Magnetic Field and ISM in the Local Galactic Disc

Transcription

1 Mon. Not. R. Astron. Soc. 000, Printed 6 January 2019 MN ATEX style file v2.2 Magnetic Field and ISM in the ocal Galactic Disc Y. Sofue 1 H. Nakanishi 2, and K. Ichiki 3, 1. Institute of Astronomy, The University of Tokyo, Tokyo , Japan 2. Graduate Schools of Sci. and Engineering, Kagoshima Univ., Kagoshima , Japan 3. Graduate School of Science, Div. Particle and Astrophys. Sci., Nagoya University, Nagoya , Japan ABSTRACT Correlation analysis is obtained among Faraday rotation measure, HI column density, thermal and synchrotron radio emissions using archival all-sky maps of the Galaxy. Useful formulas are derived for calculating physical quantities of the local ISM magnetic strength and its lineof-sight OS component, volume gas densities, and OS depth by combining the radio observational data in a hybrid way. Using determined OS depths of the ISM, we obtained all-sky maps of the strength of local magnetic field and its parallel component, which reveal details of local field reversals. Key words: galaxies: individual Milky Way ISM: general ISM: magnetic field 1 INTRODUCTION arge scale mappings of Faraday rotation measure RM of extragalactic linearly polarized radio sources have been achieved in the decades Taylor et al 2009; Oppermann et al. 2013, with which extensive analyses have been obtained to investigate structures of galactic as well as intergalactic magnetic fields review by Akahori et al ocal magnetic fields in the Solar vicinity have been also studied using these RM data and polarization observations of the Galactic radio emission Mao et al. 2010; Wolleben et al. 2010; Still et al. 2011; Sun et al. 2015; Sofue and Nakanishi 2016; iu et al. 2017; van Eck et al. 2017; Alves et al. 2018;. Synchrotron radio emission is another tool to measure the total strength of magnetic field on the assumption that the magnetic energy-density pressure is in equipartition with the thermal and cosmic ray energy densities. This method requires information about the depth of the emitting region in order to calculate the synchrotron emissivity per volume, as the intensity is an integration of the emissivity along the OS. Rotation measure is an integration of the parallel component of magnetic field multiplied by thermal electron density along the line-of-sight OS. It is related not only to thermal free-free radio emission, but also to HI column density through thermal electron fraction in the neutral ISM. Determination of the OS depth is, therefore, a key to measure the magnetic strength from synchrotron emission and the parallel magnetic component from RM. The depth is also required to estimate the volume densities of HI and thermal electrons from observed HI and thermal radio intensities. Emission measure and HI column density are useful to estimate the OS depth, given a relation between the thermal and HI gas densities is appropriately settled such as by assuming a power-law relation. In this paper, correlation analyses are obtained among various sofue@ioa.s.u-tokyo.ac.jp radio astronomical observables RM, HI column density, thermal and synchrotron radio brightness in order to determine physical quantities of the ISM such as the magnetic strength, gas densities, and OS depth. One of the major goal of the present hybrid analysis method will be to obtain whole-sky maps of the total strength and parallel component of the magnetic field in the local Galactic disk. 2 DATA AND FUNDAMENTA OBSERVABES The observational data for the Faraday rotation were taken from all-sky RM map by Taylor et al. 2009, HI data from the eiden- Argentine-Bonn AB survey by Kalberla et al. 2008, synchrotron and free-free emissions at 23 GHz from the 7-years result of Wilkinson Microwave Anisotropy Probe WMAP project by Gold et al Figure 1 shows employed map data for RM, RM, HI column density N HI, thermal free-free radio brightness temperature T ff and synchrotron radio brightness T syn at 23 GHz. Figure 2 shows plots of the same data in the maps shown in figure 1 against the latitude and cosh b. The global similarity of the latitudinal variations indicates that the four observed quantities are deeply coupled with each other. It is remarkable that all the plots depend beautifully on the cosec b relation shown by the full lines. This indicates that these observed quantities are tightly coupled with the line-of-sight depth of the galactic disk composed of a plane parallel layer in the first approximation. However, more detailed inspection reveals that, besides the global common cosec b property, there exist systematic differences in the latitudinal variations among the quantities. The rotation measure, RM, shows milder increase toward the galactic plane than the other quantities. The saturation of RM toward the galactic plane suggests that the magnetic field directions are reversing near the Galactic plane. On the other hand, synchrotron intensity has much sharper peak at the plane, and shows similar variation to the thermal and HI intensities. c 0000 RAS

2 2 Y. Sofue et al Another remarkable property is the sharper peak of the thermal emission toward the galactic plane than HI. This manifests stronger dependence of the thermal emission on the ISM density through the emission measure EM n2e than that of the HI column NHI nhi. We recall that the radio observables are related to the ISM quantities by the following equations. Faraday rotation measure RM is determined by radio polarization observations of extragalactic radio sources, and is related to the thermal electron density ne and line-of-sight OS component of the magnetic field strength B// through B// RM ne µg rad m 2 cm 3 The emission measure EM is determined by observations of the thermal free-free radio emission, and is related to ne as EM ne 2. 2 cm 3 cm 6 The HI column density NHI is measured by 21-cm HI line observations, and is defined by the OS integration of the HI density nhi as NHI 18 nhi cm 2 cm 3 The synchrotron radio brightness Σν, as observed by the brightness temperature Tsyn, is related to the volume emissivity εν and OS depth as Σν = 2kTsyn d εν [erg cm 2 s 1 Hz 1 ], dν λ2 4 where λ = c/ν is the wavelength, ν is the observing frequency, and k is the Boltzmann constant. Here, is the OS depth, and the OS averages noted by are defined as follows. We assume that the ISM quantities are smooth functions of the distance along a OS, and an average of ISM quantity f along the OS satisfies the following relation, f = 0 0 f dx dx = 0 f dx, 5 and for any quantities f and g f f 2 1/2, 6 and f g f g. Figure 1. All-sky maps of RM, RM rad m 2 Taylor et al. 2009, Tsyn and Tff mk at 23 GHz Gold et al 2011, and NHI 1021 cm 2 Kalberla et al We also assume that the Galactic disk is composed of four horizontal layers disks of HI gas, thermal electrons, magnetic fields and cosmic rays, which have the same scale thickness. This means that the OS depth of the four quantities are equal. This assumption may not be good enough, particularly for the synchrotron emission that may originate from a thicker magnetic halo, which includes bright spurs and discrete sources. We consider that the contribution of magnetic halo to RM and synchrotron emission is not so large because of its presumably weaker strength compared to that in the disk. 3 CORREATION AMONG RADIO OBSERVABES We take various correlations among the observed quantities RM, EM or Tff, Tsyn, and NHI, which are plotted in figure 2. c 0000 RAS, MNRAS 000,

3 Magnetic Field and ISM in the ocal Galactic Disc 3 Figure 2. RM rad m 2, T syn mk, 0.1 T ff mk and N HI cm 2 plotted against latitude. Points are reduced by 1 per 10. Note the beautiful dependence of the plotted quantities on cosec b relation shown by the full lines, indicating that they are tightly coupled with the line-of-sight depth of a plane parallel layer. Right panels shows the same, but against cosec b. 3.1 Free-Free to HI tight relation Figure 3a shows a plot of T ff against the square of N HI. The straight line indicates a T ff NHI 2, and plots on the log-log space well obeys this proportionality. This relation indicates that the thermal electron density n e is approximately proportional to n HI, if is not strongly variable from point to point, which is indeed the case except for the high N HI region close to the galactic plane. This correlation will be used to estimate the electron density from HI column in the following analyses. 3.2 Synchrotron to ISM relation Figure 3b is a plot of T syn against N HI. High intensity region is approximately represented by a power law of index 7/4, T syn N 7/4 HI, as expected from frozen-in magnetic field into the ISM and energydensity equipartition between the magnetic field, cosmic rays, and ISM. However, low intensity, therefore high latitude, regions show significant excess over that from the frozn-in assumption. The plot of T syn against T ff in figure 3c indicates a similar relation with a power law index of T syn T 7/8 ff, as expected from the energy-density equipartition for high intensity region. It shows also excess of synchrotron emission over the energy equipartition law in low intensity regions, hence at high latitudes. However, we here simply assume that the synchrotron component approximately represents a single non-thermal disk and is identical to the neutral ISM disk. As will be discussed later, this approximation yields uncertainty of the resulting magnetic strength at high latitude regions by a factor of RM to ISM relation Figure 4 shows plots of the absolute RM values against a N HI, b T syn and c T ff. It is impressible that the plots are more scattered than those in figure 3. This is because the rotation measure is an integrated function of the magnetic field strength along the OS including the reversal of field direction. This scattered characteristics of RM to ISM relations is useful to derive the spatial variation of the OS field direction and strength B //, as discussed later. While RM is scattered against the other ISM observables, it may be better correlated in a narrower restricted region on the sky. Figure 5 shows plots of RM, T syn and T ff against N HI in a small area in the 1st quadrant of the Galaxy at 30 l 50 and b 0 in order to demonstrate how RM is tightly correlated to N HI in a restricted area. Shown by green circles are latitudes corresponding to individual N HI data points, indicating the tight dependence of N HI on the latitude through OS depths.. Figure 6 shows plots of RM against galactic latitude and HI column in the 1st quadrant of the Galaxy. Grey dots show all points, and blue and red dots represent those in northern and southern two small regions in the same quadrant, respectively. at 30 l 60 and b 10 and at 30 l 60 and b 10. Absolute RM value increases toward the galactic plane. Near the plane, the sign of RM changes from negative to positive as the latitude increases, which indicates sudden reversal of the magnetic field direction. The bottom panel in the figure represents the same phenomenon in terms of the column density of HI gas. The linear relation of RM with HI column is evident in the small regions. However, the linearity is lost toward the galactic

4 4 Y. Sofue et al Figure 3. Correlation of T ff and N HI with a line of power index 2, T syn and N HI with a line of power 7/4, and T syn and T ff with a line of power 7/8. Vertical scaling of the lines are arbitrary. Figure 4. Correlation of RM with a HI, b FF and c Synchrotron emissions <l<50 deg 0<b<90 deg plane at b 10 with increasing N HI. This represents decrease in the OS component of magnetic strength, B //, which is caused by rapidly changing field direction near the plane. TffmK, Tsyn mk, RM rad/sqm RM Sync FF b NHI Figure 5. Correlations of RM, T ff and T syn with N HI in a restricted area on the sky 30 l 50, b 0. Shown by green circles are latitudes. b deg 4 DETERMINATION OF ISM PARAMETERS Given the four radio observables RM, T ff, T syn, and N HI, four ISM parameters averaged electron density n e x e n HI, OS depth, total magnetic intensity B, and OS component of magnetic field B // can be estimated as follows. Figure 7 illustrates the flow of the analysis, which we call the ISM hybrid. The thermal electron density is assumed to be proportional to the HI gas density as n e = x e n HI, 8 where, x e is the thermal electron fraction in the neutral ISM defined through x e = n e n e + n HI n e n HI, 9 which may be taken from the literature, e.g. x e 0.08 Foster et al. 2013, or it may be determined by plotting against latitude b through a relation between and the scale height z 1/2 of the HI disk through latitude, = z 1/2 /sin b, as will be discussed later. On these assumptions, the electron density is obtained by dividing the emission measure by the column density of electrons, which is related to the HI column density as above. Thus, we have ne cm 3 x e n HI cm 3

5 Figure 6. RM against HI column top and latitude bottom in the 1st quadrant of the Galaxy. Grey circles show all data points in the 1st quadrant, blue dots are for a small region at 30 l 60 and b 10 and red for 30 l 60 and b 10 in the same quadrant. Magnetic Field and ISM in the ocal Galactic Disc EM xe N 1 HI cm cm In a similar way, the OS component of the magnetic field can be obtained by dividing RM by the column density of electrons as B// RM xe N 1 HI µg rad m cm Using equation 10, the OS depth is approximated by xe N 2 HI EM cm 2 cm The total magnetic intensity B tot is calculated by assuming that the ISM is in energy equipartition between the magnetic and cosmic ray pressures, B 2 /8π N CR E cr, where N CR is the cosmic ray electron number density and E cr is representative energy of radio emitting cosmic rays and is related to the frequency as ν BEcr 2 e.g., Sofue Rewriting as B B tot, we obtain Btot µg ν 1/7 2/7 εν, 13 GHz where Iν dν ε ν ν 2kT syn λ 2 2kT synν 3 c We here assume that the OS depth for the synchrotron emission is equal to the HI column depth. Then, we can calculate the magnetic strength B with the aid of equations 12 and 20. We use synchrotron radio data at ν = 23 GHz. We derive some useful relation linking T syn to N HI and T ff on the assumption of the equipartition. From the above equations, we have n HI B 2 ε 4/7 ν and T syn n 7/4 HI. These relations lead to T syn N 1/4 HI T 3/4 ff. Since N HI T 1/2 ff, we have T syn T 7/8 ff, and similarly T syn N 7/4 HI. These proportionalities are indicated in figure 3b to d in comparison with the observations. It should be mentioned that the assumption of the energy equipartition between non-thermal and thermal components of ISM may not be good enough at high latitudes, as discussed above related to the plots of T syn against N HI and T ff in figure 3. In fact in low T syn high b regions the excess of T syn amounts to an order of magnitude over that expected from the power-law of equipartition. However, we may remember the weak dependence of the emissivity on the estimation of magnetic strength by equation 13, B T 2/7 syn. This eases the uncertainty due to the overestimation, so that the amount of over estimation of magnetic strength would be only about 2 at most. Such over estimation may be significant toward discrete radio sources and bright radio spurs. 5 A-SKY MAPS OF OCA INTERSTEAR MAGNETIC FIED In figure 8 we show the whole-sky maps of total magnetic intensity B tot, and OS component of magnetic intensity B // as calculated using equations 11 and 13. Figure 7. ISM hybrid for determination of physical quantities in the local disk from multiple radio observablves. 5.1 B tot map The B tot map shows a rather smooth distribution of field strength in almost entire sky, except for the GC region and discrete radio sources including radio spurs.

6 6 Y. Sofue et al Ga. a. g. Ga a g B_a G..... Ga. g. g B_ara G 6 Ga g g 6 6 Figure 8. All-sky maps of B tot and B // with contours at interval of 1 µg. Since the OS depth estimation is not accurate in the direction of the Galactic Center, where the HI disk is deficient, the high magnetic intensities around the GC and inner disk may not be appropriate for quantitative discussion, while the stronger field toward the inner Galaxy is reasonable. Similarly, the high magnetic intensity toward discrete radio sources may not be taken seriously for interstellar field estimation. As emphasized in the previous section related to figure 3 and equation 13, the magnetic strength is slightly overestimated in high-latitude regions, particularly in discrete sources including the radio spurs, where the field may be overestimated by a factor of 2 at most. Therefore, the large-scale ridges of relatively strong B, such as along the North Polar Spur, may include such a overestimation. Except for the GC, discrete sources and radio spurs, the map shows an overall distribution of local interstellar magnetic intensity within inside the local gas disk. Since the Sun is surrounded by the local field close in touch, the figure indicates a rather uniform B distribution on the sky with local magnetic strength of B tot 5 6 µg. Using the obtained B tot map, we may calculate representative values of the local magnetic strength by avoiding the region near the Galactic plane. We have thus chosen two areas at +30 b +70 region A and 70 b 30 region B, and obtained mean magnetic strengths of B tot = 5.80 ± 1.57 µg and 5.41 ± 1.44µG, respectively. Combining the two regions, we obtain B tot = 5.60 ± 1.52µG in region A + B as the representitave mean local magnetic strength in the Solar vicinity. 5.2 B // map Equation 11 is particularly simple and useful to estimate the parallel component of magnetic strength, because it includes only two observables, RM and N HI, where the OS depth has been canceled out, leaving x e as one parameter to be assumed. We here apply this relation to map the B // value on the sky assuming x e = 0.1. First, we binned the sky into 1 degree grids in longitude and

7 latitude, and calculate averages of RM and N HI values within ±1 about each grid point for b 50 region, and within ±2 for b > 50. At each point on the grids, we calculated B // with the aid of equation 11. By this procedure we obtain a meshed map of B // on the sky with a resolution of 1 2. Figure 8 shows the thus obtained all-sky map of B // as compared with the N HI and RM maps. The latter two are identical to those presented in Kalberla et al and Taylor et al. 2009, respectively. The RM map is strongly affected by the peaked line-of-sight depth near the galactic plane, causing large positive and negative values near the plane. This causes steep latitudinal gradient of RM due to the field reversal from north to south, resulting in RM singularity along the galactic plane. On the other hand, the B // map is not affected by the OS depth, so that it exhibits the field strength only, and the singularity belt along the galactic plane does not appear. The map reveals a widely extended arched region with positive magnetic strength of B // +5µG in the north from l,b 40,5 to 210,0. This arch seems to be continued by a negative strength arch with B // 5µG in the south from l,b 50, 5 to 160, 30. It may be possible to connect the positive and negative B // arches to draw a giant loop, or a shell, from l 40 to 220 with the field direction being reversed from north to south. Alternatively, the positive arch may be traced through the empty sky around the south pole in the present data Taylor et al. 2009, where the improved map shows positive RM Oppermann et al If this is the case, the RM arches may trace a sinusoidal belt from the southern hemisphere in the 1st and 2nd quadrants to northern in the 3rd and 4th quadrants, drawing an shaped belt on the sky, with the necks in the galactic plane at l 30 and 240. The arched magnetic region along the Aquila Rift from l,b 30, 10 to 300,+30 with B // +2 to 4µG could be a part of the. It is also interesting to note that both the northern and southern polar regions show positive B // with B // +1 µg, indicating that the vertical zenith field directions are pointing away from the Sun in the local interstellar space. Given B tot and B // maps, we can easily calculate the perpendicular component of the magnetic field by B B tot 2 B // 2. However, the accuracy of the above estimated B tot would be too poor to obtain a meanigful map of the perpendicular component. 6 GAS DENSITIES, OS DEPTH AND DISK THICKNESS We discuss some more relations among the ISM parameters and OS depth such as n e, n HI, x e, and z 1/2. The electron density along the line of sight can be determined by observing the emission measure and HI column density: n 2 e EM 1 cm 6 cm 6, 15 and ne cm xe N HI cm The optical depth of the free-free thermal emission is given by Oster 1961 Magnetic Field and ISM in the ocal Galactic Disc τ = Te ν 2.1 EM 10 4 K GHz cm 6, 17 and the brightness temperature is given by T ff = 1 e τ T e τt e. 18 Thus, the emission measure is related to the brightness temperature, electron temperature T e 10 4 K, and observed frequency ν = 23 GHz as EM cm 6 = Tff Te K 10 4 K Tff K 0.35 ν 2.1 GHz GHz Then, equation 12 can be rewritten in terms of T ff and N HI as xe N 2 1 HI Tff cm K 23GHz Figure 9 shows the derived plotted against N HI for x e = 0.1. It is reasonably confirmed that the plot is roughly obey the linear relation, N HI, indicated by the straight line, while the points are largely scattered because of the volume density variations on the sky. We here recall that the OS depth is related to the disk thickness z 1/2 and latitude through z 1/2 = sin b. 21 If the ISM is distributed in a plane parallel layer of a constant half thickness, equations 20 and21 may be combined to determine z 1/2. In figure 10 we plot the calculated z 1/2 against the latitude. Original data points are shown by gray dots, and averaged values in latitudinal interval of b = ±5 are shown by big dots with standard errors. Points without error bar indicate those, whose errors are greater than the averaged values. The averaged half thickness tends to a constant about z 1/2 60 at b > 40, or a full HI thickness of z FWHM 120. This value is smaller than those determined by more global fitting to the whole HI disk, indicting z FWHM 200 Nakanishi and Sofue 2016; Marasco et al The difference may be attributed either to uncertainty in the adopted thermal electron fraction x e, or to in-homogeneity of ISM density. If we take a higher value of x e 0.13, the full disk thickness in the present analysis increases to 200, agreeing with the other determinations. However, such high x e value may not be realistic in view of the current measurement of x e Foster et al. 2012, and the literature therein. Another more plausible reason will be the density inhomogeneity. The here determined represents a summation of line segments, each of which penetrates each of the gas clouds, or density irregularities, on the OS. Accordingly, the calculated densities are those of the clouds. et η be the volume filling factor of gas clouds in the disk, the physical depth,, can be related to the true geometrical distance and thickness of the disk as disk = /η 1/3. 22 In order for the true thickness z 1/2disk = z 1/2 /η 1/3 agrees with the current determinations of HI half thickness of 200, we may take η 2z 1/2 /200 3 = We may thus obtain generalized formula z1/2disk xe 2 η 1/

8 8 Y. Sofue et al G. at. deg OS Depth NHI cm G. ong. deg Figure 11. ogarithmic display of the volume density map of HI and thermal electrons. Shown is nhi [cm 3 ], which is assumed to be proportional to thermal electron density as ne xe nhi [cm 3 ] with xe Figure 9. HI depth for xe = 0.1 plotted against NHI. The straight line indicates a linear relation, NHI, with arbitrary vertical scaling xe 1 NHI 1021 cm 2 1 Tff K. 25 It must be remembered that the densities are proportional to the inverse of xe. In figure 11 we show the calculated all-sky map of the local gas densities for HI and thermal electrons. Except for interstellar clouds shown by bright clumps and the belt near the GC, the derived HI density at b > 10 has a nearly constant value around 1 2 cm 3. We emphasize that such a whole-sky map of the volume density of local HI has been obtained for the first time by the present method, which made it possible to determine OS depths in individual directions of the data points on the sky. 103 z1/2 = sin b DISCUSSION Summary 0 30 b deg Figure 10. Physical half thickness of the HI disk defined by z1/2 = sin b plotted against latitude, which is related to geometrical half thickness by z1/2 disk z1/2 /η 1/3 with η being the volume filling factor of th gas. Circles are averages of the neighboring latitude points with standard errors by bars. Points without bar indicate errors greater than the averaged values. The half thickness tends to 60 at high latitudes. In these estimations, the electron fraction xe has been assumed a priori. Alternatively, it may be able to be estimated by measuring by other means. In fact, the electron fraction xe is determined through xe = 1 1/2 1/2 ne T NHI. = 1.45 ff NHI K 1021 cm 2 24 Finally, we present a useful relation to express the local densities of the thermal electrons and neutral hydrogen with the aid of equation 20, xe nhi xe NHI ne cm 3 cm 3 The latitudinal plots in figure 2 indicate that the four observables, RM, Tsyn, Tff, and NHI, are tightly correlated with each other through their latitudinal variations. This indicates that the distributions of the sources and their physical parameters are also tightly correlated with each other. Based on this fact, we assumed that the sources of these emissions and Faraday rotation are distributed in a single local disk in the Galaxy. On this assumption, we derived useful relations to calculate the local ISM quantities such as magnetic strength, Btot, and OS component of magnetic field, B//, thermal electron density, ne, HI density, nhi, and OS depth,, or scale thickness, z1/2 or z1/2 disk, of the local galactic gas disk. It was emphasized that determination of plays an essential role in the present hybrid method to calculate the physical quantities. Confirming that the radio observables, RM, Tff EM, Tsyn and NHI, are tightly correlated to each other by plotting them against each other, we applied the derived formulas to the archival radio observational data from the literature in order to calculate the ISM physical parameters. We obtained all-sky maps of Btot and B// for the first time, which revealed a magnetic structure in the local interstellar space within the Galactic disk of thickness z1/2 disk 100 around the Sun. The mean local magnetic strength in the Solar vicinity was obtained to be Btot = 5.60 ± 1.52µ G. The magnetic direction varies sinusoidally along a giant shaped belt on the sky, changing its c 0000 RAS, MNRAS 000,

9 Magnetic Field and ISM in the ocal Galactic Disc 9 OS direction from north to south and vise versa every two galactic quadrants. A disk thickness of z 1/2disk 100, consistent with the determination from the current studies, is obtained, if the electron fraction and volume filling factor of ISM gases are taken to be x e 0.1 and η 0.22, respectively. It is stressed that the key in the present analyses is the derivation of OS depth by equation 12 and figure 9. All the derived quantities in this paper are averaged values in the OS depth. Namely, they are values averaged within a distance of 100 cosec b around the Sun. Tsynch mk Dependence on the thermal electron fraction The proportionality of the densities of thermal electrons and HI gas is confirmed through the tight correlation between N HI and T ff EM by figures 2 and 3. On this basis, we assumed a constant thermal electron fraction of x e 0.1, which is close to the current measurement on the order of 0.08 e.g. Foster et al We examine how the ISM parameters depend on x e, given the observables are fixed. From equation 12, we have xe, 2 which propagates to the other quantities as B // xe 1, B tot xe 4/7, n e xe 1, and n HI xe 2. Namely, the ISM quantities are generally proportional inversely to x e, with strongest effect on n HI and weakest on B tot. Tsynch mk e+20 1e+21 1e+22 NHI cm Uncertainty from energy equipartition An uncertain point in the present analysis is the estimation of B tot from the energy equipartition assuming that the non-thermal and thermal disks have the same scale thickness. As in figure 3, plots of T syn against N HI and T ff show an excess over that expected from frozen-in assumption by δt syn 0.1 K at low intensity regions, hence in high-latitude regions. In figure 12 we plot T syn δt syn mimicking halo-subtracted synchrotron emission. It is found that the plots are well represented by the power laws of index 7/4 and 7/8 for N HI and T ff, respectively. This fact implies that the energy-equipartition between frozen-in field and thermal gases holds reasonably well in the disk. At the same time, it is concluded that a non-thermal halo component indeed exists at a level of 0.1 mk at 23 GHz, which is not frozen into the gas disk Tff mk Figure 12. Correlation of T syn 0.1 mk with N HI and T ff, which are approximately represented by a power law of index 7/4 and 7/8, respectively. Vertical scaling of the lines are arbitrary. Original data are shown in violet. ACKNOWEDGMENTS We thank the authors of the AB HI survey Dr. Kalberla et al., all-sky rotation measure map Dr. Taylor et al., and the WMAP 7 years maps Dr. Gold et al. for the archival data. The data analyses were performed on a computer system at the Astronomical Data Center of NAOJ. 7.4 Other observables We have not taken into account the molecular gas, because the nearest molecular clouds within OS depths concerned in this paper are rather few Knude and Hog Comparison with a local bubble surrounded by dusty clouds e.g., allemant et al would be an interesting subject, although the present analysis yields only averaged values along the OS within, and hence cannot directly compared with the 3D study. However, in view of the tight cosec b relation of the used observables, the responsible ISMs are considered to be distributed rather uniformly in a layered disk. This implies that the presently determined quantities may not be directly coupled with the bubble structure edged by dense dusty clouds. Polarization data in radio and infrared observations were not used, while they are obviously useful to improve the present hybrid analysis. Inclusion of these observables would be a subject for the future, while a sophisticated analyses would be required. REFERENCES Akahori T., et al., 2018, PASJ, 70, R2 Alves M. I. R., Boulanger F., Ferrière K., Montier., 2018, A&A, 611, 5 Foster T., Kothes R., Brown J. C., 2013, ApJ, 773, 11 Gold B., et al., 2011, ApJS, 192, 15 Kalberla P. M. W., Burton W. B., Hartmann D., Arnal E. M., Bajaja E., Morras R., Pöppel W. G.., 2005, A&A, 440, 775 Knude J., Hog E., 1998, A&A, 338, 897 allement R., Welsh B. Y., Vergely J.., Crifo F., Sfeir D., 2003, A&A, 411, 447 iu W., et al., 2017, ApJ, 834, 33 Mao S. A., et al., 2012, ApJ, 755, 21 Marasco A., Fraternali F., van der Hulst J. M., Oosterloo T., 2017, A&A, 607, A106 Nakanishi H., Sofue Y., 2016, PASJ, 68, 5

10 10 Y. Sofue et al Oppermann N., et al., 2012, A&A, 542, A93 Oster., 1961, AJ, 66, 50 Purcell C. R., et al., 2015, ApJ, 804, 22 Sofue Y., 2015, MNRAS, 447, 3824 Sofue Y. 2017, in Galactic Radio Astronomy, Chap. 6. Sofue Y., Nakanishi H., 2017, MNRAS, 464, 783 Stil J. M., Taylor A. R., Sunstrum C., 2011, ApJ, 726, 4 Sun X. H., et al., 2015, ApJ, 811, 40 Taylor A. R., Stil J. M., Sunstrum C., 2009, ApJ, 702, 1230 Van Eck C.., et al., 2017, A&A, 597, A98 Wolleben M., et al., 2010, ApJ, 724, 48