# Compute observables from a model

Source: https://pianistpy.com/docs/notebooks/tutorials/observables

The line-of-sight velocity dispersion at projected radii, its average over an aperture, and the same quantities in physical units. This tutorial uses the isotropic Hernquist model, whose distribution function is known in closed form.

```ipython3
import numpy as np
import pianistpy as pp

print("pianistpy", pp.__version__)
from pianistpy.df.analytic import analytic_model

model = analytic_model("hernquist")
```

```myst
pianistpy 1.0.0
```

`sigma_los` takes projected radii, in the model’s units (internal here: the scale radius is 1).

```ipython3
r_2d = np.array([0.1, 1.0, 10.0])
pp.sigma_los(model, r_2d)
```

```myst
array([0.30974034, 0.27598986, 0.1234161 ])
```

`sigma_los_aperture` averages the squared dispersion over an aperture, weighted by the surface density: a centered circle, or a rectangle at an offset and an angle.

```ipython3
circle = pp.Aperture.circle(1.0)
slit = pp.Aperture.rectangle(2.0, 0.5, r_off=1.0, angle=0.3)
pp.sigma_los_aperture(model, circle), pp.sigma_los_aperture(model, slit)
```

```myst
(0.30288723224579633, 0.30034364264265345)
```

A `LosTable` computes the line-of-sight velocity distributions once and serves every call made from it. Its aperture dispersion is the second moment of the aperture’s velocity distribution, an independent route that agrees with the one from the model.

```ipython3
table = pp.LosTable(model)
pp.sigma_los(table, r_2d), pp.sigma_los_aperture(table, circle)
```

```myst
(array([0.30974034, 0.27598986, 0.1234161 ]), 0.3028872396228435)
```

In physical units, radii and apertures are given in parsecs (or as Astropy quantities) and the dispersions are returned in km/s. Here \(M_0 = 10^9\) solar masses and \(r_0 = 500\) pc, with a concentration of 10.

```ipython3
physical = model.to("physical", concentration=10.0, mass_0=1e9, r_0=500.0)
pp.sigma_los(physical, [50.0, 500.0, 5000.0])
```

\[[31.599921,~28.156674,~12.590995] \; \mathrm{\frac{km}{s}}\]

```ipython3
pp.sigma_los_aperture(
    table.to("physical", concentration=10.0, mass_0=1e9, r_0=500.0), pp.Aperture.circle(500.0)
)
```

\[30.900763 \; \mathrm{\frac{km}{s}}\]