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These small changes in dates do not significantly alter our results. For O2, we consider specific possible dates 15 to allow our analysis to be performed, and the likely split into two parts. For our example, Virgo is taken to join these two detectors in O2B, spanning the period from April 1 to May Considering that not all detectors will be functioning throughout these phases, we undertake separate analysis of O2B into two-detector events Section 4.

We also impose an upper bound on the zenith angle of observations, based on two principles: most telescopes cannot point arbitrarily close to the horizon, and quality of data is poor for observations at high airmass 16 high zenith angle. We do not include any constraints based on lunar phase or lunar angle. Certain site-specific conditions like weather, instrument breakdowns, etc.


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Hence, we exclude these factors from our simulations, as is often done in such studies Singer et al. The patch area of events recovered with a two-detector network ranges from a few tens to a few thousand square degrees, so we choose deg 2 as the upper bound.


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For the same reason, the distribution of probability covered improves only slightly when we go from to deg 2 coverage capability. For reference, we also consider a best-case scenario that could be attained by, say, a space telescope, free of horizon and twilight constraints.

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We note two important caveats in these analyses: we have not considered the shape of a telescope field of view, or the duration of visibility of the patch. Thus, the telescope may image, say, deg 2 , but cover only 50 deg 2 of our patch. This inefficiency is sensitive to the exact shape and size of the field of view.

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There might also be scenarios where a large part of the patch is visible from the site, but only for a short time, for instance, an event occurring overhead just before morning twilight. However, the time taken to image N square degrees to a given depth can vary drastically between different telescopes.

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This in turn may limit the fraction of the visible patch that can be imaged through the night. As our focus is to compare geographic locations rather than telescopes and instruments, we do not consider these two effects in our simulations. As discussed in Section 2. For each event, we calculate the probability observable from each location, and find that all sites have comparable performance on the day of the equinox.

As an example, in Figure 2 we show histograms of the probability of finding the counterpart by imaging the best visible 30 deg 2 from two sites—Blenheim, New Zealand and Haleakala, Hawaii—which respectively had the worst and best median performance in this category. It is seen that both extremes actually have very similar performance in this case. Figure 2. We simulated observations of GW events detected by the two-detector network O1 or O2 , with all events moved to the date of the fall equinox. We assume that each site has a telescope capable of imaging 30 deg 2 to the requisite sensitivity.

The overall histograms are comparable. The slightly poorer observability from Blenheim results from a greater distance from the equator, which makes some northern patches completely inaccessible. Upper panel: histograms of observable probabilities for all events. Lower panel: box-and-whisker plots for the same histograms.

In order to simplify visual comparisons, in the rest of this paper we use box-and-whisker plots Figure 2 , lower panel. The range between these two points is called the inter-quartile range IQR. Since we are primarily interested in properties of the distribution rather than specific outliers, we have often scaled the plots such that some of the outliers are beyond the plot limits.

The line and star inside the box show the median and mean of the distribution. As expected, we see that northern observatories perform better during the winter solstice, owing to longer nights, while southern observatories perform better during the summer solstice. Figure 3. Location-wise performance comparison between La Serena, Hanle, and Palomar, for all the O1 and O2 two-detector events.

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Events were shifted to the day of vernal equinox green , summer solstice yellow , fall equinox chocolate and winter solstice blue. The three panels' distributions of observable probabilities are calculated using telescopes that can image 3, 30 and deg 2 respectively within 24 hr of the trigger. While the overall performance of the various sites for equinox observations is similar Figure 5 , looking at the median values of observable probabilities shows some interesting trends Figure 4.

We see that on the equinoxes, sites at mid-latitudes have a few percent higher probability of finding the optical counterpart of a GW event, as compared to observatories in the temperate zones. This can be explained by a combination of two effects: i the two LIGO detectors detect more sources at mid-declinations as compared to equatorial or polar declinations, and ii sites further from the equator have a progressively smaller fraction of the sky accessible even on the equinox night.

Similar effects have also been discussed in Chen et al. Figure 4. Annual variations in median observable probability for different latitudes. The different colors and symbols show median observable probability on the days of the vernal equinox green circles , summer solstice yellow squares , fall equinox pink triangles and winter solstice blue pentagons.

Lower panel: median probability of finding the counterpart for a source localized by the three-detector HLV network, with the same telescopes limited to observing deg 2 during the night. Figure 5. Probability of finding optical counterparts for simulated two-detector events on the fall equinox.

The simulation sample includes events with O1 and O2 sensitivity. The observatories are sorted by latitude, and color-coded by continent as in Figure 1. The best-case scenario, considering only solar exclusion angle but ignoring horizon constraints, is plotted in the rightmost column. On comparing the location-wise performance for 1, 3, 10, 30, , , , and deg 2 , we see that all sites perform comparably with a very slight trend along the latitude. This annual variation stems primarily from the duration of the night, determining the fraction of the sky visible.

The effect is limited to a few percent due to the large areas and long arc-like shapes of GW events localized by just two detectors Singer et al. One would then expect the seasonal differences to be more stark if the localization improved. In the limiting case, if GW sources were pinpointed on the sky by the GW detectors, the observable probability would be governed by the latitudinal variation of detector sensitivity function Fairhurst and would vary more strongly with the fraction of the sky visible at night.

Indeed, this is the case with improved localizations provided by a network of three GW detectors. For reasons discussed in Section 4. As a specific case, we repeat our simulations using the actual dates of the first LIGO science run O1 and a set of example dates of O2. Although the dates of O2 are uncertain, we aim to give an overall perspective of how observatories at different locations may perform under these conditions.

Our two-detector sample consists of events with O1 sensitivity, and events with O2 sensitivity. The smaller number of Virgo-detected events arises from expected sensitivity and uptime of the three detectors, as discussed in Singer et al. As O1 was conducted during northern winter, one expects northern observatories to perform better than southern ones, and this expectation is borne out by simulations Figure 6.

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Mauna Kea and Haleakala have the best chance of discovering an optical counterpart, with a median probability of 0. Blenheim, the southernmost location in this study, had a median probability of 0. Incidentally, the localization of GW happened to peak in the southern skies Abbott et al. Thus, small-number statistics worked in favor of southern observatories in O1.

Figure 6. Comparisons of various observatories for O1. A total of simulated events detected with Hanford and Livingston GW detectors at O1 sensitivity were randomly distributed over actual dates of O1. The box-plots of observable probabilities are as in Figure 5. Northern observatories had a better chance of finding EM counterparts compared to southern ones.

The assumed split dates of O2 span approximately northern winter and spring, slightly favoring northern observatories in O2A and southern ones in O2B. The net result is that the timing of observing runs slightly favors northern observatories, but the overall performance of observatories is dominated by their latitude, following a similar trend as the equinoxes Figure 7. The number of two-detector detections involving Virgo in O2B is rather small, and does not alter the trends in any significant manner.

Figure 7. Comparisons of various observatories for O2. A total of simulated events detected with GW detectors at O2 sensitivity were distributed over example dates of O2 as described in Section 4. Operational periods O2A and O2B favor locations in different hemispheres; as a result, all observatories have comparable odds of finding EM counterparts of GW sources. The joint detection of any GW event by all three GW detectors drastically changes the follow-up scenario.