Evidence of Photosphere Emission Origin for Gamma-Ray Burst Prompt Emission

The physical origin of gamma-ray burst (GRB) prompt emission (photosphere or synchrotron) is still subject to debate after five decades. Here, firstly we find that many observed characteristics of 15 long GRBs, which have the highest prompt emission efficiency $\epsilon _{\gamma}$ ($\epsilon_{\gamma }\gtrsim 80\%$), strongly support the photosphere (thermal) emission origin: (1) the relation between $E_{\text{p}}$ and $E_{\text{iso}}$ is almost $E_{\text{p}}\propto (E_{\text{iso}})^{1/4}$ , and the dispersion is quite small; (2) the simple power-law shape of the X-ray afterglow light curves and the significant reverse shock signals in the optical afterglow light curves; (3) best-fitted by the cutoff power-law model for the time-integrated spectrum; (4) the consistent efficiency from observation (with $E_{\text{iso}}/E_{k}$) and the prediction of photosphere emission model (with $\eta /\Gamma $). Then, we further investigate the characteristics of the long GRBs for two distinguished samples ($\epsilon _{\gamma }\gtrsim 50\%$ and $\epsilon _{\gamma }\lesssim 50\%$). It is found that the different distributions for $E_{\text{p}}$ and $E_{\text{iso}}$, and the similar observed efficiency (from the X-ray afterglow) and theoretically predicted efficiency (from the prompt emission or the optical afterglow) well follow the prediction of photosphere emission model. Also, based on the same efficiency, we derive an excellent correlation of $\Gamma \propto E_{\text{iso}}^{1/8}E_{\text{p}}^{1/2}/(T_{90})^{1/4}$ to estimate $\Gamma $. Finally, the different distributions for $E_{\text{p}}$ and $E_{\text{iso}}$, and the consistent efficiency exist for the short GRBs. Besides, we give a natural explanation of the extended emission ($\epsilon _{\gamma }\lesssim 50\%$) and the main pulse ($\epsilon _{\gamma }\gtrsim 50\%$).

The paper is organized as follows. In Section 2, we state the data accumulation and the scaling relations predicted by the photosphere model. In Section 3, we describe the evidence from long GRBs with extremely high prompt efficiency γ ( γ 80%). Then, in Section 4, the evidence from long GRBs with γ 50% and γ 50% is shown. In Section 5, we illustrate the evidence from short GRBs. A brief summary is provided in Section 6.
2. DATA ACCUMULATION AND THE SCALING RELATIONS EXPECTED BY THE FIREBALL MODEL 2.1. Data Accumulation Generally, the radiation efficiency of the prompt emission γ is defined as E γ /(E γ + E k ). Here, E γ is the radiated energy in the prompt phase and E k is the remaining kinetic energy in the afterglow phase.
To obtain γ , the isotropic energy E iso (namely E γ ) and the L X,11h (the late-time X-ray afterglow luminosity at 11 hr) data should be accumulated for the GRB sample with the redshift z 1 . Because that, L X,11h is roughly proportional to the E k (see Appendix B.1).
(1) For 46 long bursts before GRB 110213A, we use the L X,11h data given in D' Avanzo et al. (2012). Also, E iso , the isotropic luminosity L iso , the peak spectral energy in the rest frame E p,z (or E p ), the low-energy spectral index α, and the high-energy spectral index β are taken from Nava et al. (2012).
Our spectral fitting results for the γ 80% sample are given in Table 4. And for several bursts of the γ 50% sample, the perfectly consistent observed efficiency (with E iso /E k ) and the theoretically predicted efficiency of the photosphere model (from the prompt emission) are given in Table 5.
In Table 6, for the long-GRB sample with the detection of the peak time of the early optical afterglow (namely T p or T p,op , to obtain the Lorentz factor of the outflow Γ and then (R ph /R s ) −2/3 ) in Ghirlanda et al. (2018), the consistent predicted efficiency (from the prompt emission and the optical afterglow) of the photosphere model is provided. Also, E iso , L iso , and E p,z are taken from Ghirlanda et al. (2018). In Table 7, its subsample (9 bursts) with the maximum Γ (for fixed L iso ; according to the photosphere model, γ = 50%) is provided.
In Table 8, the short-GRB sample (with L X,11h derived in this work) is given, along with that possessing extended emission. E iso and E p,z are taken from Minaev & Pozanenko (2020).
E iso is generally estimated by E iso = 4πD 2 L S γ /(1 + z), where S γ is the time-integral fluence in the 1 − 10 4 keV energy range in the rest frame (in units of erg cm −2 ) and D L is the luminosity distance. L iso is estimated as L iso = 4πD 2 L F p , where F p is the peak flux (in units of erg cm −2 s −1 ). T 90,i is calculated as T 90,i = T 90,ob /(1 + z), where T 90,ob is determined by the time range between the epochs when the accumulated net photon counts reach the 5% level and the 95% level. And E p = (1 + z) · E p,ob , where E p,ob is determined by the peak energy in the νF ν spectrum.
Typically, the afterglow peak time T p is estimated from the optical afterglow peak time T p,op , since the early Xray afterglow peak can be produced by "internal" mechanisms (such as the prolonged central engine activity) or bright flares. Also, the bursts with an early multipeaked optical light curve or an optical peak preceded by a decreasing light curve should be excluded (see Ghirlanda et al. 2018).

Scaling Relations Predicted by the Photosphere
Model For the photosphere emission model, γ 50% and γ 50% should correspond to the unsaturated acceleration case (Γ η; R ph < R s ; E iso /E k = ηM c 2 /ΓM c 2 = η/Γ 1) and the saturated acceleration case (Γ = η; When the outflow Lorentz factor Γ at the photosphere radius R ph is less than the baryon loading η (η = E/M c 2 , where E and M are the injected energy and the baryon mass at the outflow base, respectively), the photosphere emission is in the unsaturated acceleration case (or R ph < R s ; where R s = ηr 0 , and r 0 is the initial acceleration radius). In this case, Γ = R ph /r 0 .
For the unsaturated acceleration case, the observed temperature T ob = D· T comoving = Γ· (T 0 /Γ) = T 0 . Here, D is the Doppler factor, T comoving is the photon temperature in the outflow comoving frame, and T 0 is the temperature at the outflow base r 0 . Thus, E iso = E = ηM c 2 . Also, because all the thermal energy is released at the photosphere radius (where there is no remaining energy to accelerate the jet), the Lorentz factor in the afterglow phase will remain as Γ, namely E k = ΓM c 2 . So we should have E iso /E k = ηM c 2 /ΓM c 2 = η/Γ 1 (see Figure 1(b) and 2(e)), corresponding to γ 50%.
For the photosphere emission model, the peak energy of the observed spectrum E p corresponds to the temperature of the observed blackbody T ob . In the unsaturated regime (R ph < R s , γ 50%), T ob = T 0 . Since For γ 50% (R ph > R s ), the outflow performs adiabatic expansion at r > R s . Thus, the comoving temperature decreased as T comoving = (T 0 /Γ) · (r/R s ) −2/3 . The escaped photons at  The L iso − Γ correlation and the E iso /E k − η/Γ correlation for the selected extremely high-efficiency GRBs. (a) The correlation of L iso and Γ. Obviously, the tight correlation of Γ ∝ (L iso ) 0.29 (the red stars, reduced χ 2 = 0.007) is found, which is well consistent with the prediction of the neutrino annihilation from the hyperaccretion disk, Γ ∝ (L iso ) 7/27 = (L iso ) 0.26 . Thus, the jet is likely to be thermaldominated. The cyan dashed line shows the L iso − Γ correlation found in Lü et al. (2012) (reduced χ 2 = 0.038, for the large Γ sample of green circles (Xue et al. 2019)). (b) The correlation of E iso /E k and η/Γ. A significant linear correlation is found, and they are almost the same when we take E k,52 = 5 * L X,45 (reduced χ 2 = 0.138). This is well consistent with the predicted E iso /E k = ηM c 2 /ΓM c 2 = η/Γ by the photosphere emission model in the unsaturated acceleration regime.
So, E iso and E p should both decrease by the same factor of (R ph /R s ) −2/3 , compared with the E p ∝ (E iso ) 1/4 correlation for γ 50% (see Figures 4 and 5 For the γ 50% sample (saturated acceleration, E k = E), we should have and here r 2 1 r 2 0 * T 90. From the relation of log (E p ) = 2.54 + 0.25 log (E iso ) in Figure 3(b), we obtain r 1 = 8.45 × 10 8 cm. Then, from Equation (1) and Equation (2), we get The similar observed efficiency (with E iso /E k from the X-ray afterglow) and theoretically predicted efficiency (from the prompt emission or the optical afterglow) by the photosphere emission model for the bursts after GRB 110213A. (a) The distribution of E ratio (E ratio = [(Ep/2.7k) 4 * (4πr 2 1 ac)/E iso ] 1/3 , from the prompt emission) and E iso /E k for the γ 50% sample (see Table 2). They are found to be well centered around the equal-value line, and have a linear correlation (reduced χ 2 = 0.172). The dispersion is likely to be caused by the estimation error for E k , since many X-ray afterglow light curves are not the power law with a slope of −1. (b) and (c) The X-ray afterglow light curves for the bursts (7 bursts) with almost same E ratio and E iso /E k (see Table 5). We find all these light curves do have the power-law shape with a slope of ∼ −1. (d) The distribution of E ratio , E iso /E k and (R ph /Rs) −2/3 for the sample (6 bursts, see Table  2) with detections of peak time of the optical afterglow (to estimate the Γ and thus (R ph /Rs) −2/3 ). It is found that 1 burst has the almost same values for these three quantities and the other 5 bursts have the almost same values for two quantities of them (reduced χ 2 = 0.089 for all 12 markers). (e) The consistent efficiency from observation (with E iso /E k ) and the prediction of photosphere emission model (with η/Γ) for the γ 50% sample with detections of peak time of the optical afterglow (7 bursts, see Table 3; reduced χ 2 = 0.120). log(E p,z ) = 2.54 + 0.25 log(E iso ) 10 -2 10 -1 10 0 10 1 10 2 10 3 E iso ( × 10 52 erg)

(c)
Long GRBs Short GRBs Amati relation (E p,z ∝E 0.52 iso ), reduced χ 2 = 0.035 log(E p,z ) = 2.54 + 0.25 log(E iso ), reduced χ 2 = 0.009 Long GRBs Short GRBs 10 -1 10 0 10 1 10 2 10 3 L iso ( × 10 52 erg s −1 )  Table 1). (a) The E iso and calculated L X,11h (the late-time X-ray afterglow luminosity at 11 hours) distribution for the whole GRB sample used (117 bursts, after GRB 110213A) with redshift. The red stars with the smallest L X,11h represent the bursts with the highest prompt efficiency. (b) The Ep and E iso distribution for the selected 15 long GRBs. The best-fit result is log (Ep) = 2.54 + 0.25 log (E iso ), quite consistent with the Ep ∝ (E iso ) 1/4 relation predicted by the photosphere (thermal) emission model. (c) Comparison of the Ep and E iso distributions for the selected GRBs (red stars) and the large sample of long GRBs (see Figure 3 in Zhang et al. (2018b)) (yellow circles). Obviously, the dispersion for the selected GRBs is quite small (reduced χ 2 = 0.009) relative to that for the large sample (reduced χ 2 = 0.035). The black dotted line shows the Amati relation (Amati et al. 2002) for the large sample. (d) Comparison of the Ep − L iso distribution (red stars) and the Ep − E iso distribution (blue stars) for the selected GRBs. Likewise, the Ep ∝ (L iso ) 1/4 relation exists. The best-fit result is log (Ep) = 2.75 + 0.23 log (L iso ). And the dispersion (reduced χ 2 = 0.012) is found to be similar to that of Ep ∝ (E iso ) 1/4 .

thus
Combined with the abovementioned   Figure 7(b), for γ 50%, we have E ratio = (R ph /R s ) −2/3 . Considering and R s = Γ · r 0 , we have Thus, we can use the quantities of the prompt emission (E iso , E p and T 90 ) to estimate the Lorentz factor Γ, just as the obtained Γ = 10 3.33 L 0.46 iso E −0.43 p from the statistic fitting in Liang et al. (2015) (see Figures 7(c) and 8). Based on Equation (6), we derive Considering the constants, we obtain For the γ 50% case, from 10 -2 10 -1 10 0 10 1 10 2 10 3 E iso ( × 10 52 erg)  Tables 2 and 3) for the long GRBs. (a) The Ep and E iso distribution for the bursts after GRB 110213A (117 bursts, with L X,11h derived in this work). For the γ 50% (red circles) sample, the best-fit result is log (Ep) = 2.47 + 0.25 log (E iso ) (reduced χ 2 = 0.039; for the black dotted line of the typical Amati relation, reduced χ 2 = 0.079), well consistent with the predicted Ep ∝ (E iso ) 1/4 by the photosphere emission model. For the γ 50% (blue and cyan circles) sample, the up-most distribution is found well around log (Ep) = 2.54 + 0.25 log (E iso ), and the best-fit result is log (Ep) = 2.31 + 0.26 log (E iso ) (reduced χ 2 = 0.076; for the typical Amati relation, reduced χ 2 = 0.113), quite below that. (b) The Ep and E iso distribution for the bursts before GRB 110213A (46 bursts, with L X,11h given in D' Avanzo et al. 2012). The different distributions for Ep and E iso of the two distinguished samples ( γ 50% and γ 50%) are also found, similar to the distributions for the bursts after GRB 110213A.

Tightness of the Scaling Relations and Data Errors
To quantitatively measure the tightness of the scaling relations, we calculate the statistical value for each relation of lg y = a lg x − b (for example, lg E p = 0.25 lg E iso + 2.54). The reduced χ 2 = χ 2 /degrees of freedom (dof = N − 2) for each relation is given in the caption for the corresponding figure. Note that χ 2 /dof is approximate to the typical dispersion measure σ (both in units of dex). In Figure 10, for the Gaussian fit, χ 2 = (y i −ȳ i,model ) 2 is adopted.
To show how well the data follows the predicted relations, the missing errors for the considered quantities are estimated. For the compound quantities (such as E iso /E k ), the errors are estimated by error propagation: For the L X,11h derived in this work, T p,op , and T 90,i , the errors at the 90% confidence level (∼ 0.1 dex) 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 E iso ( × 10 52 erg) 10 1 10 2 10 3 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 E iso ( × 10 52 erg)   Table 8). (a) The different distributions of Ep and E iso for the two distinguished samples: γ 50% (red stars) and γ 50% (blue stars). (b) The distribution of E ratio and E iso /E k for the γ 50% sample (reduced χ 2 = 0.140). (c) The Ep and E iso distribution of the main pulse for other 7 bursts (blue circles and boxes), which have extended emission (see Minaev & Pozanenko 2020) and lack efficiency. Note that all the 5 bursts of the γ 50% sample (red stars) have extended emission, including GRB 170817A. (d) The comparison of the ratios of the Ep and the fluence (or E iso ) for the main pulse and the extended emission of a large extended emission sample (reduced χ 2 = 0.196), including the bursts without redshift (blue stars).

EVIDENCE FROM LONG GRBS WITH EXTREMELY HIGH EFFICIENCY ( γ 80%)
There is much controversy about the spectral differences between the photosphere emission model and the synchrotron emission model, after considering the more natural and complicated physical conditions (jet structure, decaying magnetic field, and so on; Uhm & Zhang 2014;Geng et al. 2018;Meng et al. 2018Meng et al. , 2019Meng et al. , 2022. Nevertheless, a crucial difference between these two models is that the photosphere emission model predicts much higher radiation efficiency γ . The synchrotron emission models mainly include the internal shock model (for a matter-dominated fireball; Rees & Meszaros 1994) and the ICMART model (internal-collision-induced magnetic reconnection and turbulence, for a Poynting fluxdominated outflow; Zhang & Yan 2011). For the internal shock model, since only the relative kinetic energy between different shells can be released, the radiation efficiency is rather low (∼ 10%; Kobayashi et al. 1997). For the ICMART model, the radiation efficiency can be much higher (∼ 50%), and it reaches ∼ 80% in the extreme case. However, the extremely high γ ( γ 80%) is unlikely to be achieved, because the magnetic reconnection requires some conditions to be triggered, so much magnetic energy is left. For the photosphere emission model, if only the acceleration is in the unsaturated regime (R ph < R s ), the radiation efficiency can be close to 100%. Thus, in this work, we select the GRBs with extremely high γ ( γ 80%; see Figure 3(a) and Table  1). Note that the two bursts (GRB 990705 and GRB 000210) that are claimed to have extremely high γ in Lloyd- Ronning & Zhang (2004) are also included. We then analyze the prompt 2 and afterglow 3 properties of these 15 long GRBs, to confirm the photosphere emission origin.

Characteristics of Prompt Emission
In Figure 3(b), we plot the E p and E iso distributions of the selected GRBs (omitting GRB 081203A and GRB 130606A, because of the large E p error), and we find The similar observed efficiency (with E iso /E k from the X-ray afterglow) and theoretically predicted efficiency (from the prompt emission) for the bursts before GRB 110213A (with L X,11h given in D' Avanzo et al. 2012). (a) The distribution of E ratio and E iso /E k for the γ 50% sample, which is found to be well centered around the equal-value line and have a linear correlation (reduced χ 2 = 0.160). (b), (c) The X-ray afterglow light curves for the bursts (7 bursts) with almost same E ratio and E iso /E k (see Table 5). We find that all these light curves do have a power-law shape with a slope of ∼ −1.
that they follow the predicted E p ∝ (E iso ) 1/4 relation (see Section 2.2.2) quite well. The best-fit result is log (E p ) = 2.54 + 0.25 log (E iso ). In Figure 3(c), we compare the E p and E iso distributions of the selected GRBs with those of the large sample of long GRBs, and find that the dispersion is quite small relative to that of the large sample.
In Figure 3(d), we plot the E p and L iso distributions of the selected GRBs, and find that they also follow the E p ∝ (L iso ) 1/4 relation well. The best-fit result is log (E p ) = 2.75 + 0.23 log (L iso ). And the dispersion is found to be similar to that of E p ∝ (E iso ) 1/4 . Furthermore, based on the best-fit E p ∼ 10 2.75 · (L iso ) 1/4 and E p = 2.7kT 0 = 2.7k(L iso /4πr 2 0 ac) 1/4 , we obtain the initial acceleration radius r 0 ∼ 3.21×10 8 cm, well consistent with the quite high mean value r 0 ∼ 10 8.5 cm deduced in Pe'er et al. (2015).
In Figure 12, we show the E p evolutions of the timeresolved spectra for 5 GRBs detected by Fermi/GBM. The E p evolutions are found to follow the evolution of the flux F quite well (intensity tracking pattern; Liang & Kargatis 1996). This positive correlation is consistent with the abovementioned unsaturated acceleration condition of the photosphere emission. From Tables 1 and 4, we can see that the best-fit spectral model of the time-integrated spectra is the CPL model, or that the high-energy spectral index β (using the BAND function to fit) is minimal. Thus, the photosphere emission model can better explain the high-energy spectra of these highefficiency GRBs. Figure 13 shows the X-ray afterglow light curves of the selected GRBs (except for GRB 990705 and GRB 000210). We find that all the X-ray afterglow light curves appear as simple power-law shapes 4 , without any plateau, steep decay (Zhang et al. 2006), or significant flare (with weak flares in the early times). In Figure 14, we show the optical afterglow light curves of 6 GRBs whose early peaks can be detected. All the optical afterglow light curves show significant reverse shock signals. The power-law shape of the X-ray afterglow and the reverse shock in the optical afterglow are the basic predictions (Paczynski & Rhoads 1993;Mészáros & Rees 1997;Sari & Piran 1999) of the classical hot fireball model of GRBs (see Appendices B.2 and B.3). Thus, the jets of these high-efficiency GRBs are likely to be thermaldominated, and the radiation mechanism of the prompt 10 -1 10 0
emission is unlikely to be the ICMART model (for Poynting flux-dominated outflow; Zhang & Yan 2011). Also, considering the high efficiency ( γ 80%), the internal shock model ( γ ∼ 10%; Rees & Meszaros 1994;Kobayashi et al. 1997) is unlikely. The prompt emission of these GRBs is likely to be produced by the photosphere emission, then.
In Figure 1(a), we show the correlation of L iso and Γ for the selected GRBs. The Γ is obtained by the tight L iso − E p − Γ correlation (Liang et al. 2015; the values are taken from Xue et al. 2019). In Section 4.3, we find that this estimation is likely to be quite accurate for these high-efficiency GRBs with E p ∝ (L iso ) 1/4 . Though 4 GRBs have detections of the peak time of the optical afterglow, we do not use them to estimate the Γ, because of the significant reverse shock signals. We find that Γ is tightly correlated with L iso , Γ ∝ (L iso ) 0.29 . This is well consistent with the prediction of the neutrino annihilation from the hyperaccretion disk (Lü et al. 2012), Γ ∝ (L iso ) 7/27 = (L iso ) 0.26 . This therefore also supports the jets of these GRBs being thermal-dominated.
In Figure 1(b), we show the correlation of E iso /E k and η/Γ (see Section 2.2.1) for the selected GRBs. According to Equation (10), along with r 0 ∼ 3.21 × 10 8 cm, as derived above, and L iso , we can use Γ to obtain the η for each burst. We find the obvious linear correlations for E iso /E k and η/Γ, and they are almost the same (aside from 3 bursts: note that we obtain 4 bursts that are almost the same when we take r 0 = 3.21 × 10 8 cm, and then use the offset from the best-fit E p ∝ (L iso ) 1/4 relation to slightly modify r 0 for the other bursts, to obtain 3 other bursts that are almost the same) when we take E k,52 = 5 * L X,45 (E k,52 = E k /10 52 , L X,45 = L X /10 45 ; this is quite close to the derivation of E k,52 = 3.7 * L X,45 described in Section 4.1, and the slight difference is likely to result from the slight error of Γ estimated by the L iso − E p − Γ correlation). Again, this result strongly supports the photosphere emission origin in the unsaturated acceleration regime for these high-efficiency GRBs.   Table 3). It is obvious that, the tight correlations of Γ = 10 −0.1 · E p /(T 90 ) 1/4 and Ep ∝ η (for the 5 bursts with higher efficiency, thus η ∝ (E iso ) 1/4 ) are found. (b) The distributions of (Ep/E ratio )/(T 90 ) 1/4 − Γ (blue circles; reduced χ 2 = 0.031) and E p /(T 90 ) 1/4 − Γ (cyan triangles) for the selected γ 50% sample with T p detection (6 bursts in Figure 2(d); see Table 2). The tight correlation of (Ep/E ratio )/(T 90 ) 1/4 ∝ Γ is found, which is in line with the E p /(T 90 ) 1/4 ∝ Γ correlation for the γ 50% case. (c) The distribution of Γ− E p /(T 90 ) 1/4 (blue circles; reduced χ 2 = 0.025) for the large sample with Γ (47 bursts; see Table 6) in Ghirlanda et al. (2018). For the γ 50% case the E p is re-derived from the log (Ep) = 2.54 + 0.25 log (E iso ) correlation (using the E iso ), and for the γ 50% case the E p is re-derived from Ep/E ratio . Obviously, we find the distribution of Γ and E p /(T 90 ) 1/4 is well centered around Γ = 10 −0.1 · E p /(T 90 ) 1/4 and have a linear correlation.
produced by the photosphere emission in the unsaturated acceleration regime. But noteworthily, from Table 1, we can see that the low-energy spectral index α is quite typical (around −1), rather than very hard. This strongly supports that the photosphere emission model having the capacity to produce the observed typical soft lowenergy spectrum. Theoretically, the probability photosphere model (with geometric broadening) and the dissipative photosphere model (with subphotospheric energy dissipation) can both achieve this. But for the dissipative photosphere model, the E p ∝ (E iso ) 1/4 relation should be violated, since the inverse Compton scattering below the photosphere radius will change the photon energy (namely T ob = T 0 ; see Appendix A). Also, the high-energy spectrum for this model should be a power law, rather than the exponential cutoff. So the characteristics of the selected high-efficiency GRBs favor the probability photosphere model. 4. EVIDENCE FROM LONG GRBS WITH γ 50% AND γ 50% 4.1. γ = 50% and Maximum Γ For the γ = 50% (R ph = R s ) case, with a fixed L iso the observed Lorentz factor Γ in the afterglow phase should be the maximum, because of the following reason. To obtain γ < 50% (R ph > R s , R ph ∝ L iso /Γ 3 and R s = Γ · r 0 ), Γ (Γ = η) should be smaller. Conversely, for γ > 50%, η should be larger. And in this case, from Equation (10), we have Γ ∝ (L iso /η) 1/3 , thus Γ should also be smaller. Note that this maximum Γ exists for the hot fireball, while the corresponding γ = 50% is the prediction of the photosphere emission origin for the prompt emission. The maximum Γ is given as (see also Equation 16 in Ghirlanda et al. 2018 Tables 1-3). The mean value is around ∼ 10 −0.2 to 10 −0.3 , thus indicating the average efficiency of γ ∼ 33% to 40%. Also, the distribution seems to consist of three Gaussian distributions (smaller χ 2 for E iso /E k 10 −1.0 ).

(b)
GRBs with efficiency of 50% log(L X ) = -0.574 + 1.00 × log(E iso ) GRBs with efficiency of 50% Fig. 11.-The L iso − Γ and E iso − L X,45 distributions for the γ = 50% sample (see Table 7). (a) The distribution of L iso and Γ for the complete sample (62 bursts) with the detection of Tp. We select the sample (9 bursts, marked by the red plus) with the maximum Γ (for fixed L iso ), to check whether their efficiency is 50% as predicted by the photosphere emission model. (b) The distribution of E iso and L X,45 for the selected sample with the maximum Γ (4 bursts with L X,45 detection, red stars). It is found that all these bursts have almost the same efficiency (with E iso ∝ L X ∝ E k , reduced χ 2 = 0.008). Thus, we think that the efficiency γ for these bursts is likely to be 50% (E iso,52 = E k,52 3.7 * L X,45 ). In Figure 11(a), we show the distribution of L iso and Γ for the complete sample (62 bursts), with the detection of the peak time of the early optical afterglow (Ghirlanda et al. 2018;obtaining Γ). Obviously, with the exception of GRB 080319B (with a strong reverse shock signal) and 4 bursts (peak time is obtained from the Fermi /LAT light curve, and the decay slope of ∼ 1.5 implies that it is likely to be produced by the radiative fireball and the Γ should be smaller by a factor of ∼ 1.6, see Ghisellini et al. (2010) and Appendix B.4), the distribution of the maximum Γ well follows the predicted L 1/4 iso correlation, and only has the difference of a constant ∼ 10 0.1 (1.27) from the prediction of Equation (15) (the dashed line, r 0 ∼ 3.21 × 10 8 cm is used based on Figure 3). Note that though the equation for calculating Γ is confirmed to act as Γ ∝ (E k ) 1/8 · [T p /(1 + z)] −3/8 , its constant is highly uncertain (see Table 2 in Ghirlanda et al. (2018)). The constant that is given in other works (with different methods) can be 1.7 (or 0.5) times that used in Ghirlanda et al. (2018). So the above difference (1.27) obtained by our work is reasonable, and may be more accurate (since it does not strongly depend on the model assumption; if r 0 is accurate, then it is likely to be accurate).
Then, we select the sample (9 bursts) with the maximum Γ (see Table 7) to check their efficiency properties. In Figure 11(b), we show the distribution of E iso and L X,45 for this sample (4 bursts with L X,45 detection). Note that we exclude GRB 081007 due to the too small E iso and GRB 080310 due to the plateau in the early optical afterglow. As expected from the photosphere emission model, all these bursts have almost the same efficiency (with E iso ∝ L X ∝ E k ). Thus, we think that the efficiency γ for these bursts is likely to be 50%  Fig. 13.-The X-ray afterglow light curves for the selected extremely high-efficiency GRBs (except for GRB 990705 and GRB 000210). All these light curves appear as a simple power-law shape, without any plateau, steep decay, or significant flare (with weak flare in the early times).

E p − E iso Distributions and Consistent Efficiency
Based on the derivation of E k,52 = 3.7 * L X,45 described above, and separated by E iso,52 = E k,52 = 3.7 * L X,45 ( γ = 50%) for the distribution of E iso,52 and L X,45 in Figure 11, we obtain two distinguished long-GRB samples ( γ 50%, see Table 2; and γ 50%, see Table 3). Note that we exclude the above high-efficiency sample ( γ 80%) and the sample with γ = 50%.
In Figure 4(a), we show that the best-fit result for the γ 50% sample is log (E p ) = 2.47 + 0.25 log (E iso ), which is consistent with the prediction of the photosphere emission model. Also, this result is almost the same as that for the above high-efficiency sample. The offset from the best-fit result is likely to be caused by a distribution of r 0 (as for the constrained results in Pe'er et al. 2015). For the γ 50% sample, E iso and E p should both decrease by the same factor of (R ph /R s ) −2/3 (see Section 2.2.3), compared with the distribution of log (E p ) = 2.54 + 0.25 log (E iso ) for the above high-efficiency sample. In Figure 4(a), we show that the upmost distri-bution for the γ 50% sample is well around log (E p ) = 2.54+0.25 log (E iso ), and that the best-fit result (much smaller) is log (E p ) = 2.31 + 0.26 log (E iso ). The decreases of E iso and E p are more obvious when we divide the γ 50% sample into two subsamples ( γ 17% and γ 17%).
In Figure 2(a), we show the distributions of E ratio (E ratio = [(E p /2.7k) 4 * (4πr 2 1 ac)/E iso ] 1/3 ) and E iso /E k (implicitly, E k,52 = 3.7 * L X,45 /((1 + z)/2) is adopted) for the γ 50% sample. They are well centered around the equal-value line and have a linear correlation. This is well consistent with the prediction from the photosphere emission model (see Section 2.2.4). The dispersion is likely to be caused by the estimation error for E k (we have excluded the bursts with large E p errors of dE p /E p ≥ 0.2), since many X-ray afterglow light curves are quite complex (with plateaus, steep decays, or significant flares). The method of using L X,11h to estimate E k should only be completely correct for X-ray afterglows with power-law shapes and slopes of −1. To check the origin of the dispersion, we select the bursts with almost the same E ratio and E iso /E k (see Table 5). As expected, we find that all these bursts (7 bursts) have a power-law X-ray afterglow light curve with a slope of ∼ −1 (shown in Figures 2(b) and (c)).
Also, according to Section 2.2.4, for the γ 50% sample, we should have E ratio = E iso /E k = (R ph /R s ) −2/3 .
GRB 061007 V band GRB 080319B R band GRB 110205A R band GRB 110205A V band 10 1 10 2 10 3 10 4 Time since BAT trigger (s) 10 -13 10 -12 10 -11 10 -10 Flux (erg cm −2 GRB 060927 V band GRB 080607 R band GRB 081203A U band Fig. 14.-The optical afterglow light curves for 6 extremely high-efficiency GRBs ( γ 80%) whose early peaks can be detected. All these light curves show significant reverse shock signals. (a) 3 GRBs considered also to have reverse shock signals (with detections of both the rapid rise f t 3 and fall f ∼ t −2 ) in other works (Gao et al. 2015;Yi et al. 2020). (b) 3 GRBs with detection of only the rapid rise f t 3 or the rapid fall f ∼ t −2 .
To check this, we select the bursts with detections of the peak time of the optical afterglow (using them to estimate the Γ and thus (R ph /R s ) −2/3 ). Note that since E ratio and E iso /E k are the average results for the whole duration, and (R ph /R s ) −2/3 ∝ L −2/3 , we use L = E iso /T 90 (rather than L iso ). In Figure 2(d), we show the distributions of E ratio , E iso /E k and (R ph /R s ) −2/3 for the selected sample (6 bursts). Similar to the above, they are well centered around the equal-value line and have a linear correlation. Also, 1 burst has almost the same values for these three quantities, and the other 5 bursts have almost the same values for two quantities. So the predicted E ratio = E iso /E k = (R ph /R s ) −2/3 from the photosphere emission model can be well reproduced.
For the γ 50% sample, similar to Figure 1(b) (for the high-efficiency GRBs), we should have E iso /E k = η/Γ. To check this, we also select the bursts with detections of the peak time of the optical afterglow (using them to estimate Γ and thus η). Note that since Γ ∝ L 1/3 and the derived Γ is likely to correspond to L iso (the maximum L), we use L = L iso when calculating η. In Figure 2(e), we show the distribution of E iso /E k and η/Γ for the selected sample (7 bursts). As expected, they are well centered around the equal-value line and have a linear correlation.
The analysis results obtained above are for the sample with L X,11h as derived in this work (for GRBs after GRB 110213A). For another sample with L X,11h presented in D'Avanzo et al. (2012) (for GRBs before GRB 110213A), we perform a similar analysis and obtain similar results. For the γ 50% sub-sample, the E p and E iso distribution is also centered around log (E p ) = 2.47 + 0.25 log (E iso ) (see Figure 4(b)). For the γ 50% sub-sample, the decreases of E iso and E p are also obvious. Also, the distributions of E ratio and E iso /E k are well centered around the equal-value line, and have linear correlations (see Figure 6(a)). For the selected bursts with almost same E ratio and E iso /E k (see Table 5), all (7 bursts) show a power-law X-ray afterglow light curve with a slope of ∼ −1 (see Figures 6(b) and (c)).

The Excellent Derived
The small burst number in Figure 2(d) is a result of obtaining both the X-ray afterglow light curve and the detection of the peak time of the early optical afterglow. To further check E ratio = E iso /E k = (R ph /R s ) −2/3 for the γ 50% case, we then analyze the complete sample with detections of the peak time of the early optical afterglow (obtaining Γ and thus (R ph /R s ) −2/3 ; Ghirlanda et al. 2018; see Table 6). Note that the used constant is 1.27 times that given in Ghirlanda et al. (2018) (see Section 4.1). Though lacking of E iso /E k for most bursts in the sample, considering the different distributions of E p and E iso for γ 50% and γ 50%, we can use the judgment of (E ratio 0.9) to roughly select the γ 50% sub-sample. Note that we do not use the bursts without the E p value in Minaev & Pozanenko (2020) and Xue et al. (2019) due to the large E p errors, and we move 4 bursts to the γ 50% sub-sample based on their detections of E iso /E k . In Figure 7(a), we show the distribution of E ratio and (R ph /R s ) −2/3 for the selected γ 50% sub-sample (24 bursts). This distribution is roughly centered around the equal-value line, and has a linear correlation. After modifying the Γ or E p based on Figure 2(d) (using E ratio = E iso /E k = (R ph /R s ) −2/3 ) for the 6 bursts there, in Figure 7(b), we show the distributions of E ratio and (R ph /R s ) −2/3 for the two sub-samples with smaller E p errors (dE p /E p ≤ 0.2) and larger E p errors (dE p /E p ≥ 0.2). Obviously, for the sub-sample with smaller E p errors, the values of E ratio and (R ph /R s ) −2/3 are almost the same. Note that we decrease the Γ of GRB 090926A by a factor of 1.6, since its peak time is obtained from the LAT light curve and the decay slope of ∼ 1.5 implies that it is likely to be produced by the radiative fireball (Ghisellini et al. 2010). And, for the large offset in Figure 7(b) (the upper offset), we check its (GRB 090618) optical afterglow light curve, and find that the reverse shock signal is significant, thus overestimating the Γ and (R ph /R s ) −2/3 .
Based on E ratio = (R ph /R s ) −2/3 , we derive Γ ∝ E 1/8 iso E 1/2 p /(T 90 ) 1/4 (see Equation (6) and Equation (7) in Section 2.2.5). In Figure 7(c), we show a comparison of the Γ obtained from the optical afterglow (for 47 bursts in Ghirlanda et al. 2018) and the Γ obtained from the prompt emission (orange triangles for Γ = 10 3.33 L 0.46 iso E −0.43 p and blue circles for Γ = 17 · E 1/8 iso E 1/2 p /(T 90 ) 1/4 ). Obviously, using these two correlations, we can give an approximate estimation for Γ, both. Furthermore, the Equation (7) derived in our work from the photosphere emission model can give a better estimation for Γ (with a smaller reduced χ 2 ).
According to Equation (7), two correlations of T p /(1 + z) ∝ [E p /(T 90 ) 1/2 ] −4/3 (see Section 2.2.6) and Γ ∝ E p /(T 90 ) 1/4 (see Section 2.2.7) are predicted, which will be tested in the following section. Figure 7(d) we show the distribution of E p /(T 90 ) 1/2 and T p /(1 + z) for 35 bursts (we delect 7 bursts with dE p /E p ≥ 0.2 and 5 bursts with LAT light curves, and we modify the E p or T p for 5 bursts based on Figure  2(d)). Just as predicted, this distribution shows a linear correlation with a slope of ∼ −4/3, and it is consistent with T p /(1 + z) ∝ (E p ) −1.25 presented in Table 1 of Ghirlanda et al. (2018). So Equation (7) is likely to be correct.
The smaller constant (16.75) for the γ 50% case is consistent with the larger T p in Figure 7(d). In Figure  7(e), we show the derived Γ using the Equation (8) and the Equation (11), and find that the distribution for the γ 50% case is better (more symmetric) than that in Figure 7(c). 4.3.2. Γ = 10 −0.1 · E p /(T90) 1/4 correlation In Figure 9(a), we do find the tight correlation of for the γ 50% case (with L X,11h detection). Note that the E p here is re-derived from the log (E p ) = 2.54+0.25 log (E iso ) correlation (using the E iso ), since the observed E p with an offset from the above line is likely to arise from the different r 0 (actually E p ∝ (E iso /r 2 0 ) 1/4 ) or the error of E p . Besides, we modify the Γ (using E iso /E k , mainly for 3 bursts) based on Figure 2(e). In Figure 9(a), we also show the distributions for E p − Γ and E p − η, here η is obtained by η/Γ = E iso /E k . It is obvious that there is a tight correlation of E p ∝ η for the sub-sample (5 bursts) with higher efficiency. This means that η ∝ (E iso ) 1/4 , which is well consistent with the prediction of the neutrino annihilation from the hyperaccretion disk (for the hot fireball).
For the γ 50% case, since (E p /E ratio ) ∝ (E iso /E ratio ) 1/4 and Γ ∝ (L iso /E ratio ) 1/4 we should have Γ ∝ (E p /E ratio )/(T 90 ) 1/4 . From Figure 9(b) we do find this correlation, which is also in line with that for the γ 50% case. Note that we modify the E p or Γ based on Figure 2(d). In Figure 9(c) we show the distribution of Γ− E p /(T 90 ) 1/4 for the large Γ sample (47 bursts) in Ghirlanda et al. (2018). Note that for the γ 50% case, the E p is re-derived from the log (E p ) = 2.54 + 0.25 log (E iso ) correlation (using the E iso ), and for the γ 50% case, the E p is re-derived from E p /E ratio . For the γ 50% case, when calculating Γ, we modify the E k based on the original η/Γ. Obviously, we find that the distribution of Γ and E p /(T 90 ) 1/4 is well centered around Γ = 10 −0.1 ·E p /(T 90 ) 1/4 and shows a linear correlation. Note that the Γ− E p correlation is also found in Ghirlanda et al. (2012). sub-sample (17 bursts) and the γ 50% sub-sample (18 bursts), we use the different derived constants. Obviously, the Equation (7) derived in our work gives a much better estimation of Γ (with a much smaller reduced χ 2 , compared with Figure 7(c)). Noteworthily, the Γ = 10 3.33 L 0.46 iso E −0.43 p correlation obtained from the statistical fitting is actually consistent with our Γ ∝ E 1/8 iso E 1/2 p /(T 90 ) 1/4 correlation derived from the photosphere emission model. Because, along with E p ∝ (E iso ) 1/4 (or E p ∝ (L iso ) 1/4 ; see Figures 3 and 4), they can be transferred to each other, as shown in the following: Here, the adopted T 90 ∝ (L iso ) −0.5 correlation is found from Figure 8(c) for the high-efficiency sub-sample ( γ 80%).

EVIDENCE FROM SHORT GRBS WITH
For the short GRBs, similar to the long GRBs, we use the judgment of E iso,52 = E k,52 = 3.7 * L X,45 ( γ = 50%) to obtain the γ 50% sample (4 bursts) and the γ 50% sample (8 bursts; see Table 8). In Figure  5(a), we show the E p and E iso distributions for these two distinguished samples. Obviously, the γ 50% sample and the up-most distribution for the large sample of short GRBs (Zhang et al. 2018b) (without L X,45 detections for most) do follow the E p ∝ (E iso ) 1/4 correlation, well consistent with the prediction of the photosphere emission model. Noteworthily, the distribution for GRB 170817A well fits the above line (log (E p ) = 3.24+0.25 log (E iso )), too. For the γ 50% sample, as predicted, the distribution is below this line (since E p and E iso are smaller). Similar to Figure 2(a) (to test whether the E p and E iso are both smaller by a factor of E iso /E k = (R ph /R s ) −2/3 ), in Figure 5(b) we show the distribution of E ratio (E ratio = [(E p /2.7k) 4 * (4πr 2 1 ac)/E iso ] 1/3 ) and E iso /E k for this γ 50% sample. Note that we exclude 3 bursts with E iso ≤ 10 50 erg and that we have r 1 = 3.4 × 10 7 cm here. Again as predicted, they are found to be almost centered around the equal-value line, and have a linear correlation.
Interestingly, we find that all the bursts of the γ 50% sample (5 bursts, including GRB 170817A) have extended emission, and the E p and E iso distribution in Figure 5(a) is for their main pulse. To further test this finding, in Figure 5(c) we show the E p and E iso distribution of the main pulse for 7 other bursts that have extended emission (Minaev & Pozanenko 2020). It is found that, except for 3 bursts only detected by Swift or HETE-2 (lacking the detections in the high-energy band), other 4 bursts do follow the log (E p ) = 3.24 + 0.25 log (E iso ) correlation of the γ 50% sample, supporting the above finding again. The true E p values for these 3 outliers are likely to be much larger. According to the above, the main pulse for the short GRBs with extended emission is likely to be produced by the photosphere emission in the unsaturated acceleration regime. Then, considering the smaller values of both the E p and E iso for their extended emission, we think that the extended emission may be produced by the transition from the unsaturated acceleration to the saturated acceleration (E p and E iso are both smaller by the same factor of E iso /E k = (R ph /R s ) −2/3 ). To test this hypothesis, in Figure 5(d), we show the comparison of the ratios of the E p and the fluence of the main pulse and the extended emission for a large extended emission sample (including the Swift/BAT bursts with redshift; Gompertz et al. 2020; and the Fermi /GBM bursts without redshift; Lan et al. 2020). As predicted, these two ratios are found to be almost centered around the equal-value line and they have linear correlations.

SUMMARY
In this work, after obtaining the prompt emission efficiency of a large GRB sample with redshift, we divide that GRB sample into three sub-samples ( γ 80%, γ 50%, and γ 50%). Then, the well-known Amati relation (Amati et al. 2002) is well explained by the photosphere emission model. Furthermore, for each subsample, the X-ray and optical afterglow characteristics are well consistent with the predictions of the photosphere emission model. Ultimately, large amounts of convincing observational evidence for the photosphere emission model are revealed for the first time. Manned Space Project with No.CMS-CSST-2021-B11, and the Program for Innovative Talents, Entrepreneur in Jiangsu. I also acknowledge the use of public data from the Fermi Science Support Center, the Swift and the Konus-Wind.

A.1. 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 (Pe'er 2008). Based on careful theoretical derivation, the probability function P (r, Ω), donating the probability for a photon to be last scattered at the radius r and angular coordinate Ω, can be given as (Pe'er 2008; Beloborodov 2011;Lundman et al. 2013) where β is the jet velocity and D = [Γ(1 − β · cos θ)] −1 is the Doppler factor.
For the probability photosphere model, the observed photosphere spectrum is the overlapping of a series of blackbodies with different temperatures, thus its lowenergy spectrum is broadened. After considering the jet with angular structure (e.g., Dai & Gou 2001;Rossi et al. 2002;Zhang & Mészáros 2002), the observed typical low-energy photon index α ∼ −1.0 (Kaneko et al. 2006;, spectral evolution and E p evolutions (hard-to-soft evolution or E p -intensity tracking; Liang & Kargatis 1996;Lu et al. 2010Lu et al. , 2012 can be reproduced (Lundman et al. 2013;Meng et al. 2019Meng et al. , 2022.

A.2. THE DISSIPATIVE PHOTOSPHERE MODEL
The dissipative photosphere model (or the subphotosphere model) considers that there is an extra energy dissipation process in the area of moderate optical depth (1 < τ < 10; the sub-photosphere). Different dissipative mechanisms have been proposed, such as shocks (Rees & Mészáros 2005), magnetic reconnection (Giannios & Spruit 2007) and proton-neutron nuclear collisions (Vurm & Beloborodov 2016;Beloborodov 2017). Then, relativistic electrons (with a higher temperature than that of the photons) are generated that upscatter the thermal photons to obtain the non-thermal (broadened) high-energy spectrum. In the context of the standard afterglow model (Paczynski & Rhoads 1993;Mészáros & Rees 1997), since at a late afterglow epoch (11 hours) the X-ray band is above the cooling frequency ν c , the late-time X-ray afterglow luminosity (L X,11h ) only sensitively depends on E k and e (the electron equipartition parameter). Furthermore, the fraction of energy in the electrons ( e ) is quite centered around 0.1, based on the large-sample afterglow analysis. Thus, E k can be well estimated by the L X,11h as following (Lloyd-Ronning & Zhang 2004): 10 52 ergs R 1.1L X,11h 10 46 ergs s −1 ( where R = [t(10 h)/t(prompt)] (17/16) e ∼ 2.27 is the radiative losses during the first 10 hours after the prompt phase. Note that the derived E k is 9.2 times larger in Fan & Piran (2006), since the ν m (the characteristic frequency corresponding to the minimum electron Lorentz factor) is about one and a half orders smaller. Previously, it had been hard to judge which constant was better. Here, using the method ( γ = 50% and the maximum Γ; see Section 4.1) in this work, our result (E k,52 3.7 * L X,45 ) is quite consistent with E k,52 2.5 * L X,45 (Fan & Piran 2006, without the inverse Compton effect).
The isotropic X-ray afterglow luminosity (in the 2-10 keV rest-frame common energy band) at 11 hours (rest frame), L X,11h , is computed from the observed integral 0.3-10 keV unabsorbed fluxes at 11 hours (F X,11h ; estimated from the Swift/XRT light curves) and the measured spectral index Γ 1 (from the XRT spectra), along with the luminosity distance D L . The equation is as follows (D'Avanzo et al. 2012): Here, F X,11h is obtained by interpolating (or extrapolating) the best-fit power law, for the XRT light curve within a selected time range including (or close to) 11 hours, to the 11 hours. The "generic" afterglow model (relativistic blastwave theory) for GRB predicts a power-law decaying multi-wavelength afterglow (Paczynski & Rhoads 1993;Mészáros & Rees 1997), due to the self-similar nature of the blastwave solution. The observed specific flux is This power-law behavior (f ∼ t −1 obs ) is well consistent with the observations of the optical and radio afterglows. But several surprising emission components (the steep decay f ∼ t −3 obs phase, the plateau f ∼ t −0.5 obs phase, and the flare) in the early X-ray afterglow are revealed by the Swift observations (Zhang et al. 2006), which are not predicted by the above standard (hot fireball) model. These extra components imply that an extra energy injection (internal or external) may exist, which can be magneticdominated.

B.3. THE REVERSE SHOCK IN THE OPTICAL AFTERGLOW PREDICTED BY THE CLASSICAL HOT FIREBALL.
For the classical hot fireball (the magnetic field in the ejecta is dynamically unimportant, namely the magnetization parameter σ ≡ B 2 /(4πn p m p c 2 ) 1), a strong reverse shock (propagating back across the GRB ejecta to decelerate it) is predicted in the early optical afterglow phase (Mészáros & Rees 1997;Sari & Piran 1999). This prediction is almost confirmed by the discovery of a very bright optical flash in GRB 990123 while the GRB is still active. Later on, many more reverse shock signals are found. The light curve of the reverse shock declines more rapidly (f ∼ t −2 ) than that of the forward shock (f ∼ t −1 ), and rises more rapidly (f t 3 ) than that of the forward shock (f ∼ t 2 ) before the peak time. For the generic afterglow model, the total energy of the fireball remains constant (the adiabatic case) after the forward shock starts to decelerate (entering the selfsimilar phase). However, there could be another case that the total energy of the fireball decreases (the radiative case), since a large fraction of the dissipated energy is radiated away (by magnetic reconnection or electronproton collisions) (Ghisellini et al. 2010). For this radiative fireball, the light curve (after the peak time) declines more rapidly (f ∼ t −10/7 ), Γ ∝ t −3/7 (Γ ∝ t −3/8 for the adiabatic case), and the peak time is much earlier (t peak = 0.44t dec , t peak = 0.63t dec for the adiabatic case; t dec is the deceleration time).