# Observables

Source: https://pianistpy.com/docs/guide/observables

`pianistpy.sigma_los()`, `pianistpy.sigma_los_aperture()`, `pianistpy.aperture_losvd()`,
`pianistpy.Aperture`, `pianistpy.LosTable`

Observables are functions of a model. Given a model, `losvd` and `aperture_losvd` build the model’s
line-of-sight table (`LosTable`) at every call. For repeated requests on one model, build
`LosTable(model)` once. A `LosTable` tabulates the line-of-sight velocity distributions of the model, and every
observable accepts the `LosTable` in place of the model.

```python
sigma = pp.sigma_los(model, [0.1, 1.0, 10.0])        # at projected radii

circle = pp.Aperture.circle(r_ape=1.0, r_off=0.0)
slit = pp.Aperture.rectangle(side_x=2.0, side_y=0.2, r_off=0.0, angle=0.0)
pp.sigma_los_aperture(model, circle)                 # averaged over an aperture
pp.sigma_los_aperture(model, slit)

table = pp.LosTable(model)                           # once, for many requests
pp.sigma_los(table, [0.1, 1.0, 10.0])
pp.sigma_los_aperture(table, circle)
pp.aperture_losvd(table, circle, [0.0, 0.1, 0.2])    # the distribution in the aperture
```

[](https://pianistpy.com/docs/_images/projection_configurations.png)

The projection configurations of a spherical system truncated at \(R_{\rm t}\) (dashed circle), from
[Chiang+(2026d)](https://arxiv.org/abs/2610.28758), Fig. 14. (a) The projected radial profile along the line of sight, as for a slit of negligible width.
(b) The average over a rectangular aperture of sides \(x\) and \(y\) (`side_x`, `side_y`), whose
center lies at the projected distance `r_off` from the center of the system, on a line at the angle
\(\theta\) (`angle`) to the rectangle’s \(x\) side, as for a slit of finite width. (c) The average
over a circular aperture, centered or offset, as for fiber-fed observations.

An `Aperture` is a circle or a rectangle, centered or offset. A rectangle also takes an `angle`, in radians,
between the rectangle’s x axis and the line through the center of the model and the center of the rectangle.
`sigma_los_aperture` and `aperture_losvd` take the same `Aperture` object. `sigma_r`, `sigma_t`, `losvd`
and `pdf_v` also take a model or the model’s line-of-sight table as the first argument. The
[Observables](https://pianistpy.com/docs/reference/api/observables) page of the API reference lists every call. The Gauss-Hermite
moments \(h_3\) and \(h_4\) of the line-of-sight velocity distribution have no top-level call. The inference
takes \(h_3\) and \(h_4\) as data ([`pianistpy.inference.Data`](https://pianistpy.com/docs/reference/api/inference#pianistpy.inference.Data)).

`ProjectedSampler` draws pairs of projected radius and line-of-sight velocity on an annulus and, if given, in a
user-specified range of line-of-sight velocity. For every anisotropy model, the draw is conditioned exactly on the
velocity range. At each projected radius, the distribution of the projected radius is weighted by the share of the
line-of-sight velocity distribution inside the velocity range. The ranges and the drawn pairs are in the unit system
of the model or the line-of-sight table given to `ProjectedSampler`, as for the other observables. The attribute
`validation` of a `ProjectedSampler` is the validation report of the draw.

```python
sampler = pp.ProjectedSampler(table, R_range=(0.1, 3.0), v_range=(-0.2, 0.3))
pairs = sampler.sample(1_000_000, seed=1)   # pairs.r_2d, pairs.v_los, pairs.mass
```

Observables over an aperture of any other shape can be estimated by Monte Carlo. Draw pairs with
`ProjectedSampler` on an annulus that encloses the aperture, give each pair a uniformly random position angle
(the model is spherical), and keep the pairs inside the aperture. Sums over the kept pairs, each with its mass
(`pairs.mass`), give the aperture’s mass, dispersion and velocity distribution, with a statistical error that
falls as the inverse square root of the number of pairs.

- [Radial velocity dispersion (sigma_r)](https://pianistpy.com/docs/guide/sigma_r)

- [Tangential velocity dispersion (sigma_t)](https://pianistpy.com/docs/guide/sigma_t)

- [Line-of-sight velocity distribution (losvd)](https://pianistpy.com/docs/guide/losvd)

- [Line-of-sight velocity distribution in an aperture (aperture_losvd)](https://pianistpy.com/docs/guide/aperture_losvd)

- [Local velocity distribution (pdf_v)](https://pianistpy.com/docs/guide/pdf_v)

- [Line-of-sight table (LosTable)](https://pianistpy.com/docs/guide/lostable)

- [End-to-end differentiability](https://pianistpy.com/docs/guide/differentiability)

- [Differentiable observables (Differentiable)](https://pianistpy.com/docs/guide/differentiable)