Unveiling the origin of steep decay in γ -ray bursts

γ -ray bursts (GRBs) are short-lived transients releasing a large amount of energy ( 10 51 − 5 10 53 erg ) in the keV-MeV energy range. GRBs are thought to originate from internal dissi-6 pation of the energy carried by ultra-relativistic jets launched by the remnant of a massive 7 star’s death or a compact binary coalescence. While thousands of GRBs have been observed 8 over the last thirty years, we still have an incomplete understanding of where and how the 9 radiation is generated in the jet. A novel investigation of the GRB emission mechanism, via 10 time-resolved spectral analysis of the X-ray tails of bright GRB pulses, enables us to discover 11 a unique relation between the spectral index and the ﬂux. This relation is incompatible with 12 the long standing scenario invoked to interpret X-ray tails, that is, the delayed arrival of pho-13 tons from high-latitude parts of the jet. We show that our results provide for the ﬁrst time 14 evidence of adiabatic cooling and efﬁcient energy exchange between the emitting particles in 15 the relativistic outﬂows of GRBs. 16


Introduction
The prompt emission of γ-ray bursts (GRBs) is typically followed by a steep decay phase 1 (tail) in the X-ray band.The duration of the steep decay is around 10 2 − 10 3 s and it is characterized by a typical temporal slope of 3 − 5. Since afterglow models cannot account for such slopes, the origin of steep decay is related to the fade-off of the emission mechanism that generates the prompt phase.In order to investigate the spectral evolution during this phase, we select a sample of GRBs from the archive of the X-ray Telescope (XRT, 0.3-10 keV) on-board the Neil Gehrels Swift Observatory (Swift) 2 .We restrict our study to a sample of GRBs (8 in total) whose brightest pulse in the Burst Alert Telescope (BAT, 15-350 keV) corresponds to the XRT peak preceding the X-ray tail (see as example the left panel in Fig. 1).We perform a time-resolved spectral analysis of the tail in the 0.5-10 keV band assuming a simple power-law model for the photon spectrum N γ ∝ E −α (see Methods).We introduce a novel method for the representation of the spectral evolution plotting the photon index α as a function of the flux F integrated in the 0.5-10 keV band, 1 Gran Sasso Science Institute, Viale F. Crispi 7, I-67100, L'Aquila (AQ), Italy 2 INFN -Laboratori Nazionali del Gran Sasso, I-67100, L'Aquila (AQ), Italy hereafter referred to as the α − F relation.The flux is normalized to the peak value of the X-ray tail.This normalization makes the result independent of the intrinsic brightness of the pulse and of the distance of the GRB.

Results
We discover a unique α − F relation for the analyzed GRBs as shown in the right panel of Fig. 1.This is consistent with a systematic softening of the spectrum (as already observed for several GRBs 3 ); the photon index evolves from a value of α ∼ 0.5 − 1 at the peak of the XRT pulse to α ∼ 2 − 2.5 at the end of the tail emission, while the flux drops by two orders of magnitude.
The initial and final photon indices are consistent with the typical low-and high-energy values found from the analysis of the prompt emission spectrum of GRBs, namely ∼ 1 and ∼ 2.3 4,5,6 , respectively.The α − F relation can be interpreted as being due to a spectral evolution in which the spectral shape does not vary in time, but the whole spectrum is gradually shifted towards lower energies while becoming progressively dimmer (see Fig. 2).The consistent spectral evolution discovered in our analysis is a clear indication of a universal physical mechanism responsible for the tail emission of GRBs and the corresponding spectral softening.

Testing High Latitude Emission
We first compare our results with the expectations from the high-latitude emission (HLE) 7,8,9,10 , which is the widely adopted model for interpreting the X-ray tails of GRBs.When the emission from a curved surface is switched off, an observer receives photons from increasing latitudes with respect to the line of sight.The higher the latitude, the lower the Doppler factor, resulting in a shift towards lower energies of the spectrum in the observer frame.Through an accurate modeling of HLE (as described in Methods) we derive the predicted α − F relation.We first consider a smoothly broken power-law (SBPL) comoving spectrum.Regardless of the choice of the peak energy, the bulk Lorentz factor or the radius of the emitting surface, the HLE predicts an α − F relation whose rise is shallower than the observed one (right panel of Fig. S1 in Supplementary materials).We additionally test the Band function, commonly adopted for GRB spectra 11 , and the physically motivated synchrotron spectrum 12 , obtaining similar results (Fig. S2 and left panel of Fig. S3 in Supplementary Materials): the HLE softening is too slow to account for the observed α − F relation.We further relax the assumption of an infinitesimal duration pulse, i.e. considering a shell that is continuously emitting during its expansion and suddenly switches off at radius R 0 13 (see Methods).The contributions from regions R < R 0 are sub-dominant with respect to the emission coming from the last emitting surface at R = R 0 , resulting in a spectral evolution still incompatible with the observations (Fig. S4).An interesting alternative is the HLE emission from an accelerating region 14 taking place in some Poynting flux dissipation scenarios 15 .Even though it can explain the temporal slopes observed in the X-ray tails, also this scenario fails in reproducing the α − F relation (see Fig. S5 in Supplementary materials).Our results on HLE are based on the assumption of a common comoving spectrum along the entire jet core.Even changing the curvature (or sharpness) of the spectrum or assuming a latitude dependence of the spectral shape, the disagreement with the data remains, unless we adopt a very fine-tuned structure of the spectrum along the jet core, which is not physically motivated (see Methods).Alternative models, such as anisotropic jet core or sub-photopheric dissipation, can hardly reproduce our results (see Methods).

Adiabatic cooling
Since the standard HLE and its modified versions, as well as alternative scenarios, are not able to robustly interpret the observed α − F relation, we consider a mechanism based on an intrinsic evolution of the comoving spectrum.The most natural process is the adiabatic cooling of the emitting particles 16 .Here we assume conservation of the entropy of the emitting system γ 3 V ′ throughout its dynamical evolution, where γ is the average random Lorentz factor of the emitting particles and V ′ ∝ R 2 ∆R ′ the comoving volume 17 .We consider both thick and thin emitting regions, i.e. a comoving thickness of the emitting shell ∆R ′ = const or ∆R ′ ∝ R, respectively.We assume a power law radial decay of the magnetic field B = B 0 (R/R 0 ) −λ , with λ > 0, and synchrotron radiation as the dominant emission mechanism.Here R 0 is the radius at which adiabatic cooling starts to dominate the evolution of the emitting particles.We compute the observed emission taking also into account the effect of HLE by integrating the comoving intensity along the equal arrival time surfaces (see Methods).In this scenario, contrary to HLE alone, the emission from the jet is not switched off suddenly, but the drop in flux and the spectral evolution are produced by a gradual fading and softening of the source, driven by adiabatic cooling of particles.The resulting spectral evolution and light curves are shown in Fig. 3.
Adiabatic cooling produces a much faster softening of α as a function of the flux decay, with respect to HLE alone, in agreement with the data.Adopting an initial peak frequency ν p = 100 keV, the α − F relation is well reproduced for values 0.2 λ 0.8, which are smaller than those expected in an emitting region with a transverse magnetic field (λ = 1 or λ = 2 for a thick or a thin shell, respectively) or magnetic field in pressure equilibrium with the emitting particles (λ = 4/3 or λ = 2 for a thick or a thin shell, respectively 17 ).Decreasing the observed initial peak frequency, the curves become steeper especially in the initial part of the decay.On the other hand, assuming a different evolution of the shell thickness, the behavior of the curves changes only marginally.For large values of λ the evolution of α flattens in the late part of the decay (see Fig. 3), indicating that the spectral evolution becomes dominated by the emission at larger angles, rather than by adiabatic cooling in the jet core.For the same values of λ, adiabatic cooling can also well reproduce the light curve of X-ray tails (right Panel of Fig. 3).For comparison, in the same plot we show the light curve given by pure HLE, adopting the same value of R 0 and Γ.The typical timescale of adiabatic cooling τ ad = R 0 /2cΓ 2 , i.e. the observed time interval during which the radius doubles, is equal to the HLE timescale 7,18 and radically affects the slope of X-ray tails.Therefore, the comparison between the model and the observed light curves allows us to constrain the size R 0 of the emitting region as in HLE 19,20 .Assuming the same range of λ derived before from the α − F relation, we find values in the range 0.8 s τ ad 8 s produce a good agreement with the data and are marginally consistent with the typical duration of GRB pulses (<1 sec 21 ).This implies a size 5 × 10 14 (Γ/100) 2 cm R 0 5 × 10 15 (Γ/100) 2 cm for the emission site.A different prescription for adiabatic cooling has been suggested in the literature 16 , in which the particle's momentum gets dynamically oriented transverse to the direction of the local magnetic field.In this case, HLE is the dominant contributor to the X-ray tail emission, which is again incompatible with the observed α − F relation.From our work we conclude that the energy of the emitting particles should be necessarily coupled at the micro-physical level with the magnetic field to provide the ideal gas prescription of a constant entropy 16 .

Extending the sample
In order to further test the solidity of the α − F relation, we extend our analysis to a second sample of GRBs (composed by 8 elements) which present directly a steep decay at the beginning of the XRT light curve, instead of an X-ray pulse (see left panel of Fig. 4), often observed in early Xray afterglows 22,23 .We require that the XRT steep decay is preceded by a pulse in the BAT light curve (the brightest since its trigger time).We add the data of this second sample to the α − F plot, estimating the peak flux by the extrapolation of the XRT light curve backwards to the peak time of the BAT pulse, under the assumption that BAT peak and XRT peak were simultaneous (see Methods).We find that these GRBs follow the overall α − F relation (right panel of Fig. 4), confirming the universal nature of the physical process governing the spectral evolution of X-ray tails.Adiabatic cooling is still capable of reproducing the data of this second sample (see left panel of Fig. S1), provided we assume a slightly softer high energy intrinsic spectrum (α ∼ 3 instead of α ∼ 2.5, as used in Fig. 3).We specify that the samples considered in this work are representative of usual prompt emission phase and their limited size is related to the adopted requirements, which are necessary for an appropriate time-resolved spectral analysis.

Conclusions
The α − F relation, discovered in our analysis, requires a mechanism that produces the X-ray tails of GRBs with a unique law of flux decay and spectral softening.We find that adiabatic cooling of the emitting particles, together with a slowly decaying magnetic field, robustly reproduces this relation.Our results suggest an efficient coupling between a slowly decaying magnetic field and the emitting particles.The adiabatic cooling should dominate over radiative cooling, otherwise most of the internal energy would be radiated away before the system expands substantially.This may be an indication of intrinsically inefficient emitting particles in GRB outflows, such as protons 24 , since the electron cooling timescale τ e − in a compact and highly magnetized ejecta is catastrophically short (τ e − ≪ 1 s 25 ).Our findings are generally in agreement with moderately fast and slow cooling regimes of the synchrotron radiation, which is able to reproduce the overall GRB spectral features 26 .However, our results disfavor models with re-acceleration/slow heating of electrons 27,28 , since when electrons are left unenergized we would again observe HLE-dominated X-ray tails.In conclusion, our results show that adiabatic cooling plays a crucial role for the collective evolution of the radiating particles in GRB outflows and consequently for the determination of spectral and temporal properties of prompt emission episodes.The coupling between particles and magnetic field ensures the intrinsic nature and hence the universality of this process, whose effects are independent of the global properties of the system, such as the luminosity of the GRB or the geometry of the jet.
x e + X S e e s 7 r l 1 7 / a 8 1 r g q m i m T I 3 x e + X S e e s 7 r l 1 7 / a 8 1 r g q m i m T I 3 x e + X S e e s 7 r l 1 7 / a 8 1 r g q m i m T I 3          The theoretical curves are computed taking also into account the effect of HLE.The value of λ specifies the evolution of the magnetic field.We adopt a SBPL as spectral shape with α s = −1/3 and β s = 1.5, an initial observed peak frequency of 100 keV and a thickness of the expanding shell that is constant in time.The dot-dashed line is the evolution expected in case of HLE without adiabatic cooling, assuming the same spectral shape and initial observed peak frequency.Right panel: Temporal evolution of normalized flux expected in case of adiabatic cooling.δt obs + 100 s is the time measured from the peak of the decay shifted at 100 s, the typical starting time of the tail emission detected by XRT.We adopt the same parameters as in the left panel, assuming R 0 = 2 × 10 15 cm and Γ = 100.The dot-dashed line is the corresponding HLE model without accounting for adiabatic cooling.δ var = R 0 /2cΓ 2 indicates the timescale of adiabatic cooling, which is the same of HLE.
s s z 9 w P r 4 B 4 z C U f A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y 6 s s z 9 w P r 4 B 4 z C U f A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y 6 s s z 9 w P r 4 B 4 z C U f A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " y 6 e j T f j / W e 0 Z M x 3 q u Q P j I 9 v p T C Y r g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z Z 8 P T 7 a q n T e H q k D Z 9 T y n e j T f j / W e 0 Z M x 3 q u Q P j I 9 v p T C Y r g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z Z 8 P T 7 a q n T e H q k D Z 9 T y n e j T f j / W e 0 Z M x 3 q u Q P j I 9 v p T C Y r g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " Z Z 8 P T 7 a q n T e H q k D Z 9 T y n Figure 4: Left panel: An example of a light curve of an X-ray tail selected for our second sample, taken from GRB 150323A.We report on the same plot the XRT (orange) and the BAT (blue) flux density at 1 keV and 50 keV, respectively.The peak flux F max is estimated extrapolating the X-ray tail back to the BAT peak.Right panel: Spectral evolution of our extended sample of GRBs, which present a steep decay at the beginning of the XRT light curve, preceded by the brightest BAT pulse since the trigger time.The evolution of α lies on the same region of the plane occupied by the original sample, indicated in grey.

Methods
Sample selection.We define the steep decay (SD) segment 1,9,22,23 as the portion of the light curve that is well approximated by a power law, F ∝ t −α with α > 2. Such criterion allow us to exclude a decay coming from a forward shock 29,30,31 .In order to determine the presence of a SD, we analyze the light curve of the integrated flux in the XRT E = 0.3 − 10 keV band.
From the Swift catalog 32 as of the end of 2019, we selected all GRBs with an XRT peak flux We selected the brightest pulses in order to have a good enough spectral quality as to perform a time resolved spectral analysis.The peak flux is computed taking the maximum of F (t i ), where F (t i ) are the points of the light curve at each time t i (the light curve data are taken at this link https://www.swift.ac.uk/xrt_curves/GRB_ID/flux.qdp, where GRB ID is the GRB observation ID).Among these GRBs, we selected our first sample according to the following criteria: 1.The XRT light curve shows at least one SD segment that is clean, i.e. without secondary peaks or relevant fluctuations.
2. If we call F 1 and F 2 the fluxes at the beginning and at the end of the SD, respectively, we require that F 1 F 2 > 10.This requirement is necessary to have a sufficient number of temporal bins inside the SD segment and therefore a well sampled spectral evolution.
3. The beginning of the SD phase corresponds to a peak in the XRT light curve, such that we have a reliable reference for the initial time.We stress that the identification of the SD starting time in XRT is limited by the observational window of the instrument.This means that, if the XRT light curve starts directly with a SD phase, with no evidence of a peak, the initial reference time is possibly located before and its value cannot be directly derived.
4. The XRT peak before the SD has a counterpart in BAT, whose peak is the brightest since the trigger time.This requirement is necessary to ensure that XRT is looking at a prompt emission episode, whose typical peak energy is above 100 keV.In a quantitative way, we define two times, t p and t stop 90 , where the first indicates the beginning of the peak that generates the SD, while the second is the end time of T 90 33 , with respect to the trigger time.We require t stop 90 > t p in order to have an overlap between the last prompt pulses (monitored by BAT) and the XRT peak that precedes the SD phase.Namely, such requirement ensures that a considerable fraction of the energy released by the burst goes into the pulse that generates the X-ray tail.
It is possible that more than one peak is present in the XRT light curve, each with a following SD.
In this case we consider only the SD after the brightest peak.If two peaks have a similar flux, we consider the SD with the larger value of F 1 F 2 .
We define then a second sample of GRBs that satisfy the first two points listed before, but have a SD at the beginning of the XRT light curve, namely no initial peak preceding the SD is present.In addition, we require that a BAT pulse precedes the XRT SD and is the brightest since the trigger time.The BAT pulse enables us to constrain the starting of the SD.
Time resolved spectral analysis.For each GRB we divided the XRT light curve in several time bins, according to the following criteria: 1.Each bin contains only data in Windowed Timing (WT) mode or in Photon Counting (PC) mode, since mixed WT+PC data cannot be analyzed as a single spectrum.
2. Each bin contains a total number of counts N bin in the E = 0.3 − 10 keV band larger than a certain threshold N 0 , which is chosen case by case according to the brightness of the source.The definition of the time bins is obtained by an iterative process, i.e. starting from the first point of the light curve we keep including subsequent points until where N (t n ) are the counts associated to each point of the light curve, while t i and t f define the starting and ending time of the bin.Then the process is repeated for the next bins, until t f is equal to the XRT ending time.Due to the large range of count rates covered during a typical XRT light curve, the choice of only one value for N 0 would create an assembly of short bins at the beginning and too long bins toward the end.Therefore we use one value of N 0 for bins in WT mode (N W T

0
) and a smaller value of N 0 for bins in PC mode (N P C 0 ).In our sample, the SD is usually in observed in WT mode, therefore we adjust N W T 0 in order to have at least 4-5 bins inside the SD.A typical value of N W T 0 is around 1500-3000, while 0 is around 500-1000.Using these values, we verified that the relative errors of photon index and normalization resulting from spectral analysis are below ∼ 30%.
3. For each couple (N i , N j ) of points inside the bin, the following relation must hold: where σ i and σ j are the associated errors.Such requirement avoids large flux variations within the bin itself.
4. The duration of the bin is larger than 5 seconds, in order to avoid pileup in the automatically produced XRT spectra, obtained from the website https://www.swift.ac.uk/ xrt_spectra/addspec.php?targ=GRB_ID, where GRB_ID is the ID number of the GRB.
It is possible that condition 3 is satisfied only for a duration of the bin T bin < T 0 , while condition 2 is satisfied for T bin > T * 0 , but T * 0 > T 0 , meaning that they cannot be satisfied at the same time.
In this case, we give priority to condition 3, provided that N bin is not much smaller than N 0 .
Due to the iterative process that defines the duration of the bins, it is possible that the last points in WT and PC mode are grouped in a single bin with a too small N bin , giving a too noisy spectrum.
Therefore, they are excluded from the spectral analysis.
Spectral modeling.The spectrum of each bin is obtained using the automatic online tool provided by Swift for spectral analysis (https://www.swift.ac.uk/xrt_spectra/addspec. php?targ=GRB_ID, where GRB_ID is the ID number of the GRB).The details of the automatic spectral analysis can be found here https://www.swift.ac.uk/xrt_spectra/docs. php#filespec.Each spectrum is analyzed using XSPEC 34 , version 12.10.1,and the Python interface PyXspec.We discard all photons with energy E < 0.5 keV and E > 10 keV.The spectra are modeled with an absorbed power law and for the absorption we adopted the Tuebingen-Boulder model 35 .If the GRB redshift is known, we use two distinct absorbers, one Galactic 36 and one relative to the host galaxy (the XSPEC syntax is tbabs*ztbabs*po).The column density N H of the second absorber is estimated through the spectral analysis, as explained below.On the other hand, if the GRB redshift is unknown, we model the absorption as a single component located at redshift z=0 (the XSPEC syntax is tbabs*po) and also in this case the value of N H is derived from spectral analysis.
For the estimation of the host N H we consider only the late part of the XRT light curve following the SD phase.At late time with respect to the trigger we do not expect strong spectral evolution, as verified in several works in the literature 37,38 .Therefore, for each GRB, the spectrum of each bin after the SD is fitted adopting the same N H which is left free during the fit.Normalization and photon index are also left free, but they have different values for each spectrum.We call N late H the value of N H obtained with this procedure.In principle the burst can affect the ionization state of the surrounding medium, but we assume that such effects are negligible and N H does not change dramatically across the duration of the burst 39 .Hence we analyzed separately all the spectra of the SD using a unique value of N H = N late H , which is fixed during the fit.Normalization and photon index, instead, are left free.
An alternative method for the derivation of N H is the fitting of all the spectra simultaneously imposing a unique value of N H that is left free.On the other hand, since N H and photon index are correlated, an intrinsic spectral evolution can induce an incorrect estimation of N H .For the same reason we do not fit the spectra adopting a free N H , since we would obtain an evolution of photon index strongly affected by the degeneracy with N H .
In this regard, we tested how our result about spectral evolution depend on the choice of N H . On average we found that the fits of the SD spectra remain good (stat/dof 1) for a variation of N H of about 50%.As a consequence, the photon index derived by the fit would change at most of 30%.Therefore the error bars reported in all the plots α −F are possibly under-estimated, but even considering a systematic error that corresponds to ∼ 30% of the value itself would not undermine the solidity of the results.
Extrapolation of F max .We explain here how we extrapolated the F max for the GRBs of the second sample, for which the XRT light curve starts directly with a SD.We consider the peak time T BAT p of the BAT pulse that precedes the SD.In the assumption that the SD starts at T BAT p , we can derive F max using the following procedure.We consider the 0.5-10 keV flux F (t i ) for each bin time t i in the SD, derived from spectral analysis.Then we fit these points with a power law High Latitude Emission.We assume that an infinitesimal duration pulse of radiation is emitted on the surface of a spherical shell, at radius R 0 from the center of the burst.The jet has an aperture angle ϑ j and it expands with a bulk Lorentz factor Γ. We assume also that the comoving spectrum is the same on the whole jet surface.The temporal evolution of the observed flux density is given by 40 : with S ν ′ (ν/D(ϑ)) the comoving spectral shape, D(ϑ) the Doppler factor and ϑ the angle measured from the line of sight, which is assumed to coincide with the jet symmetry axis.The observer time t obs is related to the angle ϑ through this formula: where t em is the emission time.Eq. ( 1) is valid for ϑ < ϑ j , while for ϑ > ϑ j the emission drops to zero.This implies that for t obs > t em (1 − β cos ϑ j ) the flux drops to zero.At each time t obs (ϑ) the observer receives a spectrum that is Doppler shifted by a factor D(ϑ) with respect to the comoving spectrum.If the comoving spectrum is curved, i.e. if d 2 dν ′2 S ν ′ = 0, then also the photon index is a function of time 41 .The shape of the resulting curve α − F is determined only by the spectral shape and the comoving peak frequency ν ′ p , while it is independent on the emission radius R 0 and the bulk Lorentz factor Γ.
We notice that the observed photon index goes from 0.5 − 1.0 up to 2.0 − 2.5, consistent with the slopes of a synchrotron spectrum before and after the peak frequency.Indeed for a population of particles with an injected energy distribution N (γ) ∝ γ −p that has not completely cooled, the expected shape of the spectrum is for ν > ν m ν c .Hereafter, if not otherwise specified, we assume a spectral shape given by a smoothly broken lower law, which well approximates the synchrotron spectrum below and above from which we derive dR In the limit of Thus, the delay time is Given an arrival time ∆t obs , this equation allow us to associate a radius R em to each angle ϑ em through the following expression: Inverting this equation, we obtain the polar equation R em (ϑ em , ∆t obs ) which defines the EATS, namely all the photons emitted on this locus of points arrive to the observer with a time delay ∆t obs with respect to the first photon coming from R = R in and ϑ = 0.The computation of flux as a function of time is again done using eq.( 16), with the only difference that now β and Γ, which appear in the Doppler factor D(ϑ), depend on R(ϑ).The light curve and the spectral evolution for values of k in the range −0.4 ≤ k ≤ 0.4 are showed in Fig. S5.
Alternative scenarios shaping the X-ray tails.In this section we explore other possible models of prompt emission which can drive the evolution during the X-ray tails.We consider an anisotropic jet core, for instance, made of mini jets 58 with angular sizes of < 1/Γ, with different comoving spectra and bulk motion 59 .In this case, the overall velocity and spectral distribution of mini-jets should be the same for all the GRBs and a fine tuning is necessary to reproduce the α − F relation.
Moreover the small aperture angle of the mini-jets would produce very early a shallow segment due to jet structure 40 , in contradiction with the typical duration of X-ray tails.Within the HLE scenario, only models which assume a dissipation occurring above the jet photosphere, such as in internal shocks 12 or in magnetic reconnection scenarios 57,60 , are able to reproduce the typical duration of X-ray tails (∼ 100 s).Photospheric models 61 , where dissipation occurs at radii R ph ∼ 10 12 cm 62 , give smaller times scales of ∼ 10 −2 s, incompatible with observations.Only a common declining activity of the central engine 63,64 and a fine-tuned intrinsic spectral softening 65 would be required to account for the α − F relation.
Adiabatic Cooling.In this section we derive the effect of adiabatic cooling of the emitting particles 66 on the light curve and the spectral evolution of X-ray tails.We assume that the emission is energy density and particle energy density, giving B 2 ∼ γ /V ∼ V −4/3 , where in the last step we used eq.( 23).In this case B ∼ R −4/3 ( λ = 4/3) for ∆R ′ = const and B ∼ R −2 ( λ = 2) for ∆R ′ ∝ R. All these predicted values of λ are larger than the range found from our analysis.Such tension can be solved, for instance, if the shell thickness decreases as the jet expands, or if the jet is not conical (e.g.paraboloidal, with r ∝ √ R).
Supplementary Materials.superimposed to the first and the second sample.The theoretical curves are computed taking also into account the effect of HLE.The value of λ specifies the evolution of the magnetic field.We adopt a SBPL as spectral shape with α s = −1/3 and β s = 2.0, an initial observed peak frequency of 100 keV and a thickness of the expanding shell that is constant in time.The dot-dashed line is the evolution expected considering only HLE, assuming the same spectral shape and initial observed peak frequency.Right panel: Spectral evolution expected for HLE from a infinitesimal duration pulse, with the assumption of a SBPL spectrum.The several colors indicate the observed peak frequency at the beginning of the decay.The magnetic field does not evolve with radius, i.e λ = 0.The adopted parameters are R in = 3 × 10 15 cm, R of f = 9 × 10 15 cm, Γ 0 = 100 and ν p = 100 keV.The adopted spectral shape is a SBPL.The peak of each curve is shifted at 100 s.Right panel: Spectral evolution in case of HLE from a finite-duration pulse, in case of not constant Γ.The magnetic field does not evolve with radius, i.e λ = 0.The adopted parameters are R in = 3 × 10 15 cm, R of f = 9 × 10 15 cm, Γ 0 = 100 and ν p = 100 keV.The adopted spectral shape is a SBPL.S2: Main information about the GRBs of the second sample.z is the redshift, when available.N H is the column density adopted in the spectral analysis of the X-ray tail.T i and T f are the central times of the initial and final bins of the spectral analysis, respectively.T BAT p (s) is the peak time of the BAT pulse preceding the X-ray tail, used for the extrapolation of F max     keV, respectively.The peak ux Fmax is estimated extrapolating the X-ray tail back to the BAT peak.Right panel: Spectral evolution of our extended sample of GRBs, which present a steep decay at the beginning of the XRT light curve, preceded by the brightest BAT pulse since the trigger time.The evolution of α lies on the same region of the plane occupied by the original sample, indicated in grey.

Flux < l a t e x i t s h a 1 _
b a s e 6 4 = " m M 5 X S + R P V 8 k u X + B v W H d G o a i 2 d r g = " > A A A B / X i c b V D L S s N A F J 3 4 r P V V d e l m s A i u S i K C L o u C u K x g H 9 C G M p n e 1 q G T S Z i 5 I y 2 h + B V u d e V O 3 P o t L v w X k 5 i F t p 7 V 4 Z x 7 u e e e I J b C o O t + O k v L K 6 t r 6 6 W N 8 u b W 9 s 5 u Z W + / Z S K r O T R 5 J CP d C Z g B K R Q 0 U a C E T q y B h Y G E d j C + y v z 2 A 2 g j I n W H 0 x j 8 k I 2 U G A r O M J W 6 P Y Q J J t f S T m b 9 S t W t u T n o I v E K U i U F G v 3 K V 2 8 Q c R u C Q i 6 Z M V 3 P j d F P m E b B J c z K P W s g Z n z M R t B N q W I h G D / J I 8 / o s T U M I x q D p k L S X I T f G w k L j Z m G Q T o Z Mr w 3 8 1 4 m / u d 1 L Q 4 v / E S o 2 C I o n h 1 C I S E / Z L g W a R d A B 0 I D I s u S A x W K c q Y Z I m h B G e e p a N N y y m k f 3 v z 3 i 6 R 1 W v P c m n d 7 V q 1 f F s 2 U y C E 5 I i f E I + e k T m 5 I g z Q J J x F 5 I s / k x X l 0 X p 0 3 5 / 1 n d M k p d g 7 I H z g f 3 6 9 q l i U = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " m M 5 X S + R P V 8 k u X + B v W H d G o a i 2 d r g = " > A A A B / X i c b V D L S s N A F J 3 4 r P V V d e l m s A i u S i K C L o u C u K x g H 9 C G M p n e 1 q G T S Z i 5 I y 2 h + B V u d e V O 3 P o t L v w X k 5 i F t p 7 V 4 Z x 7 u e e e I J b C o O t + O k v L K 6 t r 6 6 W N 8 u b W 9 s 5 u Z W + / Z S K r O T R 5 J CP d C Z g B K R Q 0 U a C E T q y B h Y G E d j C + y v z 2 A 2 g j I n W H 0 x j 8 k I 2 U G A r O M J W 6 P Y Q J J t f S T m b 9 S t W t u T n o I v E K U i U F G v 3 K V 2 8 Q c R u C Q i 6 Z M V 3 P j d F P m E b B J c z K P W s g Z n z M R t B N q W I h G D / J I 8 / o s T U M I x q D p k L S X I T f G w k L j Z m G Q T o Z Mr w 3 8 1 4 m / u d 1 L Q 4 v / E S o 2 C I o n h 1 C I S E / Z L g W a R d A B 0 I D I s u S A x W K c q Y Z I m h B G e e p a N N y y m k f 3 v z 3 i 6 R 1 W v P c m n d 7 V q 1 f F s 2 U y C E 5 I i f E I + e k T m 5 I g z Q J J x F 5 I s / k x X l 0 X p 0 3 5 / 1 n d M k p d g 7 I H z g f 3 6 9 q l i U = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " m M 5 X S + R P V 8 k u X + B v W H d G o a i 2 d r g = " > A A A B / X i c b V D L S s N A F J 3 4 r P V V d e l m s A i u S i K C L o u C u K x g H 9 C G M p n e 1 q G T S Z i 5 I y 2 h + B V u d e V O 3 P o t L v w X k 5 i F t p 7 V 4 Z x 7 u e e e I J b C o O t + O k v L K 6 t r 6 6 W N 8 u b W 9 s 5 u Z W + / Z S K r O T R 5 J C P d C Z g B K R Q 0 U a C E T q y B h Y G E d j C + y v z 2 A 2 g j I n W H 0 x j 8 k I 2 U G A r O M J W 6 P Y Q J J t f S T m b 9 S t W t u T n o I v E K U i U F G v 3 K V 2 8 Q c R u C Q i 6 Z M V 3 P j d F P m E b B J c z K P W s g Z n z M R t B N q W I h G D / J I 8 / o s T U M I x q D p k L S X I T f G w k L j Z m G Q T o Z M r w 3 8 1 4 m / u d 1 L Q 4 v / E S o 2 C I o n h 1 C I S E / Z L g W a R d A B 0 I D I s u S A x W K c q Y Z I m h B G e e p a N N y y m k f 3 v z 3 i 6 R 1 W v P c m n d 7 V q 1 f F s 2 U y C E 5 I i f E I + e k T m 5 I g z Q J J x F 5 I s / k x X l 0 X p 0 3 5 / 1 n d M k p d g 7 I H z g f 3 6 9 q l i U = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " m M 5 X S + R P V 8 k u X + B v W H d G o a i 2 d r g = " > A A A B / X i c b V D L S s N A F J 3 4 r P V V d e l m s A i u S i K C L o u C u K x g H 9 C G M p n e 1 q G T S Z i 5 I y 2 h + B V u d e V O 3 P o t L v w X k 5 i F t p 7 V 4 Z x 7 u e e e I J b C o O t + O k v L K 6 t r 6 6 W N 8 u b W 9 s 5 u Z W + / Z S K r O T R 5 J C P d C Z g B K R Q 0 U a C E T q y B h Y G E d j C + y v z 2 A 2 g j I n W H 0 x j 8 k I 2 U G A r O M J W 6 P Y Q J J t f S T m b 9 S t W t u T n o I v E K U i U F G v 3 K V 2 8 Q c R u C Q i 6 Z M V 3 P j d F P m E b B J c z K P W s g Z n z M R t B N q W I h G D / J I 8 / o s T U M I x q D p k L S X I T f G w k L j Z m G Q T o Z M r w 3 8 1 4 m / u d 1 L Q 4 v / E S o 2 C I o n h 1 C I S E / Z L g W a R d A B 0 I D I s u S A x W K c q Y Z I m h B G e e p a N N y y m k f 3 v z 3 i 6 R 1 W v P c m n d 7 V q 1 f F s 2 U y C E 5 I i f E I + e k T m 5 I g z Q J J x F 5 I s / k x X l 0 X p 0 3 5 / 1 n d M k p d g 7 I H z g f 3 6 9 q l i U = < / l a t e x i t > Time < l a t e x i t s h a 1 _ b a s e 6 4 = " x 6 D V p O x s 6 z J P v p A g P N 5 O N s j 6 8 x e + X S e e s 7 r l 1 7 / a 8 1 r g q m i m T I 3 J M T o l H L k i D 3 J A m a R N O I v J E n s m L 8 + i 8 O m / O + 8 9 o y S l 2 D s k f O B / f l o a W F Q = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x 6 D V p O x s 6 z J P v p A g P N 5 O N s 5 k g I o l d r X g y S A x 9 m g U n V r b g 6 + S L y C V F m B x q D y 1 R u G I g 5 Q k 1 B g b d d z I + o n Y E g K h b N y L 7 Y Y g Z j A G L s p 1 R C g 7 S d 5 3 B k / j i 1 Q y C M 0 X C q e i / h 7 I 4 H A 2 m n g p 5 M B 0 L 2 d 9 z L x P 6 8 b 0 + i i n 0 g d x Y R a Z I d I K s w P W W F k 2 g P y o T R I B F l y 5 F J z A Q a I 0 E g O Q q R i n B Z T T v v w 5 r 9 f J K 3 T m u f W v N u z a v 2 y a K b E D t k R O 2 E e O 2 d 1 d s M a r M k E m 7 A n 9 s x e n M R 5 d d 6 c 9 5 / R J a f Y O W B / 4 H x 8 A z E 9 l C g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / k g e U J Q 4 P T v o V q Z 7 M I u k u C R H Z k Q = " > A A A B + X i c b V B N S 8 N A E N 3 4 W e t X 1 a O X x S J 4 K o k I e i w K 4 r G C / Y C 2 l M l 2 W p d u N m F 3 I p b Q H + F V T 9 7 E q 7 / G g / / F J O a g r e / 0 e G + G e f P 8 S E l s 5 k g I o l d r X g y S A x 9 m g U n V r b g 6 + S L y C V F m B x q D y 1 R u G I g 5 Q k 1 B g b d d z I + o n Y E g K h b N y L 7 Y Y g Z j A G L s p 1 R C g 7 S d 5 3 B k / j i 1 Q y C M 0 X C q e i / h 7 I 4 H A 2 m n g p 5 M B 0 L 2 d 9 z L x P 6 8 b 0 + i i n 0 g d x Y R a Z I d I K s w P W W F k 2 g P y o T R I B F l y 5 F J z A Q a I 0 E g O Q q R i n B Z T T v v w 5 r 9 f J K 3 T m u f W v N u z a v 2 y a K b E D t k R O 2 E e O 2 d 1 d s M a r M k E m 7 A n 9 s x e n M R 5 d d 6 c 9 5 / R J a f Y O W B / 4 H x 8 A z E 9 l C g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / k g e U J Q 4 P T v o V q Z 7 M I u k u C R H Z k Q = " > A A A B + X i c b V B N S 8 N A E N 3 4 W e t X 1 a O X x S J 4 K o k I e i w K 4 r G C / Y C 2 l M l 2 W p d u N m F 3 I p b Q H + F V T 9 7 E q 7 / G g / / F J O a g r e / 0 e G + G e f P 8 S E l s 5 k g I o l d r X g y S A x 9 m g U n V r b g 6 + S L y C V F m B x q D y 1 R u G I g 5 Q k 1 B g b d d z I + o n Y E g K h b N y L 7 Y Y g Z j A G L s p 1 R C g 7 S d 5 3 B k / j i 1 Q y C M 0 X C q e i / h 7 I 4 H A 2 m n g p 5 M B 0 L 2 d 9 z L x P 6 8 b 0 + i i n 0 g d x Y R a Z I d I K s w P W W F k 2 g P y o T R I B F l y 5 F J z A Q a I 0 E g O Q q R i n B Z T T v v w 5 r 9 f J K 3 T m u f W v N u z a v 2 y a K b E D t k R O 2 E e O 2 d 1 d s M a r M k E m 7 A n 9 s x e n M R 5 d d 6 c 9 5 / R J a f Y O W B / 4 H x 8 A z E 9 l C g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " / k g e U J Q 4 P T v o V q Z 7 M I u k u C R H Z k Q = " > A A A B + X i c b V B N S 8 N A E N 3 4 W e t X 1 a O X x S J 4 K o k I e i w K 4 r G C / Y C 2 l M l 2 W p d u N m F 3 I p b Q H + F V T 9 7 E q 7 / G g / / F J O a g r e / 0 e G + G e f P 8 S E l

1 <
s 5 k g I o l d r X g y S A x 9 m g U n V r b g 6 + S L y C V F m B x q D y 1 R u G I g 5 Q k 1 B g b d d z I + o n Y E g K h b N y L 7 Y Y g Z j A G L s p 1 R C g 7 S d 5 3 B k / j i 1 Q y C M 0 X C q e i / h 7 I 4 H A 2 m n g p 5 M B 0 L 2 d 9 z L x P 6 8 b 0 + i i n 0 g d x Y R a Z I d I K s w P W W F k 2 g P y o T R I B F l y 5 F J z A Q a I 0 E g O Q q R i n B Z T T v v w 5 r 9 f J K 3 T m u f W v N u z a v 2 y a K b E D t k R O 2 E e O 2 d 1 d s M a r M k E m 7 A n 9 s x e n M R 5 d d 6 c 9 5 / R J a f Y O W B / 4 H x 8 A z E 9 l C g = < / l a t e x i t > t l a t e x i t s h a _ b a s e = " b p j M A T P M O L 9 W i 9 W m / W + 8 9 o y S p 2 D u E P r I 9 v r Y K S I g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 7 U X o O W m 4 C P R H u 8 R i H b 9 I m d O J r 6 X T v p A 9 2 b e y 8 T / v F 5 M o w s 3 k U E U E w Y i O 0 R S Y X 7 I C C 3 T D p A P p U Y i y J I j l w E X o I E I t e Q g R C r G a S m V t A 9 n / v t F 0 j 6 p O 3 b d u T m r N S 6 L Z s r s g B 2 y Y + a w c 9 Z g 1 6 z J W k y w M X t i z + z F e r R e r T f r / W e 0 Z B U 7 + + w P r I 9 v r x G S I w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " T h X r 6 X T v p A 9 2 b e y 8 T / v F 5 M o w s 3 k U E U E w Y i O 0 R S Y X 7 I C C 3 T D p A P p U Y i y J I j l w E X o I E I t e Q g R C r G a S m V t A 9 n / v t F 0 j 6 p O 3 b d u T m r N S 6 L Z s r s g B 2 y Y + a w c 9 Z g 1 6 z J W k y w M X t i z + z F e r R e r T f r / W e 0 Z B U 7 + + w P r I 9 v r x G S I w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " T h X r 6 X T v p A 9 2 b e y 8 T / v F 5 M o w s 3 k U E U E w Y i O 0 R S Y X 7 I C C 3 T D p A P p U Y i y J I j l w E X o I E I t e Q g R C r G a S m V t A 9 n / v t F 0 j 6 p O 3 b d u T m r N S 6 L Z s r s g B 2 y Y + a w c 9 Z g 1 6 z J W k y w M X t i z + z F e r R e r T f r / W e 0 Z B U 7 + + w P r I 9 v r x G S I w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " T h X

Figure 2 :
Figure 2: Illustration of the spectral evolution caused by a shift of the spectrum towards lower energies.The transition of the spectral peak through the XRT band explains the observed spectral softening.Since in the right panel we plot the flux density, the local slope in the XRT band is given by 1 − α, where α is the photon index.

F
max < l a t e x i t s h a 1 _ b a s e 6 4 = " y 6 I 4 0 and imposing that t 0 = T BAT p .Finally we derive the best fit value of F max with the associated 1σ error.The error of F max has a contribution coming from the error associated to β and another associated to t 0 , as well as from the assumption of a power law as fitting function.The value of T BAT p is obtained fitting the BAT pulse with a Gaussian profile.Since usually the BAT pulse can have multiple sub-peaks and taking also into account possible lags between XRT and BAT peaks, we adopt a conservative error associated to T BAT p equal to 5 seconds.

Figures Figure 1
Figures

Table S1 :
name zN H (10 22 cm −2 ) T i (s) T f (s) Main information about the GRBs of the first sample.z is the redshift, when available.N H is the column density adopted in the spectral analysis of the X-ray tail.T i and T f are the central times of the initial and final bins of the spectral analysis, respectively.name z N H (10 22 cm −2 ) T i (s) T f (s) T BAT