Table 2.2 of the original FEROS Design Report, tabulates the individual, theoretical efficiencies of each component of FEROS as well as the overall Instrumental efficiency. That tableis reproduced here in columns 1-8.
The column for the reflectivity of Aluminium from taken from Allen C.W., Astrophysical Quantities, 3rd ed. pg. 108 (with linear interpolation as required). The column Total Inst'+Tel' is derived from Total Instrument times the reflectivity of Aluminium squared representing two reflections (M1 + M2).
A slightly modified version of the FEROS MIDAS efficiency.prg script included in the standard ESO-MIDAS distribution is used for calculating the efficiency of Instrument+Telescope using observations of SpectroPhotometric Standard stars.
A critical factor in this calculation is the Effective Area of the telescope, which here we define as the actual imaging surface, i.e. the area of M1 minus the area of M2 occulting the beam. For the ESO-1.52m the Effective Area calculated thus (assuming dia[M1] = 1.524m and dia[M2] = 0.432m) is 1.678 square meters. For the MPG/ESO-2.20m the Effective Area calculated thus (assuming dia[M1] = 2.200m and dia[M2] = 0.887m) is 3.183 square meters. Hense assuming the same instrument efficiency and efficiency of the mirrors of the two respective telescopes, the throughput of the 2.20m should be a factor of 1.897 greater.
In practice, the Effective Area used in the efficiency.prg script for the 1.52m is 1.5 square meters. This is probably due to either the actual imaging area of M1 being slightly smaller than 1.524m, or the occultation by M2 being slightly larger than 0.432m, or a combination of the two.
Empirical measurements of the efficiency of FEROS+2.20m are made according to the FEROS Calibration Plan every night when FEROS OBs are are executed (both Visitor & Service Mode) and are plotted in the graphic below. The blue line represents the theoretical expectation tabulated above.
Complete information about the Instrument efficiency (and other performance statistics can be found in the Quality Control pages.
Historical empirical measurements of the efficiency of FEROS+1.52m can be found here.