How Are Gamma-Ray Burst Radio Afterglows Populated?

We first plot the distributions of radio afterglows for detections and 3$\sigma$ upper limits between 0 and 10 days at 8.5 GHz in top-left panel of Figure 1, where it is found that our distributions are similar to those in Chandra et al. (2012) and Hancock et al (2013), in which the upper limits are confirmed again to peak at 50-100 $\mu$Jy in and the detections peaked around 200 $\mu$Jy with a long extending tail. We also find that there is an obvious truncation at $\sim$ 400 $\mu$Jy in the VLA-based detection sample, which motivates us to examine whether the distribution of the flux densities less than 400 $\mu$Jy is associated with that of the upper-limit sample. For the purpose, we try to define the detection sample whose flux density larger than 400 $\mu$Jy as radio-loud I sample, and other detections with radio flux density less than 400 $\mu$Jy to be radio-loud II sample, temporally. It is interestingly found from the bottom panels of Figure 1 that the flux density distributions of radio-loud and radio-quiet AMI afterglows are also bimodally distributed and resemble those of the VLA-based sample. However, the AMI peak flux densities of both detections and upper limits are on average two times larger than those VLA-based ones, correspondingly.

To check if it is necessary to reclassify radio-loud GRBs into two subsamples, we display the cumulative fractions of the intrinsic duration $T_{int}$ and the $E_{\gamma,iso}$ for radio-loud I, radio-loud II and radio-quiet GRBs (upper limits) in Figure 2. As shown in Table LABEL:Table5:k-s_test, the Kolmogorov-Smirnov (K-S) tests return the statistic $D=0.48$ (0.31) and $P=7\times 10^{-4}$ (0.031) between the $T_{int}$ distributions of the radio-loud I (II) and the radio-quiet samples showing the radio-quiet bursts are different from either radio-loud I or II. Similarly, the statistic and p-value of the $E_{\gamma,iso}$ distributions are $D=0.35$ (0.38) and $P=0.037$ (0.004) for comparisons between the radio-quiet and the radio-loud I (II) samples. Surprisingly, the K-S test to the radio-loud I and the radio-loud II samples returns $D=0.27$ with $P=0.24$ for the $T_{int}$ distribution and $D=0.14$ with $P=0.92$ for the $E_{\gamma,iso}$ distribution, indicating that the two radio-loud sub-samples should be taken from the same parent distribution. In other words, dividing radio-loud bursts into two classes is not necessary. Consequently, we shall only investigate the radio-loud, the radio-quiet and the radio-none samples in the subsequent sections, and explore in statistics whether they are basically different kinds of bursts on basis of their observational properties.

Using the total sample of 206 GRBs including 151 VLA-based and 55 AMI bursts, we plot the histograms of $z$, $E_{\gamma,iso}$ and $T_{int}$ for radio-loud, radio-quiet and radio-none samples in Figure 12, where one can find that the distributions of $z$, $T_{int}$ and $E_{\gamma,iso}$ of radio-loud and radio-quiet samples are well fitted by a gaussian function, but the radio-none sample seems to be eccentric (see Appendix for a detail). The fitting results are summarized in Table 4, from which we notice that the mean values of $z$, $E_{\gamma,iso}$ and $T_{int}$ of radio-none GRBs are systematically smaller than those of the other two samples. In particular, the isotropic energies of radio-none bursts are on average one order of magnitude lower than the $E_{\gamma,iso}$ values of either radio-loud or radio-quiet bursts.

Following Hancock et al (2013) and Lloyd et al. (2019), we also analyze the cumulative fractions of the $T_{int}$ and the $E_{\gamma,iso}$ but for different radio-loud, radio-quiet and radio-none VLA-based samples in Figure 3 and Table LABEL:Table5:k-s_test, where we see that the radio-quiet samples are evidently different from the radio-loud ones in terms of the $T_{int}$ distribution, on average the radio-loud GRBs have relatively longer $T_{int}$ as found before (Hancock et al, 2013; Lloyd et al., 2019). However, the K-S test demonstrates that the $T_{int}$ distributions of radio-quiet and radio-none GRBs are indistinguishable. Regarding the $E_{\gamma,iso}$ distributions, we also perform the K-S tests to any two of the above three VLA-based samples and find from Table LABEL:Table5:k-s_test that they are drawn from different parent distributions. In addition, the median discrepancy of $E_{\gamma,iso}$ between the radio-loud and the radio-none bursts is about two orders of magnitude. With the increase of frequency, it is interestingly found that two AMI samples of radio-loud and radio-quiet GRBs at 15.7 GHz are consistent with being drawn from the same parent $E_{\gamma,iso}$ distribution.

We notice that some GRBs with radio flux densities of host galaxies in Zhang et al. (2018) were not included in our initial radio-loud, radio-quiet or none samples. To increase the reliability in statistics, we assume them to be radio-none or radio-quiet because they don’t have radio afterglows detected. In order to analyze the radio flux density of host galaxies for the three samples, we combine the data of radio-none and radio-quiet into a simple radio-faint sample. Then we plot the cumulative fractions for the radio-loud, radio-faint samples in Figure 4 where we find when the radio (flux density of the host galaxies, $F_{host}$) is less than 50 $\mu$Jy the radio-loud, the radio-faint samples share the same distribution, but when it is more than 50$\mu$Jy the Cumulative fractions of these two samples are significantly different. A K-S test shows that the probability of those two samples from the same distribution is 0.19, so that in terms of host galaxies the two samples might be taken from the same distribution.

Li et al. (2015) found the host flux density $F_{host}$ is positively correlated with the observed peak flux density ($F_{o,peak}$) or the pure flux density($F_{b,peak}$) of GRBs at a given radio frequency $\nu$ as follows

and

where $a\simeq 0.3$, $b\simeq-0.02$, and $F_{o,peak}=F_{b,peak}+F_{host}$. The Eq. (1) can be used to estimate the host flux density once the peak values of radio afterglows are measured. Figure 5 displays the relationships of $F_{host}$ with $F_{o,peak}$ or $F_{b,peak}$ for the radio-loud and radio-quiet samples. One can find that the radio flux densities of the radio-quiet GRBs and their host galaxies are relatively lower than those of the radio-loud ones. It is noticeable that the very famous nearby short GRB (sGRB) 170817A seen off-axis with an estimated viewing angle of $20^{\circ}\sim 40^{\circ}$ (Alexander et al., 2017) is the first electromagnetic counterpart of gravitational-wave event. It has peak flux densities of $\sim$84.5 $\mu$Jy and $\sim$58.6 $\mu$Jy observed correspondingly at $\nu=$3 GHz and 5.5 GHz around 130 days since the merge of double neutron stars (Li et al., 2018). Using the above Eq. (1) and (2), one can easily predict the host flux densities to be about 20.3 $\mu$Jy at $\nu=$3 GHz and 11.1 $\mu$Jy at $\nu=$5.5 GHz. Interestingly, GW 170817/sGRB170817A as a radio-loud burst has relatively weaker radio afterglows and lower host fluxes in contrast with other normal radio-loud GRBs. However, it is located near the radio-quiet bursts as shown in Figure 5, which indicates that galactic types or circum-burst environment of different radio-selected GRBs could be diverse although their dominant radiation mechanisms might be the same.

As pointed out by Chandra et al. (2012), the centimeter radio afterglow emission is the brightest for circum-burst densities from 1 to 10 cm^-3. Beyond the narrow density range, the flux density will become weak due to either a low intrinsic emission strength (for lower densities) or the increased synchrotron self-absorption (for higher densities). From the literatures, it is well known that the circum-burst medium densities ($n$) of GRBs usually span serval orders of magnitude and are hard to be determined (e.g. Wijers & Galama, 1999; Chandra et al., 2012; Fong et al., 2015; Zhang et al., 2018). In our samples, the circum-burst densities are distributed in a fairly wide scope spanning $\sim$10 orders of magnitude seen from Table LABEL:Table1:radio-loud to LABEL:Table3:radio-none. Because the number of radio-none GRBs with estimated densities is extremely limited, we thus combine the radio-quiet and the radio-none samples into a newly-formed radio faint sample in order to increase the statistical confidence level. Then we plot the cumulative fractions for the two samples in Figure 6 and apply a K-S test to get $D$= 0.55 with a probability of 0.002, which demonstrates that the radio-loud and radio faint samples are significantly incongruous with each other. In contrast, the medium densities of the radio-loud host galaxies are relatively larger than those of the radio faint ones. Furthermore, the fraction of low densities of $n\leq$ 0.1 cm^-3 for the radio faint sample is around six times more than that for the radio-loud sample. On the contrary, about 90 percent of radio-loud afterglows are surrounded by relatively denser mediums of $n\simeq 10^{-1}-10^{2}$ cm^-3.

We utilize all the GRBs with measured isotropic $\gamma$-ray energy instead of $E_{\gamma,iso}$ $\textgreater$ $10^{52}$ (Lloyd et al., 2019) only to ensure our samples to be as complete as possible. Simultaneously, we calculate the spectral peak luminosity at radio band ($L_{\nu,p}$) for the radio-loud and the radio-quiet (or upper limit) samples as (Zhang et al., 2018)

where $f_{m,radio}$ denotes the peak flux density $F_{o,peak}$ of the radio-loud afterglows or the upper limits of radio-quiet afterglows, $k$ is a $K$-correction factor determined by

where $\alpha$ $\sim$ 0 and $\beta$ $\sim$ 1/3 are assumed to be the normal temporal and spectral indexes, respectively. $D_{L}$ denoting the luminosity distance of a burst is given by

in which $c$=$3.0\times 10^{8}$ m/s is the speed of light, $H_{0}$ is the Hubble constant taken as 70 km/s/Mpc, other cosmological parameters $\Omega_{M}$=0.27 and $\Omega_{\Lambda}$=0.73 have been assumed for a flat universe (Schaefer, 2007). Consequently, the $L_{\nu,p}$ values can be obtained from Eq. (3) for the VLA-based GRBs at 8.5 GHz since most afterglows were detected at this frequency. For the AMI bursts reported in Anderson et al. (2018), their $L_{\nu,p}$ values are calculated at a frequency of 15.7 GHz. Owing to lack of measurement of the radio afterglows with the upper limits, the $L_{\nu,p}$ values of radio-quiet afterglows can be only estimated as the upper limits too. Figure 7 displays the $L_{\nu,p}$ distributions of radio-loud, radio-quiet and SN-associated GRBs respectively. On average, the peak luminosity of radio-loud bursts is relatively larger than the other two, while the mean values of radio-quiet and SN-associated GRBs are comparable. The cumulative fractions of all the above samples are shown in Figure 8. A K-S test to them shows that the luminosity distributions of radio-loud and radio-quiet GRBs are largely different for the VLA-based samples since $D=0.4$ (>$D_{\alpha=0.05}=0.25$) with $P\simeq 6.6\times 10^{-5}$ and are however consistent with each other for the AMI samples. It needs to be emphasized that the distributional consistency of $L_{\nu,p}$ for different kinds of radio-selected GRBs is similar to that of the $E_{\gamma,iso}$ distributions in Figure 3. Moreover, the actual deviation between them would become more significant since the accumulative line of the radio-quiet sample consisted of the upper limits should move leftward in a certain sense. The median $L_{\nu,p}$ of radio-quiet sample is about one order of magnitude smaller than that of radio-loud sample. Interestingly, this is similar to the one order of magnitude difference between radio fluxes of host galaxies and GRB afterglows (Zhang et al., 2018). Hence, we conclude that the majority of radio-quiet emissions should be contributed by their surrounding host galaxies.

As shown in Figures 2, 3, 7 and 8, the averaged energies of $E_{\gamma,iso}$ and $L_{\nu,p}$ of radio-loud bursts are larger than the corresponding values of radio-quiet ones. In the section, we will testify the possible correlation between the $E_{\gamma,iso}$ and the $L_{\nu,p}$ of radio-loud (N=100) and radio-quiet (N=76) GRB samples. For this purpose, the radio peak flux densities at 8.5 GHz and 15.7 GHz have been utilized. Figure 9 displays the relations of $E_{\gamma,iso}$ with $L_{\nu,p}$ for all the radio-loud/quiet VLA-based bursts including 95 long GRBs (lGRBs), 23 SN/GRBs, 2 X-Ray Flashes (XRFs) and 6 short GRBs (sGRBs), and 50 AMI GRBs. Interestingly, we find on the left panel that $E_{\gamma,iso}$ is positively correlated with $L_{\nu,p}$ with a Pearson correlation coefficient of $R=$0.76 ($SL=2.2\times 10^{-16}$) or Spearman rank correlation coefficient of 0.55 ($SL=1.08\times 10^{-7}$). The correlation function can be roughly written as $L_{\nu,p}\propto E_{\gamma,iso}^{0.41\pm 0.04}$ for the whole radio-loud sample with $\chi^{2}_{\nu}=0.23$. On the right panel, a positive correlation, $L_{\nu,p}\propto E_{\gamma,iso}^{0.48\pm 0.09}$ with a $\chi^{2}_{\nu}=0.42$, weakly exists for the radio-quiet bursts, of which the Pearson and the Spearman correlation coefficients are respectively $R=$0.62 ($SL=5.48\times 10^{-6}$) and $R=$0.47 ($SL=1.2\times 10^{-3}$) that are very close to those of the radio-loud bursts. This demonstrates that the radio peak luminosities and the prompt $\gamma$-ray energies are highly associated. It is notable that our finding here is different from Chandra et al. (2012), where they claimed no obvious correlation between $E_{\gamma,iso}$ and $L_{\nu,p}$ in their Figure 20 possibly owing to the limit of sample size. Recently, Tang et al. (2019) found that the X-ray peak luminosity is positively correlated with the $E_{\gamma,iso}$ as $L_{X}\propto E_{\gamma,iso}^{0.97}$. It is valuable to mention that the radio peak luminosities of 21 SN/GRBs in our sample and 6 SN/GRBs in Chandra et al. (2012) exhibit a consistent dependence of $E_{\gamma,iso}$. This may imply these SN/GRBs should undergo with the same processes of energy dissipations. Data points of the sGRBs and the XRFs are too limited to show if they behave a positive interdependency as the lGRBs did.

Lloyd et al. (2019) found that there was a negative correlation between $T_{int}$ and $1+z$ for the radio-loud rather than radio-quiet GRB sample. They concluded that if this negative correlation indeed exists, other than affected by the selection effect, it could reflect that the systems at higher redshift have less angular momentum or less materials accreted to the GRB disks. Recently, Zhang et al. (2018) investigated the correlations between the intrinsic peak times of radio afterglows at 8.5 GHz and the redshift factor $(1+z)$ and found that they are fully uncorrelated, which seems to conflict with the negative correlation of $T_{int}$ versus $1+z$. Meanwhile, the $T_{int}$ distribution of Swift/BAT bursts was still bimodal in that all the durations move towards to the short end once the $T_{90}$ over 1+z was considered (Zhang & Choi, 2008). It is well known that the sGRBs are usually observed at nearby universe unlike the lGRBs. Strictly speaking, the negative dependence of the $T_{int}$ on the redshift is hard to understand unless a fraction of sGRBs have extremely small redshift while parts of lGRBs have very high redshift.

As mentioned in Section 2, our current samples as an expansion of Lloyd et al. (2019) are relatively complete. Therefore, it is timely and essential to check if the correlations between $T_{int}$ and $1+z$ coexist in both radio-loud, radio-quiet and radio-none samples as plotted in Figure 10. In statistics, the Pearson correlations of $T_{int}$ vs. $1+z$ for the radio-loud and the radio-quiet samples give the R-indexes as -0.29 ($SL$=0.012), -0.35 ($SL$=0.018) and -0.15 ($SL$=0.53) for the radio-loud, the radio-quiet and radio-none lGRB samples, respectively. This demonstrates that the radio-loud GRBs do have a weaker negative correlation of $T_{int}$ with redshift, of which this result is in good agreement with Lloyd et al. (2019). Additionally, our radio-quiet sample also hold the similar anti-correlation with a 95.4% confidence level like the radio-loud GRBs. It is surprisingly found that there is very weak correlation between $T_{int}$ and $1+z$ for the radio-none sample. We notice that the sGRBs in any case of our samples are outliers of the $T_{int}$-$(1+z)$ correlation of the lGRBs and the sGRBs with smaller $T_{int}$ and $1+z$ are systematically located at the bottom-left side of plane. Particularly, the radio-loud sGRB 170817A is situated in the region of normal sGRBs. Hence, the $T_{int}$ distributions of two kinds of GRBs are well in agreement with Zhang & Choi (2008).