## Observed brightnesses

The brightness observed in direction ( ) is the flux density observed in this direction divided by the typical beam area

| (67)

where

( ) | [ ] | observed brightness,
| [ ] | actual brightness as a function of position over source,
| [sr] | typical beam area,
| [ ] | normalized antenna power pattern,
| [ ] | infinitesimal solid angle of sky .

The typical beam area is an ambiguous notion. Indeed, there are three different solid angles that can be used in this formula: The antenna beam solid angle ( ), the forward beam solid angle ( ), or the main beam solid angle ( )

|  |  | (68)
|  |  | (69)
| and |  | (70)

It is trivial to show that

| (71)

At IRAM, the observed brightness obtained after a single-dish calibration are by default computed using the forward beam solid angle.