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[ cols="^,^ " , ]      for the forward - scattering case of a disk - center observation as a function of the spatial resolution @xmath3 ( in arcseconds ) of a square @xmath4 element . the dotted lines indicate the resolution for which the average polarization signal is reduced by a factor @xmath5 with respect to the full - resolution case . , width=302 ]    the 3d model atmosphere considered in this paper is a snapshot of a radiation mhd simulation performed with the _ bifrost _ code @xcite taking into account non - equilibrium hydrogen ionization ( see details in * ? ? ? we used snapshot 385 of the simulation  en024048-hion " , which is the same simulation recently made publicly available through the interface region imaging spectrograph project ( iris ; * ? ? ? * ) at the hinode science data center europe ( http://sdc.uio.no ; see iris technical note 33 ) . this 3d model atmosphere , which has horizontal periodic boundary conditions , encompasses the upper part of the convection zone , the photosphere , chromosphere , transition region and corona , and is identical to that used by @xcite to investigate the h@xmath0 intensity profile . the model s magnetic field configuration is representative of an enhanced network region ; it has magnetic field lines connecting two clusters of photospheric magnetic concentrations with two dominant opposite polarity regions 8 mm apart , which reach chromospheric and coronal heights ( see fig . [ fig : mgf ] ) . the ly@xmath0 line results from two blended transitions between the ground level of hydrogen , @xmath6 , and the @xmath7 and @xmath8 excited levels . the ly@xmath0 scattering polarization calculations of this paper were carried out fixing the model s neutral hydrogen number density at each spatial grid point ( which were computed with the _ bifrost _ code taking into account time - dependent hydrogen ionization ) and solving the problem of the generation and transfer of polarized radiation assuming statistical equilibrium in a hydrogen model atom with only bound - bound transitions . the _ bifrost _ ionization balance for hydrogen was computed using the method described in @xcite , which assumes that the lyman transitions are in detailed balance . the hydrogen model atom we have used in the scattering polarization calculations ( see fig .  [ fig : grotr ] ) is composed of the above - mentioned levels and the @xmath9 level . the collisional coupling of this level with the @xmath10-levels may contribute to a slight depolarization of the ly@xmath0 line in regions where the density of perturbers ( protons and electrons ) is significantly larger than @xmath11 @xcite . since in models of the upper solar chromosphere the proton and electron density is typically smaller than @xmath11 , even a three - level model atom ( i.e. , without the @xmath9 level of fig . [ fig : grotr ] ) could be sufficiently realistic for computing the line - center polarization of ly@xmath0 . the only level that produces scattering polarization in the core of the ly@xmath0 line is the @xmath12 upper level . it is interesting to note that the suitability of the above - mentioned three - level model atom implies that the atomic polarization of the @xmath12 level ( population imbalances and quantum coherence between its magnetic sublevels ) is dominated by the absorption of anisotropic radiation in the resonance line transition itself and by the action of the hanle effect on such upper level , without any serious impact of collisional depolarization . the critical hanle field of the level @xmath12 is @xmath13 g and the scattering polarization of the on - disk ly@xmath0 radiation is sensitive to magnetic strengths between 10 and 100 g , approximately @xcite    to compute the emergent stokes profiles @xmath14 , @xmath15 and @xmath16 at each surface point of the atmospheric model we have to solve the following radiative transfer equations for any desired line of sight ( @xmath17 being @xmath14 , @xmath15 or @xmath16 ) @xmath18 where @xmath19 defines the monochromatic optical path length along the ray direction ( with @xmath20 the corresponding geometrical distance ) , @xmath21 is the absorption coefficient , and @xmath22 is the source function of the stokes parameter @xmath17 , with @xmath23 the emission coefficient . is the only non - zero coefficient of the propagation matrix of the stokes - vector transfer equation , because the zeeman splitting is negligible compared to the width of the hydrogen ly@xmath0 line and the angular momentum of its lower level is @xmath24 and , therefore , can not carry atomic alignment . ] while the emission coefficient @xmath25 has contributions from the two blended transitions indicated in fig . [ fig : grotr ] , @xmath26 and @xmath27 depend only on the atomic polarization of the @xmath8 upper level ( because it is the only ly@xmath0 level that can carry atomic alignment ) . we quantify the population and atomic polarization of this @xmath28 level by means of the multipolar components of its atomic density matrix ( i.e. , the @xmath29 components , with @xmath30 and @xmath31 , as quantified in the reference system whose quantization axis is along the local vertical ) or , equivalently , by the following quantities ( with the @xmath29 components normalized to the fraction of neutral hydrogen atoms that are in the ground level ) : @xmath32 . in our radiative transfer calculations with porta we have used the exact @xmath33 , @xmath34 and @xmath35 expressions , which take into account that the two ly@xmath0 transitions are not perfectly overlapped . such transitions are actually very close in wavelength ( they are only about 6 m  apart ) and , for simplicity , we give here the @xmath34 and @xmath35 expressions for the perfect overlapping case :    @xmath36 - \sin \chi { \rm im}[{s}^2_1 ] )      \nonumber   \\    & - &   \frac{\sqrt{3}}{2 } ( 1+\mu^2 ) ( \cos 2\chi   { \rm re}[{s}^2_2 ] -\sin 2\chi \ , { \rm im}[{s}^2_2 ] )    \big{\}}\ ; , \nonumber\end{aligned}\ ] ]    and @xmath37+\cos \chi { \rm im}[{s}^2_1 ] )      \label{eq : su }       \\ \nonumber    & \quad+\mu ( \sin 2\chi \ , { \rm re}[{s}^2_2 ] + \cos 2\chi \ , { \rm im}[{s}^2_2])\big{\}}\;,\end{aligned}\ ] ] where @xmath38 and @xmath39 are the inclination with respect to the solar local vertical and azimuth of the ray , respectively , and @xmath40 ( with @xmath41 and @xmath42 the continuum and the line - integrated opacities , respectively , and @xmath43 the voigt line absorption profile ) . in such equations the reference direction for stokes @xmath15 is in the plane formed by the ray s propagation direction and the vertical @xmath44-axis . we point out that eqs . ( [ eq : sq ] ) and ( [ eq : su ] ) , with their corresponding eqs . ( [ eq : rte ] ) , imply that the sources of stokes @xmath15 are the population imbalances ( @xmath45 ) and the quantum coherence ( @xmath46 , with @xmath47 ) , while the stokes @xmath16 signal is produced only by quantum coherence ( @xmath46 , with @xmath47 ) between pairs of sublevels pertaining to the @xmath8 excited level . to solve eqs . ( [ eq : rte ] ) for any desired line of sight ( los ) we need to know first @xmath21 and the @xmath48 source function components at each spatial grid point . to obtain the self - consistent values of such quantities we have to solve jointly the statistical equilibrium equations for the multipolar components ( @xmath49 ) of the atomic density matrix corresponding to each atomic level @xmath50 and the radiative transfer equations for each of the radiative transitions in the chosen model atom . the general expressions of such equations can be found in sections 7.2 and 7.13.c of @xcite , which we solved applying porta . we used a gaussian quadrature for the ray inclinations with respect to the vertical @xmath44-axis ( with 5 inclinations per octant ) and the trapezoidal rule for the ray azimuth ( with 4 azimuths per octant ) . in our radiative transfer calculations we took into account the impact of the doppler shifts , produced by the macroscopic velocity gradients of the 3d atmospheric model , on the anisotropic radiation pumping . although this physical ingredient is not very important for modeling the scattering polarization of the hydrogen ly@xmath0 line ( because the line is very broad ) , porta takes it fully into account because the dynamic state of the solar atmosphere may produce observable signatures on the scattering polarization of some chromospheric lines @xcite . our calculations with porta were carried out in the marenostrum iii supercomputer of the barcelona supercomputing center . the total computing time of the non - lte calculations carried out for the investigation of this paper exceeded one million cpu hours .    to grasp better the physics of this complex radiative transfer problem it is useful to write down a simplified set of equations , instead of the general equations for the four - level model atom of fig .  [ fig : grotr ] we have actually solved with porta to obtain all the results shown in this paper . to this end , we note that the weakly inelastic collisions with protons and electrons that couple the @xmath9 level with the @xmath51 and @xmath52 ones , which we have taken into account in our 3d radiative transfer calculations , do not really have a significant depolarizing effect on the ly@xmath0 polarization in the considered model atmosphere @xcite . taking also into account that elastic collisions with neutral hydrogen atoms can be safely neglected when modeling the line - core polarization of ly@xmath0 and that the only level contributing to the ly@xmath0 scattering polarization is the @xmath52 one , we have the following approximate expressions for the @xmath53 tensors that quantify the excitation state of the @xmath52 upper level @xcite :    @xmath54 \\\\        { { \rm im}[{s}^2_1 } ] \\\\         { { \rm re}[{s}^2_2 } ] \\\\        { { \rm im}[{s}^2_2 } ]      \end{array } \right ) & = &      { { ( 1-\epsilon)}\over{\sqrt{2 } } }     \left ( \begin{array}{r }        { j}^2_0 \\\\        { \rm re}[{j}{}^{2}_{1 } ] \\\\        -{\rm im}[{j}{}^{2}_{1 } ] \\\\        { \rm re}[{j}{}^{2}_{2 } ] \\\\ -{\rm im}[{j}{}^{2}_{2 } ]      \end{array } \right)\ , \\      & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! + & \!\!\!\!\!\!\!\!\!\!\!\ ! { \gamma_u}\ , { { ( 1-\epsilon)}}\ ,      \left ( \begin{array}{cccccc }          0 & m_{12 } & m_{13 } & 0 & 0 \\\\           m_{21 } & 0 & m_{23 } & m_{24 } & m_{25 } \\\\ m_{31 } & m_{32 } & 0 & m_{34 } & m_{35 } \\\\          0 & m_{42 } & m_{43 } & 0 & m_{45 } \\\\          0 & m_{52 } & m_{53 } & m_{54 } & 0         \end{array } \right )       \left ( \begin{array}{l }          { s^2_0 } \\\\          { { \rm re}[{s}^2_1 } ] \\\\          { { \rm im}[{s}^2_1 } ] \\\\          { { \rm re}[{s}^2_2 } ] \\\\ { { \rm im}[{s}^2_2 } ]       \end{array } \right)\ , , \nonumber\end{aligned}\ ] ]    where @xmath55 is the planck function at the ly@xmath0 line - center frequency , @xmath56 is the photon collisional destruction probablility , @xmath57 ( with the magnetic strength @xmath58 in gauss ) , and the @xmath59 coefficients of the magnetic kernel @xmath60 depend on the inclination ( @xmath61 ) of the local magnetic field vector with respect to the vertical @xmath44-axis and on its azimuth ( @xmath62 ) . the frequency averaged ( over the absorption profile @xmath43 ) radiation field tensors @xmath63 that appear in such equations quantify the mean intensity ( @xmath64 ) , the anisotropy ( @xmath65 ) , and the breaking of the axial symmetry ( the real and imaginary parts of @xmath66 and @xmath67 ) of the incident radiation field at each point within the medium ; their explicit expressions can be found in @xcite , but note that the stokes @xmath14 of the incident ly@xmath0 radiation has to be calculated taking into account that it results from two blended transitions ( see fig . [ fig : grotr ] ) . the equations for the @xmath68 ( atomic ) quantities have a clear physical meaning . in particular , note that the magnetic operator @xmath60 couples locally the @xmath69 components among them ( the hanle effect ) . the top left panel of fig . [ fig : jkq ] shows the spatial variation of the kinetic temperature across a 2d slice through the above - mentioned 3d snapshot model atmosphere ; it corresponds to the @xmath44-@xmath70 plane at @xmath71 mm in fig.[fig : mgf ] . note that the height of the tr changes drastically from one horizontal location to another , outlining a highly crumpled surface . as shown by the white solid curve , the corrugated surface that delineates the model s tr practically coincides with that defined by the ly@xmath0 line - center optical depth unity along the line of sight ( los ) . the other panels of fig . [ fig : jkq ] show the radiation field tensors @xmath72 of the ly@xmath0 radiation , normalized to @xmath73 . such quantities quantify the fractional anisotropy and the symmetry properties of the incident radiation at each point within the medium . we calculated them after obtaining with the 3d radiative transfer code porta the self - consistent solution in the 3d snapshot model illustrated in fig.[fig : mgf ] , using the four - level model atom of fig . [ fig : grotr ] and taking into account the ( very weak ) collisional coupling between the @xmath9 and the two @xmath10 levels . of particular interest is the spatial variation of the fractional anisotropy ( top right panel ) ; it is negligible in the photosphere and chromosphere of the model , but it suddenly becomes significant right at the location of the tr with negative values of about 10% . the other radiation field tensors shown in the remaining panels quantify the breaking of the axial symmetry of the ly@xmath0 radiation at each point within the model atmosphere ; note that at the atmospheric heights where the line center optical depth is unity such tensors may have sizable positive or negative values , depending on the horizontal point under consideration . the multipolar components of the atomic density matrix ( i.e. , the @xmath29 quantities or , equivalently , the @xmath53 ones ) quantify the excitation state of the atomic level under consideration . as mentioned above , the scattering polarization of ly@xmath0 is produced by the atomic polarization of its @xmath8 upper level ; therefore , in this section we provide information on the spatial variation of the level s fractional atomic polarization throughout the same @xmath44-@xmath70 plane considered in fig . [ fig : jkq ] . ( [ eq : mtx-20 ] ) indicate that for @xmath74 ( i.e. , in the unmagnetized case ) each multipolar component @xmath75 of the @xmath8 level is directly proportional to its corresponding radiation field tensor component . we can say that there is a transfer of ` order ' from the radiation field to the atomic system , and eqs . ( [ eq : rte])([eq : su ] ) show that the atoms of a polarized ( ` ordered ' ) atomic system emit linearly polarized radiation . ( [ eq : mtx-20 ] ) indicate that if all @xmath76 were zero then all @xmath46 would vanish , even in the presence of a magnetic field . through the hanle effect the magnetic field modifies the polarization of the atomic level according to the @xmath77 term of eqs . ( [ eq : mtx-20 ] ) , and this has an impact on the emergent linear polarization @xmath78 and @xmath79 signals .    fig . [ fig : rhokq ] shows the multipolar components of the density matrix corresponding to the @xmath8 level , normalized to the @xmath80 value of the same level ( which is proportional to its overall population ) . note that the @xmath81 panel ( which quantifies the population imbalances among the sublevels of the @xmath8 level ) shows mainly negative values at the atmospheric heights where the ly@xmath0 line center optical depth is unity along the los , while each of the quantum coherence shown in the other panels shows both positive and negative values . this section considers the case of spatially resolved observations . it shows maps of the line - center scattering polarization , which we have obtained by calculating the emergent stokes profiles of the ly@xmath0 radiation after obtaining the self - consistent solution of the above - mentioned 3d radiative transfer problem . of particular interest is to investigate the spatial variability of the emergent @xmath78 and @xmath79 line - center signals and their sensitivity to the model s magnetic field . to this end , we solved the 3d radiative transfer problem of resonance line polarization for the following two cases : ( 1 ) taking into account the hanle effect produced by the magnetic field of the 3d snapshot model atmosphere and ( 2 ) assuming @xmath82 g at each spatial grid point of the 3d model . we show the results for two line of sights ( los ) defined by @xmath83 ( with @xmath2 the heliocentric angle ) : @xmath84 ( corresponding to a disk center observation ) and @xmath85 ( corresponding to another on - disk observation , but at about 80  seconds of arc from the solar limb ) . in all our calculations of the emergent stokes profiles the azimuth of the corresponding los was @xmath86  that is , the los was contained in the @xmath17-@xmath44 plane . in all the following figures we have chosen the @xmath70-axis as the positive reference direction for stokes @xmath15 , and for any los with @xmath87 we chose this @xmath70-direction as the parallel to the nearest limb . [ fig : dco ] shows , for the forward scattering case of a disk center observation ( los with @xmath84 ) , the intensity ( top left panel ) , the total fractional linear polarization ( @xmath88 , top right panel ) , and the @xmath78 ( bottom left panel ) and @xmath79 ( bottom right panel ) signals at the center of the emergent ly@xmath0 intensity profile . the calculated ly@xmath0 line - center intensity variations across the surface of the 3d atmospheric model show fibril - like structures , reminiscent of those observed at 1  arcsecond resolution by the vault sounding rocket ( e.g. , * ? ? ? although we do not show illustrations here , we point out that the effects of 3d radiative transfer do have an impact on the emergent line - center intensities , as expected from the smoothing effect produced by horizontal radiative transfer under non - lte conditions ( e.g. , * ? ? ? * ) ; the so - called 1.5d approximation gives a larger line - center intensity contrast across the field of view . the same applies to other spectral lines such as hydrogen h@xmath0 for which this effect is even more significant due to the lower opacity of the line @xcite . it is interesting to note that at high spatial resolution the linear polarization signals of the emergent ly@xmath0 line - center radiation are very significant ( see fig .  [ fig : dco ] ) . at first sight this result might appear surprising , given that in fig . [ fig : dco ] we are considering the forward scattering case of a disk - center observation . however , in this scattering geometry zero linear polarization should be expected only for the idealized case of an unmagnetized and horizontally homogeneous stellar atmosphere model . as soon as there is a magnetic field inclined with respect to the symmetry axis of the incident radiation field ( hereafter , inclined field ) and/or horizontal atmospheric inhomogeneities ( e.g. , in the gas kinetic temperature and/or density ) the ensuing symmetry breaking effects can produce significant line scattering polarization signals ( see * ? ? ? * ; * ? ? ? * and more references therein ) . the linear polarization signals of fig . [ fig : dco ] are due to the breaking of the axial symmetry of the ly@xmath0 radiation at each spatial point within the 3d model atmosphere , as quantified by the real and imaginary parts of the @xmath89 and @xmath90 radiation field tensors , as well to the magnetic field of the atmospheric model ( see eqs .  [ eq : mtx][eq : mtx-20 ] ) . the @xmath78 and @xmath79 line - center signals shown in the bottom panels of fig .  [ fig : dco ] have similar values that fluctuate in sign across the field of view , as expected from the particularization of eqs . ( [ eq : sq ] ) and ( [ eq : su ] ) to the case of a los with @xmath84 . in particular , the image corresponding to the total linear polarization at the line center ( see the top right panel of fig . [ fig : dco ] ) shows an impressive detail of fine - scale structuring . as explained above , the forward - scattering polarization signals are caused by both the model s horizontal atmospheric inhomogeneities in temperature , density , and macroscopic velocity , and by the magnetic field . as shown in the top left panel of fig . [ fig : diff ] , there are very significant forward scattering polarization signals in the absence of magnetic fields . such zero - field linear polarization signals are solely due to the symmetry breaking caused by the horizontal atmospheric inhomogeneties ; therefore , their very existence indicates that the so - called 1.5d approximation is unsuitable for modeling high spatio - temporal resolution observations of the scattering polarization in ly@xmath0 . [ fig : diff ] shows that there is a significant difference between the true total fractional linear polarization ( calculated taking into account the hanle effect produced by the magnetic field of the 3d model atmosphere ) and that corresponding to the zero - field reference case . as expected , the action of the hanle effect in forward scattering geometry is found mainly along the diagonal of the image seen in fig . [ fig : diff ] , where the magnetization of the model atmosphere is the largest . however , as seen in the figure , the model s magnetic field tends to depolarize the emergent radiation , and this occurs mainly in the strongly polarized regions of the unmagnetized case . this contrasts with the case of a plane - parallel model atmosphere , where in the absence of magnetic fields the symmetry axis of the incident radiation coincides with the vertical direction and the hanle effect of an inclined magnetic field creates linear polarization in forward scattering geometry but depolarization in close to the limb observations . in our 3d model atmosphere the symmetry axis of the radiation field that illuminates each of the points of the tr does not in general coincide with the local vertical . in fig . [ fig : trnorm ] , we attempt to visualize the geometrical complexity of the present 3d problem by showing , at each point of the field of view , the inclination of the vector that is normal to the tr surface . in the unmagnetized case the strongly polarized regions are typically those for which the inclination of the local normal vector to the tr surface is larger , and depolarization by the hanle effect is statistically more likely . in regions where the local normal vector to the tr surface is predominantly vertical the forward scattering signals of the unmagnetized case are typically lower and the hanle effect produced by the model s magnetic field tends to be more similar to that found in the case of a plane - parallel model atmosphere . these two situations can be understood by analogy with the scattering polarization signals calculated in a 1d plane - parallel model near the limb and at the disk center , respectively ( cf . we now consider a los with @xmath85 , which corresponds to the case of an observation relatively close to the solar limb . [ fig : clo ] shows maps of the line - center intensity ( top left panel ) , of the total fractional linear polarization ( top right panel ) , as well as of the @xmath78 ( bottom left panel ) and @xmath79 ( bottom right panel ) line - center signals . the particularization of eqs . ( [ eq : sq ] ) and ( [ eq : su ] ) to the los under consideration ( or to any los with @xmath91 ) indicates that @xmath27 ( the emissivity in stokes @xmath16 at each point along the los ) depends only on the quantum coherence ( @xmath92 and @xmath93 ) of the @xmath8 excited level , while @xmath26 ( the emissivity in stokes @xmath15 at each point along the los ) depends also on the population imbalances between the sublevels of the excited level . as indicated by eq . ( [ eq : mtx-20 ] ) , the population imbalances , quantified by @xmath45 , are determined mainly by the radiation field tensor @xmath94 , which at the crumpled surface that delineates the model s tr tends to be negative ( as in the 1d atmospheric models considered by * ? ? ? we therefore expect that the @xmath78 signals of the emergent ly@xmath0 radiation across the surface of the 3d model have mainly negative sign . for the @xmath79 signals we expect the occurrence of positive and negative values to be equally likely , following the fact ( see eq . [ eq : mtx-20 ] ) that the radiation field tensors @xmath89 and @xmath90 ( which quantify the symmetry breaking of the incident radiation field ) play a key role on the quantum coherence quantified by @xmath95 and @xmath96 . the upper panel of fig . [ fig : histo ] shows the histogram of the line center polarization amplitudes of the emergent ly@xmath0 radiation calculated in forward scattering geometry ( los with @xmath84 ) assuming @xmath82 g . most of the linear polarization signals have values between 0 and 1% , but there is a clear tail of stronger polarization signals extending up to about 3% . in this forward scattering geometry the mean fractional linear polarization amplitude at the maximum spatial resolution of the 3d model is @xmath97    the bottom panel of fig . [ fig : histo ] shows an analogous histogram , but for the case of a los with @xmath85 . as in the @xmath84 case ( see the upper panel of fig . [ fig : histo ] ) , almost all the surface points are significantly polarized . in contrast , in this close to the limb scattering geometry there is only a minority of surface points with @xmath98% and there is a significantly larger fraction of points of the field of view showing strong polarization signals with @xmath99% . as a result , the mean fractional linear polarization amplitude at the maximum spatial resolution of the 3d model is now @xmath100 . [ fig : clprofs ] shows , for the case of a close to the limb los with @xmath85 , the synthetic @xmath78 and @xmath79 profiles computed at each @xmath70-point without accounting for any spatial degradation ; the @xmath70-direction here can be assumed to be along the spectrograph s slit , assuming it to be oriented parallel to the nearest limb ( i.e. , the @xmath70 axis ) . of great interest is to compare the top panels ( the zero - field reference case ) with the bottom panels ( where the hanle effect of the model s magnetic field has been taken into account ) . in both cases the @xmath78 signals are mainly negative , as expected from the behavior of the radiation field anisotropy ( @xmath101 ) illustrated in fig . [ fig : jkq ] . however , the polarization amplitudes are larger in the absence of magnetic fields , with a sizable factor of difference in the atmospheric regions where the magnetic field of the 3d model is relatively strong and inclined ( e.g. , see the region around @xmath102 mm ) . the action of the hanle effect can also be clearly seen in the @xmath79 signals ( right panels ) . the results shown above indicate that at high spatial resolution we should expect conspicuous linear polarization signals in the ly@xmath0 line core , with amplitudes often larger than @xmath103% . these @xmath78 and @xmath79 signals are produced by anisotropic radiation pumping in the upper solar chromosphere and they are modified , via the hanle effect , by the vector magnetic field of the model s tr . in the following section , we turn our attention to the case of spatially averaged observations . signal that results from spatially averaging the stokes @xmath14 and @xmath15 line - center signals at each @xmath104 location , averaged over all the los azimuths @xmath39 , with ( solid curve ) and without ( dotted curve ) the hanle effect produced by the model s magnetic field . the dashed curve corresponds to the hanle effect saturation regime . the dash - dotted curve shows the calculated clv in the semi - empirical model c of @xcite . , width=302 ]     the figure shows the spatially averaged @xmath78 amplitude vs. the mean magnetic field strength of the model s transition region . the arrow indicates the @xmath78 signal corresponding to the original 3d model . , width=302 ]    the solar atmosphere is a highly inhomogeneous and dynamic medium , much more complex than the idealization of a 1d , static , plane - parallel semi - empirical model atmosphere . in a 1d model atmosphere with axial symmetry around the local vertical the only possibility to break the axial symmetry of the incident radiation field at each point within the medium is by the presence of a magnetic field inclined with respect to the solar radius vector through the point under consideration ( see * ? ? ? * and more references therein ) . in the real solar atmosphere we have , in addition , the symmetry breaking produced by the horizontal inhomogeneities of the atmospheric plasma , in its density , velocity and kinetic temperature . these ( non - magnetic ) causes of symmetry breaking produce @xmath78 and @xmath79 signals which could be confused with those resulting from the hanle effect .    in this paper we represent the complexity of the tr of the sun by a single snapshot taken from the three - dimensional mhd simulation described in section  [ sec : formulation ] . one possible way to obtain quantitative information on the magnetization of the solar tr is to study the statistical properties of the ly@xmath0 polarization signals , and their relation with the physical quantities that describe the 3d atmospheric model . the first step towards this goal is to consider the average polarization signals and how they can be exploited to infer information on the average magnetic field strength in the tr . in this section , we study the sensitivity of the emergent @xmath78 and @xmath79 line - center signals to the model s magnetic field , considering spatially averaged synthetic observations . as shown below , such spatially - averaged signals can be used to obtain quantitative information on the average magnetization of the solar tr . if not stated otherwise , the results of this section are shown for a los with azimuth @xmath105 , i.e. , with the @xmath70-axis parallel to the nearest solar limb which is also our reference direction for stokes @xmath15 . in order to simplify the notation , in what follows the spatially averaged quantities @xmath106 , @xmath107 , and @xmath108 will be denoted by @xmath78 , @xmath79 , and @xmath10 . if the spatial resolution of the observation is reduced , the emergent stokes profiles @xmath14 , @xmath15 , and @xmath16 are averaged over the corresponding resolution element . at the disk center , the @xmath78 , @xmath79 , and @xmath10 signals are therefore generally degraded because of cancellation of the positive and negative signals coming from different regions of the atmosphere . let us consider a square resolution element of size @xmath4 located at the solar disk center . in fig . [ fig : char ] we show the variation with @xmath3 of the total polarization signal @xmath10 at the ly@xmath0 line center . this result has been obtained statistically , after computing the emergent stokes profiles for many possible realizations of the resolution element position . as seen in the figure , at @xmath109 the total polarization line - center signal decreases by a factor two with respect to the maximum resolution value . three seconds of arc can therefore be regarded as a characteristic spatial scale of the polarized structures seen at the center of the ly@xmath0 line . we can therefore argue that unless the spatial resolution of the observation is significantly better than 3 arcseconds , the polarization of the emergent radiation will be a mixture of signals coming from locations having significantly different plasma structures . likewise , we can expect that for inferring the average properties of the solar tr , the spatial resolution of a single observation should be much worse than 3 arcseconds , so that the influence of the local inhomogeneities is sufficiently reduced .    for the case of a disk - center observation , the line - center signal @xmath10 tends to zero in the limit of no spatial resolution . for any los with @xmath87 , the spatially averaged signals will be generally non - zero and converging , in the limit of no spatial resolution , to the values given by the center - to - limb variation ( clv ) . this applies to the unmagnetized case , but as shown in fig . [ fig : clvav ] it happens also when we take into account the hanle effect produced by the weak magnetic field of the chosen 3d model atmosphere . the clv of the emergent spectral line radiation contains useful information on the structuring of the solar atmosphere . in quiet regions of the solar surface the observed clv of the stokes-@xmath14 profile of the hydrogen ly@xmath0 line is known to be negligible ( e.g. , * ? ? ? however , radiative transfer calculations in the semi - empirical model c of @xcite ( hereafter , fal - c model ) , show that the clv of the @xmath78 line - center signal is rather significant ( cf . the fig .  2 of * ? ? ? * ) . here we study the clv of the spatially - averaged @xmath78 line center signals in the chosen 3d model . the results are shown in fig . [ fig : clvav ] for the case of azimuthally averaged line of sights of inclination @xmath104 . only the @xmath78 line - center signal is shown because , as expected , the spatially averaged @xmath79 signal is practically zero for all heliocentric angles . interestingly , the behavior of the resulting clv of the line - center @xmath78 signal is qualitatively similar to that found in the fal - c model ( see the dash - dotted line in fig . [ fig : clvav ] ) . the largest fractional polarization amplitude is found around @xmath110 and it decreases towards larger heliocentric angles . in contrast to the clv behavior found in 1d models , the decrease of the @xmath78 line - center amplitude when approaching the limb of the 3d atmospheric model is not so significant and in the chosen 3d model the @xmath78 signal is still sizable at near - limb locations .    without spatio - temporal resolution , the resulting @xmath78 signals drop below the 1% level , with amplitudes similar to those found in 1d semi - empirical models . therefore , their measurement requires advanced instrumentation , capable of reaching a polarimetric sensitivity of at least 0.1% . via the hanle effect , the spatially - averaged @xmath78 signals are clearly sensitive to the model s magnetic field ( see fig .  [ fig : clvav ] ) .      following the approach introduced in section  4 of @xcite , we study the magnetic sensitivity of the spatially - averaged polarization amplitudes in the 3d snapshot model atmosphere . we define the quantity @xmath111 as the average magnetic field strength at the points where the optical depth at the ly@xmath0 line center is unity for the case of a disk - center observation . given the  height of formation " of the ly@xmath0 line center , @xmath111 can also be understood as the average magnetic field strength in the tr . we follow the approach of @xcite and multiply the magnetic field strength at each grid point of the 3d model by a dimensionless factor @xmath112 . in this way , we can study the dependence of the line polarization on the average field strength @xmath113 . the result for the @xmath78 line center amplitude is shown in fig . [ fig : bsens ] for a near - limb observation with @xmath85 . is not shown because this quantity remains practically zero for all values of @xmath114 . ] as expected , the maximum polarization signal is found in the unmagnetized case ( @xmath115 ) . the arrow in the figure indicates the actual value of the average tr magnetization ( the @xmath116 case ) which is @xmath117 g . the @xmath78 polarization signal gradually decreases as the magnetic field increases . in contrast to the 2d result shown in fig .  4 of @xcite , in the 3d model the @xmath78 signal is lower and its maximum amplitude is more similar to that found in the fal - c unmagnetized model . amplitude on the averaging length along the spectrograph s slit , with the slit located at @xmath85 and oriented parallel to the model s solar limb . the figure shows the polarization signal modified by the hanle effect of the model s magnetic field ( red solid curve ) , the solution without the magnetic field ( dash - dotted curve ) , and the solution obtained with the magnetic field strength increased by a factor 4 at every point of the atmospheric model ( green solid curve ) . the error bars indicate the standard deviation of the @xmath78 amplitudes calculated from the statistics of randomly positioned slits . , width=302 ]     signal , integrated along the @xmath118 slit , as a function of the averaged magnetic field strength @xmath119 of the model s transition region . blue curve : slit oriented radially and extending from @xmath120 to @xmath121 . red curve : slit oriented tangentially with the slit center located at @xmath85 . , width=302 ]    the chromospheric ly@xmath0 spectropolarimeter ( clasp ) has been designed for measuring the core of the stokes @xmath14 , @xmath15 and @xmath16 profiles using a spectrograph with a slit of @xmath118 , with a spatial resolution of about @xmath122 , a spectral resolution of 0.1  and a polarimetric sensitivity of @xmath123 at the @xmath124 level ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? in order to reach this level of polarimetric sensitivity it will be necessary to significantly sacrifice the temporal resolution and to partially average the observed stokes @xmath14 , @xmath15 , and @xmath16 signals along the direction of the spectrograph s slit , in order to increase the signal - to - noise ratio . nevertheless , in addition to the important goal of measuring for the first time scattering polarization in a fuv line ( hydrogen ly@xmath0 ) whose core originates in the solar tr , clasp can still aim at constraining the average magnetization of the tr by partial spectrum averaging along the spectrograph slit .    an important question to consider is whether the spectrograph s slit , which selects solar light along a distance of @xmath125 on the solar disk ( i.e. , almost one half of the apparent solar radius ) , should be oriented radially or tangentially with respect to the solar limb , in order to maximize the chances of detection of the expected scattering polarization in ly@xmath0 and the scientific output of the clasp experiment . we dedicate this section to analyze the pros and cons of both options . although both options seem to be suitable for detecting the scattering polarization signals , we have some reasons to prefer the radial option ( see below ) . we begin our analysis by considering the case of tangential orientation of the spectrograph s slit , with its center located at the heliocentric angle @xmath85 , i.e. , at an on - disk position where the spatially - averaged @xmath78 signal is maximal ( cf . fig .  [ fig : clvav ] ) . we have studied the dependence of the spatially - averaged @xmath78 line - center signal on the length @xmath126 ( in arc seconds ) along the slit s direction over which we average the computed stokes @xmath14 , @xmath15 and @xmath16 signals . we have done this analysis by averaging the individual signals corresponding to a large number of randomly positioned virtual slits on top of the 3d model grid . this has been done by selecting randomly the @xmath127 $ ] point that corresponds to the first point of the slit . the average is then obtained by integration of the @xmath14 , @xmath15 , and @xmath16 signals along the randomly selected slit . if the virtual slit crosses the model s boundary , then the integration continues along the same direction but starting at the opposite boundary , with a random position along this boundary . this way , it is possible to obtain slit - integrated signals for a large number of slits that are longer than the model s dimensions ( which is about @xmath128 ) . we have applied this procedure for averaging lengths between @xmath129 and @xmath130 . the results are shown in fig . [ fig : bdet ] for three different degrees of the model s magnetization . it is important to emphasize that the spatialy - averaged @xmath78 signal depends on @xmath126 : stronger signals are found , on average , for smaller lengths @xmath126 . this is due to the fact that @xmath78 depends on correlations between @xmath15 and @xmath14 at particular pixels . in our 3d model , the regions with lower line intensity are , on average , more polarized than the high - intensity regions . consequently , stronger signals are obtained for smaller lengths @xmath126 . the spread of the resulting @xmath78 signals is indicated by the error bars in the figure , which correspond to the standard deviation of each @xmath78 value for any given length @xmath126 . as expected , the larger @xmath126 , the smaller the spread of the values . averaging over the whole @xmath118 distance of the clasp spectrograph s slit would provide a @xmath131 test for detection of the zero magnetization versus the actual model magnetization ( @xmath132 g ) of the 3d model . similarly , @xmath133 would be sufficient for a test distinguishing between a non - magnetized and a strongly magnetized ( @xmath134 g ) transition region , with a similar level of confidence . [ fig : bdettara ] shows , both for the radial and tangential slit options , how the spatially - averaged ( including averaging along the whole length of the spectrograph s slit ) @xmath78 line - center signal of the emergent ly@xmath0 radiation from the 3d model varies with the average magnetization value of the model s tr . for each average magnetization , @xmath113 , a very large number of @xmath78 signals were obtained by a method analogous to the one described in the previous paragraph . in the radial - slit option case , the fact that the @xmath78 signal varies with the heliocentric angle ( see fig .  [ fig : clvav ] ) has been taken into account . the amplitude of the @xmath78 signal predicted for the tangential configuration is slightly larger than in the radial case , but the difference is not very important and both slit orientations appear to be suitable for detecting the ly@xmath0 scattering polarization .    our reasons in favor of the radial slit option for a sounding rocket experiment such as clasp are the following . first , in the case of the tangential slit orientation we would have data only for a small interval of @xmath104 values , for example with @xmath135 at the central slit point ( where the polarization amplitude is the largest ) . obviously , any significant pointing error would compromise the measurement . in contrast , the radial slit option provides much more information on the clv of the scattering polarization signals , and having such clv data would facilitate reaching more reliable conclusions via detailed confrontations with radiative transfer modeling . note that with a @xmath118 radially - oriented slit one can have information from the solar limb till about @xmath121 . with the radial slit option such clv information would be available both at the line center ( where the hanle effect in ly@xmath0 operates ) and in the nearby wings of the @xmath78 profile ( where effects of partial frequency redistribution and @xmath136-state interference come into play ; see @xcite ) . as shown by these authors , having in addition information on the clv of the @xmath78 wing signals may help to constrain ( together with the observed stokes @xmath14 profiles ) the thermal structure of the upper chromosphere . another reason to prefer the radial slit option is related with the fact that with the clasp ( which will measure the ly@xmath0 radiation @xmath137   from the line center ) it will be difficult to determine with sufficient precision the zero offset of the linear polarization scale , and having the above - mentioned clv information may help to end up with reliable conclusions in terms of the hanle effect . clasp will be observing the solar atmosphere for approximately five minutes during which the structure of the dynamical tr will change to some extent . in our numerical experiments , we have only considered an instantaneous state of the solar model atmosphere corresponding to a particular snapshot of the time - dependent simulation . furthermore , we have assumed an infinitely narrow spectrograph s slit and we have neglected any possible drift of the slit during the simulated observation . because of the fine structuring and dynamical behavior of the solar chromosphere , clasp can not aim at determining the magnetic field at high spatial resolution . however , the lack of spatio - temporal resolution might actually help us to end up with observables useful to determine the mean field strength of the solar tr . therefore , the standard deviations shown in figs .  [ fig : bdet ] and [ fig : bdettara ] should be understood as being the most pessimistic estimates and the real observation will be of better accuracy . the outer solar atmosphere is a highly structured and dynamic plasma , much more complex than a uniform , plane - parallel 1d configuration . very likely , the height of the tr varies from one place in the atmosphere to another , delineating a highly crumpled surface ( e.g. , such as in the top left panel of fig . [ fig : jkq ] ) . strong spectral lines originate in this type of spatially complex plasmas , where the horizontal atmospheric inhomogeneities break the axial symmetry of the radiation field . this symmetry breaking produces changes in the scattering polarization signals , which may compete with those caused by the hanle effect . therefore , the interpretation of spectropolarimetric observations in chromospheric and tr lines requires solving a complex radiative transfer problem , namely the generation of spectral line polarization in realistic 3d models of the solar atmosphere . likewise , since the intensity profiles are practically insensitive to the magnetic field vector , the true way to validate or refute the models themselves is by confronting synthesized and measured stokes profiles , because the magnetic field information is encoded in the spectral line polarization .    in this paper we have extended to the full 3d case previous radiative transfer investigations of the scattering polarization in the hydrogen ly@xmath0 line , which were carried out using 1d @xcite and 2d @xcite models of the solar atmosphere . the 3d model of the extended solar atmosphere chosen for this investigation is representative of the physical conditions of an enhanced network region , with magnetic field lines that reach chromospheric and coronal heights ( see fig . [ fig : mgf ] ) . using the radiative transfer code porta developed by @xcite we solved the full 3d radiative transfer problem of the generation and transfer of scattering polarization in the hydrogen ly@xmath0 line , and carried out numerical experiments to understand its magnetic sensitivity due to the hanle effect . the first noteworthy point is that contained in fig . [ fig : jkq ] , which shows the radiation field tensors that quantify the symmetry properties of the ly@xmath0 radiation . the fractional anisotropy of the ly@xmath0 radiation ( see the top right panel ) is fully negligible throughout the whole photosphere and chromosphere , but it suddenly becomes significant ( of the order of 10% ) and predominantly negative right at the tr . that the model s tr practically coincides with the corrugated surface where the optical depth is unity along the los . ] the other panels of fig . [ fig : jkq ] show that the radiation field tensors that quantify the breaking of the axial symmetry of the ly@xmath0 radiation are also negligible below the tr ; i.e. , in the regions where the optical depth at the line center is larger than unity . as the optical depth at the ly@xmath0 line center becomes smaller than unity at the tr , such radiation field tensors become significant . as a result , the emergent ly@xmath0 radiation shows conspicuous scattering polarization signals , with sizable @xmath78 and @xmath79 line - center amplitudes all over the solar disk , even in the unmagnetized reference case .    at full spatial resolution the most likely fractional polarization amplitude is 1.5% for simulated observations at @xmath85 ( close to the limb scattering geometry ) , while it is 0.7% at @xmath84 ( forward scattering geometry ) . the polarization signals drop below the 1% level when the spatial resolution is degraded . for example , for a square resolution element of 3 arc seconds the most likely forward scattering polarization signal decreases by a factor two . in the limit of no spatial resolution we find that the clv of the line - center @xmath78 signal is qualitatively similar to that found in semi - empirical 1d models of the solar atmosphere , with the largest polarization amplitudes around @xmath85 and with @xmath138 at @xmath84 . although the average magnetic field strength of the model s tr is only @xmath139 g , the hanle effect in the hydrogen ly@xmath0 line ( whose critical hanle field is 53 g ) produces a significant depolarization ( see fig . [ fig : bsens ] ) . for stronger magnetic fields the ensuing depolarization is larger , until it reaches the hanle saturation limit for @xmath140 g . as shown in fig . [ fig : bdettara ] , for the radial slit case the @xmath141 line - center amplitudes are slightly smaller than in the tangential slit case , but they are equally measurable and show a similarly significant hanle depolarization . in section  [ ssec : tara ] we give our arguments in favor of the radial slit option when observing the ly@xmath0 scattering polarization with the clasp sounding rocket experiment . another important conclusion is that the hanle effect in forward scattering geometry ( i.e. , when observing the solar disk center ) does not necessarily create linear polarization in the presence of an inclined magnetic field , as it happens in the idealized case of 1d model atmospheres . the fact that in a 3d model of the solar atmosphere the radiation field that illuminates each point of the tr plasma does not have axial symmetry with respect to the local vertical direction implies that even in the absence of a magnetic field forward - scattering processes can produce significant linear polarization signals in ly@xmath0 . interestingly , we have found and understood that in such a spatially - complex plasma the hanle effect of an inclined magnetic field tends instead to destroy the forward - scattering polarization , as it happens when observing close to the edge of the solar disk where one reaches the case of @xmath142 scattering geometry . this and the previously summarized results are important for a reliable interpretation of future observations of the ly@xmath0 polarization produced by optically pumped hydrogen atoms in the chromosphere - corona transition region of the sun .    financial support by the grant agency of the czech republic through grant and project , as well as by the spanish ministry of economy and competitiveness through projects ( solar magnetism and astrophysical spectropolarimetry ) and consolider ingenio csd2009 - 00038 ( molecular astrophysics : the herschel and alma era ) is gratefully acknowledged . this research was also supported by the research council of norway through the grant `` solar atmospheric modelling '' and by the european research council under the european union s seventh framework programme ( fp7/2007 - 2013 ) / erc grant agreement nr . the radiative transfer simulations presented in this paper were carried out with the marenostrum supercomputer of the barcelona supercomputing center ( national supercomputing center , barcelona , spain ) . we gratefully acknowledge the computing grants , technical expertise and assistance provided by the barcelona supercomputing center . we are also grateful to the european union cost action mp1104 ( polarization as a tool to study the solar system and beyond ) for financing short - term scientific missions at the iac that facilitated the development of this research project .
probing the magnetism of the upper solar chromosphere requires measuring and modeling the scattering polarization produced by anisotropic radiation pumping in uv spectral lines . 
 here we apply porta ( a novel radiative transfer code ) to investigate the hydrogen ly@xmath0 line in a 3d model of the solar atmosphere resulting from a state of the art mhd simulation . at full spatial resolution 
 the linear polarization signals are very significant all over the solar disk , with a large fraction of the field of view showing line - center amplitudes well above the 1% level . via the hanle effect 
 the line - center polarization signals are sensitive to the magnetic field of the model s transition region , even when its mean field strength is only 15 g. the breaking of the axial symmetry of the radiation field produces significant forward - scattering polarization in ly@xmath0 , without the need of an inclined magnetic field . 
 interestingly , the hanle effect tends to decrease such forward - scattering polarization signals in most of the points of the field of view . when the spatial resolution is degraded , the line - center polarization of ly@xmath0 drops below the 1% level , reaching values similar to those previously found in 1d semi - empirical models ( i.e. , up to about 0.5% ) . 
 the center to limb variation of the spatially - averaged polarization signals is qualitatively similar to that found in 1d models , with the largest line - center amplitudes at @xmath1 ( @xmath2 being the heliocentric angle ) . 
 these results are important , both for designing the needed space - based instrumentation and for a reliable interpretation of future observations of the ly@xmath0 polarization .