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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)_⊙ = (r₀,0,0), or equivalently at (r,θ,φ)_⊙ = (r₀,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/r₀). 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 r₀ 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/r₀) this reads 2arctan(^ξ/r₀) = 0.707. The multiplication of an assumed value of ρ₀ = 1.5 × 10^-4 M_⊙ pc^-3 (Fuchs & Jahreiß 1998) with these three integrals gives M_unev = 3.6 × 10⁷ 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 × 10⁸. 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 × 10⁷, from which follows (B.3)where (B.4)To obtain actual numbers instead of an upper limit, we assume that the low-mass 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 × 10⁷ M_⊙, we find B = 2.2 × 10⁷. 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 low-mass part of the mass function is also Salpeter with (B.13)thus C = 1.1 × 10⁷, and (B.14)Finally, for the top-heay 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 main-sequence 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 two-component model, with 24% of the stars belonging to a low-metallicity population with a peak at [Fe/H] = − 2.33 (i.e. their calibration model). If this population of low-metallicity 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 top-heavy IMF follow again from (B.16)

From the standard integral (B.17)it now follows that D_upper = 1.4 × 10⁹ and E_upper = 4.3 × 10⁷. Consequently, the number of evolved stars with (B.20)If the suggested top-heavy IMF holds in the low-mass regime, (B.21)where we used the standard integral: (B.22)Combining Eqs. (B.15) and (B.21), we find D = 3.4 × 10⁸ and E = 1.0 × 10⁷, as well as where (B.27)

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