Radio, X-ray and gamma-ray surface brightness profiles as powerful diagnostic tools for non-thermal SNR shells

Distributions of nonthermal surface brightness of supernova remnants (SNRs) contain important information about the properties of magnetic field and acceleration of charged particles. In the present paper, the synchrotron radio, X-ray, and inverse-Compton (IC) gamma-ray maps of adiabatic SNRs in uniform interstellar medium and interstellar magnetic field are modeled and their morphology is analyzed, with particular emphasis to comparison of azimuthal and radial variations of brightness in radio, X-rays, and gamma-rays. Approximate analytical formulae for the azimuthal and radial profiles of the synchrotron radio and X-ray as well as the IC gamma-ray brightness are derived. They reveal the main factors which influence the pattern of the surface brightness distribution due to leptonic emission processes in shells of SNRs and can account for some non-linear effects of acceleration if necessary. These approximations provide observers and theorists with a set of simple diagnostic tools for quick analysis of the non-thermal maps of SNRs.


INTRODUCTION
Non-thermal images of SNRs are rich sources of information about the properties of interstellar magnetic field (ISMF) behavior and kinetics of charged particles in vicinity of the strong non-relativistic shocks. Despite of their importance, images of SNRs -in contrast to broad-band spectra -are not well studied. Synchrotron X-ray brightness profiles were used as diagnostic tools for the estimate of the post-shock magnetic field in some SNRs (e.g. Berezhko & Völk 2004). Radio azimuthal profiles were used for determination of some properties of SN 1006 (Petruk et al. 2009c, hereafter Paper I) and X-ray radial profiles were used to detect the shock precursor in SN 1006 and prove particle acceleration (Morlino et al. 2010).
A detailed approach to modeling the synchrotron images of adiabatic SNRs in uniform ISMF and uniform interstellar medium (ISM) is developed by Reynolds (1998). Fulbright & Reynolds (1990); Reynolds (1998Reynolds ( , 2004 use modeled synchrotron maps of SNRs to put constraints on properties of accelerated particles. Properties of the inverse-Compton (IC) γ-ray maps are investigated and compared to radio images in Petruk et al. (2009a, hereafter Paper II).
The influence of nonuniform ISM and/or nonuniform ISMF on the thermal X-ray morphology of adiabatic SNRs are studied in Hnatyk & Petruk (1999), on the radio maps in Orlando et al. (2007) and on the synchrotron X-ray and IC γ-ray images in Orlando et al. (2010).
All studies of SNR maps assume classic MHD and testparticle theory of acceleration. Though they neglect effects of the back-reaction of the efficiently accelerated particles, they are able to explain general properties of the distribution of the surface brightness in radio, X-rays and γ-rays. This is because the classic theory, in contrast to the non-linear one, is able to deal with oblique shocks, that is vital for synthesis of SNR images.
At present, the theory which considers effects of accelerated particles on the shock and on acceleration itself is developed for the initially quasi-parallel shocks only. One may therefore model the only radial profiles of brightness, in the rather narrow region close to the shock (in order to be certain that obliquity does not introduce prominent modifications). Effects of non-linear acceleration on the radial profiles of brightness are considered in ; Cassam-Chenaï et al. (2005); Lee et al. (2008); Zirakashvili & Aharonian (2010).
Future studies on SNR morphology should take into account the NLA effects. Nevertheless, the classic approach is able to reveal the general properties of SNR maps dec 2010 RAS termined by MF behavior and particle acceleration. Beside that, it is important to know the properties of the 'classic' images because they create the reference base for investigation of the efficiency of NLA effects in the surface brightness distribution of SNRs.
In Paper I, we introduced a method to derive an aspect angle of ISMF from the radio brightness of SNR. In Paper II, we synthesized radio and IC γ-ray maps and concluded that coincidence of the position of the γ-ray and radio limbs is not a common case in theoretical models, because different parameters are dominant in determintion of the radio and γ-ray brightness variations. On the other hand, radio, Xray and γ-ray observations (Petruk et al. 2009c;Miceli et al. 2009;Acero at al. 2010) show that radio, X-ray and γ-ray limbs coincide in SN 1006. As discussed in Paper II, such coincidence might be attributed to a combination of obliquity dependences of magnetic field and properties of emitting particles, as well as orientation versus the observer.
In the present paper, we make a step forward extending our model of non-thermal leptonic emission of Sedov SNRs in uniform medium to the X-ray band. We also derive brightness profiles for representative parameters which are suited for the comparison with adiabatic SNRs. Moreover, we derive analytical approximations of the azimuthal and radial profiles of radio, X-ray and γ-ray brightness which are extremely useful to demonstrate their dependence on the acceleration parameters, magnetic field orientation and the viewing geometry; they can also be directly and very easily fitted to SNR images to derive estimations for the best-fit quantities. While the analytical approximation cannot substitute the more accurate numerical simulations, we show that they retain enough accuracy to represent an effective diagnostic tool for the study of non-thermal SNR shells.

General considerations
The present paper is continuation of the study in Paper II. The model and numerical realisation are therefore similar to those used in the Paper II; the reader is referred to this paper for more details. In short, in order to investigate the properties of the leptonic emission of Sedov SNR, we use the Sedov (1959) solutions for dynamics of fluid as well as the Reynolds (1998) description of the MF behavior downstream of the shock. The use of analytical solutions allows us to reduce the computational time considerably. We follow Reynolds (1998) also in calculation of the evolution of the distribution function N (E) of relativistic electrons downstream of the shock (see Appendix A for details). The ISMF orientation versus observer is described by the aspect angle φo, an angle between ISMF and the line of sight. Let us define also the obliquity angle Θo as the angle between ISMF and the normal to the shock, and the azimuth angle ϕ in the projection plane which is measured from the direction of the component of ISMF in the plane of the sky (see Fig. 1 in Paper II).
The compression factor for ISMF at the shock front σB is modulated from unity at parallel shock to σ = (γ +1)/(γ − 1) at perpendicular one (where γ is the adiabatic index), in agreement with Reynolds (1998).
At the shock, the energy spectrum of relativistic electrons is taken to be N (E) = KE −s exp (−(E/Emax) α ), where Emax is the maximum energy of electrons, s the spectral index, K the normalization, free parameter α regulates the rate of the spectrum decrease around the high-energy end 1 . Evolution of the 'electron injection ability' of the shock is represented as K ∝ V −b where V is the shock velocity, and b is a constant. The variation of the distribution N (E) over the surface of the SNR and its evolution downstream of the shock are calculated as described by Reynolds (1998). Following Reynolds (1998), we synthesize the synchrotron and IC emission, considering each of the three cases of variation of electron injection efficiency with the shock obliquity (quasi-perpendicular, isotropic, and quasi-parallel particle injection) and each of the three alternatives for time and spatial dependence of Emax (time-limited, loss-limited and escape-limited with a gyrofactor η as a parameter; it is a ratio of the mean free path of a particle to its gyroradius). Both the synchrotron and the IC losses are included as channels for the radiative losses of relativistic electrons, though the IC losses are quite small comparing to synchrotron due to large MF considered.
The surface brightness is calculated by integrating emissivities along the line of sight within the SNR. The emissivity of electrons is given by where ε is the photon energy and p is the spectral distribution of synchrotron or IC radiation power of electron. We assume that information about orientation of B inside SNR is lost because of turbulence, in practice, we use an average aspect angle downstream. The distribution p is calculated with the use of the analytical approximation developed by Petruk (2009, see also Paper II).

Images
The resulting synchrotron radio images and the IC γ-ray images synthesized by our model have been already presented in Paper I and Paper II. Therefore, we present here only the X-ray images (see Fig. 1), adding appropriate references to the previous work to let the reader do a quick comparison. The pattern of synchrotron X-ray brightness of SNR is in general similar to the radio one. In most cases, the bright X-ray limbs or other features are located in the same azimuth as in the radio images. The only differences appear 1 A number of observations (Ellison et al. 2000(Ellison et al. , 2001Lazendic et al. 2004;Uchiyama et al. 2003) reveal evidence of broadening of the high-energy cut-off, i.e. α < 1. Such broadening should be attributed to the physics of acceleration (Petruk 2006), rather than to the effect of superposition of spectra in different conditions along the line of sight as it was suggested initially by Reynolds (1996). From other side, theoretical model of Zirakashvili & Aharonian (2007) demonstrates that it should be α = 2 for the loss-limited models, whereas in practice, taking the time evolution into account it could be between 1 and 2 (Schure et al. 2010). Also Kang & Ryu (2010) suggest that α > 1. due to radiative losses which modify downstream distribution of the electrons emitting in X-rays (thus the features of brightness are radially thinner) and due to variation of Emax over the SNR surface. In the radio (see Fig. 4 in Paper II) as also in the X-ray band, the remnant shows two symmetric bright lobes (for φo = 90 o ) in all the injection models with the maxima in surface brightness coincident in the two bands. The maxima are located at perpendicular shocks in the quasi-perpendicular and isotropic models (i.e. where B is higher), and at parallel shocks in the quasi-parallel model (i.e. where emitting electrons are only presented). The lobes are much radially thinner in X-rays than in radio because of the large radiative losses at the highest energies that make the X-ray emission dominated by radii closest to the shock. The X-ray morphology of SNR is different for different aspect angles (Fig. 1, cf. with Figs. 4, 5 and 6 of Paper II for radio and γ-ray images, respectively). In the case of quasi-perpendicular injection, the morphology is bilateral (two lobes) for large aspect angles (φo > 60 0 , i.e. the component of ISMF in the plane of the sky is larger than that along the line of sight) and almost ring-like for low aspect angles (φo < 30 0 ; see Fig. 1) with intermediate morphology between 30 0 and 60 0 . In the case of quasi-parallel injection, the remnant morphology in the radio band is known to be bilateral for large aspect angles and characterized by one or two eyes for low aspect angles (Fulbright & Reynolds 1990;Orlando et al. 2007). On the other hand, it is worth noting that the remnant morphology in X-rays is in general bilateral for aspect angles φ > 30 0 and centrally bright for very low angles, indeed a rather limited set of possible cases (lower panels in Fig. 1). This happens because the non-thermal X-ray emission originates from a very thin shell behind the shock making the effect of limb brightening in X-rays more important than in the radio band. In addition, we note that centrally bright X-ray (and radio) SNRs are expected to be much fainter than bilateral SNRs (see lower panels in Fig. 1) and consequently much more difficult to be observed. The above considerations may affect the statistical arguments generally invoked against the quasi-parallel injection (i.e. the fact that this model produces morphology which is not observed; e.g. Fulbright & Reynolds 1990;Orlando et al. 2007). Fig. 1 are calculated assuming a dependency of Emax on the obliquity angle which corresponds to the time-limited model with η = 1 as introduced by Reynolds (1998), namely Emax ∝ Emax(Θo)Bo, where Emax(Θo) is a function describing smooth variations of Emax versus obliquity, and Bo is the pre-shock ISMF strength (in such a model no dependency on the shock velocity is present; Reynolds 1998). With this particular choice, Emax is quite similar for different obliquities, namely E max⊥ /E max = 1.3. Larger values of η always provide E max⊥ /E max > 1 in this model, thus the character of azimuthal variation of brightness would be similar.

Images on
The efficiency of variation of Emax with obliquity in modification of the azimuthal distribution of X-ray synchrotron brightness depends obviously on the photon energy: if the maximum contribution to the emission at given photon energy is from electrons with energy much less than Emax then this effect is negligible. It is useful to introduce the reduced photon energy, asε = ν/νc(E max , Bo) where  . Effect of radiative losses (represented by ǫ f ) on the radial profiles of X-ray surface brightness, at two azimuth. The reduced fiducial energy ǫ f = 0.3, 1, 3 (from below); φo = π/2, ε = 1, b = 0.
νc(E, B) = c1 sin φ E 2 B is the synchrotron characteristic frequency, c1 = 6.26 × 10 18 cgs, or where ε keV is the photon energy in keV. Ifε = 0.29 then most of the contribution to the synchrotron X-ray emission is from electrons with energy E = Emax. 2 Figs. 1 is calculated forε = 2.8, i.e. images shown are mainly due to emission of electrons with energy few times higher than E max ; the role of variation of Emax is therefore clearly visible in the images.

Brigthness profiles
In the present subsection, Emax is assumed to be constant in time and the same for any obliquity; in addition, isotropic injection, s = 2 and α = 1 in the energy spectrum of relativistic electrons are assumed.
The radial thickness of features in the X-ray images is sensitive to the photon energy: the larger the energy the thinner the limbs (Fig. 2). This is because radiative losseṡ E of electrons with energy E is efficient for more energetic electrons,Ė ∝ E 2 . Ifε > 0.29 then most of the contribution to the synchrotron X-ray emission is from electrons with energies E > Emax where the radiative losses are of the main importance.
An important factor for emission of highly energetic electrons is the fiducial energy, which reflects the importance of radiative losses in modification of the electron distribution. It is defined as ǫ f = 637 B 2 s tEmax −1 (Reynolds 1998), ( Radiative losses are important for ǫ f < 1 and minor for ǫ f > 1. Fig. 3 demonstrates how the value of ǫ f affects the radial profiles of X-ray brightness: the smaller ǫ f the thinner the rim. Our model does not include consistently the effects on shock dynamics due to back-reaction of accelerated CRs. However, we may approach the effect of shock modification by considering different values of the adiabatic index γ which is expected to drop from the value of an ideal monoatomic gas. In particular, Fig. 4 considers the cases of γ = 5/3 (for an ideal monatomic gas), γ = 4/3 (for a gas dominated by relativistic particles), and γ = 1.1 (for large energy drain from the shock region due to the escape of high energy CRs). The shock modification results in more compressible plasma and, therefore, in the radially-thinner features of the nonthermal images of SNRs. A small distance between the forward shock and contact discontinuity (Cassam-Chenaï et al. 2008;Miceli et al. 2009) could also be attributed to γ < 5/3. Effect of the index γ on the profiles of hydrodynamical pa-rameters downstream of the adiabatic shock is widely studied (e.g. Sedov 1959): smaller γ makes the shock compression factor higher, σ = 4 for γ = 5/3, 7 for γ = 4/3, 21 for γ = 1.1, (4) and the gradient of density downstream stronger (e.g. Appendix B), n(r) ≈r κnr , κnr = 12 for γ = 5/3, 25 for γ = 4/3, 88 for γ = 1.1, (wheren = n/ns, ns is the density immediately postshock andr = r/R ≤ 1). In addition to that, the X-ray (and also TeV γ-ray) brightness is modified by increased radiative losses of emitting electrons. Really, the larger compression leads to the higher post-shock MF and thus to increased losses,Ė ∝ B 2 , which results in turn in the thinner radial profiles of brightness. It is unknown how the injection efficiency (the fraction of nonthermal particles) depends on the properties of the shock. We parameterized its evolution as Ks ∝ V −b where b is a constant. Effect of the parameter b on the radial profiles of the surface brightness is demonstrated on Fig. 5. The smaller b the thicker the profiles, because there are more emitting electrons in deeper layers, which were injected at previous times. This property affects the nonthermal emission in all bands. However, the effect is less prominent in Xrays (and in TeV γ-rays) if radiative losses are quite effective to dominate it (see Fig. 5, lines for different photon energies). This is in agreement with finding of Parizot et al. (2006) and Vink et al. (2006) who showed that, for E > Emax, the width of the synchrotron limbs does not depend on the shock velocity in the loss limited case. Instead, profiles of the radio brightness may be used to put limitations on the value of b.
In a similar fashion, the X-ray and γ-ray radial profiles are affected also by the time evolution of the maximum energy, Emax ∝ V q . However, it seems impossible to determine q from such profiles because contribution of other factors is often dominant.
An interesting feature of the synchrotron images of SNRs is apparent from Fig. 6. The maxima of the radial profiles of brightness for different azimuth are located almost at the same distance ρ from the center of projection, for radio and X-rays. Thus, the best way to analyze the azimuthal profiles of the surface brightness is to find the position ρ of the maximum for one azimuth and then to trace the azimuthal profile of brightness Sρ(ϕ) for fixed ρ.

EXPLICIT APPROXIMATE ANALYTICAL FORMULAE FOR SURFACE BRIGHTNESS PROFILES OF NON-THERMAL SNR SHELLS
The rigorous model discussed in the previous section is able to predict the non-thermal emission of Sedov SNR under a variety of conditions. However, in practical applications, it may be rather time-consuming to perform an extensive parameter space exploration, and the crucial dependence on the relevant parameters of the acceleration processes may be hidden. In order to understand how the properties of MF, electron injection and acceleration influence the brightness distribution, we derived analytic approximate formulae for the azimuthal and radial profiles of the surface brightness of adiabatic SNR. In this way, we can easily see what are the main factors which determine the pattern of the nonthermal images of SNRs, and which of them are mostly responsible for the azimuthal variation of the surface brightness and which for the radial one. This turns to be extremely useful in guiding the comparison with real observations. The analytical formulae are valid close to the shock only, but are adequate to describe azimuthal and radial variations of brightness around maxima which are located close to the edge of SNR shells.

Radio profiles
Let the evolution and obliquity variation of the electron injection efficiency be denoted as V (t) −b ς(Θo) and of the obliquity variation of MF compression/amplification as σB(Θo); for the sake of generality we assume ς(Θo) and σB(Θo) to be some arbitrary smooth functions. Properties of the azimuthal and radial profiles of the radio brightness is determined mostly by (Appendix E) ̺ = ρ/R, σ is the shock compression ratio, κr = 3b/2 + (2 + s)κ ad + 1/σ + s, κ ad is close to unity for γ = 1.1 ÷ 5/3 (Appendix B). The effective obliquity angle Θ o,eff is related to azimuth ϕ and aspect φo as the azimuth angle is measured from the direction of ISMF in the plane of the sky. Eq. (6) is a generalisation of the approximate formula derived in Paper I.
Eq. (6) shows that the azimuthal variation of the radio surface brightness S̺(ϕ) at a fixed radius ̺ of projection, is mostly determined by the variations of the magnetic field compression (and amplification, if any) σB and by the variation of the electron injection efficiency ς. The radial profile Sϕ(̺) ∝ Ir(̺) is determined mostly by σ, b and s. Adiabatic index γ affects the radial and azimuthal profiles mostly through the compression factor σ = (γ + 1)/(γ − 1) because κ ad weakly depends on γ.

Synchrotron X-ray profiles
Let us assume that the maximum energy is expressed as Emax(Θo, t) ∝ V (t) q Emax(Θo), where q is a constant and Emax(Θo), for the sake of generality, is some arbitrary function describing the smooth variation of Emax versus obliquity. The synchrotron X-ray brightness close to the forward shock is approximately (Appendix C) Figure 5. Evolution of injection efficiency (represented by b) and radial profiles of surface brightness in radio (left), X-rays (middle) and γ-rays (right). The parameter in relation Ks ∝ V −b is b = −3/2, 0, 2 (from above). Other parameters are φo = π/2, ϕ = 0 (for radio and X-rays) and ϕ = π/2 (for γ-rays), ǫ f = 1. X-ray and γ-ray profiles are shown for two photon energies: where the most contribution is from electrons with E = Emax (dashed lines) and three times larger (solid lines); in case of X-rays, solid lines correspond toε = 1. Figure 6. Radial profiles of the radio (left), X-ray (middle) and γ-ray (right) surface brightness of SNR for different azimuth: 0, π/6, π/3, π/2 (marked near lines). The aspect angles are π/2 (solid lines) and π/3 (dashed lines); b = −3/2, ǫ f = 1,ε = 1, γray profiles are for photons with energy 0.1εγ(Emax). with The parameter is responsible for the adiabatic (the first term) and radiative (the second term) losses of emitting electrons and the time evolution of Emax on the shock (the third term). The reduced electron energy which gives the maximum contribution to emission of photons with energyε is it varies with obliquity (since MF varies; electrons with different energies contribute to the synchrotron emission atε). Parameters ψ, ǫm, Emax depend on Θ o,eff and, therefore, on the aspect angle φo and the azimuth angle ϕ.
If ǫm ≪ 1 then Eq. (10) for the X-ray brightness transforms to Eq. (6) for the radio brightness.
The thickness of the hard X-ray radial profile is used to estimate the post-shock strength of MF in a number of SNRs (e.g. Berezhko & Völk 2004). The absolute value of MF is present in Eq. (10) throughε and ǫ f , Eqs. (2), (3) which appear in ψ and ǫm, Eqs. (13), (14). In both cases, Bs is in combination with Emax (thus, the value of the electron maximum energy may affect the estimations). The idea of the method bases on the increased role of losses in X-rays due to larger MF, i.e. on the role of the second term in ψ, Eq. (13). Really, the influence of κr (i.e. of s and b) is minor in X-rays (middle panel on Fig. 5), if radiative losses affect the electron evolution downstream of the shock (i.e. forε > ∼ 0.29, ǫ f < ∼ 1). The role of the first and the third terms in ψ are also minor in most cases (q = 0 for the timelimited and escape-limited models and unity for the losslimited one) because the second term > ∼ 10. However, the adiabatic index makes an important effect on the thickness of the profile, mostly through σ which appears in σB and in Ix. Being smaller than 5/3 (that is reasonable especially in the case of efficient acceleration, which is actually believed to be responsible for the large MF), the index may compete to some extend the role of losses, used in the method for estimation of MF (see e.g. Fig. 4) that might lead to smaller estimates of MF strength.
The expression for ψ is the same as (13) but ǫm is different: where ε is the γ-ray photon energy, T the temperature of the seed photon field, γmax the Lorentz factor of electrons with energy Emax, or Eq. (15) is a generalisation of the approximate formula derived in Paper II. The azimuthal variation of the IC γ-ray brightness depends mostly on the injection efficiency. The role of variation of Emax is prominent only if obliquity dependence of injection is not strong. Parameter α, being smaller than unity, results in smaller azimuthal contrasts of synchrotron X-ray or IC γ-ray brightness comparing to model with purely exponential cut-off in N (E). The radial distribution of IC brightness is determined mostly by σ, ǫm, ǫ f , b and, to the smaller extend, by s and q.

Accuracy of the formulae
The approximations presented above do not require long and complicate numerical simulations but restore all the properties of nonthermal images discussed in the previous sections as well as in Papers I and II, including dependence on the aspect angle. Therefore, they may be used as a simple diagnistic tool for non-thermal maps of SNRs.
The formulae are rather accurate in description of the brightness distribution close to the shock. They do not represent centrally-brightened SNRs. Instead, they may be used in SNR shells for those azimuth ϕ where ǫm < ∼ 1 and ǫ f > ∼ 0.1, in the range of̺ from (1−2∆̺m) to 1, where ∆̺m = 1−̺m, ̺m is the radius (close to the shock) where the maximum in the radial profile of brightness happens. Approximations are compared with numerical calculations in respective Appendices and their applicability is discussed in details on example of the IC emission in Appendix D1.

DISCUSSION
Analysis of azimuthal profiles of brightness in different bands allows one to put limitations on models of injection, MF, Emax. In most cases, the best way to estimate the azimuthal variation Sρ(ϕ) of the surface brightness is the following. An approximate radial profile Sϕ(ρ) of the brightness should be produced for azimuth where the largest losses occur (i.e. where ǫ f ∝ Emaxσ 2 B −1 is smaller; e.g. at ϕ = 90 o ). This allows us to find̺m which should be used in Sρ(ϕ) in order to estimate the azimuthal variation of brightness.̺m keeps us at maxima in the radial brightness profiles for different azimuth (Fig. 6).
Energy of electrons evolve downstream of the shock as E(ā) = EiE ad E rad , where Ei is initial energy,ā = a/R, a the Lagrangian coordinate. Adiabatic and radiative losses of electrons in a given fluid element a are represented by functions E ad (ā) ≤ 1, E rad (ā) ≤ 1 respectively (Appendix A). Close to the shock, they are approximately E ad ≈ā κ ad (κ ad depends on the adiabatic index γ only and is close to . The latter expression clearly shows that the fiducial energy ǫ f is important parameter reflecting the 'sensitivity' of the model to the radiative losses, as it is shown by Reynolds (1998): the larger the fiducial energy the smaller the radiative losses. In fact, E rad = 1 means no radiative losses at all. Another fact directly visible from this approximation is that radiative losses are much more important at the perpendicular shock (where σB = σ) than at the parallel one (where σB = 1). In addition, the radiative losses depends rather strongly on the index γ: σ 2 B = 16 for γ = 5/3 but σ 2 B = 49 for γ = 4/3. Note, that in the analysis above, the difference between the parallel and perpendicular shocks are only due to the compression factor σB which may be treated as "compression-plus-amplification" factor. Therefore, our consideration may also be applied in case of the very turbulent and amplified pre-shock field (when information about obliquity is lost) once this factor is known.
TeV γ-ray image of SN 1006 demonstrates good correlations with X-ray image smoothed to the H.E.S.S. resolution (Acero at al. 2010). We mean here both the location and sizes of the bright limbs. Let us consider the polar-caps model of SN 1006. The shock is quasi-parallel around the limbs, MF azimuthally increases (in ≈ 4 times). The injection efficiency decreases (in > 10 −3 times) out of the limbs; the number of electrons emitting in X-rays and TeV γ-rays is therefore dramatically low at perpendicular shock comparing to the parallel, that is in agreement with no TeV emission at NW and SE regions of SN 1006. However, the azimuthal sizes of the limbs in X-rays and γ-rays is expected to be different, in the polar-caps model: they should be larger in X-rays. Really, azimuthal variation of Emax is smaller than variations of ς and σ 3/2 B ; therefore, from Eqs. (10) and (15), azimuthal variations of brightness are mostly Sx ∝ ςσ 3/2 B while Sic ∝ ς (see Fig. 7 for a comparison of remnant morphologies in the radio, X-ray and γ-ray bands). We hope that How back-reaction of accelerated particles may modify nonthermal images of SNRs? Our formulae can restore some of these effects. In our approximations, s, in general, is allowed to vary with E, e.g. to be s(E) = s+δs(E). The index s reflects the 'local' slope of the electron spectrum around ǫm. Therefore, if s(E) = const, the index s has to vary with azimuth because ǫm varies, Eq. (14). Generally speaking, such approach allows one to estimate the role of the nonlinear 'concavity' of the electron spectrum in modification of the nonthermal images. However, we expect that this effect is almost negligible because δs(E) is very slow function, at least within interval of electron energies contributing to images. Another effect of efficient acceleration consists in the adiabatic index γ smaller than 5/3. Our approximations are written for general γ. Namely, the index γ affects S(ϕ,̺) through σ. Cosmic rays may also cause the amplification of the seed ISMF. In our formulae, σB(Θo) represents the obliquity variation of the ratio of the downstream MF to ISMF strength, Bs/Bo. Therefore, it may account for both the compression and amplification of ISMF; for such purpose, σB should be expressed in a way to be unity at parallel shock.

COMPARISON WITH OBSERVATIONS: A WORKING EXAMPLE
The main reason for the derivation of the explicit approximations of Sect. 3 is to highlight the factors which are the most efficient in the formation of the pattern of surface brightness. However, sometimes the analytical formulae may help in estimates some of the remnant parameters. We would like to present two examples. Let us consider SN 1006, s = 2 ), γ = 5/3. The injection efficiency is isotropic if one assumes that SN 1006 evolves in the uniform ISMF and uniform ISM (Paper I). The best-fit value of the aspect angle found from the approximate Eq. (6) is 68 o ± 4.0 o (Fig. 8) while the detailed numerical calculations give 70 o ± 4.2 o (Paper I).
The same approximate formula shows that the radial profile of radio brightness depends only on the value of b (which shows how the injection efficiency evolve with the shock velocity). Unfortunately, the differences between profiles for b = −1 and b = 1 are comparable with accuracy of the approximate formulae and of the experimantal data; therefore, the approximation may not be used for estimations of b.
The sharpest radial profile of X-ray brightness (from Fig. 4A in Long et al. 2003) was used to estimate the strength of the post-shock MF in ∼ 100 µG (Berezhko et al. 2003). Fig. 9 shows approximate profiles for three values of MF in comparison with the Chandra profile and detailed numerical simulations (Petruk et al., in preparation). One can also see from the approximation that B ∼ 100 µG is the most appropriate value in agreement with the value found in the literature.

CONCLUSIONS
The present paper extends analysis of properties of the surface brightness distribution of spherical adiabatic SNRs started in Paper I (radio band) and Paper II (IC γ-rays) to the nonthermal X-rays. It also generalizes the method of approximate analytical description of the azimuthal and radial profiles of brightness introduced in these papers.
Synchrotron images of adiabatic SNR in X-rays are synthesized for different assumptions about obliquity variations of the injection efficiency, MF and maximum energy of accelerated electrons. We analyze properties of these images. Different models of electron injection (quasi-parallel, isotropic and quasi-perpendicular) as well as models of the electron maximum energy (time-limited, loss-limited and escape-limited) are considered.
The azimuthal variation of the synchrotron X-ray and IC γ-ray brightness is mostly determined by variations of ς, σB and Emax, of the radio brightness by ς and σB only. In general, higher B increases X-ray and decreases IC γ-ray brightness. Really, higher MF is a reason of larger losses of emitting electrons (i.e. decrease of their number) and thus of the smaller brightness due to IC process. In contrast, X-rays are more efficient there because Sx ∝ B 3/2 .
The radial profiles of brightness depend on a number of factors. It is quite sensitive to the adiabatic index: γ < 5/3 makes plasma more compressible. Therefore, the brightness profile is thinner due to larger compression factor, larger gradient of density and MF downstream of the shock and larger radiative losses.
The role and importance of various factors on the surface brightness in radio, synchrotron X-rays and IC γ-rays are demonstrated by the approximate analytical formulae. They accurately represent numerical simulations close to the shock and are able to account for some non-linear effects of acceleration if necessary. This makes the approximations a powerful tool for quick analysis of the surface brightness distribution due to emission of accelerated electrons around SNR shells. The application of the approximate formulae to the case of SN1006 yields measures of the aspect angle and the post-shock MF in good agreement with more accurate analysis found in the literature.

APPENDIX A: EVOLUTION OF THE ELECTRON ENERGY SPECTRUM DOWNSTREAM OF THE ADIABATIC SHOCK
Relativistic electrons evolving downstream of the shock suffer from adiabatic expansion and radiative losses due to synchrotron and inverse Compton processes. Electrons are considered to be confined to the fluid element which removed them from the acceleration site (Reynolds 1998). 3 Let the fluid element with Lagrangian coordinate a was shocked at time ti. If energy of electrons was Ei at ti, it becomes later (Reynolds 1998 where the first summand in the denominator reflects adiabatic losses, the second one is due to radiative losses, n = n/ns, index 's' means 'immediately post-shock', the fiducial energy for parallel shock E f = 637/(B 2 eff,s t) cgs. The effective magnetic field is B 2 eff = B 2 +B 2 CMB , B(a) takes into account its evolution downstream (Reynolds 1998), BCMB = 3.27 µG strength of magnetic field with energy density equal to energy density of CMBR. BCMB is introduced in order to account for the inverse-Compton losses (Reynolds 1998), therefore it is constant everywhere. The synchrotron channel dominates inverse-Compton losses if Bs ≫ BCMB.
The dimensionless function I accounts for evolution of fluid during time from ti to t; it was initially defined as integral over time (Reynolds 1998). In case of Sedov shock, I may be written in terms of spatial coordinate that is more convenient for simulations than original representation in terms of time. Namely, for uniform ISM: Eq. (A1) results in relations where the adiabatic and radiative losses are represented by Shocks of different strength are able to accelerate electrons to different Emax. Let Emax ∝ V q where V is the shock velocity, q = 1, 2, 0 for loss-limited, time-limited and escape-limited models respectively (Reynolds 1998). If shock accelerates electrons to Emax at present time t, then, at some previous time ti when fluid element a ≡ R(ti) was shocked, the shock was able to accelerate electrons to whereā = a/R(t) and we used Sedov solutions. The obliquity variation of the maximum energy is given by Emax = E max, Emax(Θo) with Emax(Θo) independent of time.
3 Such an assumption means that we consider the advection only, not the diffusion. However, for X-ray emitting electrons, the lengthscales for advection l adv and diffusion l dif are comparable: l adv /l dif = 0.8 − 1.1 (Ballet 2006). Therefore, the results of the present paper are robust.
Let us assume that, at time ti, an electron distribution has been produced at the shock where α is constant. The obliquity variation of Ks is given by Ks(Θo) = K s K(Θo) with K(Θo) also independent of time.
Conservation equation (where σ = ns/no) and continuity equation no(a)a 2 da = n(a, t)r 2 dr shows that downstream Therefore, in general, where the profileK(ā) is independent of obliquity. Note, that K(a, t, Θo) is not affected by the radiative losses, therefore it behaves in the same way also for radio emitting electrons. Once s is close to 2, the radiative losses influence the shape of N (E) mostly through the exponential term in Eq. (A8). In other words, they are effective only around the high-energy end of the electron spectrum as it is shown by Reynolds (1998).
The above formulae are also valid if the spectral index s depends on E, e.g. s(E) = s + δs(E) like it would be in case of the nonlinear acceleration. In addition, no specific value of the adiabatic index γ is assumed here. It influences the downstream evolution of relativistic electrons throughn(ā) which depends on γ (Sedov 1959).
Magnetic field is approximatelȳ β(Θ o,eff ) = β cos 2 Θo + β ⊥ σ 2 sin 2 Θo cos 2 Θo + σ 2 sin 2 Θo , Approximation for normalization K follows from the definition (Appendix A) and approximation forn. Adiabatic losses are accounted with E ad (a) which is defined by (A4). Its approximation is therefore it is valid forr > 0.8 with error less than few per cent. The value of κ ad is close to unity for γ = 1.1 ÷ 5/3 (Table B1). In order to approximate E rad defined by (A4), we substitute (A2) with approximations forn andB. Then we use the property for function of the form f (a) = 1 − ca x (1 − a y ). In this way, This expression is good forr > 0.94, with error of few per cent. It depends on γ through σ in σB which is (Reynolds 1998) σB = 1 + σ 2 tan 2 Θo 1 + tan 2 Θo Note that dependence on the absolute value of the magnetic field strength B s is present in (B13): s . This approximation clearly shows that the fiducial energy E f is important parameter reflecting the 'sensitivity' to the radiative losses, as it shown by Reynolds (1998): the larger the fiducial energy the smaller the radiative losses. In fact, E rad = 1 means no radiative losses at all, see (A4). Another fact directly visible from Eq. (B13) is that radiative losses are much more important at the perpendicular shock (σB = σ) than at the parallel one (σB = 1). Radiative losses depends rather strongly on the index γ: σ 2 B = 16 for γ = 5/3 but σ 2 B = 49 for γ = 4/3. The values of parameters in approximations for different adiabatic index γ are presented in Table B1. A formula obtained here may be useful in situations when an approximate quantitative estimation for the azimuthal variation of the synchrotron X-ray surface brightness is sufficient.
1. The emissivity due to synchrotron emission is Spectral distribution of the synchrotron radiation power of electrons with energy E in magnetic field of the strength B is where ν is frequency, νc = c1BE 2 the characteristic frequency. Most of this radiation is in photons with energy εp = 0.29hνc. In the 'delta-function approximation', the special function F is substituted with With this approximation, (C1) becomes q(ε) = 4πe 3 sin φ ε 1/2 B 1/2 9mec 2 0.29c where Em is the energy of electrons which give maximum contribution to synchrotron emission at photons with energy ε: Em = ε 1/2 (0.29hc1B) −1/2 . 2. Let the energy of relativistic electrons is E in a given fluid element at present time. Their energy was Ei at the time this element was shocked. These two energies are related as where E ad accounts for the adiabatic losses and E rad for the radiative losses (Appendix A). There are approximations valid close to the shock (Appendix B): whereā = a/R, a is Lagrangian coordinate of the fluid element, E f, is the fiducial energy for parallel shock, κ ad depends on γ and is given by (B11); κ ad = 1 for γ = 5/3 (for other γ see Table B1). The factor σB represents compression in the classical MHD (Reynolds 1998) but may be interpreted also as amplification-plus-compression factor. In the latter case, it should be written in a way to be unity at parallel shock. The downstream evolution of K in a Sedov SNR is (Appendix A) where ς is injection efficiency. With the approximations (C7) and s close to 2, the distribution N (E) may be written from (A8) as and s is allowed to vary with E.
3. Let us consider the azimuthal profile of the synchrotron X-ray brightness S̺ at a given radius ̺ from the center of the SNR projection.
Like in Paper II, we consider the 'effective' obliquity angle Θ o,eff which, for a given azimuth, equals to the obliquity angle for a sector with the same azimuth in the plane of the sky (see details in Paper II). The relation between the azimuthal angle ϕ, the obliquity angle Θ o,eff and the aspect angle φo is as simple as cos Θ o,eff (ϕ, φo) = cos ϕ sin φo (C11) for the azimuth angle ϕ measured from the direction of ISMF in the plane of the sky. The surface brightness of SNR projection at distance ̺ from the center and at azimuth ϕ is whererā is the derivative ofr(ā) in respect toā. The azimuthal variation of the synchrotron X-ray brightness is approximately reflects the dependence on ρ, Ems is Em forB = 1: Note, that Ems ∝ ε 1/2 , i.e. S̺ depends in our approximation on the energy ε of observed X-ray photons. 4. Let us approximate Irx. First, we use the approximationsā ≈r σ ,KB (s+1)/2rā ≈ā κr /σ, which are valid close to the shock (Appendix B), σ is the shock compression ratio. Next, we expandr/ r 2 −̺ 2 in powers of the small parameter (r − 1) and consider the only first term of the decomposition: The exponential term in the integral expands in powers of the small parameter (1 − a): In addition, Ems is used instead of Em.
Close to the shock, the integral of interest is therefore where The parameter is responsible for the losses of emitting electrons and the time evolution of Emax on the shock. The value of κ ad is rather close to unity for γ = 1.1 ÷ 5/3 (Table B1); unless radiative losses (the second term in ψ) are negligible, one may use κ ad ≈ 1 for any γ. Other parameters are β is given by Eq. (B8), Parameters ψ, ǫm, σB, Emax and β depend on Θ o,eff and therefore on the aspect angle φo and the azimuth angle ϕ.
The parameter β reflects differences between MF distribution downstream the shock of the different obliquity. It varies from β at parallel shock to β ⊥ at perpendicular one, Eq. (B8). In the approximate formulae, it appears in the combination β/2; the role of β ∈ [β ; β ⊥ ] is minor in modification of the approximate azimuthal and radial profiles. Therefore, in order to simplify the approximation, we may take β/2 ≈ 1.
The index s in (C9), in general, is allowed to vary with E, e.g. to be s(E) = s + δs(E). In our approximation, due to (C3), s reflects the 'local' slope of the electron spectrum appropriate to ǫm. Therefore, if one assumes s(E) = const, the index s(ǫm) may vary with azimuth because ǫm varies, Eq. (C22).

The final formula is
where only Ix depends on̺ and ǫ f . The formula Eq. (C24) gives us the possibility to approximate both the azimuthal and the radial brightness profiles of X-ray brightness for̺ close to unity. It may be used (with a bit larger errors compared to the case of IC emission; Fig. C1, cf. Fig. D1), for those azimuth ϕ where ǫm < ∼ 1 and ǫ f > ∼ 0.1, in the range of̺ from 1 − 2∆̺m to 1, where ∆̺m = 1 −̺m,̺m is the radius where the maximum in the radial profile of brightness happens. We have in mind the maximum which is close to the shock, say̺m > 0.95; therefore, in order to determine̺m, one should look for the azimuth with the largest radiative losses. This is discussed in details on example of the IC emission in Sect. D.

APPENDIX D: APPROXIMATE FORMULA FOR THE IC GAMMA-RAY SURFACE BRIGHTNESS IN SEDOV SNR
In Paper II, we have developed an analytic approximation for the azimuthal variation of the surface brightness of Sedov SNR in γ-rays due to the inverse-Compton process, for regions close to the forward shock. The approximation, Eq. (11), in the cited paper accounts to zeroth order. However, like in the case of the X-ray brightness, the fall of the γ-ray emissivity downstream of the shock is quite strong in case of the efficient radiative losses of electrons. Therefore, in such cases of the efficient losses we need to consider the next order of approximation.
Adopting the approach from the Appendix C to IC emission (see also some details in Paper II), we come to the approximation where the energy Em of electrons which gives maximum contribution to IC emission at photons with energy ε is (e.g. Petruk 2009) T is the temperature of the black-body photons. The factor where κic and σ comes from the approximationsā ≈r σ , Krā ≈ā κ ic /σ, The final formula is It gives us the possibility to approximate both the azimuthal and the radial brightness profiles for̺ close to unity.
D1 Accuracy of the approximation Fig. D1 demonstrates accuracy of the approximation (D8) (left and middle panels show in fact the variation of Iic because both ς and Emax are constant there). Our calculations may be summarized as follows: this approximation may be used, with errors less than ∼ 30%, for those azimuth ϕ where ǫm < ∼ 1 and ǫ f > ∼ 0.1, in the range of̺ from 1−2∆̺m to 1, where ∆̺m = 1 −̺m,̺m is the radius (close to the shock) where the maximum in the radial profile of brightness happens; in addition, approximation may not be used for̺ < ∼ 0.9. If for some azimuth, the above conditions on ǫm and ǫ f do not hold, the accuracy of approximation gradually decreases because the role of the exponent in N (E) and of the radiative losses may not be described by the first terms in the decompositions used for derivation of the formula.
Let's consider Fig. D1. The photon energy ǫm does not change with azimuth for IC process. On the left panels, the reduced fiducial energy ǫ f (ϕ) = ǫ f /(Emaxσ 2 B ) > ∼ 0.1 for any azimuth: ǫ f = 3 at the parallel shock and ǫ f⊥ = 0.19 at the perpendicular shock. The approximation is accurate for any azimuth, for 0.98 < ∼̺ ≤ 1 at ϕ = 90 o and for a wider range of ̺ at ϕ = 0 o . Middle panels on Fig. D1 show the same case except of ǫ f = 1. At parallel shock (i.e. ϕ = 90 o ), the range for̺ is smaller, 0.99 < ∼̺ ≤ 1 (lower panel). Therefore, the approximation of the azimuthal profile for̺ = 0.98 is inaccurate (upper panel, blue line), especially for ϕ > 45 o where ǫ f decreases; it is ǫ f⊥ = 0.06. The azimuthal profile is however accurate for̺ = 0.99 (black line). Similar situation is for variable Emax (right panels on Fig. D1). E max⊥ /E max = 3.25 for considered model, therefore ǫ f /ǫ f⊥ = 52. Therefore, in order to obtain a representative approximation, the lowest possible ǫ f should be about 0.1 × 52 = 5.2. We see from the figure that accuracy decreases toward smaller ǫ f (i.e. where the role of radiative losses are very efficient in modification of the electron distribution) and for smaller̺.
In general, the accuracy of the approximation is better for larger ǫ f and smaller ǫm. With decreasing of the aspect angle φo, the accuracy of the approximations for the azimuthal profile increases at the beginning (because contrasts in σB, ǫ f and Emax are lower) and then decreases again, for the case of the quasi-parallel injection, because SNR becomes centrally-brightened while our approximation is developed for regions close to the edge of SNR. Figure C1. Azimuthal (upper panels) and radial (lower panels) profiles of the X-ray surface brightness Sx (solid lines) and its approximations (C24) (dahsed lines). Calculations are done for φo = 90 o , b = 0, isotropic injection, γ = 5/3, s = 2, α = 1. Models of Emax: Emax = const (left and middle panels) and time-limited one with η = 1.5 (right panels). The reduced electron energy is ǫm = 1 and the reduced fiducial energy is ǫ f = 3 (left), ǫ f = 1 (middle), ǫ f = 5 (right panels). Figure D1. Azimuthal (upper panels) and radial (lower panels) profiles of the IC surface brightness S ic (solid lines) and its approximations (D8) (dahsed lines). Calculations are done for φo = 90 o , b = 0, isotropic injection, γ = 5/3, s = 2, α = 1. Models of Emax: Emax = const (left and middle panels) and time-limited one with η = 1.5 (right panels). The reduced electron energy is ǫm = 1 and the reduced fiducial energy is ǫ f = 3 (left), ǫ f = 1 (middle), ǫ f = 5 (right panels). Figure D2. Radial profile of the radio brightness S̺ (solid lines) and its approximation (C19) (dashed lines) for azimuth ϕ = 0 o (black lines) and ϕ = 90 o (blue lines), φo = 90 o . For smaller aspects, φo < 90 o , the approximation agree better with the numerical profiles.

APPENDIX E: APPROXIMATE FORMULA FOR THE RADIO SURFACE BRIGHTNESS IN SEDOV SNR
An analytic approximation for the azimuthal variation of the radio surface brightness of Sedov SNR (Paper I) may be extended to allow also for a description of the radial variation close to the forward shock. Namely, the correction consists in a factor Ir: where Ir is the same as for the X-ray approximation (C18). Accuracy of this approximation for the radial profile of brightness is demonstrated on Fig. D2 and on Fig. 2 in Paper I for the azimuthal profiles. Ir varies with azimuth less than 10% (cf. e.g. black and blue dashed lines on Fig. D2). This variation is due only to β(Θo). Thus, β may be taken constant with a good choice β/2 = 1 (see also Appendix C). The smaller φo, the smaller differences between the radial profiles for azimuth ϕ = 0 o and 90 o (black and blue solid lines approach one another with decrease of the aspect angle).