Optimal number of ON per OFF measurements

This section is just a reformulation of the original demonstration by Ball (1976).

Let's assume that we are measuring \ensuremath{n_\ensuremath{\mathrm{on/off}}} independent on-positions for a single off. The same integration time ( \ensuremath{t_\ensuremath{\mathrm{on}}}) is spent on each on-position and the off integration time is

  $\displaystyle \ensuremath{t_\ensuremath{\mathrm{off}}}= \alpha \, \ensuremath{t_\ensuremath{\mathrm{on}}},
$ (53)
where $\alpha$ can be varied. Using eq. [*] and $\ensuremath{t_\ensuremath{\mathrm{onoff}}}= \ensuremath{n_\ensuremath{\mathrm{o...
..._\ensuremath{\mathrm{on/off}}}+\alpha)\,\ensuremath{t_\ensuremath{\mathrm{on}}}$, it can be shown than
  $\displaystyle \ensuremath{t_\ensuremath{\mathrm{onoff}}}= \frac{\ensuremath{T_\...
...}+\alpha+\frac{\ensuremath{n_\ensuremath{\mathrm{on/off}}}}{\alpha} \right) }.
$ (54)
Differenciating with respect to $\alpha$, we obtain
  $\displaystyle \frac{d\ensuremath{t_\ensuremath{\mathrm{onoff}}}}{d\alpha} \propto 1-\frac{\ensuremath{n_\ensuremath{\mathrm{on/off}}}}{\alpha^2}
$ (55)
Setting the result to zero then gives that the minimum elapsed time to reach a given rms noise is obtained for
  $\displaystyle \alpha = \sqrt{\ensuremath{n_\ensuremath{\mathrm{on/off}}}} \quad...
...h{n_\ensuremath{\mathrm{on/off}}}} \, \ensuremath{t_\ensuremath{\mathrm{on}}}.
$ (56)