= Time-of-flight timing offset
{title2=$t_0$}
A <time-of-flight timing offset>, $t_0$, is an additive time-origin parameter in a model connecting recorded ion arrival time with physical flight time. In a simple <time-of-flight mass spectrometer> calibration,
$$
x=k_0(t-t_0)^2,
$$
where $x$ is the numerical <mass-to-charge ratio> in declared units. A 2017 instrument-analysis paper uses this equation and determines instrument-dependent constants from identified isotope peaks.<Fully automatic and precise data analysis developed for time-of-flight mass spectrometry by Stefan Meyer and others>[Page 582, equation 2 and isotope-fit description]{ref}
An original error calculation shows why the offset matters. Suppose the physical flight contribution is $20\,\mathrm{\mu s}$ and $t_0=3\,\mathrm{\mu s}$, so the recorded time is $23\,\mathrm{\mu s}$. With $k_0$ held fixed, ignoring the offset multiplies the inferred coordinate by
$$
\frac{23^2}{20^2}=1.3225,
$$
a $32.25\%$ overestimate. This deliberately large example isolates the effect; it does not describe a measured instrument error.
With $k_0$ fixed, a small error in the fitted offset propagates approximately as
$$
\frac{\Delta x}{x}\approx-\frac{2\,\Delta t_0}{t-t_0}.
$$
An offset uncertainty of $0.05\,\mathrm{\mu s}$ at a corrected flight time of $20\,\mathrm{\mu s}$ therefore contributes about $0.5\%$ relative uncertainty in this approximation. The same absolute offset error has greater relative impact on earlier arrivals.
This parameter concerns timing, whereas <time-lag focusing> is a physical delayed-extraction technique affecting the ion packet. Fitting an offset does not by itself sharpen a peak. More detailed instruments can require additional calibration terms; the stated equation is a model, not a universal transfer function.
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