Star Distribution in the Galaxy

To calculate the ratio S/N ratio, one has to answer two questions:

1. What is the fraction of supernovae in the Galaxy within the visibility range of our detector?
2. How often do stellar collapses occur?

Notice that actual Supernovae (optically bright) must not be confused with the occurrence of stellar collapses. Stellar collapses are optically faint, but potential neutrino ``bombs'', since neutrinos carry away almost the total energy (Type Ib and II), while optically bright Supernovae (Type Ia) are not important neutrinos producers. To answer the first question, the results of the Galaxy model computed by Bahcall and Soneira have been used [#!soneira!#], which describe the expected distribution of stars in our Galaxy. The star density in the Galactic disk, $\rho_{d}$ is expected to be:

$$\rho_{d}(x,z,M) \approx {\rm exp}{\left[-z/H(M)-(x-r_{0})/h\right]}$$ (38)

where and $M$ are the distance from the Sun and the mass of the star, respectively. The variables $z$ and $x$ are the transversal and perpendicular coordinates in the Galactic plane. $r_{0}$ is the distance of the Sun from the Galactic center, $H = 352$ pc the scale height and $h = 3.5$ kpc the scale length [#!soneira!#].

A more elaborated version of the model in Eq. describes the expected fraction $F(r)$ of potential SN progenitor stars in the Galactic plane, that lie within a distance of the Sun [#!piran!#], as is shown in Fig. left. The fraction $F(r)$ is the probability distribution within a given distance of stellar collapses in the Milky Way. This answers the first question. The scatter plot in Fig. gives an idea how the approximation with the exponential function in Eq. for the stellar distribution in our Galaxy looks, neglecting the perpendicular contribution ($z=0$). The derivative of $F(r)$ is shown in Fig. right. It is the star density distribution starting from the location of the Sun, as illustrated schematically in Fig.. The huge peak around distance 9 kpc represents the dense cluster of stars enclosed in the Galactic bulge.

Figure: (Left) Derivative function of the fraction of stars $F(r)$ within a distance from the Sun in the Galactic plane. The zero correspond to the Sun position and the huge peak represents the high star density within the Galaxy bulge. (Right) Fraction of Galactic stars $F(r)$ for the distance in kpc according with the model by Bahcall & Piran [#!piran!#]. From that distribution it is easy to see that 90% of stars are enclosed within 17 kpc.

Figure: Schematic scatter plot of star distribution in the Galaxy. The distribution was generated with a MC following Eq.. Visibility radii of 3 and 8 kpc around AMANDA are indicated.

However the central bulge has a diameter of 2 kpc only (4% of the diameter of the Galaxy) [#!bradley!#], more than 20% of all stars are contained in its spheroid. The function decreases fast up to 17 kpc then drops at the edge of the Galaxy at $\sim$ 30 kpc from the Sun.

The question about the frequency of stellar collapses is more difficult to answer due to the large uncertainty in the input data of different models. Some estimates for the rate of Galactic Type II supernova are in the range of 1 per 30 years to 1 per 80 years [#!bahcall!#]. A recent estimate [#!tammann!#] gives a rate of 1 SN per $47 \pm 15$ years, ($2.1 \pm 0.7$ per century, see also Sec.). The most conservative estimates are 1 SN/11 years [#!bahcall!#] and 1 SN/100 years [#!suzuki!#].