Mid-infrared Studies of Dusty Sources in the Galactic Center

Mid-infrared Studies of Dusty Sources in the Galactic Center Harshitha K. Bhat 1,2 , Nadeen B. Sabha 1,3 , Michal Zajaček 1,4 , Andreas Eckart 1,2 , Rainer Schödel 5 , S. Elaheh Hosseini 1,2 , Florian Peißker 1 , and Anton Zensus 2 1 I.Physikalisches Institut der Universität zu Köln, Zülpicher Str. 77, D-50937 Köln, Germany 2 Max-Planck-Institut für Radioastronomie (MPIfR), Auf dem Hügel 69, D-53121 Bonn, Germany 3 Institut für Astround Teilchenphysik, Universität Innsbruck, Technikerstrasse 25/8, A-6929 Innsbruck, Austria 4 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlár ská 2, 611 37 Brno, Czech Republic 5 Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía s/n, E-18008 Granada, Spain Received 2021 November 29; revised 2022 March 21; accepted 2022 March 23; published 2022 April 27 Abstract Mid-infrared (MIR)images of the Galactic center show extended gas and dust features along with bright infrared sources (IRS). Some of these dust features are a part of ionized clumpy streamers orbiting Sgr A * , known as the mini-spiral. We present their proper motions over a 12 yr time period and report their flux densities in N-band filters and derive their spectral indices. The observations were carried out by VISIR at the ESO’s Very Large Telescope. High-pass filtering led to the detection of several resolved filaments and clumps along the mini-spiral. Each source was fit by a 2D Gaussian profile to determine the offsets and aperture sizes. We perform aperture photometry to extract fluxes in two different bands. We present the proper motions of the largest consistent set of resolved and reliably determined sources. In addition to stellar orbital motions, we identify a stream-like motion of extended clumps along the mini-spiral. We also detect MIR counterparts of the radio tail components of the IRS 7 source. They show a clear kinematical deviation with respect to the star. They likely represent Kelvin–Helmholtz instabilities formed downstream in the shocked stellar wind. We also analyze the shape and orientation of the extended late-type IRS 3 star that is consistent with the Atacama Large Millimeter/submillimeter Array submillimeter detection of the source. Its puffed-up envelope with a radius of ∼2×10 6 R e could be the result of the red-giant collision with a nuclear jet, which was followed by tidal prolongation along the orbit. Unified Astronomy Thesaurus concepts: Galactic center (565);Infrared photometry (792);Galaxy kinematics (602) 1. Introduction High-angular-resolution observations of the vicinity of the compact radio source Sgr A * in the Galactic center (GC),which is associated with the supermassive black hole (SMBH; Eckart et al. 2002; Genzel et al. 2010; Falcke & Markoff 2013;Eckart et al. 2017), showed the presence of a ring of dense clumpy molecular and neutral gas and warm dust, called the circumnuclear disk (CND), extending from ∼1.5 to ∼7pc (Vollmer et al. 2004; Christopher et al. 2005; Mills et al. 2013; Hsieh et al. 2021). The CND surrounds an ionized central cavity which has a much lower mean gas density within ∼1–1.5 pc radius with a total mass of ∼60 M e (Lo & Claussen 1983; Blank et al. 2016). At the Bondi radius, r kT0.14 1.3 keV pc BB 1 ~- () , the mean electron number density is n f26 cm eV 12 3 =--(Baganoff et al. 2003),wheref V is the filling factor of 1.3 keV plasma, while the CND density reaches ∼10 6 –10 8 cm −3 in molecular cores (Jackson et al. 1993; Shukla et al. 2004; Christopher et al. 2005).Mossoux&Eckart(2018)found a depression in the X-ray surface brightness at the position of the CND, which could be attributed to the CND acting as a barrier for the hot and diluted plasma in the central cavity. A system of orbiting ionized clumpy streamers and gas filaments extending inwards from the inner edge of the CND is denoted as the mini-spiral (Lo & Claussen 1983; Nitschai et al. 2020, and references therein).The western arc of the mini-spiral appears to have a circular orbit similar to the neutral gas in the CND. However, the northern and the eastern arms penetrate deep into the ionized cavity on eccentric orbits and reach up to a few arcseconds from Sgr A * , possibly colliding in the Bar region (Becklin et al. 1982;Jackson et al. 1993; Christopher et al. 2005;Zhaoetal.2009). There have been several studies on the kinematics of the mini-spiral. Vollmer & Duschl (2000)derived a 3D kinematic model of gas streams that describes the bulk motion of the mini-spiral in three different planes based on a data cube of the [Ne II]line (12.8 μm), with the main plane coinciding with the inner rim of the CND. Paumard et al. (2006)performed a kinematic study and showed that the northern arm consists of a weak continuous surface that is drawn into a narrow stream near Sgr A * . Using L¢ -band data (3.8 μm),Mužićet al. (2007) provided proper motions of a number of thin dusty filaments along the mini-spiral and considered a central, partially collimated outflow as the possible explanation for their formation and motion along the mini-spiral, with some deviations from a purely Keplerian rotation. Based on radio observations using the H92αand H30αlines, Zhao et al. (2009)and Zhao et al. (2010)determined the 3D velocity field of the mini-spiral. The ionized streamers can dynamically be modeled as a system of three bundles of quasi-Keplerian orbits in the potential dominated by the central mass of 10 7 M e , i.e., dominated by Sgr A * (Zhao et al. 2009,2010). The orbital planes of the northern arm and the western arc are nearly coplanar, while the eastern arm plane is perpendicular to them. In the central parsec, there is a nuclear stellar cluster (NSC)that consists predominantly of the nearly spherical old cluster of latetype stars as well as the cusp of ∼100 massive young OB/Wolf– Rayet (WR)stars that supply about 3 ×10 −3 M e yr −1 in the form of stellar winds (Najarro et al. 1997; Moultaka et al. 2004; The Astrophysical Journal, 929:178 (21pp), 2022 April 20 https://doi.org/10.3847/1538-4357/ac6106 © 2022. The Author(s). Published by the American Astronomical Society. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s)and the title of the work, journal citation and DOI. 1 Schödeletal.2014). However, less than 1% of the Bondi accretion rate is accreted by Sgr A * (Bower et al. 2003; Marrone et al. 2006;Wangetal.2013). Blandford & Begelman (1999) proposed a solution for the low accretion rate with the adiabatic inflow-outflow model, in which most of the gas that is accreted has positive energy and is lost through winds and only a small fraction is accreted onto the SMBH. Observational evidence for the outflow first emerged with the detection of the “mini-cavity” region on radio maps by Yusef-Zadeh et al. (1990). The detection of a strong [Fe III]bubble surrounding Sgr A * (Eckart et al. 1992; Lutz et al. 1993)has led to the conclusion that the fast wind originating within the central few arcseconds blows into the orbiting streamers creating an expanding gas bubble. This has also been supported by Peißker et al. (2020b), who found that some identifieddustysourceslocatedtothe west within the S cluster exhibit Doppler-shifted [Fe III]multiplet lines that could be excited by the collimated wind outflowingintheirdirection toward the mini-cavity. The extended region of low excitation centered on Sgr A * (Schödel et al. 2007)and the mass-losing envelope, along with the extended tail of IRS 7 (Yusef-Zadeh & Melia 1992),couldalsobeinfluenced by a strong central wind coming from the central few arcseconds. Closer to Sgr A * ,the orientation of infrared-excess, comet-shaped sources X7, X3, and X8 suggests the presence of a fast, collimated outflow that originates in the Sgr A * accretion flow or in the collective wind of OB/WR stars (Mužićet al. 2010;Peißkeretal.2019;YusefZadeh et al. 2020; Peißker et al. 2021). The nuclear outflow that nearly balances the inflow is also consistent with the flat number density profile of hot plasma, n e ∝r −0.5 , inside the Bondi radius, as inferred from the analysis of the X-ray bremsstrahlung surfacedensity profile (Wang et al. 2013). In a broader context, the past active jet of Sgr A * could have contributed to the depletion of bright red giants in the GC due to the intense stripping and truncation of their extended envelopes (Zajaček et al. 2020a,2020b; Karas et al. 2021). To fully understand the dynamics of these processes, it becomes important to get proper motions or tangential velocities of the various parts of the CND as well as the mini-spiral. The dynamics as well as the physical properties of gaseous-dusty structures within the sphere of the gravitational influence of Sgr A * are crucial for the understanding of the mass and the momentum transport from larger to smaller scales all the way to Sgr A * . In this work, we analyze mid-infrared (MIR)images of the central parsec at two wavelengths in the Nband, the 8.59 (PAH1 filter)and the 13.04 μm(Ne II_2 filter), over the course of 12 yr. This allows us to study the proper motions of infraredexcess sources of the NSC as well as of identified extended objects in the mini-spiral region. The photometric information at two wavelengths enables us to infer the spectral indices that in turn shed light on the properties of the identified sources. We manage to identify MIR components associated with the circumstellar material of two late-type stars, IRS 7 and IRS 3, which manifest their interaction with the circumnuclear medium. The paper is structured as follows. In Section 2, we describe the data set used and the imaging tools that were applied. Subsequently, in Section 3, we describe the main results of the analysis, including the MIR differential map, tangential velocities, and the photometry, including spectral indices. In particular, we focus on the identification of the MIR components of IRS 7, the extended circumstellar structure of IRS 3, and the general characteristics of the identified dusty sources. We discuss the results in Section 4, and subsequently conclude with Section 5. 2. Data and Observations 2.1. VISIR Observations were carried out at the ESO’s Very Large Telescope (VLT, UT3)using the VLT Imager and Spectrometer for the mid-InfraRed (VISIR)at five different epochs covering a total time from 2006 to 2018 in the N-band PAH1 (8.59 μm)and Ne II_2 (13.04 μm). Depending on the atmospheric conditions, VISIR provides 0 25–0 4 angular resolution imaging with high sensitivity. The data used here were obtained as a part of a larger survey (N. B. Sabha et al. 2022, in preparation). The field of view (FOV)and spatial pixel size at each epoch is listed in Table 1. To increase the signal-to-noise ratio (S/N), jittered images with different offsets were added to create a mosaic with a slightly larger FOV. To reduce the bright and varying MIR background, differential observations with the chop/nod mode were executed. 2.2. High-pass Filter MIR images generally show dusty structures, dust-embedded sources, stellar sources, and even emission from the minispiral. Overlapping wings of point-spread functions (PSFs)can create artificial sources and together with noise can complicate the identification of extended objects of interest, especially in a crowded FOV as in the central few arcseconds of the GC. In order to obtain high-angular-resolution information and to highlight the structures of the extended sources, we produce high-pass filter maps. High-pass filters can be used as a sharpener and to resolve the objects that are close to the detection limit while preserving the shape of the extended objects. This filtering technique and its significance are described in detail in Mužićet al. (2007); see also Peißker et al. (2020a,2021). First, the Gaussian-smoothed (3–9 pixel Gaussian, corresponding to 0 375–0 405)version of the input image is subtracted from itself. After removing the negatives, the resulted image is smoothed again using a Gaussian whose size is adjusted depending on the required angular resolution and sensitivity. In Figure 1, we show the effectiveness of this technique for isolating the signals and thereby reducing the chance of confusion between nearby objects. 3. Results 3.1. Differential Map To get a general idea of the direction of the bulk motion, we produce a differential map using the smooth-subtracted images from the 2006 and 2018 epochs. As illustrated in Figure 2, Table 1 Details of the Observed VISIR Images at Each Epoch Year Pixel Scale FOV of a Single Image 2006 0 075 19 2 ×19 2 2007 0 075 19 2 ×19 2 2010 0 127 32 5 ×32 5 2016 0 045 38 0 ×38 0 2018 0 045 38 0 ×38 0 2 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. subtracting the images separated by large timescales essentially represents the subtraction of two slightly shifted Gaussian profiles. The result is a crude but clear indication of the direction of the bulk motion. We first scaled (rebinned)and transformed all PAH1 image frames into the common coordinate system of the 2018 epoch using the positions of IRS 10EE, IRS 9, and IRS 7 (and Ne II_2 image frames using the positions of IRS 10EE, IRS 7, IRS 12N, IRS 15NE, and IRS 17)to calculate the transformation matrix. These positions were corrected for stellar velocities, as reported in the Kband by Schödel et al. (2009)and Genzel et al. (2000). The resulting image (differential map)is presented in Figure 3along with the derived proper motions of the stellar sources, which are discussed in subsequent subsections. 3.2. Proper Motions We first identified the infrared sources (IRS)in N-band images by comparing them with the Kand L′bands. To make sure the proper motions that we calculate are reliable, we make Figure 1. Central 24″×24″region of the Milky Way in the PAH1 filter (8.59 μm)in the top panel and in the Ne II_2 filter (13.04 μm)in the bottom panel. The north is up and the east is to the left. The right panel displays the high-pass-filtered version of the left image. All the dusty-filament-like structures and infrared-excess stars are resolved and enhanced through the high-pass filter. Figure 2. Difference of shifted Gaussian profiles as an indicator of the direction of motion. 3 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. similar calculations for IRS sources in the Kband and compare them both with the velocities reported by Schödel et al. (2009) and Genzel et al. (2000). We chose IRS 10EE, IRS 9, IRS 12N, and IRS 7 as calibrators for PAH1 images (and IRS 10EE, IRS 7, IRS 12N, IRS 15NE, and IRS 17 for Ne II_2 images)as they were unambiguously identifiable in both Nand Kbands and had velocities, v α and v δ , from Schödel et al. (2009).We measure the offsets of each source of interest from these IRS sources. We calculate relative velocities w.r.t. each of them and then add their respective velocities (as given by Schödel et al. 2009), before averaging them to get the proper motions w.r.t. Sgr A * . By doing so, we are minimizing the errors introduced during the process of offset measurements. As an example, we show in Figure 4the v α and v δ plots for the source IRS 1W. Multiplying the proper motions with the distance to the GC, D GC =8.1 kpc, gives the tangential velocity. We compare tangential velocities in the Nband and Kband with Schödel et al. (2009)and Genzel et al. (2000)in Appendix A. Table 5in Appendix Acontains the calculated velocities in various bands along with the literature values. Figures 12 and 13 in Appendix Adepict the deviations from Schödel et al. (2009). It is important to note that the uncertainties in the fainter sources could be larger than in the bright sources due to the possibility of unresolved background sources blending with our target sources. While the uncertainties of the Gaussian fits for the positions is typically of the order of a few hundredth of a pixel, we conservatively assume an uncertainty of 0.25 pixels for that quantity. The mean absolute difference between the velocities in the two filters is about 100 km s −1 and standard deviation of the angle difference is about 40°. We then move on to calculate the tangential velocities of all the extended sources in the FOV, concentrating especially on the inner edge of the northern arm. We determine the positions of each resolved source by fitting an elliptical Gaussian using the data visualization tool QFitsView (Ott 2012). We derive tangential velocities as described above and tabulate them in Table 2. We only report velocities of those sources that are free from confusion in at least three epochs. The uncertainties in the combined R.A./decl. velocities are about ±120 km s −1 and the uncertainties in the flight direction are about ±30°. Figure 5 shows the source labels and the apertures used in the left panel and their derived proper motion vectors in the right panel. We have removed the velocity vectors belonging to the IRS sources to get a clear indication of any bulk motion. We see a clear stream-like motion along the inner edge of the northern arm toward the southwest direction, which changes direction sharply as it crosses the Sgr A * to move in the northwest direction (blue dashed arrow in Figure 6). We assume that within the area of the mini-spiral’s northern arm we find N NA sources that truly belong to the northern arm. The number of sources in that region which belong to the underlying cluster is N C . We assume that N C is a fraction of N NA , i.e., NfN.1 CNA =´ () The total number of sources detected within the area of the mini-spiral’s northern arm is then given by NNN N f1. 2 Ctot NA NA =+=´+() () Therefore, even if all the N NA sources move downstream one would not expect that all sources observed in that region follow that trend. While we assume that all the N NA sources in the northern arm will be headed downstream along the mini-spiral arm, we also assume in a very simplistic way that half of the cluster sources, N C , in that area will be moving downstream and half of them upstream. This implies that the ratio, R, between the sources moving upand downstream is RNN N Nf N Nf 0.5 0.5 0.5 0.5 21. 3 C C CC C NA =+=+=+ () As can be seen in Figure 5,wefind that in the mini-spiral region 11 sources move downstream and four sources move upstream, implying R=11/4=2.75. Hence, we find that Rf 212.75. 4=+= () Therefore, that fraction, f, turns out to be just above unity, with f2 2.75 1 1.14, 5=-=() just as expected for an additional contribution of dusty infraredexcess sources due to the northern arm on top of a cluster contribution with conceivably almost the same number density in dusty, stellar infrared-excess sources. For an equal contribution from the northern arm and the cluster in the region of the northern arm, we would have expected R=3. 3.3. Photometry Even though there have been many photometric studies on stars in the GC in various other wavelengths, they are limited in the MIR (e.g., Blum et al. 1996; Ott et al. 1999; Tanner et al. 2002; Viehmann et al. 2005,2006). We carry on these studies on the extended dusty sources that we have identified. Viehmann et al. (2006)report flux densities in various filters including the Nband. In the MIR regime, they performed relative aperture photometry while relying on Tanner et al. (2002)for flux calibration. Contrary to what is mentioned in Viehmann et al. (2006), it appears that the reported N-band flux densities are not extinction-corrected. For example, Figure 3. Differential map produced by subtracting the PAH1 2006 image from the PAH1 2018 image. The arrows correspond to velocities obtained by measuring offset positions and velocities w.r.t. IRS 10EE, 9, and 7. 4 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. Viehmann et al. (2006)report IRS 21’sflux density as 4.56 Jy at 8.6 μm, which is consistent with 3.6 Jy by Stolovy et al. (1996) at 8.7 μm and 3.6 by Tanner et al. (2002)at 8.8 μm. Stolovy et al.’s(1996)results are based on the zero-point measurements made with SpectroCam-10 on the 200 inch Hale telescope. As both of these previous studies did not report dereddened fluxes, if Viehmann et al.’s(2006)values were, in fact, extinctioncorrected we would expect it to be about 5 times higher. This finding is further confirmed by two other independent studies by N. B. Sabha et al. (2022, in preparation)and R. Schödel et al. (2022, in preparation; through private communication), both based on zero-point measurements provided by the ESO. The flux densities were extracted via aperture photometry using elliptical apertures to match the shape of extended or filamentary sources (see Figure 5). Aperture sizes were determined by fitting a 2D Gaussian along the semimajor and semiminor axes of the source. We selected IRS 5NE, IRS 10W, IRS 7, and IRS 1W as calibrators using a source aperture and background similar to Viehmann et al. (2006). As all the elliptical apertures we use do not have the same area, we measured the surface density of background contribution at various apertures in the uncrowded regions of the image and multiplied it by the area of each ellipse to get their background contributions. An extinction correction of A λ ∼2.04 for PAH1 and A λ ∼1.34 for Ne II (Fritz et al. 2011)was applied to obtain dereddened flux densities. Table 6in Appendix Blists the flux density values by Viehmann et al. (2006), our results when we use similar apertures and background as Viehmann et al. (2006), and our results when we use aperture sizes determined by the FWHM of the 2D Gaussian that we fit(our results in the table are before extinction correction). The table shows that our calibration approach results in source flux densities that are in good agreement with the results from Viehmann et al. (2006). It is to be noted that the central wavelength of the Ne II_2 filter used for our images is 13.04 μm(Ne II_2), while that of Viehmann et al.’s(2006)images is 12.81 μm(Ne II). For α s ±1 this difference corresponds to a 2% variation in flux densities. Table 3lists both the reddened (F)and dereddened (F′)flux densities of all the reliable sources at both PAH1 (8.59 μm)and Ne II_2 (13.04 μm), along with their spectral indices, which were calculated using FFlog log 8.59 13.04 ,6 s13.04 8.59 a=¢¢() () () i.e., using the convention F s nµa+or Fs lµa-. In particular, hotter sources (nonembedded stars)are characterized by a positive spectral index in this convention, while dustenshrouded stars, colder dusty filaments, or potential compact objects powered by nonthermal synchrotron emission (neutron stars)exhibit a steep power-law spectrum with a negative spectral index in the MIR domain. In Figure 7, we depict the spectral indices of each source. The color and the size of each source indicate the spectral index and the flux density at the PAH1 band, respectively. In Figure 7, it is apparent that the brightest infrared-excess sources are also warm and blue, while the fainter and colder dust sources are along the mini-spiral. 3.4. Cometary Tail of IRS 7 IRS 7 (about 5 5 north of Sgr A * )is one of the brightest IR sources in the region and is classified as a pulsating M1/M2 red supergiant (Carr et al. 2000; Paumard et al. 2014; Gravity Collaboration et al. 2021). Radio (Yusef-Zadeh & Morris 1991; Yusef-Zadeh & Melia 1992)and MIR (Serabyn et al. 1991) observations of IRS 7 have revealed a bow-shock feature toward the north and a cometary-tail-like structure pointed directly away from Sgr A * . Recently, Tsuboi et al. (2020)also reported the shell-like structure surrounding IRS 7 and its northern extension in the H30αrecombination line. They measured line-of-sight velocities and concluded that the tail is the gas stream flowing from the shell around IRS 7. As IRS 7 moves southward, the pulsating release of gas as a stellar wind (Paumard et al. 2014)is left behind and is ionized by far-UV radiation coming from the NSC. Figure 4. Proper motion, v α and v δ , of an example source IRS 1W. Relative velocities w.r.t. IRS 10EE, 7, 12N, and 9 were determined using offset measurements. Error bars correspond to 0.25 pixels. 5 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. Table 2 List of Tangential Velocities PAH1 Ne II_2 Source Name ΔαΔδv α v δ v α v δ (arcsec)(arcsec)(all velocities are in km s −1 ) 1 IRS 5NE 12.71 10.55 −374 243 −167 −167 2 IRS 5E 10.87 9.27 −232 155 −216 −211 3 IRS 5S 9.07 7.97 −218 268 −75 97 4 IRS 5 8.53 9.67 −303 167 −71 −145 5 10.60 6.87 −223 286 −226 1 6 IRS 17 12.97 5.45 −351 198 −238 −83 7 6.75 12.08 −151 −345 −187 −279 8 6.68 9.95 −350 −472 −55 −52 9 5.00 8.10 −210 −96 10 IRS 15NE 1.17 11.04 −67 −133 −6−308 11 IRS 10EE 7.60 4.07 −157 54 −186 142 12 IRS 10W 6.45 5.08 −118 331 120 148 13 IRS 7 0.03 5.34 27 −125 232 −250 14 IRS 3 −2.26 3.70 282 −2 412 −13 15 4.90 4.41 −77 −3−7−182 16 4.69 2.97 −180 −132 −215 −215 17 6.10 0.24 159 178 20 −51 18 IRS 1W 5.21 0.57 −113 307 53 275 19 3.67 0.22 −221 −337 301 −218 20 2.34 −0.40 −360 −330 163 −205 21 3.54 −1.67 −136 −162 −25 −303 22 IRS 21 2.29 −2.78 −137 8 −65 3 23 1.47 −2.07 38 −93 196 17 24 −1.57 −0.53 −116 218 −170 39 25 −0.78 −1.61 −108 207 −180 255 26 −0.33 −2.44 −61 −225 −395 −33 27 10.00 −8.83 −21 −181 −62 −159 28 7.10 −7.83 −184 −80 173 −277 29 5.70 −7.86 −74 −120 −259 −455 30 5.60 −5.89 −16 −138 63 −175 31 4.82 −6.39 −89 −228 316 37 32 14.38 −5.71 −128 61 −72 −161 33 11.07 −5.41 −191 −160 −44 −267 34 10.47 −5.97 103 −313 −135 −7 35 6.61 −1.80 −142 −126 −141 −195 36 6.35 −2.96 20 −157 64 −193 37 2.84 −5.14 −126 63 −185 7 38 3.66 −5.29 −53 −70 55 −3 39 0.26 −3.57 −51 −209 −189 −198 40 −1.14 −4.07 −98 −52 −57 −353 41 −1.41 −3.06 −139 166 6 −209 42 −2.47 −2.42 86 180 76 175 43 0.58 −1.94 −50 5 −295 −7 44 IRS 29 −1.50 1.22 256 −186 585 −205 45 −3.54 2.05 −71 374 −89 502 46 −3.19 −1.67 41 144 93 122 47 IRS 2L −3.39 −3.94 173 −2 293 191 48 −5.44 2.50 198 92 362 74 49 −5.39 1.39 36 138 699 92 50 −5.90 0.33 −74 104 418 92 51 −6.00 −1.03 202 −87 100 185 52 −4.34 −1.70 −61 −21 185 180 53 −5.24 −0.11 −27 45 546 −85 54 −7.12 −1.73 329 206 368 −79 55 −7.75 −0.19 203 −81 273 158 56 −8.30 1.38 −297 20 98 181 57 −9.82 0.67 −32 438 123 179 58 −7.36 1.28 96 234 15 160 59 2.66 −4.41 −46 93 −84 357 60 IRS 2S −3.78 −5.59 180 −124 82 −230 61 −2.81 −5.53 27 −168 −255 −15 62 −2.23 −5.60 −230 −137 168 −139 6 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. Table 2 (Continued) 63 −3.10 −4.81 −79 −13 331 36 64 −2.45 −4.47 30 166 −71 287 65 −1.63 −5.04 −49 −149 −366 145 66 −4.24 −7.01 −222 132 −100 −24 67 IRS 12N −3.24 −6.99 80 −133 −5−106 68 −3.01 −6.11 −39 −50 −127 123 69 −1.97 −6.56 91 −310 −326 −66 70 −4.52 −5.99 −191 −52 1 343 71 4.42 1.53 351 259 132 −271 72 3.55 1.47 −72 −68 −150 −244 73 3.85 3.01 −373 −419 334 −96 74 3.26 3.35 48 220 −21 −593 75 3.77 7.10 −16 188 76 3.60 6.23 −22 261 −146 −338 77 3.24 5.16 47 105 108 173 78 0.02 8.46 302 231 79 −0.09 7.73 297 211 516 126 80 −0.27 7.16 268 −301 990 250 81 −2.30 8.38 −193 340 970 −529 82 10.44 1.18 −90 189 83 8.87 0.49 −27 207 84 4.25 1.91 41 −273 85 IRS 16NE 2.87 0.81 −83 −224 86 1.27 0.85 −289 −79 87 IRS 16C 0.93 0.51 −513 358 88 IRS 16NW 0.12 1.13 351 54 89 IRS 29NE −1.15 1.93 −435 7 90 IRS 9 5.66 −6.40 97 −22 91 1.59 −0.53 −227 −289 92 1.13 −1.39 88 102 93 −1.89 −2.02 −95 42 94 −2.53 −1.41 150 220 95 −2.92 −0.88 114 247 96 IRS 34 −3.91 1.56 119 166 97 −6.37 2.97 −36 112 98 −6.98 2.90 237 −133 99 −6.72 2.42 −60 118 100 1.69 −4.39 −88 −190 −100 −91 101 0.87 −3.12 −335 −538 102 −0.39 −3.15 −258 81 −105 −43 103 −0.69 −3.69 −342 −47 104 −1.90 −3.22 24 166 −205 13 105 1.35 −6.69 80 −113 112 100 106 1.30 −7.13 −65 −157 107 IRS 14NE 0.95 −8.22 223 −130 153 −91 108 −1.25 −8.69 −47 −546 109 3.66 −11.41 −3−456 110 4.24 −8.74 5 −164 111 4.28 −7.38 −20 −195 112 3.75 −7.31 82 −316 113 3.00 −7.70 −213 −235 114 2.68 −7.26 −367 −5 115 3.07 −6.62 −24 −84 −458 338 116 3.08 −5.82 77 18 117 IRS 30E −5.52 5.53 435 −59 118 IRS 30W −6.50 5.94 −125 384 119 IRS 6E −5.07 0.76 294 234 398 181 120 AF/AHH −6.46 −6.93 152 −85 121 IRS 14SW −0.18 −8.88 −162 −80 Note. The positions are offsets from Sgr A * (in 2018)in arcsecs and all the velocities are in km s −1 . We chose a conservative 0.25 pixel uncertainty for proper motion calculations, which corresponds to about 45 km s −1 . 7 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. In the right panel of Figure 8,weshowthePAH1(8.59 μm) image of IRS 7 and its extended tail in comparison with the λ=2 cm contour map from Yusef-Zadeh & Melia (1992), which is on the left. Contours of individual substructures of the tail in both wavelengths seem to match well with each other, which is an indication of the quality of our data and filtering techniques. The tail extends from 1″to 3 6northofIRS7andhas threee major substructures. Blue arrows mark the proper motions and gray lines mark their error boundaries. The slight discrepancy in the direction of the transverse velocity in the MIR/near-IR bands (see Appendix A), and by using SiO maser astrometry (Reid et al. 2003; Borkar et al. 2020), could be caused by the fact that they probe different layers of the star. The proper-motion measurements of the substructures of the tail reveal the dominant influences on each of them. The northern two, which have detached from IRS 7 earliest, have almost similar velocities. This could mean that they are driven by a nuclear outflow coming from the inner few arcseconds. In addition, the proper-motion distribution of the three components reflects the combination of the downstream fluid motion in the IRS 7 tail and the development of hydrodynamic instabilities, such as Kelvin–Helmholtz (KH)instabilities, which are manifested as propagating waves moving transverse along the south–north direction of the flow, i.e., they introduce an additional turbulent velocity field to the predominant south–north downstream bow-shock flow. The development of the KH instability is expected due to the velocity shear between the shocked stellar wind of IRS 7 and the surrounding hot medium. Assuming that the MIR–radio tail clumps formed due to the KH instability, we adopt their typical length scale, λ tail ∼0 5 ∼0.02 pc, from Figure 8. The density and the temperature of the tail components, n tail ∼6×10 4 cm −3 and T tail ∼4650 K, respectively, can be inferred from H30αrecombination-line observations (Tsuboi et al. 2020). These tail components are approximately in pressure equilibrium with the surrounding hot medium, under the assumption of the extrapolation of the Bondi-radius values, T a ∼10 7 Kandn a ∼26 cm −3 (Baganoff et al. 2003),i.e., n tail T tail ∼n a T a . This implies a density ratio between the ambient medium and the IRS 7 tail of r=n a /n tail ∼4.3 ×10 −4 .Forthe shear velocity, we take the mean of the IRS 7 stellar motions, v shear ;v å ∼180 km s −1 , according to Table 5. The growth timescale of KH instabilities of the size λ tail can be estimated as v r r 1111 yr. 7 KH tail shear tl ~+~() Since the crossing timescale of the IRS 7 for the total length of the tail, l tail ∼3″,isτ cross ∼l tail /v å ∼652 yr τ KH , the KH Figure 5. Left: the PAH1 image with the identification for each source and the apertures used to perform photometry. Right: proper motion of all the labeled sources. Arrows corresponding to IRS sources are removed for a better visualization of the stream-like motion. Figure 6. Presence of a stream-like motion along the inner edge of the northern arm toward the southwest direction, which changes direction sharply as it crosses the Sgr A * to move in the northwest direction (blue dashed arrow). 8 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. Table 3 List of Reliable Flux Densities in PAH1 and Ne II_2 Filters Source Name F 8.59 ΔF 8.59 F8.59 ¢F8.59 D¢ F 13.04 ΔF 13.04 F13.04 ¢F13.0 4 D¢ α s Δα s (Jy)(Jy)(Jy)(Jy)(Jy)(Jy)(Jy)(Jy) 1 IRS 5NE 0.57 0.09 3.75 0.58 1 0.09 3.43 0.3 0.21 0.43 2 IRS 5E 0.49 0.08 3.22 0.50 1.99 0.17 6.84 0.58 −1.80 0.42 3 IRS 5S 0.61 0.09 4.01 0.62 2.35 0.2 8.07 0.69 −1.68 0.42 4 IRS 5 3.92 0.60 25.64 3.90 4.54 0.38 15.58 1.32 1.19 0.42 5 0.54 0.08 3.50 0.54 1.83 0.16 6.3 0.54 −1.41 0.42 6 IRS 17 0.02 0.01 0.11 0.02 0.27 0.02 0.94 0.09 −5.14 0.49 7 0.33 0.05 2.18 0.34 3.47 0.29 11.91 1.01 −4.07 0.43 8 1.38 0.21 9.02 1.39 6.33 0.54 21.73 1.84 −2.11 0.42 9 0.21 0.03 1.38 0.22 5.36 0.46 18.43 1.56 −6.21 0.43 10 IRS 15NE 0.04 0.01 0.26 0.05 0.09 0.01 0.3 0.03 −0.34 0.52 11 IRS 10EE 0.46 0.07 3.02 0.47 1.22 0.1 4.19 0.36 −0.78 0.43 12 IRS 10W 8.33 1.27 54.51 8.28 11.61 0.98 39.87 3.36 0.75 0.42 13 IRS 7 1.15 0.18 7.54 1.15 2.04 0.17 7 0.6 0.18 0.42 14 IRS 3 8.65 1.31 56.59 8.60 11.75 0.99 40.37 3.4 0.81 0.42 15 4.00 0.61 26.17 3.99 9.39 0.79 32.26 2.72 −0.50 0.42 16 1.64 0.25 10.73 1.64 9.27 0.78 31.83 2.69 −2.60 0.42 17 6.67 1.01 43.64 6.63 11.87 1 40.76 3.44 0.16 0.42 18 IRS 1W 16.14 2.45 105.66 16.05 21.97 1.85 75.47 6.36 0.81 0.42 19 3.99 0.61 26.15 3.98 12.64 1.07 43.44 3.67 −1.22 0.42 20 3.03 0.46 19.85 3.02 5.28 0.45 18.14 1.53 0.22 0.42 21 3.54 0.54 23.19 3.53 12.48 1.05 42.86 3.62 −1.47 0.42 22 IRS 21 4.79 0.73 31.39 4.77 6.92 0.58 23.77 2.01 0.67 0.42 23 2.37 0.36 15.52 2.36 11.58 0.98 39.79 3.36 −2.26 0.42 24 0.49 0.08 3.24 0.50 1.49 0.13 5.11 0.44 −1.09 0.42 25 0.81 0.12 5.28 0.81 2.34 0.2 8.02 0.68 −1.00 0.42 26 1.86 0.28 12.18 1.86 6.16 0.52 21.15 1.79 −1.32 0.42 27 0.42 0.07 2.74 0.43 2.78 0.24 9.54 0.81 −2.99 0.43 28 0.33 0.05 2.13 0.33 1.78 0.15 6.12 0.52 −2.53 0.42 29 0.74 0.11 4.82 0.74 3.61 0.31 12.39 1.05 −2.26 0.42 30 1.10 0.17 7.18 1.10 2.95 0.25 10.14 0.86 −0.83 0.42 31 2.26 0.35 14.81 2.26 6.09 0.52 20.92 1.77 −0.83 0.42 32 0.60 0.09 3.91 0.61 4.45 0.38 15.28 1.3 −3.27 0.43 33 0.49 0.08 3.19 0.50 3.44 0.29 11.83 1.01 −3.14 0.43 34 0.18 0.03 1.15 0.18 1.89 0.16 6.48 0.55 −4.14 0.43 35 0.59 0.09 3.87 0.60 2.99 0.25 10.28 0.87 −2.34 0.42 36 0.59 0.09 3.86 0.60 2.65 0.23 9.11 0.78 −2.06 0.43 37 1.32 0.20 8.65 1.32 3.72 0.32 12.79 1.08 −0.94 0.42 38 3.05 0.47 19.98 3.04 8.05 0.68 27.67 2.34 −0.78 0.42 39 0.96 0.15 6.29 0.96 4.24 0.36 14.56 1.23 −2.01 0.42 40 0.58 0.09 3.78 0.58 2.25 0.19 7.72 0.66 −1.71 0.42 41 0.81 0.12 5.30 0.81 2.05 0.17 7.04 0.6 −0.68 0.42 42 1.06 0.16 6.92 1.06 2.26 0.19 7.77 0.66 −0.28 0.42 43 2.33 0.36 15.24 2.33 6.6 0.56 22.68 1.92 −0.95 0.42 44 IRS 29 0.13 0.02 0.85 0.14 0.14 0.01 0.48 0.05 1.37 0.47 45 0.14 0.02 0.94 0.15 0.61 0.05 2.09 0.18 −1.91 0.43 46 2.41 0.37 15.80 2.41 4.08 0.35 14 1.19 0.29 0.42 47 IRS 2L 3.28 0.50 21.49 3.27 5.52 0.47 18.98 1.6 0.30 0.42 48 0.64 0.10 4.20 0.65 1.48 0.13 5.09 0.44 −0.46 0.42 49 0.57 0.09 3.72 0.57 2.5 0.21 8.6 0.73 −2.01 0.42 50 1.33 0.20 8.71 1.33 4.11 0.35 14.14 1.2 −1.16 0.42 51 1.60 0.25 10.49 1.61 5.88 0.5 20.21 1.71 −1.57 0.42 52 0.97 0.15 6.34 0.97 7.58 0.64 26.04 2.21 −3.38 0.42 53 0.46 0.07 3.01 0.46 1.51 0.13 5.17 0.44 −1.30 0.42 54 0.34 0.05 2.24 0.35 1.39 0.12 4.79 0.41 −1.82 0.43 55 0.55 0.09 3.61 0.56 2.09 0.18 7.19 0.61 −1.65 0.42 56 0.82 0.13 5.38 0.83 3.65 0.31 12.55 1.07 −2.03 0.42 57 0.03 0.01 0.22 0.04 0.47 0.04 1.61 0.14 −4.77 0.48 58 4.07 0.62 26.65 4.06 7.26 0.61 24.95 2.11 0.16 0.42 59 0.66 0.10 4.34 0.67 2.77 0.23 9.5 0.81 −1.88 0.42 60 IRS 2S 1.93 0.29 12.64 1.93 3.67 0.31 12.61 1.07 0.01 0.42 61 1.68 0.26 11.00 1.68 4.4 0.37 15.1 1.28 −0.76 0.42 62 1.37 0.21 8.97 1.37 5.63 0.48 19.36 1.64 −1.84 0.42 63 1.73 0.26 11.36 1.73 3.83 0.32 13.17 1.12 −0.35 0.42 9 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. Table 4 Spearman Correlation Coefficients between Quantities that Characterize Compact MIR Dusty Sources Length Width Angle Flux (PAH1)Flux (Ne II_2)Spectral index Ellipticity Length L0.064 (0.524)−0.098 (0.326)0.260 (0.008)0.404 (2.310 ×10 −5 )−0.136 (0.171)0.715 (2.04 ×10 −17 ) Width 0.064 (0.524)L−0.091(0.361)0.146(0.140)0.235(0.017)−0.078(0.432)0.048(0.630) Angle −0.098 (0.326)−0.091(0.361)L−0.074(0.457)−0.180(0.069)0.089 (0.372)−0.106(0.287) Flux (PAH1)0.260 (0.008)0.146(0.140)−0.074(0.457)L0.850(7.36 ×10 −30 )0.510(3.65 ×10 −8 )0.307(0.0016) Flux (Ne II_2)0.404 (2.310 ×10 −5 )0.235(0.017)−0.180(0.069)0.850(7.36 ×10 −30 )L0.081(0.419)0.382(6.70 ×10 −5 ) Spectral Index −0.136 (0.171)−0.078(0.432)0.089 (0.372)0.510(3.65 ×10 −8 )0.081(0.419)L−0.073(0.466) Ellipticity 0.715 (2.04 ×10 −17 )0.048(0.630)−0.106(0.287)0.307(0.0016)0.382(6.70 ×10 −5 )−0.073(0.466)L Note. In parentheses, we include the p-value. Five significant positive correlations were found: length flux (Ne II_2), length–ellipticity, flux (PAH1)–flux (Ne II_2),flux (PAH1)–spectral index, and flux (Ne II_2)– ellipticity. 16 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. the relation based on Rieke et al. (1978)and Kunneriath et al. (2012), MFD BT a Q, 4 3,15 dGC 2 d d n nn r=() ()() () where F(ν)is the measured flux density and B(ν,T d )is the Planck function calculated for a dust temperature of T d ∼200 K (Cotera et al. 1999). The dust is characterized by the mean values of the radius, a=0.1 μm, the mass density, ρ d =2gcm −3 , and the emissivity, Q≈10 −3 –10 −2 close to 10 μm(Aannestad 1975;Riekeetal.1978).The distance to the GC is set to D GC =8.1 kpc. Assuming a gasto-dust ratio of ∼100, we obtain a mean gas mass of the minispiral clump, MMPAH1 0.046 0.46 g ~() – and MNe ii_2 g ~() M 0 .015 0.15  –, as based on 85 dusty filaments that are not associated with stellar sources. These values are consistent within a factor of 3. Considering the peak width of the clumps, w∼2000 au, and the peak length, l∼3000 au, we obtain the characteristic clump volume of V clump ∼3.2 × 10 49 cm 3 , which yields a filament number density in the range of n clump ∼1.1 ×10 6 −3.4 ×10 7 cm −3 , which is at least one order of magnitude more than the electron number density of the ionized component, n e =(3–21)×10 4 cm −3 (Zhao et al. 2010).Thefilaments could thus be overdense regions that are either pressure-confined by the stellar winds of OB/WR stars or they could stand for KH instabilities that got denser due to radiative cooling. This supports the multiphase nature of the mini-spiral streamers, with denser filaments embedded within a more diluted ionized gas (Różańska et al. 2014).Denser dusty filaments could also be the sites of the waterand COice features and hydrocarbons detected within the central parsec (Moultaka et al. 2015a,2015b). If the mean clump gas mass is in the range of ∼0.01–0.1 M e and we have ∼100 filaments, then their total gas mass of ∼1–10 M e is consistent with the total ionized gas mass of ∼60 M e within the central cavity (Lo & Claussen 1983).Thedenserfilaments are currently not massive and dense enough to form stellar and substellar objects. As discussed in the previous section, they are transient features formed via the KH instability along the streaming motion and they evaporate on a timescale of ∼100–1000 yr. The filaments are also expected to be tidally elongated along the streaming motion of the mini-spiral during their lifetime. Mužićet al. (2007)show that the shape and motion of the mini-spiral filaments do not agree with a purely Keplerian motion of gas in the potential of the SMBH at the position of Sgr A * . The authors involve additional mechanisms that are responsible for the formation and the motion of these filaments. They assume that the filaments are affected by an outflow from the disk of young mass-losing stars around Sgr A * . In addition, an outflow from the Sgr A * black hole region itself may be responsible for the elongated shape and the motion of the filaments. 5. Summary We studied MIR images of the central parsec of the GC in the Nband (8.6 and 13.04 μm). As the MIR emission is dominated by dust and extended regions around the central SMBH, we applied a high-pass filter on the images to resolve and identify the sources. We present the proper motions of these extended objects over a 12 yr time period. There are two distinct types of the observed motion: one related to infraredexcess sources of the central stellar cluster and the other a stream-like motion of extended objects along the mini-spiral streamer. We also present the flux densities of all the sources using elliptical apertures. Using the spectral indices, we infer that the MIR region is dominated by dust-enshrouded stars or colder dusty filaments and the temperature of ∼200 K (Cotera et al. 1999)is at least the lower limit of infrared-excess sources within the mini-spiral. We detect a bow-shock feature and tail components of IRS 7 that are pointed away from Sgr A * . The proper-motion distribution of the individual tail components can be interpreted with a combination of downstream fluid motion and the development of KH instabilities. We detect and resolve the brightest MIR source in the region, IRS 3. The extended structure of the star is likely a result of its atmosphere’s perturbation followed by tidal prolongation. We also report on the nature of all the dusty sources and delve into their possible origins. This work was supported in part by SFB 956, “Conditions and Impact of Star Formation.”H.B. and E.H. are members of the International Max Planck Research School for Astronomy and Astrophysics at the Universities of Bonn and Cologne. We thank the Collaborative Research Centre 956, subproject A02, funded by the Deutsche Forschungsgemeinschaft (DFG), project ID 184018867. N.B.S. acknowledges financial support from the Austrian National Science Foundation through a FWF standalone grant No. P31154-N27. M.Z. acknowledges financial support by the GAČR EXPRO grant No. 21-13491X, “Exploring the Hot Universe and Understanding Cosmic Feedback.”R.S. acknowledges financial support from the State Agency for Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa”award for the Instituto de Astrofisica de Andalucia (grant No. SEV-2017-0709)and financial support from national project PGC2018-095049-BC21 (MCIU/AEI/FEDER, UE). Appendix A Tangential Velocity Comparisons between MIR and Kband Velocities A comparison between the K-band tangential velocity (Schödel et al. 2009; Genzel et al. 2000)and our MIR velocities for the point sources (see Table 5)looks favorable. The mean absolute difference of around ±100 km s −1 is most likely affected by the larger PSF in the MIR, the limited baseline in time, and the fact that even the point sources may show some IR excess/extension or they are located on background emission that is spatially structured/variable on the scales of the PSF. Figures 12 and 13 in depict the deviations from Schödel et al. (2009). 17 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. Figure 12. Comparison velocities from the Kband and from Schödel et al. (2009)and Genzel et al. (2000). Figure 13. Angle deviations from Schödel et al. (2009)and Genzel et al. (2000)values in all the wavelengths. Mean absolute deviation in velocity (R.A.)is about 102 km s −1 , in velocity (decl.)is about 76 km s −1 , and the standard deviation of the angle difference is about 40°. 18 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. Appendix B N-band Flux Density Comparison Table 6presents a comparison of our measured flux densities with Viehmann et al. (2006), using both circular apertures and elliptic apertures determined by their FWHM. Table 5 Comparison of Stellar Tangential Velocities in PAH1 and Ne II_2 Bands with those in the KBand PAH1 Ne II K S09 G00 Source v α v δ v α v δ v α Δv α v δ Δv δ v α Δv α v δ Δv δ v α Δv α v δ Δv δ IRS 5 −303 167 −71 −145 −267 6 84 12 IRS 10W −118 331 120 148 −29 6 115 10 IRS 10EE * −157 54 −186 142 −44 6 −16 10 −15 7 −60 7 IRS 1W −113 307 53 275 −9 6 131 10 IRS 16NE −83 -224 104 6 −281 10 199 65 −279 21 IRS 16C −513 358 −211 6 108 10 −330 39 353 34 IRS 21 −138 8 −65 3 −16−43 10 −159 65 64 38 IRS 9 * 97 −22 176 6 90 10 127 10 116 10 IRS 2S 180 −124 82 −230 352 8 −368 14 IRS 2L 173 −2 293 191 174 8 −273 10 IRS 29 256 −186 585 −205 165 6 −261 10 IRS 29NE −435 7 −215 6 −151 10 IRS 34 119 166 −24 8 −335 10 IRS 6E 294 234 398 −206 141 6 −39 10 IRS 3 282 −2 412 −13 84 8 −137 10 170 40 115 45 IRS 7 * 27 −125 232 −250 −23 8 −193 12 −29−176 9 100 67 −118 35 IRS 12N * 80 −133 −5−106 −62 8 −107 8 IRS 15NE * −67 −133 −6−308 −58 6 −223 6 IRS 17 * −351 198 −238 −83 −65 5 −40 5 Note. All velocities are in kilometers per second. S09 is the data from Schödel et al. (2009). G00 is the data from Genzel et al. (2000). Sources used as calibrators are marked with * . We chose a conservative 0.25 pixel uncertainty for PAH1 and Ne II_2 proper motions, which corresponds to about 45 km s −1 . 19 The Astrophysical Journal, 929:178 (21pp), 2022 April 20 Bhat et al. ORCID iDs Harshitha K. Bhat https://orcid.org/0000-0002-4408-0650 Nadeen B. Sabha https://orcid.org/0000-0001-7134-9005 Michal Zajaček https://orcid.org/0000-0001-6450-1187 Andreas Eckart https://orcid.org/0000-0001-6049-3132 Rainer Schödel https://orcid.org/0000-0001-5404-797X S. Elaheh Hosseini https://orcid.org/0000-0002-3004-6208 Florian Peißker https://orcid.org/0000-0002-9850-2708 Anton Zensus https://orcid.org/0000-0001-7470-3321 References Aannestad, P. A. 1975, ApJ,200, 30 Aitken, D. K., Smith, C. H., Moore, T. J. T., & Roche, P. F. 1998, MNRAS, 299, 743 Baganoff, F. 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