Made-to-measure modeling

MPE   OPINAS   Dynamics Group

Syer & Tremaine (1996) introduced a new modeling technique in which an N-particle system is tailored to reproduce galaxy observations, rather than superposing orbits as, is done in Schwarzschild's method. The basic concept of the method is to take an N-particle system and to assign each particle an individual adjustable weight. Then, the particles are integrated along their trajectories in a gravitational potential and their weights are continuously changed in such a way that the difference between the model observables and the target observables (observational data) are minimized. The observables of a particle system are given by

Eqn 6

where wi are the particle weights and the kernel Kj gives the contribution of a particle to the observable under consideration. The heart of the algorithm is given by the recipe how to change the particle weights according to the mismatch between model and observations, the so-called force of change (FOC) equation (see below).
A first practical application of the Syer & Tremaine (1996) method was made by Bissantz, Debattista & Gerhard (2004). These authors modeled the face-on mass distribution of the Milky Way and used their particle model to explore the event timescale distribution of microlensing events. The dynamical model matched the long duration wing of the microlensing event timescale distribution (ETD).
We have extended Syer and Tremaine's made-to-measure (M2M) method to properly account for observational errors (cf. De Lorenzi et al., 2007) and have incorporated kinematic observables. The goodness of fit is measured using a standard Χ2 statistics:

Eqn 7

where σ(Yj) is the error of the observations Yj. If we set the FOC to

Eqn 8

once the weights have converged (i.e. dw/dt=0), then the function

Eqn 9

is maximized. Here S is some form of profit function, such as the entropy, which removes the ill-conditioning in the FOC (for μ=0) when the number of particles exceeds the number of observational data constraints. We have implemented this Χ2M2M algorithm in a parallel code NMAGIC (N-particle Made-to-measure Al Gorithm mInimizing Chi squared).

Flow chart

Figure 1: A high level flowchart describing NMAGIC. The main Χ2M2M algorithm is contained in the dashed block, the remainder is an optional potential solver and code for moving the particles, both of which are exchangeable. In our tests, Χ2M2M is generally applied only after a number of position/velocity updates.