Bridge between the two estimates

The pointing-list method also gives two useful diagnostics that bridge it to the area-based estimate. First, the imaged area is computed directly from the non-blanked pixels:

$$\ensuremath{A_\ensuremath{\mathrm{img}}}(\ensuremath{\nu}) = N_\ensuremath{\mathrm{valid}}(\ensuremath{\nu})\,\Omega_\ensuremath{\mathrm{pix}}.$$ (69)

In dual-band mode, $\ensuremath{A_\ensuremath{\mathrm{img}}}(\ensuremath{\nu}_1)$ and $\ensuremath{A_\ensuremath{\mathrm{img}}}(\ensuremath{\nu}_2)$ are reported independently. If $\ensuremath{\nu}_1 < \ensuremath{\nu}_2$, the primary beam is larger at $\ensuremath{\nu}_1$ than at $\ensuremath{\nu}_2$. For the same pointing offsets and the same primary-beam truncation level, the valid area at the highest frequency is therefore included in the valid area at the lowest frequency. The common overlap is thus simply the highest-frequency imaged area,

$$\ensuremath{A_\ensuremath{\mathrm{overlap}}}= \ensuremath{A_\ensuremath{\mathrm{img}}}(\ensuremath{\nu}_2).$$ (70)

The overlap can then be reported as percentages of each band area:

$$P_{1\cap2\vert 1} = 100\,\frac{\ensuremath{A_\ensuremath{\mathrm{... ...nsuremath{\mathrm{img}}}(\ensuremath{\nu}_1)}, \quad P_{1\cap2\vert 2} = 100\%.$$ (71)

The common-overlap quantities are not used to define the median sensitivity, which remains a per-band quantity computed on each band's own valid area.

A purely geometrical number of independent primary beams can then be formed from this imaged area:

$$n_\ensuremath{\mathrm{beam,geom}}(\ensuremath{\nu}) = \frac{\ensu... ...ensuremath{\nu})}{\ensuremath{A_\ensuremath{\mathrm{beam}}}(\ensuremath{\nu})}.$$ (72)

This quantity is useful for comparing the pointing-list footprint to the area originally requested by the user, but it is not, by itself, a sensitivity-equivalent number of beams.

The sensitivity-equivalent bridge is obtained by asking which value of $n_\ensuremath{\mathrm{beam}}$ would make the area-based formula reproduce the median sensitivity of the actual pointing pattern at frequency $\nu$, for the same observing setup. Equating the expressions  and  yields

$$\boxed{% \ensuremath{n_\ensuremath{\mathrm{beam,eq}}}(\ensuremat... ...th{n_\ensuremath{\mathrm{point}}}\,\tilde{\ensuremath{f}}_\ensuremath{\nu}^2. }$$ (73)

The quantity $n_\ensuremath{\mathrm{beam,eq}}$($\nu$) is the number of independent primary beams that would make the area-based TSE formula give the same median sensitivity as the actual pointing list at frequency $\nu$. It is different from $n_\ensuremath{\mathrm{beam,geom}}(\ensuremath{\nu})$: the former is defined by median sensitivity, whereas the latter is defined only by the imaged area.

In summary, the pointing-list estimate can also be understood as a minimal modification of the nine-step summary of the area-based estimate above. The observing overhead part of the computation is intentionally kept unchanged. In particular, Steps 3 to 8 keep the same equations once the map area and the number of pointings have been defined. The PMS-specific changes are confined to a new step 0 plus modified steps 1, 2, and 9.

0.

Before applying the observing-time recipe, the actual pointing offsets are used to compute the dimensionless mosaic noise factor at the reference frequency of each receiver band

$$\boxed{% \ensuremath{f}_\ensuremath{\nu}(p) = \frac{1}{\sqrt{\su... ...}^{\ensuremath{n_\ensuremath{\mathrm{point}}}} B_{i,\ensuremath{\nu}}^2(p)}}. }$$ (74)

Here $B_{i,\ensuremath{\nu}}(p)$ is the primary beam of pointing $i$, normalized to unity at its center and evaluated at sky position $p$. Pixels outside the primary-beam support defined by the adopted truncation level are blanked. This defines the valid imaged area

$$\boxed{% \ensuremath{A_\ensuremath{\mathrm{img}}}(\ensuremath{\n... ...suremath{\mathrm{valid}}(\ensuremath{\nu})\,\Omega_\ensuremath{\mathrm{pix}}, }$$ (75)

and the typical noise factor

$$\boxed{% \tilde{\ensuremath{f}}_\ensuremath{\nu}= \ensuremath{\m... ...thcal{A}_\ensuremath{\nu}} \left[ \ensuremath{f}_\ensuremath{\nu}(p) \right], }$$ (76)

where $\mathcal{A}_\ensuremath{\nu}$ is the set of non-blanked pixels at frequency $\ensuremath{\nu}$. The minimum and maximum values of $\ensuremath{f}_\ensuremath{\nu}(p)$ over the same area are also reported to describe the sensitivity variation between the best-covered part of the mosaic and its edges.

1.

The user-provided area $A_\ensuremath{\mathrm{map}}$ used by the TSE estimate is replaced by the imaged area measured from the pointing-list sensitivity map

$$\boxed{% \ensuremath{A_\ensuremath{\mathrm{map}}}\longrightarrow \ensuremath{A_\ensuremath{\mathrm{img}}}(\ensuremath{\nu}). }$$ (77)

This area is frequency dependent because the primary beam and thus the non-blanked support depend on the reference frequency of the receiver band. It is this area that is used whenever the PMS pointing-list estimate needs a map area, for instance in the scan-time condition.

2.

The number of pointings is no longer inferred from the standard hexagonal sampling relation. It is the actual number of offsets in the PMS pointing list

$$\boxed{% \ensuremath{n_\ensuremath{\mathrm{point}}}= \mbox{numbe... ...math{\mathrm{point}}}}{\ensuremath{n_\ensuremath{\mathrm{track}}}} \right) }. }$$ (78)

The relation

$$\ensuremath{n_\ensuremath{\mathrm{point}}}= \ensuremath{n_\ensuremath{\mathrm{beam}}}\ensuremath{\displaystyle\left( \frac{7}{4} \right) }^2$$ (79)

is therefore only used by the quick area-based estimate, not by the pointing-list estimate.

9.

The rms is first computed for one pointing of the actual mosaic

$$\boxed{% \ensuremath{\Delta t_\ensuremath{\mathrm{on,point}}}= \... ...math{\mathrm{setup}}}}{\ensuremath{n_\ensuremath{\mathrm{point}}}} \right) }, }$$ (80)

and

$$\boxed{% \ensuremath{\sigma_\ensuremath{\mathrm{Jy,point}}}(\ens... ...emath{\mathrm{pol}}}\,\ensuremath{\Delta t_\ensuremath{\mathrm{on,point}}}}}. }$$ (81)

The typical mosaic sensitivity is then obtained by multiplying this per-pointing rms by the median noise factor of the actual pointing pattern

$$\boxed{% \ensuremath{\sigma_\ensuremath{\mathrm{Jy,typ}}}(\ensur... ...athrm{Jy,point}}}(\ensuremath{\nu})\,\tilde{\ensuremath{f}}_\ensuremath{\nu}. }$$ (82)

Equivalently, for comparison with the area-based formula, this last step can be written by replacing $n_\ensuremath{\mathrm{beam}}$ in Eq.  by the sensitivity-equivalent number of beams

$$\boxed{% \ensuremath{n_\ensuremath{\mathrm{beam,eq}}}(\ensuremat... ...th{n_\ensuremath{\mathrm{point}}}\,\tilde{\ensuremath{f}}_\ensuremath{\nu}^2. }$$ (83)

This equivalent form is only a bridge to the TSE formulation; the pointing-list computation itself keeps the explicit separation between the per-pointing rms and the mosaic noise factor.