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Astron. Astrophys. 335, 488-499 (1998)

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2. Model interstellar medium

An axisymmetric model for the ISM was constructed by Ferrière (1998), based on the best-to-date observational data on the interstellar gas, cosmic rays, and magnetic fields, on the Galactic rotation curve and vertical gravitational acceleration, and on the parameters of the SN distribution. The available observational information, which contains several important gaps (mainly with regard to the hot gas component), was complemented with the requirement that the ISM be in hydrostatic equilibrium and with theoretical considerations on the radial dependence of the hot gas column density and scale height. It was thus fit into a comprehensive picture that represents, at best, the true ISM.

In this picture, the ISM is composed of a molecular medium, a cold neutral medium (CM), a warm neutral medium (WNM), a warm ionized medium (WIM), and a hot ionized medium (HM), respectively denoted by the subscripts [FORMULA] and [FORMULA]. Hydrogen is completely neutral in the CM and in the WNM, while it is completely ionized in the WIM and in the HM. Helium, which represents 9% of hydrogen by number, is everywhere fully neutral except in the HM where it is doubly ionized.

The space-averaged number density of hydrogen nuclei in the different media is given as a function of Galactic radius, [FORMULA], and height, [FORMULA], by

[EQUATION]

[EQUATION]

with

[EQUATION]

[EQUATION]

[EQUATION]

with

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and

[EQUATION]

with

[EQUATION]

and [FORMULA]  kpc being the IAU recommended value of the Galactocentric radius at the Sun.

The space-averaged total interstellar mass density is obtained by summing the partial densities contributed by the different media:

[EQUATION]

where [FORMULA] is the proton rest mass and the factor 1.36 accounts for the 9% by number of helium. The [FORMULA] -dependence of [FORMULA] is plotted in Fig. 1a at three different Galactic radii: [FORMULA]  kpc (dotted line), [FORMULA] (solid line), and [FORMULA]  kpc (dashed line). To give a better idea of how the interstellar mass is radially distributed, we also plotted, in Fig. 1b, the R -dependence of the total column density of interstellar hydrogen nuclei through the Galactic disk, [FORMULA].

[FIGURE] Fig. 1. a (left panel) Space-averaged number density of interstellar hydrogen nuclei as a function of Galactic height, at Galactic radii [FORMULA]  5 kpc (dotted line), [FORMULA]  kpc (solid line), and [FORMULA]  12 kpc (dashed line). The space-averaged interstellar mass density is given by [FORMULA] (see Eq. (6)). b (right panel) Total number of interstellar hydrogen nuclei per unit area on the Galactic disk, [FORMULA], as a function of Galactic radius.

The space-averaged interstellar thermal pressure can be written as a sum over the four true ISM phases (which do not include the molecular gas):

[EQUATION]

[EQUATION]

where [FORMULA] is Boltzmann's constant, [FORMULA] are the temperatures, assumed to be uniform with the standard values [FORMULA]  K, [FORMULA]  K and [FORMULA]  K, and the numerical factors include the helium abundance and the ionization fractions given above.

The space-averaged interstellar turbulent pressure arising from random bulk motions is

[EQUATION]

with the one-dimensional turbulent velocities in the molecular and cold neutral media equal to [FORMULA]  km s-1 and [FORMULA]  km s-1.

The interstellar CR and magnetic pressures are

[EQUATION]

and

[EQUATION]

where the factor inside the brackets reflects the spatial dependence of the synchrotron emissivity, and the exponents [FORMULA] and [FORMULA] are given by the implicit equations

[EQUATION]

and

[EQUATION]

in particular, [FORMULA], and [FORMULA].

In Fig. 2, we show the space-averaged total interstellar pressure,

[EQUATION]

at our three reference radii, and in Fig. 3, we show the average interstellar signal speed defined by

[EQUATION]


[FIGURE] Fig. 2. Space-averaged interstellar pressure (Eq. (11)) as a function of Galactic height, at Galactic radii [FORMULA]  5 kpc (dotted line), [FORMULA]  kpc (solid line), and [FORMULA]  kpc (dashed line).

[FIGURE] Fig. 3. Average interstellar signal speed (Eq. (12)) [FORMULA]. Galactic height, at Galactic radii [FORMULA]  5 kpc (dotted line), [FORMULA]  kpc (solid line), and [FORMULA]  12 kpc (dashed line).

Regarding the Galactic rotation curve, we assume that the rotation velocity, V, increases linearly from the center to [FORMULA]  kpc, varies quadratically between 3 and 5 kpc, and remains equal to 220 km s [FORMULA] beyond 5 kpc. The corresponding rotation rate,

[EQUATION]

and shear rate,

[EQUATION]

are displayed in Fig. 4 as a function of [FORMULA].

[FIGURE] Fig. 4. Large-scale rotation rate, [FORMULA] (Eq. (13)), and minus the large-scale shear rate, [FORMULA] (Eq. (14)), against Galactic radius.

For the vertical component of the Galactic gravitational acceleration we take

[EQUATION]

With our adopted rotation law, the last term vanishes and we obtain the vertical profiles of Fig. 5.

[FIGURE] Fig. 5. Minus the vertical component of the Galactic gravitational acceleration (Eq. (15)) as a function of Galactic height, at Galactic radii [FORMULA]  5 kpc (dotted line), [FORMULA]  kpc (solid line), and [FORMULA]  kpc (dashed line).

The last important ingredient of the model is the SN distribution. Type Ia SNs (hereafter denoted by SNI) arise from old degenerate stars and have a rate per unit volume

[EQUATION]

Type Ibc and Type II SNs (hereafter denoted by SNII) result from the core collapse of young massive stars. 40% of them are isolated and have a rate per unit volume

[EQUATION]

[EQUATION]

leading to a total isolated SN rate per unit volume

[EQUATION]

The other 60% are clustered and partake to the formation of SBs at a rate per unit volume

[EQUATION]

where [FORMULA] is the molecular scale height, given below Eq. (1). The number, [FORMULA], of SNs contributing to one SB varies between [FORMULA] and [FORMULA], with an average of [FORMULA], and is distributed according to

[EQUATION]

Fig. 6a displays the vertical dependence of the SN and SB rates per unit volume at the three reference Galactic radii, while Fig. 6b gives the radial dependence of the corresponding rates per unit area, [FORMULA], and [FORMULA]. The Galactic frequency of isolated SNs, obtained from an integration of Eq. (18) across the Galactic disk, is 1/(101 yr). For comparison, a spatial integration of Eq. (19) yields a SB frequency of 1/(2600 yr), corresponding to a clustered SN frequency of 1/(87 yr). Hence, it appears that isolated SNs are almost as frequent as their clustered counterparts.

[FIGURE] Fig. 6. a (left panel) Isolated SN rate per unit volume, [FORMULA] (thin line; Eq. (18) together with Eqs. (16) and (17)), and SB rate per unit volume, [FORMULA] (thick line; Eq. (19)), [FORMULA]. Galactic height, at Galactic radii [FORMULA]  5 kpc (dotted line), [FORMULA]  kpc (solid line), and [FORMULA]  12 kpc (dashed line). b (right panel) Isolated SN rate per unit area, [FORMULA] (thin line), and SB rate per unit area, [FORMULA], multiplied by [FORMULA] (thick line), vs. Galactic radius. The SB rate was multiplied by the averaged number of clustered SNs per SB in order to yield the clustered SN rate, which can then be easily compared to the isolated SN rate.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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