Tracing Milky Way scattering by compact extragalactic radio sources

MNRAS 000, 1–14 (2022) Preprint 13 January 2022 Compiled using MNRAS LATEX style file v3.0

 Tracing Milky Way scattering by compact extragalactic radio sources
 T. A. Koryukova,1★ A. B. Pushkarev,2,1 A. V. Plavin,1,3 Y. Y. Kovalev1,3,4
 1 LebedevPhysical Institute of the Russian Academy of Sciences, Leninsky prospekt 53, 119991 Moscow, Russia
 2 Crimean Astrophysical Observatory, Nauchny 298688, Crimea, Russia
 3 Moscow Institute of Physics and Technology, Institutsky per. 9, Dolgoprudny 141700, Russia
 4 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
arXiv:2201.04359v1 [astro-ph.GA] 12 Jan 2022

 Accepted XXX. Received YYY; in original form 2021 December 30

 ABSTRACT
 We used archival very long baseline interferometry (VLBI) data of active galactic nuclei (AGN) observed from 1.4 GHz
 to 86 GHz to measure the angular size of VLBI radio cores in 9525 AGNs. We analysed their sky distributions, frequency
 dependencies and created the distribution map of large-scale scattering properties of the interstellar medium in our Galaxy for
 the first time ever. Significant angular broadening of the measured AGN core sizes is detected for the sources seen through the
 Galactic plane, and this effect is especially strong at low frequencies (e.g. at 2 GHz). The scattering screens containing electron
 density fluctuations of hot plasma are mainly concentrated in the Galactic plane and manifest clumpy distribution. The region of
 the strongest scattering is the Galactic centre, where the Galactic bar and the compact radio source Sagittarius A∗ are located.
 We have also found the enhancement of scattering strength in regions of the Cygnus constellation, supernova remnants Taurus A,
 Vela, W78 and Cassiopeia A, and the Orion Nebula. Using multi-frequency observational data of AGN core sizes, we separated
 the contribution of the intrinsic and scattered sizes to the measured angular diameter for 1546 sources. For the sources observed
 through the Galactic plane, the contribution of the scattered size component is systematically larger than for those seen outside
 the Galactic plane. The derived power-law scattering indices are found to be in a good agreement with theoretical predictions
 for the diffractive dominated scattering of radio emission in a hot turbulent plasma.
 Key words: galaxies: active – galaxies: jets – galaxies: ISM – Galaxy: structure

 1 INTRODUCTION the brightness temperatures of AGNs, which is crucial for theoretical
 models of relativistic jets (Johnson et al. 2016). At the same time,
 The interstellar medium is a rarefied medium which fills the space
 radiation scattering contains the most important information about
 between stars in galaxies. The ISM includes interstellar gas (molec-
 properties of the turbulent medium, and a detailed study of this effect
 ular, atomic, ionized), dust, electromagnetic fields and cosmic rays.
 allows to investigate the properties of the ISM in our Galaxy (e.g.,
 All the components of the ISM are closely related to each other
 Pushkarev & Kovalev 2015). The AGN VLBI radio core (or shortly
 due to the constant circulation of matter and energy in the galaxy.
 AGN core) that is generally observed in the region where the jet
 The ISM is highly turbulent (Falceta-Gonçalves et al. 2014), with
 stops being opaque to synchrotron radiation is compact enough to
 high Reynolds numbers. Turbulence can be caused by processes on a
 probe the scattering properties of the ISM. The advantage of these
 wide range of scales (from pc to kpc), e.g., the interaction of cosmic
 sources over pulsars is that they are (i) more numerous, (ii) uniformly
 rays and the interstellar plasma, stars and the interstellar medium, as
 distributed over the sky, and (iii) their emission passes through the
 well as rotation of the Galaxy, collisions between the stellar and gas
 entire depth of the scattering screens.
 components (Elmegreen & Scalo 2004), etc.
 When a radio wave passes through turbulent ionized gas of the
 According to the studies of the radio emission scattering based
 ISM, it encounters stochastic free-electron density fluctuations on its
 on pulsar observations, it has been shown that scattering screens in
 way, which cause fluctuations in the refractive index. Thus, wave-
 our Galaxy consist of two main components: (1) a nearly uniform
 fronts will be randomly distorted, resulting scattering of radio emis-
 medium with a characteristic scale of about 500 pc and (2) a clumped
 sion (Ferrière 2020). Scattering may distort an image of compact
 medium with a scale of 100 pc, with an approximate size of one
 radio sources in a number of ways (e.g., Savolainen & Kovalev 2008;
 clump about 1 pc (Cordes et al. 1985, 1986). The typical size of
 Pushkarev et al. 2013; Gwinn et al. 2014; Johnson et al. 2018), this
 such a substructure is important since it directly determines how the
 complicates the interpretation of their observations. The study of
 scattering manifests itself. For example, large-scale inhomogeneities
 radio emission scattering effects makes it possible to reconstruct the
 lead to the refractive scattering effects, e.g., angular wandering of
 intrinsic characteristics of a scattered source. It is important to con-
 the apparent source position (Clegg et al. 1998), extreme scattering
 sider the phenomena of radio emission scattering when measuring
 effects (Fiedler et al. 1987; Pushkarev et al. 2013), ‘slow’ intensity
 variations (Rickett et al. 1984). Fluctuations at smaller scales lead
 ★ E-mail: tatyana.koryukova@gmail.com to diffractive scattering effects, e.g., angular broadening (Duffett-

 © 2022 The Authors

2 Koryukova et al.
Smith & Readhead 1976), ‘fast’ intensity scintillations in time and calibration and self-calibration effects are not accounted here. Specif-
frequency (Jauncey et al. 2020). ically, we drop the measurements with uncertainty in the core size
 There are two competing models of scattering screens: the Gaus- exceeding 50 per cent of the value itself. We find that the unresolved
sian screen model (Booker et al. 1950; Ratcliffe 1956) and the model sources are also removed by this criterion and we do not perform any
with a power-law spectrum of electron density fluctuations (Lovelace additional selection. Table 1 contains eight randomly selected AGN
et al. 1970; Cordes et al. 1985). The brightness distribution of a point for which the core size was measured.
source seen through the Gaussian scattering screen is Gaussian (unre-
solved core surrounded by a halo) with an angular size proportional
to wavelength squared, i.e., ∝ , where = 2.0 (Goodman &
Narayan 1985; Cordes et al. 1986). A power-law spectrum of tur- 3 ANGULAR BROADENING OF THE AGN CORE SIZE
bulence assumes that the spectrum of free electron density fluctua- To study the scattering properties of the interstellar medium in our
tions can be approximated as a Kolmogorov power-law (Kolmogorov Galaxy, we have analyzed the VLBI core size measurements de-
1941; Cordes et al. 1986; Rickett 1990; Armstrong et al. 1995). In scribed in Section 2. There are 60 683 measurements of 9 525 active
this case, the scattering angle of a point source is proportional to galactic nuclei which pass our filtering criterion. These AGNs are
wavelength to the power of 2.2, i.e., ∝ , where = 2.2. located over the entire sky, except for the region of far southern lat-
 If there is no intermediate scattering screen, then the observed itudes of the celestial sphere due to the lack of robust observational
angular size of a compact background source will coincide with its data in this area.
intrinsic size. For example, in the case of an apparent jet base in Electron density fluctuations in the ISM lead to angular broadening
active galactic nuclei, an observed size will be proportional to , of radio sources. Scattering material spreads throughout the Galaxy,
where = 1 (Blandford & Königl 1979; Königl 1981). but it mostly concentrates in a thin disc of about 100 pc in width,
 In this work we further develop the study made by Pushkarev & likely associated with HII regions, stellar wind bubbles and supernova
Kovalev (2015). For that, we use a larger sample of the observed explosions (Geldzahler & Shaffer 1981; Cordes et al. 1984; Spangler
AGNs, and introduce new methods for modelling the apparent jet et al. 1986). Figure 1 shows the AGN core sizes as a function of the
base structure of active galactic nuclei and a completely new approach absolute value of the Galactic latitude | | at all observing frequencies
to study the scattering effects in our Galaxy. between 1.4 GHz and 86 GHz. These figures demonstrate that the
 median size of cores increases significantly when a source is seen
 through the Galactic plane (| | < 10◦ ). This effect is most noticeable
2 AGN VLBI CORE SIZE MEASUREMENTS at low frequencies 2, 5, or 8 GHz, being most sensitive to scattering
 of radio emission and, accordingly, to angular broadening of the
Our analysis is based on the VLBI observations of AGN jets at
 observed AGN core size.
frequencies ranging from 1.4 to 86 GHz compiled in the Astrogeo
 The isotropic distribution of the investigated sources over the sky
database1 . We rely on the measured interferometric visibilities and
 allowed us to map out a distribution of the source sizes at different
do not analyse the corresponding restored images. The Astrogeo
 frequencies to visualise the angular broadening effect (Figure 2). We
database collects geodetic VLBI observations (Petrov et al. 2009;
 measured the angular diameter for 3968, 5198, and 7660 AGN cores
Pushkarev & Kovalev 2012; Piner et al. 2012), the VLBA2 calibrator
 at 2, 5, 8 GHz, respectively. At other frequencies, the number of
surveys (VCS; Beasley et al. 2002; Fomalont et al. 2003; Petrov et al.
 sources is considerably smaller. To create these maps, we excluded
2005, 2006; Kovalev et al. 2007; Petrov et al. 2008), the MOJAVE
 all the sources with core sizes greater than 20 mas for 2 GHz data,
VLBA survey (Lister et al. 2018 and references therein) and other
 greater than 15 mas for 5 GHz data and greater than 10 mas for 8
VLBI observations (Helmboldt et al. 2007; Lee et al. 2008; Petrov
 GHz data. The sources with very large sizes introduce a lot of noise
et al. 2011a,b; Petrov 2011, 2012, 2013; Schinzel et al. 2015; Shu et al.
 to the resulting map. Each 1◦ × 1◦ pixel of the map contains the
2017; Jorstad et al. 2017; Petrov et al. 2019; Nair et al. 2019; Petrov
 weighted by the Gaussian function average observed core size of
2021; Popkov et al. 2021). This dataset contains 14 483 sources
 AGN which falls into a circular area of 10◦ radius around a given
observed from 1994 to 2020, comprising more than 80 000 individual
 pixel. The empty area around the South pole is the region where we
observations. The majority of them were performed at 2, 5, 8, or
 do not have enough observational data yet.
15 GHz.
 The distributions of AGN core sizes over the sky for the three
 The AGN VLBI core is the apparent jet origin. We apply the
 frequencies are shown in Figure 2. A characteristic increase of the
model-fitting of interferometric visibilities approach and describe
 average value of measured sizes in the Galactic plane for all frequen-
the apparent structure with two Gaussian components: the core and
 cies is observed. It is also notable that the average sizes of the sources
the extended jet emission. We use nested sampling to make fitting
 in the Galactic plane at lower frequencies are much larger, (e.g., the
completely automatic and independent of initial guesses. The bright-
 map for 2 GHz, Figure 2, top panel) than at higher frequencies (e.g.,
est component of the two is first selected as the core; if the jet
 8 GHz, Figure 2, bottom panel). This effect is a result of a power-
orientation at a lower frequency is opposite to that at a higher fre-
 law frequency dependence of the observed AGN core size ∝ − ,
quency, we switch the two components. Plavin et al. (2022, ApJS,
 where is expected to be 1 for the unscattered sources (Blandford
submitted) present and discuss this in details as well as evaluate this
 & Königl 1979; Königl 1981) and close to 2 for the scattered ones
fitting approach used to measure jet directions. Nested sampling pro-
 (Cordes et al. 1986; Rickett 1990; Armstrong et al. 1995). Table 2
vides principled uncertainty estimates on all parameters, including
 shows the median sizes of sources in the Galactic plane and outside
the core component size. We find these uncertainties directly useful
 of it derived for different frequencies. The difference between the
for our analysis even though they are fundamentally underestimated:
 median size values is especially noticeable at low frequencies, while
 at high frequencies there is almost no difference between the median
1 http://astrogeo.org/vlbi_images/ sizes within and outside the Galactic plane.
2 Very Long Baseline Array of the National Radio Astronomy Observatory, The red color on the map (Figure 2) corresponds to the largest
Socorro, NM, USA observed sizes of AGN cores. These sources are concentrated within

MNRAS 000, 1–14 (2022)

3
 10 10
 (mas)

 (mas)
 1.0 1.0

 1.4 GHz 2 GHz
 0.1 0.1
 1 10 90 1 10 90
 |b| (deg) |b| (deg)
 10 10
 (mas)

 (mas)
 1.0 1.0

 5 GHz 8 GHz
 0.1 0.1
 1 10 90 1 10 90
 |b| (deg) |b| (deg)
 1.0 1.0
 (mas)

 (mas)

 0.1 0.1

 15 GHz 24 GHz
 0.01 0.01
 1 10 90 1 10 90
 |b| (deg) |b| (deg)
 1.0 1.0
 (mas)

 (mas)

 0.1 0.1

 43 GHz 86 GHz
 0.01 0.01
 1 10 90 1 10 90
 |b| (deg) |b| (deg)

Figure 1. The AGN VLBI core sizes as a function of the absolute value of the Galactic latitude measured at frequencies ranging from 1.4 to 86 GHz. Each
dot represents an individual source for which the median size for all epochs was taken. The red curve is the running median which demonstrates an increase of
the median sizes of the AGN cores as they approach the Galactic plane. The running median was taken over a range of 5◦ of | | at 2 – 8 GHz and 10◦ for 15,
24 GHz. The blue shaded area shows the standard deviation of the median value of the angular size.

 MNRAS 000, 1–14 (2022)

4 Koryukova et al.

 75°
 60° 2 GHz 6.3
 45° 150° 120° 90° 60° 30° 0° -30° -60° -90° -120° -150° 5.0
 30°
 4.0
 Galactic Latitude

 15°

 (mas)
 0°
 3.2
 -15° 2.5
 -30° 2.0
 -45°
 1.6
 -60°
 -75° 1.3
 Galactic Longitude

 75°
 60° 5 GHz 4.0
 45° 150° 120° 90° 60° 30° 0° -30° -60° -90° -120° -150° 3.2
 30°
 2.5
 Galactic Latitude

 15°

 (mas)
 0° 2.0
 -15°
 1.6
 -30°

 -45°
 1.3
 -60° 1.0
 -75°
 Galactic Longitude

 75°
 60° 8 GHz 1.6
 45° 150° 120° 90° 60° 30° 0° -30° -60° -90° -120° -150° 1.3
 30°
 Galactic Latitude

 15° 1.0
 (mas)

 0°
 0.8
 -15°

 -30°
 0.6
 -45°
 0.5
 -60°
 -75° 0.4
 Galactic Longitude

Figure 2. The observed AGN core size distribution maps. The colour of each pixel reflects the average size of sources at this location. The average size was
estimated based on the sources which fall into a circular area of 10◦ radius around this central point (pixel). The red colour corresponds to the larger observed
AGN core sizes (dominance of scattering), and the dark blue colour means the smaller values of the observed AGN core sizes (weak or no scattering). All the
spheres presented in this work are shown in the Galactic coordinates over the celestial sphere in the Mollweide equal-area projection.

MNRAS 000, 1–14 (2022)

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Table 1. Apparent angular sizes of the AGN VLBI core measured at frequencies ranging from 1.4 to 86 GHz with separate records for every epoch.

 Name Epoch core err
 core 
 (GHz) (mas) (mas) (deg) (deg)
 (1) (2) (3) (4) (5) (6) (7)

 J0006−0623 1.35 2010-08-23 2.612 0.007 −66.65 93.50
 J0825+0831 2.24 2005-07-20 1.478 0.070 24.66 215.92
 J1842+7946 4.12 2018-08-29 0.886 0.032 27.07 111.44
 J0158+2124 8.44 2014-08-09 0.688 0.006 −38.86 142.95
 J1337−1257 15.4 2006-10-06 0.104 0.001 48.38 320.02
 J1048−1909 24.2 2003-09-13 0.091 0.002 34.90 266.75
 J2258−2758 42.9 2003-09-13 0.089 0.002 −64.92 24.37
 J1658+0741 86.2 2002-10-24 0.027 0.003 28.65 26.61

 The columns are as follows: (1) source name in the J2000.0 notation; (2) central observing frequency; (3) epoch; (4) measured AGN core size; (5) the formal
 model depended error of the source size fitting; (6) Galactic latitude; (7) Galactic longitude. The table is published in its entirety in the machine-readable
 format. Eight randomly selected records are shown here for guidance regarding its form and content.

Table 2. Median sizes of the measured AGN VLBI cores as presented at Figure 4, and (ii) a larger beam of the sources located in the region
Figure 1. of high southern latitudes.
 It is widely known that a distinctive feature of AGNs is their high
 med ( | | < 10◦ ) med ( | | > 10◦ ) intrinsic variability across the whole electromagnetic spectrum on
 (GHz) (mas) (mas) scales from hours to years (e.g., Zensus 1997; Blandford et al. 2019;
 (1) (2) (3) (4) (5) Plavin et al. 2019). To search for the characteristic time scales of the
 external variability of scattering properties, we used our measured
 1.4 2.123 ± 0.643 13 1.636 ± 0.126 126 multi-epoch data of the AGN VLBI core sizes at 2 GHz and 8 GHz.
 2 1.857 ± 0.082 570 1.175 ± 0.012 3398
 We calculated the size variability at a given frequency, which exceeds
 5 0.998 ± 0.036 695 0.772 ± 0.010 4503
 8 0.416 ± 0.008 1093 0.361 ± 0.003 6567
 the variability allowed by the estimated error. In Figure 3 we show
 15 0.184 ± 0.023 66 0.146 ± 0.004 621 the size variability at 2 GHz. Namely, the difference between two
 24 0.147 ± 0.098 87 0.117 ± 0.005 266 measured source sizes at all available time intervals Δ . All possible
 43 0.074 ± 0.010 12 0.068 ± 0.003 116 time periods were used for each source. Approximately 8 per cent
 86 0.202 ± 0.104 2 0.045 ± 0.006 16 of all calculated amplitudes at 2 GHz and 6 per cent at 8 GHz were
 excluded from the analysis, because they do not exceed the estimated
 The columns are as follows: (1) frequency band; (2) median AGN core error for this amplitude. Results can be summarized as follows. We
 size within the Galactic plane and its error estimated with the bootstarp observe the same general properties of the core sizes at 2 GHz and
 method; (3) number of sources used to estimate (2); (4) median AGN core
 8 GHz: (i) variability on long time scales is much higher than on short
 size outside the Galactic plane and its error estimated with the bootstarp
 time scales; (ii) variability outside the Galactic plane greatly exceeds
 method; (5) number of sources used to estimate (4).
 the variability within the Galactic plane for Δ more than a few years;
 (iii) on average, variability does not exceed 20 per cent for timescales
a narrow band in the center of the map, where the Galactic longitude more than a year; for time scales less than a year, median amplitude
( ) is in a range of −120◦ < < 120◦ for 2 GHz. The Galactic retains a constant value. This indicates that the observed variability is
centre region is located approximately in the range of longitudes of dominated by internal AGN effects rather than scattering. We did not
−20◦ 6 6 20◦ . As expected, the smallest number of sources manage to reveal the characteristic time of variability of scattering
with large observed sizes is in the regions of longitudes > 150◦ in the Galaxy.
and < − 150◦ — this is the Galactic anti-centre, which contains
much less scattering material. It is also clearly seen that the sources
at high Galactic latitudes (| | > 10◦ ) are not subject to significant 4 SINGLE POWER-LOW APPROXIMATION OF THE
angular broadening. As it was demonstrated, the unscattered sources -INDEX
fill almost the entire sphere, with the exception of some regions in
the Galactic plane. This means that there are certain Galactic regions The diffraction phenomena associated with the radiation scattering
which cause strong scattering at this frequency range. in a turbulent interstellar medium cause the angular broadening of a
 The area of the map outside the Galactic plane with distant background radio source. Thus, the observed distribution of
100◦ < < 130◦ and 15◦ < < 60◦ is the region of a special the source brightness is a convolution of the intrinsic structure of the
interest because it extends far beyond the Galactic plane. It demon- source with the scattering function. The measured angular diameter
strates the enhancement of scattering strength at 2 GHz (Figure 2, of the AGN core should correspond to a convolution of the intrinsic
top) and could be caused by a nearby scattering screen of a moderate and scattered components of the size (Lazio et al. 2008):
power. In addition to this, a similar area is located approximately 2 2 2
 obs = int + scat , (1)
at ( , ) = (−150◦ ± 5◦ , −15◦ ± 5◦ ). At 5 GHz and 8 GHz, these
areas are not that noticeable. Additionally, once can see indications
 int = int1 · − int , (2)
of scattering near the edge of the empty region. However, as we will
see later in Section 4, the data near the edge have poorer accuracy
due to the following reasons: (i) a smaller number of sources, see scat = scat1 · − scat , (3)

 MNRAS 000, 1–14 (2022)

6 Koryukova et al.

 0.40 75°
 60°
 0.35 2 GHz 45°

 0.30 |b|10° -15°
 log t2|

 0.25 -30°
 -45°
 0.20 -60°
 t1

 0.15 -75°
 Galactic Longitude
 |log

 0.10
 10 20 30 40 50 60
 0.05 Number of sources
 0.00 0
 10 101 102 103 104
 t (days) Figure 4. Top panel: Sky distribution of the measured AGN core sizes. The
 red dots represent 3405 AGNs for which the angular size was simultaneously
Figure 3. Amplitude of the variability of the AGN core size at 2 GHz over a measured at 2 GHz and 8 GHz. The grey triangles are 7660 AGNs measured
wide range of time covered by observations (days to tens of year). Each dot only at 8 GHz. The bottom panel shows the averaged density distribution map
in these plots represents the relative amplitude of a change of the AGN core for simultaneously measured AGN core sizes at 2 GHz and 8 GHz. Each pixel
sizes at 2 GHz. The running median (the red curve) was taken over a range of the map reflects the number of the sources which fall into a circular area
of 100 days. of 10◦ radius around this central point (pixel).

where obs is the measured AGN core size at the observing frequency, 514 and 2891 sources seen through the Galactic plane (Figure 5, left)
 int and scat are the intrinsic and scattered AGN core sizes at the and outside of it (Figure 5, right), with the median values 1.27
observing frequency, int1 and scat1 are the intrinsic and scattered and 1.01, respectively. We fitted the obtained distribution of index
AGN core sizes at 1 GHz, is the frequency of observation (GHz), using the Gaussian function. The left histogram (Figure 5) was fitted
 int is the power-law index from the frequency dependence of the with two Gaussian curves, while the right one was fitted with one
intrinsic core size and scat is the power-law index from the frequency Gaussian curve. The obtained parameters of the Gaussian curves are
dependence of the scattered AGN core size. presented in Table 3.
 If we consider a simple approximation to Equations 1–3, it can be If we combine the obtained distributions of the indices of sources
expressed as obs ∝ − . It is expected that the index is approxi- in the Galactic plane and outside of it as shown in Figure 6, and also,
mately equal to 2, if scattering dominates and is equal to 1 when the comparing the left and right histograms in Figure 5, then we can
radiation coming from a source is not scattered. This power-law clearly see that values of the on the right histogram in Figure 5
index reflects the strength of scattering of the source and will allow cluster around a value of = 1.01 ± 0.01 and the first Gaussian
us to analyse its distribution over the sky, similar to the distribution curve on the left histogram in Figure 5 with a peak at = 0.97 ± 0.02
of the observed AGN core sizes at different frequencies, which we describe the distributions of the same kind of sources, i.e., the un-
showed in Figure 2. scattered ones. Notably, the obtained peak values of the indices
 To estimate the index , we can use the data of simultaneous multi- are in good agreement with the theoretically expected = 1 for the
epoch measurements of the AGN core sizes at two frequencies, 2 GHz frequency dependence obs ∝ − of the intrinsic AGN core sizes
and 8 GHz. We calculated the index for 3405 AGNs. If a source has (Blandford & Königl 1979; Königl 1981). Note that this value is
more than one observation epoch, the median value of the index also in a good agreement with the often observed core-shift depen-
was used. Figure 4 (top panel) demonstrates the sky distribution of dence ∝ −1 , see for details Sokolovsky et al. (2011); Kutkin et al.
these sources (red dots). We also plotted in Figure 4 (top panel) the (2014). The second Gaussian curve on the left histogram with a peak
sources measured only at 8 GHz (gray triangles). Figure 4 (bottom at = 1.65 ± 0.02 describes the contribution of the scattered sources
panel) also shows a source density distribution map based on the data to the distribution of the indices in the Galactic plane. The result-
from simultaneous observations at 2 GHz and 8 GHz. Each pixel of ing value differs significantly from the theoretically expected value
the map reflects the average number of sources that fall into a circular ≈ 2 for the scattered sources. The index value close to = 2 is
area of 10◦ radius around this central pixel. observed for a small number of sources in the Galactic plane. We
 The derived index values are shown in Figure 5, separately for used a simple approximation for the frequency dependence of the

MNRAS 000, 1–14 (2022)

7

 0.8 = 0.97, = 0.32 = 1.01, = 0.38
 = 1.65, = 0.33 1.0
 0.7
 |b| > 10°
 |b| < 10° 0.8
 Probability density

 Probability density
 0.6
 0.5
 0.6
 0.4
 0.3 0.4
 0.2
 0.2
 0.1
 0.0 0.0
 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
 k k
Figure 5. Histograms of the indices, derived from simultaneous measurements of the AGN core sizes ∝ − at 2 GHz and 8 GHz. The left histogram is the
distribution of indices in the Galactic plane ( | | < 10◦ ), fitted with two Gaussian curves representing unscattered and scattered sources. The right histogram
is the distribution of indices outside the Galactic plane ( | | > 10◦ ), fitted by a single Gaussian curve. All the parameters of the fitting functions are listed in
the legend of the figure, where is the mathematical expectation and is the distribution width. The red curve presented on the left histogram shows the
distribution formed by the sum of the first (solid) and the second (dashed) Gaussian curves.

Table 3. Fitting parameters of the Gaussian curves for the distributions of 1.0 |b| > 10°
the indices derived from the data of simultaneous observations of the AGN Normalized probability density |b| < 10°
core sizes at 2 GHz and 8 GHz.
 0.8
 | | 2015 2015
 (1) (2) (3) (4) (5) 0.6
 < 10◦ 0.97 0.32 0.91 0.33
 1.65 0.33 1.76 0.28 0.4
 > 10◦ 1.01 0.38 0.90 0.44

 The columns are as follows: (1) | | is the absolute range of the Galactic
 0.2
 latitude (◦ ); (2) is the mathematical expectation obtained in this work;
 (3) is the distribution width obtained in this work; (4) 2015 is the math- 0.0
 ematical expectation obtained in Pushkarev & Kovalev (2015); (5) 2015 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
 is the distribution width obtained in Pushkarev & Kovalev (2015). k

 Figure 6. Amplitude-normalised histogram of the indices, derived from
observed AGN core size for these calculations. It is the main reason simultaneous measurements of the AGN core sizes at 2 GHz and 8 GHz in
for underestimating this value of the index for the scattered sources. the Galactic plane (grey histogram) is plotted on top of the histogram of the
All of the obtained results of the are very similar to those obtained outside the Galactic plane (blue histogram). This figure is the combination
 of two histograms from Figure 5.
by an earlier study by Pushkarev & Kovalev (2015).
 The presence of two modes in the indices distribution in the
Galactic plane indicates that a significant number of sources are radio-emitting gas cloud approximately 1.8 pc in diameter (Downes
not scattered even though lie at low latitudes and the signal from & Martin 1971);
these sources passes through the most densely populated part of our (ii) Cygnus supernova remnant is located at
Galaxy. Thus we can expect many scatter–free regions in the Galactic ( , ) = (74.0◦ , −8.6◦ );
plane. (iii) Cassiopeia A is a supernova remnant located at
 Using the obtained values of the indices calculated for 3405 ( , ) = (111.7◦ , −2.1◦ );
sources uniformly distributed over the sky, we created the distribution (iv) Vela supernova remnant is located at ( , ) = (−96.5◦ , −2.8◦ );
map of scattering power in the Galaxy (Figure 7). To create this map, (v) Taurus A supernova remnant is located at
we excluded the sources outside the interval 0.5 6 6 2.5, because ( , ) = (−175.4◦ , −5.8◦ ) in the constellation of Taurus
too small and too large index values introduce needless noise. The and also an area of moderate scattering in a direction to the
resulting index distribution map replicates the pattern of the AGN Orion Nebula M42, the closest region of massive star formation at
core size distribution map at low frequencies, for example, at 2 GHz ( , ) = (−151.0◦ , −19.4◦ ).
(compare with Figure 2, top). According to the map, we can highlight
the regions of strong scattering in the Galactic plane (| | < 10◦ ): We also derived the indices using the two-frequency method
 mentioned above and the data from non-simultaneous observations
 (i) the region at −20◦ < < +20◦ encompasses the Galactic of the AGN core sizes at frequency pairs 2 GHz and 8 GHz (3695
centre and Galactic bar. This central region also contains Sagittar- sources), 2 GHz and 5 GHz (3968 sources), 5 GHz and 8 GHz (4222
ius A∗ the hosting a supermassive black hole surrounded by a hot sources). A common feature for all three cases is that the index

 MNRAS 000, 1–14 (2022)

8 Koryukova et al.

 75°
 k 1.8
 60°

 45° 150° 120° 90° 60° 30° 0° -30° -60° -90° -120° -150° 1.7
 30° 1.6
 Galactic Latitude

 15°
 1.5
 0°
 1.4
 Tau
 -15° Cas-A 1.3
 -30°
 Vela M42
 Cygnus Galactic 1.2
 -45°
 region centre
 -60°
 1.1
 -75° 1.0
 Galactic Longitude k
 75°
 60°
 0.09
 45° 150° 120° 90° 60° 30° 0° -30° -60° -90° -120° -150°
 30° 0.08
 Galactic Latitude

 15°
 0.07
 0°

 -15° 0.06
 -30°
 0.05
 -45°

 -60° 0.04
 -75°
 Galactic Longitude kerr
Figure 7. Top: Distribution map of the power-law index derived from the AGN core sizes ∝ − simultaneously measured at 2 GHz and 8 GHz over the
sky. The magnitude and colour of each pixel of the map reflect the average over 10◦ area value of the index at this location. The red colour corresponds to
higher values of the index (dominance of scattering), the dark blue colour corresponds to the lower values of the index (unscattered sources). Bottom: The
distribution map of the standard deviation of the estimated average index . The darker the map area, the greater the error of the estimated average value of the
 index in this direction of the sky.

Table 4. Results of the correlation Kendall’s -test between values obtained in this work and the other measured characteristics of the Galaxy.

 | | < 10◦ | | > 10◦ All sky
 
 – 0.126 ± 0.030 514 1.9 × 10−5 0.061 ± 0.012 2891 1.6 × 10−6 0.126±0.111 3405 1.8 × 10−22
 – NE2001 0.183 ± 0.028 514 2.6 × 10−10 0.072 ± 0.012 2891 1.5 × 10−9 0.149±0.011 3405 1.3 × 10−41
 – H 0.256 ± 0.027 514 4.2 × 10−18 0.093 ± 0.012 2891 4.7 × 10−14 0.161±0.012 3405 5.8 × 10−45
 obs,2 GHz – H 0.374 ± 0.027 570 9.4 × 10−41 0.071 ± 0.012 3398 6.3 × 10−10 0.161±0.011 3968 4.1 × 10−52

 Note: is the power-law index calculated by the two-frequency method (see Section 4 for details) using the simultaneous observational data at 2 GHz and
 8 GHz; is the rotation measure; NE2001 is the scattered AGN core size at 1 GHz obtained based the NE2001 model; H is the intensity of the H 
 radiation in the Galaxy; obs,2 GHz is the AGN core sizes measured at 2 GHz. is the Kendall’s correlation coefficient, is the number of the sources, is
 the probability of chance correlation.

MNRAS 000, 1–14 (2022)

9
distribution for the AGNs in the Galactic plane (| | < 10◦ ) does Table 5. Scattering indices scat derived from different sets of the measured
not have a clear two-peak structure and cannot be reliably fitted by AGN core sizes at the Galactic plane.
two Gaussian curves. The distributions of for the sources outside
the Galactic plane obtained from the non-simultaneous measure- Subgroup scat
ments look similar and almost do not differ from the distribution (1) (2) (3)
represented in Figure 5, left. The peak of all obtained histograms is
concentrated around the value = 1. We conclude that simultaneous All bands 215 1.94+0.07
 −0.09
measurements are required to properly address the joint case of in- 2, 5, 8 GHz 154 2.03+0.16
 −0.05
trinsic and external effects of the core size. For this reason, we prefer 2, 8, 15 GHz 60 2.03+0.03
 −0.17
the results from the simultaneous observations at two frequencies for
 2, 5, 8, 15 GHz 35 1.95+0.11
the further analysis in this paper. −0.24

 Columns are as follows: (1) frequency bands over which the subgroups of
 the sources were selected for fitting; (2) the number of the sources used
 to estimate the scattering index value; (3) the scattering index value.
5 CONNECTION BETWEEN SCATTERING PROPERTIES
 AND ROTATION MEASURE, ELECTRON DENSITY, within 100◦ < < 130◦ and 15◦ < < 60◦ , it can be a local screen
 AND H DISTRIBUTIONS IN THE GALAXY with modest scattering close to the Earth.

The strength of scattering of radio emission which passes through
clumpy distributed plasma is closely related to the density of free 6 TWO-COMPONENT MODEL OF THE
electrons along the line of sight. Therefore, we can expect an en- FREQUENCY-DEPENDENT AGN CORE SIZE
hancement of the scattering strength in regions with a high concen-
tration of free electrons. For this reason, in this section, we will test This section presents Monte-Carlo simulations of the intrinsic and
the connection between the distribution of the scattering strength and scattered components of the observed source size obs . Also, we
the parameters measured in our Galaxy which are related to free elec- present two methods for estimating the characteristic value of the
trons concentration, namely the rotation measure (RM, Taylor et al. scattering index scat and then compare all the obtained results with
2009), the scattered size distribution based on the NE2001 model the theoretical predictions.
( NE2001 , Cordes & Lazio 2002) and the distribution of the H radi-
ation intensity in the Galaxy (Finkbeiner 2003). The distribution of
 6.1 Multi-frequency fitting of observed core sizes
the scattering strength in the Galaxy, as we have already noted, can
be reflected by the distribution of the AGN core sizes measured at Given a large amount of experimental data, we determined (Section 4)
low frequency, for example, at 2 GHz (see Section 3 for details), and that the exponent for the unscattered sources equals 1 with a high
also by the distribution of the power-law indices from the frequency accuracy. Therefore, we can set int in Equations 1–3 equal to 1
dependence of the AGN core sizes (see Section 4 for details). and determine the unknown parameters of these equations for each
 We used Kendall’s non-parametric rank correlation coefficient selected source, namely scat , int1 , scat1 .
to assess the magnitude of the correlation between the compared pa- Figure 9 shows how the AGN core size changes depending on the
rameters. The errors of are estimated using the bootstrap method. observing frequency for sources used as two clearly different exam-
The results of Kendall’s tests are shown in Table 4. The strongest ples: J0238+1636, which is unscattered, and J0359+5057, which is
correlation is traced between the AGN core sizes obs,2 GHz obtained a highly scattered. As discussed and shown above, the observational
from the observations at 2 GHz and the H radiation intensity dis- data for the unscattered source can be fitted by the size-frequency
tribution in the Galaxy. Thus we can conclude that there is a direct dependence obs ∝ −1 . The frequency dependence for the scattered
relationship between the regions of a high H intensity and the dis- source hear is best fitted by a sum of two spectral components with
tribution of scattering screens in the Galaxy. different slopes, int = 1 and scat = 2 in the logarithmic scale.
 Figure 8 shows the distribution map of the H radiation intensity Using the multi-frequency observational data of the AGN core
in the Galaxy (H , Finkbeiner 2003). On top of H , we plotted con- sizes in the Galactic plane (| | < 10◦ ) according to Equations 1–3,
tours of the average observed AGN core sizes measured at 2 GHz, we can estimate the characteristic value of the scattering index scat ,
log obs, 2 GHz . For the contours, we used the sources with the core which will correspond to the best fit of the intrinsic int1 and scattered
sizes less than 20 mas to avoid unnecessary noise introduced by scat1 sizes at 1 GHz setting int = 1. For each source, the average
large individual sources. These contours largely repeat the locations observed core size at a given frequency was taken. We performed
of high-intensity H clouds. This result confirms the expectation be- fitting for four sub-groups listed in Table 5.
cause the H emission observed in spiral galaxies is a direct indicator The obtained values of the scattering index scat are summarised
of a hot ionised interstellar medium (HII, Reynolds 1983). in Table 5. All of them are consistent with a power-law index 2.0
 As we already noted in Section 3, there are regions outside the predicted in the Gaussian screen model, and only one, for the sub-
Galactic plane in which we observe an increase of average AGN core group based on 2, 8 and 15 GHz data, with scat = 2.2 within the
sizes. These are areas such as 100◦ < < 130◦ , 15◦ < < 60◦ and errors expected for a screen with the Kolmogorov turbulence. We
(−150◦ ± 5◦ , −15◦ ± 5◦ ) and also the area near the edge of the empty also note a tendency for scat to increase with progressive excluding
(white) region, where we don’t have enough measurements. As we the data at high-frequency bands from the analyses. This indicates a
can see from Figure 8, the ( , ) = (−150◦ ± 5◦ , −15◦ ± 5◦ ) region possible bias brought by high frequency data which are less sensitive
is also observed according to H data, and as we saw in Section 4, to scattering and suggests that the scattering screens do have a Kol-
almost in this region, the Orion Nebula and massive star formation mogorov spectrum of inhomogeneities. To distinguish between these
region are located. Thus these objects could potentially cause the two competing models, more data is needed at frequencies lower than
strong scattering of radio emission in this direction. As for the region 2 GHz. It is also possible that there are scattering screens of various

 MNRAS 000, 1–14 (2022)

10 Koryukova et al.

 75°
 60
 60°
 H
 45° 150° 120° 90° 60° 30° 0° -30° -60° -90° -120° -150° 50
 30°
 Galactic Latitude

 15° 40

 Intensity (R)
 0°
 30
 -15°
 20
 -30°

 -45° 10
 -60°
 -75° 0
 Galactic Longitude

Figure 8. Distribution map of the H radiation intensity in the Galaxy, on top of which the observed AGN core size distribution log obs, 2 GHz (mas) contours
are plotted. The colour of the map reflects intensity in Rayleighs (1 = 106 photons cm−2 s−1 sr−1 ). The bottom contour is shown at log obs, 2 GHz = 0.2, and
the rest contours are plotted at an increasing power of 2.

 30 Table 6. The intrinsic and scattered sizes of 1546 AGN VLBI cores at 1 GHz.
 obs 1
 J0359+5057
 10 int 1 err err
 Name int1 int1 scat1 scat1
 scat 2 (deg) (mas) (mas) (mas) (mas)
 3.0 (1) (2) (3) (4) (5) (6)
(mas)

 J0238+1636
 J0122−0056 −62.78 1.99 0.10 < 0.10
 1.0
 ...
 J0359+6005 5.28 1.17 0.03 5.92 0.08
 obs

 J0544+2655 −1.32 4.25 0.15 12.16 0.59
 0.3 J0814−2521 5.23 5.42 0.23 21.47 0.70
 J0832+4913 36.38 2.04 0.03 < 0.10 ...
 0.1 J0902+4310 41.55 1.70 0.02 < 0.10 ...
 J1838−1833 −5.62 1.89 0.09 9.14 0.34
 0.0 J2110−0126 −31.23 3.42 0.28 4.49 2.02
 1 10 100 J2257+0743 −45.52 1.80 0.04 1.78 0.27
 (GHz) J2327+0940 −47.95 1.12 0.01 < 0.10 ...

 Columns are as follows: (1) name of source; (2) Galactic latitude; (3) the
Figure 9. Example of the frequency dependence of the AGN core size for two value of calculated intrinsic size of the source; (4) the intrinsic size
sources: (1) the unscattered source J0238+1636 (blue open circles) located calculation error estimated using the Monte-Carlo method; (5) the value
at ( , ) = (156.8◦ , −39.1◦ ) and (2) the scattered source J0359+5057 (red of calculated scattered size of the source; (6) the scattered size calculation
circles) located at ( , ) = (150.4◦ , −1.6◦ ). The size measurement error is error estimated using the Monte-Carlo method. The full table is published
not shown in the plot, since its value does not exceed the radius of the source in its entirety in the machine-readable format. A portion is shown here
markers (red and blue circles). The dotted lines represent a fit according to for guidance regarding its form and content.
the equations mentioned in the legend.

 estimated using the largest sample of sources, and this value has one
structures in the Galaxy and only a dedicated analysis of separate of the smallest estimated errors. The sources from the entire sky
screens can find the dominance of one of the models or show the that have more than three observing frequencies were selected for
possibility of an existence of both options. Since this work is devoted this analysis. The intrinsic, int1 , and scattered, scat1 , sizes were
to the study of a large-scale properties of scattering in our Galaxy, estimated at 1 GHz for 1546 AGN. The results are listed in Table 6.
we do not focus here on the properties of individual screens. The obtained results are also presented in Figure 10.
 Using the obtained values of scat from Table 5, we can separate the The distribution of the intrinsic sizes (Figure 10, left) has one
contribution of the intrinsic, int1 , and scattered, scat1 , component peak. The median value of the intrinsic sizes in the Galactic plane
size to the observed one, obs . To find int1 and scat1 , we put the (blue contour histogram) and outside of it (solid gray histogram)
found value of the scattering index into Equations 1-3. For these is 2.75 and 2.31 mas, respectively. There is a slight shift between
calculations, the value scat = 1.94+0.08
 −0.07
 was used because it was the medians of the blue and gray histograms. We also analysed the

MNRAS 000, 1–14 (2022)

11

 1.8 0.6
 |b| > 10° #1331 |b| > 10° #1331
 1.6 |b| < 10° #215 |b| < 10° #215
 kscat = 1.94 all #1546
 0.5
 all #1546
 1.4

 Probability density
 Probability density

 1.2 0.4
 kscat = 1.94
 1.0
 0.3
 0.8
 0.6 0.2
 0.4
 0.1
 0.2
 0.0 1 0.0
 10 10 0.5 100 100.5 101 101.5 102 10 8 10 6 10 4 10 2 100 102 104
 int1 (mas) scat1 (mas)
Figure 10. Histograms of the intrinsic ( int1 , left) and scattered ( scat1 , right) sizes of the AGN cores at 1 GHz obtained from the simulation. The blue contour
histogram shows the size distribution for the sources in the Galactic plane. The grey solid histogram shows the size distribution outside the Galactic plane. The
black dashed histogram shows the size distribution of the sources from the entire sky. The values given in the legend after the ‘#’ sign indicate the number of
the sources for this set of the data.

 30 30

 10 10
 (mas)
 (mas)

 3.0 3.0
 scat1
 int1

 1.0 1.0

 0.3 0.3
 0.1 1 10 90 0.1 1 10 90
 |b| (deg) |b| (deg)
Figure 11. Intrinsic ( int1 , left) and scattered ( scat1 , right) AGN core sizes at 1 GHz obtained from the simulation, depending on the absolute value of the
Galactic latitude. Each dot represents an individual source and is plotted only for AGN cores with the significant scattered component. The running median (the
red curve) was taken over a range of 10◦ of | |. The blue shaded area shows the standard deviation of the median value of the angular size.

dependence of the obtained intrinsic sizes on the absolute value of is systematically larger than that of outside the Galactic plane, and
the Galactic latitude, which is shown in Figure 11, left. As seen, the this difference is significant. Figure 11 shows the dependence of the
running median retains its value through the whole range of | | even obtained scattered sizes on the absolute value of the Galactic latitude.
within the Galactic plane. It is assumed that the intrinsic sizes do not We plotted only those sources for which the scattered component of
depend on the Galactic latitude. the observed size is significant. The running median demonstrates
 The distribution of the obtained scattered sizes of the AGN cores a large increase in the median scattered sizes as it approaches and
presented in Figure 10, right. For this distribution, a completely dif- crosses the Galactic plane (| | = 10◦ ).
ferent picture is observed. The histogram has a two-peak structure. We determined that for about 40 per cent of AGN in the Galactic
The left peak is populated by the sources for which the scattered com- plane the contribution of the scattered component of the observed
ponent of the measured size scat1 is estimated to be very close to core size is insignificant. It means that this fraction of the observed
zero within the errors. This means that the contribution of scattering sources in the Galactic plane is not subject to scattering. Figure 12
to the observed angular diameter is negligible; these are unscattered demonstrates that the sources with scat1 ≈ 0 are concentrated in the
sources. The right peak of the distribution corresponds to the sources areas where the unscattered sources ( ≈ 1) are located. We note that
for which the contribution of scattering is dominant or at least signifi- the results presented on two panels in Figure 12 being obtained using
cant. For this peak, there is a significant shift between the distribution different methods and data sets, are in good agreement with each
of the scattered sizes in the Galactic plane and outside of it. Com- other. Therefore, we can conclude that most of the unscattered sources
pare blue and gray histograms in Figure 10, right. The median values are located in the region of the Galactic anti-centre (| | > 150◦ ).
of scattered sizes in the Galactic plane (blue contour histogram) and The free electron density distribution in the Galaxy model NE2001
outside of it (grey solid histogram) in Figure 10 are 12.0 and 4.4 mas, (Cordes & Lazio 2002) is mainly based on radio observations of pul-
respectively. Therefore, we conclude that the contribution of the scat- sars which provide the information about the properties of the local
tered component to the observed AGN core size in the Galactic plane ionized interstellar medium. Pulsars being Galactic radio sources are

 MNRAS 000, 1–14 (2022)

12 Koryukova et al.

 k
 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
 10°
Latitude
Galactic

 0°
 -10°
 Number of sources

 6 scat1 0
 4
 2
 0
 180° 150° 120° 90° 60° 30° 0° -30° -60° -90° -120° -150° -180°
 Galactic Longitude
Figure 12. Histogram of the Galactic longitude for the sources with an insignificant scattered component of the observed AGN core size (bottom) against the
map of the index distribution (top panel) corresponds to Figure 7, cropped at latitudes (| | = 10◦ ) and projected onto the plane. The axes of the Galactic
longitudes ( ) in both panels coincide.

 between the compared sizes. The red dots are scattered around the
 |b| < 10° #205 line of equality scat1 = NE2001 . Nearly a half of the sources (761)
 100 |b| > 10° #1254 lie outside the plot, we mark them along the y-axis. These are the
 sources for which the scattering does not dominate according to our
 calculations, scat1 ≈ 0, but the NE2001-model sizes are distributed
 from 0.1 mas to about 8 mas. Thus we conclude that the scattering
 of these sources predicted by the NE2001 model was not confirmed
 (mas)

 10 by the results of our estimates made on the basis of multi-frequency
 VLBI AGN data.
 NE2001

 761
 6.2 Modeling observed core size distributions at 2 GHz and
 1.0 8 GHz
 In this section we describe an alternative method for estimating the
 scattering index scat using the AGN VLBI core size measurements
 at two frequencies. For simplicity, we here assume that sources seen
 outside the Galactic plane (| | > 10◦ ) are not subject to scattering at
 0.1
 0.1 1.0 10 100 all. In Figure 5 (left), we showed that the distribution of the observed
 AGN core sizes in the Galactic plane (| | < 10◦ ) contains both scat-
 scat1 (mas) tered and unscattered (about 40 per cent) sources. Therefore, the size
 distribution of the sources from this area contains the contribution
Figure 13. Comparison of the results obtained from our modelling of the of both types of sources.
scattered AGN core sizes scat1 with the sizes predicted by the NE2001 We also assume that the observed size distribution of the scattered
model at 1 GHz. See Section 5 for details. The green arrow points to 761
 sources could be fitted with a generalization of half-Gaussian distri-
missing sources lying outside the plot (their y-axis positions are marked).
 bution or half-Student’s distribution. We performed an iteration over
The values given in the legend after the ‘#’ sign indicate the number of the
sources for this dataset. all parameters of these distributions. For example, for the Student’s
 function: and are degrees of freedom and width of the dis-
 tribution, respectively. We generated a random sample of sizes, and
not optimal for studying large-scale scattering properties. The pul- then added in quadrature to the observed sizes of the sources outside
sar data can provide reliable information only in the Galactic plane. the Galactic plane (the unscattered sources), that is int according to
Nevertheless, we performed a comparison of the estimated scattered Equation 1. The result of these steps is the distribution fit , which we
sizes scat1 obtained in our work, with the scattered sizes obtained will compare with the distribution obtained from the observational
on the basis of the NE2001 model ( NE2001 ) at 1 GHz. In Figure 13 data for the sources in the Galactic plane obs (| | < 10◦ ). If these
we show the sources lying in the Galactic plane and outside of it distributions are similar, the selected analytical distribution with its
in different colors. The scattering sizes determined on the basis of parameters is an appropriate description of scat distribution at a
the NE2001 model in the region | | > 10◦ are mainly concentrated given frequency.
around NE2001 ≈ 1 mas (black circles), but according to our results We minimize the Kolmogorov-Smirnov (KS) test statistic between
the scat1 sizes of these sources cover the range from 0.1 mas to the observed and fitted size distributions. The smallest values of these
more than 10 mas. The extragalactic objects, the path to which lies statistics, i.e., the greatest similarity of distributions, is obtained
through the Galactic plane (red triangles), have a better agreement when we used the Student’s distribution to fit the scattered size scat .

MNRAS 000, 1–14 (2022)

13

 1.0 ing is in a strong regime, with the diffractive scale is smaller than the
 Students distribution dissipation scale of the turbulence. In this case, the source angular
 obs (|b| < 10°) #570 size follows the 2 scaling regardless of a spatial spectrum slope. In
 0.8 obs (|b| > 10°) #3398
 our case, the sources are more weakly scattered, and the obtained
 fit = obs (|b| > 10°) + Student s
 2 2 0 2
Probability Density

 value of indicates that the spatial spectrum of inhomogeneities is
 0.6 steeper ( ' 4) than that for a Kolmogorov turbulence.
 dof = 0.90, = 0.93
 0.4 2 GHz

 0.2 7 SUMMARY
 We used the largest to date number of experimental VLBI data from
 0.0 multi-frequency observations of AGNs to study the ISM scattering
 0 1 2 3 4 5 properties in the Galaxy. We analyzed the dependencies of the ob-
 (mas) served AGN core sizes on the Galactic latitude and found significant
 3.0 angular broadening of the measured core sizes for the sources seen
 Students distribution through the Galactic plane. This effect is especially strong at low
 obs (|b| < 10°) #1093
 2.5 frequencies, particularly at 2 GHz. We established that scattering
 obs (|b| > 10°) #6567 screens containing density fluctuations of hot plasma are concen-
 fit = obs (|b| > 10°) + Students
Probability Density

 2 2 2
 2.0 trated mainly in the Galactic plane. Outside the Galactic plane, we
 did not detect regions with strong scattering. We created the sky dis-
 1.5 dof = 0.90, = 0.07 tribution maps of the measured AGN core sizes at 2, 5, and 8 GHz,
 marking the distribution of scattering screens in the Galaxy.
 8 GHz
 We calculated the power-law index from the frequency depen-
 1.0
 dence of the AGN core size ∝ − derived from simultaneous
 observations at 2 GHz and 8 GHz for 3405 sources. The mean value
 0.5 = 1.01 ± 0.01 derived for the sources outside the Galactic plane
 is in good agreement with the theoretical prediction ∝ −1 for
 0.0
 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 the unscattered AGN cores with synchrotron self-absorption. For the
 (mas) scattered sources, we obtained a characteristic value = 1.65 ± 0.02
 ignoring the contribution of the intrinsic structure, hence it is less
 than the predicted ' 2. Using the -index values calculated be-
 Figure 14. Histograms of the observed AGN core sizes obs . The gray his- tween 2 GHz and 8 GHz, we constructed the first detailed sky dis-
 togram represents the AGNs in the Galactic plane while the green contour tribution map of scattering properties in the Galaxy based on VLBI
 histogram shows the distribution outside the plane. The blue curve is the experimental data.
 distribution of the scattered sizes at a given frequency scat (see Equation 1) The regions of the Galaxy characterized by a high H radiation
 applying half-Student’s distribution. The fitting parameters for half-Student’s intensity show a significant spatial correlation with the areas of strong
 function are shown on the right side of the plots; is degrees of freedom,
 scattering. One of them is positionally associated with the Cygnus
 is the distribution width. The black contour histogram depicts total dis-
 constellation region, which contains active star-forming regions and
 tribution of the source sizes fit , which is the squared sum of half-Student’s
 distribution and the distribution of the unscattered sources ( | | > 10◦ ). The the supernova remnant W78. In the locations of Taurus A, Vela and
 values given in the legend after the ‘#’ sign indicate the number of the sources Cassiopeia A supernova remnants, as well as in the location of Orion
 for a given dataset. nebula (M42), we also found an increase of the scattering strength
 of the ISM. The region with the strongest scattering is the Galactic
 centre, which extends in the Galactic plane at −20◦ 6 6 20◦ .
 Figure 14 shows the data fitting results at two frequencies, 2 GHz Using the AGN VLBI core sizes derived from multi-frequency
 and 8 GHz (black contour histograms). data, we separated the contribution of the intrinsic and scattered
 We searched for the best widths at 2 GHz and 8 GHz with a fixed sizes to the observed angular diameter for 1546 AGN. As expected,
 value of the degree of freedom for both frequencies. Thus, the the contribution of scattered components of the observed size for the
 found parameters of for 2 GHz and 8 GHz can be used to derive the sources in the Galactic plane is systematically larger than for those
 value of the scattering index, similarly to the two-frequency method observed outside the Galactic plane. We found that about 40 per
 (see Section 4 for details). We reached the maximum similarity with cent of the AGN observed in the Galactic plane are not subject to
 the following parameters of the Student’s function: = 0.90, scattering. This reflects that the interstellar medium of our Galaxy is
 = 0.93 for 2 GHz, = 0.07 for 8 GHz. Thus, using this alternative highly inhomogeneous. Most of the unscattered sources are located
 method, the scattering index scat = 1.96 ± 0.13 was inferred. in the direction of the Galactic anti-centre.
 Applying different methods of deriving the power-law scattered in-
 dex scat , we found it ranging from 1.94+0.08 through 2.03+0.16 that
 6.3 The special case of Sagittarius A∗ −0.07 −0.05
 agrees with both the Gaussian screen and Kolmogorov turbulence
 Johnson et al. (2018) have shown that for Sagittarius A∗ , the most models. But taking into account that refractive scattering, which is
 heavily scattered source on the sky, the power-law index for den- present to some degree as well, tends to decrease the scat value due
 sity fluctuations < 3.47, which is shallower than expected for a to steepening the spatial spectrum of free-electron density irregu-
 Kolmogorov spectrum ( = 11/3). At the same time they found 2 larities, the turbulence model can be even favoured. New targeted
 dependence of the angular size ( = 2.0) at > 1 cm, as the scatter- observational campaigns on studying the main scattering screens in-

 MNRAS 000, 1–14 (2022)