Associated noise

When mapping, the noise per independent resolution element in the map ( ) is given by

$$\ensuremath{\sigma_\ensuremath{\mathrm{beam}}}= \frac{\ensu... ...math{n_\ensuremath{\mathrm{beam}}^\ensuremath{\mathrm{bol}}}},$$ (23)

where is the total integration time per independent resolution element, is the number of repeated coverages and is the number of independent resolution elements per bolometer of the array. This number is easily obtained through

$$\ensuremath{n_\ensuremath{\mathrm{beam}}^\ensuremath{\mathr... ...emath{\mathrm{grid}}}\,\pi\,\ensuremath{\theta}^2}{4\,\ln(2)},$$ (24)

where is the total number of valid bolometers (equal to the maximum number of bolometers minus the number of dead ones), is the solid angle of the resolution element in the map (assumed to be a Gaussian), is the factor due to gridding and is the telescope full width at half maximum. Using the equations derived from the mapping strategy, it is easy to obtain

$$\ensuremath{\sigma_\ensuremath{\mathrm{beam}}}= \frac{\ensu... ...mathrm{\Vert}}}}{\ensuremath{n_\ensuremath{\mathrm{cover}}}}}.$$ (25)

In the case of MAMBO,

$$\frac{\ensuremath{\mathrm{NEFD}}}{\sqrt{\ensuremath{n^\ensu... ...beam}}}}} \simeq 0.2\,\mbox{(mJy/Beam)}\,\sqrt{\mbox{s}}\,/''.$$ (26)

However, this noise value stands before any data processing. The restoration will multiply the above noise level by the square root of the number of chopper throws used to compute the brightness at any given position. This makes the actual noise value dependent on the total map size in the scanning direction. Moreover, the methods to restore the source are not perfect¹. Altogether, we thus use

$$\ensuremath{\sigma_\ensuremath{\mathrm{actual}}} = \ensurem... ...mathrm{\Vert}}}}{\ensuremath{n_\ensuremath{\mathrm{cover}}}}},$$ (27)

where $\ensuremath{\alpha}(\ensuremath{\Delta^\ensuremath{\mathrm{\Vert}}_\ensuremath{\mathrm{sou}}})$ value is deduced from the experience of the previous bolometer pools with MAMBO at the IRAM-30m.