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it is widely accepted that gamma - ray burst ( grb ) afterglows originate from the interaction of an ultrarelativistically expanding shell into the circumburst medium ( cbm ) . differences exists on the detailed kinematics and dynamics of such a shell ( see e.g. refs .   and refs . therein ) .    due to the ultrarelativistic velocity of the expanding shell ( lorentz gamma factor @xmath0 ) , photons emitted at the same time in the laboratory frame ( i.e. the one in which the center of the expanding shell is at rest ) from the shell surface but at different angles from the line of sight do not reach the observer at the same arrival time . therefore , if we were able to resolve spatially the grb afterglows , we would not see the spherical surface of the shell . we would see instead the projection on the celestial sphere of the equitemporal surface ( eqts ) , defined as the surface locus of points which are source of radiation reaching the observer at the same arrival time ( see e.g. refs . and refs . therein ) . the knowledge of the exact shape of the eqtss is crucial , since any theoretical model must perform an integration over the eqtss to compute any prediction for the observed quantities ( see e.g. refs .   and refs . therein ) . one of the key problems is the determination of the angular size of the visible region of each eqts , as well as the distribution of the luminosity over such a visible region . in the current literature it has been shown that in the latest afterglow phases the luminosity is maximum at the boundaries of the visible region and that the eqts must then appear as expanding luminous `` rings '' ( see e.g. refs .   and refs . therein ) . such an analysis is applied only in the latest afterglow phases to interpret data from radio observations @xcite or gravitational microlensing @xcite . the shell dynamics is usually assumed to be fully adiabatic and to be described by a power - law @xmath1 , following the blandford - mckee self similar solution@xcite , where @xmath2 and @xmath3 are respectively the lorentz gamma factor and the radius of the expanding shell . such a power - law behavior has been extrapolated backward from the latest phases of the afterglow all the way to the prompt emission phase .    in refs . there have been presented the analytic solutions of the equations of motion for grb afterglow , compared with the above mentioned approximate solutions , both in the fully radiative and adiabatic regimes , and the corresponding analytic expressions for the eqtss . it has been shown that the approximate power - law regime can be asymptotically reached by the lorentz gamma factor only in the latest afterglow phases , when @xmath4 , and only if the initial lorentz gamma factor @xmath5 of the shell satisfies @xmath6 in the adiabatic case or @xmath7 in the radiative case . therefore , in no way the approximate power - law solution can be used to describe the previous dynamical phases of the shell , which are the relevant ones for the prompt emission and for the early afterglow . starting from these premises , in this paper we present the distribution of the extended afterglow luminosity over the visible region of a single eqtss within the `` fireshell '' model for grbs . such a model uses the exact solutions of the fireshell equations of motion and assumes a fully radiative dynamics ( see refs . therein for details ) . we recall that within the fireshell model the peak of the extended afterglow encompasses the prompt emission . we focus our analysis on the prompt emission and the early afterglow phases . our approach is therefore complementary to the other ones in the current literature , which analyze only the latest afterglow phases , and it clearly leads to new results when applied to the prompt emission phase . for simplicity , we consider only the bolometric luminosity@xcite , since during the prompt phase this is a good approximation of the one observed e.g. by bat or gbm instruments@xcite . the analysis is separately performed over different selected eqtss . the temporal evolution of the luminosity distribution over the eqtss visible region is presented . as a consequence of these results , we show the novel feature that at the beginning of the prompt emission the most luminous regions of the eqtss are the ones closest to the line of sight . on the contrary , in the late prompt emission and in the early afterglow phases the most luminous eqts regions are the ones closest to the boundary of the visible region . this transition in the emitting region may lead to specific observational signatures , i.e. an anomalous spectral evolution , in the rising part or at the peak of the prompt emission . we also present an analytic expression for the temporal evolution , measured in arrival time , of the apparent radius of the eqtss in the sky . we finally discuss analogies and differences with other approaches in the current literature which assumes an adiabatic dynamics instead of a fully radiative one . for the case of a spherically symmetric fireshell considered in this letter , the eqtss are surfaces of revolution about the line of sight . the general expression for their profile , in the form @xmath8 , corresponding to an arrival time @xmath9 of the photons at the detector , can be obtained from ( see e.g. ref .  ): @xmath10 where @xmath11 is the initial size of the expanding fireshell , @xmath12 is the angle between the radial expansion velocity of a point on its surface and the line of sight , @xmath13 is its equation of motion , expressed in the laboratory frame , and @xmath14 is the speed of light .    in the case of a fully radiative regime , the dynamics of the system is given by the following solution of the equations of motion ( see e.g. refs .   and refs . therein ) : @xmath15}{\gamma_\circ^{-1}+\left(m_\mathrm{cbm}/m_b\right)\left(1+\gamma_\circ^{-1}\right)\left[1+\left(1/2\right)\left(m_\mathrm{cbm}/m_b\right)\right]}\ , , \label{gamma_rad}\ ] ] where @xmath2 is the lorentz gamma factor of the fireshell , @xmath16 is the amount of cbm mass swept up within the radius @xmath3 and @xmath5 and @xmath17 are respectively the values of the lorentz gamma factor and of the mass of the fireshell at the beginning of the extended afterglow phase . correspondingly , the exact analytic expression for @xmath18 is@xcite : @xmath19 \\[6pt ] & + \tfrac{{r_\circ \sqrt c } } { { 12cm_i^\circ a^2 } } \ln \left\ { { \tfrac{{\left [ { a + \left(r / r_\circ\right ) } \right]^3 \left(a^3   + 1\right)}}{{\left[a^3   + \left ( r / r_\circ \right)^3\right ] \left ( { a + 1 } \right)^3 } } } \right\ } + t_\circ + \tfrac{m_i^\circ r_\circ } { 8c\sqrt c } \left(\tfrac{r^4-r_\circ^4}{r_\circ^4}\right)\ , , \end{split } \label{analsol}\ ] ] where @xmath20{(m_b - m_i^\circ)/m_i^\circ}$ ] , @xmath21 , @xmath22 is the value of the time @xmath23 at the beginning of the extended afterglow phase and @xmath24 . inserting eq.([analsol ] ) into eq.([ta_g ] ) we have the analytic expression for the eqts in the fully radiative regime@xcite : @xmath25 \\[6pt ] & + \tfrac{m_i^\circ r_\circ } { 8r\sqrt{c}}\left(\tfrac{r^4-r_\circ^4}{r_\circ^4}\right)+\tfrac{{r_\circ \sqrt{c } } } { { 12rm_i^\circ a^2 } } \ln \left\ { { \tfrac{{\left [ { a + \left(r / r_\circ\right ) } \right]^3 \left(a^3   + 1\right)}}{{\left[a^3   + \left ( r / r_\circ \right)^3\right ] \left ( { a + 1 } \right)^3 } } } \right\ } + \tfrac{ct_\circ - ct_a + r^\star}{r}. \end{split } \label{eqts_g_dopo}\ ] ]    instead , the corresponding equations in the adiabatic regime are@xcite : @xmath26 @xmath27 @xmath28    ) , and in the adiabatic regime ( blue line ) , given by eq.([gamma_ad ] ) . ]    a comparison between the lorentz gamma factors in the two regimes is presented in fig . [ gamma_rad_ad ] . here and in the following we assume the same initial conditions as in ref .  , namely @xmath29 , @xmath30 cm , @xmath31 s , @xmath32 cm , @xmath33 particles/@xmath34 , @xmath35 g. for simplicity , and since we are interested in the overall behavior of the luminosity distribution , we assume a constant cbm density , neglecting the inhomogeneities which are responsible of the temporal variability of the prompt emission @xcite . +   +   +     within the fireshell model , the grb extended afterglow bolometric luminosity in an arrival time @xmath36 and per unit solid angle @xmath37 is given by ( details in refs .   and refs . therein ) : @xmath38 where @xmath39 is the energy density released in the interaction of the abm pulse with the cbm measured in the comoving frame , @xmath40 is the doppler factor , @xmath41 is the surface element of the eqts at arrival time @xmath9 on which the integration is performed , and it has been assumed the fully radiative condition . we are here not considering the cosmological redshift of the source , which is constant during the grb explosion and therefore it can not affect the results of the present analysis . we recall that in our case such a bolometric luminosity is a good approximation of the one observed in the prompt emission and in the early afterglow by e.g. the bat or gbm instruments@xcite . we are now going to show how this luminosity is distributed over the eqtss , i.e. we are going to plot over selected eqtss the luminosity density @xmath42 . the results are represented in fig . [ eqts_multi ] . we chose eight different eqtss , corresponding to arrival time values ranging from the prompt emission ( @xmath43 seconds ) to the early ( @xmath44 hour ) afterglow phases . for each eqts we represent also the boundaries of the visible region due to relativistic collimation , defined by the condition ( see e.g. refs .   and refs . therein ) : @xmath45 we obtain that , at the beginning ( @xmath46 seconds ) , when @xmath2 is approximately constant , the most luminous regions of the eqts are the ones along the line of sight . however , as @xmath2 starts to drop ( @xmath47 seconds ) , the most luminous regions of the eqtss become the ones closest to the boundary of the visible region . this transition in the emitting region may lead to specific observational signatures , i.e. an anomalous spectral evolution , in the rising part or at the peak of the prompt emission . from sec .  [ bld ] we obtain that within the fireshell model the eqtss of grb extended afterglows should appear in the sky as point - like sources at the beginning of the prompt emission but evolving after a few seconds into expanding luminous `` rings '' , with an apparent radius evolving in time and always equal to the maximum transverse eqts visible radius @xmath48 which can be obtained from eqs.([ta_g],[vis ] ) : @xmath49 where @xmath13 is given by eq.([analsol ] ) . with a small algebra we get : @xmath50 where @xmath51 is given by eq.([gamma_rad ] ) and @xmath52 is given by eq.([analsol ] ) , since we assumed the fully radiative condition . eq.([rbotfin ] ) defines parametrically the evolution of @xmath53 , i.e. the evolution of the maximum transverse eqts visible radius as a function of the arrival time . in fig . [ eqts_multi ] we saw that such @xmath48 coincides with the actual value of the eqts apparent radius in the sky only for @xmath47 s , since for @xmath54 s the most luminous eqts regions are the ones closest to the line of sight ( see the first three plots in fig . [ eqts_multi ] ) . therefore , in fig .  [ rmaxvsta_rad ] we plot @xmath48 given by eq.([rbotfin ] ) in the fully radiative regime together with the actual values of the eqts apparent radius in the sky taken from fig . [ eqts_multi ] in the three cases in which they are different . it is clear that , during the early phases of the prompt emission , even the `` exact solution '' given by eq.([rbotfin ] ) can be considered only an upper limit to the actual eqts apparent radius in the sky .     in the fully radiative regime . the red line represents the maximum transverse eqts visible radius @xmath48 following eq.([rbotfin ] ) . the three blue points represents the actual eqts apparent radius in the sky in the three cases @xmath55 s , @xmath56 s and @xmath57 s in which , as shown in fig . [ eqts_multi ] , it is smaller than @xmath48 . the blue line is a linear interpolation of such points . ]    in the current literature ( see e.g. refs .  ) there are no analogous treatments , since it is always assumed an adiabatic dynamics instead of a fully radiative one and only the latest afterglow phases are addressed . it is usually assumed the blandford - mckee self similar solution@xcite for the adiabatic dynamics @xmath1 . a critical analysis of the applicability to grbs of this approximate dynamics , instead of the exact solutions in eqs.([gamma_ad],[analsol_ad],[eqts_g_dopo_ad ] ) , has been presented in ref . , as recalled in the introduction . the most widely applied formula for the eqts apparent radius in the above mentioned current literature is the one proposed by sari@xcite : @xmath58^{5/8 } \ , \mathrm{cm } , \label{gps99}\ ] ] where @xmath59 is the initial energy of the shell in units of @xmath60 ergs , @xmath61 is the cbm density in units of @xmath44 particle/@xmath34 , @xmath62 is the arrival time at the detector of the radiation measured in days , and @xmath63 is the source cosmological redshift . waxman et al.@xcite however derived a numerical factor of @xmath64 instead of @xmath65 . eq.([gps99 ] ) can not be compared directly with eq.([rbotfin ] ) , since they assume two different dynamical regimes . therefore , we first plot in fig . [ rmaxvsta_rad_ad ] the exact solution for @xmath48 given by eq.([rbotfin ] ) both in the fully radiative case , using eqs.([gamma_rad],[analsol ] ) , and in the adiabatic one , using eqs.([gamma_ad],[analsol_ad ] ) . we see that they almost coincide during the prompt emission while they start to diverge in the early afterglow phases , following the behavior of the corresponding lorentz gamma factors ( see fig . [ gamma_rad_ad ] and details in ref .  ) .     computed with eq.([rbotfin ] ) in the adiabatic ( blue line ) and in the fully radiative ( red line ) cases respectively . ] now , in fig . [ rmaxvsta_ad ] and for the same initial conditions assumed in previous figures , we plot the maximum transverse eqts visible radius @xmath48 given by eq.([rbotfin ] ) in the adiabatic case , using eqs.([gamma_ad],[analsol_ad ] ) , together with eq.([gps99 ] ) , proposed in ref .  , and with the corresponding modification in the numerical factor proposed in ref . from such a comparison , we can see that the approximate regime @xmath66 overestimates the exact solution for @xmath48 during the prompt emission and the early afterglow phases . it is asymptotically reached only in the latest afterglow phases ( @xmath67 s ) , in the very small region in which the approximate power - law dynamics starts to be applicable ( see details in ref .  ) . however , there is still a small discrepancy in the normalization : the constant numerical factor in front of eq.([gps99 ] ) should be @xmath68 instead of @xmath69 or @xmath70 to reproduce the behavior of the exact solution for large @xmath9 . moreover , we must emphasize that , in analogy with what we obtained in the fully radiative case ( see fig . [ rmaxvsta_rad ] ) , in the early phases of the prompt emission , when the fireshell lorentz @xmath2 factor is almost constant , the maximum transverse eqts visible radius @xmath48 given by eq.([rbotfin ] ) can only be considered an upper limit to the actual value of the eqts apparent radius in the sky . in such phases the approximation implied by eq.([gps99 ] ) can therefore be even worse that what represented in fig . [ rmaxvsta_ad ] .     in the adiabatic regime . the red line represents the maximum transverse eqts visible radius @xmath48 following eq.([rbotfin ] ) . the green line represents eq.([gps99 ] ) proposed by sari . the blue line represents the corresponding modification in the numerical factor proposed by waxman et al . these last two lines are almost coincident on the scale of this plot . ] within the fireshell model , using the exact analytic expressions for the fireshell equations of motion and for the corresponding eqtss in the fully radiative condition , we analyzed the temporal evolution of the distribution of the extended afterglow luminosity over the eqts during the prompt emission and the early afterglow phases . we find that , at the beginning of the prompt emission ( @xmath46 seconds ) , when @xmath2 is approximately constant , the most luminous regions of the eqts are the ones along the line of sight . as @xmath2 starts to drop ( @xmath47 seconds ) , the most luminous regions of the eqtss become the ones closest to the boundary of the visible region . this transition in the emitting region may lead to specific observational signatures , i.e. an anomalous spectral evolution , in the rising part or at the peak of the prompt emission . the eqtss of grb extended afterglows should therefore appear in the sky as point - like sources at the beginning of the prompt emission but evolving after a few seconds into expanding luminous `` rings '' , with an apparent radius evolving in time and always equal to the maximum transverse eqts visible radius @xmath48 . we also derive an exact analytic expression for @xmath48 , both in the fully radiative and in the adiabatic conditions , and we compared it with the approximate formulas commonly used in the current literature in the adiabatic case . we found that these last ones can not be applied in the prompt emission nor in the early afterglow phases . even when the asymptotic regime is reached ( @xmath67 s ) , it is necessary a correction to the numerical factor in front of the expression given in eq.([gps99 ] ) which should be @xmath68 instead of @xmath69 or @xmath70 . r.  ruffini , a.  g. aksenov , m.  g. bernardini , c.  l. bianco , l.  caito , p.  chardonnet , m.  g. dainotti , g.  de barros , r.  guida , l.  izzo , b.  patricelli , l.  j.  r. lemos , m.  rotondo , j.  a.  r. hernandez , g.  vereshchagin and s.  xue , the blackholic energy and the canonical gamma - ray burst iv : the `` long , '' `` genuine short '' and `` fake - disguised short '' grbs , in _ xiii brazilian school on cosmology and gravitation _ , eds . m.  novello and s.  perez  bergliaffa , american institute of physics conference series , vol . 1132 ( 2009 ) , pp . 199266 .
due to the ultrarelativistic velocity of the expanding `` fireshell '' ( lorentz gamma factor @xmath0 ) , photons emitted at the same time from the fireshell surface do not reach the observer at the same arrival time . in interpreting gamma - ray bursts ( grbs ) it is crucial to determine the properties of the equitemporal surfaces ( eqtss ) : the locus of points which are source of radiation reaching the observer at the same arrival time . in the current literature 
 this analysis is performed only in the latest phases of the afterglow . here 
 we study the distribution of the grb bolometric luminosity over the eqtss , with special attention to the prompt emission phase . 
 we analyze as well the temporal evolution of the eqts apparent size in the sky . 
 we use the analytic solutions of the equations of motion of the fireshell and the corresponding analytic expressions of the eqtss which have been presented in recent works and which are valid for both the fully radiative and the adiabatic dynamics . 
 we find the novel result that at the beginning of the prompt emission the most luminous regions of the eqtss are the ones closest to the line of sight . on the contrary , in the late prompt emission and in the early afterglow phases the most luminous eqts regions are the ones closest to the boundary of the visible region . 
 this transition in the emitting region may lead to specific observational signatures , i.e. an anomalous spectral evolution , in the rising part or at the peak of the prompt emission . 
 we find as well an expression for the apparent radius of the eqts in the sky , valid in both the fully radiative and the adiabatic regimes . 
 such considerations are essential for the theoretical interpretation of the prompt emission phase of grbs .