unperturbed evolution. The main difference in comparison with the analysis of Amaro-Seoane et al. (2020) is their trend toward larger magnitudes (stars become fainter), while we see a small gradual increase in brightness. This difference is due to our simplifying assumption of a constant radius after the series of collisions with the jet, while in reality the radius should evolve, especially after the jet ceases to be active. Because ${L}_{\mathrm{IR}}\propto {R}_{\star }^{2}{T}_{\star }$, the near-infrared luminosity grows linearly with increasing temperature for the fixed stellar radius. This motivates further exploration of the effect of star–jet collisions using a modified stellar evolutionary code.

For asymptotic giant branch stars, an extreme transition from a red, cool luminous giant to a hot and faint white dwarf is possible when it is completely stripped of its envelope. This was studied by King (2020) for tidal stripping close to the SMBH but cannot be excluded also for asymptotic giant branch stars and jet collisions for a case when the giant star is at milliparsec separation from Sgr A* and less, in which case the stagnation radius is typically a fraction of the solar radius. In fact, an active jet can enlarge the volume around the SMBH where asymptotic giant branch stars are turned into white dwarfs. Considering Equation (6), we can derive that in order for R_stag to be of the order of a white-dwarf radius ${R}_{\mathrm{wd}}\sim 0.01{R}_{\odot }$, the giant needs to orbit the SMBH at z ≈ 0.15 mpc so that the jet with ${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ can truncate it down to the white-dwarf size. The stellar interior that is not affected by tidal forces is characterized by the Hill radius

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{Hill}} & \approx & z{\left(\displaystyle \frac{{m}_{\star }}{3{M}_{\bullet }}\right)}^{\tfrac{1}{3}}=29\left(\displaystyle \frac{z}{0.15\,\mathrm{mpc}}\right)\\ & & \times {\left(\displaystyle \frac{{m}_{\star }}{1{M}_{\odot }}\right)}^{\tfrac{1}{3}}{\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{-\tfrac{1}{3}}\,{R}_{\odot },\end{array}\end{eqnarray} \tag{ 23 }$$

from which we see that tidal forces alone will not truncate the giant down to its white-dwarf core because r_Hill > R_stag at z.

In summary, the jet–star interaction could have affected the appearance of late-type giants in the central arcsecond by making them warmer or bluer in terms of color and hence fainter in the near-infrared domain.

We estimate the number of late-type stars, i.e., stars that form a cusp, that could have passed and interacted with the jet during its estimated lifetime of ∼0.1–0.5 Myr (Guo & Mathews 2012). For the density distribution of late-type stars in the inner 0.5 pc, we adopt a cusp-like power-law distribution ${n}_{\mathrm{RG}}\approx {n}_{0}{\left(z/{z}_{0}\right)}^{-\gamma }$, with ${n}_{0}\simeq 52\,{\mathrm{pc}}^{-3}$, ${z}_{0}\simeq 4.9\,\mathrm{pc}$, and γ ≃ 1.43 (Gallego-Cano et al. 2018). The expected number of late-type stars within a certain distance z_out is

$$\begin{eqnarray}&&{N}_{\star }(\lt {z}_{\mathrm{out}})={\int }_{0}^{{z}_{\mathrm{out}}}{n}_{\mathrm{RG}}(z)4\pi z{{\prime} }^{2}{dz}^{\prime} =4\pi {n}_{0}\displaystyle \frac{{z}_{0}^{\gamma }{z}_{\mathrm{out}}^{3-\gamma }}{3-\gamma }\end{eqnarray} \tag{ 24 }$$

giving N_⋆ ≈ 610 and 25.8 inside 0.3 and 0.04 pc, respectively. The number of stars inside the jet at any time is given by the jet covering factor f_j in a spherical volume ${V}_{\star }=(4/3)\pi {z}_{\mathrm{out}}^{3}$. By considering a conical jet and a counter-jet with the total volume ${V}_{{\rm{j}}}=2/3\pi {R}_{{\rm{j}}}^{2}{z}_{\mathrm{out}}$, the covering factor is ${f}_{{\rm{j}}}={V}_{{\rm{j}}}/{V}_{\star }\sim {\left({R}_{{\rm{j}}}/{z}_{\mathrm{out}}\right)}^{2}$ and the number of red giants inside the jet is ${N}_{{\rm{j}}}(\lt {z}_{\mathrm{out}})\,={f}_{{\rm{j}}}{N}_{\star }(\lt {z}_{\mathrm{out}})\sim {N}_{\star }(\lt {z}_{\mathrm{out}}){\tan }^{2}(\theta )/2$. Then, the average number of red giants that are simultaneously inside the jet is N_j ≈ 14.8 and 0.62 inside 0.3 pc and 0.04 pc, respectively.

To estimate the number of stars crossing the jet per orbital timescale, we first calculate the jet-crossing rate per unit of time. Because we focus on the region z ≲ 0.5 pc, which is well inside the radius of influence of Sgr A*, r_h ∼ 2 pc (Merritt 2013), we approximate the stellar velocity dispersion by the local Keplerian velocity, ${\sigma }_{\star }\sim {v}_{\mathrm{orb}}=\sqrt{{{GM}}_{\bullet }/z};$ see, e.g., Šubr & Haas (2014). With the jet cross section ${S}_{\mathrm{jet}}\approx {{zR}}_{{\rm{j}}}\sim {z}^{2}\tan \theta$, the number of late-type stars entering both the jet and counter-jet per unit of time is

$$\begin{eqnarray}&&{\dot{N}}_{\mathrm{RG}}\approx 2{n}_{\mathrm{RG}}{\sigma }_{\star }{S}_{\mathrm{jet}}\simeq 2{n}_{0}{z}_{0}^{\gamma }\sqrt{{{GM}}_{\bullet }}\tan \theta {z}^{\tfrac{3}{2}-\gamma }.\end{eqnarray} \tag{ 25 }$$

The number of red giants crossing the jet per orbital timescale is

$$\begin{eqnarray}&&{N}_{\mathrm{RG}}={\dot{N}}_{\mathrm{RG}}{P}_{\mathrm{orb}}=4\pi {n}_{0}{z}_{0}^{\gamma }\tan \theta {z}^{3-\gamma },\end{eqnarray} \tag{ 26 }$$

where the orbital period in the sphere of influence of the SMBH follows from the third Keplerian law, ${P}_{\mathrm{orb}}=2\pi {z}^{3/2}/\sqrt{{{GM}}_{\bullet }}$. In Figure 7, we plot N_RG. The average number of crossing giants per orbital period in the region with an outer radius z_out is

$$\begin{eqnarray}&&{\overline{N}}_{\mathrm{RG}}=\displaystyle \frac{4\pi }{4-\gamma }{n}_{0}{z}_{0}^{\gamma }\tan \theta {z}_{\mathrm{out}}^{3-\gamma }.\end{eqnarray} \tag{ 27 }$$

In particular, ${\overline{N}}_{\mathrm{RG}}\,\simeq \,3.5$ and ${\overline{N}}_{\mathrm{RG}}\,\simeq \,82$ when ${z}_{\mathrm{out}}\,=0.04\,\mathrm{pc}$ (S cluster) and ${z}_{\mathrm{out}}=0.3\,\mathrm{pc}$, respectively (see Figure 7).

Zoom In Zoom Out Reset image size

In this case, ${\overline{N}}_{\mathrm{RG}}$ represents all late-type stars that cross the jet sheath per orbital period on average. The fraction of giants whose envelopes could have been stripped off by the jet can be estimated by comparing the radii of stars with the corresponding stagnation radius at a certain distance from Sgr A*. The basic condition for the ablation is that R_⋆ ≳ R_stag at a given z from Sgr A*. In particular, for z = 0.04 pc and the jet luminosity of ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, the minimum stellar parameters for ablation are R_⋆ = 27 R_⊙, μ_c = 0.3, L_⋆ = 129 L_⊙, T_⋆ = 3743 K, and m_abl = 11.7 mag, where m_abl denotes the upper magnitude limit, below which stars are expected to be affected by the jet. Using the K-band luminosity function approximated by the power law, $d\mathrm{log}N/{{dm}}_{{\rm{K}}}=\beta$ with β ≃ 0.3 for late-type stars (Buchholz et al. 2009; Pfuhl et al. 2011) between 12 and 18 mag, we can estimate the fraction of ablated stars as $\eta ={N}_{\mathrm{abl}}/{N}_{\mathrm{tot}}\times 100={10}^{\beta ({m}_{\mathrm{abl}}-{m}_{\max })+2} \%$, where m_max is the limiting magnitude, which we set to 18 mag according to Pfuhl et al. (2011). Then, for z = 0.04 pc and ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, we get η = 1.27%. For the larger distance z = 0.5 pc and ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, we obtain the minimum parameters of ablated stars as follows, R_⋆ = 338 R_⊙, μ_c = 0.6, L_⋆ = 6081 L_⊙, T_⋆ = 2775 K, m_abl = 6.95 mag with η = 0.05%. The limiting values and the percentage of ablated giants are quite sensitive to the jet luminosity. Increasing L_j to ${10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, we get R_⋆ = 2.7 R_⊙, μ_c = 0.16, L_⋆ = 3.97 L_⊙, T_⋆ = 4960 K, m_abl = 16.1 mag with η = 26.5% for z = 0.04 pc and R_⋆ = 33.8 R_⊙, μ_c = 0.31, L_⋆ = 181 L_⊙, T_⋆ = 3645 K, m_abl = 11.3 mag with η = 0.95% for z = 0.5 pc. We summarize the relevant values in Table 2. Although the fraction of affected late-type stars is small, it significantly affects brighter stars with smaller magnitudes—stars brighter than 14 mag constitute ∼6.3% of the total observed sample and stars brighter than 12 mag constitute only 1.6%. We explicitly show the change in the projected brightness distribution for brighter stars in Section 5.

Table 2.  Limiting Minimal Stellar Radii, Maximum Apparent K-band Magnitudes (dereddened), and the Fraction of Ablated Giants for Two Distances from Sgr A* (0.04 and 0.5 pc) and Two Luminosities of Its Jet (${10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and ${10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$)

Distance	${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$	${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$
0.04 pc	R⋆ = 27 R⊙, mabl = 11.7, η = 1.27%	R⋆ = 2.7 R⊙, mabl = 16.1, η = 26.5%
0.5 pc	R⋆ = 338 R⊙, mabl = 6.95, η = 0.05%	R⋆ = 33.8 R⊙, mabl = 11.3, η = 0.95%

Download table as:  ASCIITypeset image

The jet ablation could partially have contributed to the inferred four to five missing late-type giants in the region with z_out = 0.04 pc (Habibi et al. 2019) as well as to the ∼100 missing late-type giants in the larger region with z_out = 0.3 pc (Gallego-Cano et al. 2018), especially for higher luminosities of the jet.

The number of stars that can interact with the jet is increased via dynamical processes in the dense nuclear star cluster. In particular, the vector resonant relaxation (VRR) changes the direction of the orbital angular momentum (Alexander 2005; Merritt 2013), and therefore, stars that were not passing through the jet can do so on the resonant relaxation timescale. More precisely, in the sphere of influence of the SMBH, stars move on Keplerian ellipses, and the gravitational interactions between stars are correlated. Given the finite number of stars, there is a nonzero torque on a test star. During the time interval ${\rm{\Delta }}t$, for which ${P}_{\mathrm{orb}}\ll {\rm{\Delta }}t\ll {T}_{\mathrm{coh}}$ and the coherence timescale

$$\begin{eqnarray}&&{T}_{\mathrm{coh}}=\displaystyle \frac{{P}_{\mathrm{orb}}}{2\pi \sqrt{{N}_{\star }}}\displaystyle \frac{{M}_{\bullet }}{{m}_{\star }}=\sqrt{\displaystyle \frac{3-\gamma }{4\pi {{Gn}}_{0}{z}_{0}^{\gamma }}}\displaystyle \frac{{M}_{\bullet }^{\tfrac{1}{2}}}{{m}_{\star }}{z}^{\tfrac{\gamma }{2}}\end{eqnarray} \tag{ 28 }$$

is inversely proportional to the square root of the number of enclosed stars, the angular momentum of a test star changes linearly with time.

The inclination of stellar orbits would change by ∼π only when ${T}_{\mathrm{coh}}(z)\lesssim {t}_{{\rm{j}}}\sim 0.1\mbox{--}0.5\,\mathrm{Myr}$ and hence essentially all late-type stars could interact with the jet during its lifetime. The estimated number of enclosed stars in the S cluster (z_out ∼ 0.04 pc) is 26 < N_⋆ < 10,000, where the lower limit considers only late-type stars according to the analysis by Habibi et al. (2019) and the upper limit stands for all the stars including compact remnants. The upper limit is supposed to be closer to the actual number of stellar objects because the number of old neutron stars and stellar black holes in the central arcsecond could be of that order of magnitude (Morris 1993; Deegan & Nayakshin 2007; Zhu et al. 2018). The total number of massive objects naturally affects the coherence timescale by more than an order of magnitude. In Figure 8, we plot T_coh. We see that when N_⋆ ∼ 1000 and more, T_coh is comparable to the lifetime of the jet in the inner parts of the S cluster. In summary, the coherent resonant relaxation makes the number of affected giants bigger and the estimates per orbital timescale ${\overline{N}}_{\mathrm{RG}}$ can be considered as a lower limit. Another more hypothetical effect that can enlarge the number of affected giants is the jet precession (see Section 6.1).

Zoom In Zoom Out Reset image size

The VRR can affect the number of encounters, n_cross; see Equation (13). In case T_coh > t_jet, i.e., for a smaller number of enclosed objects (N_⋆ ≲ 100), n_cross is still mainly determined by t_jet; see Equation (13). However, then the mean number of interacting giants ${\overline{N}}_{\mathrm{RG}}$ is also not significantly enlarged. On the other hand, if T_coh ≲ t_jet, then n_cross is reduced approximately by a factor of 2θ/π, which assumes that the angular momentum vector shifts linearly with time during T_coh. In that sense, the interaction timescale with the jet is ${t}_{\mathrm{int}}\sim {T}_{\mathrm{coh}}(2\theta /\pi )$. Considering T_coh ∼ t_jet, the number of crossings is

$$\begin{eqnarray}&&{n}_{\mathrm{cross}}^{\mathrm{RR}}\sim 2800\left(\displaystyle \frac{{T}_{\mathrm{coh}}}{0.5\,\mathrm{Myr}}\right){\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{\displaystyle \frac{1}{2}}{\left(\displaystyle \frac{z}{0.01\mathrm{pc}}\right)}^{-\tfrac{3}{2}},\end{eqnarray} \tag{ 29 }$$

which is smaller by an order of magnitude in comparison with n_cross.

To assess the observational signatures of the jet ablation of late-type stars, we generate a mock spherical cluster of stars. Their initial spatial distribution follows ${n}_{\mathrm{RG}}={n}_{0}{\left(z/{z}_{0}\right)}^{-\gamma }$ with ${n}_{0}=52\,{\mathrm{pc}}^{-3}$, z₀ = 4.9 pc, and γ ∼ 1.43 (Gallego-Cano et al. 2018). This spatial profile suggests that there are in total ∼4000 late-type stars inside the inner 1 pc, which we generate using the Monte Carlo approach to form a mock NSC; see Figure 9 (left panel) for illustration.

Zoom In Zoom Out Reset image size

Each star is assigned its mass in the range from 0.08 M_⊙ to 100 M_⊙ following the initial mass function (IMF) according to Kroupa (2001), i.e.,

$$\begin{eqnarray}&&\zeta ({m}_{\star })={m}_{\star }^{-\alpha },\,\mathrm{where}\,\left\{\begin{array}{l}\alpha =0.03,{m}_{\star }\lt 0.08\,{M}_{\odot },\\ \alpha =1.3,0.08\,{M}_{\odot }\lt {m}_{\star }\lt 0.5\,{M}_{\odot },\\ \alpha =2.3,{m}_{\star }\gt 0.5\,{M}_{\odot }.\end{array}\right.\end{eqnarray} \tag{ 30 }$$

The Chabrier/Kroupa IMF is a good approximation for the observed mass distribution of the late-type NSC population (Pfuhl et al. 2011).

In the next step, we assigned the core mass to each star of the mock cluster. Here, we fix the ratio between the core mass and the stellar mass to μ_c/m_⋆ = 0.4, which is in between the value inferred from the Schönberg–Chandrasekhar limit¹⁰ and the final phases of the stellar evolution, where the white-dwarf core constitutes most of the mass for solar-type stars. For more precise simulations, core masses from the stellar evolution of the NSC should be adopted; however, here we are interested in the first-order effects of the jet activity on the surface-brightness distributions.

To construct the surface-brightness profiles of the late-type population after the active jet phase in different magnitude bins, we followed these steps:

• 1.  
  We calculated L_⋆(μ_c) and ${R}_{\star }({\mu }_{{\rm{c}}})$ using Equation (19).
• 2.  
  If the jet was set active with a certain luminosity L_j, we compared R_⋆ and R_stag for a given distance z of the star. If R_⋆ ≥ R_stag, then we set R_⋆ = R_stag. In this case, we also implicitly assumed that at a given distance, all of the stars, for which R_⋆ ≥ R_stag, are eventually ablated by the jet; hence, the resonant relaxation was assumed to be efficient and hence T_coh < t_jet.
• 3.  
  We estimated the effective temperature of a star using ${T}_{\star }={T}_{\odot }{\left({L}_{\star }/{L}_{\odot }\right)}^{1/4}{\left({R}_{\star }/{R}_{\odot }\right)}^{-1/2}$.
• 4.  
  From the Planck function ${B}_{\nu }({T}_{\star })$, we calculated the monochromatic flux in the K_s band (2.2 μm) ${F}_{\nu }({R}_{\star },{T}_{\star })={\left({R}_{\star }/{d}_{\mathrm{GC}}\right)}^{2}\pi {B}_{\nu }({T}_{\star })$ and the corresponding apparent magnitude m_K (dereddened).

The initial relation between the K_s-band magnitude and the stellar radius is shown in Figure 9 in the right panel. We calculate the projected stellar density using concentric annuli with mean radius R and width of ΔR, ${\sigma }_{\star }={N}_{\star }/(2\pi R{\rm{\Delta }}R)$, where N_⋆ is the number of stars in an annulus. We estimate the uncertainty of the stellar number counts as ${\sigma }_{{\rm{N}}}\approx \sqrt{{N}_{\star }}$. Subsequently, we construct the surface stellar profiles in two-magnitude bins starting at m_K = 18 mag up to m_K = 10 mag, i.e., in total, four bins; see Figure 10. In the top-left panel of Figure 10, we plot the nominal projected distribution without considering the effect of the jet. The brightness profile for all four magnitude bins can be approximated by simple power-law functions, $N(R)={N}_{0}{\left(R/{R}_{0}\right)}^{-{\rm{\Gamma }}}$, whose slopes are listed in Table 3. Hence, the initial cluster distribution is cusp like. In the top-right panel, we show the case with an active jet with the luminosity of ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. We see that the profile for the brightest stars in the 10–12 mag bin becomes flat in the inner arcsecond and can be described as a broken power-law function, $N(R)={N}_{0}{\left(R/{R}_{\mathrm{br}}\right)}^{-{\rm{\Gamma }}}{\left[1+{\left(R/{R}_{\mathrm{br}}\right)}^{{\rm{\Delta }}}\right]}^{({\rm{\Gamma }}-{{\rm{\Gamma }}}_{0})/{\rm{\Delta }}}$, where ${R}_{\mathrm{br}}$ is a break radius, Γ is a slope of the inner part, Γ₀ marks the slope of an outer part, and Δ denotes the sharpness of the transition. The larger surface-brightness values for this magnitude bin may be interpreted as an extra input of ablated giants with initial magnitudes m_K < 10 mag that after the ablation fall into bins with a larger magnitude. Finally, we increase the jet luminosity to ${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, which makes the flattening of the brightest giants even more profound. The stars in the 12–14 mag bin also exhibit a flatter profile inside the inner arcsecond for this case, which shows the significance of the jet luminosity in affecting the observed surface profile of the NSC. We list the power-law slopes for both a simple and a broken power-law function and the break radii, where available, for all the magnitude bins and the three jet-activity cases in Table 3.

Zoom In Zoom Out Reset image size

Table 3.  Summary of the Power-law Slopes (for Both Single and Broken if Relevant) for the Four Magnitude Bins (2 mag Intervals Starting at 18 mag up 10 mag in the Near-infrared K_s Band; Magnitudes are Dereddened) and Three Jet-activity Cases (No Jet, ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, ${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$)

Magnitude Bin	No Jet	Jet ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$	Jet ${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$
18–16 mag	single: Γ = 0.9	single: Γ = 0.8	single: Γ = 1.0
16–14 mag	single: Γ = 0.6	single: Γ = 0.6	single: Γ = 0.7
14–12 mag	single: Γ = 0.6	single: Γ = 0.6	broken: Γ = 0.2, Γ0 = 0.9, Rbr = 1''
12–10 mag	single: Γ = 0.7	broken: Γ = −0.08, Γ0 = 1.4, Rbr = 03	broken: Γ = 0.04, Γ0 = 2.2, Rbr = 66

Download table as:  ASCIITypeset image

Figure 10 demonstrates a potential signature of the jet activity on the surface profile of the NSC. It is important to study differential profiles, i.e., the distribution in different magnitude bins, because the lower-luminosity jet starts affecting the profile of bright stars (smaller magnitudes), while with an increasing jet luminosity, the fainter stars become affected as well, starting at smaller projected radii (<1''). Our Monte Carlo simulation suggests that the active jet phase with ${L}_{{\rm{j}}}\lesssim {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ likely affected late-type stars with m_K ≤ 14 mag that exhibit the flat profile inside the inner arcsecond. The fainter stars of m_K > 14 mag can still keep a cusp-like profile after the jet ceased its enhanced activity.

For better quantitative comparisons with observations, it is necessary to include the stars of different ages and hence different core masses. This is rather complex as there were recurrent star formation episodes in the NSC, with 80% of the stellar mass being formed 5 Gyr ago, the minimum in the star formation rate close to 1 Gyr, and the renewed star formation in the last 100–200 million years, although with the 10× lower star formation rate than at earlier episodes (Pfuhl et al. 2011). These findings were confirmed by Schödel et al. (2020), who estimate that 80% of stars formed 10 Gyr ago or earlier, then about 15% formed 3 Gyr ago, and the remaining fraction in the last 100 Myr. Furthermore, the dynamical effects such as the mass segregation and the relaxation processes could also have played a role in shaping the final observed profile of the NSC, and in addition, we might expect other bright-giant depletion processes (Alexander 2005). Despite these difficulties in comparing theoretical and observed profiles, the basic trend shown in Figure 10, in particular for ${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (bottom panel), is consistent with the observational findings of Schödel et al. (2020), who found cusp-like profiles for all magnitude bins apart from the m_K = 14.5–14 mag bin (including foreground field extinction), which shows a flat/decreasing profile in their analysis. In our panels in Figure 10, this corresponds to the dereddened 10–12 and 12–14 mag bins, whose profiles become affected for ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ starting from the projected radii below 1''. However, Schödel et al. (2020) also note that precise surface profiles for late-type stars are difficult to construct due to the contamination at all magnitude bins by an unrelaxed population of young stars. Habibi et al. (2019) report a cusp-like profile for late-type stars of m_K < 17 mag (including foreground extinction), which approximately corresponds to our bin of 14–16 mag (yellow points) that maintains a cusp-like profile even for the largest jet luminosity (bottom panel). In conclusion, the expected trend of preferential depletion of bright late-type stars by the jet is confirmed.

We investigated the effects of a jet during an active phase of Sgr A* in the last million years on the appearance of late-type giant stars with atmosphere radii more than 30R_⊙. We found that especially in the innermost arcsecond of the Galactic center (the S cluster), the upper layers of the stellar envelope could be removed by the jet ram pressure. Hence, the jet–red giant interactions during the active phase of Sgr A* could have contributed to the depletion of bright late-type stars. In other words, the atmosphere ablation by the jet would alter the red giant appearance in a way that would make them look bluer and fainter in the near-infrared bands (mainly K' and L bands), in which stars in the Galactic center region are generally monitored. In the following, we outline several additional effects that could be associated with the jet/red giant interaction.

6.1. Enlarging the Number of Affected Stars by the Jet Precession

Jet precession is a phenomenon that accompanies the launching of jets during the evolution of galaxies and stellar binaries. It is caused by perturbations due to the misalignment of the accretion flow and the black hole spin, the so-called Lense–Thirring precession, or by a secondary black hole. The jet precession was proposed to explain a long-term flux variability in radio galaxies, e.g., OJ 287 (Britzen et al. 2018), 3C 84 (Britzen et al. 2019b), 3C 279 (Abraham & Carrara 1998), the neutrino emission from TXS 0506+056 (Britzen et al. 2019a), as well as in X-ray binaries (Monceau-Baroux et al. 2015; Miller-Jones et al. 2019).

For the Galactic center, the Lense–Thirring precession of the hot thick accretion flow was analyzed by Dexter & Fragile (2013) in relation to the near-infrared and millimeter variability of Sgr A*. This effect would also translate to the precession of the jet under the assumption that it is coupled to the disk via the launching mechanism (Blandford–Payne mechanism; see Blandford & Payne 1982). The jet precession is also suggested by wide UV ionization cones with the opening angle of 60◦ (Bland-Hawthorn et al. 2019), which is larger by a factor of a few expected for the jet opening angle of ∼25° (Li et al. 2013).

We estimate the factor by which the volume of the affected red giants is enlarged. We adopt the jet precession half-opening angle of Ω_p ≈ 30° based on the scale of UV ionization cones (Bland-Hawthorn et al. 2019). During the precession motion, the jet circumscribes a cone with the radius ${R}_{{\rm{p}}}=z\sin {{\rm{\Omega }}}_{{\rm{p}}}$. The factor by which the volume at given distance z enlarges is given by

$$\begin{eqnarray}&&{f}_{\mathrm{prec}}=\displaystyle \frac{2\pi {R}_{{\rm{p}}}}{2{R}_{{\rm{j}}}}=\pi \displaystyle \frac{\sin {{\rm{\Omega }}}_{{\rm{p}}}}{\tan \theta }\sim 7.1,\end{eqnarray} \tag{ 31 }$$

when θ = 125 and Ω_p = 30°. The number of affected late-type stars would then increase by the same factor to N_j ≈ 105 and 4.4 within 0.3 and 0.04 pc, respectively, which is comparable to the number of missing bright red giants at these scales—100 at z < 0.3 pc (Gallego-Cano et al. 2018) and 4 at z < 0.04 pc (Habibi et al. 2019). The factor derived in Equation (31) should be treated as an upper limit on the volume enlargement as it assumes that the precession period is comparable to or less than the jet lifetime, but it could also be longer. On the other hand, a larger volume means that the stars are affected correspondingly less (over a shorter period of time) by the jet action, which is spread in different directions with the precession duty cycle.

6.2. High-energy Particle Acceleration and Jet Mass Loading due to Jet/Star Interactions

6.3. Chemically Peculiar Stars as Remnant Cores of Ablated Red Giants

6.4. A Collimated Jet or a Broad-angle Disk Wind?

6.5. Recurrent Seyfert-like Activity and TDEs

6.6. Comparison with Other Mechanisms—the Region of Efficiency

The jet-induced alternation of the population of late-type stars is not necessarily an alternative to other proposed mechanism, listed in the introductory, Section 1. In reality, it could have coexisted simultaneously during the past few million years with other previously proposed mechanisms, in particular the tidal disruption of red giant envelopes as well as direct star–disk interactions. This follows from the fact that these mechanisms have different length scales of their efficiency, as we further outline in the paragraphs below.

First, the tidal disruption of red giant envelopes takes place on the smallest scales—less than 1 mpc from Sgr A*—as given by the tidal radius, ${r}_{{\rm{t}}}\approx {R}_{\star }{\left(2{M}_{\bullet }/{m}_{\star }\right)}^{1/3}$,

$$\begin{eqnarray}\begin{array}{rcl}{r}_{{\rm{t}}} & \lesssim & 6000\left(\displaystyle \frac{{R}_{\star }}{30\,{R}_{\odot }}\right){\left(\displaystyle \frac{{m}_{\star }}{1{M}_{\odot }}\right)}^{-\tfrac{1}{3}}{\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}{M}_{\odot }}\right)}^{\tfrac{1}{3}}\,{R}_{\odot }\,\\ & = & 0.14\,\mathrm{mpc}=354{R}_{\mathrm{Schw}}.\end{array}\end{eqnarray} \tag{ 32 }$$

The resonant relaxation process (discussed in Section 4), in particular the scalar resonant relaxation, can cause an increase in orbital eccentricities and thus effectively induce the tidal disruption of giants as their orbital distance decreases below r_t close to the pericenter of their orbits. This could have contributed to the dearth of brighter red giants in the inner ∼0.1 pc (Madigan et al. 2011), which is a larger scale than given by r_t but still smaller than the total extent of 0.5 pc of the red giant hole.

Then, the jet-induced mass removal is clearly the most efficient in the S-cluster region, $z\lesssim 0.04\,\mathrm{pc}$, according to the cumulative mass removal distance profile in Figure 5. Another way to constrain the region of the maximum efficiency is to use the relation in Equation (6) for the stagnation radius, from which we derive the distance z. It follows that for the maximum jet luminosity of ${L}_{{\rm{j}}}={10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and the stagnation radius range of ${R}_{\mathrm{stag}}=4\mbox{--}30\,{R}_{\odot }$, we obtain the distance range ${z}_{\mathrm{jet}}=0.06\mbox{--}0.44\,\mathrm{pc}$. Hence, within the S cluster, $z\lesssim 0.04\,\mathrm{pc}$, R_stag would be effectively below 4 R_⊙, which corresponds to K ∼ 16 mag stars for a typical age of 5 Gyr. Therefore, the jet luminosities close to the Eddington limit for Sgr A* are required to truncate the atmospheres of late-type stars of K ≲ 16 mag. For the moderate jet luminosity of ${L}_{{\rm{j}}}={10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, the distance range decreases by an order of magnitude to z_j = 0.006–0.04 pc; hence, only brighter giants with R_⋆ = 30 R_⊙ (K ∼ 13.5 mag) would be effectively truncated within the S cluster, while the smaller and fainter giants with K ∼ 16 mag would remain largely unaffected by the jet.

Finally, the star–clumpy disk collisions are the most efficient for the disk surface densities ${\rm{\Sigma }}\gt {10}^{4}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ typical of self-gravitating clumps (Kieffer & Bogdanović 2016; Amaro-Seoane et al. 2020), which can form at larger distance scales where the condition for gravitational instability is met as given by the Toomre instability criterion (Milosavljević & Loeb 2004). In the Galactic center, this region likely corresponds to 0.04 ≲ z ≲ 0.5 pc, where the disk population of young massive stars is observed, and they are believed to have formed in situ in a massive gaseous disk (Levin & Beloborodov 2003).

Hence, we speculate that the dearth of bright red giants for z ≲ 0.5 pc is due to the combination of the three processes—tidal stripping, jet-induced atmosphere ablation, and star–disk interactions—that operated the most efficiently at complementary length scales up to ∼0.5 pc.

6.7. Observational Signatures and Falsifiability

We presented a novel scenario to explain the lack of bright red giants in the inner regions of the Galactic center in the sphere of influence of the currently quiescent, but previously active, radio source Sgr A*. Taking this increased activity into account, we focused on the effect of the jet on late-type stars at ≲0.3 pc. By adopting the scenario of the recent active period of Sgr A^⋆, we considered the interaction of red giants with a jet of a typical active Seyfert-like nucleus with the expected kinetic luminosity ${L}_{{\rm{j}}}={10}^{41}\mbox{--}{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Given that red giants have a very slow wind, the jet can significantly ablate the stellar envelope down to at least ∼30 R_⊙ within the S cluster (z ≲ 0.04 pc) after repetitive encounters. Specifically, at z = 0.02 pc, the stagnation radius is 4 ≤ R_stag/R_⊙ ≤ 30 for $2.0\times {10}^{41}\leqslant {L}_{{\rm{j}}}/\mathrm{erg}\,{{\rm{s}}}^{-1}\leqslant 1.1\times {10}^{43}$. Hence, the higher luminosity end that corresponds to the less frequent events that formed the Fermi bubbles can ablate the stellar atmospheres of late-type giants by a factor of ∼7.5 more than the more frequent, less energetic outbursts of Sgr A*.

This truncation is accompanied by the removal of a large fraction of matter, reaching as much as ${\rm{\Delta }}{M}_{1}\approx 3\times {10}^{-5}\,{M}_{\odot }\,\approx 10{M}_{\mathrm{Earth}}$ for red giants with radii R_⋆ > 200 R_⊙ at distances smaller than 0.01 pc for a single encounter. After at least a thousand red giant–jet encounters, we expect the cumulative mass loss of at least ΔM ≈ 10⁻⁴ M_⊙ at the orbital distance of 0.01 pc. This is comparable to the values inferred from red giant–accretion-clump simulations. The proposed mechanism can thus help to explain the presence of late-type stars with the maximum atmosphere radius of ∼30R_⊙ within the S cluster as inferred from the near-infrared observations.

The reduction in the mass and radius of the red giant atmosphere after repetitive jet–star crossings will produce an estimated decrease in the near-infrared K-band magnitude by 1.9, 5.3, and 8.9 mag with respect to the normal evolution at 0.1, 0.01, and 0.001 pc from Sgr A*, respectively. Simultaneously, the color index would decrease to negative values, i.e., the stars should appear bluer with a higher effective temperature. The mean expected number of red giant–jet crossings per orbital period is 3.5 within 0.04 pc, and 82.5 within 0.3 pc, respectively. For the jet kinetic luminosity of ${10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, ∼26.5% of currently detectable late-type stars located at z = 0.04 pc (S cluster) with radii larger than 2.7 R_⊙ and K-band magnitudes smaller than ∼16 mag could be affected by the jet ablation. The estimated numbers of interacting giants can be considered as lower limits because various dynamical effects, such as the coherent resonant relaxation within the nuclear star cluster as well as a potential jet precession would enlarge the number of affected giants.

Constructed surface-brightness profiles of the mock NSC affected by the jet with the luminosity of ${10}^{42}\mbox{--}{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ show that profiles of mainly brighter late-type stars with m_K < 14 mag (dereddened, <16 mag with the foreground extinction included) are flattened by the jet inside the inner arcsecond (≲0.04 pc). Fainter stars keep the initially assumed cusp-like projected profile.

In summary, the interaction of red giants with the jet of Sgr A* during its enhanced activity could contribute to the observed lack of bright red giants and hence affect their surface-brightness profile in the central parts of the NSC. More likely, this mechanism operated in parallel with other previously proposed mechanisms, such as the star–disk interactions, star–star collisions, and TDEs that have different spatial scales of efficiency. Detailed numerical computations of red giant–jet interactions in combination with a modified stellar evolution will help to verify our analytical estimates.

We thank the referee for constructive comments that helped us improve our manuscript. M.Z., B.C., and V.K. acknowledge the continued support by the National Science Center, Poland, grant No. 2017/26/A/ST9/00756 (Maestro 9), and the Czech-Polish mobility program (MSMT 8J20PL037). A.A. and V.K. acknowledge the Czech Science Foundation under grant GACR 20-19854S titled "Particle Acceleration Studies in Astrophysical Jets," and the European Space Agency PRODEX project eXTP in the Czech Republic. This work was carried out partly also within the Collaborative Research Center 956, sub-project [A02], funded by the Deutsche Forschungsgemeinschaft (DFG) project ID184018867. The work on this project was partially carried out during the short-term stay of M.Z. within the Polish-Czech bilateral exchange program supported by NAWA under agreement PPN/BCZ/2019/1/00069. M.Z. also acknowledges the NAWA financial support under agreement PPN/WYM/2019/1/00064 to perform a three-month exchange stay at the Astronomical Institute of the Czech Academy of Sciences in Prague.