deletions | additions       diff --git a/Scattering times.tex b/Scattering times.tex  index 65b6f17..b198213 100644  --- a/Scattering times.tex  +++ b/Scattering times.tex 
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\begin{equation}  \tau_s \propto \frac{D^2}{\nu^\alpha}  \end{equation}  where $\alpha$ depends on the structure of turbulence in the ISM and is approximated as $\alpha \sim 4$. The scattering timescale can also be expressed in terms of dispersion measure (DM). Here we will use the empirical model of \cite{Bhat_2004}, that relates $\tau_s$ to DM via \begin{equation}  \log_{10} \tau_s(\nu,{\rm DM}) = -9.46 + 0.154 \log_{10} {\rm DM}+ 1.07 (\log_{10} {\rm DM})^2 - 3.86 \log_{10}{\nu},  \end{equation} 
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\log_{10} \tau_{s_{\rm best}}(\nu,{\rm DM}) = \log_{10} \tau_s(\nu,{\rm DM}) - 4.  \end{equation}  The point at which the scattering timescale exceeds the timescale implied by new versions of SKA1-SYS\_REQ-2961 and SKA1-SYS\_REQ-2962 (see above) ultimately determines the maximum DM for which the SKA1 pulsar timing instrumentation must support phase-coherent dispersion removal. Beyond this DM, the computational cost of coherent dedispersion can be decreased by trading time resolution for frequency resolution. By rearranging equation \ref{eqn:best_case_scattering} we can determine the limiting DM (DM$_{\rm limit}$) above which it is no longer possible to achieve a given timing resolution ($t_{\rm res}$) due to scattering: \begin{equation}  \label{eqn:limiting_dm} 
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t_{\rm res} = \max \left(t_{\rm max}, \frac{P_0}{\min (n_{\rm bins},2048)} \right).  \end{equation}  Here, $t_{\rm max}$ is the maximum effective time resolution as described by requirements 2961 and 2962 (200~ns and 2.5~$\mu$s for SKA1\_Mid and SKA1\_Low, respectively) and $n_{\rm bins}$ is user defined on a per-observation basis (although for practical reasons, such as compatibility with legacy software, this may be limited to a power of 2 in size). Figure \ref{fig:limiting_dm_p} shows the scatter-limited DM for time resolutions between 200 ns and 10 ms for a range of SKA1 observing frequencies. Above the limiting DM it is no longer necessary for the SKA1 pulsar timing instrumentation to support coherent dedispersion.