Significant Cocoon Emission and Photosphere Duration Stretching in GRB 211211A: A Burst from a Neutron Star−Black Hole Merger

The radiation mechanism (thermal photosphere or magnetic synchrotron) and the progenitor of gamma-ray bursts (GRBs) are under hot debate. Recently discovered, the prompt long-duration (∼10 s, normally from the collapse of massive stars) property of GRB 211211A strongly conflicts with its association with a kilonova (normally from the merger of two compact objects, NS–NS, NS–BH, or NS–WD, duration ≲2 s). In this paper, we find that the probability photosphere model with a structured jet can satisfactorily explain this peculiar long duration, through the duration stretching effect (∼3 times) on the intrinsic longer (∼3 s) duration of an NS–BH merger, the observed empirical 2SBPL spectrum (with soft low-energy index α of ∼−1), and its evolution. In addition, much evidence of the NS–BH merger origin is found, especially the good fit of the afterglow-subtracted optical−near-IR light curves by the significant thermal cocoon emission and the sole thermal “red” kilonova component. Finally, a convincing new explanation for the X-ray afterglow plateau is revealed.

The probability density function can be calculated by (see Ref. 33 ) where β means the jet velocity, D means the Doppler factor, and R ph is the photospheric radius and depends on the angular coordinate Ω.
Furthermore, the GRB relativistic jet launched by the center engine will be collimated by the surrounding material 34 .Then, the break-out jet should have angular profiles outside the isotropic core θ c , for both the luminosity L and Lorentz factor Γ. This is called the structured jet (see Methods).Previously, θ c,Γ = θ c,L is typically assumed.Here, θ c,Γ means the isotropic angular width for the Lorentz factor, and θ c,L means the isotropic angular width for luminosity.
Based on the simulation results of GRB jet [34][35][36][37][38] , we now propose that the structured jet with θ c,Γ < θ c,L is likely to be commonly obtained.(see Figure 1a).This is due to the enhanced material in the outside part, with decreasing velocity and almost constant energy (achieving the pressure equilibrium) within θ c,L (see Extended Data Figure 1 and further discussions in Methods).In our previous photosphere spectrum calculations for structured jet, the constant luminosity case 23 and the θ c,Γ = θ c,L case 24 are both explored.Here, we further study the θ c,Γ < θ c,L case.
We find the luminosity structure has a negligible effect to the spectrum calculation when the viewing angle θ v satisfies θ v < θ c,L , since relativistic emission is within 5/Γ.Also, θ v < θ c,L should be common due to the large θ c,L and the decreasing luminosity outwards.In the following calculations and fitting, we adopt θ c,L = 0.1, the typical constrained value of the jet opening angle θ jet .On the contrary, the Lorentz factor structure could have significant influence, since θ c,Γ may be quite small (θ c,Γ • Γ ∼ a few) or larger θ v (θ v θ c,Γ , our considered case for GRB 211211A) is obtained.The adopted Lorentz factor (or the baryon loading η, normally Γ η) structure is as following: where θ is the angle from the jet axis, η 0 is the constant baryon loading parameter in the core with the width of θ c,Γ , p describes how the baryon loading decreases outside the core.The constant of 1.2 donates the minimum Lorentz factor of the relativistic jet (velocity v ∼ 0.6 c, where c is the speed of the light; corresponding to the highest velocity of the outside non-relativistic cocoon, see Extended Data Figure 1).
The duration stretching effect for the saturated-acceleration photosphere.For the photosphere emission, the saturated-acceleration regime is defined when R ph is larger than the radius R s , where the maximum acceleration is achieved (Γ = η, R s = η • R 0 , R 0 is the initial acceleration radius; see further discussions in Methods).The adiabatic cooling happens during R s < r R ph , thus decreasing the observed temperature (or frequency) and luminosity.
For the injected luminosity history L w ( t) of the center engine ( t is the injection time), we use the exponential model 23,39 to approximate it, with following form: The jet structure and the duration stretching effect for the saturated photosphere.a, The considered structured jet with θ c,Γ < θ c,L (Γ ≃ η).The considered viewing angle θ v for GRB 211211A is a bit larger than θ c,Γ .b, The photosphere luminosity L ph profiles for different Lorentz factor structures, with the same injected duration of T 90 ∼ 3 s (see also Figure 16 in Ref. 24 ).The observed duration can be stretched to T 90 ∼ 6 s, for the strongly saturated cases (the blue and cyan markers).Note that, the L ph profiles have all been normalized to L w,p = 10 52 erg s −1 at 2.4 s.Actually, for all the saturated cases (except for the orange box), the luminosity is 1-10 times smaller.c, The L ph profile for θ c,Γ • Γ = 4, θ v = 1.5 • θ c,Γ (red squares).The light curve of GRB 211211A (blue plus) can be well reproduced, including the T 90 ∼ 10 s and the double pulses.
3 Figure 1: The jet structure and the duration stretching effect for the saturated photosphere.a, The considered structured jet with θ c,Γ < θ c,L (Γ η).The considered viewing angle θ v for GRB 211211A is a bit larger than θ c,Γ .b, The photosphere luminosity L ph profiles for different Lorentz factor structures, with the same injected duration of T 90 ∼ 3 s (see also Figure 16 in Ref. 24 ).The observed duration can be stretched to T 90 ∼ 6 s, for the strongly saturated cases (the blue and cyan markers).Note that, the L ph profiles have all been normalized to L w,p = 10 52 erg s −1 at 2.4 s.Actually, for all the saturated cases (except for the orange box), the luminosity is 1-10 times smaller.c, The L ph profile for θ c,Γ • Γ = 4, θ v = 1.5 • θ c,Γ (red squares).The light curve of GRB 211211A (blue plus) can be well reproduced, including the T 90 ∼ 10 s and the double pulses.
where ts means the start time, τ 1 and τ 2 are the characteristic quantities indicating the rise and decay timescale, respectively.L w,p is the peak luminosity at tp , and tp = ts + (τ 1 • τ 2 ) 1/2 .
In Figure 1b, with the injected duration T 90 of ∼ 3 s (τ 1 = 32, τ 2 = 0.5, and ts = −1.6 are adopted; tp = 2.4 s), we show the observed photosphere luminosity L ph profiles for different Lorentz factor structures (see also Figure 16 in Ref. 23 ).Obviously, for the strongly saturatedacceleration cases (R ph is larger, corresponding to the luminosity around the peak time), whether Γ = 200 (smaller), L w,p = 10 53 erg s −1 (higher) or larger θ v ( θ c,Γ ), the observed duration can be stretched to T 90 ∼ 6 s (two times larger).While for the strongly unsaturated case, R 0 = 10 9 cm, the observed luminosity profile is almost the same as the injected profile, thus obtaining a similar duration.
The reason for the above results is that, for saturated cases (actually for the higher luminosity around the peak time), the observed luminosity around the peak time will be significantly decreased (due to the adiabatic cooling), while the observed luminosity for earlier and later times will not suffer from this drop (with lower luminosity, R ph is smaller and the unsaturated condition is obtained).In total, we discover the duration stretching effect for photosphere emission, which means that the injected luminosity duration T 90 of the center engine can be stretched to an observed duration of ∼ 2-3 T 90 , when R s < R ph (saturated acceleration) is satisfied around L w,p .
Note that, for the long GRBs, this photosphere duration stretching effect takes part for the saturated case is confirmed by comparing the duration distributions of the γ 50% and γ 50% samples (see Extended Data Figure 2a, data is taken from Ref. 25 ).Here, γ = E iso /(E iso + E k ), standing for the prompt efficiency.E iso means the isotropic energy for prompt emission, and E k means the energy for afterglow.Theoretically, γ 50% should correspond to the saturated acceleration (smaller γ due to the adiabatic cooling, see Methods), and its duration is indeed found ∼ 1.6 times longer than that for the γ 50% sample (with similar distribution profile).This duration stretching effect for the saturated photosphere may also contribute greatly to the peculiar long duration of GRB 211211A (∼ 10 s) and GRB 060614 (∼ 6 s).Since, according to their observed αθ jet distribution (see Extended Data Figure 3a and discussions in Methods), the larger θ v ( θ c,Γ ; thus Γ is smaller, see Figure 4) is likely to be satisfied.
In Extended Data Figure 3b, we compare the peak energy E p and E iso distributions of the main pulse (MP) and extended emission (EE) of GRB 211211A and GRB 060614, with other short GRBs with EE.For these two special bursts, the E p is smaller of ∼ 2-3 times for both the MP and EE.This further supports the above saturated scenario (due to the adiabatic cooling).The larger θ v is preferred, since in Ref. 25 we find the normal short GRB sample with EE almost has γ 50% (thus intrinsically unsaturated; with large θ c,Γ , see discussions below and Extended Data Figure 1).Indeed, in Figure 1c, with θ v = 1.5 • θ c,Γ and θ c,Γ • Γ = 4, and injected duration T 90 ∼ 3 s, we find the light curve of GRB 211211A can be well explained by the photosphere emission, including the T 90 ∼ 10 s and the double pulses (stronger decrease around L w,p ).The photosphere explanation for the empirical 2SBPL spectrum and the spectral evolution.In Extended Data Figure 4a, we show the theoretical photosphere spectral evolution, for larger 1c.The photosphere spectra could reproduce the observed soft α ∼ -1 of GRB 211211A.
In Extended Data Figure 4b, we show the spectral evolution of GRB 211211A for the empirical 2SBPL (two smoothly broken power-law) model fitting, taking the best-fit parameters in Ref. 13 (accounting for the X-ray data, the 2SBPL is likely to be the better empirical model, see Methods).This evolution is quite similar to the photosphere spectral evolution in Extended Data Figure 4a, transferring from the early 2SBPL spectrum (with high-energy exponential cutoff, which may be better, see Methods) to the late-time SBPL spectrum (due to the high-altitude effect, see Methods).
The spectral fitting using the physical photosphere model.Generally, we need to convolve the model spectra with the instrumental response, the Detector Response Matrix (DRM), to compare the model spectra with the observational spectra (adopting the statistical value of BIC, Bayesian information criterion).We use the McSpecFit package which accepts flexible user-defined spectral model 40 to perform this.The comparisons of the observed and model-convolved count spectra are shown in the bottom panels of Figure 2, and the de-convolved observed and modeled photon number spectra are in the top panels.
The spectral fitting results of 2.0 -3.0 s (around the peak), using the above photosphere model with structured jet (constant injected luminosity is assumed), are shown in Figure 2, Extended Data Figure 11 (parameter constraints) and Extended Data Table 2 (best-fit parameters).Obviously, the photosphere model can give a rather well fit (see the residuals distribution, BIC/dof = 354/359), showing the exponential high-energy cutoff (E p ∼ 2000 keV, low-energy power-law index α 2 ∼ -1) combined with a smoothly broken power law in the low-energy end (the break energy E b ∼ 30 keV, low-energy power-law index α 1 ∼ 0).
In addition, as shown in Figure 2, Extended Data Figures 12 and 13, and Extended Data Table 2, we give acceptable fit for the late-time spectra (12.0 -14.0 s and 40.0 -45.0 s) after considering  NS-BH-merger evidence and significant cocoon emission in GRB 211211A.According to Extended Data Figure 2b (see discussions in Methods), for the short GRBs, we propose that the γ 50% sample (without EE) is likely to come from the NS-NS merger, with shorter intrinsic duration of ∼ 0.1 -0.2 s 42 .The γ 50% sample (with EE) comes from the NS-BH merger, with longer intrinsic duration of ∼ 0.8 s and wider duration distribution of ∼ 0.05 s -3 s 43 .
Besides, there are several other supports for this point.First, the low-energy extended emission is likely to originate from the fallback accretion of the BH, considering the comparable energy for MP and EE (especially after correcting the efficiency of EE, see Figure 5(d) in Ref. 25 ), and the existence of the time gap between them (disfavoring the magnetar spin-down scenario).Moreover, the luminosity of the fallback accretion from the NS-BH merger is theoretically an order of magnitude larger than that from the NS-NS merger 30,44 .Second, the short GRBs with EE typically have a small offset from their host galaxies (∼ a few kpc; including GRB 211211A and GRB 060614, see Ref. 11 ), quite compatible with the low kick velocity for NS-BH merger 45 .Third, in Extended Data Figure 8, we find that all the X-ray afterglow light curves for the short GRB sample with EE show a power-law shape, while the significant plateau appears in the light curves for the sample without EE.Also, the evidence that the X-ray plateaus result from θ v θ c,Γ is revealed below (see Extended Data Figures 7 and 9, and Figure 4).Then, the power-law shape and plateau well correspond to the jet structures for the NS-BH merger (larger θ c,Γ ) and NS-NS merger (smaller θ c,Γ ; due to much dynamical ejecta in jet propagating direction 30 and thus stronger jet-ejecta interaction) (see Extended Data Figure 1).The outliers of GRB 211211A and GRB 060614, which have EE but show plateau, can be well explained by the rare regime of θ v θ c,Γ (with relatively large θ v , see Extended Data Figure 1) for NS-BH merger.
In addition, in Figure 3, we find that the afterglow-subtracted optical-NIR light curves of GRB 211211A (data is taken from Tables 1 and 2 in Ref. 10 ) are well fitted by the combination of the dominated early cocoon emission and the late "red" kilonova component (with a mass of M ejr 0.030M ), after treating more cocoon parameters than Ref. 10 .The lack of significant "blue" (may have similar mass to the cocoon mass M eco ∼ 0.001M ) or "purple" kilonova components strongly supports the NS-BH merger origin of GRB 211211A (see Extended Data Figure 1 and Methods).
Note that we better explain the late-time i-band data, which is obviously over-estimated in Ref. 10 .The parameter constraints from the fitting (see details in Methods) are shown in Extended Data Figure 10.Also, with an energy distribution for different velocity  10 are well fitted by the combination (solid lines) of the early cocoon emission (dotted lines) and the late "red" kilonova component (dot-dashed lines).Except for the NIR band, the dotted lines are overlapped with the solid lines, indicating that the cocoon emission well explains the early optical light curves.Besides, the existence of a sole "red" kilonova component, without significant "blue" or "purple" kilonova components, strongly supports 30 the NS-BH merger origin of GRB 211211A (see Extended Data Figure 1).
) and afterglow data.For the GRBs with X-Ray plateau (may result from large θ v , then smaller Γ; see Extended Data Figure 7), the Γ in the light of sight may be estimated by the cutoff time of the plateau T c, X (taken from Ref. 48 ; the circles), just as the peak time of the early optical afterglow T p,op (the gray plus) 25 .For GRB 211211A and GRB 060614, these two almost consistent estimates indicate small Γ ( 100), which is compatible with the large θ v consideration (θ v θ c,Γ ) in the above light curve and spectral analyses.The rare rate for such events within the short GRB sample with EE implies a large θ c,Γ , consistent with the expected jet structure of NS-BH merger (see Extended Data Figure 1).
1, the typical value from cocoon numerical simulation 28 , the observed evolutions of bolometric luminosity L bol , effective temperature T eff , and photospheric radius R ph (data is taken from Figure 2 in Ref. 12 ) can be well explained by the cocoon emission (see Extended Data Figure 6), without the need of peculiar higher L bol and velocity (∼ 0.6 c) in the early time (for the kilonova explanation) 12 .
Previously, the optical cocoon signature in long GRBs was discovered 29 by analyzing the early spectra of the supernova.For short GRBs, only weak evidence (or signature) was found in the optical counterpart of GRB 170817A (Swope Supernova Survey 2017a, SSS17a; Refs 46,47 ).This is due to the intense "blue" and "purple" kilonova emissions in GRB 170817A, produced by the NS-NS merger.Here, for GRB 211211A, which is likely to originate from the NS-BH merger, we discover the significant cocoon emission in short GRBs for the first time.
According to the above results, we consider that GRB 211211A and GRB 060614 (with EE) come from the NS-BH merger with intrinsic long duration (∼ 3 s).In addition, the larger θ v (θ v θ c,Γ , rare event; see Extended Data Figure 1) makes them saturated (smaller Γ, see Figure 4; γ 50%, see Extended Data Figure 9), resulting in the photosphere duration stretching (reaching ∼ 6 -10 s duration, see Figure 1), softer α (see Figure 2 and Extended Data Figure 3a), and more minor E p (see Extended Data Figure 3b).The new explanation for the X-ray afterglow plateau: the structured jet with θ c,Γ < θ c,L and θ v > θ c,Γ .In Ref. 25 (see Figure 7(c) there), we find p /(T 90 ) 1/4 , derived from the photosphere model, is an excellent Γ estimate for the bursts with the peak time of the early optical afterglow T p,op (also shown by the gray plus in Figure 4).In this work, as shown in Extended Data Figure 7 (the theoretical calculation, see Methods), we find the X-ray afterglow plateau can be explained by the structured jet with θ c,Γ < θ c,L and θ v > θ c,Γ .This condition means that the Γ in the light of sight (LOS) is much smaller, and the large cutoff time of the plateau T c, X should correspond to this smaller here z is the redshift, see Ref. 41 ).In Figure 4, for the large sample with plateau and T c, X (taken from Ref. 48 ), we do find consistent Γ results from the photosphere estimate (Γ = 17 ), regardless of long and short GRBs (see Methods).Furthermore, the obtained Γ is indeed much smaller (the circles; Γ 100, with T c, X > 1000 selection for significant plateau) than that for the bursts with early peak (the gray plus; without plateau).In Extended Data Figure 9, for this sample with significant plateau, E iso /E k 1 is found, indicating a small prompt efficiency γ 50% (resulting from the smaller Γ and the photosphere model, ).These consistent Γ estimates, the smaller Γ, and the lower efficiency strongly support the above explanation for the X-ray afterglow plateau.
Both the X-ray afterglow plateau exists in GRB 211211A and GRB 060614 (see Ref. 11 ).
As shown in Figure 4, above two Γ estimates obtain consistently smaller Γ for these two bursts (Γ 70 for GRB 211211A; quite close to the constrained Γ = 73 from afterglow modeling in Ref. 10 , see Table 3 there).This further supports the θ c,Γ < θ c,L and θ v θ c,Γ consideration above (the off-core case may be named), from the light curve (Figure 1), the spectral fitting (Figure 2) and the NS-BH merger origin (Figure 3 and Extended Data Figure 1; larger θ c,Γ and rare rate for In physics research, a pure blackbody is generally expected for thermal emission.But, recent studies (especially this work) for the GRB spectrum reveal that, in the relativistic condition, a pure spherical shell photosphere should change to a probability photosphere, obtaining a multi-color blackbody (α ∼ 0).This relativistic probability photosphere should be treated in many other astrophysics and physics regions.Also, the GRB photosphere emission is normally powered by the cooling of the high-temperature accretion disk, through the neutrino annihilation (for neutrinodominated accretion flow, namely NDAF 49 ).Thus, our study (achieving the parameters from the fitting) provides a new and vital path to studying the accretion disk.
The relativistic jet widely exists in astrophysics objects (AGN, TDE, high-mass X-ray binaries), whose structure contains rich physics information and is crucial to the observed characteristics.Recent studies (especially this work) indicate that θ c,Γ < θ c,L is likely to exist commonly, with which the typical GRB prompt emission (α ∼ −1) and afterglow properties (the X-ray plateau, see Extended Data Figure 7 and Methods) can both be reproduced.Our findings should provide an important base for the X-ray and gamma-ray polarization studies since the polarization degree strongly depends on the jet structure.In addition, the structured jet should be accompanied by the outward non-relativistic cocoon.In GRB 211211A, we reveal the significant optical cocoon emission, for the first time in short GRBs.
The gravitational wave from the NS-BH coalescence has been detected (see Methods).And it is long believed that, the short GRBs should consist of two subsamples, originating respectively from the NS-NS merger and NS-BH merger.Here, based on the duration distribution (wider range of ∼ 0.05 s -3 s, and a longer typical value of ∼ 0.4 s), the sore "red" kilonova component and other arguments, we consider that the short GRB sample with extended emission (along with GRB 211211A and GRB 060614) originates from the NS-BH merger.Our claimed distinguished duration distributions and kilonova property could be further tested for a larger sample in the future.Meanwhile, future gravitational wave observation can also check our opinion.

Methods
The gravitational wave observation for the NS-BH coalescence.The NS-BH coalescence has been confirmed recently by gravitational wave observation (GW200105 and GW200115), in LIGO's and Virgo's third observing runs 50 .For GW200105, the masses of the BH and NS are 8.9 +1.2 −1.5 M and 1.9 +0.3 −0.2 M , respectively.While for GW200115, the masses are 5.7 +1.8 M and 1.5 +0.7 −0.3 M , respectively.Unfortunately, no electromagnetic counterpart (GRB or kilonova) has been identified.The reason may be the low projected-aligned spin for the BH, based on GW observations.Namely, these two mergers are plunging events 51,52 .However, the BHs in some observed high-mass X-ray binaries, which could be the progenitors of the NS-BH system, are claimed to have a high spin 53 .So far, whether the GRB can be produced by NS-BH coalescence is still uncertain.More detected NS-BH coalescences in the future will surely settle down this.In this work, for GRB 211211A, we find convincing evidence that it (and other short GRBs with EE) comes from the NS-BH merger.
The kilonova thermal emissions from the NS-NS merger and NS-BH merger.The neutron-rich ejecta, from the NS-NS merger or NS-BH merger, will undergo rapid neutron capture (r-process 54 ) nucleosynthesis, producing the heaviest elements like platinum and Lanthanide.Then, the radioactive decay of these unstable nuclei will power a rapidly evolving (∼ days), supernova-like transient in NIR or optical band, known as "kilonova" 14,30 .
For the NS-NS merger, the kilonova emission consists of the "red" component (the opacity κ = 10 cm 2 g −1 , from lanthanide-rich dynamical ejecta in the equatorial plane), the "blue" component (κ = 0.5 cm 2 g −1 , from lanthanide-free dynamical ejecta in the polar directions, produced by shock heating in the contact interface between the merging stars; the dynamical shock enriches the electron fraction and thus lowers the neutron fraction), and the "purple" component (κ ∼ 3 cm 2 g −1 , from intermediate-opacity isotropic disk wind ejecta; moderate neutrino irradiation from the longer-lived NS remnant lowers the lanthanide fraction) (see Ref. 30 and Extended Data Figure 1).
For the NS-BH merger, the kilonova emission only contains the "red" components from the dynamical ejecta (in the equatorial plane, by tidal forces that disrupt the NS) and the disk wind ejecta.
The cocoon thermal emission.When the jet propagates crosses the stellar envelope (for long GRBs) or the dynamical ejecta (for short GRBs), the dense material in the envelope or ejecta will be spilled sideways, forming a cocoon 28 that engulfs the jet and collimates it to make the structured jet.The GRB jet dissipates significant (may comparable) energy of itself to the cocoon through the shock.The adiabatic cooling of shock-heated material (outside the struc-tured jet, see Extended Data Figure 1) produces a transient quite similar to the blue kilonova (at optical band, peaks at ∼ 1 day or much earlier).
Cocoon plus "red" kilonova model fitting for the afterglow-subtracted optical-NIR data 10 .We fit the afterglow-subtracted photometry with the modified model (cocoon plus "red" kilonova) using the MOSFiT (Modular Open Source Fitter for Transients 55 ) platform.The fitting goodness for EMCEE sampling is given by the WAIC ("Watanabe-Akaike information criterion" 56 ).Here, WAIC ∼ 21 is obtained for 49 data points.For the nested sampling with DYNESTY, the total model evidence is ln(Z) ∼ 21, almost the same as that obtained in Ref. 10 (ln(Z) ∼ 20-25).The shown results are for the EMCEE sampling.
The mass of the "red" kilonova component, M ejr 0.030M , is consistent with that in Ref. 10 (M ejr 0.025M ).For DYNESTY sampling, higher M ejr is preferred (M ejr 0.05M ), s may also be higher.A significant improvement for our fitting, using the cocoon component instead of the "blue" and "purple" kilonova components 10 , is the better explanation of the late-time (1.68 days, 2.68 days and 5.11 days) i-band data.
The photosphere thermal model for γ-ray prompt emission.The existence of the photospheric emission is the basic prediction of the classical fireball model 57,58 for GRB, because the optical depth τ at the jet base is much larger than unity 59 .As the fireball expands and the optical depth drops down, the internally trapped thermal photons finally escape at the photosphere (τ = 1).Indeed, based on the spectral analysis, a quasi-thermal component has been found in several Fermi GRBs (especially in GRB 090902B 60 ).But whether the typical observed Band function 61 (smoothly joint broken power law) or cutoff power law (CPL) can be explained by the photosphere emission, namely the photosphere emission model, remains unknown.If this scenario is true, the quasi-thermal spectrum should be broadened.Theoretically, two different broadening mechanisms have been proposed: subphotospheric dissipation (namely the dissipative photosphere model [62][63][64] ) or geometric broadening (namely the probability photosphere model 20-24, 33, 65 ).
Previous observed supports for the photosphere emission model.First, a quasi-thermal component has been found in a great amount of BATSE GRBs 66 and several Fermi GRBs (especially in GRB 090902B and GRB 170817A) 2,16,60 .Second, lots of bursts have a low-energy spectral index α harder than the death line (or the maximum value, α = −2/3) of the basic synchrotron model, especially for the short GRBs and the peak-flux spectrum 67,68 .Third, the spectral width is found to be quite narrow for a significant fraction of GRBs 69 .Fourth, for a half or more of the GRBs, the cutoff power law is the best-fit empirical model, indicating that the photosphere emission model can naturally interpret their high-energy spectrum.Fifth, recently in Ref. 25 , by dividing the GRB sample into three sub-samples based on the prompt efficiency γ , the observed E p -E iso distribution 70 of each sub-sample can be perfectly explained by the photosphere emission model.Also, for each subsample, the characteristics of the X-ray afterglow and optical afterglow are well consistent with the predictions of the photosphere emission model.
The probability photosphere model.For the traditional photosphere model, the photosphere emission is all emitted at the photospheric radius R ph , where the optical depth for a photon propagating towards the observer is equal to unity (τ = 1).But, if only there is an electron at any position, the photon should have a probability to be scattered there.For an expanding fireball, the photons can be last scattered at any place in the fireball with a certain probability.Thus, the traditional spherical shell photosphere is changed to a probability photosphere, namely the probability photosphere model 20 .
For the probability photosphere model, the observed photosphere spectrum is the overlapping of a series of blackbodies with different temperatures, thus its low-energy spectrum is broadened.After considering the jet with angular structure, the observed typical low-energy photon index α ∼ −1.0 67,68 , spectral evolution and E p evolutions (hard-to-soft evolution or E p -intensity tracking 71 ) can be reproduced 23,24,33 , from the theoretical perspective.
The structured jet with θ c,Γ < θ c,L .For GRBs, the jet launched by the center engine will be collimated, by the gas envelope of the progenitor star for long GRBs 34 , and the dynamical ejecta for short GRBs 72,73 .Thus, the structured jet 74,75 should both exist.Noteworthily, based on the unusual performance of the prompt emission and the afterglow of GRB 170817A (the first joint detection of short GRB and gravitational wave), a structured jet in it is strongly favored 76,77 .
Normally, for a structured jet, θ c,Γ = θ c,L is assumed for simplicity.But whether this is the real situation is quite uncertain and without proper reasons.Considering that θ c,Γ = θ c,L is true, the material density within (or beyond) the core should be rather isotropic.This may conflict with the collimation condition.Theoretically, the progenitor envelope (or dynamical ejecta) is matter-dominated and should make the shocked jet have an enhanced material density at larger angle when collimation happens.Then, because Γ(θ) ∝ L/M (θ), the Lorentz factor should start to decrease even when the L remains constant (see Figure 3 in Ref. 35 and similar discussion in Ref. 24 ).Besides, θ c,Γ < θ c,L is supported in many simulations, including both hydrodynamical ones 34,35,38 and magnetohydrodynamical ones (MHD, Refs. 36,37 .
On the other hand, as shown in Extended Data Figure 1, the jet energy E = ΓM c 2 (mainly for the inner isotropic core, M jet ∼ 10 −6 − 10 −5 M 41 , and Γ ∼ 100 − 1000) is comparable to the outside cocoon energy E M v 2 (M cocoon ∼ 10 −3 − 10 −1 M , and v ∼ 0.1 − 0.6c; Ref. 28 ).Thus, the large θ c,L , may extending to the non-relativistic cocoon region, is reasonable (may available for most bursts, and at least for a part).Besides, numerical simulations for the jet-cocoon interaction 28 suggest roughly constant energy within the cocoon (including the jet region, especially the mildly relativistic region; see Figure 2 Saturated and unsaturated acceleration, γ 50% and γ 50% 25 respectively.For the photosphere emission model, γ 50% should correspond to the saturated acceleration Here, R ph ∝ L iso /Γ 3 .The adiabatic cooling exists at r R s . While, γ 50% should correspond to the unsaturated acceleration (R ph R s ; Γ η) case, since E iso /E k = ηM c 2 /ΓM c 2 = η/Γ 1.The adiabatic cooling does not exist.
The duration distribution test for long and short GRBs.As mentioned above, the photosphere duration stretching expects that the bursts with γ 50% (saturated) should have a longer duration than that with γ 50%.In Extended Data Figure 2a, we compare the duration distributions of the γ 50% and γ 50% samples (see Ref. 25 ) for the long GRBs.As expected, the γ 50% sample has a much longer (∼ 0.2 dex, namely ∼1.6 times) duration.Also, apart from the mean values, these distributions are similar.Surprisingly, for the short GRBs (see Extended Data Figure 2b), the γ 50% sample has a much shorter (∼ 0.1 dex, namely ∼ 1.3 times) duration.Furthermore, the distribution of the γ 50% sample is less extended.Considering these two properties, we think these two distinguished samples may originate from different sources.According to the simulations of NS-NS 42 and NS-BH mergers 26,27,43 , the short GRBs from NS-NS merger typically have a smaller duration of ∼ 0.1 -0.2 s 42 , entirely consistent with the result of the γ 50% sample (Note that the mean value ∼ 10 −0.5 should be decreased by 1.6 -2 times due to the photosphere duration stretching effect).While, the short GRBs from the NS-BH merger typically have a larger duration of ∼ 0.8 s 43 , more consistent with the result of the γ 50% sample (for the MP).
Besides, for the NS-BH merger, since the disk mass M disk (crucial to the duration) depends on many parameters of the NS and BH, as follows (see Ref. 26 ): Here, M NS and R NS are the mass and radius of the NS, C = M NS /R NS means the NS compactions.R = M BH /M NS and M BH is the mass of the BH.And, R ISCO is the radius of the innermost stable circular orbit, which strongly depends on the BH spin a BH .The duration distribution for the NS-BH merger is expected to be more extended, just similar to the distribution of the γ 50% sample (could be smaller to ∼ 0.05 s, and larger to ∼ 3 s).
The theoretically predicted αθ c,Γ distribution and observed αθ jet distribution.As mentioned above, the observed overlapped photosphere spectrum should be broadened.Previous studies 23,24,33 show that, for a uniform jet α can be softened to α ∼ 0. Whereas, for the structured jet considered here, α could be much softer, reproducing the observed typical α ∼ -1.Because that, according to Equation (1), the extra low-energy contribution within angle of is the angle aparted from the line of sight).When smaller Γ is obtained for θ LOS > 0, the extra low-energy component will be greatly enhanced.
In Extended Data Figure 3a, we show the schematically theoretically-predicted αθ c,Γ distribution (the dashed lines) 23,33 .Two categories are obviously seen: smaller θ c,Γ and larger θ c,Γ .Notice that, acturally θ c,Γ < θ c,L θ jet is likely expected for structured jet.But, we think these three quantities are likely to change with the same trend.Namely, with larger θ jet , θ c,Γ and θ c,L are both larger.For smaller θ c,Γ (θ c,Γ • Γ ∼ 1), regardless of the θ v , the decreasing-Γ component always exists within θ LOS 5/Γ.Thus, the α is quite soft, clustered around ∼ -1.
For larger θ c,Γ , θ v has great effect.Normally, θ v < θ c,Γ is expected.Then, within θ LOS 5/Γ, the isotropic-Γ component dominates and the decreasing-Γ component contributes less.Thus, the α should be quite hard, ranging from 0 to ∼ -0.7.However, when θ v θ c,Γ is obtained with lower chance, the decreasing-Γ component dominates, α can range from ∼ -1 to -2.Interestingly, the above theoretically predicted αθ c,Γ distribution seems to be consistent with the observed αθ jet distribution (θ jet is taken from Ref. 78 ), shown also in Extended Data Figure 3a.For smaller θ jet (may θ c,Γ also), the α is clustered around ∼ -1.When θ jet is larger, the α becomes much harder (reaching ∼ 0), except for a few bursts (including GRB 211211A and GRB 060614).These outliers are likely to be due to the larger θ v ( θ c,Γ ).The photosphere explanation for the α-flux-tracking evolution 11,79 .As shown in Extended Data Figure 5b, the softer α can be obtained in the earlier time (0.3 s).Unlike the origin of the softer α in the later time (due to the delayed low-energy high-altitude emission emitted around the peak time, see discussions in Refs. 23,79 , it is due to the comparability of the peak energies for the isotropic-Γ component and the decreasing-Γ component.These two peak energies both strongly depend on the luminosity and have distinguished dependences (since Γ and θ v are different) if any one is in the saturated-acceleration regime.For the strongly saturated (at the peak time) case, such as our adopted parameters and GRB 211211A, in the earlier time (0.3 s) the peak energy of the decreasing-Γ component is smaller (but not extremely, ∼ 10 times) than that of the isotropic-Γ component, thus contributing a great part to the low-energy spectrum and making the α softer.So, combined with the softer α after the peak time, the special αflux-tracking evolution 11,79 in a few bursts (including GRB 211211A) is explained.
The 2SBPL spectra in GRB 211211A.For the spectra of GRB 211211A, just as stated in Ref. 13 (with low-energy BAT data), the 2SBPL model is likely to be the better empirical model (statistic/dof = 480/353 for 2SBPL model, statistic/dof = 642/355 for Band function, of 3.0 ± 1.0 s).In Ref. 11 , similar conclusion is drawn.For the fit with the CPL model, the residuals show obvious rising trend in the low-energy (∼ 8 -30 keV) end, indicating a much harder (α ∼ -0.5 to 0) component there (pgstat/dof = 440.78/362, of 3.4 -3.5 s).The low-energy hard component may exist in all the bursts (with almost fixed energy break E b ∼ 20 keV, as discussed in the following for the photosphere model).But considering the high redshift (z ∼ 1 -2) for most bursts, the energy break will decrease to ∼ 8 keV (beyond the Fermi energy range, thus not obvious).For GRB 211211A, with z = 0.076, the low-energy hard component is significant.
The drawbacks for the synchrotron origin 13 of the 2SBPL spectrum.First, as shown in Figure 3 and Table 1 of Ref. 13 , α 1 is all harder than the synchrotron death line α = − 2/3.Second, the high-energy index β is almost close to ∼ -3, indicating a cutoff in the highenergy end.Third, α ∼ -1 according to Ref. 11 , while α 2 ∼ -3/2 is obtained in Ref. 13 .This conflict is likely to come from the inaccurate power-law formular in the high-energy end for the 2SBPL model (thus the statistic of the CPL model is comparable to that of the 2SBPL model, even though its low-energy fitting is bad).Since for the CPL spectrum, towards the higher energy, the α will gradually change from -1 to -Inf, mimicking a much softer α ∼ -3/2 when the power-law formular is taken.In total, the best-fit empirical model for GRB 211211A is likely to be the CPL high-energy spectrum combined with a smoothly broken power law in the low-energy end, which resembles the photosphere spectrum in Extended Data Figure 4a.
The low-energy break E b explanation by the photosphere model.Notice that, for the photosphere model, the low-energy break is naturally expected.Since the low-energy blackbody component comes from much earlier injection (with larger θ LOS , then arriving later and obtaining lower energy due to the Doppler effect).In real GRB and our calculation, limited injection duration (∼ a few seconds) is expected.So, the lowest blackbody energy (mainly depends on the injection duration) should correspond to the injection of ∼ seconds earlier, and regardless of the prompt injection.Then, an almost fixed low-energy break is predicted (for ∼ 1 second, E b ∼ 10 keV, see the red and green lines of Figure 2 in Ref. 23 ).The photosphere low-energy break (∼ 10 keV) has been shown in Figures 6 -9 of Ref. 23 (especially the red lines).For GRB 211211A, as shown in Figure 3 and Table 1 of Ref. 13 , the almost fixed low-energy break (∼ 20 -30 keV) indeed exists, strongly favoring the photosphere origin.
The possible photosphere explanation for the extended emission lasting ∼ 100 s.As mentioned above, the observed late-time (a few tenths seconds) low-energy (∼ keV) SBPL spectrum could be produced by the photosphere high-altitude effect (see Extended Data Figure 4a).But, whether the theoretical flux decay can match the flux decay of observed EE is very crucial.In Extended Data Figure 5a, we compare the theoretical flux decay with that of GRB 211211A.Obviously, the observed decay could be reproduced.Notice that, an extra injection at ∼ 20 -30 s is considered, which may come from the fallback accretion of the black hole, for the NS-BH merger.The photosphere high-altitude effect extends this ∼ 10 s duration to ∼ 100 s.
The smoother decay of the X-ray afterglow for the structure jet with θ c,Γ < θ c,L .As shown in Refs. 74,75  for the structured jet with off-axis θ v (θ v > θ c,L , and θ c,Γ = θ c,L ), the observed smoother component (close to a plateau, flux f ∼ t −0.5 obs ) before normal decay (f ∼ t −1 obs ) in most X-ray afterglows 31 can be explained.However, the off-axis condition should be unusual, which conflicts with the usual appearance of the smoother decay.
In this work, as shown in Extended Data Figure 7, we find the usual on-axis condition (θ v < θ c,L , see Figure 1a) can also reproduce the observed smoother decay, when θ c,Γ < θ c,L is adopted.This is quite reasonable, since the peak time of the afterglow T p strongly depends on the Γ (Γ ∝ (E k ) 1/8 • [T p /(1 + z)] −3/8 ) 41 .The significant Γ structure within θ c,L will surely result in the overlapping of the afterglow components with different arriving times.
For θ v θ c,Γ considered here, the Γ in the light of sight is smaller (T p is larger), contributing to the normal decay at the later time.The inner jet (with larger Γ, larger θ LOS ) will contribute to the earlier component.The flux will be suppressed due to the larger θ LOS , causing the observed smoother decay (or the plateau of GRB 211211A and GRB 060614, see Ref. 11 ).
. For the estimate from T c, X in Figure 4, a basic constant of 1.27×[17×9×9/(2 10 ×3.14×4)] 1/8 is adopted, as for the T p,op in Ref. 25 (the gray plus; homogeneous medium with density n = 1 cm −3 and E k = 5 • E iso are still assumed).But another factor of ∼ 2.0 is needed for the long GRBs to match these two equations completely.We think it is due to the slight difference between the defined T c, X and the real peak time for the observed smaller Γ (T c, X may be ∼ 4.0 times larger), since the turning after the plateau is quite smooth.The real peak time may be the earliest turning time (T c, X donates the middle), and the smooth turning component may originate from the weak emission of the most outside jet region (with the smallest Γ, arriving later).Other scenarios, such as smaller density, wind medium or different constant of the photosphere equation, for this off-core (large θ v ) case may also be possible.
For the short GRBs (the constant of the photosphere equation should be ∼ 17/1.8),adopting the above factor of ∼ 2.0, an extra constant of ∼ 2.5 is needed.We think it is due to the smaller density n ∼ 2.5 −8 ∼ 0.001 cm −3 (indeed found for the short GRBs 80,81 ) or the smaller photosphere constant than 17/1.8.
Obviously, the bursts with internal plateau (all are long GRBs, considering the factor of ∼ 2.0) significantly deviate from the other samples.This indicates that they do have a different origin, likely from the late-time injection of the magnetar 82,83 .
Extended Data Figure 1: The schematic diagram of distinguished kilonova components and jet structures for the NS-BH and NS-NS mergers.For the NS-NS merger, the kilonova emission 30 consists of the "red" component (from lanthanide-rich dynamical ejecta in the equatorial plane), the "blue" component (from lanthanide-free dynamical ejecta in the polar directions, produced by shock heating), and the "purple" component (from intermediate-opacity isotropic disk wind ejecta).For the NS-BH merger, the kilonova emission only contains the "red" components from the dynamical ejecta (in the equatorial plane) and the disk wind ejecta.As much dynamical ejecta exists in the jet propagating direction, for the NS-NS merger, the structured jet (mainly the outer mild-relativistic part) should be more significant, namely with a smaller isotropic core θ c,Γ .For the NS-BH merger, θ c,Γ should be larger.The rare rate of θ v θ c,Γ is consistent with that for GRB 211211A and GRB 060614.Moreover, the large θ v naturally explains the observations of kilonova and cocoon emissions (see Figure 3).Extended Data Figure 2: The duration distributions of the ϵ γ ≲ 50% (saturated, red) and ϵ γ ≳ 50% (unsaturated, green) samples, for long and short GRBs.a, The duration distribution for long GRBs.As expected by the photosphere duration stretching effect, the ϵ γ ≲ 50% sample has a much longer duration (∼ 1.6 times).b, The duration distribution for short GRBs, including GRB 211211A and GRB 060614.Considering the longer duration (∼ 0.8 s) and more extended distribution (∼ 0.05 -3.0 s), the ϵ γ ≳ 50% sample (possessing EE; along with GRB 211211A and GRB 060614) may originate from the NS-BH merger 26,43 .The even longer duration of GRB 211211A and GRB 060614 is likely to be caused by the larger θ v (transforming to the saturated regime, and thus ϵ γ ≲ 50%).

30
Extended Data Figure 2: The duration distributions of the γ 50% (saturated, red) and γ 50% (unsaturated, green) samples, for long and short GRBs.a, The duration distribution for long GRBs.As expected by the photosphere duration stretching effect, the γ 50% sample has a much longer duration (∼ 1.6 times).b, The duration distribution for short GRBs, including GRB 211211A and GRB 060614.Considering the longer duration (∼ 0.8 s) and more extended distribution (∼ 0.05 -3.0 s), the γ 50% sample (possessing EE; along with GRB 211211A and GRB 060614) may originate from the NS-BH merger 26,43 .The even longer duration of GRB 211211A and GRB 060614 is likely to be caused by the larger θ v (transforming to the saturated regime, and thus γ 50%).
Extended Data Figure 4: The comparison of the spectral evolutions of the theoretical photosphere model and GRB 211211A (for the empirical 2SBPL model fitting 13 ).a, The spectral evolution of theoretical photosphere model (θ c,Γ • η = 4, θ v = 1.5 • θ c,Γ ; and τ 1 = 32, τ 2 = 0.5, ts = −1.6).The spectrum transfers from the early 2SBPL spectrum (with high-energy exponential cutoff) to the late-time SBPL spectrum.b, The spectral evolution of GRB 211211A./ bol > HUJV @ / bol W V W 7HPSHUDWXUH> .@7 eff W V V W 'D\VDIWHUEXUVW 5DGLXV> FP@ 5 ph W V V W Extended Data Figure 6: The cocoon explanation of the evolutions of bolometric luminosity, effective temperature, and photospheric radius for the afterglow-subtracted optical-NIR data of GRB 211211A.The data points (markers) are taken from Ref. 12 .The solid lines represent the theoretical power-law evolutions of cocoon emission 46 , for s = −d ln E/d ln v = 1 (the typical value from cocoon numerical simulation 30 ).Obviously, the observed evolutions can be well reproduced by the cocoon emission.

a b
Extended Data Figure 8: Comparison of the X-ray afterglow light curves for the short GRB samples with (left) and without (right) extended emission.a, The power-law shapes of the Xray afterglow light curves for the short GRB sample with EE. b, The X-ray afterglow light curves show a significant plateau for the short GRB sample without EE.If the plateaus do result from θ v ≳ θ c,Γ (see Extended Data Figure 7), the power-law shape and plateau, for bursts with and without EE, well correspond to the jet structures for the NS-BH merger (larger θ c,Γ ) and NS-NS merger (smaller θ c,Γ ) (see Extended Data Figure 1).
Extended Data Figure 8: Comparison of the X-ray afterglow light curves for the short GRB samples with (left) and without (right) extended emission.a, The power-law shapes of the Xray afterglow light curves for the short GRB sample with EE. b, The X-ray afterglow light curves show a significant plateau for the short GRB sample without EE.If the plateaus do result from θ v θ c,Γ (see Extended Data Figure 7), the power-law shape and plateau, for bursts with and without EE, well correspond to the jet structures for the NS-BH merger (larger θ c,Γ ) and NS-NS merger (smaller θ c,Γ ) (see Extended Data Figure 1).ORJE iso E k 1XPEHUV $OOEXUVWV %XUVWVZLWKSODWHDX Extended Data Figure 9: Comparison of the E iso /E k distributions for the sample with significant plateau (purple) and the whole sample (green).The distribution for the whole sample is taken from Figure 10 in Ref. 25 .And the sample with plateau is taken from Ref (T c, X 1000 is adopted, to omit the weak plateau).Obviously, E iso /E k 1 is obtained for the sample with significant plateau, indicating a small prompt efficiency ( γ 50%).This is quite consistent with the smaller Γ (see Figure 4) and larger θ v (see Extended Data Figure 7), under the photosphere framework for prompt emission ( γ ∝ Γ 8/3 ).

1 )cFigure 1 :
Figure1: The jet structure and the duration stretching effect for the saturated photosphere.a, The considered structured jet with θ c,Γ < θ c,L (Γ ≃ η).The considered viewing angle θ v for GRB 211211A is a bit larger than θ c,Γ .b, The photosphere luminosity L ph profiles for different Lorentz factor structures, with the same injected duration of T 90 ∼ 3 s (see also Figure16in Ref.24 ).The observed duration can be stretched to T 90 ∼ 6 s, for the strongly saturated cases (the blue and cyan markers).Note that, the L ph profiles have all been normalized to L w,p = 10 52 erg s −1 at 2.4 s.Actually, for all the saturated cases (except for the orange box), the luminosity is 1-10 times smaller.c, The L ph profile for θ c,Γ • Γ = 4, θ v = 1.5 • θ c,Γ (red squares).The light curve of GRB 211211A (blue plus) can be well reproduced, including the T 90 ∼ 10 s and the double pulses.

Figure 2 : 6 Figure 2 :
Figure 2: The spectral fitting using the physical photosphere model, for the time-resolved spectra of GRB 211211A measured from 2.0 -3.0 s (a and d), 12.0 -14.0 s (b and e), 40.0 -45.0 s (c and f).a, b and c, De-convolved (black error bars) and modeled (red line) photon number spectra.d, e and f, Observed (bars) and modeled (lines) photon count spectra.

Figure 3 :
Figure3: Cocoon emission and NS-BH-merger evidence in GRB 211211A.The afterglowsubtracted optical-NIR light curves of GRB 211211A (the markers)10 are well fitted by the combination (solid lines) of the early cocoon emission (dotted lines) and the late "red" kilonova component (dot-dashed lines).Except for the NIR band, the dotted lines are overlapped with the solid lines, indicating that the cocoon emission well explains the early optical light curves.Besides, the existence of a sole "red" kilonova component, without significant "blue" or "purple" kilonova components, strongly supports30 the NS-BH merger origin of GRB 211211A (see Extended Data Figure1).

Figure 4 :
Figure 4: Consistent estimates of Γ from photosphere model (with the prompt emission data,

1 ) 5 : 32 Extended Data Figure 5 :
Γ * Γ =4, p=4, θ v =1.5 θ c,Γ θ c,Γ * Γ =10, p=4, θ v =1.4 θ c,Γ The photosphere explanations for the extended emission (lasting ∼ 100 s) and the α-flux-tracking evolution79 .a, Comparison of the late-time light curve (extended emission, blue plus) of GRB 211211A and the theoretical photosphere flux decay (red and green squares).The green squares consider an extra injection at ∼ 20 -30 s, which may come from the fallback accretion of the black hole, for the NS-BH merger.b, Comparison of the theoretical photosphere spectra for the peak time (2.4 s, red line, without the double pulses for θ c,Γ • Γ 0 = 10,θ v = 1.4 • θ c,Γ) and the earlier time (0.3 s, orange line).The softer α can be obtained in the earlier time (0.3 s).The photosphere explanations for the extended emission (lasting ∼ 100 s) and the α-flux-tracking evolution79 .a, Comparison of the late-time light curve (extended emission, blue plus) of GRB 211211A and the theoretical photosphere flux decay (red and green squares).The green squares consider an extra injection at ∼ 20 -30 s, which may come from the fallback accretion of the black hole, for the NS-BH merger.b, Comparison of the theoretical photosphere spectra for the peak time (2.4 s, red line, without the double pulses for θ c,Γ • Γ 0 = 10, θ v = 1.4 • θ c,Γ ) and the earlier time (0.3 s, orange line).The softer α can be obtained in the earlier time (0.3 s).

Figure 11 :
Parameter constraints of our photosphere model fitting for the time-resolved spectrum measured from T0 + 2.0 s to T0 + 3.0 s.Histograms and contours show the likelihood map.Red crosses illustrate the best-fit values and 1-sigma error bars.Extended Data Figure 12: Parameter constraints of our photosphere model fitting for the time-resolved spectrum measured from T0 + 12.0 s to T0 + 14.0 s.For the luminosity injection, τ 1 = 32, τ 2 = 0.5, ts = 3.4 is adopted.The luminosity peaks at tp = ts + (τ 1 • τ 2 ) 1/2 = 7.4 s, corresponding to the second pulse.

Table 1 :
Extended Data Best-fit parameters (see the meanings in Extended Data Figure10) using cocoon plus "red" kilonova model for the afterglow-subtracted optical-NIR light curves in GRB 211211A