4.4.1. Subsample of ‘observable’ GRBs

By construction, the method presented above includes the generation of GRB events that cannot be observed by the nano-satellite constellation (either because they are too faint or occur outside the FOV of the constellation), or that are otherwise unsuitable for localisation if too few satellites detect the burst. For these reasons, and to aid in the clarity of our results, we define the following additional conditions that must be met in order to include a GRB in our sample of $10^4$ events that we use as base for our analysis:

• The GRB must occur within the FOV of one of the co-pointing groups in the constellation:

  \begin{equation*} \text{Burst incident angle } \unicode{x03B8} \leq 80^{\circ}. \end{equation*}

  Since the nano-satellite constellation is divided into two co-pointing groups each pointing in opposite directions the burst can only ever be detected by one co-pointing group at a time.

• The observed flux of the GRB must be higher than the nominal flux limit of the HERMES instrument:

  \begin{equation*} F_{\text{obs}} = F_{\text{GRB}} \times \cos{\unicode{x03B8} } \geq F_{\text{lim}}. \end{equation*}

• At least three satellites in the constellation must directly detect the burst:

  \begin{equation*} N_{\text{det}} \geq 3. \end{equation*}

  In order to localise the burst using triangulation techniques at least three independent detections are needed (see Section 4.5). Note that we do not enforce that every satellite in the co-pointing group detects the GRB, since Earth may reduce the effective field of view of some elements, and a non-detection in this context can still contribute to the burst localisation (see Section 4.5.3 for further discussion).

• Every satellite in the co-pointing group must be operational at the time of the burst (i.e., not travelling through a region of high particle flux). We impose this restriction because the GRB localisation capabilities of an n-satellite constellation with one satellite missing are approximately equivalent to localisation statistics reported by an (n-1) satellite constellation (not taking into account the FOV and orbital position of each satellite). Therefore, to avoid such overlap in our results we enforce that every co-pointing satellite is operational at the time of the burst.

• In the case of the SpIRIT + HERMES-TP/SP constellation (i.e., a constellation with only one satellite in Polar orbit) we impose one additional condition that SpIRIT is part of the co-pointing group that detects the GRB. Without this condition, the localisation capabilities presented for the SpIRIT + HERMES-TP/SP constellation would consist both of bursts that are observed by SpIRIT + 3 HERMES satellites, as well as bursts that are observed by only 3 HERMES satellites, the latter of which would be identical to the localisation capabilities presented for the HERMES-TP/SP constellation alone. Therefore we ensure that SpIRIT is part of the co-pointing group that detects the GRB, which is only true for ${\sim}$ 50% of bursts (since SpIRIT can only point towards one of the two fields at any given time). Note that we do not enforce that SpIRIT detects the GRB; we only impose that it is co-pointing with other satellites that detect the GRB.

If a simulated GRB meets these conditions, then we include it in our sample and perform the localisation technique described in the following section.

4.5.1. Triangulation method

In this work we follow the localisation method used in Sanna et al. (Reference Sanna2020), which utilises the arrival time difference of photons at each detector in order to triangulate the position of the GRB. This technique has been used for decades by the IPN to triangulate the positions of GRBs (see e.g., Hurley et al. Reference Hurley, Briggs, Kippen, Kouveliotou, Meegan, Fishman, Cline and Boer1999, Reference Hurley2013), and as such is appropriate for our estimation purposes, despite some limitations discussed more in Section 4.5.2. Here, we provide a brief outline of the method, referring the reader to the work referenced above for full details.

As GRBs occur at cosmological distances, one can represent the gamma ray photons from the burst as a narrow plane wave travelling through space. Given a network of n detectors distributed throughout space (i.e., a satellite constellation), this plane wave will pass over each detector i at a different global time $t_i$ depending on the coordinates of each detector $\vec{r}_i$ .

If the GRB occurred at right ascension $\alpha$ and declination $\delta$ , its Cartesian direction can be represented by unit vector $\hat d$ :

(3) \begin{equation} \hat d(\alpha, \delta) = \{ \cos{\alpha}\cos{\delta}, \sin{\alpha}\cos{\delta}, \sin{\delta} \}. \end{equation}

Defining $t_0$ as the time the GRB signal reaches the origin of the chosen reference frame, the arrival time of photons at detector i is then given by

(4) \begin{equation} t_i = t_0 - \frac{\vec{r_i} \cdot \hat d}{c}, \end{equation}

Where $\vec{r_i}$ represents the position vector of the i-th detector. The time delay between two satellites will then be:

(5) \begin{equation} \Delta t_{ij}(\hat d) = \frac{(\vec{r_j} - \vec{r_i})\cdot \hat d}{c}. \end{equation}

In Equation (5), $\Delta t_{ij}$ represents the difference in arrival time of the signal at each instrument due purely to satellite geometry. The measured value of time delay $\Delta \tau_{ij}$ will depend on this intrinsic time delay plus some offset due to measurement uncertainty:

(6) \begin{equation} \Delta \tau_{ij}(\hat d_{GRB}) = \Delta t_{ij}(\hat d_{GRB}) + \mathcal{N}(0,\sigma_{cc}), \end{equation}

where the term $\mathcal{N}(0,\sigma_{cc})$ represents the uncertainty on the measurement of the arrival time difference. This timing uncertainty is discussed in detail in Sections 4.5.2 and 4.5.3.

To estimate the true position of the GRB, we compare the measured time delay pairs $\Delta \tau_{ij}(\hat d_{GRB})$ from each satellite that detected the GRB to the time delays that would be measured if the GRB were coming from some guess direction, $\Delta t_{ij}\left(\hat d_{guess}\right)$ . In this work we perform this comparison using a non-linear least squares method: we define the $\chi^2\left(\hat d_{guess}\right)$ function as the sum of the squares of the differences between the expected and measured time delays divided by the uncertainty on the signal alignment after cross-correlation:

(7) \begin{equation} \chi^2\left(\hat d_{guess}\right) = \sum_{i=0}^{n-2}\sum_{j=i+1}^{n-1} \frac{\left(\Delta \tau_{ij}\left(\hat d_{GRB}\right)\right. - \Delta t_{ij}\left(\hat d_{guess}\right)^2}{\sigma_{\text{cc}}^2}. \end{equation}

The coordinates of the GRB can then be estimated by finding the value of $\hat d_{guess}$ which minimise the $\chi^2$ function. Equation (7) takes the form of a ‘minimum $\chi^2$ ’ function with two degrees of freedom (right ascension and declination), and so we estimate the 68% and 90% confidence region of the source localisation as the set of all values of $\alpha, \delta$ where (Avni Reference Avni1976):

(8) \begin{equation} \chi^2(\alpha, \delta) - \chi^2_{\text{min, sky}} <= \begin{cases} 2.3 , \qquad\!\!\! 68\%\\[4pt] 4.61 , \quad 90\% \end{cases} \end{equation}

Note that Equation (8) is a linear approximation for determining the 68% and 90% confidence intervals. In fact this equation is only asymptotically correct for linear models, and as triangulation is a non-linear problem, the degrees of freedom are not necessarily exactly 2. In fact, the exact value of the degrees of freedom depends on data quality and predictiveness of the fit parameters (e.g., Janson, Fithian, & Hastie Reference Hurley2013).

Finally, we consider the sky coverage and LOS of each satellite in the co-pointing group, which influences GRB localisation depending on whether each satellite directly detects the GRB:

• If a satellite detects the GRB, we exclude all directions $\hat d_{guess}$ from the localisation region that are either outside that satellites FOV or obstructed by the Earth from that satellite’s perspective.

• If one satellite in a co-pointing group does not detect the GRB, but other satellites in that co-pointing group detect the burst, then we only consider possible burst directions $\hat d_{guess}$ that were obstructed by Earth from the non-detecting satellite’s perspective.

4.5.2. Signal cross-correlation uncertainty

The triangulation method for GRB localisation relies heavily on an accurate estimation of $\Delta \tau_{ij}$ —the measured time difference between the signal arriving at each detector in the network. Classically, this time difference has been computed by binning the light curves observed by each detector with some temporal resolution (which usually depends on the flux of the GRB—brighter bursts can be binned with finer resolution), and then using a cross-correlation algorithm to determine the time offset between the two observed signals (Hurley et al. Reference Hurley, Briggs, Kippen, Kouveliotou, Meegan, Fishman, Cline and Boer1999, Reference Hurley2013; Sanna et al. Reference Sanna2020). Note that Equation (7) therefore presumes that the only source of timing uncertainty is in the signal cross-correlation, which is a valid approximation in our case as the uncertainty on cross-correlation is of the order ${\sim}$ 1 ms (Sanna et al. Reference Sanna2020) whereas the positional uncertainty on the satellite is $<$ 30 m (translating to a temporal accuracy of $<$ 20 ns; Fiore et al. Reference Fiore2020) and the detector has an absolute timing accuracy lower than $0.4\,\mu$ s (Evangelista et al. Reference Evangelista2020).

It has been shown by Burgess et al. (Reference Burgess, Cameron, Svinkin and Greiner2021) that this ‘classical’ triangulation method has limitations, especially for faint bursts when the number of photons per time bin is small. When compared to a full Bayesian treatment of statistical uncertainties, Burgess et al. (Reference Burgess, Cameron, Svinkin and Greiner2021) shows that the classic cross-correlation approach results in an overconfidence of the $1\sigma$ uncertainty on time delay by a factor of ${\sim}$ $2-3$ , which translates directly into an overconfidence on the burst localisation. Nevertheless, given this paper is a continuation of the work performed in Sanna et al. (Reference Sanna2020) and given the computational benefits in our Monte Carlo simulation approach, we use the classic cross-correlation approach, and note that our results may contain a systematic uncertainty of 2-3 times our reported statistical localisations for the faintest bursts.

In order to determine the impact that GRB flux has on time delay uncertainty, we adopt the work of Sanna et al. (Reference Sanna2020), where the standard deviation in the cross-correlation uncertainty, $\sigma_{\text{cc}}$ is reported as a function of average GRB flux for a random sample of 100 long and 100 short GRBs observed by Fermi GBM. Specifically, in this work we use the ‘Flnc_Band_Phtfluxb’ data type from the Fermi GBM catalogue (Gruber et al. Reference Gruber2014; von Kienlin et al. Reference von Kienlin2014, Reference von Kienlin2020; Bhat et al. Reference Bhat2016), which is a quantity related to the burst photon flux between 50 and 300 keV. The results from Sanna et al. (Reference Sanna2020) are reported here in Figure 4 as black data points. In this figure, each data point represents the results from 1 000 Monte-Carlo (MC) simulations which calculated the difference between the true arrival time delay and the value inferred from cross-correlation methods. The resulting distribution of time-offset values is then fit with a Gaussian function with a mean of zero and a standard deviation $\sigma_{cc}$ . To perform 1 000 MC trials for each burst, the GRB light curve observed by Fermi GBM is used as a template from which unique GRB light curves can be generated by both rescaling the detector effective area to match that of the HERMES instrument and applying a Poissonian randomisation of the counts contained in each bin of the template.

Figure 4. Black points: Standard deviation in the cross-correlation uncertainty ( $\sigma_{\text{cc}}$ ) calculated for a sample of 100 long and 100 short GRBs from Sanna et al. (Reference Sanna2020). Red line: Best-fit power-law to the black data points, where the $1\sigma$ interval is shown in light red.

To generalise the results of Sanna et al. (Reference Sanna2020) we construct a power-law fit to this data (Figure 4, red lines), assuming that the residuals follow a Gaussian distribution. The parameters for our fits are given in Table 1.

We use these relations to compute the uncertainty on signal cross-correlation, $\sigma_{cc}$ , given the observed flux of a burst, which depends not only on the flux of the GRB as seen from LEO, but also on the incident angle of the burst to the detector ( $F_{obs} = F_{GRB} \cos \unicode{x03B8}$ ).

Figure 5 plots the results from $10^4$ Mote-Carlo trials using this method to calculate the standard deviation on cross-correlation uncertainty. As expected, the satellites in our simulated constellation observe a lower GRB flux on average than those bursts used by Sanna et al. (Reference Sanna2020) (black points/curves), since we take into account the HERMES instruments’ approximate cosine dependence on photon incident angle. Otherwise we find good agreement with the results presented in Sanna et al. (Reference Sanna2020).

Figure 5. Scatter plot showing the relation between the observed flux of the GRB ( $F_{obs} = F_{GRB}\cos\unicode{x03B8} $ ) and the expected standard deviation on timing uncertainty after cross-correlation $\sigma_{cc}$ from $10^4$ MC trials. The top panel plots the distribution of observed average GRB fluxes, and right hand panel show histograms for the resulting distribution of cross-correlation uncertainty as derived from the given flux distributions. For comparison, in black we also plot the 100 long GRBs (circles, solid histograms) and 100 short GRBs (triangles, dashed histograms) from Sanna et al. (Reference Sanna2020) which were used as a basis for our model. We have arbitrarily scaled the histograms of the Sanna et al. (Reference Sanna2020) data by eye so that they match the scale of our simulated data.

Note that this method contains several assumptions, justified by our aim to forecast performance of a future constellation rather than a detailed characterisation of the localisation algorithm applied to actual data. Firstly, we are assuming that average GRB flux is as accurate predictor of cross-correlation uncertainty. For a more realistic simulation, one could follow the same procedure as performed in Sanna et al. (Reference Sanna2020) and perform the cross-correlation of GRB light curves to determine the ‘measured’ arrival time difference.

Secondly, our simulation assumes that the uncertainty on signal cross-correlation between two detectors $\sigma_{cc,ij}$ is uncorrelated between pairs of detectors. In reality, we would expect this quantity to have some correlation between pairs of satellites, e.g., $\sigma_{cc,ij}$ should have some relation to $\sigma_{cc,ik}$ since both involve the signal observed by detector i. However, understanding these correlations is non-trivial, and beyond the scope of this work.

4.5.3. Simulating GRB localisation

For each GRB that meets the conditions defined in Section 4.4 we perform the triangulation procedure described in Section 4.5.1. We start by constructing a set of ‘measured’ time-difference pairs $\Delta \tau_{ij} = \Delta t_{ij}(\hat d_{GRB}) + \mathcal{N}(0,\sigma_{cc})$ between each satellite which detected the event, where $\mathcal{N}(0,\sigma_{cc})$ represents a random variable sampled from a Gaussian distribution with a mean of zero and standard deviation $\sigma_{cc}$ , as defined above.

Using these values we calculate Equation (7) for each point on the sky, where we represent the whole sky using a grid of $660048$ points uniformly distributed on the celestial sphere (giving each point an area of ${\sim}$ $0.0625$ deg $^2$ ), with positions generated by the Fibonacci lattice method (González Reference González2009). For computational efficiency, in cases where the localisation is not expected to reach $<$ 1 deg $^2$ (e.g., when localising short GRBs, or when testing constellations with fewer satellites in Polar orbit), a grid of 41 253 points (1 deg $^2$ resolution), or $165012$ points ( $0.25$ deg $^2$ resolution) is used.

Finally, we determine the 68% or 90% localisation confidence regions using the values defined in Equation (8), excluding those points that are disallowed by LOS arguments as defined in Section 4.5.1.

We note here that when calculating the time-difference pairs $\Delta \tau_{ij}$ and the timing uncertainty on the signal cross-correlation $\sigma_{cc}$ , we presume that the satellites are stationary for the duration of the burst. In reality, the relative motion of each satellite introduces a small difference in the arrival time of photons at each detector. We estimate that as a worst-case scenario—a satellite in LEO moving at 10 km s $^{-1}$ in the direction of the burst—would detect the final photon from a typical long GRB (approximately ${\sim}$ 10 s) $0.3$ ms before a stationary satellite, meaning that the moving satellite would effectively observe the signal of the GRB ‘compressed’ in time by $0.3$ ms, which is comparable to the cross-correlation accuracy achieved for the brightest ${\sim}$ 20% of long GRBs. However, given the position and velocity of each satellite in the SpIRIT + HERMES constellation are known to a high accuracy, in principle this effect is entirely deterministic and can be accounted for in a sufficiently sophisticated cross-correlation algorithm. For this reason, we choose to neglect the relative motion of each satellite in this work, leaving its inclusion to a future simulation of the HERMES constellation’s localisation capabilities, and note that this assumption may introduce some systematic uncertainty into our results for the best localised long GRBs.

5.2.1. Long GRBs

Figure 8 compares the cumulative distributions of the $1\sigma$ localisation region for $10^4$ long GRBs localised by HERMES-TP/SP and SpIRIT + HERMES-TP/SP. For comparison we also plot the 68% statistical sky area uncertainties reported in the fourth Fermi/GBM catalogue (von Kienlin et al. Reference von Kienlin2020). Note here that we are comparing the statistical GRB localisation capabilities only—while both instruments also have an associated systematic uncertainty, we cannot estimate these for the nano-satellite constellation since to do so would require in-flight data, which is impossible since the constellation has not been launched yet. For reference, the systematic uncertainty for Fermi GBM is approximately ${\sim}$ $2-4^{\circ}$ (Goldstein et al. Reference Goldstein2020), and it will likely be comparable to this for the nano-satellite constellation.

Figure 8. Cumulative probability of localising a long GRB within a given $1\sigma$ region on the sky for n SpIRIT-like satellites in Polar orbit + HERMES-TP/SP. The $n_{\text{polar}} = 0$ constellation (blue line) represents the HERMES-TP/SP constellation operating alone.

Figure 9. Cumulative probability of localising a short GRB within a given area on the sky for n SpIRIT-like satellites in Polar orbit + HERMES-TP/SP. The $n_{\text{polar}} = 0$ constellation (blue line) represents the HERMES-TP/SP constellation operating alone. Left: 68% confidence intervals. Right: 90% confidence intervals. The dashed lines represent the highest expected localisation capabilities of the advanced LIGO Hanford and Livingstone, advanced VIRGO and KAGRA GW detectors to localise BNS coalescences in O3 and O4, respectively (note that for O3, detector sensitivities were taken to be representative of the first 3 months of observations for aLIGO Hanford and Livingston, and AdV, and the highest expected O3 sensitivity for KAGRA). See Abbott et al. (Reference Abbott2020) Figure 6 for details). The two vertical grey lines represent the prompt reported 90% confidence regions for the two binary neutron star coalescences observed to date; GW170817, and GW190425.

Figure 10. Localisation histograms for the sample of $10^4$ GRBs detected by the SpIRIT + HERMES-TP/SP Constellation. The blue curve represents the full sample of simulated GRBs, while the orange and green curves compare the localisation capabilities of the constellation when SpIRIT either detects or does not detect the burst. Left: Long GRB sample. Right: Short GRB sample.

The inclusion of SpIRIT greatly enhances the accuracy of GRB localisations compared to the HERMES-TP/SP alone. SpIRIT + HERMES-TP/SP is able to localise 60% of long GRBs within ${\sim}$ 28 deg $^2$ , which corresponds to an uncertainty radius on the sky of ${\sim}$ $3^{\circ}$ (presuming the uncertainty region is circular), and 90% of long GRBs within 150 deg $^2$ ( ${\sim}$ $7^{\circ}$ radius). In reality the uncertainty regions are irregular in shape, but a circle is a good approximation when the burst is relatively well localised. This represents an improvement by a factor of ${\sim}$ 5 over the HERMES-TP/SP alone, which is only able to localise ${\sim}$ 50% of GRBs to within 100 deg $^2$ on the sky.

Figure 8 shows that the combined SpIRIT + HERMES-TP/SP constellation will have statistical GRB localisation capabilities comparable to those reported by Fermi GBM. This result demonstrates the potential cost-effectiveness of using modular nano-satellite astronomy to achieve specific science goals, as a constellation of several relatively inexpensive nano-satellites is expected to achieve a localisation accuracy similar to the substantially more complex and expensive Fermi GBM. In fact, estimating an approximate cost of US $150M for the Fermi GBM instrument ( ${\sim}$ 25% of the total cost of the mission (Fermi Gamma-ray Space Telescope Q&A on the GLAST MissionFootnote ^f)) and of US $2.5M for each element of the nano-satellite constellation, there is a possible factor 10x in savings using a distributed-aperture solution for localising long GRB afterglows. We note however that the disadvantage to using a HERMES-like distributed-aperture solution with the current design is sky coverage: the Fermi GBM can observe over half the sky at any time (i.e., the whole sky not obstructed by the Earth Meegan et al. Reference Meegan2009), whereas the FoV of a single HERMES instrument only achieves ${\sim}$ 25% sky coverage at FWHM, meaning that achieving all-sky coverage with a nano-satellite constellation with overlapped FoV’s would partially offset the cost savings (although there would be further gains from mass production of the constellation elements). Furthermore, it is non-trivial to achieve maximal overlap of the FOV of each nano-satellite element, as it requires continual optimisation of the pointing directions of each satellite to account for orbital drift and different orbital configurations (Colagrossi et al. Reference Colagrossi, Prinetto, Silvestrini and Lavagna2020).

5.2.2. Short GRBs

Figure 9 compares the localisation capabilities of the HERMES-TP/SP and combined SpIRIT + HERMES-TP/SP constellations to both the Fermi GBM (where we have plot the Fermi localisation uncertainty using same method as discussed in Section 5.2.1) and to the theoretical current and future localisation capabilities of the Advanced LIGO Hanford and Livingston, Advanced VIRGO and KAGRA gravitational wave detectors (Abbott et al. Reference Abbott2020; 90% confidence regions, Figure 9b). Note that this reflects the predicted performance of the four gravitational wave detectors operating concurrently during O3 and O4—in reality KAGRA did not participate in O3, but is planned for joint operation in O4. We have also included the reported prompt localisations for both of the BNS coalescences observed by GW detectors; GW170817 (Abbott et al. Reference Abbott2020) and GW190425 (Abbott et al. Reference Abbott2021). While the error box for GW190425 was later refined, here we compare the prompt localisation capability to ensure a fair comparison to a nano-satellite constellation. Note that the HERMES-TP/SP constellation is less effective at localising short GRBs compared to long GRBs due to the increased difficulty in precise signal cross-correlation for short GRBs, although a future instrument redesign with increased collecting area would mitigate this current limitation (Sanna et al. Reference Sanna2020).

The addition of SpIRIT to HERMES-TP/SP is able to substantially improve the localisation of short GRBs. HERMES-TP/SP alone has a 90% $1\sigma$ localisation range of ${\sim}$ 215–13000 deg $^2$ , while the SpIRIT + HERMES-TP/SP constellation’s 90% $1\sigma$ range is ${\sim}$ 50–8450 deg $^2$ , where we have defined the 90% $1\sigma$ localisation interval as the range of $1\sigma$ confidence regions spanning 5–95% of the cumulative distribution.

Despite this improvement in the constellation’s ability to localise short GRBs, in terms of multi-messenger observations of short GRBs/BNS coalescences the combined SpIRIT + HERMES-TP/SP constellation is not expected to dramatically improve upon existing capabilities to localise these sources. Not only is the Fermi GBM more effective by a factor of ${\sim} 5-10\times$ at the EM localisation of short GRBs, but current-generation GW instruments are theoretically capable of localising $>$ 50% of short GRBs within ${\sim}$ 200 deg $^2$ (and within ${\sim}$ 40 deg $^2$ in the future O4 observing runs) from the GW signal alone (Abbott et al. Reference Abbott2020), which exceeds the localisation capabilities of the nano-satellite constellation. Therefore, in the domain of multi-messenger astronomy the utility of the SpIRIT + HERMES-TP/SP constellation will be in the constellation’s sky coverage ( ${\sim}$ 50%) and flux sensitivity, which gives it a high probability of detecting the electromagnetic counterpart to a compact binary event observed by GW detectors. Such dual detections are critically needed to improve our understanding of these compact binary mergers, as currently only one simultaneous detection of the GW signal from a BNS merger (GW170817) and a short GRB (GRB170817A) has been recorded to date (Abbott et al. Reference Abbott2017a).

5.2.3. Utilising SpIRIT non-detections

The inclusion of SpIRIT into the HERMES-TP/SP enables improvements to the GRB localisation capabilities of the HERMES-TP/SP both when SpIRIT directly detects the burst, but also when SpIRIT does not directly detect the burst. The improvement in the latter case is a product of SpIRIT’s high inclination orbit, since when SpIRIT is over the poles then a considerable section of the HERMES triplet’s FOV is obstructed by the Earth from SpIRIT’s perspective. This means that a SpIRIT non-detection can impose relatively strong constraints on possible source location under these conditions, assuming that SpIRIT is co-pointing at the same field as the HERMES triplet that detects the burst, and that the noise level is low enough for SpIRIT that we would expect to detect the GRB signal.

Figure 10 compares the size of the $1\sigma$ confidence region for long and short GRBs, where the results have been divided depending on whether SpIRIT detects the GRB or not. Note that in all cases, SpIRIT is co-pointing with the HERMES triplet that detects the GRB. In both cases, we find that SpIRIT makes a direct detection of the GRB for ${\sim}$ 85% of bursts in our ‘observable’ sample, which represents the average fraction of the HERMES triplet’s FOV that is simultaneously observed by SpIRIT.

In the case of long GRBs we find that the size of the $1\sigma$ localisation region is always smaller when SpIRIT directly detects the burst compared to when SpIRIT does not detect the GRB. This is to be expected, since for HERMES-TP/SP the typical size of the long GRB localisation region is much lower than the sky coverage of the triplet, meaning that a SpIRIT non-detection will only improve the source localisation in the unlikely case that the burst occurs at coordinates close to the edge of Earth as seen from SpIRIT’s perspective.

Conversely, in the case of short GRBs we find that in some cases it is possible to achieve more accurate GRB localisations when SpIRIT does not detect the burst. These cases arise for faint short GRBs where the typical uncertainty on signal cross-correlation is large, meaning that the typical $1\sigma$ localisation region is comparable to the sky coverage of the constellation ( ${\geq}$ $10^4$ deg $^2$ ). In such cases, SpIRIT non-detections enable more accurate GRB localisations as they restrict the possible location of the GRB to those regions of sky that are obstructed by the Earth at the time of the burst, which is why there are practically no bursts with $1\sigma$ localisation regions larger than ${\sim}$ $5\times10^3$ deg $^2$ in the green curve in Figure 10 (right panel).

5.3.1. Long GRBs

Figure 8 demonstrates that a constellation of four Polar satellites plus HERMES-TP/SP would improve the localisation capabilities of the constellation by a factor of ${\sim}$ 2 over the case of a constellation with only one Polar satellite and would be capable of localising 60% of long GRBs to within ${\sim}$ 10 deg $^2$ of random uncertainty on the sky ( ${\sim}$ $2^{\circ}$ radius). A constellation of six Polar satellites would reflect a further improvement by an additional factor of ${\sim}$ $1.5$ (or equivalently a $4\times$ improvement over the case of a single Polar satellite) and could enable the localisation of ${\sim}$ 50% of long GRBs to within a radius of $1^{\circ}$ on the sky.

Despite the substantial improvement over the case of a single Polar-orbiting satellite, GRB localisations with an error radius of ${\sim}$ $1^{\circ}$ are not accurate enough to enable reliable afterglow follow-up observations from existing ground or space-based observatories, most of which have a FoV of the order $10 \times 10$ arcmin $^2$ , e.g., GROND: $10' \times 10'$ (Greiner et al. Reference Greiner2008), Swift UVOT: $17' \times 17'$ (Troja Reference Troja2020), Gemini Multi-Object Spectrograph (GMOS): $5'.5 \times 5'.5$ (Hook et al. Reference Hook, Jørgensen, Allington-Smith, Davies, Metcalfe, Murowinski and Crampton2004). Taking effective advantage of these GRB detections at high energy for infrared follow-up may, however, be possible in the future with $\gtrsim$ 1 deg $^2$ FoV space-based telescopes with rapid repointing capabilities, such as the SkyHopper nano-satellite concept (Thomas et al. Reference Thomas2022).

5.3.2. Short GRBs

As Figure 9 demonstrates, launching a constellation of 4 or 6 Polar orbiting satellites into Polar orbit can improve short GRB localisations by a factor of ${\sim}$ 2–4 over a constellation with a single Polar satellite, achieving a 90% $1\sigma$ short GRB localisation range between ${\sim}$ 20 and 4100 deg $^2$ (4 Polar) and 12–2850 deg $^2$ (6 Polar), respectively. These improvements enable the statistical localisation of short GRBs with an accuracy close to that reported by Fermi GBM for a small fraction of the best-localised bursts, meaning that such an expanded nano-satellite constellation could be valuable in the near real-time localisation of EM counterparts to BNS coalescences, especially in cases where a short GRB is observed by both Fermi GBM and the nano-satellite constellation (which is likely, given Fermi GBM observes such a large fraction of the sky).

At the time of writing, real time triggers and $1\sigma$ localisations for short GRBs can be obtained with accuracies $\lt 10$ ’s of square degrees (e.g., Fermi GBM Goldstein et al. Reference Goldstein2020, the Neil Gehrels Swift Observatory Gehrels et al. Reference Gehrels2004; Troja Reference Troja2020, the Gravitational wave high-energy Electromagnetic Counterpart All-sky Monitor (GECAM; Zhao et al. Reference Zhao2021). To localise a substantial fraction of short GRBs to this accuracy with the current design of the HERMES instrument (and under the assumption that the satellites are uniformly distributed around equatorial and Polar orbits), at least 20 satellites divided evenly between equatorial and Polar orbit all observing the same field are required (i.e., combined detection from at least 10 HERMES instruments with large baselines in both the equatorial and Polar planes). On the other hand, the performance forecast for the localisation of long GRBs indicates that with an increase in the effective area of the detector, a constellation with a smaller number of elements would have a strong potential to provide state-of-the-art and cost effective all-sky monitoring and localisation of short GRBs/gravitational wave counterparts. A detailed investigation on the trade off between detector collecting area and number of constellation elements in the HERMES constellation is beyond the scope of this work (though the reader is directed to Greiner et al. Reference Greiner, Hugentobler, Burgess, Berlato, Rott and Tsvetkova2022, where a similar study for a hosted instrument on the Galileo constellation is presented).