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Astron. Astrophys. 318, 700-720 (1997)

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3. The magneto-ionic medium in M 51

In order to interpret a polarization pattern in a galaxy, one should know certain parameters of the interstellar medium such as the scale heights of the thermal and synchrotron disks, electron volume density and filling factor. Furthermore, we should assess in advance the importance of depolarization effects in order to estimate the depth visible in polarized emission. In this section we discuss how this information can be extracted from data on synchrotron and thermal radio emission.

3.1. The nonthermal disk in M 51

3.1.1. The scale height of the nonthermal emission

The exponential scale heights of the synchrotron emission given in Table 1 were estimated from those in the Milky Way by scaling the latter values obtained at 408 MHz (Beuermann et al. 1985) with frequency as [FORMULA], as observed for NGC 891 (Hummel et al. 1991a) and M 31 (Berkhuijsen et al. 1991). As M 51 and the Milky Way are galaxies of a similar type (Sc and Sbc, respectively) and have about the same linear dimensions, the scale height in the Solar neighbourhood ([FORMULA]) was assumed to apply at [FORMULA] in M 51.


[TABLE]

Table 1. The synchrotron disk in M 51


3.1.2. The magnetic field strength

The total field strength B (including the regular and turbulent components) can be evaluated from the intensity of the nonthermal emission using, for example, the standard assumption of equipartition of energy density between the magnetic field and the cosmic rays (see, e.g., Krause et al. 1984). The nonthermal emission was obtained from the total emission by subtracting the thermal component derived by Klein et al. (1984).

Using the total nonthermal intensity one first estimates the strength of the transverse magnetic field (i.e. the projection of B on the plane of the sky). From the polarized intensity one obtains the strength of the transverse regular magnetic field [FORMULA], from which one gets [FORMULA] by deprojection assuming that the field lies in the galaxy's plane. The turbulent magnetic field strength is found from its transverse component by multiplication by [FORMULA] assuming statistical isotropy. As Faraday effects are negligible at [FORMULA] 2.8 cm, the degree of polarization at this wavelength yields the best estimate for the strength of the regular magnetic field. Therefore we evaluated B, [FORMULA] and [FORMULA] from the [FORMULA] 2.8 cm data.

In Table 1 we show the averaged strengths of regular and turbulent magnetic field inferred from the observed intensity of the total nonthermal emission at [FORMULA], [FORMULA], and the degree of polarization at [FORMULA], [FORMULA], using the assumption of energy equipartition for a nonthermal spectral index [FORMULA] (Klein et al. 1984; [FORMULA]) and the standard ratio of relativistic proton to electron energy density of 100. The full thickness of the emission layer was adopted as [FORMULA].

We may note that the magnetic field strengths derived only weakly depend on both the scale height and the ratio of relativistic proton to electron energy density (as the power [FORMULA]). For example, a 50% increase in [FORMULA] would lower the field strengths by 10%. However, the choice of pressure equilibrium between cosmic ray particles and magnetic field instead of energy equipartition would lower the field strengths by 25%. As [FORMULA] the measurement errors in [FORMULA] and [FORMULA] have a negligible effect on the derived field strengths. Therefore the uncertainty in the derived field strengths given in Table 1 is about 30%.

We stress that the estimates of [FORMULA] and [FORMULA] from the synchrotron emission are used only to assess the role of Faraday depolarization effects in Sect. 3.3. The analysis of the polarization pattern performed in Sect. 4 yields independent estimates of [FORMULA] which are in agreement with those given in Table 1.

3.2. The thermal disk in M 51

In this Section we estimate parameters of the disk of thermal electrons in M 51. For this purpose we used the distribution of the thermal flux density [FORMULA] of M 51 (Klein et al. 1984). At [FORMULA] in M 51 the thermal flux density is [FORMULA] mJy/beam. This is close to the value of [FORMULA] mJy/beam for the Solar neighbourhood (see Sect. 3.2.3), which supports our use of an analogy with the Milky Way, where necessary.

3.2.1. Volume density of the thermal electrons

Klein et al. (1984) derived the radial dependence of the thermal radio emission in M 51 at [FORMULA]. For each ring the average thermal flux density [FORMULA] is given in Table 2. To calculate the average electron density we used the formulae for the thermal flux density (in terms of emission measure) of [FORMULA] regions of Mezger & Henderson (1967) in the form given by Israel et al. (1973) for an unresolved source. In our case we have

[EQUATION]

where [FORMULA] is the average electron density in the thermal ionized gas layer of exponential scale height [FORMULA], f is the volume filling factor of the electron density defined by [FORMULA], and C is a certain constant depending on distance, resolution and electron temperature. For a distance of 9.7 Mpc, a resolution of [FORMULA], and a temperature of [FORMULA] we have [FORMULA].

Using the values of f and [FORMULA] derived below we found the average electron densities presented in Table 2.


[TABLE]

Table 2. Thermal disk and halo in M 51


3.2.2. The scale height of the thermal electrons

The scale height [FORMULA] is not known for M 51. Again we can make an estimate using the values known for the Milky Way. Observations indicate that [FORMULA] varies with radius.

For the inner Galactic region we used the observation of Reich & Reich (1988) that the full halfwidth of the thermal emission at [FORMULA] kpc is 2:O0-2:O5, which yielded a scale height of about 400 pc at a mean radius of [FORMULA]. This scale height was adopted for the ring 3-6 kpc in M 51.

For the Solar neighbourhood Reynolds (1991a) found that the distribution of the electron density perpendicular to the Galactic plane could be described by two components: a thin disk of scale height 70 pc containing the giant and classical [FORMULA] regions, and a thick disk of diffuse gas with a scale height of 900 pc. However, for our present purpose it is sufficient to consider one thermal disk described by a single exponential fitted to the sum of the thin and the thick disk (see Sect. 3.2.4). After scaling to [FORMULA] Reynolds' distribution becomes [FORMULA]. This scale height, [FORMULA], was taken for 6-9 kpc in M 51.

For the outer regions of M 51 we scaled the values derived from a comparison of [FORMULA] and [FORMULA] observations of the Milky Way. The [FORMULA] layer in the Milky Way becomes thicker at large radii, and the same may hold for the ionized gas. Dickey & Lockman (1990) derived a constant [FORMULA] scale height of 165 pc (the cool plus warm components) between 4 and 8 kpc, and it increases considerably beyond the Solar circle (Henderson et al. 1982). Near the Sun the [FORMULA] scale height is about 200 pc, a factor of 3 smaller than the scale height of the ionized gas. Assuming this ratio to be constant for [FORMULA], and taking the solar-neighbourhood values at a radius of 9 kpc in M 51, we found the scale heights of the thermal gas for the outer rings as given in Table 2.

We refer to Fig. 2 for a sketch of M 51 illustrating the spatial distribution of the various components.

[FIGURE] Fig. 2. Sketch of the various layers in M 51 seen in a vertical plane through the centre: disk of thermal electrons with scale height [FORMULA], synchrotron disk at [FORMULA] with scale height [FORMULA], and the elliptical halo. The depth in the thermal disk [FORMULA] from which polarized emission at [FORMULA] 20.5 cm is observed, is indicated. Z is the extent of the halo at [FORMULA]. For clarity the thermal and synchrotron disk are shown thicker than to scale

3.2.3. The filling factor

The filling factor of the free electrons in the thermal disk of M 51 was also adopted from the Solar neighbourhood. It was calculated by comparing the flux density of the thermal radio emission with that expected from the disk of thermal electrons (Berkhuijsen, in preparation).

The thermal flux density at 2.8 cm of the Solar neighbourhood, scaled to the distance of M 51 and seen with a beam width of [FORMULA], is [FORMULA]  mJy/beam. Using Eq. (1), with C taken for the distance of M 51, [FORMULA] and [FORMULA] pc (as applicable near the Sun) we obtained [FORMULA]. This value is in good agreement with other estimates, being halfway between the filling factor of the diffuse warm gas ([FORMULA], Reynolds 1991b) and that of giant [FORMULA] regions ([FORMULA], Güsten & Mezger 1983).

As no information is available on how f varies with r in the Milky Way, we adopted [FORMULA] for all 4 rings in M 51.

3.2.4. Thin and thick thermal disk

In order to check how the approximation of one thermal disk influences the results in Table 2, we also considered a thermal disk in M 51 consisting of two separate components: a thin disk containing the discrete [FORMULA] regions and a thick disk of diffuse emission, as observed in the solar neighbourhood. An estimate of the relative contributions of the two disks to the thermal emission and the rotation measures is then possible.

Using the distribution of electron density of Reynolds (1991a) scaled to [FORMULA] kpc Berkhuijsen (in prep.) estimated the filling factors of the thin and thick disk to be [FORMULA] and [FORMULA], respectively. In this case 20% of the thermal emission is coming from the thick disk.

Assuming the same values of [FORMULA] and [FORMULA] for M 51 we can now use Eq. (1) for each of the components separately. Compared to the results in Table 2 we then find [FORMULA], [FORMULA], [FORMULA] and [FORMULA], where [FORMULA] and [FORMULA] are the scale heights of the thin and thick disk, respectively.

As we have seen above the thin disk produces 80% of the thermal emission [FORMULA]. The thick disk, however, causes 95% of the rotation measure [FORMULA]. Therefore it is reassuring that the scale height and the electron density of the thick disk differ only 15% from the values in Table 2 which we used to calculate model magnetic field strengths (see Sect. 5). Clearly in the present context the approximation of the two-component thermal disk by a single exponential disk is fully acceptable.

3.3. Depolarization

In order to estimate the depth in the disk of M 51 visible in polarized emission, we discuss the various depolarization mechanisms and their significance in M 51.

3.3.1. Wavelength-independent depolarization

At a wavelength as short as [FORMULA] Faraday rotation and Faraday depolarization effects are negligible. Hence only the wavelength-independent depolarization can reduce the degree of linear polarization. This depolarization is caused by a tangling of the magnetic field lines in the emission region both across the beam and along the line of sight. The data at [FORMULA] enables to distinguish between a magnetic field component which is uniform in the beam cylinder (yielding the observed polarized emission) and a nonuniform component (that significantly reduces the degree of polarization, see Table 1).

Observations of various spiral galaxies with different beam sizes (ranging from about 3 kpc to 250 pc) show a regular magnetic field [FORMULA] on scales exceeding 1-3 kpc. But even polarized emission observed at a resolution as high as 250 pc at short wavelengths at intermediate radii is significantly depolarized (e.g., in IC 342 - Krause 1993) and the degree of polarization is about the same for different beam sizes in the above range. The latter is evidence for a turbulent magnetic field component [FORMULA] with a correlation length d significantly smaller than 250 pc. This also implies that the galactic magnetic fields have a two-scale structure with the regular and turbulent magnetic fields being separated from each other by a gap in the wavenumber space. (The term "turbulent" is loosely applied here to any magnetic field tangled on scales smaller than the beamwidth without any reference to the turbulent cascade.) Also in the Milky Way the magnetic field has a regular component with a scale exceeding about 1 kpc and a random field at significantly smaller scales (see Rickett 1990; Ohno & Shibata 1993).

Another source of beam depolarization is unresolved curvature of the regular magnetic field. However, with our beam size of about 3 kpc in M 51 the curvature of, e.g., a circular regular magnetic field would reduce the degree of polarization by only about 10% in the inner part of the galaxy and even less at larger radii, so that this effect can be neglected. Therefore, the observed degree of linear polarization at [FORMULA] provides a good measure of the ratio [FORMULA].

3.3.2. Wavelength-dependent depolarization (Faraday depolarization)

Both the regular and the turbulent magnetic field cause significant Faraday depolarization at [FORMULA] and 20.5 cm. The regular field component along the line of sight inside the source causes depolarization by differential Faraday rotation, [FORMULA]. The turbulent field inside the source causes depolarization [FORMULA] due to dispersion in Faraday depth both along and perpendicular to the line of sight. Furthermore, also a turbulent field in a layer with thermal electrons in front of the source depolarizes due to dispersion in Faraday depth across the beam, [FORMULA]. Below we discuss how significant each of these depolarization mechanisms is in M 51, and we estimate from which layer in the disk the observed polarized emission is coming.

First we discuss the external depolarization. Because of the high Galactic latitude of M 51 ([FORMULA]) [FORMULA] caused in the Milky Way appears to be negligible. Horellou et al. (1992) showed that the structures of [FORMULA] and depolarization across M 51 are closely related to features in the disk. This implies that depolarization in the halo of M 51 must be small.

We may estimate [FORMULA] using the results of Burn (1966) or Tribble (1991), depending on the correlation length of the turbulent cells, d. Burn's formula applies if d is much smaller than the beamwidth of 3.5 kpc, thus if, say, [FORMULA]  pc. Then [FORMULA], where [FORMULA] is the observed dispersion in [FORMULA] which we estimated from our maps as about [FORMULA] between [FORMULA] 18.0 cm and [FORMULA] 20.5 cm. But this value includes the contribution from both the disk and the halo. We assume that the halo contribution is smaller than that of the disk, i.e. [FORMULA]. For [FORMULA]  cm this yields [FORMULA]. However, Dumke et al. (1995) found correlation lengths [FORMULA] at [FORMULA] in spiral galaxies seen edge-on. In this case we may estimate the standard deviation of [FORMULA] from Eq. (20) of Tribble (1991) as [FORMULA] at [FORMULA] 20.5 cm. Using [FORMULA] we find that it exceeds 0.8, which will be the typical value of [FORMULA]. We conclude that [FORMULA] is insignificant compared with the depolarization caused within the disk.

We now discuss the depolarization mechanisms within the synchrotron disk as described by Burn (1966).

The depolarization by differential Faraday rotation in a slab with uniform magnetic field and electron density is given by the well-known expression

[EQUATION]

where [FORMULA] is the degree of polarization at wavelength [FORMULA], [FORMULA] the intrinsic degree of polarization and [FORMULA] the observed rotation measure ([FORMULA] with [FORMULA] measured in [FORMULA], [FORMULA] in [FORMULA], [FORMULA], the line-of-sight regular magnetic field, in µG, and L, the line-of-sight in the Faraday active and emitting region, in pc).

Burn gives the expression for the internal Faraday dispersion as

[EQUATION]

where [FORMULA] with d being the correlation scale of [FORMULA].

In the lower part of the halo, in the region [FORMULA], where synchrotron emission and thermal halo gas occur together, differential Faraday depolarization (2) and internal Faraday dispersion (3) may play a role. However, with the values for [FORMULA] and [FORMULA] in Tables 1, 2 and 5 these mechanisms appear to be negligible ([FORMULA]).

In what follows we assume that the Faraday depolarization occurs entirely in the thermal disk of M 51.

In order to take out the effect of the wavelength-independent depolarization at [FORMULA] we shall use the relative depolarization between [FORMULA] and [FORMULA], denoted as [FORMULA]. As Faraday effects at [FORMULA] are negligible, this ratio is essentially a measure of the Faraday depolarization at the longer wavelengths. The observed values are given in Table 3.


[TABLE]

Table 3. The observed relative depolarization between [FORMULA] and [FORMULA], [FORMULA], the vertical extent [FORMULA] of the upper layer of the thermal disk visible in polarized emission at [FORMULA], and the corresponding values of [FORMULA] and [FORMULA] at [FORMULA] calculated from Eq. (7).


The Faraday depolarization in the disk is caused by both differential Faraday rotation and internal Faraday dispersion. Using equations (2) and (3) with the values in Tables 1, 2 and 3 we find that in the disk of M 51 each of these effects is strong enough to significantly depolarize the emission at [FORMULA]. Due to internal Faraday dispersion, only an upper layer of the disk is visible. As we estimate below, this layer is only about 200-300 pc deep at [FORMULA] -9 kpc. It can be easily seen that depolarization due to differential Faraday rotation across this depth is relatively weak.

The fact that only polarized emission from an upper layer is observed is evident from the much smaller rotation measures observed between [FORMULA] 20.5 and 18.0 cm (Horellou 1990) than between [FORMULA] 6.3 and 2.8 cm (Neininger 1992b), at which wavelenghts Faraday rotation is negligible. For the two inner rings [FORMULA]. As the disk is transparent at short wavelengths, this indicates directly that only a part of it is seen in polarized emission at [FORMULA]. The visible depth in the disk is then estimated as [FORMULA] -300 pc for [FORMULA] -9 kpc. In the case of field reversals in the part of the disk invisible at [FORMULA] 20.5, 18.5 cm this value is an upper limit to [FORMULA]. Horellou et al. (1992), using different arguments, also concluded that at [FORMULA] only the upper part of the polarized disk is observed, and Beck (1991) found the same for NGC 6946.

As the data at [FORMULA] are not complete for the two outer rings, we cannot make the above comparison for [FORMULA] -15 kpc. Instead we propose the following estimate of the minimum visible depth. Let us define [FORMULA] as the depth in the thermal disk from which polarized emission is observed (see Fig. 2). Then the layer in the synchrotron disk, which produces the observed polarized emission at [FORMULA] 20.5 cm, has the thickness [FORMULA]. If no Faraday depolarization occurred in the visible layer, then the fraction [FORMULA] of the polarized emission at [FORMULA] would come from a layer [FORMULA] deep. Since this depth must be equal to the former value, this yields

[EQUATION]

As some depolarization actually occurs within [FORMULA], the true visible depth must be larger than this. The values of [FORMULA] thus calculated are given in Table 3. These values are remarkably close to the upper limits derived above from the [FORMULA] s observed in the two wavelength ranges. We note that in the radial range 12-15 kpc the thermal disk is completely transparent to polarized emission at [FORMULA].

3.3.3. Qualitative analysis of Faraday rotation in a two-layer system

The polarization angle of the polarized emission is given by

[EQUATION]

where [FORMULA] is the foreground Faraday rotation measure produced mainly within the Milky Way. RM is the intrinsic Faraday rotation measure produced by the magnetic field within the galaxy considered; [FORMULA] is the wavelength and [FORMULA] is the intrinsic polarization angle.

At the longer wavelengths a complete depolarization occurs in certain localized regions in M 51 (Horellou et al. 1992), and then Eq. (5) no longer holds. However, this hardly affects the averages over the sectors, so that Eq. (5) is well applicable to sector averages; we directly checked this by fitting Eq. (5) to the available values of [FORMULA] for each sector (see Sect. 2.2).

In order to distinguish between the contributions of the disk and the halo (see Sect. 4.3), we write

[EQUATION]

where [FORMULA] and [FORMULA] are the Faraday rotation measures produced across the disk and the halo, respectively, if both are fully transparent to polarized emission. [FORMULA] is defined here in terms of the disk scale height. (The full thickness of the disk is [FORMULA].) However, it is more convenient to define [FORMULA] in terms of the vertical extent of the halo, Z, i.e., the distance along z between [FORMULA] and the upper boundary of the halo (see Fig. 2). [FORMULA] is the line-of-sight component of the regular magnetic field. Since not the whole disk (or even halo) may be visible in polarized emission at a given wavelength, we introduce factors [FORMULA] and [FORMULA]. As follows from above [FORMULA] and [FORMULA] depend on the wavelength. We assume that both [FORMULA] and [FORMULA] are the same at [FORMULA] and 6.2 cm, and also at [FORMULA] and 20.5 cm.

We can see from Tables 1 and 2 that the synchrotron disk at [FORMULA] and 6.2 cm is about as thick as or thinner than the thermal one. Thus there is only little synchrotron emission originating in the halo and the halo magnetic field can be detected mainly via Faraday rotation in the near half. At [FORMULA] and 20.5 cm, where [FORMULA], the disk is not transparent to polarized emission at [FORMULA] where the halo is present. As a result, it is impossible to determine the structure of the magnetic field in the part of the halo lying beyond the thermal disk from observations of the intrinsic polarized emission.

Now we express [FORMULA] and [FORMULA] in terms of the scale heights of the thermal and synchrotron disk, [FORMULA] and [FORMULA], and [FORMULA] in a given wavelength range. One should take into account that, if synchrotron emission and Faraday rotation occur in the same region, the observed Faraday rotation measure of a transparent layer is equal to [FORMULA], whereas that produced in a foreground Faraday screen (i.e., a magneto-ionic layer devoid of relativistic electrons) is [FORMULA]. Assume that [FORMULA], which inequality is true in the case of M 51 for [FORMULA]. The Faraday rotation measure observed from the disk is given by [FORMULA]. The contribution of the halo to the observed Faraday rotation measure is [FORMULA], where the first term is due to the synchrotron-emitting region, and the second one is the contribution of the rest of the halo which acts as a foreground Faraday screen. Thus, we have

[EQUATION]

[FORMULA] is undefined for [FORMULA] whereas [FORMULA] in this case.

At [FORMULA] the galaxy is transparent, so that [FORMULA] and we obtain [FORMULA] ; furthermore, [FORMULA] because at these wavelengths [FORMULA] differs insignificantly from [FORMULA] for [FORMULA] -9 kpc and [FORMULA] for [FORMULA] -12 kpc. However, [FORMULA] strongly differs from unity at [FORMULA]. The dependence of [FORMULA] and [FORMULA] on [FORMULA] is due to the [FORMULA] -dependence of [FORMULA] and [FORMULA]. The values of [FORMULA], [FORMULA] and [FORMULA] given in Table 3 refer to [FORMULA]. The halo is transparent for polarized emission at all the wavelengths considered, and [FORMULA] differs from unity only because some synchrotron emission originates within the halo (at [FORMULA]), whereas the remaining part of the halo acts as a foreground screen.

For the sake of completeness and having in mind possible applications to other galaxies, we also give expressions for [FORMULA] and [FORMULA] for the case that [FORMULA]. When calculating [FORMULA], it is useful to distinguish two physically different cases: [FORMULA] (i.e., the synchrotron disk is visible at the longer wavelengths) and [FORMULA]. In the former case, the total Faraday rotation measure produced in the disk is given by [FORMULA], where the first term is due to the synchrotron-emitting region, and the second one is the contribution of the rest of the thermal disk which acts as a foreground Faraday screen.

For [FORMULA], when the synchrotron disk in not visible at a given wavelength, we have the [FORMULA] produced in the disk as [FORMULA]. Thus,

[EQUATION]

We have [FORMULA] in this case.

Representation (6) and the values of [FORMULA] and [FORMULA] given in Eq. (7) were used when fitting the observed distributions of polarization angles.

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Online publication: July 3, 1998
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