Centering

Although BFEs are inherently non-parametric, practical use of a BFE requires a truncation of an infinite series of terms, and therefore the reconstructed fields depend on the expansion center [1]. To help with this issue, every Component has the option of centering the expansion on a fraction of the most bound particles, which we call \(E\)-centering. Similarly, the basis may be rotated to realign the vertical axis with the current \(L_z\) axis, which we call \(J\)-centering. Both may be used together.

The algorithms for both centering types rely on the same logic as follows. The user specifies the desired number of particles for specifying the center or orientation, called nEJwant. On the first call, all particles are ranked by binding energy. The nEJwant values are retained and the center-of-mass and/or angular momentum values are computed from this central component. The maximum potential (minimum binding energy) in this group is recorded for subsequent steps, \(E_{max}\). On subsequent steps, particles are tested for membership if the EJ particle group is not full or has energy value \(E<E_{max}\). If a candidate member has a lower potential value than one currently on the list, it is accepted for membership. Subsequent calls are offset by the time interval EJdT. The default value is 0, which implies an EJ computation on every step. This value can be set to a preassigned time larger than the step size to reflect the interesting dynamical times for the user’s system.

The center-of-mass position for this new EJ group may be used to recenter the expansion or rotate the distribution to best align the vertical direction with \(L_z\). The rotation is performed by the Euler matrix whose that makes the \(L_z\) direction the transformed \(z\) axis with the new \(x\) axis along the line of nodes.

The naive application of these transformations is jittery. To reduce the jitter, we only perform the transformation after first logging nEJkeep steps. We use a linear least-squares solution of temporal trajectory of both the center and the \(L_x, L_y, L_z\) axes to predict the current value of the center and axes. We retain the last nEJkeep data points for all subsequent steps. We provide the user the option to damp the linear solution by interpolating to an earlier time point by setting the EJdamp value less than its default value of one.

The EJ algorithm is controlled by the EJ bit flag in the Component configuration. A value 1, 2 enables axis-orientation and centering, respectively. For example, both may be enabled by setting EJ: 3. Diagnostic information about the EJ center computation is enabled by setting EJdiag: true. We often use the EJ centering computation as an internal diagnostic without changing the expansion center by using EJdryrun: true along with EJdiag: true. We strongly recommend doing this for simulations of isolated galaxies. See What is YAML? for an example of setting EJ parameters in the component stanza of the YAML config file.

Footnotes

[1]

Of course, a non-truncated one does not, owing to the translation property of harmonics. In other words, the center would be irrelevant with a arbitrarily large number of terms in the expansion.