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Appendix A: Testing the code

We group in this section a few technical tests to control the code stability and its convergence and to compare them with other literature results.

A.1. Convergence tests and code scalability

We present here the test of the convergence of the results of integrating the system of equations proposed in Eqs. (4)and a few comments on a remarkable result we obtained with the parallelization of the code. As explained in the main text, our problem is to single out the volume occupied by the orbits (differentiable functions x:R → R³ of the real line (i.e., the time) on the configuration space Q) that minimizes a χ² function-difference between an observed and a synthetic CMD (convolved with distance modulus and errors). This function is clearly surjective (i.e. “onto”) so that, once the volume of the parameter space is huge and multi-dimensional, the possibility of rapidly converging to a best-fit solution becomes an important issue. The structure of the code we have developed is extremely promising thanks to the scalability of the system of Eq. (4). For its integration, the classical Runge-Kutta method with adaptive stepsize was adopted (e.g., Press et al. 1993), but the mass classes were spread over several processors. Clearly, to optimize/balance the computational load, we needed to split the equations over the rank size by minimizing the latency over the available processors: with N_p the number of processors and N_c the number of classes. The plot is as in the Fig. A.1.

Once the performance of the code was understood, we performed a few convergence tests on the astrophysical results. Depending on the available computational resources, say the N_p, we could choose the best number of processors on which to perform our integration by looking at Fig. A.1. We had the possibility of testing the lower minimum peak to the right of Fig. A.1 (for N_p = 6500). For this high-resolution test, the system of equations of Eqs. (4)was integrated with as defined in Sect. 2.1. The results are presented in Fig. A.2 where four lines are plotted, for the integration of the system of Eqs. (4)with 7, 65, 650, and 6500 molecular cloud classes on an E050 orbit. The test shows how the approximation adopted in our study, for α = 1/100, i.e., 650 molecular classes (the blue line), gives a robust consistent star formation rate with the higher resolution simulation (the red line): red and blue lines overlap almost perfectly.

On 1 May 2011 the simulation ran on 6500 processors at SGI Altix 4700 at the National Supercomputer HLRB-II, Munich (Germany) on a dedicated queue recording a peak performance of 23Tflops/second.

	Fig. A.1 Plot of theoretical load balance between number of processors Np and number of equations in the system (4).
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	Fig. A.2 Plot of theoretical load balance between number of processors Np and number of equations in the system (4).
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A.2. Tests on the integro-differential system of Eqs. (4)

Equations (4)is an integro-differential system of equations used to deduce the star formation rate of a galaxy from the total pressure P. While this system is conceptually stable because it is based on the fundamental principles of stellar structure and evolution (e.g., Bertelli et al. 2008, 2009), its integration is not trivial, and the numerical approaches adopted needs to be tested. Its integration on the independent variable t is the most computationally time-consuming part of the code, and it was performed with distributed-memory parallelization standard techniques (message-passing interface, MPI). A serial integration of a similar system has been already performed by Fujita (1998). Therefore, we can prepare a case test to reproduce the same result posted in Fujita (1998) by omitting the pressure determination with our Eq. (10)and instead by “injecting” into our code the synthetic pressure profile of Fig.1a (dashed line) in Fujita (1998). Thus, we eliminate the code section devoted to our original pressure determination, and only the parallel integrator of the system of Eqs. (4)is tested against the serial-independent determination originally presented in Fujita (1998). The pressure is defined piece-wise as (A.1)We see in Fig. A.3 that our results are almost identical to the dashed line of Fig. 1b in Fujita (1998). Here the integration is performed with the Runge-Kutta method with adaptive stepsize control (e.g., Press et al. 1993) and distributed over 650 processors.

	Fig. A.3 Star formation rate produced by the pressure injected from Eq. (A.1).
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A.3. Pressure from Eq. (10) and speed limits

We assume here that the dwarf galaxy is falling in a general intra-cluster medium (ICM) or that a dwarf galaxy is passing through the MW gaseous disk. In these situations the motion of the dwarf can easily be in the supersonic regime, i.e. Mach number M_pre > 1, or equivalently that the flow is impacting on dwarf galaxy at supersonic velocity v_g > v_s,ICM with v_s,ICM sound speed of the flow. Moreover, we keep this exercise general by assuming a polytrophic equation of state with adiabatic index γ for the gas flowing. The presence of a shock increases density (and pressure) by compression, while the velocity field from the supersonic becomes subsonic (we limit our arguments to normal shocks) whence Eq. (10)holds. Nevertheless, it is simple to prove that the pressure of the gas before the bow shock, P_pre, and the stagnation point P_s can be very different and Eq. (10)has to be “clothed” with supersonic formalism. We call the pressure after the surface of discontinuity (the thin shock), P_post. In this case the relations between pressures pre- and post-shock are already known (e.g., Landau & Lifshitz 1959), (A.2)and we obtain the pressure between P_post and P_s as (A.3)With and from Eq. (A.2) we can write the relation between as (A.4)whose plot is the same as in Fig. A.4.

	Fig. A.4 Trend between of the ratio between pressure at the stagnation point and pre-shock pressure as a function of the Mach number of the gas where the dwarf galaxy is moving.
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As is evident, the pre-shock pressure and the pressure at the stagnation point are not of the same order of magnitude even if the post shock velocity field is subsonic because the flow is compressed and its pressure (and density) increases passing through the shock and after the shock before reaching the stagnation point. As seen in Fig. A.4 this pressure remains about the same only for subsonic or weakly supersonic regime, which represent the limit of validity of Eq. (10), before supersonic correction has to be implemented.

Appendix B: Star formation efficiency and disruption time relation

The connection between star formation processes and molecular clouds is a very fertile research topic encompassing observations (e.g., Bigiel et al. 2008), theory (e.g., Krumholz & McKee 2005), and experimental/numerical works (e.g., Klessen 2011). In our approach we revisited a work by Elmegreen & Efremov (1997) chosen for its simplicity. Other literature results can be similarly implemented. Let M_i be the cloud’s initial mass as defined in Sect. 2.1, for which we assumed a constant star formation rate ς_i in the integration interval dt (see also Elmegreen 1989; Krumholz & McKee 2005; Krumholz & Tan 2007). Following Surdin (1989), the erosion rate is proportional to the luminosity L_s of embedded stars with total mass M_star,i divided by the specific mass binding energy M_iσ² with σ dispersion velocity. The equation for the rate of change of gas mass in the cloud considered is (B.1)and the luminosity is (B.2)where is the luminosity-to-mass ratio of a population of stars generated by the cloud class at the instant t that we obtained as explained in Sect. 3. We drop for the moment the subscript i to simplify the notation. From the previous Eqs. (B.1)and (B.2)we get (B.3)which can be integrated to give (B.4)where the double integral is more easily numerically computed as (B.5)to give (B.6)for every i. We can also find a quicker approach based on an interpolation function in the work of Girardi et al. (1995, their Fig. 13) as recently in Elmegreen et al. (2006), which we can also adopt when necessary to speed up the code. By using a power law like , λ ∈ [0,1[, the integrals in Eq. (B.6)can be carried out analytically. We observe that the destruction time τ of the molecular clouds can be defined in an implicit way as the instant (B.7)where we exploited the following definitions: (B.8)

and (B.9)with t₀ ~ 10⁷ yr as a fixed parameter from Girardi et al. (1995) and λ ~ 0.6 for t > t₀ and λ = 0 for t < t₀ where the solution is analytical (a quadratic equation). Apart from the difference in the mathematical formalism laid out here, the contents then follow exactly as in Elmegreen & Efremov (1997) to which we refer the reader for an extended discussion. We point out here that we can define the efficiency as (B.10)The two functions defined in (B.8)are of interest due to their dependence on pressure impacting the orbital mass of the dwarf galaxies in our LG. We assume the following functional dependence (Elmegreen 1989; Elmegreen & Efremov 1997): (B.11)with dimensionless constants α₀ = 0.1, β₀ = 180, P_⊙ = 3 × 10⁴k_B   cm^-3   K, and k_B as the Boltzmann constant.

© ESO, 2012