7. Estimation of S/N ratios

As it is apparent from Fig. 1 (click here), there is a clear correlation between the measured relative error and the (S/N-ratio)/Å. Although such a correlation arises naturally, quantitative estimates of errors as a function of (S/N-ratio)/Å are also expected to depend on index values.

We have shown that the use of approximate formulae for the computation of errors can lead to misleading error estimates. However, performing appropriate simplifications, it is possible to derive simple expressions to estimate the absolute index error as a function of the mean (S/N-ratio)/Å. It can be shown that
 
where, to simplify, we have defined SN(Å) as

(the summation extends over the three bandpasses, i.e. N pixels). The two constants c₁ and c₂ are defined as follows



being , and the bandpass widths.

Similarly
 
where

Equations (41 (click here)) and (45 (click here)) can be easily employed to predict the required (S/N-ratio)/Å to achieve a fixed index error. In Fig. 5 (click here) we show the predictions of these expressions for some particular indices compared with the actual error measurements (from Eqs. (33 (click here)) and (37 (click here))) in a star sample. It is interesting to note that the absolute error of molecular indices does not depend, in a first approximation, on the absolute index value, although the contrary is true for the atomic indices. This is the reason why a larger scatter is apparent in panel (a), where in Eq. (41 (click here)) has been replaced by its corresponding arithmetic mean in the sample. Numerical values for c₁, c₂ and c₃, which obviously depend on the considered index, are given in Table 1.

  
Figure 6: Absolute atomic (panel a)) and molecular (panel b)) line-strength errors measured in the sample of 40 stars from the Lick library as a function of the mean (S/N-ratio)/ Å. The predictions of Eqs. (41 (click here)) and (45 (click here)) are plotted as dashed lines

Following the same procedure with the D₄₀₀₀ index:

where SN(Å)_b and SN(Å)_r, the mean (S/N-ratio)/Å in the blue and red band respectively, will attain, in general, different values (given the large wavelength coverage of the break). In this case, the relative D₄₀₀₀ error does not depend on the absolute D₄₀₀₀ value.