Issue 
A&A
Volume 569, September 2014



Article Number  A42  
Number of page(s)  16  
Section  Galactic structure, stellar clusters and populations  
DOI  https://doi.org/10.1051/00046361/201424195  
Published online  16 September 2014 
Online material
Appendix A: Appendix A
Cartesian coordinates are related to spherical coordinates by with radius r, polar angle θ and azimuth angle φ. The Sun is assumed to be at position (x,y,z)_{⊙} = (r_{0},0,0), or equivalently at (r,θ,φ)_{⊙} = (r_{0},π/ 2,0). We define the primed coordinates such that the local halo density (Eq. (5)) can be expressed independent of a polar angle and azimuth angle: (A.7)We note that in the Galactic plane, z = 0, thus primed radius r′ = r and the primed polar angle θ′ = θ = ^{π}/ 2. At the Galactic pole, θ′ = θ = 0, and r′ = z′ = qz = qr. In all other cases, the relation between the r′, θ′ and their spherical equivalents is given by Since we assume an oblate stellar halo (q < 1), it follows from Eqs. (A.8) and (A.9) that θ′ ≥ θ and r′ ≤ r for any given point in the spheroid. Because we want a sphere with radius ξ around the Sun to be contained in our simulated area, we set the boundary conditions, with δ ≤ arctan(^{r0q}/ξ) and ϵ ≤ arctan(^{ξ}/r_{0}). These set the limits of integration in our determination of the stellar halo mass: (A.13)In order to solve the integral over θ, we now first make an estimation of δ. With the assumed values of ξ, q and r_{0} mentioned in the main text, we find δ ≤ 0.334 π. Thus, we take δ = π/ 3. The integral over θ can now be expressed as the hypergeometric function . Again with q = 0.64 and n = −2.8 for consistency with Jurić et al. (2008), we find . Because this value of n ≠ −3, the integral over r can also be evaluated: (A.14)The integral over φ yields 2ϵ, thus after choosing ϵ = arctan(^{ξ}/r_{0}) this reads 2arctan(^{ξ}/r_{0}) = 0.707. The multiplication of an assumed value of ρ_{0} = 1.5 × 10^{4} M_{⊙} pc^{3} (Fuchs & Jahreiß 1998) with these three integrals gives M_{unev} = 3.6 × 10^{7} M_{⊙}.
Appendix B: Appendix B
In case φ(m) is a single power law function between the upper and lower mass boundary of unevolved stars in our simulation box m_{high,unev} and m_{low,unev}, the total mass in unevolved stars (B.1)Given the mass in unevoloved stars M_{unev} which was derived in Appendix A, γ_{unev} = −1, m_{high,unev} = 0.8 and m_{low,unev} = 0.1, this results in a normalization constant belonging to the lower limit on the number of unevolved (single) stars N_{unev} in our simulation box A_{lower} = 1.1 × 10^{8}. When substituted into Eq. (4), this yields (B.2)We derive an upper limit on the number of evolved stars N_{ev} in our simulation box, for the three different IMFs that we investigate in this paper by determining their normalization constants from the IMF at m_{high,unev}. For example, writing the normalization constant for the upper limit on the number of evolved stars in case of a Kroupa IMF as B_{upper}, the relation φ(m_{high,unev}) = A_{lower} = B_{upper} (m_{high,unev})^{2.2} leads to B_{upper} = 7.0 × 10^{7}, from which follows (B.3)where (B.4)To obtain actual numbers instead of an upper limit, we assume that the lowmass part of the IMF is correctly given by Eq. (2), with normalization constant B, (B.5)again using the calculated total mass in unevolved stars M_{unev} = 3.6 × 10^{7} M_{⊙}, we find B = 2.2 × 10^{7}. Now because (B.6)we find Assuming that the Salpeter IMF holds for masses m> 0.8 results in the same way into an upper limit on the number of evolved stars, whereas assuming that it is for the entire mass range 0.1 <m < 100 gives the expected number of evolved stars. Since (B.9)the upper limit on the number of evolved stars in the case of a Salpeter IMF immediately follows from the normalization constant , (B.10)The expected number of stars in our simulation box if the lowmass part of the mass function is also Salpeter with (B.13)thus C = 1.1 × 10^{7}, and (B.14)Finally, for the topheay IMF we derive the normalization constants for the Komiya IMF (indicated by the letter D) and the Salpeter IMF (indicated by the letter E) simultaneously, using the MDF of the halo described by An et al. (2013), who studied halo mainsequence stars with masses between 0.65 M_{⊙} and 0.75 M_{⊙} in the Sloan Digital Sky Survey. These authors found that the halo can be described by a twocomponent model, with 24% of the stars belonging to a lowmetallicity population with a peak at [Fe/H] = − 2.33 (i.e. their calibration model). If this population of lowmetallicity stars is born according to a Komiya IMF, we have (B.15)which holds for and D and E, as well as for D_{upper} and E_{upper}. The normalization constants for the upper limit on the number of evolved stars in case of a topheavy IMF follow again from (B.16)
From the standard integral (B.17)it now follows that D_{upper} = 1.4 × 10^{9} and E_{upper} = 4.3 × 10^{7}. Consequently, the number of evolved stars with (B.20)If the suggested topheavy IMF holds in the lowmass regime, (B.21)where we used the standard integral: (B.22)Combining Eqs. (B.15) and (B.21), we find D = 3.4 × 10^{8} and E = 1.0 × 10^{7}, as well as where (B.27)
© ESO, 2014
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