Definitions

divtel works entirely in geometry and solid angle. It contains no shower physics, no effective area, no night-sky background and no energy threshold - for these a shower simulation has to take over. This page sets out the quantities it computes, since the rest of this section is written in them. For the package’s actual interface, see the User guide.

Pointing

A telescope is a ground position, a focal length \(f\) and a camera radius \(r\). The two camera numbers enter only through their ratio: a camera of radius \(r\) at the focus subtends a disc of angular radius \(\rho = \arctan(r/f)\) on the sky. Arrays are read from files giving these per telescope, and the official CTAO prod6 Alpha layouts ship with the package, so every number in this section can be reproduced without assembling an array by hand.

Each telescope carries its own direction, and any telescope can be aimed anywhere in the sky. Four ways of setting the directions cover everything used here:

• Parallel — every telescope aimed at one sky position, conventional pointing.

• Divergent — every telescope spread by a single parameter, div (How divergent pointing is built shows the construction).

• Sub-arrays — telescopes grouped, each group pointed independently. Not covered in this section yet.

• Per-telescope — a direction given for each telescope from a table, which turns the package from a divergence calculator into a general planning tool. Not covered in this section yet.

A telescope can also be aimed at a named object or a sky coordinate rather than an alt-azimuth direction: Tracking a source covers tracking a real source across a night.

What the array sees

Each telescope sees a disc of angular radius \(\rho\) centred on its pointing. Pointed in parallel these discs coincide and the array sees one disc. Pointed apart, the discs separate and their boundaries cut the union into patches, every point of which is seen by the same number of telescopes. That number is the multiplicity \(m\), and the field \(m(\hat{n})\) over the sky is the primary output of the package. From it,

\[A(m_{\min}) = \sum_{i \,:\, m_i \ge m_{\min}} A_i\]

is the hyper field of view: the total solid angle seen by at least \(m_{\min}\) telescopes. \(A(1)\) is the sky the array watches. \(A(2)\) is the sky it can use, because a shower recorded by a single telescope yields no stereoscopic direction or impact point. Sky seen once is not sky observed, so quoting \(A(1)\) for a divergent array overstates what it delivers — every coverage figure in this documentation is \(A(2)\) unless stated otherwise.

The second summary is the area-weighted mean multiplicity. Each camera contributes its own solid angle to every patch it covers, so

\[\langle m \rangle = \frac{\sum_i m_i A_i}{\sum_i A_i} = \frac{\Omega}{A(1)}, \qquad \Omega = \sum_{\mathrm{telescopes}} \pi \rho^2,\]

with \(\Omega\) the array’s total camera solid angle. This equation is a conservation law: no pointing arrangement creates camera area, it only redistributes a fixed \(\Omega\) over whatever sky the array chooses to watch, and \(\langle m \rangle\) falls as that sky grows. Every trade-off in The divergence spread and ceiling follows from it. The variance of \(m\) over the same patches measures how evenly the coverage is spread, and it is not a detail — see The divergence spread and ceiling for why.

Both quantities are computed on a Lambert azimuthal equal-area projection centred on the array’s mean pointing, so patch areas are true solid angles and there is no coordinate singularity at the zenith. The array and the sky it sees below shows the two views the package draws from this: the array on the ground with its pointing directions, and the sky it covers shaded by multiplicity.

The array and the sky it sees

CTAO-North at div = 0.03, 70° altitude: four LSTs inside nine MSTs, each with its own camera size.

Left, the array on the ground, with each telescope’s pointing direction as an arrow and the two camera sizes distinguished. Right, the sky the array covers, shaded by multiplicity. The array watches \(A(1) = 178\) deg², of which \(A(2) = 117\) deg² is stereoscopic, at a mean multiplicity of 2.66. The outer ring seen by a single telescope is already visible at this modest div — The divergence spread and ceiling is about how far it can be pushed before that ring is most of what is left.

How divergent pointing is built

Divergent pointing aims every telescope along the line joining it to a point \(G\) placed behind the array, so telescopes on opposite sides tilt in opposite directions and the array fans out like an umbrella. \(G\) sits on the mean pointing direction, a distance \(|BG|\) behind the barycentre \(B\), set by one dimensionless parameter div:

\[|BG| = \frac{D}{\tan(\arcsin(\mathrm{div}))}, \qquad D = 100\,\mathrm{m},\]

equivalent to \(\mathrm{div} = \sin\theta_D\), with \(\theta_D\) the divergence acquired by a telescope at perpendicular offset \(D\) from the barycentre. At div = 0, \(G\) recedes to infinity and the array points in parallel; at div = 1 it sits at the barycentre and the telescopes point radially outward.

Telescopes sit on the ground line through the barycentre \(B\); each aims along the line from its own position through the aim point \(G\). A telescope at perpendicular offset \(D\) from \(B\) acquires divergence \(\theta_D\). Moving \(G\) closer to \(B\) steepens every angle at once; moving it to infinity brings every telescope back to parallel. This is a schematic in the plane containing the array and the mean pointing direction, at an exaggerated div = 0.12 for clarity; a real array spreads in three dimensions and divergent_pointing() handles that directly.