On the investigation of the closure relations for Gamma-Ray Bursts observed by Swift in the post-plateau phase and the GRB fundamental plane
Gokul Prem Srinivasaragavan, Maria Giovanna Dainotti, Nissim Fraija, Xavier Hernandez, Shigehiro Nagataki, Aleksander Lenart, Luke Bowden, Robert Wagner
DDraft version November 20, 2020
Typeset using L A TEX preprint style in AASTeX63
On the investigation of the closure relations for Gamma-Ray Bursts observed by Swiftin the post-plateau phase and the GRB fundamental plane
G.P. Srinivasaragavan, M.G. Dainotti ∗ ,
2, 3, 4, 5
N. Fraija, X. Hernandez, S. Nagataki,
7, 8
A. Lenart, L. Bowden, and R. Wagner Cahill Center for Astrophysics, California Institute of Technology, 1200 E. California Blvd. Pasadena, CA 91125,USA Interdisciplinary Theoretical & Mathematical Science Program,RIKEN (iTHEMS), 2-1 Hirosawa, Wako, Saitama,Japan 351-0198 Physics Department, Stanford University, 382 Via Pueblo Mall, Stanford, USA Space Science Institute, Boulder, Co Obserwatorium Astronomiczne, Uniwersytet Jagiello´nski, ul. Orla 171, 31-501 Krak´ow, Poland. Instituto de Astronomia, Universidad Nacional Autonoma de Mexico, Apartado Postal 70264, C.P. 04510, MexicoD.F., Mexico RIKEN Cluster for Pioneering Research, Astrophysical Big Bang Laboratory (ABBL), 2-1 Hirosawa, Wako,Saitama, 351-0198, Japan Interdisciplinary Theoretical & Mathematical Science Program,RIKEN (iTHEMS), 2-1 Hirosawa, Wako, Saitama,351-0198, Japan Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Krak´ow, Poland Astronomy Department, Cornell University, 616-A Space Sciences Building, Ithaca, USA. Physics Department, The College of New Jersey, 2000 Pennington Rd, Ewing, NJ, USA (Dated: November 20, 2020)
ABSTRACTGamma-ray Bursts (GRBs) are the most explosive phenomena in the universe afterthe big bang. A large fraction of GRB lightcurves (LCs) shows X-ray plateaus. Weperform the most comprehensive analysis of all GRBs (with known and unknown red-shifts) with plateau emission observed by The Neil Gehrels Swift Observatory fromits launch until 2019 August. We fit 455 LCs showing a plateau and explore whetherthese LCs follow closure relations, relations between the temporal and spectral indicesof the afterglow, corresponding to two distinct astrophysical environments and coolingregimes within the external forward shock (ES) model, and find that the ES modelworks for the majority of cases. The most favored environments are a constant-densityinterstellar or wind medium with slow cooling. We also confirm the existence of thefundamental plane relation between the rest-frame time and luminosity at the end ofthe plateau emission and the peak prompt luminosity for this enlarged sample, and testthis relation on groups corresponding to the astrophysical environments of our knownredshift sample. The plane becomes a crucial discriminant corresponding to these en- *corresponding [email protected], First and second author share the same contribution a r X i v : . [ a s t r o - ph . H E ] N ov vironments in terms of the best-fitting parameters and dispersions. Most GRBs forwhich the closure relations are fulfilled with respect to astrophysical environments havean intrinsic scatter σ compatible within 1 σ of that of the “Gold” GRBs, a subset of longGRBs with relatively flat plateaus. We also find that GRBs satisfying closure relationsindicating a fast cooling regime have a lower σ than ever previously found in literature. Keywords: cosmological parameters - gamma-rays bursts: general, radiation mecha-nisms: nonthermal INTRODUCTIONGamma-ray bursts (GRBs) are short-lived bursts of γ -ray photons originating from high-energyastrophysical phenomena, and are spectacular events due to their energy emission mechanism. TheNeil Gehrels Swift Observatory (Gehrels et al. 2004), launched in 2004 November, has observed GRBswithin a wide range of redshifts. More specifically, Swift, with its onboard instruments - the BurstAlert Telescope; (BAT; 15 - 150 keV; Barthelmy et al. (2005)), the X-ray Telescope (XRT; 0.3 - 10keV; Burrows et al. (2005b)) and Ultra-Violet and Optical Telescope (UVOT; 170 - 650 nm; Rominget al. (2005)) – provides rapid localization of many GRBs and enables fast follow-up of the afterglowsin several wavelengths. The afterglow of GRBs is likely due to the external forward shock (ES)where the relativistic ejecta impacts the external medium (Paczynski & Rhoads 1993; Katz & Piran1997; Meszaros & Rees 1997). It has already been shown that Swift GRB lightcurves (LCs) havemore complex features than a simple power law (PL, Tagliaferri et al. (2005); Nousek et al. (2006);O’Brien et al. (2006); Zhang et al. (2006a); Sakamoto et al. (2007); Zhao et al. (2019). O’Brien et al.(2006) and Sakamoto et al. (2007) discovered the existence of a flat part in the X-ray LCs of GRBs,called the plateau, which is present soon after the decaying phase of the prompt emission. TheSwift plateaus generally last from hundreds to a few thousands of seconds (Willingale et al. 2007).Physically, this plateau emission has been associated either due to continuous energy injection fromthe central engine (Dai & Lu 1998; Rees & M´esz´aros 1998; Sari & M´esz´aros 2000; Zhang & M´esz´aros2001; Zhang et al. 2006a; Liang et al. 2007), with millisecond newborn spinning neutron stars (e.g.,Zhang & M´esz´aros 2001; Troja et al. 2007; Dall’Osso et al. 2011; Rowlinson et al. 2013, 2014; Reaet al. 2015; Beniamini & Mochkovitch 2017; Toma et al. 2007; Metzger et al. 2018; Stratta et al. 2018;Fraija et al. 2020) or with mass fall-back accretion onto a black hole (Kumar et al. 2008; Cannizzo& Gehrels 2009; Cannizzo et al. 2011; Beniamini & Giannios 2017; Metzger et al. 2018).In previous literature, some theoretical models ascribe the X-ray plateau to the continuous, long-lasting, energy injection into the ES (Zhang & M´esz´aros 2001; Zhang et al. 2006b; MacFadyen 2001).The study of the ES emission within the standard fireball model has already been tested in X–rays byWillingale et al. (2007) through a set of relationships called closure relations (Zhang & M´esz´aros 2004;Gao et al. 2013). These relations are theoretical relationships between the temporal PL index of theafterglow ( α ) and the spectral index derived from the electron spectral index of synchrotron emission( β ). Each relation indicates different possible astrophysical environments and cooling regimes forGRBs. The first study related to the closure relationships dates back to 2004: Zhang & M´esz´aros(2004) collected the available closure relations (and derived some more) in a comprehensive review.Their review was later used by several authors including Racusin et al. (2009), whose work on theserelations is highly relevant to this current paper, and Gao et al. (2013), who has collected the mostcomplete set of closure relations in literature. Here we test an extended set of closure relations takenfrom Racusin et al. (2009) with a large sample of GRBs observed by Swift from 2005 January until2019 August, an analysis of 15 yr of observations.GRBs are fascinating events not only for the challenge of understanding their emission mechanism,but also because they are observed at very high redshifts ( z ) up to z = 10 (Cucchiara et al. 2011).Thus, they have the potential to be used as standard candles, with known redshifts much largerthan the most accredited standard candles, Supernova Ia, with the furthest one observed at z = 2 . Swift has observed GRBs in such a wide range of redshifts, it hascontributed enormously to this process of possibly making GRBs standard candles. However, withtheir prompt luminosities ranging over eight orders of magnitudes, the process of standardizingGRBs has proven to be a challenging task. Classifying GRBs into their type-specific classes hasbeen one of the first steps in starting this process for the afterglow emission, as detailed in Dainottiet al. (2010). The mixing of GRBs of different classes leads to comparing observables and relationsbetween phenomena with fundamentally different intrinsic physics, which is why it is not advisableto standardize GRBs as a whole. Thus, it is better to instead hunt specific classes in order to workwith observationally homogeneous samples that could lead to tighter correlations (Cardone et al.2009, 2010; Dainotti et al. 2016, 2017a,b; Dainotti & Del Vecchio 2017).This new information has helped with the search for finding meaningful correlations between in-trinsic parameters of GRBs. Dainotti et al. (2016) made use of the discovery of the plateau in orderto isolate a sub-class of GRBs detailed as the “Gold” class that defines a very tight 3D fundamentalplane relation, called the gold fundamental plane (the so-called Dainotti 3D relation). This relation isan extension of the 2D relation given by L X − T X (the so-called Dainotti 2D relation) and previouslydiscovered by Dainotti et al. (2008, 2010, 2011, 2013, 2015a,b, 2017b). The “Gold” class presents thesmallest intrinsic scatter among all of the subclasses, showing it has the highest potential to be usedas a standard candle. In this paper, we define a slightly different “Gold” class, called the “Gold 2”,to see if the tight relations found in Dainotti et al. (2017b) continue to hold despite using a differentfitting of prompt spectral parameters, a cutoff power law (CPL) model when possible rather than aPL model, and with the addition of two extra years of GRB measurements. Furthermore, to the bestof our knowledge, there has not been an analysis done on this fundamental plane relation with re-spect to the astrophysical environments of GRBs derived from their fulfillment of the aforementionedtheoretical closure relations, which is another main aim of our paper.In the current analysis, we investigate the following questions:1. How many GRBs observed by Swift showing a plateau (similar to the one presented in Figure1) obey the closure relations in the post-plateau phase, and what can we infer about theirastrophysical environments?2. Does the 3D fundamental plane relation hold with the addition of two extra years of GRBmeasurements?3. Do the closure relations reveal new fundamental planes which show tighter correlations thanthe ones already found? - - - - - - - ( s ) l og f l u x ( e r g c m - s - ) GRB 090510 Swift BAT + XRT
Figure 1.
Flux vs. time in the observer frame of GRB 090510 fitted by the W07 model defined in Equation1 with the blue solid line by employing the BAT+XRT data. The black dot shows the best-fit parametersof the end time of the plateau and its respective flux within the same model.
The paper has the following structure: in § § § § § § SAMPLE SELECTIONWe analyze 222 GRBs detected by
Swift from 2005 January up to 2019 August with known red-shifts, and 233 GRBs from the same time range with unknown redshifts, giving us a total sample of455 GRBs. All these GRBs have a well-defined plateau in the afterglow phase. The LCs have beendownloaded from the
Swift webpage repository , and have a signal to noise ratio of 4:1 with the Swift
XRT bandpass ( E min , E max ) = (0.3,10) keV. We then fit these GRBs according to the phenomenolog- ical W07 model. These GRBs all satisfy the fitting procedure and satisfy the Avni χ prescriptions(see the XSPEC manual) at the 1 σ confidence interval. The 222 GRBs with known redshifts havetheir spectroscopic or photometric redshifts available through Xiao & Schaefer (2009), on the Greinerwebpage , and have a known redshift range of 0 . ≤ z ≤ .
4. Due to the fitting, each GRB hasan associated time, T a , and flux, F a , at the end of the plateau phase, as well as a PL index for thetemporal evolution of the afterglow after the plateau, α .We divide the 222 GRBs with redshift in accordance to Dainotti et al. (2017b) into their respectivesubclasses: Short (sGRBs), Short with Extended Emission (SEE), Supernovae (SNe), X-ray Flashes(XRFs), Ultra Long (UL), and Long (lGRBs). sGRBs have T ≤
2s (Kouveliotou et al. 1993). Someof these sGRBs also have an extended emission hereafter called the SEE category (Norris & Bonnell2006; Levan et al. 2007; Norris et al. 2010). The GRB-SNe is a class of GRBs that has an associatedsupernova which has been clearly detected. XRFs are peculiar GRBs with an X-ray fluence (2 - 30keV) > γ - ray fluence (30 - 400 keV). GRBs with T ≥ T > s that do notfall into any of the other classes are categorized as purely lGRBs, with the aims of creating a singleobservationally homogeneous class. METHODOLOGYWe fit the
Swift
LCs with the W07 model, which is able to reveal the presence of a plateau and isversatile enough to fit both the prompt and afterglow LCs. Following the phenomenological W07model, each function f i ( t ) can be written as: f i ( t ) = F i exp (cid:18) α i (cid:18) − tT i (cid:19)(cid:19) exp (cid:16) − τ i t (cid:17) for t < T i F i (cid:18) tT i (cid:19) − α i exp (cid:16) − τ i t (cid:17) for t ≥ T i , (1)which contains four parameters for the prompt emission ( T p , F p , α p , τ p ), and four parameters for theafterglow emission ( T a , F a , α a , τ a ), α i is the late time PL decay index, τ i is the initial rise timescale and T i gives the plateau duration, which is well defined when T i (cid:29) τ i . The subscript i = ( p, a ) denoteseither the prompt emission, p , or the afterglow emission, a . When there is a paucity of data we set τ i = 0 or T i = T p , where T p is the end of the prompt emission.Regarding the plateau in X-rays, we compute the rest-frame luminosity, L a , in units of erg s − atthe end of the afterglow plateau as follows: L a = 4 πD L ( z ) F a ( E min , E max , T a ) ∗ K , (2)where D L is the distance luminosity calculated assuming a flat ΛCDM cosmological model withΩ m = 0 . H = 0 . − Mpc − ; F a is the flux given at the end of the plateau determined bythe fitting according to the W07 model; ( E min , E max ) is the given energy band; and K is the correction http://heasarc.nasa.gov/xanadu/xspec/manual/XspecSpectralFitting.html factor that accounts for cosmic expansion, which is given by following Bloom et al. (2001): K = (cid:82) / (1+ z )0 . / (1+ z ) Φ( E ) dE (cid:82) . Φ( E ) dE , (3)where Φ( E ) is the functional form for the spectrum, which, in our case is either a simple PL or a CPL.Given the definition of the K -correction, the energy band depends on the redshift of the GRB. Sincethe energy band of XRT is narrow, we also have added the calculation of the bolometric luminosityfor which the K -correction is usually computed over the total energy range between 1 and 10 keV,see Schaefer (2007): K = (cid:82) / (1+ z )1 / (1+ z ) Φ( E ) dE (cid:82) Φ( E ) dE . (4)A comparison of the results with the bolometric and the K-corrected luminosities in the XRT energyrange is given in § T a until the end of the LC, T end . This analysis has been performed making use of the Swift
BAT+XRTonline repository (Evans et al. 2009). In our calculations, the photon index is calculated by firsttaking values from the windowed timing mode and photon counting mode of the XRT. We thenaverage the two values and propagate their errors. However, if one of the indices does not exist,has larger error bars than the index itself, or has an extremely high value (a photon index greaterthan 6), we do not consider that index for the analysis. This way of computing the photon indicesleads the majority of them to be consistent within 1 σ . This differs from the calculation of spectralparameters from Dainotti et al. (2016) and Dainotti et al. (2017b), where only the photon countingmode was taken into account. The spectral index β , which is used in calculating the closure relationsdetailed in §
4, is the photon index subtracted by one.The peak prompt luminosity L peak , also in units of erg s − , is computed over a 1 s time interval,with the exception of GRBs 150821A and 170405A, whose luminosities are computed over a fulltime-averaged interval due to a lack of a 1 s peak flux. L peak is computed as: L peak = 4 πD L ( z ) F peak ( E min , E max , T X ) ∗ K , (5)where F peak is the measured energy flux over the 1 s interval (erg cm − s − ). Dainotti et al. (2017b)considered GRBs whose spectrum computed at 1 s have a smaller χ value for a simple PL fit ratherthan a CPL. The difference between the two calculations lies in their K corrections, due to thedifference in the functional forms of the spectrum. Here, we instead always use a CPL fit whennecessary parameters are available, since the criterion presented in Sakamoto et al. (2011) states thatif the ∆ χ difference between the PL and CPL fittings is less than 6, the fitting results are equivalent.In total, there are 65 Swift GRBs with known redshifts fitted with a CPL, and 157 with a PL. Tocheck if this new procedure leads to an increase in the errors of luminosities, we also check the scatterof the fundamental plane relation, and find that it slightly increases (more details are presented in § The definition of the “Gold” and “Gold 2” samples
The “Gold 2” class is defined here for the first time in literature, and differs from the “Gold” inDainotti et al. (2016, 2017b). The “Gold” must have at least five points in the beginning of theplateau and a plateau angle of < ◦ , while the “Gold 2” considers LCs that have at least one datapoint in the beginning of the plateau and a plateau angle of < ◦ , allowing for a larger sample size.We here stress that the identification of both the “Gold” and “Gold 2” samples rely on a robustphenomenological analysis on the number of data points shown in the beginning of the plateau. The“Gold” sample classes are created from lGRBs and on the analysis of the LCs, in order to check ifthe correlations found considering these different samples are more robust using a phenomenologicalapproach based on the LCs rather than an analysis based on type-specific classes. We show explicitlyin § THE CLOSURE RELATIONSThe closure relations are relationships between the temporal and spectral indices, respectively ( α and β ), that test the effectiveness of the ES fireball model (Cavallo & Rees 1978; Goodman 1986;Paczynski 1986), assuming that synchrotron radiation is the dominant mechanism in the afterglow.The parameter α is taken from the W07 fitting for the afterglow detailed in §
3, while the parameter β is the spectral index (details on its calculation are presented in § I − IV ), and maystem from the sporadic emission from the central engine (Burrows et al. 2005a; Zhang et al. 2006a;Falcone et al. 2007). In our analysis we only consider phase III for the LCs that present plateauemission. We also include LCs with flaring activities, since we manually remove flares in our fits,thereby not influencing the determination of the α or β parameters.The closure relations assume that breaks in the LCs are sharp, when in reality they are oftensmooth. However, this simplified assumption is necessary because smooth spectral breaks are verydifficult to measure. They are derived through assuming that F ν ∝ t − α ν − β , where F ν is the fluxseen at a particular frequency (Sari et al. 1998; Granot & Sari 2002), and both α and β are relatedto the spectral index p of the electron distribution, varying depending on the different astrophysicalenvironments GRBs originate from. The electron spectral index is related to the spectral index β used in the closure relations through β = ( p − / ν m < ν < ν c , and β = p/ ν m , ν c ) < ν where ν m and ν c are the characteristic and the cooling spectral breaks of the synchrotron emissionfrequencies (Sari et al. 1998; Zhang & M´esz´aros 2004; Zhang et al. 2006b; Racusin et al. 2009). Inthe ES model, the electron distribution is usually described through a PL function dn e dγ e ∝ γ − pe , where p is the spectral index and γ e is the electron Lorentz factor, which must be greater than γ m . Theterm is represented as γ m (cid:39) (cid:15) e γ , (6)and indicates the minimum Lorentz factor needed to create a PL distribution of electrons (Sari et al.1998). In this paper, we specifically investigate a subset of cases given in Racusin et al. (2009). Itis worth noting that differently from Racusin et al. (2009), we do not consider jet models in the ES,since these relations are only relevant when the post-jet-break phase occurs (Phase IV). Indeed, inorder to apply such models, the segment should follow a “normal decay phase” as defined by Zhanget al. (2006b). The transition from the normal decay phase (Phase III) to the post-jet-break phase(Phase IV) is where the jet break occurs. In rare cases, a plateau phase (Phase II) may be followedimmediately by Phase IV. In this scenario, one must invoke that the energy injection ending timecoincides with the jet break time. We also do not consider energy injection into the ES for therelations that we test, and focus on determining the GRB during Phase III of its LC, and whether itis either in a constant-density ISM or wind environment with slow or fast cooling during this phase.We here stress that though we test the closure relations after the end of the plateau emission (PhaseIII), we are aware of the importance of testing the closure relations for the plateau phase (Phase II)as done by L¨u & Zhang (2014) for lGRBs, by L¨u et al. (2015) for sGRBs, and by Wang et al. (2015)for both groups. However, this analysis is out of the scope of the current paper, and we plan toperform an analogous analysis of the closure relations using Phase II of our LCs in a forthcomingpaper.Below we describe the environments and regimes we test:1. Early GRB models that assumed the relativistic ejecta from the blast wave expanding into aconstant-density ( ∝ R ) ISM (Sari et al. 1998) were highly compatible with LC observationsup to the early 2000s. However, it was determined afterwards that some GRBs have massivestar progenitors, and may be the result of core collapse supernovae, which ascertains that therelativistic ejecta expands into the stellar wind environment ( ∝ R − ) of the progenitor source(Chevalier & Li 2000).2. In both these environments, the cooling regime can be either fast or slow. The calculations ofthe closure relations depend on the particle energy distribution, in particular, whether electronshave gone through significant cooling (Sari et al. 1998). Electrons are significantly cooled viasynchrotron radiation if they have a Lorentz factor γ e > γ c , where γ c is the critical Lorentzfactor where synchrotron cooling becomes significant. Therefore, two different regimes exist inrelation to γ m . When γ m > γ c , all electrons in the shocked ejecta will be able to cool downto γ c . This regime is called the fast cooling regime. In contrast, if γ m < γ c , then only someelectrons whose γ e > γ c will be able to cool. This leaves the electrons in the region where γ m < γ e < γ c unaffected. This region of the spectrum is where the majority of electrons in theshock lies, and they therefore will not be able to cool in a given t , leading to the slow coolingregime (Sari et al. 1998).In order to answer question 1) from § ν ) range (see Table 1), through recreating the top panels ofFigure 3 in Uhm & Zhang (2014), which are theoretical models of the afterglow spectra assumingthe ES model (without taking into account curvature effects or an energy injection mechanism), atdifferent observational times and cooling regimes. We convert the flux from units of mJy in Figure 3in Uhm & Zhang (2014) to ergs cm − s − . We check the regime during which F a (derived from ourfitting procedures) is found, and then categorize every GRB into either ν m < ν < ν c or ν > ν c forslow cooling, or ν c < ν < ν m or ν > ν m for fast cooling. Then, through using the relations between p and β depending on the regime we are in, we determine whether p > < p < ν and p range for every GRB in oursample, through plotting the α and β parameters along with their error bars at a 1 σ level, as well asthe equations of the closure relations. We group together relations characterized by the same p range,astrophysical environment, and cooling regime on the same plot to create the so-called “gray-region”,a zone between two relations within the same environment. GRBs that lie between the two linesshould be regarded as consistent cases, see Figure 2, and 3, along with Tables 2 and 3. In theseFigures, we detail the specific closure relationships calculated for the time range T a to T end in theafterglow, along with the error bars and the lines corresponding to the closure relation equations.We only consider GRBs with δ x /x < x indicates either α or β and δ x indicates the errormeasurement. In addition, we also discard GRBs for which the errors on the closure relations aregreater than the values of the closure relations themselves. Furthermore, histograms detailing thedistribution of the α and β parameters are presented in Figure 4. Table 1.
Closure relations (Part of the table is taken from Racusin et al. (2009))
No Energy Injection ν range β ( p ) α ( β ) α ( β )( p >
2) (1 < p < ν m < ν < ν c p − α = β α = β +3)16 ν > ν c p α = β − α = β +58 ISM, Fast Cooling ν c < ν < ν m α = β α = β ν > ν m p α = β − α = β +58 Wind, Slow Cooling ν m < ν < ν c p − α = β +12 α = β +98 ν > ν c p α = β − α = β +34 Wind, Fast Cooling ν c < ν < ν m α = − β α = − β ν > ν m p α = β − α = β +34 β α β / ( β - )/ β α ( ( β + ))/ ( β + )/ β α β / ( β - )/ β α β / ( β + )/ β α ( β + )/ ( β - )/ β α ( β + )/ ( β + )/ β α ( - β )/ ( β - )/ β α ( - β )/ ( β + )/ Figure 2.
Closure relations extracted from the ES for the GRBs with known redshifts from T a to T end .GRB sets are color-coded (blue or red) according to their frequency range given in Table 1 and the closurerelations. The equality line corresponds to the two different relations and are color-coded in the same color.Relations corresponding to the same environment and electron spectral index ( p ) range are grouped together. β α β / ( β - )/ β α ( ( β + ))/ ( β + )/ β α β / ( β - )/ β α β / ( β + )/ β α ( β + )/ ( β - )/ β α ( β + )/ ( β + )/ β α ( - β )/ ( β - )/ β α ( - β )/ ( β + )/ Figure 3.
Closure relations extracted from the ES for the GRBs with unknown redshifts with the samescheme as Figure 2. ν Range p Range Closure Relation GRBs Total GRBs Satisfying Relation Percentage ν m < ν < ν c p > β/ ν > ν c p > β − / ν m < ν < ν c < p < β + 3)) /
16 3 1 33.3% ν > ν c < p < β + 5) / ν c < ν < ν m p > β/ ν > ν m p > β − / ν c < ν < ν m < p < β/ ν > ν m < p < β + 5) / ν m < ν < ν c p > β + 1) / ν > ν c p > β − / ν m < ν < ν c < p < β + 9) / ν > ν c < p < β + 3) / ν c < ν < ν m p > − β ) / ν > ν m p > β − / ν c < ν < ν m < p < − β ) / ν > ν m < p < β + 3) / Table 2.
Closure relations for the 222 redshift GRBs.. ν Range p Range Closure Relation GRBs Total GRBs Satisfying Relation Percentage ν m < ν < ν c p > β/ ν > ν c p > β − / ν m < ν < ν c < p < β + 3)) /
16 4 2 50.0% ν > ν c < p < β + 5) / ν c < ν < ν m p > β/ ν > ν m p > β − / ν c < ν < ν m < p < β/ ν > ν m < p < β + 5) / ν m < ν < ν c p > β + 1) / ν > ν c p > β − / ν m < ν < ν c < p < β + 9) / ν > ν c < p < β + 3) / ν c < ν < ν m p > − β ) / ν > ν m p > β − / ν c < ν < ν m < p < − β ) / ν > ν m < p < β + 3) / Table 3.
Closure relations for the 233 GRBs with unknown redshifts. α N (a) α for GRBs with known redshifts α N (b) α for GRBs with unknown redshifts β N (c) β for GRBs with known redshifts from T a to T end β N (d) β for GRBs with unknown redshifts from T a to T end Figure 4.
Histograms of α and β parameters. INTERPRETATION OF THE CLOSURE RELATIONSThrough our analysis, we see that overall, the ES model is quite successful in modeling the afterglowsof GRBs (see Table 2 and Table 3). Wang et al. (2015) tested the ES model with a set of closurerelations by using a sample of 85 GRBs with both X-ray and optical afterglow data, and concludedthat the ES model is able to account for at least half of GRB afterglows. Through our analysis,we see that even with a greater number of GRB X-ray LCs used (455), this result still holds. Ourresults are discrepant with that of Willingale et al. (2007), who through an analysis of 107 SwiftLCs, reached to the conclusion that the ES model works for less than 50% of Swift GRBs. Thisdifference is due to the fact that we took into account the “gray-region”, differently from Willingaleet al. (2007). Furthermore, through their analysis of 318 Swift LCs, Evans et al. (2009) came to theconclusion that closure relations corresponding to the ES model without energy injection are fulfilledby a reasonable amount of GRBs, though there are also a number of GRBs for which energy injectionmechanisms are needed. If there is a source of continuous energy injection, then the forward shockwill continue to be “refreshed” with continuous bursts of energy such that the fireball decelerates ata slower timescale than it would in the normal ES scenario (Zhang et al. 2006a). A previous analysisin relation to this energy injection mechanism and its relation to the 2D Dainotti relation has beeninvestigated in Del Vecchio et al. (2016).We can interpret the results of the closure relations within these main scenarios:(i) Out of the 16 closure relation groups for the standard fireball model tested for both the knownand unknown redshift data sets, 11 have at least 50% of their GRBs fulfilled. The closurerelations are a quick check to assess the reliability of the ES scenario, thus it is possible thatfor cases where the closure relations are not fulfilled, more complex physical processes must betaken into account.(ii) The W07 model is an empirical model for the plateau and afterglow emission, but in its currentformulation, we have removed flares manually. Thus, we cannot control how much this manualremoval of flares influences results for cases where they are present in the plateau emission.(iii) According to Table 2 and Table 3, the most favored closure relation set for the known redshiftdata set is (2 β + 9) / β + 3) /
4, which corresponds to a wind environment with slowcooling and 1 < p <
2, with all possible GRBs in the correct ν and p range fulfilling theserelations. However, there are only three GRBs within these ranges. Another set of note thathas a high fulfillment rate (82.7%) in the known redshift data set is (3 β + 1) / β − / < p <
2. The mostfavored sets for the unknown redshift data set are β/ β − /
2, as well as (1 − β ) / β − /
2, which correspond to a constant-density ISM or wind environment with fastcooling respectively, also both with p >
2. The fulfillment rate of both these sets is 90.9%.However, it is important to note that these percentages are relative to the number of GRBsthat satisfy the ν ranges for each cooling regime. As seen in Table 2 and Table 3 there areless GRBs that satisfy the fast cooling regime than do the slow cooling, which has an effect onthese percentages.With regards to the actual astrophysical environments and cooling regimes of the GRBs in oursample with known redshifts, the most fulfilled environments are a wind and constant-densityISM environment with slow cooling. There are a total of 127 GRBs that satisfy a closure5relation indicating a wind environment with slow cooling, and 110 that indicate a constant-density ISM environment with slow cooling. The environments that are fulfilled the leastare a wind and constant-density ISM environment with fast cooling, with 32 and 31 GRBsfulfilling those environments, respectively. Furthermore, regarding the sample of GRBs withunknown redshifts, the most fulfilled environments again are either a wind or constant-densityISM environment with slow cooling. There are 117 GRBs that indicate a wind environmentwith slow cooling, and 110 pointing toward a constant-density ISM environment with slowcooling. Again, the environments that are fulfilled the least are a wind and constant-densityISM environment with fast cooling, with 43 GRBs fulfilling both groups respectively. Therefore,we can conclude that a constant-density ISM or wind environment with slow cooling is the mostlikely scenario for our sample of GRBs during Phase III of their LCs, and that the fast coolingregime is disfavored.This interpretation (i-iii) constitutes the answer to question 1) in §
1. The ES model seems to bea good explanation of the high-energy LCs presenting a plateau emission. However, we are open toexploring new possibilities which allow us to take into account cases that do not follow the ES model.These cases include more complex physical processes such as nonlinear particle acceleration, or anenergy injection mechanism such as the one obtained with a magnetar or mass accretion onto a blackhole.Given that it is necessary to explain the minority of GRBs for which standard closure relations arenot a viable explanation, more complex evolution of afterglows, alternative models, and correspondingclosure relations have been so far investigated (M´esz´aros 1998; Sari et al. 1998; Chevalier & Li 2000;Dai & Cheng 2001; Zhang & M´esz´aros 2004; Zhang et al. 2006b). Furthermore, in our analysis, weconsider linear particle acceleration, although the nonlinear particle acceleration scenario (Warrenet al. 2017) cannot be ruled out. Warren et al. (2017) studied the time evolution of LCs of afterglowsby taking into account the effects of non-linear particle acceleration for the first time. They foundthat the temporal and spectral evolution is much different from the simplistic formulation of theafterglow model mentioned above. They also showed that very high energy γ -rays can be produced bysynchrotron self-Compton emission, especially at the early phase of the afterglow (Zhang & M´esz´aros2001). Analyzing more data can help us shed more light on these results in the near future.In Section 6, we connect the astrophysical environments of GRBs which accounts for phase IIIof the LCs involving the decay phase after the plateau emission, α , and the 3D fundamental planerelation, which considers T a and L a at the end of phase II of the LCs. FUNDAMENTAL PLANE CORRELATIONUsing the classifications from §
3, we update the 3D relation (log( T a ), log( L peak ), log( L a )), with aplane fitted to the data for all 222 GRBs with redshift. We see the 3D relation still holds with theupdated data set. The equation of the plane is given by:log L a = C o + a log T a + b log L peak , (7)where C o = C ( θ, φ, σ int ) + z , represents the normalization of the plane with respect to θ and φ , aswell as σ int , the intrinsic scatter of the sample. z o is a normalization parameter and C is the co-variance function. Furthermore, a and b are both functions of the variables θ and φ , where a ( θ, φ ) = − cos( φ ) tan( θ ) and b ( θ, φ ) = − sin( φ ) tan( θ ). Our best-fit plane to the data has C o = 8 . ± . a = − . ± .
06, and b = 0 . ± .
05 with σ = 0 . ± . (a) Edge-on View (b) Plane View Figure 5.
222 GRBs in the log( L a ) − log( T a ) − log( L peak ) space with a plane fitted to the data withthe following classifications: GRB-SNe (black cones), XRFs (blue spheres), SEE (cuboids), lGRBs (blackcircles), UL GRBs (green icosahedrons). The same color coding, but darker colors indicate data points abovethe plane, while lighter colors indicate GRBs below the plane, except for UL GRBs which are all denotedby bright green truncated icosahedrons. We show in Figure 5 a 3D projection of the relation with the classes of GRBs presented in differentshapes and colors: GRB-SNe (black cones), XRFs (blue spheres), SEE (red cuboids), lGRBs (blackcircles), and UL GRBs (truncated icosahedron). Darker colors indicate GRBs above the plane, whilelighter colors show those below the plane, except for UL GRBs which are all the same shade of green.We test the “Gold” class using both the old definition from Dainotti et al. (2016, 2017b) alongwith our new definition for the “Gold 2” class in order to check if the correlations we derive are morerobust than seen in previous literature. There are a total of 69 GRBs in the “Gold” sample and100 GRBs in the “Gold 2” sample, in comparison to 45 GRBs in the “Gold” set in Dainotti et al.(2017b). Using the data points for the “Gold 2” and the best-fit plane, we are able to derive the R adj correlation coefficient for the sample. The “Gold 2” has a R adj = 0 .
73, which is 9.9% lower than thatof the “Gold” in Dainotti et al. (2017b) ( R adj = 0 . σ = 0 . ± . σ = 0 . ± .
04. The current “Gold” sample inthis analysis has σ = 0 . ± .
04, making it compatible in 1 σ with “Gold 2”, where the analysishas been done using the K -correction computation in the different rest-frame bands of GRBs. Itis important to note that the analysis of “Gold 2” did not increase the σ , while also increasing thesample size by 45%. When we redo the analysis using the bolometric luminosities instead, we obtain σ = 0 . ± .
04 for the “Gold” sample and σ = 0 . ± .
04 for “Gold 2”. This analysis clearlyshows that regardless of the different K -corrections, the fundamental plane relation results are the7same within 1 σ , strengthening our findings toward the use of this relation as a standard candle. Thecontour plots of the best-fit parameters generated through the D’Agostini (1995) method is shownin Figure 6. Figure 6.
Figure showing the contour plots generated by the D’agostini statistical method for the the best-fit parameters and intrinsic scatter for the “Gold” and “Gold 2” classes (using the bolometric K -correction). Furthermore, in order to determine whether the σ we obtain is truly characteristic of the samplesrather than due to by chance, we draw a random sample of 69 and 100 GRBs, respectively, corre-sponding to the “Gold” and “Gold 2” classes, out of the total sample of 222 GRBs. We then calculatethe best-fit parameters and intrinsic scatter of these samples, and bootstrap the sample 10,000 times,creating histograms representing the distribution of σ from the random samples. We see that outof the 10,000 samples, 296 have a σ < .
40– therefore the probability that we randomly obtain a σ < .
40 corresponding to the “Gold” class is 3.0%. Similarly, out of the 10,000 samples for the“Gold 2” class, 450 have a σ < .
41, corresponding to a probability of 4.9% of achieving this scatterat random. Because both classes have probabilities of less than 5% of obtaining their respectiveintrinsic scatters randomly, it further supports the robustness of our correlations. Histograms of thedistributions of the samples are detailed in Figure 7.8
Figure 7.
Histograms detailing the random σ distributions taken for a set of 69 GRBs (left panel) and 100GRBs (right panel) 10,000 times. The red bins mark those that are ≤ σ obtained through analyzing the“Gold” and “Gold 2” classes. The Fundamental Planes According to the Closure Relations
We also group GRBs in terms of their astrophysical environments in relation to Table 1, and plot thesame 3D relation (log( T a ), log( L peak ), log( L a )) to investigate whether grouping GRBs in accordanceto their astrophysical environments rather than their classes results in more tighter correlations. Wegroup GRBs with respect to their astrophysical environments taken from Table 1, as well as groupingtogether GRBs that follow a constant-density ISM and wind environment regardless of their coolingregime. The plots of the respective planes are shown in Figure 9. The R adj correlation coefficientswith respect to the planes, the best-fit parameters, and intrinsic scatters are presented in Table 4,along with the contour plots in Figure 10.With the exception of the two fast cooling groups, every other group’s σ is consistent with the onesof the “Gold” samples at the 1 σ level. This is a revealing discovery, since the procedure to createthe groups differs enormously. The groups pertaining to the astrophysical environments are derivedthrough checking if they fulfill the theoretical closure relations corresponding to the ES model inPhase III of the LCs, whereas the “Gold” groups were extracted phenomenologically through fittingthe LCs. Furthermore, the GRBs in the “Gold” groups all display similar intrinsic physical processes,since they are subsamples of lGRBs, whereas GRBs originating from the same environment may notnecessarily all have similar intrinsic physical processes. This finding shows that GRBs grouped intotheir astrophysical environments should continue to be pursued as possible standard candles, as thereare intrinsic consistencies in 1 σ of observable parameters within some of their particular subsamples.When looking at the ISM fast cooling and wind fast cooling groups, we obtain a σ lower thanpreviously seen in literature, including the “Gold” class in Dainotti et al. (2017b), of σ = 0 .
29. Inorder to verify that these results are not drawn by chance, we again check the probability of obtainingsuch a σ or lower by using the same bootstrapping method that we use previously for the “Gold”classes. Through this, we determine that the probability of randomly obtaining a σ < .
29 is 1.76%for the ISM fast cooling sample size, and 1.35% for the wind fast cooling sample. The histogramsfor the distributions are presented in Figure 8. Thus, the σ we obtain is indeed robust, leading usto conclude that the fast cooling groups hold the highest potential to eventually be used as standardcandles.9 Sample a b c σ R adj N All ISM -0.73 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 4.
Table Indicating the Best-Fit Parameters, Intrinsic Scatter, and Number of GRBs with RedshiftSatisfying at Least One Relation from Table 1 and the “Gold 2” Class.
Figure 8.
Histograms detailing the random σ distributions taken for a set of 31 GRBs (left panel) and 32GRBs (right panel) 10,000 times. The red bins mark those that are ≤ σ obtained through analyzing theISM fast cooling and ISM slow cooling classes. (a) 125 GRBs fulfill all ISM environments (b) 142 GRBs fulfill all wind environments(c) 110 GRBs fulfill ISM slow cooling (d) 31 GRBs fulfill ISM fast cooling(e) 127 GRBs fulfill wind slow cooling (f) 32 GRBs fulfill wind fast cooling Figure 9.
The fundamental planes according to astrophysical environments with same classifications asFigure 5. (a) All ISM environments (b) All wind environments(c) ISM slow cooling environments (d) ISM fast cooling environments(e) Wind slow cooling environments (f) Wind fast cooling environments Figure 10.
Contour plots generated by the D’agostini statistical method of the best-fit parameters and σ scatter of closure relation groups from Table 1. SUMMARY AND CONCLUSIONSIn summary, we test whether a set of closure relations corresponding to two distinct astrophysicalenvironments and cooling regimes are fulfilled for 455 Swift X-ray LCs from 2005 January to 2019August that show a plateau. We employ the closure relations as a quick test on the reliability of theES emission, and also to infer information about their astrophysical environments. We also confirmthe existence of an updated 3D fundamental plane relation between the rest-frame time at the endof the plateau emission, log T a , the prompt peak luminosity, log L peak , and luminosity at the endof the plateau emission, log L a , with two additional years of Swift observations. We introduce anew definition of the “Gold” class, called the “Gold 2” and compare it to the “Gold” mentionedin previous papers. Finally, we analyze the 3D fundamental plane for a set of GRBs that fulfillthe closure relations and thus reveal peculiar astrophysical environments. In conclusion, we haveanswered the main queries stated in § Swift
GRBs. We find that the “Gold 2” classcomposed of 100 GRBs has a smaller R adj = 0 .
73 and a larger intrinsic scatter σ = 0 . . σ for both the “Gold” and “Gold2” classes are equivalent within 1 σ of Dainotti et al. (2016), Dainotti et al. (2017b), showingthat regardless of the K -correction used, the correlations are compatible within 1 σ .3. We compute the 3D fundamental plane relation with respect to the astrophysical environmentsand cooling regimes of the GRBs in our data set, and see that the majority of the groups havea consistent intrinsic scatter ( σ ) with one another, as well as with the “Gold” classes. Since themethods we used to determine these groups varies significantly (a theoretical approach for theastrophysical environments and a phenomenological approach for the “Gold” classes), this alsostrengthens the argument for pursuing GRBs grouped into their astrophysical environments andcooling regimes as possible standard candles. Furthermore, the two groups of GRBs that satisfythe fast cooling CRs have the lowest σ obtained thus far in literature, further strengtheningthis argument above. Further investigation of these groups will be the topic of further papers,as they have the highest potential to eventually be used as standard candles.3 ACKNOWLEDGEMENTSG.S. is grateful for the support of the United States Department of Energy in funding the ScienceUndergraduate Laboratory Internship (SULI) program. M.G. Dainotti is grateful to MINIATURA2,grant No. 2018/02/X/ST9/03673 and the American Astronomical Society Chretienne Fellowship.N.F. acknowledges the support from UNAM-DGAPA-PAPIT through grant IA102019. X.H. ac-knowledges support from DGAPAUNAM PAPIIT IN104517 and CONACyT. S.N. acknowledges the”JSPS Grant-in-Aid for Scientific Research “KAKENHI” (A) with grant No. JP19H00693, the “Pio-neering Program of RIKEN for Evolution of Matter in the universe (r-EMU)”, and “InterdisciplinaryTheoretical and Mathematical Sciences Program of RIKEN (iTHEMS)”. The authors are grateful forthe help of Ray Wynne and Zooey Ngyuen, undergraduate students at the Massachusetts Instituteof Technology and University of California Los Angeles, for their help in the fitting of the lightcurves.Authors are particularly grateful to Giuseppe Sarracino and Stefano Savastano for the help in writingthe python code for deriving the best-fit parameters of the fundamental plane relation. This workmade use of data supplied by the UK Swift Science Data Centre at the University of Leicester.REFERENCES
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