CGRO/BATSE DATA SUPPORT THE NEW PARADIGM FOR GRB PROMPT EMISSION AND THE NEW ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ RELATION

The νF${}_{\nu }$ spectra resulting from the Band-only and C_nTh + C_Th fits to the time-resolved data of GRBs 941017, 970111, and 990123 are overplotted in Figures 15–17, respectively.

For only one additional free parameter, the Cstat values obtained when fitting C_nTh + C_Th to the data are overall much lower than those resulting from the fit to a Band function alone (see Tables 4–6.) Our simulation procedure indicates that C_nTh + C_Th is a better description of the data than Band alone in ∼100% of the time intervals for the three GRBs.

The similarities of BATSE and Fermi results are striking when comparing Band and C_nTh + C_Th. Indeed, ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}$ is systematically lower than ${\alpha }_{{\rm{i}}}^{\mathrm{Band}}$ (panels (a) of Figures 7–9) and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ is systematically higher than ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ (panels (c) of Figures 7–9) as reported in Guiriec et al. (2011a, 2013, 2015a, 2015b).¹⁰

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The ${\alpha }_{{\rm{i}}}^{\mathrm{Band}}$ values exhibit strong evolution and discontinuities. While usually incompatible with synchrotron scenario limits during the first few tens of seconds, they become consistent with synchrotron emission from electrons in the slow (i.e., ${\alpha }_{{\rm{i}}}^{\mathrm{Band}}\;\lt$ −2/3) and even in the fast (i.e., ${\alpha }_{{\rm{i}}}^{\mathrm{Band}}\;\lt$ −3/2) cooling regimes at late times. Similarly to the results presented in Guiriec et al. (2013, 2015b), the values of ${\alpha }_{{\rm{i}}}^{\mathrm{Band}}$ are clearly positive and incompatible with non-thermal processes during the first ∼10 s of GRB 970111; this also corresponds to the time period during which C_Th overpowers C_nTh and therefore the spectral parameters of Band alone are biased toward those of C_Th (see the several first time-resolved spectra of GRB 970111 in Figure 16).

Conversely to ${\alpha }_{{\rm{i}}}^{\mathrm{Band}}$, the values of ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}$ are always compatible with at least the synchrotron slow cooling scenario (i.e., ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}\;\lt$ −2/3). As we will see in the next section, the discontinuity of ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}$ for GRB 990123—whose values drop from ∼−0.8 down to less than −1.5 after ∼T₀ + 40 s—is consistent with the presence of an intense additional PL at late times (see Figure 9(a)).

Overall, the temperature of C_Th (i.e., kT ∼ ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$/2.7; we choose to plot ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ instead of kT for an easier comparison to ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$) varies much less than ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ (see panels (c) of Figures 7–9.) While the temperature kT of the C_Th varies between ∼40 and ∼120 keV for GRB 990123, its evolution is much more limited for GRBs 941017 and 970111 in which kT varies from ∼30 to ∼70 keV and from ∼20 to ∼65 keV, respectively (see panels (e) of Figures 7–9.).

${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ is usually lower than ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$. As noted in Guiriec et al. (2011a, 2013, 2015a, 2015b), the discrepancies between ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ are strongest at early times when the intensity of C_Th is the highest; at later times, when the C_Th contribution decreases, the values of ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ become extremely similar. This is particularly striking in the case of GRB 970111 (see Figure 8(c)): from ∼T₀ to ∼T₀ + 6 s, ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ evolves like ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ and not like ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$, while after ∼T₀ + 6 s ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ are perfectly identical. It is also interesting to note that the values of ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ seem to fall below those of ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ after ∼T₀ + 20 s, as also reported in Guiriec et al. (2015a) for Fermi GRBs; this trend will be clearer in the C_nTh + C_Th + PL analysis presented in the next section.

The νF${}_{\nu }$ spectra resulting from the C_nTh + C_Th + PL fits to the time-resolved data of GRBs 941017, 970111, and 990123 are displayed in Figures 18, 19, and 20, respectively—together with the Band-only fit results. Our simulation procedure indicates that C_nTh + C_Th + PL is a better description of the data than Band alone in ∼100% of the time intervals for the three GRBs.

In Guiriec et al. (2015a), we showed that the distributions of ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}$ and ${\alpha }_{{\rm{i}}}^{\mathrm{PL}}$ peaked at ∼−0.7 and at ∼−1.5, respectively, for a sample of Fermi GRBs fitted to C_nTh + C_Th + PL. When fixing ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}$ and ${\alpha }_{{\rm{i}}}^{\mathrm{PL}}$ of C_nTh + C_Th + PL to these typical values for all the time intervals of our current analysis of BATSE data, C_nTh + C_Th and C_nTh + C_Th + PL result in similar Cstat values, overall, for the same number of free parameters (see Tables 4–6). However, our simulation procedure indicates that C_nTh + C_Th + PL is a better description of the data than C_nTh + C_Th in ∼80% of the time intervals for the three GRBs; the time intervals in which C_nTh + C_Th + PL and C_nTh + C_Th are equivalent mostly correspond to GRB 970111 in which the contribution of the additional PL is the weakest.

When C_nTh + C_Th is fitted to the data of the three GRBs without the additional PL, the values of ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}$ cluster around −1, that is, an average of ∼−0.7 and ∼−1.5 (see panels (a) of Figures 7–9); this result is similar to the one reported in Guiriec et al. (2015a) which showed the impact of the extension at low energy of the additional PL on the values of α. For GRB 970111, which exhibits a weaker additional PL component (see Section 4.1), the ${\alpha }_{{\rm{i}}}^{\mathrm{nTh}}$ values remains closer to ∼−0.7 with C_nTh + C_Th.

As also reported in Guiriec et al. (2015a), the values of ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ obtained when fitting C_nTh + C_Th + PL to the data are usually lower than those resulting from the C_nTh + C_Th fits (see panels (b) of Figures 7–9); this is particularly striking for GRB 941017 as well as for GRB 990123 after ∼T₀ + 45 s. This can easily be explained if we consider that the contribution of the additional PL at high energy (i.e., $\gt \sim 1$ MeV) is included in C_nTh when fitting C_nTh + C_Th to the data and makes, therefore, the spectra of C_nTh harder and their ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ values higher.

We mentioned in Section 4.1 the difficulty with clearly identifying the additional PL in the time-integrated spectrum of GRB 970111. It is interesting to note that the C_Th + PL fit—with ${\alpha }_{{\rm{i}}}^{\mathrm{PL}}$ = −1.5 and without any C_nTh component—to the data of GRB 970111 provides an adequate description of the spectra during the first few seconds of the burst. This has two important consequences: first, the use of C_Th + PL at early time results in an initial monotonic cooling of the thermal-like component, C_Th, while that was not the case for the first time interval when fitting C_nTh + C_Th (see Figure 8(e)); second, this would indicate that similarly to the cases of GRBs 080916C and 090926A published in Guiriec et al. (2015a), the additional PL would be initially strong in GRB 970111 while C_nTh is either weak or absent. The additional PL would then be present from the very beginning or even before the other components conversely to previous reports where the additional PL was considered to start with a delay compared to the other main non-thermal component (see, for instance, Ackermann et al. 2010, 2011).

Overall, C_nTh overpowers the other components in the BATSE energy range from 20 keV to 2 MeV for the three GRBs (see Figures 10–12). Its contribution is usually higher than 50% of the total energy released between 20 keV and 2 MeV during the most intense part of the prompt emission (see Figure 13). The hardness of C_nTh strongly evolves with time during each burst; indeed, ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ can vary from tens of keV up to several MeV in only a few seconds (see panels (d) of Figures 7–10).

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The light curves of the C_nTh component usually exhibit multiple intensity peaks whose hardness is correlated with the flux; this is clearly evident with the ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ values tracking the photon fluxes as shown in panels (b) of Figures 10–12. The correlation between the energy flux of C_nTh, ${F}_{{\rm{i}}}^{\mathrm{nTh}}$, and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ will be studied in detail in Section 4.4.

The contribution of the C_Th component to the total energy between 20 keV and 2 MeV is globally $\leqslant$20% of the total emission during most of the burst duration, which is consistent with the results reported from Fermi data in Guiriec et al. (2011a, 2013, 2015a, 2015b). C_Th is usually more intense at early times. The temperature of C_Th—approximated with ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ for an easier comparison with ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$—does not vary over the same amplitude as ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ (see panels (b) of Figures 10–12.); indeed, kT_i varies between ∼30 and ∼80 keV, between ∼20 and ∼140 keV, and between ∼40 and ∼110 keV for GRBs 941017, 970111, and 990123, respectively (see panels (e) of Figures 7–9). Although a limited correlation may exist between the flux of C_Th and its temperature, kT_i remains more or less constant with time for GRBs 941017 and 990123; this result is similar to the one reported from the observations of GRBs 080916C and 090926A in Guiriec et al. (2015a).

The behavior of GRB 970111 is peculiar compared to the two other GRBs of this study but very similar to Fermi GRB 131014A published in Guiriec et al. (2015b). Indeed, GRB 970111 exhibits an initial nearly purely thermal episode¹¹ —despite the presence of the additional PL at early times—with a monotonic cooling from ∼120 keV down to ∼45 keV during the first five seconds of the burst (see Figures 8(e) and 11(b)). After ∼T₀ + 5 s, kT_i continues to cool slowly from ∼45 down to ∼25 keV during the remaining ∼40 s of the studied emission period although there is very slight reheating consistent with the light-curve peaks of the thermal-like component.

Despite some similarities in the light-curve structures, there is no evidence for a perfect correlation between either the flux variations of C_nTh and C_Th or the variations of ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ (see panels (d) of Figures 7–9 and panels (b) of Figures 10–12). It is particularly striking that while ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ is usually much higher than ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ at early times for all three GRBs, its values drop below ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Th}}$ at late times.

The overall contribution of the additional PL to the total energy between 20 keV and 2 MeV is roughly a few tens of percent although it can be >70% at early and late times (see Figure 13).

It has been reported many times from Fermi data that the additional PL usually extends from a few keV up to tens to hundreds of MeV and maybe even up to GeVs; the additional PL usually overpowers C_nTh below a few tens of keV and above several MeVs (Abdo et al. 2009b; Ackermann et al. 2010, 2011; Guiriec et al. 2010, 2015a). Because of the limited energy range of BATSE, the additional PL is mostly subdominant compared to C_nTh over the whole observed energy band, and it only starts to be the dominant component at the very high end of the spectrum (see Figures 18–20). Although C_nTh clearly outshines the additional PL below 20 keV in GRB 970111 (see Figure 11), there is evidence for an extension of the additional PL at low and high energies with a flux higher than in the other components in GRBs 941017 and 990123 (see Figures 10 and 12). For a better comparison to the Fermi data, we extrapolated the C_nTh + C_Th + PL model derived from the 20 keV–2 MeV BATSE data down to 5 keV—to mimic the low end of the energy range of the GBM Na idetectors—and beyond 2 MeV—to mimic the high energy coverage of GBM BGO detectors up to 40 MeV, as well as the 20 to 100 MeV energy range of the Fermi/Large Area Telescope (LAT). If Fermi had observed GRBs 941017 and 990123—without considering the instrumental sensitivities or possible <100 MeV breaks in the additional PL as suggested in Guiriec et al. (2015a)—it would have detected an additional PL outshining the other components at low and high energies similarly to the other bursts with additional PL detected by Fermi:

• 1.  
  If observed with Fermi, the additional PL in GRB 941017 would correspond to a broad pulse in the light curves with a fast rise and exponential decay type shape (see Figure 10); despite the similarities in the timing of the most intense part of the additional PL with the other components, their evolutions with time are quite different. The additional PL emission would be the most intense between ∼T₀ + 10 s and ∼T₀ + 30 s, then the emission would quickly decay but it would remain present until at least T₀ + 100 s; this is in perfect agreement with the >1 MeV light curves of the data from the Total Absorption Shower Counter, which is the calorimeter of EGRET on board CGRO, presented in Figure 1 in González et al. (2003). The broad shape of the additional PL light curve is similar to the one reported for Fermi GRB 090902B in Abdo et al. (2009b).
• 2.  
  The case of GRB 990123 is particularly interesting with respect to the additional PL. In Guiriec et al. (2015a) we reported that the additional PLs identified in Fermi GRBs 080916C and 090926A were, overall, low-intensity components that sometimes strongly outshone the other ones during short periods of time of the order of a few seconds—short duration excesses associated with the additional PL were also reported in Ackermann et al. (2011) and González et al. (2012) for Fermi GRB 090926A and BATSE GRB 980923. The same behavior is observed in GRB 990123 (see Figure 12). Indeed, if Fermi had observed this GRB, it would have detected a strong intensity peak at low energies below ∼10 keV around T₀ + 40 s and lasting ∼5 s, perfectly correlated in time with an intense peak in the light curve above a few MeV. As shown in Figure 12, this intensity peak in the light curves would be mostly related to the additional PL. However, the contribution of the additional PL would be mostly hidden by the intense C_nTh component between a few tens of keV and several MeV as in GRB 080916C (Guiriec et al. 2015a).

When fitting the γ-ray prompt emission of both short and long GRBs to a Band function, a correlation is often observed between the evolution of the γ-ray energy flux, ${F}_{{\rm{i}}}^{\mathrm{Band}}$, and the evolution of the corresponding ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ within each burst (i.e., ${F}_{{\rm{i}}}^{\mathrm{Band}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ relation—see, for instance, Golenetskii et al. 1983; Borgonovo & Ryde 2001; Liang et al. 2004; Guiriec et al. 2010, 2015a, 2015b; Ghirlanda et al. 2011a, 2011b; Lu et al. 2012). Although this correlation globally follows a PL, there is often large scatter in the data of each burst and the PL indices can be dramatically different from burst to burst. More importantly, there are many bursts that do not exhibit this correlation.

In Guiriec et al. (2013, 2015a, 2015b) we introduced a new relation between the γ-ray energy flux and the spectral hardness. Indeed, by isolating the non-thermal component of the spectra from the thermal-like one—in fitting C_nTh + C_Th or C_nTh + C_Th + PL to the data—we showed that a very strong correlation, intrinsic to the non-thermal component only, appears between the evolution of its energy flux, ${F}_{{\rm{i}}}^{\mathrm{nTh}}$, and the evolution of its νF${}_{\nu }\;\gamma$-ray spectral peak, ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ (i.e., ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ relation). Conversely to the ${F}_{{\rm{i}}}^{\mathrm{Band}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ relations, the ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ relations have extremely similar indices for all GRBs when fitted to PLs.

Moreover, in the central engine frame, the ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ relations translate into a unique ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ relation—with the same index as the ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ relation—which seems to be universal for all short and long GRBs (Guiriec et al. 2013, 2015a, 2015b).

Panel (f) of Figures 7–9 shows the ${F}_{{\rm{i}}}^{\mathrm{Band}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ and ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ relations for the three GRBs using Band, C_nTh + C_Th, and C_nTh + C_Th + PL models. As already reported in Guiriec et al. (2013, 2015a, 2015b) for other GRBs, the energy flux and E_peak values are typically anti-correlated at very early times in the burst, which also usually corresponds to the highest values of E_peak. The indices of the PLs fitted to these relations—after excluding the anti-correlated early phase—are reported in Table 2. Apart from GRB 970111, the indices of the ${F}_{{\rm{i}}}^{\mathrm{Band}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ relations are inconsistent with the ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ ones; the indices of the ${F}_{{\rm{i}}}^{\mathrm{Band}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ relations are also dramatically different from one burst to the other (i.e., from +1.4 to +3.2). Conversely, the indices of the ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ relations are all clustered when using the C_nTh + C_Th + PL model (between +1.3 and +1.5—see Figure 14(a)); these values are perfectly consistent with the indices of +1.33 ± 0.06, +1.38 ± 0.04 and +1.43 ± 0.03 reported in Guiriec et al. (2013, 2015a, 2015b), respectively.

Table 2.  Power-law Indices of the ${F}_{{\rm{i}}}^{\mathrm{Band}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ or ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ Relations Obtained When Fitting Band-only, C_nTh +C_Th, or C_nTh + C_Th + PL to the Time-resolved Spectra of GRBs 941017, 970111, and 990123 (See Section 4.4)

	${F}_{{\rm{i}}}^{\mathrm{Band}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{Band}}$ or ${F}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh}}$ Relation PL indices For
	Band	CnTh + CTh	CnTh + CTh + PL
GRB 941017	+3.07 ± 0.12	+1.35 ± 0.06	+1.40 ± 0.10
GRB 970111	+1.43 ± 0.04	+1.61 ± 0.32	+1.33 ± 0.23
GRB 990123	+2.68 ± 0.09	+1.72 ± 0.09	+1.28 ± 0.05

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In our three-BATSE-burst sample, GRB 990123 has an accurate redshift measurement of z ∼ 1.61. There are no redshift measurements available for GRB 941017 and GRB 970111, although host galaxies possibly associated with GRB 970111 resulted in an initial redshift range of $0.2\;\leqslant$ z $\leqslant$ 1.4 (Gorosabel et al. 1998). Assuming that the ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ relation is universal¹² —as suggested in Guiriec et al. (2013, 2015a, 2015b)¹³ —we estimated the distance of GRBs 941017 and 970111 using GRB 990123 as the reference; the ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ relation for GRB 990123 is displayed in Figure 14(b) (dashed blue line). We varied the redshift to determine the values that minimize the distances between the dashed blue line and the data of GRBs 941017 and 970111. Using the same technique as described in Guiriec et al. (2015b), we performed a linear fit in the log–log space to account for the uncertainties on both the ${L}_{{\rm{i}}}^{\mathrm{nTh}}$ and ${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ quantities; the resulting ${\chi }^{2}$ profiles are displayed in panels (c) and (d) of Figure 14 for GRBs 941017 and 970111, respectively. The redshift estimates using this technique are reported in Table 3. The uncertainties on the best ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ relation of GRB 990123 do not affect the redshift estimates at more than the second figure after the decimal point and can therefore be neglected. Interestingly, the distance estimate for GRB 970111 (i.e., z = 1.18 ± 0.06) is compatible with the redshifts of the galaxies (i.e., $0.2\;\leqslant$z$\leqslant$ 1.4) identified as possible hosts for this burst in Gorosabel et al. (1998); however, this result should neither be interpreted as a proof of the technique nor as a proof that the host galaxy is one of the objects identified between z = 0.2 and z = 1.4 since it may be a coincidence. The redshift estimate for GRB 941017 (i.e., z = 1.79 ± 0.07) is a typical value for a long GRB (i.e., $\langle {\rm{z}}\rangle$ ∼ 2.16—Jakobsson et al. 2012).

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Table 3.  Measured and Estimated Redshifts (z) for GRBs 941017, 97011, and 990123 Using GRB 990123 with z = 1.61 as a Reference (see Section 4.4)

	Measured z	z Estimates
GRB 941017	⋯	1.79 ± 0.07
GRB 970111	$0.2\;\leqslant$ z $\leqslant$ 1.4a	1.18 ± 0.06
GRB 990123	∼1.61	⋯

Note.

^aThis photometric redshift estimate, based on possible host-galaxy identifications reported in Gorosabel et al. (1998), is not secure because fainter objects are present in this region of the sky and one of them may be the actual host galaxy of GRB 970111.

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Figure 14(b) shows the ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ relations for the three bursts using the redshift measurement for GRB 990123 and the redshift estimates for the two other bursts. The PL fits to each burst are displayed in dashed colored lines. The solid black line is the fit to the whole data together, and it results in a PL index of +1.31 ± 0.03, which is in perfect agreement with the results published in Guiriec et al. (2013, 2015a, 2015b) using Fermi data.

Once again, the aim of this article is not to validate the ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ relation nor to test its robustness to estimate GRB distances; such a study is an ongoing project using a large sample of GRBs with known redshifts, and it will be the topic of a future paper. Here, we simply show that the ${L}_{{\rm{i}}}^{\mathrm{nTh}}$–${E}_{\mathrm{peak},{\rm{i}}}^{\mathrm{nTh},\mathrm{rest}}$ relation obtained in this article using BATSE data is perfectly compatible with our previous results using Fermi data (Guiriec et al. 2013, 2015a, 2015b) and that the redshift estimates that we obtained using this relation seem to be reasonable based on knowledge of GRB redshifts.