Thermal Electrons in Mildly-relativistic Synchrotron Blast-waves

Numerical models of collisionless shocks robustly predict an electron distribution comprised of both thermal and non-thermal electrons. Here, we explore in detail the effect of thermal electrons on the emergent synchrotron emission from sub-relativistic shocks. We present a complete `thermal + non-thermal' synchrotron model and derive properties of the resulting spectrum and light-curves. Using these results we delineate the relative importance of thermal and non-thermal electrons for sub-relativistic shock-powered synchrotron transients. We find that thermal electrons are naturally expected to contribute significantly to the peak emission if the shock velocity is $\gtrsim 0.2c$, but would be mostly undetectable in non-relativistic shocks. This helps explain the dichotomy between typical radio supernovae and the emerging class of `AT2018cow-like' events. The signpost of thermal electron synchrotron emission is a steep optically-thin spectral index and a $\nu^2$ optically-thick spectrum. These spectral features are also predicted to correlate with a steep post-peak light-curve decline rate, broadly consistent with observed AT2018cow-like events. We expect that thermal electrons may be observable in other contexts where mildly-relativistic shocks are present, and briefly estimate this effect for gamma-ray burst afterglows and binary neutron star mergers. Our model can be used to fit spectra and light-curves of events and accounts for both thermal and non-thermal electron populations with no additional physical degrees of freedom.


INTRODUCTION
Synchrotron emission from relativistic electrons energized in astrophysical shock waves is observed in a wide variety of astrophysical sources. It is typically assumed that strong collisionless shocks accelerate a non-thermal power-law distribution of relativistic electrons via diffusive-shock (first-order Fermi) acceleration (Bell 1978;Blandford & Ostriker 1978;Blandford & Eichler 1987). This idea is directly supported both theoretically-by first-principles particle-in-cell (PIC) simulations (e.g. Spitkovsky 2008; Sironi & Spitkovsky 2009Park et al. 2015)-and observationally-via power-law synchrotron emission that is observed ubiquitously in astrophysical sources. Radio emission from non-relativistic shocks (β sh 1; where β sh c is the shock velocity) in interacting supernovae (SNe) has long been interpreted within a framework of non-thermal electron acceleration and shock amplified magnetic fields (e.g. * NASA Einstein Fellow Chevalier 1982;Weiler et al. 2002). It is also understood that synchrotron self-absorption (SSA) plays a crucial role in such events (Chevalier 1998). These ideas have also been successfully applied to ultra-relativistic explosions (Γ sh 1, where Γ sh = 1/ 1 − β 2 sh is the shock Lorentz factor). Multi-wavelength observations of gamma-ray burst (GRB) afterglows are wellinterpreted as synchrotron emission produced by a nonthermal power-law distribution of relativistic electrons (Sari et al. 1998).
Recently, a new and enigmatic class of transients whose prototype is AT2018cow has been discovered (Prentice et al. 2018;Margutti et al. 2019;Ho et al. 2019Ho et al. , 2020Coppejans et al. 2020;Perley et al. 2021;Ho et al. 2021;Bright et al. 2021). These events, discovered via optical transient surveys, are characterized by fast evolving bright optical emission, peculiar X-ray properties, and unusually luminous radio and millimeter (mm) emission. There is currently no consensus model that easily explains all aspects of these events, but it is typically thought that circumstellar interaction must play an important role. In particular, the radio-mm data is usually interpreted within a SSA framework similar to radio SNe models (Chevalier 1998). The shock velocities inferred from such modelling place these events in an unusual mildly-relativistic regime, with 0.1 β sh 0.5. Ho et al. (2021) presented a detailed analysis of the radio and mm observations of AT2020xnd (see also Bright et al. 2021). The well sampled mm spectral-energy distribution (SED) of this event showed an unusually steep optically-thin spectral index α ≈ −2 (where F ν ∝ ν α ) that is difficult to explain within the standard model of shock-powered synchrotron emission. Steep spectra were also observed for AT2018cow and CSS161010 at early epochs, potentially implicating a common physical mechanism. This led Ho et al. (2021) to propose that the observed steep spectra were due to synchrotron emission from a thermal population of electrons. Using this model, Ho et al. (2021) showed that the SEDs could be well-fit if the peak (SSA) frequency is ∼ O(100) times larger than the frequency at which 'typical' thermal electrons emit. For such parameters, the model could simultaneously explain both the steep optically-thin and shallow optically-thick slopes of observed events (see e.g. Fig. 11 in Ho et al. 2021). The success of this modelling makes the thermal electron scenario compelling, but it also raises many new questions: Why would thermal electrons be observed for AT2018cow-like events but not in radio SNe or GRB afterglows? How is this seeminglypeculiar property related to the unusually-bright and prolonged mm emission in such events? Is it fine-tuned that the inferred SSA frequency is ∼ O(100) times above the characteristic frequency at which most thermal electrons emit?
Separate from the specific observational motivation given above, numerical models of shock acceleration predict that most of the shock energy resides in the thermal population and that the non-thermal tail only contains a small fraction of the total post-shock energy (e.g. Park et al. 2015;Crumley et al. 2019). Why then, should the non-thermal particles dominate the observed emission? And are there physical conditions under which the thermal population is more observationally significant than has typically been assumed?
Here we address these questions by considering in greater detail the effect of thermal electrons on observed properties of synchrotron emission from sub-relativistic blast-waves. The problem of synchrotron emission from a combined thermal + non-thermal electron distribution has previously been studied for ultra-relativistic shocks in GRBs (Eichler & Waxman 2005;Giannios & Spitkovsky 2009;Ressler & Laskar 2017;Warren et al. 2018), and separately in the context of hot accretion flows (e.g. Özel et al. 2000). Here we study the case of sub-relativistic shocks (Γ sh β sh < 1) relevant to AT2018cow-like events, radio SNe, radio flares from binary neutron star (BNS) mergers, and GRBs at late times. We show that the shock velocity is the most important parameter that governs whether thermal electrons have an appreciable impact, and in particular that thermal electrons are naturally expected to dominate peak emission for mildly-relativistic shocks similar to those inferred for AT2018cow-like events.
This paper is organized as follows: we begin in §2 by presenting the basic formalism and our model assumptions. This applies the results of Mahadevan et al. (1996) to the astrophysical setting of sub-relativistic shocks (analogous toÖzel et al. 2000 who applied these results to hot accretion flows). We extend these results by considering the effects of synchrotron self-absorption and the fast-cooling regime for thermal electron synchrotron emission ( §2.1). In §3 we discuss the synchrotron spectrum resulting from this model ( Fig. 1) and derive expressions for relevant break frequencies in the problem (3.1). We subsequently present the landscape of sub-relativistic shock-powered synchrotron transients, illustrating a qualitative dichotomy between non-relativistic and mildly relativistic events ( §4). We provide a brief discussion of the expected light-curves and temporal evolution in §5, and conclude by discussing broader implications of our results ( §6).

THE THERMAL + NON-THERMAL MODEL
We consider a sub-relativistic strong shock propagating with velocity β sh c into a medium of density n. For simplicity, we assume a constant shock compression ratio of 4, formally valid if the total post-shock energy density is dominated by non-relativistic particles. 1 The downstream electron number density is then n e = 4µ e n, where µ e 1.18 for Solar composition. PIC simulations show that, even when a non-thermal power-law tail of electrons is accelerated at the shock front, the majority of post-shock electrons do not participate in diffusive shock acceleration and instead occupy a quasi-thermal distribution. Denoting the dimensionless temperature of these electrons as Θ ≡ k B T e /m e c 2 , the post-shock electron energy density is u e = a (Θ) n e Θm e c 2 , where is an approximation that is good to within ∼ 2% (Gammie & Popham 1998;Özel et al. 2000). Assuming that ions govern the shock jump conditions (so that the effective adiabatic index is 5/3 regardless of whether or not thermal electrons are relativistic), the total postshock thermal energy density is u = (9/8)nµm p (β sh c) 2 , where µ 0.62 for Solar composition. If the electron "thermalization efficiency" is T 1 such that u e = T u ) then the post-shock electron temperature is, and β −1 ≡ β sh /0.1. The thermal electron population occupies a Maxwell-Jüttner distribution where γ is the electron Lorentz factor and is a correction-term that is only relevant in the nonrelativistic regime (f ≈ 1 for Θ 1; f 1 for Θ 1). We additionally consider a non-thermal electron population which we model as a power-law distribution ∝ γ −p (where p > 2) that is terminated at some minimal Lorentz factor γ m . We choose γ m (Θ) = γ th = 1 + a(Θ)Θ equal to the mean Lorentz factor of thermal electrons γ th , such that γ m ≈ 3Θ (≈ 1) for Θ 1 (Θ 1). This choice is somewhat ad-hoc, but motivated by the fact that only supra-thermal electrons are capable of undergoing diffusive shock acceleration (the so-called 'injection problem'; see e.g. Blandford & Eichler 1987). Note that the exact value of γ m does not affect our results and is in any case degenerate with the assumed energy in the non-thermal tail. The non-thermal electron distribution is then fully specified with only one additional parameter. We choose this to be δ-the ratio of energy in the non-thermal electron population to that of thermal electrons (in terms of conventional e notation, δ ≡ e / T ). The non-thermal power-law distribution is therefore where is a correction factor that is only important in the nonrelativistic regime (g ≈ 1 for Θ 1). Finally, we assume that plasma instabilities amplify magnetic fields in the downstream region with "efficiency" B , such that B 2 /8π = B u. This implies where n 5 = n/10 5 cm −3 is the upstream density, and The angle-averaged (assuming isotropic pitch angle distribution) synchrotron emissivity of the thermal electron population is where x ≡ ν/ν Θ and is a scaling frequency that corresponds to the synchrotron frequency of electrons near the thermal peak when Θ 1. Similarly, the absorption coefficient of thermal electrons is given by In eqs. (10,12) above, the function is an approximation of the frequency dependence when Θ 1 (Mahadevan et al. 1996). Although Mahadevan et al. (1996) provide different fitting coefficients as a function of Θ 1, we find it unnecessary to include these corrections here. As we later show, only high frequencies x O(10 2 ) are generally of interest (especially when Θ 1). In this regime x 1 and eq. (13) is exact (Petrosian 1981).
The pitch-angle averaged synchrotron emissivity of non-thermal electrons at frequencies x (γ m /Θ) 2 is 2 j ν,pl = C j e 3 n e Bδ m e c 2 g(Θ)x − p−1 2 (14) 2 At lower frequencies the emissivity and absorption coefficient are affected by the power-law termination at γm, such that asymptotically j ν,pl ∝ x 1/3 and α ν,pl ∝ x −5/3 . We include this in our numeric calculations for completeness, but remark that this has no affect on any of our results, and can therefore be neglected. where The non-thermal absorption coefficient is similarly Note that pitch-angle averaging has often been neglected in many previous applications of non-thermal synchrotron emission (e.g. Chevalier 1998). We include this order-unity correction here because it is expected in the standard scenario where magnetic fields are turbulently amplified, and because thermal electrons are more appreciably affected by such averaging (Mahadevan et al. 1996).

Fast Cooling
Observed emission depends on line-of-sight integrals of the emissivity and absorption coefficients. Synchrotron emitting electrons whose Lorentz factor exceeds will radiate most of their energy over a timescale that is short compared to the dynamical time, t = 100 d t 100 . Such fast-cooling electrons would only reside within a fractional depth ∼ (γ/γ cool ) −1 1 behind the shock front, so that the effective line-of-sight averaged distribution function is ∂n/∂γ ∼ (∂n/∂γ) min(1, γ cool /γ). The emissivity and absorption coefficient are proportional to this distribution function and therefore similarly affected, introducing a frequency-dependent correction to eqs. (10,12,14,12) above.
For power-law electrons, emission/absorption at frequency x is contributed predominantly by electrons of Lorentz factor γ/Θ ∼ x 1/2 . This implies the standard fast-cooling correction where x cool,pl ≡ (γ cool /Θ) 2 . For thermal electrons however, this no longer applies. At frequencies x 1 (of main interest here), emission/absorption samples the high-frequency tail of comparatively lower-energy electrons (which vastly outnumber electrons at higher Lorentz factors). In this regime, the synchrotron frequency x is instead related to electrons whose characteristic Lorentz factor is γ/Θ ∼ (2x) 1/3 , 3 and the coolingcorrected emissivity and absorption coefficients are The total thermal + non-thermal emissivity is simply j ν = j ν,th + j ν,pl , and the combined absorption coefficient is similarly additive, so that α ν = α ν,th + α ν,pl . The emergent specific luminosity is then where R is the characteristic size of the emitting region, and the effective absorption and emission coefficients are given by eqs. (10,12,14,16,19,20) above.

SPECTRUM
The synchrotron spectrum that results from the 'thermal + non-thermal' model (eq. 21) typically peaks at the SSA frequency ν a . At frequencies ν < ν a emission is selfabsorbed and the spectrum rises as a function frequency, whereas above it emission is optically-thin and the spectral luminosity decreases with frequency. There are two distinct regimes that are of particular interest: (i) emission near the SSA frequency is dominated by thermal electrons; or (ii) emission near this frequency is instead dominated by the power-law electron distribution. Figure 1 shows representative SEDs in these two cases. Solid (dashed) light-grey curves show the opticallythin thermal (power-law) electron emission, while solid black curves show the combined spectrum including selfabsorption (eq. 21). Vertical dotted curves show relevant break frequencies. These are listed in Table 1 and discussed in greater detail in §3.1. The left panel shows a "thermal spectrum" where peak emission is governed by thermal electrons. At low frequencies ν < ν a the SED follows the Rayleigh-Jeans limit ∝ ν 2 . This is shallower than the canonical ν 5/2 SED of optically-thick powerlaw synchrotron emission (Rybicki & Lightman 1979). At frequencies slightly above peak the SED follows the optically-thin thermal emissivity and the spectral slope 10 0 10 1 10 2 10 3 10 4 x / 10 8 10 9 10 10 10 11 10 12 10 13 (Hz) Left: A "thermal spectrum", where peak emission is dominated by thermal electrons (νa < νj). Solid (dashed) light-grey curves show the optically-thin contribution of thermal (power-law) electrons to the SED, while the black curve shows the combined emergent spectral luminosity, including synchrotron self-absorption (eq. 21). Vertical dotted curves show characteristic break frequencies (see Table 1): the thermal synchrotron frequency νΘ (eq. 11); the frequency νm corresponding to the minimal Lorentz factor of power-law electrons; the synchrotron self-absorption frequency νa (eqs. 28,29,30); the frequency νj (να) at which emission (absorption) transitions from being dominated by thermal electrons to power-law electrons (eqs. 25,26); and the synchrotron cooling frequency ν cool (eqs. 24,16). The spectral slope (stated above each segment) can be especially steep in the optically-thin thermal regime, νa < ν < νj (eq. 22). Right: Same as left panel, but for a "non-thermal spectrum" where peak emission is dominated by the power-law electron distribution (νa > νj). The SED follows the standard non-thermal spectrum, except at very low frequencies < να (eq. 26) where the SSA spectrum softens. The thermal electron population would be mostly unobservable in this regime, even though it is energetically (and by number) dominant. Both panels show cases where ν cool falls above other relevant frequencies, but alternative orderings may be possible and are fully accounted for in §3.1). The left panel is calculated using β sh = 0.45, n = 10 3 cm −3 , and t = 25 d. The right panel is for β sh = 0.1, n = 10 4 cm −3 , and t = 200 d. In both cases we assume δ = 0.01, p = 3, B = 0.1, and T = 1.
can be extremely steep. The spectrum does not follow a power-law-form in this regime, however we can characterize the slope steepness via the frequency-dependent spectral index α th ≡ d ln j ν,th /d ln ν, The expression above applies in the typical setting where frequencies of interest are ν Θ (eq. 11), and the two cases (whose spectral slope differs by 1/3) depend on whether the observing frequency is below or above the fast-cooling break frequency ν cool .
The spectral index implied by eq. (22) becomes increasingly steep at higher frequencies, and is a unique feature of the thermal electron model. However, above some frequency ν j , emission by power-law electrons will come to dominate the thermal-electron emission. This transition frequency (red dotted curve in Fig. 1) depends primarily on the relative number of power-law and thermal electrons, which is ∝ δ in our model. Lower values of δ imply a smaller fraction of power-law electrons and a higher transition frequency (in §3.1 we provide approximate expressions for this dependence). Fig. 1 illustrates the significance of ν j . At frequencies ν > ν j emission is governed by non-thermal electrons and the spectrum follows standard results for powerlaw synchrotron emission-the SED is ∝ ν −(p−1)/2 (∝ ν −p/2 ) in the slow-(fast-) cooling optically-thin regimes. If a given event is only observed at frequencies ν > ν j then the thermal-electron contribution would go undetected and this would be indistinguishable from a purely non-thermal electron model. This is further illustrated by the right-hand panel of Fig. 1, which shows a "non-thermal spectrum" where the peak (SSA) frequency is ν a > ν j . In this case peak emission is dominated by the power-law electron distribution, the usual ν 5/2 SSA optically-thick spectrum applies below peak, and the entire optically-thin SED follows the standard power-law spectrum. Thermal electrons-though present and energetically dominant in this model-would not affect the observed emission except at very low frequencies ν < ν α ∼ ν j where the optically-thick spectrum is expected to soften (at ν < ν α the optical depth becomes dominated by thermal electrons). These frequencies are usually observationally inaccessible so that our 'thermal + non-thermal' model would be indistinguishable from purely non-thermal synchrotron models that are typically used to model observations. We also note that the low frequency spectrum is sensitive to geometric effects (related to the spatial distribution of emitting electrons) and can be susceptible to scintillation, further complicating potential identification of a break frequency at ν α .
In the following subsection we discuss the various break frequencies shown in Fig. 1 in greater detail. Readers interested primarily in our main results may wish to skip forward to §4, while those interested in understanding the origin of different regions and scaling with physical parameters are welcome to continue to §3.1.

Estimates of Break Frequencies
As illustrated by Fig. 1, the resulting SED of the thermal + non-thermal model depends on several characteristic frequencies. The first is the 'thermal' frequency ν Θ given by eq. (11). Many other relevant frequencies scale in some well-determined way with ν Θ . For example, the frequency ν m that corresponds to the minimum Lorentz factor of power-law electrons is simply ν m = (γ m /Θ) 2 ν Θ (and is ν m ≈ 9ν Θ for Θ 1).
The synchrotron cooling frequency ν cool can also affect the observed SED. As discussed in §2.1, this frequency is related to γ cool (eq. 18) as for power-law emitting electrons, and for the thermal electron population. The observed cooling break therefore depends on whether emission is dominated by power-law or thermal electrons. This is governed by the frequency ν j at which the thermal and power-law emissivities are equal, j ν,th = j ν,pl . The transcendental equation for ν j does not permit a closed form analytic solution, but is easily solvable numerically. In general, the solution depends on δ, p, and Θ, however the Θ dependence is suppressed for Θ 1. An accurate fitting function to the solution is given by where x j ≡ ν j /ν Θ following our standard notation. This is accurate to within 3% for 10 −6 ≤ δ ≤ 1/3, p = 3, and any Θ 1, but is also reasonably accurate for Θ 1 or other values of 2.2 ≤ p ≤ 3.4 (17% accuracy). For a fiducial δ = 0.01, we find that x j ≈ 540. An alternative approximation that is accurate to within 19% between 10 −5 ≤ δ ≤ 0.1 is given by x j ≈ 150δ −0.25 .
A related frequency ν α is defined by equating the absorption coefficients of the two populations such that α ν,th = α ν,pl at ν α . This frequency is typically a factor 2 greater than ν j , and we find that the approximation is accurate to within several percent throughout the parameter range considered above.
Finally, the SED peak is set by the SSA frequency ν a . If the absorption coefficient is dominated by thermal electrons (x < x α ) then the SSA frequency is determined by the condition α ν,th R = 1 (eqs. 12,20). In the slow-cooling regime, this is governed by an optical-depth parameter that describes the (thermal-contribution to the) optical depth at frequency ≈ ν Θ . Note the extreme sensitivity of τ Θ to the shock velocity and the fact that τ Θ 1 for typical parameters. This implies that one would not expect to see "bare"(unabsorbed) Maxwellian SEDs that peak at ∼ ν Θ . We elaborate on this in §4.
Using the optical-depth parameter τ Θ (eq. 27), we find that the thermal SSA frequency is well approximated by the fitting function (28) that is accurate to within 3% over many orders of magnitude in optical depth, 30 ≤ τ Θ ≤ 10 12 . The SSA frequency varies between x a,th ∼ 10 − 10 3 over this range of τ Θ . An alternative simpler approximation that is accurate to within 18% for 5 × 10 3 ≤ τ Θ ≤ 2 × 10 11 is given by x a,th ≈ 7.1τ 0.2 Θ . In the fast cooling regime x a,th > x cool,th the fitting functions above are comparably accurate when transformed as If the absorption coefficient is instead dominated by power-law electrons then the SSA frequency is determined by the condition α ν,pl R = 1. This results in the  (11) characteristic synchrotron frequency of thermal electrons νm eq. (6) characteristic synchrotron frequency of power-law electrons, νm = (γm/Θ) 2 νΘ ν cool eqs. (23,24) synchrotron cooling frequency νa eq. (33) synchrotron self-absorption (SSA) frequency, αν R = 1 at νa νj eq. (25) frequency above which power-law > thermal emissivity, j ν,th = j ν,pl at νj να eq.( 26) frequency above which power-law > thermal absorption, α ν,th = α ν,pl at να a Note that normalized frequency x ≡ ν/νΘ is used interchangeably with ν throughout the text.
analytic solution Noting that g(Θ) ≈ (p − 1)(3Θ) −(p−1) when Θ 1 and g(Θ) ≈ 1 for Θ 1, we can express the power-lawdominated SSA frequency as ν a,pl (< ν cool,pl ) ≈ in the fast-cooling case, and we have chosen a fiducial p = 3 for the estimates above. In total, the SSA frequency is related to eqs. (28,29,30), The different frequencies discussed in this section are also summarized in Table 1.

PHASE-SPACE OF TRANSIENTS
In the previous section we showed that the ordering of characteristic frequencies (primarily the self-absorption frequency ν a and the frequency ν j at which thermal and non-thermal electrons have comparable emissivity) determines whether thermal electrons contribute appreciably to observed emission. This is illustrated by the different spectra in the left and right hand panels of Fig. 1. A natural question subsequently arises-what type of shock-powered transients might be expected to show signatures of a thermal electron distribution? Figure 2 addresses this question by showing the parameter-space of sub-relativistic shock-powered synchrotron transients. This phase space is determined by the upstream ambient density n, the shock velocity β sh c, and the size of the emitting region R. We relate the size to the shock velocity as R = β sh ct such that t is an effective dynamical time (fixed to 100 d in Fig. 2). This corresponds to the true time-since-explosion only if the shock velocity is temporally constant. If the shock decelerates then this time parameter would be larger than the actual observing epoch (non-spherical geometry can also affect this). The electron temperature and magnetic field are directly related to β sh , n through eqs. (2,3,9), and we adopt fiducial values T = 1, B = 0.1, δ = 0.01, and p = 3.
Blue contours in Fig. 2 show the frequency at which the SED peaks. Throughout nearly the entire illustrated parameter space the thermal optical-depth is τ Θ 1 so the peak frequency is ≈ ν a (eq. 33) as set by SSA. Black contours show the peak specific luminosity at this frequency (eq. 21). Grey shaded regions show the parameter space in which optically-thin emission is set entirely by the power-law electron distribution. This is determined by the condition ν j < ν a (eqs. 25,30) that implies a spectrum similar to the right panel in Fig. 1. Within the light-grey region ν α ν a /5 and the presence of thermal electrons may still be discernible through their effect on the self-absorbed spectrum: between ν α < ν < ν a the SSA spectrum follows the canonical ν 5/2 scaling of a power-law electron distribution, but at frequencies ν < ν α this softens to a thermal SSA spectrum ∝ ν 2 (see right panel in Fig. 1). In the dark shaded grey region ν α ν a so that this softening would occur well below the SED peak and would be more difficult to detect.
Finally, we also plot in Fig. 2 contours of the spectral index just above the SED peak (at frequency ν ≈ 2ν a ). Within the light (dark) grey shaded region emission (and absorption) are dominated by the power-law electron distribution and thermal electrons would not affect the observations (see Fig. 1, right panel). Outside these regions emission near the peak (SSA) frequency is instead dominated by thermal electrons (see left panel of Fig. 1). This may be observationally distinguishable from a purely power-law electron model by the unusually steep optically-thin spectral slope (yellow shaded region). The shock velocity is the most important parameter that governs whether a steep "thermal" or a standard "non-thermal" spectrum would be observable (eq. 34). The dichotomy between radio SNe and AT2018cow-like events can therefore be naturally understood as an artefact of the non-relativistic (β sh 1) vs mildly-relativistic (β sh 0.2) velocities inferred for these events (see §2).
For power-law electrons with our canonical p = 3 this spectral index would be −(p − 1)/2 = −1 in the slowcooling regime and −p/2 = −1.5 in the fast-cooling case. The transition between fast and slow cooling regimes is apparent through the kink in the spectral-index contours. Alternatively, if emission near ν a is dominated by the thermal electron population then the spectral index can be significantly steeper (eq. 22; left panel, Fig. 1). We highlight this with the yellow shaded area in Fig. 2, which shows regions where the spectral index is steeper than would be expected for purely power-law electron emission (< −1.5). Figure 2 shows a clear dichotomy between shockpowered synchrotron transients with mildly relativistic velocities 0.2 β sh 1 and those with non-relativistic velocities β sh 1. In the former case, peak emission is dominated by thermal electrons and a steep optically-thin spectrum can be attained, whereas the latter are governed entirely by the non-thermal powerlaw electron distribution. This dichotomy almost exclusively depends on shock velocity with only very weak dependence on density. This is because the thermal optical-depth parameter τ Θ scales strongly with velocity (eq. 27). Specifically, in order for thermal electrons to contribute to the optically-thin emission, the frequency at which emission transitions from thermal to nonthermal electrons must fall above the self-absorption frequency, i.e., ν a,th < ν j must be satisfied. For our fiducial δ = 0.01 this implies (eqs. 25,28,29) τ Θ < 1.7 × 10 9 (or Θτ Θ /γ cool < 2 × 10 10 in the fast-cooling regime), and therefore that is required for thermal electrons to dominate the SED peak. The top case corresponds to the slow-cooling regime while the bottom case applies in the fast-cooling regime (ν a,th < ν cool,th ). Smaller (larger) values of δ would imply a lower (higher) threshold velocity. Specifically, using the rough scalings x j ∝ δ −0.25 and x a ∝ τ 0.2 Θ (see text below eqs. 25,28) we find that the critical shock velocity (eq. 34) scales as δ 0.125 in the slow cooling regime, and δ 0.089 in the fast-cooling case.
Condition (34) also reflects the thermal electron temperature Θ, shown with the right vertical axis in Fig 2. When Θ 1 thermal electrons are relativistic and produce copious synchrotron emission, whereas the majority of thermal electrons are non-relativistic if Θ 1 and only a small fraction are capable of contributing to emission at frequencies ν ν Θ of relevance. The threshold velocity (eq. 34) therefore depends on the electron thermalization efficiency T . Lower efficiencies (smaller T ) would increase the threshold shock velocity and push the region where thermal electrons dominate peak emission to higher shock velocities.
The strong dependence on shock velocity evident in Fig. 2 helps explain why non-relativistic shocks in radio SNe are well modelled by a power-law electron distribution and do not show any clear evidence for thermal electrons, whereas the emerging class of AT2018cowlike events that have mildly-relativistic inferred velocities β sh 0.2 exhibit steep spectra consistent with a contribution from thermal electrons . Red points in Fig. 2 show shock properties inferred by Ho et al. (2021) for AT2018cow at an epoch of 10 d Ho et al. 2019), AT2020xnd at 40 d ), CSS161010 at 99 d (Coppejans et al. 2020), and AT2018lug at 81 d (the 'Koala'; Ho et al. 2020). In comparison, typical radio SNe have velocities of order β sh ∼ 0.03 and densities n ∼ 10 5 − 10 6 cm −3 at timescales of ∼ 100 d post-explosion (Weiler et al. 2002).
In addition to providing a natural explanation for why steep thermal-electron spectra would be seen in AT2018cow-like events but not in standard radio SNe, Fig. 2 may also help explain the unusually bright and prolonged millimeter emission observed in AT2018cow and AT2020xnd. At a fixed ambient density-shocks with higher velocities produce more luminous emission that peaks at higher frequencies (especially above β sh 0.1 where the blue contours kink to the left). In par-ticular, there is a large swath of parameter-space where emission peaks in the millimeter band. This is especially pronounced considering potential selection biases towards detecting the most luminous events.

TEMPORAL EVOLUTION
In Fig. 2 we presented the phase-space of synchrotronpowered transients as a function of shock velocity and ambient density, at a fixed epoch t. Here we briefly discuss the temporal evolution of transients within this phase-space, as time (and potentially upstream density, shock velocity) progresses. There is a rich phenomenology of possible light-curves depending on the time evolution of various quantities of interest. Here we focus on the specific case where the upstream follows a wind density profile, n ∝ r −2 . Figure 3 shows example light-curves resulting from our model. These are calculated assuming that a blastwave of initial velocity β sh = 0.4 and total energy 10 50 erg is driven into an ambient wind whose density is n = 10 5 cm −3 r/10 16 cm −2 . 4 The shock dynamics are integrated assuming a spherical thin-shell model that is accurate in both the relativistic and non-relativistic regimes (Huang et al. 1999;Pe'er 2012;see Schroeder et al. 2020 for details on this implementation). This results in a gradually decelerating shock and a timedependent shock velocity cβ sh (t), radius R(t), and upstream density n[R(t)]. We calculate the light-curves from eq. (21) using these time-dependent quantities and adopting fiducial T = 1, B = 0.1, δ = 0.01, and p = 3. The middle panel of Fig. 3 shows resulting lightcurves at different frequencies (labeled). At a given frequency, the light-curve peaks when the SSA frequency passes through the band. Shortly after peak, the highfrequency light-curves exhibit a sharp drop. This is directly related to the steep thermal spectrum at frequencies ν > ν a,th (eq. 22) and is a unique property of the thermal electron model. The top panel of Fig. 3 shows snapshots of the spectrum at different epochs and illustrates the steep optically-thin SED that is obtained at early times. The correlation between the unusually steep spectrum and steep light-curve decline rate is further illustrated by the bottom panel in this figure, which shows the spectral index (d ln L ν /d ln ν; solid curves) and temporal index (d ln L ν /d ln t; dot-dashed curves) at different frequencies (different colors). Both the temporal and spectral indices obtain steep (negative) values shortly after peak at 90 GHz.  Example light-curves for a decelerating shock-wave in a wind density profile. At a fixed frequency (labeled), the light-curve peaks when the SSA frequency passes through the band. The high-frequency peak is dominated by thermal electrons. This implies a steep post-peak decline, qualitatively consistent with observed AT2018cow-like events (eq. 38). At later times the light-curve samples power-law electrons and the declinerate softens. Top: SED snapshots of the same model at different epochs. At early times the SED peaks at highfrequencies showing tell-tale signs of thermal electronsa steep optically-thin spectrum (eq. 22) and a ∝ ν 2 selfabsorbed slope. Shock deceleration causes the observed contribution of thermal electrons to drop with time. By 180 d the SED lacks clear signatures of thermal electrons and is instead governed by power-law electrons. Bottom: The spectral (solid) and temporal (dot-dashed) indices at different frequencies (following color-scheme of middle panel) as a function of time. At high-frequencies, both spectral and temporal indices attain steep (negative) values shortly after light-curve peak. This is a unique feature of the thermal-electron model. Figure 3 illustrates another important feature: if there is enough mass in the surrounding CSM, shock deceleration will eventually cause initially mildly-relativistic shocks that satisfy eq. (34) to violate this condition at late times. This implies that the relative contribution of thermal electrons to the observed emission will decay as a function of time, and that at late enough epochs the light-curves and spectra will revert to the standard power-law electron distribution picture. This can be seen from the SED snapshots in the top panel of Fig. 3.
At early times, the spectrum exhibits the tell-tale ν 2 self-absorbed rise and steep optically-thin decline that are characteristic of thermal electrons (see left panel, Fig. 1). At later epochs the optically-thin (opticallythick) slope flattens (steepens) and is eventually governed entirely by non-thermal electrons. This is also imprinted in the late-time low-frequency light-curves, that no longer show the steep post-peak decline apparent at higher-frequencies. We note that this agrees with modeling of AT2018cow and AT2020xnd, which suggested that the late-time data was well-fit within the standard power-law synchrotron framework Ho et al. 2021). The spectrum in this power-law dominated regime follows the right panel in Fig. 1, and the temporal evolution can be derived using eqs. (21,30). This reverts to the results of Chevalier (1998) in the slow-cooling regime, and to the results presented in Appendix C of Ho et al. (2021) for the fast-cooling regime.
We can understand the results presented in Fig. 3 more quantitatively by estimating the light-curve scalings in the case where thermal electrons dominate the emission (as particularly relevant at high-frequencies and early epochs). We pursue this by denoting the temporal-scaling of the shock radius as R ∝ t m . In general (and in our numerical model) the radius does not follow a power-law evolution and the exponent m should instead be interpreted as the instantaneous expansion rate d ln R/d ln t. In typical cases we expect m = 1 at early epochs before significant shock deceleration, and lower values of m at later times (the Sedov-Taylor solution for a wind medium sets a lower limit of m ≥ 2/3). This scaling implies that β sh ∝ t m−1 and n ∝ t −2m . If the electron temperature and magnetic field are determined by eqs. (2,3,9) then we additionally have Θ ∝ t 2(m−1) , and B ∝ t −1 .
In the slow-cooling case this is bound between L a,th ∼ const and L a,th ∝ t −2.53 . The above equations imply that-for a wind density medium-the peak (SSA) frequency of thermal electrons drops moderately as a function of time, while the peak flux is sensitive to the shock deceleration parameter m. If emission is dominated by thermal electrons, then ν < ν a,th prior to the light-curve peak and the luminosity is L ν ≈ (ν/ν a,th ) 2 L a,th . From eqs. (35,36) this in both the slow-and fast-cooling regimes, and that the light-curve rises to peak as ∼ t 2/3 −t 2 . This is consistent with our numerical results shown in Fig. 3. Following the light-curve peak, unusually steep decays were observed at high frequencies for AT2018cow and AT2020xnd, qualitatively consistent with the thermal electron model. For example, in Fig. 3 we show the ∼ t −4 scaling inferred for AT2020xnd to guide the eye . A crude analytic estimate of the temporal slope in this regime can be derived using eqs. (22,35,36). Shortly after peak the luminosity is roughly L ν ∼ (ν/ν a,th ) α th (ν a,th ) L a,th and therefore L ν (t peak) ∝    t −7.6(1−m)−(1.8m−2.8)α th , ν a,th < ν cool t 6.8m−6.4−(1.4m−2.2)α th , ν a,th > ν cool .
(38) For example, if the spectral index is α th = −2, we obtain that L ν ∝ t 11.2m−13.2 in the slow-cooling regime. Even for very mild deceleration this implies a very steeply declining light-curve (e.g. L ν ∝ t −3 for m ≈ 0.9). Steeper spectral indices and/or stronger deceleration yield lightcurves that decay more abruptly.

DISCUSSION
In this work we studied the implications of a thermal electron population on sub-relativistic shock-powered synchrotron transients. The existence of a thermal electron population is a natural expectation in shock scenarios, yet has garnered little attention in the context of synchrotron transients. We find that neglecting thermal electrons is reasonable only for non-relativistic shocks where β sh 1. Much of the canonical synchrotron transient literature was derived for radio SNe where this is applicable, however the situation is markedly different for mildly-relativistic shocks (Fig. 2). If domi-nant, thermal electrons can be discerned by their telltale steep optically-thin spectrum (eq. 22) and a comparatively shallow ν 2 self-absorbed spectrum (Fig. 1). Another general prediction of the thermal electron model is a steep decay of the light-curve shortly after peak, and a correlation between the spectral and temporal indices ( Fig. 3; eq. 38). In typical settings, these effects should be most prominent at early times and at highfrequencies.
The physical processes that determine the post-shock electron temperature are a matter of ongoing investigation, but it is generally recognized that plasma instabilities must mediate electron-ion energy exchange. 5 PIC simulations of both relativistic and sub-relativistic electron-ion shocks generically show a quasi-thermal downstream electron population that shares an orderunity fraction of the downstream energy ( T ∼ 1) and that exceeds the energy in the diffusive-shockaccelerated power-law tail (δ 1; Sironi & Spitkovsky 2011;Park et al. 2015;Crumley et al. 2019;Tran & Sironi 2020). In our present work we have assumed that this quasi-thermal population can be modelled by a relativistic Maxwellian (eq. 4), i.e. that it is "perfectly" thermal. We expect that modest deviations from a pure Maxwell-Jüttner distribution would not affect our main conclusions, but may quantitatively change various estimates. In particular, our results are sensitive to the high-energy tail of the thermal distribution, which contributes most to emission at frequencies ν Θ of interest. For example, if we generalize eq. (4) to (dn/dγ) th ∝ e −(γ/Θ) n where n = 1 for a standard Maxwellian, then the high frequency thermal synchrotron spectrum would scale as L ν ∝ e −Anx n/(n+2) with A n = [1 + (n/2) 2/n ](2/n) n/(n+2) (compare with eq. 13). This may affect quantitative values of ν j , ν α , ν a,th , and ν cool,th 6 (eqs. 24-29) but should not change our overall findings.
Our study was motivated by steep spectra and lightcurves observed in several AT2018cow-like events, and by the work of Ho et al. (2021) that first suggested a thermal-electron interpretation of this data. Here we addressed several questions that arise from such an interpretation. We showed that thermal electrons are naturally expected to govern peak emission for mildlyrelativistic shocks with β sh 0.2 ( Fig. 2 and eq. 34). 5 The timescale for electron-ion equilibration through Coulomb collisions is typically too slow, t ei ∼ 200 yr n −1 5 Θ b (ln Λ/30) −1 where ln Λ is the Coulomb logarithm, b = 3/2 for Θ 1 (Spitzer 1956) and b = 1 for 1 Θ mp/me (Stepney 1983). 6 The cooling frequency of the thermal population would in this case be x cool,th ∼ (n/2)(γ cool /Θ) n+2 , affecting eqs. (20,24).
Conversely, thermal electrons would be subdominant for non-relativistic shocks. This explains the dichotomy between AT2018cow-like events and typical radio SNe (Fig. 2). Furthermore, we showed that the synchrotron optical depth at frequency ∼ ν Θ at which most thermal electrons emit is 1 for sub-relativistic shocks (eq. 27). This implies that a "bare" (unabsorbed) Maxwellian should not be observed for such shocks, and helps explain why the SED peak in AT2018cow-like events is inferred to be ∼ O(100) times above ν Θ ). The thermal optical depth α ν,th R ∝ τ Θ e −1.8899x 1/3 is exponentially sensitive to frequency, which acts to regulate the self-absorption frequency to x a,th ∼ O(100) over many orders of magnitude in τ Θ (eq. 28).
In addition to shock velocity, a second important parameter that determines the contribution of thermal electrons to observed emission is the ambient density. At a fixed shock velocity, the density sets the downstream magnetic field (eq. 9) and therefore governs the frequency ν j below which thermal electrons dominate observed emission (Fig. 1). Using the rough approximation x j ≈ 150δ −0.25 (see text below eq. 25) and eq. (11), this critical frequency is At frequencies ν > ν j emission is dominated by nonthermal electrons and the presence of thermal electrons would be undetectable.
Eq. (39) shows that ν j falls in the GHz-mm band for high-density high-velocity shocks relevant to AT2018cow-like events. Although this class of events has been our primary focus in this work, our results would apply to sub-relativistic shocks in any other astrophysical setting as well. For example, BNS mergers eject ∼ 10 −2 M of material at velocities 0.1c, and it has been suggested that the forward shock between this ejecta and the ambient interstellar medium (ISM) would produce detectable synchrotron radio emission (e.g. Nakar & Piran 2011;Margalit & Piran 2015Hajela et al. 2021). This has typically been studied using standard power-law electron models, howeveras we have shown-thermal electrons may contribute appreciably for mildly-relativistic shocks. This contribution is limited to low frequencies ν < ν j ≈ 300 MHz 1/2 B,−1 n 1/2 0 (β sh /0.2) 5 (eq. 39) that may hinder detectability prospects, especially considering that the ambient ISM density is likely to be significantly lower than the optimistic value assumed above (e.g. Fong et al. 2015;Hajela et al. 2019). Nevertheless, our estimate motivates late-time follow-up of BNS mergers at particularly low frequencies in order to test the thermalelectron hypothesis. Furthermore, if the merger man-ages to produce a "long-lived" magnetar remnant then energy injection may accelerate the BNS-merger ejecta to trans-relativistic velocities (Metzger & Bower 2014;Horesh et al. 2016;Fong et al. 2016;Margalit & Metzger 2019;Schroeder et al. 2020) implying much higher frequencies up to which thermal electrons may contribute noticeably (eq. 39).
GRB afterglows could show signs of thermal electrons, as first discussed by Giannios & Spitkovsky (2009). GRB outflows are initially ultra-relativistic and collimated, however shock deceleration leads the outflow to a quasi-spherical sub-relativistic state at sufficiently late times. At such epochs, our current formalism applies (see Ressler & Laskar 2017 for treatment of the ultra-relativistic regime) and we estimate ν j ≈ 2.7 GHz 1/2 B,−1 E 50 n −1/2 0 (t/yr) −3 for δ = 0.01, and for a GRB of total energy E = E 50 10 50 erg that is deep within the Sedov-Taylor regime. Observations at low frequencies ν < ν j would be required to potentially distinguish the thermal-electron model from a purely power-law electron distribution. Since the critical frequency ν j drops rapidly with time, the most opportune window would be to observe shortly after the shock enters the mildly relativistic regime (the earliest epoch at which our sub-relativistic results apply). Thermal electrons may also be relevant to other astrophysical settings in which mildly-relativistic shocks are present. This may apply to jetted tidal-disruption events such as the prototypical Swift J1644 (Bloom et al. 2011;Burrows et al. 2011;Zauderer et al. 2011;Eftekhari et al. 2018), to lowluminosity GRBs (e.g. Kulkarni et al. 1998;Tan et al. 2001;Barniol Duran et al. 2015), and perhaps to more exotic scenarios such as outflows from accretion-induced collapse (e.g. Dessart et al. 2006;Darbha et al. 2010).
The most important quantity that governs thermal electron emission is the post-shock electron temperature Θ. In our current model, this is set uniquely by the shock velocity (eqs. 2,3), however additional processes may impact this result. For example, inverse-Compton scattering off external (or self-produced) photons could potentially cool post-shock electrons (e.g. Katz et al. 2011;Margalit et al. 2021), though this would have to act over extremely short timescales to compete with kinetic instabilities and regulate Θ. 7 We have worked here under the simplest hypothesis that the thermal and non-thermal electrons can be adequately described by fixed values of the parameters T , δ, and p. The microphysics that sets these param-eters may in reality be more complex. For example, the strength of the power-law component δ (which is ∝ e in standard notation 8 ) may itself be affected by the shock velocity. PIC simulations have found that e ∼ 0.1 for ultra-relativistic electron-ion shocks (Sironi & Spitkovsky 2011) while e ∼ 10 −4 for non-relativistic shocks (Park et al. 2015) and a trend of increasing e with shock velocity has been suggested (e.g. Crumley et al. 2019). If this trend is correct, then the parameter space in Fig. 2 where thermal electrons contribute appreciably would expand and encompass even lowervelocity shocks. This would improve prospects for detecting emission from such thermal electrons, but may already be at odds with observations of events straddling the two regions.
The shock Mach-number and magnetic field orientation can further affect diffusive-shock acceleration and impact δ and/or p. For example, shocks where the magnetic field is perpendicular to the shock velocity are thought to be less efficient at accelerating non-thermal particles (e.g. Sironi & Spitkovsky 2009) and may produce more prominent thermal downstream distributions, i.e. δ 1 (although see Xu et al. 2020;Kumar & Reville 2021). This would again expand the range of parameters where thermal electrons must be considered.
The above uncertainties in microphysics (that are in any case not considered in typical modelling of synchrotron transients) should not be considered detrimental to the thermal + non-thermal model. In fact, we view these as an important opportunity-our model provides a direct means of measuring the acceleration efficiency δ and therefore constraining microphysical processes using observations. Such direct measurement is possible if one observes the transition frequency ν j (eq. 39) at which emission changes from thermal to power-law dominated (or similarly by detecting ν α in the SSA regime). Ho et al. (2021) have already used this method to infer that δ < 0.2 for AT2020xnd.
Finally, we note that the thermal + non-thermal synchrotron model presented here includes no additional physical parameters compared to standard non-thermal synchrotron models that are typically used to model observations. At a given epoch, the frequency-dependent spectral luminosity is fully specified by three physical parameters, R, β sh , and n, and three microphysical parameters, B , δ, and p (insofar as T ∼ 1, which is well-motivated in the simplest version of this model). The same parameters are also required in standard nonthermal synchrotron modelling (with e replacing δ).
The fact that this model is capable of fitting more complex spectra (see Ho et al. 2021) with no additional degrees of freedom is another strength of this scenario. In the Appendix we provide analytic expressions (that are applicable in a subset of the parameter space) and a link to the code used in our analysis (applicable for any subrelativistic shocks). These may be convenient for future studies, and in particular for fitting observed data to the model presented in this work.