Cavalier Fabien on behalf LAL group Orsay GWDAW 10 December, 14 th 2005 Reconstruction of Source Location using the Virgo-LIGO network Presentation of the method Toy Simulation Influence of timing resolution and Effects of systematic errors Geometrical Properties of Virgo-LIGO network Effects of the Beam-Pattern functions

With n measured arrival times t i with associated errors  i, compute the best estimation of  and  Use of  2 Minimization  2 defined as: where t 0 is the arrival time of the signal at the center of the Earth and  t i Earth ( ,  ) is the delay between the i th ITF and the center of the Earth Reconstruction Method

Symmetrical Definition No reference detector to compute timing differences  How to choose it ? detector with lower error detector leading to highest time delays detector which gives the best relative errors on time delays … Uncorrelated Errors Easily expendable to any set of detectors Reconstruction Method

Put a source at a given location Computes arrival times t True [i] Computes timing errors in each detector set  [i] = s in all detectors (no beam-pattern effect) set = 10 (beam-pattern effect) and use  [i] =   / SNR[i] ms (  0 = 1 ms will be used by default) t Measured [i] = t True [i] + GaussianRandom *  [i] Compute the angular distance (Angular Error in the following plots) between reconstructed location and true one Toy Simulation

Example of Reconstruction for Galactic Center Estimated errors (through covariance matrix) in agreement with RMS values

Effect of Systematic Errors on Arrival Time t Measured [i] = t True [i] + GaussianRandom *  [i] + Bias (for only one ITF) bias in the angular reconstruction when the timing bias and  [i] have the same order of magnitude bias in the angular reconstruction proportional to the timing bias no significant difference between the three ITF Influence of Timing Resolution

Reconstruction for GC (no beam-pattern) Mean Angular Error: 1.9 Median Angular Error: 0.95 o Minimal Angular Error:0.8 o Maximal Angular Error:4.3 o

Reconstruction for  =0 o (no beam-pattern) Mean Angular Error: 1.8 o Median Angular Error: 1.5 o Minimal Angular Error:1.3 o Maximal Angular Error:3.1 o

Error increase when source crosses 3-detector plane 2 detectors located at (±d/2,0,0) source in the (x,y) plane defined by its angle  with x axis t 21 = d/c cos(  ) (1) if error  t on t 21, then error  on the angle:  = c/d  t / |sin(  )| when  approaches 0, due to statistical errors, Eq.(1) cannot be inverted    = 1/  t 2 (t 21 measured – d/c cos(  )) 2    minimal and equal to 0 for  = acos(t 21 measured *c/d) if |t 21 measured *c/d|≤1    minimal and  1 for  = 0 if |t 21 measured *c/d|>1 Similar effect in the 3-detector case

Reconstruction for full sky (no beam-pattern) Mean Angular Error: 1.6 o Median Angular Error: 1.1 o Minimal Angular Error:0.7 o Maximal Angular Error:4.5 o

Reconstruction for GC (beam-pattern effect included)

All events: Mean Angular Error: 4.0 o Median Angular Error: 1.8 o Minimal Angular Error:0.7 o Events with all SNR > 4.5 (56%): Mean Angular Error: 1.8 o Median Angular Error: 1.25 o Minimal Angular Error:0.7 o

Reconstruction for  =0 o (beam-pattern effect included)

All events: Mean Angular Error: 3.0 o Median Angular Error: 2.2 o Minimal Angular Error:1.2 o Events with all SNR > 4.5 (79%): Mean Angular Error: 2.1 o Median Angular Error: 1.7 o Minimal Angular Error:1.2 o

Reconstruction for full sky (beam-pattern effect included) All events: Mean Angular Error: 2.7 o Median Angular Error: 1.7 o Minimal Angular Error:0.7 o Events with all SNR > 4.5 (60%): Mean Angular Error: 1.8 o Median Angular Error: 1.3 o Minimal Angular Error:0.7 o

Conclusion  2 minimization works properly for position reconstruction In the case of 3 detectors, the method is able to find the two possible solutions The method is easily extendable to any ITF network Errors on parameters provided by the minimization procedure Accuracy of one degree can be achieved with good timing reconstruction (error proportional to timing error) Bias in timing estimator can easily introduce bias in reconstructed angles