Technical notes

The Technical notes detail the analytical and numerical treatments implemented in PIANISTpy, building upon the high-level overviews provided in Chiang+(2026d). The Technical notes follow and refer to the notations, equations, and results of Chiang+(2026d). The Technical notes are organized by the objects of the pipeline, from upstream to downstream, and each object-specific section first treats the analytical theory (whenever applicable) and then the accompanying numerical implementations. Section T:0 states the conventions and notations used; we then discuss the input potential-density models (Section T:1), the distribution function (Section T:2), the truncation (Section T:3), the linear superposition of components (Section T:4), the particle sampling (Section T:5), the velocity profiles and projections (Section T:6), and the JAX-based gradients (Section T:7). Section T:8 documents the accuracy and validation tests and related information.

This document was last updated on October 21, 2026. Section, equation and table numbers of the Technical notes carry the prefix T: (Section T:2.3, Eq. (T:2.4), Table T:2.1). The Functions by subpackage page gives the form in which the docstrings cite the Technical notes.

• T:0 Principles and conventions
  • T:0.1 Notation
  • T:0.2 Thresholds derived from the inputs
  • T:0.3 Closed forms, symbolic derivatives and arbitrary precision
  • T:0.4 Limits of double precision and of the energy grid
  • T:0.5 Accuracy tiers, settings levels, and devices
• T:1 The potential–density pair and the auxiliary density
  • T:1.1 The potential–radius relation as P → P₀ and P → 0
  • T:1.2 Derivatives of the auxiliary density
  • T:1.3 Interpolation of the tabulated truncated profile
• T:2 The distribution function
  • T:2.1 The two forms of the inversion and the forward relation
  • T:2.2 Asymptotic behavior of the DF
  • T:2.3 Evaluating the inversion at each tabulated energy
  • T:2.4 The energy grid
  • T:2.5 The distribution function outside the tabulated energies
  • T:2.6 Method II: inversion of a truncated potential–density pair
  • T:2.7 A distribution function supplied as input
• T:3 Truncation
  • T:3.1 Type II models beyond the anisotropy radius
  • T:3.2 The truncated DF and its density
  • T:3.3 Solving the truncated Poisson equation
  • T:3.4 Completing the truncated profile toward the center and the truncation radius
  • T:3.5 The lowest admissible truncation value
  • T:3.6 Finding the truncation value for a target mass or truncation radius
  • T:3.7 The closed-form profile of an untruncated model
  • T:3.8 Checks of the self-consistency of a model
• T:4 Linear superposition of Method II DFs
• T:5 Particle sampling
  • T:5.1 The velocity distribution at a given radius factorizes
  • T:5.2 Tabulated inverse CDFs
  • T:5.3 Drawing particles by inverse-CDF sampling
  • T:5.4 Checks of the drawn particles and of the written files
• T:6 Velocity distributions and dispersion profiles
  • T:6.1 3D velocity dispersion profiles
  • T:6.2 Surface density and line-of-sight velocity dispersion
  • T:6.3 Aperture-averaged velocity dispersions
  • T:6.4 The line-of-sight velocity distribution
  • T:6.5 Tabulating the line-of-sight velocity distribution over projected radius
  • T:6.6 The aperture-averaged line-of-sight velocity distribution
  • T:6.7 Gauss–Hermite moments
  • T:6.8 The local velocity distribution
  • T:6.9 Velocity distributions and dispersions of a linear superposition
• T:7 Gradients
  • T:7.1 Derivatives of the DF with respect to the anisotropy parameters
  • T:7.2 Anisotropy models and parameters with derivatives, and the refused cases
  • T:7.3 Derivatives of the velocity dispersions and distributions
  • T:7.4 The likelihood
• T:8 Verification
  • T:8.1 The checks of each validation report
  • T:8.2 Tests against independent references
  • T:8.3 The roundtrip test
  • T:8.4 The accuracy tables run at every release
• References