SYNCHROTRON-LOSS SPECTRAL BREAKS IN PULSAR-WIND NEBULAE AND EXTRAGALACTIC JETS

1.1. Spectral Breaks in Jets and Pulsar-wind Nebulae

1.2. Synchrotron Losses

The first widely known calculation of the behavior of a distribution of electrons subject to synchrotron losses is that of Kardashev (1962). While most of these results are well known, it is important to recall the particular conditions under which each is applicable, so I shall beg the reader's indulgence for a brief review, which can also serve to fix notation. Kardashev solved the kinetic equation for a distribution N(E, t) of electrons subject to gains by first- and second-order Fermi acceleration, and losses due to radiation or adiabatic expansion, with and without the assumption of new-particle injection and stationarity. He writes the energy-loss rate from a single electron as

$$\begin{equation} \dot{E} = -b B_\perp ^2 E^2, \end{equation} \tag{ 1 }$$

where B_⊥ ≡ Bsin θ, b ≡ (2/3)(e⁴/m⁴_ec⁷) = 2.37 × 10⁻³ cgs (e.g., Pacholczyk 1970), and θ is the electron pitch angle between its velocity vector and the magnetic field. Kardashev pointed out that in a uniform magnetic field in the absence of scattering, electrons preserve their value of θ, since they radiate a beam pattern that is symmetric with respect to their velocity vector. An electron injected into B at t = 0 with energy E₀ has an energy E after time t given by the well known result

$$\begin{equation} E(t) = {E_0 \over 1 + E_0 b B_\perp ^2 t}. \end{equation} \tag{ 2 }$$

An initially infinitely energetic electron is reduced after time t to energy E_max(t, θ) = 1/bB²_⊥t. A power-law energy distribution of electrons N(E₀) = KE^−s₀, with a single value of pitch angle, will be cut off at E_max(t, θ). The electron distribution N(E) will evolve according to N(E)dE = N(E₀)dE₀, so that

$$\begin{equation} N(E) = K[E_0(E)]^{-s} {dE_0 \over dE}, \end{equation} \tag{ 3 }$$

with E(E₀) given by Equation (2). If s < 2, the electrons initially above E_max(t) are sufficiently numerous to pile up in a "bump" just below E_max(t), while if s > 2, the "bump" disappears. (Numerical solutions to Equation (3) are shown below for inhomogeneous models, illustrating the "bump" effect.)

An initially isotropic distribution of electrons suffers unequal radiation losses, with electrons with large pitch angles being more rapidly depleted. For an initial isotropic power-law distribution, after a time t one finds no electrons with pitch angles greater than θ_max given by sin²θ_max = 1/(bEB²t). Since for synchrotron radiation, an individual electron's radiation pattern has an angular width of order 1/γ where γ is the individual Lorentz factor, and γ ≳ 10³ for radio emission and higher frequencies, we can approximate electrons as radiating exactly in their directions of motion. Then θ is also the angle between the line of sight and the local magnetic field. An initially isotropic distribution of electrons injected at t = 0 into a source with a uniform magnetic field making an angle χ with the line of sight, observed through its synchrotron emission, would then disappear abruptly at time t(χ). More realistically, one might expect that the source has a tangled magnetic field, so that all values of χ are achieved in some part of the source. One should then (for an unresolved source) perform an angular integration over the electron distribution. The result is an electron distribution that steepens by one power of E, i.e., N(E) ∝ E^{−s − 1}, above the characteristic energy E_b = 1/(bB²t). (The synchrotron emission from this distribution steepens above ν(E_b) by more than the value of 0.5 in spectral index α because, in this time-dependent case without continuous injection, a correlation exists between E and θ such that more efficiently radiating electrons are depleted most rapidly.)

This situation is still relatively unrealistic, as it ignores any processes by which electrons could change their pitch angles. If electrons scatter in pitch angle on timescales much shorter than the synchrotron-loss timescale, one should simply average the energy-loss rate over angles:

$$\begin{equation} \dot{E} = 1.57 \times 10^{-3} B^2 E^2 \equiv a B^2 E^2, \end{equation} \tag{ 4 }$$

where a ≡ b〈sin²θ〉 = (2/3)b. Then the electron distribution will remain isotropic, and will simply cut off at E_max(t); a source synchrotron spectrum would then cutoff exponentially above ν(E_max(t)).

However, the result we all remember from graduate school is neither of these. A source that turns on at t = 0 with continuous, uniformly spatially distributed injection of a power-law distribution of electrons q(E) ≡ J₀E^−s electrons erg⁻¹ s⁻¹ cm⁻³, develops a break at energy E_b = 1/(aB²t), where the electron spectrum steepens by one power in s. This corresponds to a steepening of the synchrotron spectrum by one-half power in α at ν_b = c₁B⁻³t⁻² with c₁ = 1.12 × 10²⁴ cgs. This relation is frequently used to deduce a magnetic field strength in a synchrotron source of known age.

The result that synchrotron losses (or inverse-Compton losses, which have the same dependence on electron energy) result in the steepening of the particle spectrum by one power and the steepening of the emitted synchrotron spectrum by a half-power, is a widely held idea. It is this application that will be generalized below. It is important to remember that the standard derivation assumes a distributed injection of electrons in a homogeneous source. For other conditions, it is not correct, as we shall see.

Energy-loss breaks from a synchrotron source in which relativistic electrons are advected systematically away from an injection region (such as a wind-termination shock) can be described as being due to shrinking of the effective source size with frequency. At a high enough observing frequency, electrons drop below the energy required to emit at that frequency before they reach the edge of the object, hence limiting its effective size at that frequency. Thus, all results depend on the critical energy an electron may have after suffering synchrotron losses and convecting at speed v. Equation (4) gives the synchrotron-loss rate from a single electron. In a nonrelativistic, constant-density flow (i.e., neglecting adiabatic losses), but allowing the possibility of spatially varying B, we generalize the homogeneous results slightly to obtain

$$\begin{equation} E_c = \left[\int a \,B^2\,{dr \over v}\right]^{-1} \end{equation} \tag{ 5 }$$

for the energy an initially infinitely energetic electron would have after moving at v through a field B for a distance r. So any injected electron distribution must cut off at this energy. For a one-dimensional flow of electrons streaming at constant speed v in a constant magnetic field B, the effective length r(E) of the source is found from

$$\begin{equation} E(r) = \left({v \over a B^2} \right) r^{-1}. \end{equation} \tag{ 6 }$$

An electron of this energy radiates chiefly at frequency ν = c_mE²B = c_m(v²/a²B³)r⁻² where c_m = 1.82 × 10¹⁸ cgs (e.g., Pacholczyk 1970). Then at any frequency ν, the source has an effective length

$$\begin{equation} r(\nu) = c_{\rm m}^{1/2} \left(v \over a B^{3/2} \right) \nu ^{-1/2}. \end{equation} \tag{ 7 }$$

For synchrotron emission with a spectral index α, the observed flux density would then vary as

$$\begin{equation} S_\nu \propto \nu ^{-\alpha } r(\nu) \propto \nu ^{-\alpha -1/2} \end{equation} \tag{ 8 }$$

—the famous steepening by one-half power in the spectral index. (Note that at no position r in the source is there a break in the electron energy distribution N(r, E) to a new power law, although such a distribution does result after integrating over the entire source volume to obtain N(E).)

This kind of argument can be generalized to inhomogeneous sources, for which the steepening may be larger or smaller than 0.5. For instance, if the synchrotron emissivity increases with distance from the center, then as the effective source size shrinks, more emission will be lost than if the source were homogeneous, and the steepening can be greater than 0.5.

Here, we consider a simple model in which electrons are injected at some initial radius r₀ in a "jet" whose full width w rises with dimensionless distance l ≡ r/r₀ as l: w = w₀l. A conical jet (or piece of spherical outflow) then has = 1; a confined jet has < 1. Then, the cross-sectional area increases as A_⊥ ∝ l² (see Figure 1). We shall assume ad hoc power-law dependences of quantities on dimensionless length l, ignoring any transverse variations, making the problem one-dimensional. Let the l-dependence of basic quantities be given by

$$\begin{equation} B = B_0 l^{m_b}, \qquad v = v_0 l^{m_v}, \qquad \rho = \rho _0 l^{m_\rho }. \end{equation} \tag{ 9 }$$

While this is completely general, physical constraints may couple the m's. For instance, in the absence of some mechanism such as mass loading (Lyutikov 2003) or entrainment to alter the effective density ρ, conservation of mass gives

$$\begin{equation} {\hbox{Mass conservation:}} \quad \rho v A_\perp = {\rm const} \Rightarrow m_\rho + m_v = -2\epsilon. \end{equation} \tag{ 10 }$$

In the absence of turbulent amplification or reconnection, magnetic flux will be conserved, giving different relations for components of B parallel and perpendicular to the jet axis:

$$\begin{equation} {\hbox{Longitudinal (radial) field:}} \quad B_\parallel{:}BA_\perp = {\hbox{const}} \Rightarrow m_b = -2\epsilon, \end{equation} \tag{ 11 }$$

$$\begin{eqnarray} &&{\hbox{Transverse (toroidal) field:}} \quad B_\perp\,{:}\,Bwv = {\rm const} \Rightarrow m_b \nonumber\\ &&= -m_v - \epsilon = m_\rho + \epsilon,\hspace{1.8pc} \end{eqnarray} \tag{ 12 }$$

where the last form for transverse B involves invoking mass conservation. Of course, unless the magnetic field is exactly radial, any toroidal component will eventually dominate, barring extremely peculiar and probably unphysical behaviors (e.g., m_ρ < −3). At any rate, here our emphasis is on the greatest generality; any particular set of dynamical assumptions will then confine a model to a subregion of the parameter space of the m's.

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In this formalism, Equation (5) implies $E_c \propto l^{-(1 + 2m_b - m_v)}$ in the absence of other energy-loss mechanisms such as adiabatic expansion losses (nonconstant density). In the presence of adiabatic losses, it can be shown (e.g., Kennel & Coroniti 1984a; Reynolds 1998) that the critical energy is given by

$$\begin{eqnarray} &&E_c = \rho ^{1/3}\left[ \int _1^l {a B^2 \over v} \rho ^{1/3} r_0 dl\right]^{-1} \hphantom{aaaaaaaa}\nonumber\\ &&\hspace{3pt}= l^{m_\rho /3} \left[ \int _1^l a v_0^{-1} r_0 B_0^2 l^{2m_b -m_v + m_\rho /3} dl \right]^{-1}\hspace{-3pt} \end{eqnarray} \tag{ 13 }$$

$$\begin{equation} \hspace{6pt}={v_0 \over ar_0 B_0^2} (1 + 2m_b - m_v + m_\rho /3) l^{-(1 + 2m_b - m_v)} \equiv A_E l^{m_E}, \end{equation} \tag{ 14 }$$

where we have assumed the source is long enough such that l(E) ≫ 1. Note that the effects of adiabatic losses have canceled out (except for a small change in the integration constant), leaving E_c with the same l-dependence as in the constant-density case. We have also made the assumption that the integral in Equation (13) is dominated by the upper limit at l, demanding that m_c ≡ 2m_b − m_v + m_ρ/3> − 1. Otherwise, $E_c \propto l^{m_\rho /3}$ and radiative losses play no role in the spectral behavior, so that the calculation is not self-consistent. Even if this condition is met, we still wish to exclude situations in which the gradients conspire to arrange adiabatic gains of electrons as they convect, i.e., m_E > 0. This situation appears both unphysical and unlikely. Note that m_c = m_ρ/3 − m_E − 1. These two conditions rule out some volume in the parameter space of (, m_i) and must be checked for any particular choices of those parameters. The constraints are related; m_c > − 1 requires m_E < m_ρ/3, so that we ultimately require m_E < min(0, m_ρ/3).

Electrons with energy E_c (at position l where the magnetic field strength is B(l)) radiate chiefly at a frequency

$$\begin{eqnarray} \nu (E_c) &=& c_{\rm m} \left(v_0^2 \over a^2 r_0^2 B_0^3\right)(1\,{+}\,2m_b\,{-}\,m_v + m_\rho /3)^2 \nonumber\\ &&\times\, l^{m_b-2(1 + 2m_b - m_v)}\equiv A_\nu l^{m_\nu }, \end{eqnarray} \tag{ 15 }$$

defining A_ν and m_ν. Note m_ν = m_b + 2m_E = −2 − 3m_b + 2m_v. Then,

$$\begin{equation} l(\nu) = A_\nu ^{-1/m_\nu } \nu ^{1/m_\nu } \equiv A_l \nu ^{1/m_\nu }. \end{equation} \tag{ 16 }$$

We are focusing on conditions such that the source size shrinks with increasing frequency, i.e., m_ν < 0. It can be easily shown that the condition on m_c above is equivalent to m_ν < (m_ρ + m_v) − 1. If mass conservation is assumed, then m_ρ + m_v = −2 and m_ν < 0 always. Otherwise, this condition must be checked; but for reasonable values of the parameters (such as those in all examples described below), it is always fulfilled.

Now to discuss the synchrotron flux, we assume a power-law electron distribution N(E) = KE^−s between energies E_l and E_h ≫ E_l (we take s > 1). As the flow expands, conservation of electron number gives

$$\begin{eqnarray} n_e &=& {K \over s-1} E_l^{1-s} \propto \rho \Rightarrow K \propto \rho E_l^{s-1} \propto \rho ^{1 + (s-1)/3} \nonumber\\ &=& \rho ^{(s+2)/3} \equiv \rho ^{(2\alpha + 3)/3}, \end{eqnarray} \tag{ 17 }$$

since adiabatic losses give individual-particle energies varying as E ∝ ρ^1/3, at least near E_l, an energy we assume to be relativistic, but too low for radiative losses to be important. We have taken s enough larger than 1 such that E^1−s_l ≫ E^1−s_h. Then, we can write the synchrotron emissivity (following Pacholczyk 1970) as

$$\begin{eqnarray} j_\nu &=& c_j K B^{1+ \alpha } \nu ^{-\alpha } = c_j K_0 B_0^{1+\alpha } l^{(2\alpha + 3)m_\rho /3 + (1 + \alpha)m_b} \nu ^{-\alpha }\nonumber\\ &\equiv& A_j l^{m_j} \nu ^{-\alpha }. \end{eqnarray} \tag{ 18 }$$

Here c_j(s) ≡ c₅(s)(2c₁)^α in the notation of Pacholczyk; for s = 1.5, c_j = 1.34 × 10⁻¹⁸ cgs, and we have defined another important index, m_j ≡ (2α + 3)m_ρ/3 + (1 + α)m_b. Assume for the time being that we view the jet directly perpendicular to the axis. Then the line-of-sight path length through the jet at any position l is just w(l) = w₀l (through the center; averaged over lines of sight intersecting a circular cross section, we obtain a mean line-of-sight path of (π/4)w). We recall that we are assuming all jet quantities to be constant in cross section, that is, along these lines of sight. So if the source is at a distance D, the integrated flux density S_ν is given by

$$\begin{eqnarray} S_\nu &=& \int I_\nu d\Omega = {1 \over D^2}\int _1^{l_\nu } dA \int _0^w {\pi \over 4} j_\nu ds \nonumber\\ &=& {\pi A_j \over 4 D^2}\int _1^{l_\nu } w \ r_0 dl (w \ l^{m_j})\nu ^{-\alpha } \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} &=&{\pi A_j \over 4 D^2}\int _1^{l_\nu } r_0 w_0^2 \ l^{2\epsilon + m_j} \nu ^{-\alpha }\ dl\nonumber\\ &=&{\pi A_j r_0 w_0^2 \over 4 D^2(1 + 2\epsilon + m_j)} l_\nu ^{1 + 2\epsilon + m_j}\nu ^{-\alpha }\hphantom{aa\,\,} \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \hphantom{aaaaaa\,}&=&{\pi c_j K_0 B_0^{1+\alpha }r_0 w_0^2 \over 4 D^2 [1 + 2\epsilon + (2\alpha + 3)m_\rho /3 + (1 + \alpha)m_b]}\nonumber\\ &&\times A_l^{(1 + 2\epsilon + (2\alpha + 3)m_\rho /3 + (1 + \alpha)m_b)} \nu ^{- \alpha - \Delta }, \end{eqnarray} \tag{ 21 }$$

where the last equation defines Δ, the amount of spectral steepening:

$$\begin{eqnarray} \Delta &=& - {1 + 2\epsilon + m_j \over {m_b + 2m_E}} = {1 + 2\epsilon + m_j \over |m_\nu |} \nonumber\\ &=& {1 + 2\epsilon + (2\alpha + 3)m_\rho /3 + (1 + \alpha)m_b \over 2 + 3m_b -2m_v}. \end{eqnarray} \tag{ 22 }$$

We have assumed that m_ν < 0, and that the flux integral depends on the outer, not the inner, limit of integration, that is,

$$\begin{equation} 1 + 2\epsilon + m_j > 0. \end{equation} \tag{ 23 }$$

This latter condition can be restrictive.

While we have assumed a jet seen from the side, since the emission is optically thin the observed flux density should be independent of viewing angle. This can be shown explicitly for the case of jets seen exactly end-on, for which the flux integral Equation (19) becomes

$$\begin{equation} S_\nu = {1 \over D^2} \int _0^{w_{\rm max}} {\pi \over 2} w dw \int _{l_i(w)}^{l(\nu)} c_j K_0 B_0^{1 + \alpha }r_0 \nu ^{-\alpha } l^{m_j} dl. \end{equation} \tag{ 24 }$$

Here, the line of sight is parallel to the jet axis, beginning at a value of l ≡ l_i dependent on w (if w < w₀, l_i = 1). The upper limit w_max is just the value of w at which l_i = l_ν, i.e., w_max = w(l(ν)) ≡ w₀[l(ν)]. Now we need a slightly more restrictive condition for the emission to be dominated by the outer limit of integration l(ν): 1 + m_j > 0. If this is the case, the two integrals in Equation (24) decouple, giving

$$\begin{equation} S_\nu = {\pi \over 2 D^2} c_j K_0 B_0^{1 + \alpha } r_0 \nu ^{-\alpha } {w_{\rm max}^2 \over 2} {1 \over 1 + m_j} \left[ l(\nu) \right]^{1 + m_j}. \end{equation} \tag{ 25 }$$

But w_max = w₀[l(ν)], so

$$\begin{eqnarray} S_\nu &=& {\pi \over 2 D^2} c_j K_0 B_0^{1+\alpha } r_0 {w_0^2 \over 2} {1 \over 1 + m_j} A_l^{1 + 2\epsilon + m_j} \nu ^{(1 + 2\epsilon + m_j)/m_\nu - \alpha}\nonumber\\ &&\propto \nu ^{-\Delta - \alpha}, \end{eqnarray} \tag{ 26 }$$

the same as Equation (21) except for a factor (1 + m_j)/(1 + 2 + m_j).

These power-law spectra can hold only over a frequency range related to the size range of the source by l(ν). For instance, for conical, constant-velocity, mass-conserving flow with tangential magnetic field, we have = 1, m_ρ = −2, and m_b = −1, giving l(ν) ∝ ν⁻¹. Thus, a source showing the corresponding value of Δ (in this case, Δ = 7α/3) between frequencies ν₁ and ν₂ must shrink over a range of sizes given by r₁/r₂ = ν₂/ν₁. In general, sources whose spectra are set by effective variations of size with frequency are predicted to have sizes varying as

$$\begin{equation} l(\nu) \propto \nu ^{1/m_\nu } \equiv \nu ^{1/(-2 - 3m_b + 2m_v)}. \end{equation} \tag{ 27 }$$

Equation (22) relates the size exponent 1/|m_ν| to Δ: 1/|m_ν| = Δ/(1 + 2 + m_j). (Thus, a source with observed Δ will have a smaller rate of shrinkage with frequency for a larger value of 1 + 2 + m_j.) The source subtends a solid angle on the sky of

$$\begin{equation} \Delta \Omega = {1 \over D^2} \int _1^{l(\nu)} (w_0 l^\epsilon) (r_0 dl) = {1 \over D^2} {r_0 w_0 \over 1 + \epsilon } [l(\nu)]^{1 + \epsilon }. \end{equation} \tag{ 28 }$$

If the source is only marginally resolved, one may consider an "average" angular size 〈θ〉 defined by

$$\begin{equation} \langle \theta \rangle \equiv (\Delta \Omega)^{1/2} = {\sqrt{r_0 w_0} \over D} {1 \over \sqrt{1 + \epsilon }} [l(\nu)]^{(1 + \epsilon)/2}, \end{equation} \tag{ 29 }$$

so that $\langle \theta \rangle \propto \nu ^{(1 + \epsilon)/2m_\nu }$. For spherical or conical flows, i.e., = 1, $\langle \theta \rangle \propto \nu ^{1/m_\nu }$ as before, but for confined flows ( < 1), the average angular size decreases more slowly with frequency. If a confined jet is seen end-on, the size variation with frequency is reduced even further, as the apparent diameter is now proportional to w_max ∝ [l(ν)] instead of l(ν):

$$\begin{equation} \theta \propto \nu ^{\epsilon /m_\nu } = \nu ^{\epsilon /(m_b + 2m_E)} =\nu ^{\epsilon /(-2 - 3m_b + 2m_v)}. \end{equation} \tag{ 30 }$$

This may result in a significantly reduced size effect, since the primary change in emitting volume is shrinkage along the line of sight.

We can examine a few special cases. First, in the case of a plane constant-velocity flow, we have = 0, m_v = 0, m_ρ = 0, and m_b = 0, and we recover Δ = 1/2. Next, consider tangential magnetic field and conical outflow ( = 1) with constant density and assuming mass and flux conservation. This corresponds to the inner parts of a Kennel & Coroniti (1984a, 1984b, hereafter KC) MHD spherical flow. Then m_ρ = 0, m_v = −2, m_b = m_ρ + = 1, and we have

$$\begin{equation} \Delta = {4 + \alpha \over 9}, \end{equation} \tag{ 31 }$$

which was derived in Kennel & Coroniti (1984b, Equation (4.11b)). This already indicates that values of Δ ≠ 0.5 can be obtained from reasonable situations. It also suggests that obtaining values very different from 0.5 may be difficult. The size effect predicted above is quite weak: m_ν = −2 − 3m_b + 2m_v = −9, so l(ν) ∝ ν^−1/9, as shown in Kennel & Coroniti (1984b, Equation (4.10b)). This weak effect accounts for the very small deviation of Δ from the homogeneous value of 0.5.

Now KC models can be divided into two regions: an inner one as above, and an outer one, a constant-velocity flow with ρ ∝ r⁻² (that is, m_v = 0 and m_ρ = −2) and tangential field (m_b = m_ρ + 1 = −1). Mass conservation is assumed (m_v = −2 − m_ρ). At lower frequencies, the burnoff radius moves in through the outer region, giving

$$\begin{equation} \Delta _{\rm outer} = {7 \alpha \over 3}. \end{equation} \tag{ 32 }$$

However, this is not realized actually, as neither condition m_E < 0 nor 1 + 2 + m_j > 0 is met. First, m_E ≡ −(3 + 2m_b + m_ρ) = +1, indicating that E_c rises with l. Further, the m_j condition gives 3 − 2(2α + 3)/3 − 1 − α = −7α/3, so the flux is dominated by the inner parts of the nebula. At high enough frequencies or photon energies such that the burnoff radius is at the transition point and moves into the v ∝ r⁻² region, we obtain the above result

$$\begin{equation} \Delta _{\rm inner} = {4 + \alpha \over 9}. \end{equation} \tag{ 33 }$$

Here, 1 + 2 + m_j = 4 + α>0, so the consistency condition is met—the flux is dominated by regions near l(ν). For KC's model of the Crab, α = 0.6, so Δ_inner = 0.51.

It is straightforward to directly show that spherically symmetric sources obey the same scaling laws with = 1—that is, the assumptions of thin jets perpendicular to the line of sight still give the correct scaling for spheres. Let the dimensionless radius be R ≡ r/r₀, with injection at R = 1 and burnoff at

$$\begin{equation} R(\nu) = A_R \nu ^{1/m_\nu } \equiv A_R \nu ^{1/(-2 -3m_b + 2m_v))} \end{equation} \tag{ 34 }$$

—that is, the same expression as for l(ν). (Sphericity should not change the expression, only, perhaps, the values of the m's.) Similarly, we should have the same dependence of j_ν(R) in the spherical case as we had for j_ν(l) in the jet case:

$$\begin{equation} j_\nu (R) = c_j K_0 B_0^{1+\alpha } \nu ^{-\alpha } R^{m_j} \end{equation} \tag{ 35 }$$

with

$$\begin{equation} m_j \equiv {2\alpha + 3 \over 3}m_\rho + (1 + \alpha) m_b. \end{equation} \tag{ 36 }$$

Then the total flux from this spherically symmetric, optically thin source is just

$$\begin{eqnarray} S_\nu &=& {4\pi \over D^2} \int _1^{R(\nu)} 4\pi j_\nu (R) R^2 dR\nonumber\\ &=& {4\pi \over D^2}\left(4\pi c_j K_0 B_0^{1+\alpha }\nu ^{-\alpha }\right) {[R(\nu)]^{m_j + 3} \over {m_j +3}}, \end{eqnarray} \tag{ 37 }$$

which gives

$$\begin{equation} \Delta \equiv - {m_j + 3 \over m_\nu } = + {(2\alpha + 3)m_\rho /3 + (1+\alpha)m_b + 3 \over 2 + 3m_b - 2m_v}. \end{equation} \tag{ 38 }$$

This is the same result as can be obtained by setting = 1 in the previous expression for Δ, Equation (22).

Obtaining values of Δ considerably greater than 0.5, as seems to be required by observations of most PWNe, requires relaxing some of the assumptions. The relation from mass conservation of m_ρ = −2 − m_v might not hold if some form of mass loading occurs, for instance, by evaporation of material from thermal filaments, or entrainment of material from a confining medium. Flux freezing for the magnetic field will not hold in the presence of either turbulent amplification of magnetic field or of reconnection. As an example, consider (for algebraic simplicity) the case α = 0, but conical, constant-density flow (so m_v = −2), still conserving mass. Now in this case, m_j = m_b and m_E = −(3 + 2m_b), so

$$\begin{equation} \Delta = - {{3 + m_b} \over {m_b -2(3 + 2m_b)}} = {3 + m_b \over 6 + 3m_b} \ \Rightarrow \ m_b = -{6 \Delta - 3 \over 3\Delta - 1}. \end{equation} \tag{ 39 }$$

Now we can obtain Δ = 2/3 with the value m_b = −1 in this case: the magnetic field drops as the first power of distance (faster than if frozen-in and tangential, slower than if longitudinal). This does not seem like an unreasonable possibility. Note that the conditions are met: 1 + 2 + m_j = 3 + m_b = 2>0, m_E = −1 < 0, and m_c ≡ 2m_b − m_v + m_ρ/3 = 0> − 1. The source size would obey

$$\begin{equation} l(\nu) \propto \nu ^{1/(m_b + 2m_E)} = \nu ^{1/(-1 - 2(1))} = \nu ^{-1/3}. \end{equation} \tag{ 40 }$$

To obtain weaker size effects, one is driven to larger values of 1 + 2 + m_j, for the same observed Δ.

For practical use, it is convenient to consider m_b a dependent variable, in terms of the other quantities. First, solve the equation for Δ for m_b, in terms of Δ, m_ρ, , and α, with no presumed relation between m_b and m_ρ. The result is

$$\begin{equation} m_b = {\Delta (2m_v - 2) + (2\alpha + 3)m_\rho /3 + 1 + 2\epsilon \over 3\Delta - 1 - \alpha }. \end{equation} \tag{ 41 }$$

Two of the conditions are satisfied if m_E < min(0, m_ρ/3), or

$$\begin{equation} 1 + 2m_b - m_v > {\rm max}(0, -m_\rho /3). \end{equation} \tag{ 42 }$$

Assuming mass conservation, this becomes

$$\begin{equation} 1 + 2\epsilon + 2m_b + m_\rho > {\rm max}(0, -m_\rho /3). \end{equation} \tag{ 43 }$$

In addition, we still require 1 + 2 + m_j > 0.

An application to a particular source (i.e., an object of known α and Δ, presumably) can then be made by inserting those values. Various possibilities for and m_ρ can be tried; for each value of m_b obtained in this way, the condition on m_E must be checked by hand. For example, consider B0540-693, with a radio spectrum of α = 0.25 (Manchester et al. 1993), and a break with Δ ∼ 1 at around 20 μm (Williams et al. 2008). A constant-density spherical (or conical) outflow cannot produce this Δ. Inserting the values of α and Δ into Equation (41), and assuming for simplicity = 1, we find

$$\begin{equation} m_b = {1 + 2m_v + 1.2m_\rho \over 1.7}. \end{equation} \tag{ 44 }$$

Then at the price of abandoning mass conservation, we can choose m_ρ = 1 and m_v = −2, giving m_b = −1.06, or about −1. The consistency conditions are all met: m_c = 1/3> − 1, m_E ≡ −(1 + 2m_b − m_v) = −1 < 0, and 1 + 2 + m_j = 2.9>0. The source effective radius decreases as

$$\begin{equation} r \propto \nu ^{1/(m_b + 2m_E)} = \nu ^{-0.33}, \end{equation} \tag{ 45 }$$

which may be a serious problem, since we need the slope of −α − Δ ≅ −1.2 to hold from about 20 μm to somewhere in the blue or near UV—say 0.2 μm, requiring that the source shrink between these two wavelengths by a factor of 4.6—perhaps unlikely. (Though what is shrinking is really the region containing the dominant flux; there could be a faint halo contributing a small amount of flux in which a brighter, shrinking core is embedded). A numerical calculation of this model is shown in Figure 2, along with observations. The physics which could cause these values of m_ρ and m_b is, of course, completely unknown.

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Figures 3 and 4 plot m_b versus Δ for Equation (39) and its generalizations to the pairs (α, ) = (0.3, 1), (0, 0.5), and (0.3, 0.5), respectively. Mass conservation is still invoked. The consistency condition on m_c places upper limits on Δ (lower limits on m_b) shown as the squares on curves on each plot. (The condition 1 + 2 + m_j > 0 is met for all curves shown.) It is difficult to obtain values of Δ>0.7; flows with rapidly dropping density (such as m_ρ = −2, for constant-velocity, mass-conserving flows) seem unable to do so. Substantial deceleration seems to be required, as well as rapid decreases in the magnetic field strength. While there is some parameter space available for accomplishing this, especially for sources with very flat radio spectra, the most physically reasonable way to bring about the required deceleration seems to be mass loading, which also considerably expands the available parameter space of source gradients.

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Energy-loss spectral breaks are commonly used to infer source magnetic field strengths in PWNe. One requires a source age t; for PWNe in SNR shells, one can use modeling of the shell emission to estimate an age, while in some cases, a source size divided by a mean flow speed (estimated one way or another) can give an age estimate. Then one simply assumes a homogeneous source for which E_c = (aB²t)⁻¹ = 637/B²t and from Equation (5),

$$\begin{equation} B_h = \left(c_{\rm m} \over a^2\right)^{1/3} \nu _b^{-1/3} t^{-2/3} = 0.90 \ \nu _{\rm GHz}^{-1/3} \ t_{\rm yr}^{-2/3} \ {\rm G} \end{equation} \tag{ 46 }$$

(where we have averaged over pitch angles). It is of interest to compare this to the magnetic field that would be inferred for a flow model assuming (incorrectly) that the source is homogeneous. Let the source have a size L and break frequency ν_b (so that L = r₀l(ν_b)). For a flow model, we can deduce the initial magnetic field B₀ from Equation (15):

$$\begin{equation} \left(L \over r_0\right)^{m_\nu } = {a^2 \over c_{\rm m}} \left(r_0^2 \over v_0^2\right) {1\over f^2} B_0^3 \nu _b, \end{equation} \tag{ 47 }$$

where f ≡ 1 + 2m_b − m_v + m_ρ/3. This implies

$$\begin{equation} B_0 = \left(L \over r_0 \right)^{m_\nu /3} \left(c_{\rm m} \over a^2 \right)^{1/3} \nu _b^{-1/3} \left(v_0^2 \over r_0^2\right)^{1/3} f^{2/3}. \end{equation} \tag{ 48 }$$

Then

$$\begin{equation} {B_h \over B_0} = \left(r_0 \over v_0 t\right)^{2/3} f^{-2/3} \left(L \over r_0\right)^{-m_\nu /3}. \end{equation} \tag{ 49 }$$

Of course, with substantial magnetic field gradients, B_h/B₀ can range widely either below or above 1. Some source properties, such as the initial ratio of energy input in magnetic field to that in particles (KC's σ parameter), require knowledge of B₀. For those properties, estimation of source magnetic field from the homogeneous assumption can lead to significant error. However, it is possible to show that B_h does give a good approximation to the mean magnetic field averaged over the lifetime of a particle moving with the flow, as of course it must. The total magnetic field energy in a flow model is

$$\begin{equation} U_B = {1 \over 8\pi }\int _1^{L/r_0} r_0 dl \ \pi \left({w(l)\over 2}\right)^2 B_0^2 l^{2m_b}\hphantom{aaa} \end{equation} \tag{ 50 }$$

$$\begin{equation} = {1 \over 32} \ \left[{w_0^2 B_0^2 r_0 \over 1 + 2\epsilon + 2m_b} \right] \ \left(L \over r_0\right)^{1 + 2\epsilon + 2m_b} \end{equation} \tag{ 51 }$$

where we have assumed 1 + 2 + 2m_b > 0 and L ≫ r₀. The total volume in the flow is

$$\begin{equation} V = \int _1^{L/r_0} \pi \left({w(l)\over 2}\right)^2 r_0 dl = {\pi r_0 w_0^2 \over 4(1 + 2\epsilon)} \left(L \over r_0\right)^{1 + 2\epsilon }. \end{equation} \tag{ 52 }$$

Then the mean magnetic energy density 〈u_B〉 is

$$\begin{equation} \langle u_B \rangle \equiv {U_B \over V} = {B_0^2 \over 8 \pi } {(1 + 2\epsilon) \over 1 + 2\epsilon + 2m_b} \left(L \over r_0 \right)^{2m_b}, \end{equation} \tag{ 53 }$$

and the homogeneous energy density u_B(hom) ≡ B²_h/8π satisfies

$$\begin{equation} {u_B({\rm hom}) \over \langle u_B \rangle } = f^{-4/3} \left(r_0 \over v_0 t\right)^{4/3} \left(1 + 2\epsilon + 2m_b \over 1 + 2\epsilon \right) \left(L \over r_0\right)^{4(1 - m_v)/3}, \end{equation} \tag{ 54 }$$

where the exponent of L/r₀ has been rewritten using m_ν = 2m_v − 2 − 3m_b.

If the value of t used in the homogeneous relation Equation (46) is the actual transit time t_trans of an electron from r₀ to L, one obtains a similar result. That t_trans is given by

$$\begin{equation} t_{\rm trans} \equiv \int {r_0 dl \over v} = {r_0 \over v_0} \int _1^{L/r_0} l^{-m_v} dl = {r_0 \over v_0} {1 \over 1 - m_v} \left(L\over r_0\right)^{1 - m_v}. \end{equation} \tag{ 55 }$$

Then

$$\begin{equation} \left(r_0 \over v_0 t_{\rm trans}\right)^{4/3} \left(L \over r_0\right)^{4(1 - m_v)/3} = (1 - m_v)^{4/3} \end{equation} \tag{ 56 }$$

independent of physical parameters. (We are assuming m_v ⩽ 0, i.e., we exclude accelerating flows.) This means that all factors in Equation (54) are of order unity, so that there is no large discrepancy between the true mean magnetic field energy density and that inferred under the assumption that the source is homogeneous. However, if the source lifetime is used for t (which may differ substantially from t_trans), or an estimate of t_trans is made from the measured expansion velocity of the outer boundary of the PWN, serious errors may be made in inferring B.

These results can easily be confirmed by numerical integration of the appropriate equations. In particular, Equation (3) can be used to find the detailed particle distribution, accounting for the pileup of particles at energies just below E_max(t) as they migrate down in energy (for s < 2). This pileup can produce a detectable "bump" in the spectrum just below ν_b. The numerical calculations can also show how sharp a break can be achieved in practice. (Similar results were presented in Reynolds 2003.)

I illustrate these effects with several models. The example parameters for B0540-693 mentioned above (α = 0.25, = 1, m_ρ = 1, m_b = −1, and m_v = −2) produce distribution functions N(E, l) at various points in the flow shown in Figure 2, at positions l ≡ r/r₀ = 10, 30, 50, 70, and 90. (The energies are in units of a fiducial energy E_f ≡ aB²₀r₀/v₀, the energy an initially infinitely energetic electron would have after radiating for a time r₀/v₀ in a magnetic field B₀.) The sharp cutoff energy E_c, decreasing down the flow, is apparent, as is the spike just below it of electrons formerly above E_c. The rising density produces adiabatic gains in the density of electrons of too low energy to be subject to radiative losses. Spatial integration over these electron distributions produces the model spectral-energy distribution also shown in Figure 2, reproduced from Williams et al. (2008), which fits the data surprisingly well, apart from the anomalous X-ray flux. (A technical problem, "pileup" in the Chandra detectors due to the bright X-ray pulsar in B0540-693, makes the absolute determination of the X-ray flux of the nebula difficult; see Petre et al. 2007.) For the relatively steep low-frequency spectrum of B0540-693 (α = 0.25), the "bump" from integrating over the spikes of Figure 2 is barely noticeable, but it is much more obvious for a flatter input spectrum. The model assumes a source radius L ∼ 1.3 pc (Williams et al. 2008), but r₀ is a free parameter. If r₀ is the pulsar wind termination shock, we might expect v₀ = c/3 (Kennel & Coroniti 1984a). The break frequency (Equation (15)) constrains the remaining combination r²₀B³₀. The model of Figure 2 takes r₀ = L/100 and B₀ = 2.4 × 10⁻³ G. The radio flux (Equation (21)) then sets the combination w²₀K₀; the model takes w₀ = r₀/10 and K₀ = 2.1 × 10⁻⁸ cm⁻³ erg^0.5. The break frequency calculated from Equation (15) is about 6 × 10¹² Hz, about a factor of 5 lower than the intersection of the extrapolations from low and high frequencies. This is due to the approximation that electrons radiate entirely at their peak frequency ν_m, an approximation not made in the numerical calculations; in general, break frequency predictions will be low by a factor of several, depending somewhat on the value of s.

Chevalier (2005) summarizes spectral indices for several PWNe, including Kes 75 (α = 0 ⇒ s = 0, Δ = 1) and MSH 15–52 (α = 0.2 ⇒ s = 1.4, Δ = 0.85). Such large values of Δ typically require relaxing either mass or flux conservation, or both. For Kes 75, the values = 1, m_ρ = 1, m_v = −2, and m_b = −1 (the same as for the B0540-693 model, except α = 0) predict Δ = 1.0. The consistency condition 1 + 2m_b − m_v > max(0, − m_ρ/3) is met. Figure 5(left) illustrates the integrated spectrum. The "bump" is quite prominent; the flux at the peak around 10^13.5 Hz is 4.4 times that at 1 GHz. The slope above the break is 0.96 between 0.4 and 4 keV, close to the analytic value of 1.0. Equation (27) gives the frequency dependence of the source size as l_max ∝ ν^−1/3, sufficiently slow that it might be hard to detect. For MSH 15–52, Figure 5 (right) shows a calculation for s = 1.4, = 1, m_ρ = 1, m_v = −2.22, and m_b = −1, predicting Δ = 0.85. The consistency condition is again met. The "bump" is still perceptible. The predicted value of Δ is reproduced exactly. The size effect is even slighter: l_max ∝ ν^−0.29. A factor of 11 frequency range would be required to see the source shrink by a factor of 2. For an actual well-resolved source, the "size" would need to be measured with some relatively coarse quantity such as the 50% enclosed power radius or FWHM, as cited for the Crab by Kennel & Coroniti (1984b).

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For an AGN jet example, Stawarz et al. (2007) quote hot spots A and D in Cygnus A as having break frequencies around 3 GHz, with low-frequency spectral indices α₁ = 0.28 and 0.21, respectively, steepening with Δ ≅ 0.9 in both cases. If we assume these hot spots result from post-shock emission in a shocked relativistic flow, and approximating the flow as nonrelativistic (not an egregious approximation), we can obtain Δ = 0.9 with s = 1.6 for hot spot A, and m_ρ = 1 (entraining material; again, without such assumptions, large values of Δ are problematic), m_v = −1 (decelerating flow), and = 0.5 (confined flow, as suggested by mass entrainment). Inserting these values in Equation (41) gives m_b = −0.29 or about −0.3. These values satisfy all consistency conditions, and give a size effect of θ ∝ ν^−0.32. The available data have a range of angular resolutions, but at least qualitatively, hot-spot sizes do decrease between 5 and 230 GHz (Wright & Birkinshaw 2004).

Here, I collect the principal results and consistency requirements. The basic result is the expression for α₂ − α₁ ≡ Δ, the amount of spectral steepening, Equation (22):

$$\begin{equation} \Delta = - {1 + 2\epsilon + m_j \over {m_b + 2m_E}} = {1 + 2\epsilon + (2\alpha + 3)m_\rho /3 + (1 + \alpha)m_b \over 2 + 3m_b -2m_v}. \end{equation} \tag{ 57 }$$

This expression holds if several consistency requirements are met: m_E < min(0, m_ρ/3), where m_E ≡ −1 − 2m_b + m_v, so that the burnoff energy at position l depends on l, and E_c drops with l; and 1 + 2 + (2α + 3)m_ρ/3 + (1 + α)m_b > 0, so that the integrated flux density S_ν depends on the outer limit of integration. Finally, the effective source size should shrink with frequency: m_ν < 0, a condition always met in the presence of mass conservation, and almost always met for reasonable parameters otherwise. If the conditions are met, the source size (some measure of the region from which the bulk of the emission originates) decreases with frequency as

$$\begin{equation} l(\nu) \propto \nu ^{1/m_\nu } \equiv \nu ^{-1/(2 + 3m_b - 2m_v)} \end{equation} \tag{ 58 }$$

if seen more or less from the side; if the flow is nearly along the line of sight, the expression becomes

$$\begin{equation} \theta \propto \nu ^{\epsilon /m_\nu } = \nu ^{-\epsilon /(2 + 3m_b - 2m_v)} \end{equation} \tag{ 59 }$$

(which is the same for spherical or conical flows where = 1).

My basic conclusion is just that synchrotron-loss spectral breaks differing from 0.5 can be produced naturally in inhomogeneous sources. I have treated the inhomogeneities resulting from flows, which seem most natural, using simple power-law parameterizations, but more complex functional dependencies can be treated the same way. Other types of inhomogeneities may be possible as well. These results are most straightforwardly applied to PWNe or knots in extragalactic jets, but may have applications wherever bulk flows of relativistic material are involved. In particular, energy-loss breaks seen in gamma-ray burst afterglows (e.g., Sari et al. 1998; Galama et al. 1998; and much later work) may provide opportunities for the application of these results. For nearby sources, the simplest test of the models is the detection of the size effect; every model predicts both a particular Δ and some rate of decrease of size with frequency (really of volume, since the size decrease may take place along the line of sight). For the same Δ, some variation in the strength of the size effect is possible.

Fortunately, assuming that an inhomogeneous source is actually homogeneous does not drastically alter the inferred mean magnetic field (averaged over the history of a fluid element); but since there are by assumption large gradients of most quantities, local values of the magnetic field strength, such as those at the injection radius, may depart substantially from the mean values. This may affect inferences of the KC magnetization parameter σ. Furthermore, the inference requires knowledge of the actual flow time across the source—knowledge that may be hard to come by in the presence of large velocity gradients.

I gratefully acknowledge the hospitality of the Arcetri Observatory of the University of Florence, where this work was begun. This work was also supported by NASA through Spitzer Guest Observer grants RSA 1264893 and RSA 1276758.