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\fancyhf{} % Clears all fields.
\lhead{\scshape Pety \& Bardeau, 2011: \texttt{CLASS}{} spectral axis}
\rhead{\scshape \nouppercase{\rightmark}}   %
\fancyfoot[C]{\bfseries\thepage {}}
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{
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   \fancyhead[C]{Original version at \url{www.iram-institute.org/medias/uploads/class-spectral-axis-v12.pdf}}
   \fancyfoot[C]{\bfseries\thepage {}}
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%
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%
\providecommand{\kms}{\ensuremath{\, \ensuremath{\mathrm{km\,s^{-1}}}}}%
\providecommand{\mm}{\ensuremath{\, \ensuremath{\mathrm{mm}}}}%
\providecommand{\kHz}{\ensuremath{\, \ensuremath{\mathrm{kHz}}}} 

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%
\providecommand{\ic}[1][]{\ensuremath{i_{\ensuremath{\mathrm{#1}}}}}%
\providecommand{\freq}[2][]{\ensuremath{f_\ensuremath{\mathrm{#2}}^{\ensuremath{\mathrm{#1}}}}}%
\providecommand{\wave}[2][]{\ensuremath{\lambda_\ensuremath{\mathrm{#2}}^{\ensuremath{\mathrm{#1}}}}}%
\providecommand{\velo}[2][]{\ensuremath{v_{\ensuremath{\mathrm{#2}}}^{\ensuremath{\mathrm{#1}}}}}%
\providecommand{\doppler}[2][]{\ensuremath{d_{\ensuremath{\mathrm{#2}}}^{\ensuremath{\mathrm{#1}}}}} 

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%
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\providecommand{\fmeas}[1][]{\ensuremath{f_\ensuremath{\mathrm{#1}}^{\ensuremath{\mathrm{meas}}}}}%
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\providecommand{\flsr} [1][]{\ensuremath{f_\ensuremath{\mathrm{#1}}^{\ensuremath{\mathrm{lsr}}}}} 

%
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\providecommand{\dfmeas}[1][]{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{#1}}^{\ensuremath{\mathrm{meas}}}}}}%
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\providecommand{\dvlsr}[1][]{\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{lsr}}}}}} 

%
\providecommand{\vmeas}[1][]{\ensuremath{v_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{meas}}}}}%
\providecommand{\vobs}[1][]{\ensuremath{v_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{obs}}}}}%
\providecommand{\vlsr}[1][]{\ensuremath{v_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{lsr}}}}} 

%
\providecommand{\dmeas}[1][]{\ensuremath{d_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{meas}}}}}%
\providecommand{\dobs}[1][]{\ensuremath{d_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{obs}}}}}%
\providecommand{\dlsr}[1][]{\ensuremath{d_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{lsr}}}}} 

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\providecommand{\Dvobs}[1][]{\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{#1}}}^{\ensuremath{\mathrm{obs}}}}}} 

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\providecommand{\clight}{\ensuremath{c}} 

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\providecommand{\class}{\texttt{CLASS}}%
\providecommand{\mira}{\texttt{MIRA}}%
\providecommand{\mrtcal}{\texttt{MRTCAL}} 

%
\providecommand{\TabHeader}{%
  \begin{table}
    \centering
    \caption{\texttt{CLASS}{} header parameters used to describe the
      frequency/velocity axes and their translation in this document.}
    \begin{tabular}{cccc}
      \hline
      \hline
      \texttt{CLASS}{} name & Unit & Translation & Notation \\
      \hline
      \texttt{nchan}   &    --- & Number of channel           & ---               \\
      \texttt{rchan}   &    --- & Reference channel           & \ensuremath{i_{\ensuremath{\mathrm{0}}}}            \\
      \texttt{restf}   &    MHz & Tuned signal rest frequency & \ensuremath{f_\ensuremath{\mathrm{tuned,sig}}^{\ensuremath{\mathrm{rest}}}} \\
      \texttt{image}   &    MHz & Tuned image rest frequency  & \ensuremath{f_\ensuremath{\mathrm{tuned,ima}}^{\ensuremath{\mathrm{rest}}}} \\
      \texttt{fres}    &    MHz & Observatory channel spacing & \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{meas}}}}}{}         \\
      \texttt{vres}    & \ensuremath{\, \ensuremath{\mathrm{km\,s^{-1}}}}{} & Velocity channel spacing    & \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{meas}}}}}{}         \\
      \texttt{voff}    & \ensuremath{\, \ensuremath{\mathrm{km\,s^{-1}}}}{} & source systemic velocity    & \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{meas}}}}       \\
      \texttt{doppler} &    --- & Doppler factor              & \ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{meas}}}}       \\
      \hline
    \end{tabular}
    \label{tab:header}
  \end{table}} 



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$\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}=+1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7724}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7726}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}} \ensuremath{\equiv}\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\, (\ensuremath{f_\ensuremath{\mathrm{RFtuned}}^{\ensuremath{\mathrm{obs}}}} -\ensuremath{f_\ensuremath{\mathrm{LO}}^{\ensuremath{\mathrm{obs}}}})
                 \ensuremath{\equiv}\ensuremath{\mathrm{cste}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6096}%
\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7729}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{RF}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = (\ensuremath{f_\ensuremath{\mathrm{LO}}^{\ensuremath{\mathrm{obs}}}}+\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}) 
                 + \ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\, (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6116}%
\ensuremath{f_\ensuremath{\mathrm{RFtuned}}^{\ensuremath{\mathrm{obs}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6118}%
\ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{obs}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6120}%
\ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{obs}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7735}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{obs}}}} + \ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\, (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}
  \quad \mbox{with} \quad
  \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{obs}}}} \ensuremath{\equiv}\ensuremath{f_\ensuremath{\mathrm{LO}}^{\ensuremath{\mathrm{obs}}}}+\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7737}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{obs}}}} - \ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\, (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}
  \quad \mbox{with} \quad
  \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{obs}}}} \ensuremath{\equiv}\ensuremath{f_\ensuremath{\mathrm{LO}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7196}%
$\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6184}%
\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6190}%
\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7202}%
$\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}} > 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7748}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{obs}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}
  \quad \mbox{with} \quad
  \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{obs}}}} \ensuremath{\equiv}\ensuremath{f_\ensuremath{\mathrm{LO}}^{\ensuremath{\mathrm{obs}}}}+\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7750}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{obs}}}} - (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}
  \quad \mbox{with} \quad
  \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{obs}}}} \ensuremath{\equiv}\ensuremath{f_\ensuremath{\mathrm{LO}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7752}%
$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}} > 0.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7755}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7757}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}} - (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7759}%
$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}}= \frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{obs}}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{obs}}}}}
  \quad \mbox{and} \quad
  \ensuremath{d_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{obs}}}} = -\frac{\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}}.  
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6276}%
\ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6278}%
\ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7763}%
$\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{obs}}}}\ensuremath{\equiv}\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\ensuremath{\left[ \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{obs}}}} \right] }.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7765}%
$\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{\mathrm{cste}} = 2\,\ensuremath{f_\ensuremath{\mathrm{IFtuned}}^{\ensuremath{\mathrm{obs}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7767}%
$\displaystyle \frac{\ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}}} = 1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}{}
  \quad \mbox{} \quad
  \frac{\ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}}} = 1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}{}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7769}%
$\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{rest}}}}\ensuremath{\equiv}\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\ensuremath{\left[ \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}}-\ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}} \right] }.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7771}%
$\displaystyle \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{rest}}}}\,\ensuremath{\left[ 1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}} \right] },
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7773}%
$\displaystyle
  \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}}-\frac{\ensuremath{SB_{\ensuremath{\mathrm{sep}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\,\ensuremath{\left[ 1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}} \right] }}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7778}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}= \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}+ \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7780}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \ensuremath{\left(  1-\frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}} \right) }
  \quad \mbox{and} \quad
  \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}= \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \ensuremath{\left(  1-\frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}} \right) }.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7204}%
$\ensuremath{\left[ \ensuremath{SB_\ensuremath{\mathrm{sign}}^{\ensuremath{\mathrm{}}}}\, \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\,
  (\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}})/\ensuremath{c} \right] }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7783}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{lsr}}}} = \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{obs}}}} + \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} \, \frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7785}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) + \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} \, \frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7787}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) + \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) \, \frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7789}%
$\displaystyle \Delta\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}})\,\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}}\, \frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7206}%
$\ensuremath{i_{\ensuremath{\mathrm{}}}}= \ensuremath{i_{\ensuremath{\mathrm{0}}}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6424}%
\ensuremath{f_\ensuremath{\mathrm{tuned,sig}}^{\ensuremath{\mathrm{rest}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6426}%
\ensuremath{f_\ensuremath{\mathrm{tuned,ima}}^{\ensuremath{\mathrm{rest}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6428}%
\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{meas}}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6430}%
\ensuremath{\, \ensuremath{\mathrm{km\,s^{-1}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6432}%
\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{meas}}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6436}%
\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{meas}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6438}%
\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{meas}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7845}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7847}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{ima,tuned}}^{\ensuremath{\mathrm{rest}}}} - (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7849}%
$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}}= \frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{meas}}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{meas}}}}}
  \quad \mbox{and} \quad
  \ensuremath{d_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{meas}}}} = -\frac{\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{meas}}}}}{\ensuremath{c}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6476}%
\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{meas}}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6478}%
\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{meas}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7853}%
$\displaystyle \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{meas}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{meas}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{meas}}}}}\quad \mbox{with} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{meas}}}}}= -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{meas}}}}}}{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7856}%
$\displaystyle \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}\quad \mbox{with} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}= -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}}}{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7858}%
$\displaystyle \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}\quad \mbox{with} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}= -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}}}{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7860}%
$\displaystyle \frac{\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}-\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}}{\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}} = \frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}}}{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}}}-1 = \frac{1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}}-1.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7208}%
$v/\ensuremath{c}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7863}%
$\displaystyle \frac{\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}-\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}}{\ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}} = \frac{\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}} \sim 10^{-4}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7210}%
$\sim250\ensuremath{\, \ensuremath{\mathrm{km\,s^{-1}}}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6570}%
\ensuremath{\, \ensuremath{\mathrm{mm}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6572}%
\ensuremath{\, \ensuremath{\mathrm{kHz}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7868}%
$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}}= \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}}\, \frac{1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}},
  \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}= -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{lsr}}}}}}{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}},
  \quad \mbox{and} \quad
  \ensuremath{d_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}= 1-\frac{\ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}}.
$%
\lthtmlindisplaymathZ
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\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7872}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7873}%
$\textstyle =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7874}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,old}}^{\ensuremath{\mathrm{rest}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,old}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7876}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,new}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7878}%
$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}}= \frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{meas}}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{meas}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7880}%
$\displaystyle \ensuremath{i_{\ensuremath{\mathrm{0,new}}}}
  = \ensuremath{i_{\ensuremath{\mathrm{0,old}}}}
  + \frac{\ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}-\ensuremath{f_\ensuremath{\mathrm{sig,old}}^{\ensuremath{\mathrm{rest}}}}}{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{IF}}^{\ensuremath{\mathrm{meas}}}}}} \, \ensuremath{\left(  1+\ensuremath{d_{\ensuremath{\mathrm{\ensuremath{\mathrm{sys}}}}}^{\ensuremath{\mathrm{meas}}}} \right) }
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7883}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7885}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,old}}^{\ensuremath{\mathrm{rest}}}} - (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,old}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7887}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}} - (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,new}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7889}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7891}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,old}}^{\ensuremath{\mathrm{rest}}}} + (\ensuremath{i_{\ensuremath{\mathrm{0,new}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,old}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7892}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7894}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,old}}^{\ensuremath{\mathrm{rest}}}} - (\ensuremath{i_{\ensuremath{\mathrm{0,new}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,old}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7896}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}} + \ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{sig,old}}^{\ensuremath{\mathrm{rest}}}}+\ensuremath{f_\ensuremath{\mathrm{ima,old}}^{\ensuremath{\mathrm{rest}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7899}%
$\displaystyle \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}} =  - \ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{meas}}}}}}{\ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7903}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{meas}}}}= (1+\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}}) \, \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{rest}}}} = (1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}) \, \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{rest}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7905}%
$\displaystyle \ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}} = -\frac{\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{meas}}}}}{\ensuremath{c}}
  \quad \mbox{and} \quad
  \ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}} = -\frac{\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{meas}}}}}{\ensuremath{c}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7907}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{rest}}}} \, \frac{1+\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7910}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}(i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7912}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,new}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{rest}}}}}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7914}%
$\displaystyle \ensuremath{\left[ \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} +  (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,old}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{rest}}}}} \right] }
\, \frac{1+\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7917}%
$\displaystyle \ensuremath{i_{\ensuremath{\mathrm{0,new}}}} = \ensuremath{i_{\ensuremath{\mathrm{0,old}}}} 
             + \frac{\ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}}}{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{meas}}}}}} \, \ensuremath{\left[ \ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}-\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}} \right] },
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7919}%
$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{rest}}}}} = \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{rest}}}}} \, \frac{1+\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}},
  \quad \mbox{{\em i.e.}{},} \quad
  \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}} = \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{meas}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7922}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}}(i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7924}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,tuned,new}}^{\ensuremath{\mathrm{rest}}}} - (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,new}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{rest}}}}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7926}%
$\displaystyle \ensuremath{\left[ \ensuremath{f_\ensuremath{\mathrm{ima,tuned,old}}^{\ensuremath{\mathrm{rest}}}} -  (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0,old}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{rest}}}}} \right] }
\, \frac{1+\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7928}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}}
  = \ensuremath{f_\ensuremath{\mathrm{ima,old}}^{\ensuremath{\mathrm{rest}}}} \, \frac{1+\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}}
  - \ensuremath{f_\ensuremath{\mathrm{sig,tuned}}^{\ensuremath{\mathrm{rest}}}} \, \frac{\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}-\ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}}}{1+\ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7931}%
$\displaystyle \ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}} = -\frac{\ensuremath{v_{\ensuremath{\mathrm{lsr}}}^{\ensuremath{\mathrm{meas}}}}+\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}}
  \quad \mbox{and} \quad
  \ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}} = -\frac{\ensuremath{v_{\ensuremath{\mathrm{lsr}}}^{\ensuremath{\mathrm{meas}}}}+\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}}.  
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7933}%
$\displaystyle \ensuremath{d_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}} = \ensuremath{d_{\ensuremath{\mathrm{old}}}^{\ensuremath{\mathrm{meas}}}} + \frac{\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{lsr}}}}-\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{lsr}}}}}{\ensuremath{c}}.  
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}

%
\providecommand{\vpara}{\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}%


%
\providecommand{\vperp}{\ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}}}%

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7942}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \frac{\sqrt{\ensuremath{c}^2-{\ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}^2}}{\ensuremath{c}+\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7212}%
${\ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}^2 = {\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}^2+{\ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}}}^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6850}%
\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6852}%
\ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7214}%
$\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}\gg \ensuremath{v_{\ensuremath{\mathrm{\perp}}}^{\ensuremath{\mathrm{obs}}}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7954}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \sqrt{\frac{\ensuremath{c}-\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}+\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7957}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \ensuremath{\left(  1-\frac{\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}} \right) }.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7959}%
$\displaystyle \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{obs}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}\quad \mbox{with} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{}}}^{\ensuremath{\mathrm{obs}}}}}= -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}}}{\ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7962}%
$\displaystyle \ensuremath{\lambda_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{\lambda_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \sqrt{\frac{\ensuremath{c}+\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}-\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7964}%
$\displaystyle \ensuremath{\lambda_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}= \ensuremath{\lambda_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}\, \ensuremath{\left(  1+\frac{\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}} \right) }.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7966}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}= \frac{\ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}}{1+\frac{\ensuremath{v_{\ensuremath{\mathrm{\parallel}}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{c}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7968}%
$\displaystyle \ensuremath{v_{\ensuremath{\mathrm{opt}}}^{\ensuremath{\mathrm{obs}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{v_{\ensuremath{\mathrm{sys,opt}}}^{\ensuremath{\mathrm{obs}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{opt}}}^{\ensuremath{\mathrm{obs}}}}}
  \quad \mbox{with} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{opt}}}^{\ensuremath{\mathrm{obs}}}}} = +\ensuremath{c}\, \frac{\delta\ensuremath{\lambda_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{obs}}}}}{\ensuremath{\lambda_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsubsection}

%
\providecommand{\vconv}[1]{\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{#1}}}}}^{\ensuremath{\mathrm{}}}}}%


%
\providecommand{\Doppler}[2]{\ensuremath{D_{\ensuremath{\mathrm{#1}}}(#2)}}%

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6960}%
\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7974}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{tuned,conv1}}^{\ensuremath{\mathrm{obs}}}} = \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}\,\ensuremath{D_{\ensuremath{\mathrm{1}}}(\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}})},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6968}%
\ensuremath{D_{\ensuremath{\mathrm{1}}}()}%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7977}%
$\displaystyle D_\ensuremath{\mathrm{opt}}(\ensuremath{v_{\ensuremath{\mathrm{opt}}}^{\ensuremath{\mathrm{}}}}) = \frac{1}{1+\frac{\ensuremath{v_{\ensuremath{\mathrm{opt}}}^{\ensuremath{\mathrm{}}}}}{\ensuremath{c}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6978}%
\ensuremath{f_\ensuremath{\mathrm{tuned,1}}^{\ensuremath{\mathrm{obs}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6982}%
\ensuremath{D_{\ensuremath{\mathrm{2}}}()}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7982}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{}}^{\ensuremath{\mathrm{rest}}}}(\ensuremath{i_{\ensuremath{\mathrm{}}}}) = \ensuremath{f_\ensuremath{\mathrm{apparent}}^{\ensuremath{\mathrm{rest}}}} + (\ensuremath{i_{\ensuremath{\mathrm{}}}}-\ensuremath{i_{\ensuremath{\mathrm{0}}}}) \, \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{apparent}}^{\ensuremath{\mathrm{rest}}}}},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7984}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{apparent}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}\,\frac{\ensuremath{D_{\ensuremath{\mathrm{1}}}(\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}})}}{\ensuremath{D_{\ensuremath{\mathrm{2}}}(\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}})}},
  \quad \mbox{and} \quad
  \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{apparent}}^{\ensuremath{\mathrm{rest}}}}} = \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{tuned}}^{\ensuremath{\mathrm{rest}}}}}\,\frac{\ensuremath{D_{\ensuremath{\mathrm{1}}}(\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}})}}{\ensuremath{D_{\ensuremath{\mathrm{2}}}(\ensuremath{v_{\ensuremath{\mathrm{\ensuremath{\mathrm{1}}}}}^{\ensuremath{\mathrm{}}}})}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7989}%
\includegraphics[width=\linewidth]{class-spectral-axis-v12-optic-vs-radio}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7220}%
$10^{-5}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7995}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{sig,old}}^{\ensuremath{\mathrm{rest}}}}\,\frac{D_\ensuremath{\mathrm{optic}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{obs}}}})}{D_\ensuremath{\mathrm{radio}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{obs}}}})},
  \quad
  \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{ima,old}}^{\ensuremath{\mathrm{rest}}}}\,\frac{D_\ensuremath{\mathrm{optic}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{obs}}}})}{D_\ensuremath{\mathrm{radio}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{obs}}}})},
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7997}%
$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}} = \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{meas}}}}}\,\frac{D_\ensuremath{\mathrm{optic}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{obs}}}})}{D_\ensuremath{\mathrm{radio}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{obs}}}})},
  \quad \mbox{and} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}} = -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}}}{\ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7062}%
\ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7064}%
\ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7066}%
\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}}%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8007}%
$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{sig,old}}^{\ensuremath{\mathrm{rest}}}}\,\frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{obs}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{obs}}}})},
  \quad
  \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{ima,old}}^{\ensuremath{\mathrm{rest}}}}\,\frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{obs}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{obs}}}})},
$%
\lthtmlindisplaymathZ
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$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}} = \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{meas}}}}}\,\frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{obs}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{obs}}}})},
  \quad \mbox{and} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}} = -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}}}{\ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}}.
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$\displaystyle \frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{obs}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{obs}}}})}
  = \frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{lsr}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{lsr}}}})}.
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$\displaystyle \ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{sig,old}}^{\ensuremath{\mathrm{rest}}}}\,\frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{lsr}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{lsr}}}})},
  \quad
  \ensuremath{f_\ensuremath{\mathrm{ima,new}}^{\ensuremath{\mathrm{rest}}}} = \ensuremath{f_\ensuremath{\mathrm{ima,old}}^{\ensuremath{\mathrm{rest}}}}\,\frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{lsr}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{lsr}}}})},
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$\displaystyle \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}} = \ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{old}}^{\ensuremath{\mathrm{meas}}}}}\,\frac{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,old}}}^{\ensuremath{\mathrm{lsr}}}})}{D_\ensuremath{\mathrm{relat}}(\ensuremath{v_{\ensuremath{\mathrm{sys,new}}}^{\ensuremath{\mathrm{lsr}}}})},
  \quad \mbox{and} \quad
  \ensuremath{\delta \ensuremath{v_{\ensuremath{\mathrm{new}}}^{\ensuremath{\mathrm{meas}}}}} = -\ensuremath{c}\frac{\ensuremath{\delta \ensuremath{f_\ensuremath{\mathrm{new}}^{\ensuremath{\mathrm{meas}}}}}}{\ensuremath{f_\ensuremath{\mathrm{sig,new}}^{\ensuremath{\mathrm{rest}}}}}.
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\ensuremath{v_{\ensuremath{\mathrm{sys}}}^{\ensuremath{\mathrm{lsr}}}}%
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\end{document}
