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\begin{document}
\title{LIGO-India observatory coordinate system for GW analyses}
\author{Shivaraj kandhasamy, Sukanta Bose}
\maketitle
   The purpose of this document is to provide LIGO-India observatory 
global and local coordinate details needed for GW searches. We note here that 
these are design values. If needed, these values will be updated once the 
observatory's construction is finished. 

  The design GPS coordinates of the beam splitter and the two end-mirrors, as 
well as their heights, \footnote[2]{These are design Above Mean Sea Levels
(AMSL) of floor of beam tube enclosures (BTE) at the middle of the arms. We 
haven't accounted for the distance between the BTE floor and beamline, and 
also any effect of curvature of the earth. So these height values expected to 
have systematic uncertainties of a few meters.} are given in Table \ref{table1},

\begin{table}[h!]
\begin{tabular}{|c|c|c|c|}
  \hline
  Location & Latitude & Longitude & Height (m) \\
  \hline
  Beam splitter & $19^{\circ} \, 36' \, 47.9017''$ N & $77^{\circ} \, 01' \, 51.0997''$ E & 440.0 \\ [0.25cm]
  End-X & $19^{\circ} \, 38' \, 43.1430''$ N & $77^{\circ} \, 00' \, 47.4656''$ E & 440.0 \\ [0.25cm]
  End-Y & $19^{\circ} \, 35' \, 47.5998''$ N & $76^{\circ} \, 59' \, 49.4969''$ E & 440.0 \\
  \hline
\end{tabular}
    \caption{GPS coordinates of main optics (taken from \cite{civil_dbr})}
    \label{table1}
\end{table}
  
Based on the above values below we obtain the following detector constants 
for LIGO-India observatory to be used in the LALDetectors.h \cite{LALheader}.

\begin{verbatim}
LAL_LIO_4K_DETECTOR_NAME                 "LIO_4k" 
LAL_LIO_4K_DETECTOR_PREFIX               "I1"    
LAL_LIO_4K_DETECTOR_LONGITUDE_RAD        1.34444215058 
LAL_LIO_4K_DETECTOR_LATITUDE_RAD         0.34231676739  
LAL_LIO_4K_DETECTOR_ELEVATION_SI         440.0
LAL_LIO_4K_DETECTOR_ARM_X_AZIMUTH_RAD    5.80120119264   
LAL_LIO_4K_DETECTOR_ARM_Y_AZIMUTH_RAD    4.23039066080   
LAL_LIO_4K_DETECTOR_ARM_X_ALTITUDE_RAD   0.0   
LAL_LIO_4K_DETECTOR_ARM_Y_ALTITUDE_RAD   0.0  
LAL_LIO_4K_DETECTOR_ARM_X_MIDPOINT_SI    2000.00000000000        
LAL_LIO_4K_DETECTOR_ARM_Y_MIDPOINT_SI    2000.00000000000        
LAL_LIO_4K_VERTEX_LOCATION_X_SI          1.34897115479e+06       
LAL_LIO_4K_VERTEX_LOCATION_Y_SI          5.85742826577e+06       
LAL_LIO_4K_VERTEX_LOCATION_Z_SI          2.12756925209e+06      
LAL_LIO_4K_ARM_X_DIRECTION_X             0.38496278183  
LAL_LIO_4K_ARM_X_DIRECTION_Y             -0.39387275094  
LAL_LIO_4K_ARM_X_DIRECTION_Z             0.83466634811
LAL_LIO_4K_ARM_Y_DIRECTION_X             0.89838844906 
LAL_LIO_4K_ARM_Y_DIRECTION_Y             -0.04722636126  
LAL_LIO_4K_ARM_Y_DIRECTION_Z             -0.43665531647  
\end{verbatim}


Further details of those individual constants are given below.
\subsubsection*{Detector prefix:}
\begin{verbatim}
  LAL_LIO_4K_DETECTOR_PREFIX           "I1"
\end{verbatim}

The official data channel prefix for LIGO-India observatory is ``I1''. The three letter designation for the observatory is ``LIO'' (LIGO India Observatory) \cite{nomenclature}.

\subsubsection*{Detector location:}
\begin{verbatim}
  LAL_LIO_4K_DETECTOR_LONGITUDE_RAD        1.34444215058
  LAL_LIO_4K_DETECTOR_LATITUDE_RAD         0.34231676739
\end{verbatim}

These are longitude and latitude of the beam splitter location in radians. The 
location of the beam splitter is sometimes considered as the detector location.

\subsubsection*{Detector elevation:}
\begin{verbatim}
  LAL_LIO_4K_DETECTOR_ELEVATION_SI         440.0
\end{verbatim}

This is the height of the vertex floor from sea level. In principle this height 
parameter should also include the distance from floor to the center of the 
beam splitter ($\sim 2-3$ meter). Currently that is not included. This will be 
updated with as-built values after the construction and installation. However this does not change the results in any appreciable way.

\subsubsection*{Azimuths of arms:}
\begin{verbatim}
  LAL_LIO_4K_DETECTOR_ARM_X_AZIMUTH_RAD    5.80120119264
  LAL_LIO_4K_DETECTOR_ARM_Y_AZIMUTH_RAD    4.23039066080
\end{verbatim}

These are angles of X and Y arms measured clock-wise (east of north) from the 
true north. This is usually measured. But one can also calculate the 
azimuth of a line from the GPS coordinates of the two ends of the line. This 
is an estimate and expected to approach the true value when the model of the 
earth used is perfect. For this estimate we have used WGS 84 model (latest) 
of the earth. These values were calculated using the website 
\url{https://geodesyapps.ga.gov.au/vincenty-inverse}

\subsubsection*{Altitude of arms:}
\begin{verbatim}
  LAL_LIO_4K_DETECTOR_ARM_X_ALTITUDE_RAD   0.0
  LAL_LIO_4K_DETECTOR_ARM_Y_ALTITUDE_RAD   0.0
\end{verbatim}

These are angles that the arms make with the local horizontal. By design, they 
arms are expected to be flat for LIO and hence these angles are set to zero.

\subsubsection*{Midpoint of arms:}
\begin{verbatim}
  LAL_LIO_4K_DETECTOR_ARM_X_MIDPOINT_SI    2000.00000000000
  LAL_LIO_4K_DETECTOR_ARM_Y_MIDPOINT_SI    2000.00000000000
\end{verbatim}

By design they are expected to be half of the length of each arm (which is 
4000m).

\subsubsection*{Detector coordinate in earth-fixed frame:}
\begin{verbatim}
  LAL_LIO_4K_VERTEX_LOCATION_X_SI          1.34897099624e+06
  LAL_LIO_4K_VERTEX_LOCATION_Y_SI          5.85742757731e+06
  LAL_LIO_4K_VERTEX_LOCATION_Z_SI          2.12756900034e+06
\end{verbatim}

These are the coordinates of the vertex in the earth-fixed Cartesian 
coordinate system. 
In an oblate ellipsoidal model of the earth, they are defined by the equations 
\cite{T980044},
\[ {\rm X_E} = (R + h) \, \cos(\phi) \, \cos(\lambda) \]
\[ {\rm Y_E} = (R + h) \, \cos(\phi) \, \sin(\lambda) \]
\[ {\rm Z_E} = \left(\frac{b^2}{a^2} \, R + h\right) \, \sin(\phi) \] 
where $a$ is the semi-major axis, $b$ is the semi-minor axis and 
$R$ is the local radius of curvature given by the expression,
\[ R = \frac{a^2}{\sqrt{a^2 \cos{\phi}^2 + b^2 \sin{\phi}^2}} \]
In the latest WGS 84 earth model, $a$ = 6378137 m and $b$ = 6356752.314 m.

\subsubsection*{Orientation vectors of arms:}
\begin{verbatim}
  LAL_LIO_4K_ARM_X_DIRECTION_X             0.38496278183  
  LAL_LIO_4K_ARM_X_DIRECTION_Y             -0.39387275094  
  LAL_LIO_4K_ARM_X_DIRECTION_Z             0.83466634811
  LAL_LIO_4K_ARM_Y_DIRECTION_X             0.89838844906 
  LAL_LIO_4K_ARM_Y_DIRECTION_Y             -0.04722636126  
  LAL_LIO_4K_ARM_Y_DIRECTION_Z             -0.43665531647 
\end{verbatim}
These are unit orientation vectors of the arms in the earth-fixed Cartesian 
frame. These are given by the equations \cite{T010110},
\begin{align*}
 {\rm X_{arm}}  = [&-\cos(\psi_x) \cos(\omega_x) \sin(\lambda) - \sin(\psi_x)
   \cos(\omega_x) \cos(\lambda) \sin(\phi) + \sin(\omega_x) \cos(\phi) 
   \cos(\lambda), \\ 
   & \cos(\psi_x) \cos(\omega_x) \cos(\lambda) - \sin(\psi_x) 
   \cos(\omega_x) \sin(\lambda) \sin(\phi) + \sin(\omega_x) \cos(\phi) 
   \sin(\lambda), \\
   &\sin(\psi_x) \cos(\omega_x) \cos(\phi) + \sin(\omega_x) 
   \sin(\phi)] \\ \\
 {\rm Y_{arm}}  = [&-\cos(\psi_y) \cos(\omega_y) \sin(\lambda) - \sin(\psi_y)
   \cos(\omega_y) \cos(\lambda) \sin(\phi) + \sin(\omega_y) \cos(\phi)
   \cos(\lambda), \\
   & \cos(\psi_y) \cos(\omega_y) \cos(\lambda) - \sin(\psi_y)
   \cos(\omega_y) \sin(\lambda) \sin(\phi) + \sin(\omega_y) \cos(\phi)
   \sin(\lambda), \\
   &\sin(\psi_y) \cos(\omega_y) \cos(\phi) + \sin(\omega_y)
   \sin(\phi)]
\end{align*} 
where $\psi_{x,y}$ are angles arms make with the (North of) East and 
$\omega_{x,y}$ are angles arms make with local horizontal. $\psi$ is related 
to azimuth by the expression $\psi$ = mod(450 - azimuth, 360), and 
$\omega_{x,y}$ = altitude in radian. For LIO, 
\[ \psi_x = 117.6157^{\circ} \]
\[ \psi_y = 207.6165^{\circ} \]
and 
\[ \omega_{x,y} = 0. \]
The $\psi_{x,y}$ values correspond to \verb|xarm_azimuth| and 
\verb|yarm_azimuth| used in {\tt Bilby} software packge for 
detector configuration. 

\subsection*{Acknowledgement:}
   We thank Tarun Souradeep, Varun Bhalerao, Javed Rana, and Tathagata Ghosh 
for checking some of the data listed here and for their useful comments. We 
also thank members of the LIGO-India Lab for helpful discussions.

\begin{thebibliography}{9}
    \bibitem{civil_dbr}
        LIGO Document - M1900064 (\url{https://dcc.ligo.org/LIGO-M1900064})
    \bibitem{LALheader}
        \url{https://lscsoft.docs.ligo.org/lalsuite/lal/_l_a_l_detectors_8h_source.html}
    \bibitem{nomenclature}
        LIGO Document - L2500030 (\url{https://dcc.ligo.org/LIGO-L2500030})
    \bibitem{T980044}
        LIGO Document - T980044 (\url{https://dcc.ligo.org/LIGO-T980044/public})
    \bibitem{T010110}
        LIGO Document - T010110 (\url{https://dcc.ligo.org/LIGO-T010110/public})
\end{thebibliography} 
\end{document} 
