AT2018cow: A Luminous Millimeter Transient

Although the sky is regularly monitored across many bands of the electromagnetic spectrum (as well as in gravitational waves and energetic particles) the dynamic sky at millimeter to submillimeter wavelengths (0.1–10 mm) remains poorly explored. There has only been one blind transient survey specific to the millimeter band¹¹ (Whitehorn et al. 2016); millimeter facilities are usually only triggered after an initial discovery at another wavelength. Even when targeting known transients, the success rate for detection is low, and only a few extragalactic transients¹² have well-sampled, multifrequency light curves to date. This sample includes supernovae (SNe; Weiler et al. 2007; Horesh et al. 2013), tidal-disruption events (TDEs; Zauderer et al. 2011; Yuan et al. 2016), and gamma-ray bursts (GRBs; de Ugarte Postigo et al. 2012; Laskar et al. 2013; Perley et al. 2014; Urata et al. 2014).

The paucity of millimeter transient studies can be attributed in part to costly receiver and electronics systems and the need for excellent weather conditions, but it also reflects challenges intrinsic to millimeter-wave transients themselves: most known classes are either too dim (SNe, most TDEs) to detect unless they are very nearby, or too short-lived (GRBs) to detect without very rapid reaction times (<1 day, and even in these circumstances the emission may only be apparent from low-density environments; Laskar et al. 2013).

An evolving technical landscape, together with rapid follow-up enabled by high-cadence optical surveys, present new opportunities for millimeter transient astronomy. Lower-noise receivers and ultra-wide bandwidth capability have greatly increased the sensitivity of submm facilities (e.g., the Submillimeter Array or SMA; Ho et al. 2004), and the Atacama Large Millimeter Array (ALMA), a flagship facility, recently began operations. Optical surveys are discovering new and unexpected classes of transient events whose millimeter properties are unknown—and possibly different from previously known types—motivating renewed follow-up efforts.

AT2018cow was discovered on 2018 June 16 UT as an optical transient (Prentice et al. 2018; Smartt et al. 2018) by the Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018). It attracted immediate attention because of its fast rise time (${t}_{\mathrm{peak}}\lesssim 3$ days), which was established by earlier non-detections (Fremling 2018; Prentice et al. 2018), together with its high optical luminosity (${M}_{\mathrm{peak}}\sim -20$) and its close proximity ($d=60$ Mpc).

UV/optical/IR (UVOIR) observations (Perley et al. 2018; Prentice et al. 2018) revealed unprecedented photometric and spectroscopic properties. Long-lived luminous X-ray emission was detected with the Neil Gehrels Swift Observatory/X-ray Telescope (Swift/XRT; Rivera Sandoval & Maccarone 2018), International Gamma-Ray Astrophysics Laboratory (INTEGRAL) (Ferrigno et al. 2018; Savchenko et al. 2018), and the Nuclear Spectroscopic Telescope Array (NuSTAR) (Grefenstette et al. 2018; Margutti et al. 2018a). Early radio and submillimeter detections were reported by Northern Extended Millimeter Array (NOEMA) (de Ugarte Postigo et al. 2018), JCMT (Smith et al. 2018), Arcminute Microkelvin Imager (AMI) (Bright et al. 2018), and by us using the Australia Telescope Compact Array (ATCA) (Dobie et al. 2018a, 2018b). The source does not appear to be a GRB, as no prompt high-energy emission was detected in searches of Swift/Burst Alert Telescope (BAT) (Lien et al. 2018), Fermi/GBM (Dal Canton et al. 2018), Fermi/Large Area Telescope (LAT) (Kocevski & Cheung 2018), and AstroSat Cadmium Zinc Telluride Imager (CZTI) (Sharma et al. 2018).

Perley et al. (2018) suggested that AT2018cow is a new member of the class of rapidly rising (${t}_{\mathrm{rise}}\lesssim 5$ days) and luminous (${M}_{\mathrm{peak}}\lt -18$) blue transients, which have typically been found in archival searches of optical surveys (Drout et al. 2014; Pursiainen et al. 2018; Rest et al. 2018). The leading hypothesis for this class was circumstellar interaction of a supernova (SN; Ofek et al. 2010), but this was difficult to test because most of the events were located at cosmological distances, and not discovered in real time. AT2018cow presented the first opportunity to study a member of this class up close and in real time, but its origin remains mysterious despite the intense ensuing observational campaign. Possibilities include failed SNe and TDEs, but although AT2018cow shares properties with both of these classes, it is clearly not a typical member of either (Kuin et al. 2018; Perley et al. 2018; Prentice et al. 2018).

Given the unusual nature of the source, we were motivated to undertake high-frequency observations. We began an extensive monitoring campaign with the SMA at 230 and 340 GHz and carried out supporting observations with the ATCA from 5–34 GHz. To our surprise AT2018cow was very bright and still rising at submillimeter wavelengths (and optically thick in the centimeter band) days after the discovery. Our SMA observations represent the first millimeter observation of a transient in its rise phase.

This finding led us to seek Director's Discretionary Time (DDT) with ALMA at even higher frequencies, which enabled us to resolve the peak of the spectral energy distribution (SED). A technical highlight of the ALMA observations was the detection of the source at nearly a terahertz frequency (Band 9). We present the submillimeter, radio, and X-ray observations in Section 2, and our modeling of the radio-emitting ejecta in Section 3. In Section 4, we put our velocity and energy measurements in the context of other transients (Section 4.1), attribute the high submm luminosity of AT2018cow to the large density of the surrounding medium (Section 4.2), and discuss some problems with the synchrotron model parameters (Section 4.3). In Section 5, we attribute the late-time X-ray emission to a powerful central engine. We look ahead to the future in Section 6.

2.1.1. The SMA

2.1.2. The ATCA

2.1.3. ALMA

2.2.1. Swift/XRT

Swift (Gehrels et al. 2004) has been monitoring AT2018cow since June 19, with both the Ultraviolet-Optical Telescope (Roming et al. 2005) and the XRT (Burrows et al. 2005). The transient was well detected in both instruments (e.g., Rivera Sandoval et al. 2018).

We downloaded the Swift/XRT data products (light curves and spectra) using the web-based tools developed by the Swift-XRT team (Evans et al. 2009). We used the default values, but binned the data by observation. To convert from count rate to flux, we used the absorbed count-to-flux rate set by the spectrum on the same tool, $4.26\times {10}^{-11}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{\mathrm{ct}}^{-1}$. This assumes a photon index of ${\rm{\Gamma }}=1.54$ and a Galactic N_H column of $6.57\times {10}^{20}\,{\mathrm{cm}}^{-2}$.

Table 1.  Flux-density Measurements for AT2018cow

${\rm{\Delta }}t$ (days)	Facility	Frequency (GHz)	Flux Density (mJy)
5.39	SMA	215.5	15.14 ± 0.56
5.39	SMA	231.5	16.19 ± 0.65
6.31	SMA	215.5	31.17 ± 0.87
6.31	SMA	231.5	31.36 ± 0.97
7.37	SMA	215.5	40.19 ± 0.56
7.37	SMA	231.5	41.92 ± 0.66
7.41	SMA	330.8	36.39 ± 2.25
7.41	SMA	346.8	30.7 ± 1.99
8.37	SMA	215.5	41.19 ± 0.47
8.37	SMA	231.5	41.44 ± 0.56
8.38	SMA	344.8	26.74 ± 1.42
8.38	SMA	360.8	22.79 ± 1.63
9.26	SMA	243.3	35.21 ± 0.75
9.26	SMA	259.3	36.1 ± 1.0
9.28	SMA	341.5	22.85 ± 1.53
9.28	SMA	357.5	25.84 ± 2.5
10.26	SMA	243.3	36.6 ± 0.81
10.26	SMA	259.3	31.21 ± 0.92
10.26	SMA	341.5	19.49 ± 1.47
10.26	SMA	357.5	17.42 ± 2.8
11.26	SMA	243.3	22.14 ± 1.05
11.26	SMA	259.3	20.02 ± 1.28
13.3	SMA	215.5	35.67 ± 0.81
13.3	SMA	231.5	32.94 ± 1.01
14.36	SMA	344.8	26.85 ± 2.22
14.36	SMA	360.8	26.13 ± 2.77
14.37	SMA	215.5	42.05 ± 0.5
14.37	SMA	231.5	38.71 ± 0.58
15.23	SMA	225.0	30.82 ± 2.41
15.23	SMA	233.0	28.64 ± 4.0
15.23	SMA	241.0	27.41 ± 3.21
15.23	SMA	249.0	15.4 ± 4.74
17.29	SMA	234.6	36.57 ± 1.55
17.29	SMA	250.6	34.04 ± 1.81
18.4	SMA	217.5	52.52 ± 0.55
18.4	SMA	233.5	49.32 ± 0.65
19.25	SMA	193.5	59.27 ± 1.49
19.25	SMA	202.0	56.03 ± 1.5
19.25	SMA	209.5	55.09 ± 1.39
19.25	SMA	218.0	54.54 ± 1.33
20.28	SMA	215.5	50.6 ± 1.69
20.28	SMA	231.5	49.16 ± 1.84
20.28	SMA	267.0	41.69 ± 1.62
20.28	SMA	283.0	37.84 ± 1.63
24.39	SMA	215.5	55.57 ± 0.53
24.39	SMA	231.5	53.2 ± 0.6
24.4	SMA	333.0	23.98 ± 1.39
24.4	SMA	349.0	28.46 ± 1.37
26.26	SMA	215.6	38.83 ± 1.2
26.26	SMA	231.6	34.1 ± 1.33
31.2	SMA	230.6	36.76 ± 1.12a
31.2	SMA	246.6	31.41 ± 1.42a
35.34	SMA	215.5	21.59 ± 0.89
35.34	SMA	231.5	20.63 ± 1.04
36.34	SMA	215.5	24.32 ± 1.19
36.34	SMA	231.5	20.79 ± 1.42
39.25	SMA	217.0	18.34 ± 1.65
39.25	SMA	233.0	19.74 ± 1.76
39.26	SMA	264.0	17.61 ± 2.79
39.26	SMA	280.0	8.27 ± 2.93
41.24	SMA	217.0	12.58 ± 1.5
41.24	SMA	225.0	8.91 ± 1.9
41.24	SMA	233.0	15.08 ± 1.73
41.24	SMA	241.0	9.64 ± 2.13
44.24	SMA	230.6	9.42 ± 1.61
44.24	SMA	234.6	8.04 ± 2.51
44.24	SMA	246.3	10.43 ± 2.13
44.24	SMA	250.6	10.06 ± 3.24
45.23	SMA	217.0	8.28 ± 2.24
45.23	SMA	233.0	10.55 ± 2.39
45.23	SMA	264.0	8.35 ± 3.27
45.23	SMA	280.0	5.7 ± 3.49
47.24	SMA	230.6	11.47 ± 2.81
47.24	SMA	234.6	10.81 ± 4.39
47.24	SMA	246.6	11.65 ± 3.76
47.24	SMA	250.6	5.6 ± 5.37
48.31	SMA	217.5	7.63 ± 1.11
48.31	SMA	233.5	5.73 ± 1.32
76.27	SMA	215.5	1.33 ± 0.55
76.27	SMA	231.5	0.61 ± 0.63
76.27	SMA	335.0	−2.27 ± 1.87
76.27	SMA	351.0	−0.32 ± 1.76
10.48	ATCA	5.5	<0.15
10.48	ATCA	9.0	0.27 ± 0.06
10.48	ATCA	34.0	5.6 ± 0.16
13.47	ATCA	5.5	0.22 ± 0.05
13.47	ATCA	9.0	0.52 ± 0.04
13.47	ATCA	16.7	1.5 ± 0.1
13.47	ATCA	21.2	2.3 ± 0.3
13.47	ATCA	34.0	7.6 ± 0.5
17.47	ATCA	5.5	0.41 ± 0.04
17.47	ATCA	9.0	0.99 ± 0.03
19.615	ATCA	34.0	14.26 ± 0.21
28.44	ATCA	34.0	30.59 ± 0.2
34.43	ATCA	34.0	42.68 ± 0.19
81.37	ATCA	34.0	6.97 ± 0.09
14.03	ALMA	336.5	29.4 ± 2.94
14.03	ALMA	338.5	29.1 ± 2.91
14.03	ALMA	348.5	28.49 ± 2.85
14.03	ALMA	350.5	28.29 ± 2.83
14.14	ALMA	398.0	26.46 ± 2.65
14.14	ALMA	400.0	26.21 ± 2.62
14.14	ALMA	410.0	25.69 ± 2.57
14.14	ALMA	412.0	25.95 ± 2.6
22.02	ALMA	90.5	91.18 ± 4.6
22.02	ALMA	92.5	92.31 ± 4.6
22.02	ALMA	102.5	93.97 ± 4.7
22.02	ALMA	104.5	93.57 ± 4.7
22.04	ALMA	138.0	85.1 ± 4.3
22.04	ALMA	140.0	84.58 ± 4.2
22.04	ALMA	150.0	80.62 ± 4.0
22.04	ALMA	152.0	79.71 ± 4.0
23.06	ALMA	671.0	31.5 ± 6.3

Notes. Time of detection used is mean UT of observation. SMA measurements have formal uncertainties shown, which are appropriate for in-band measurements on a given night. However, for night-to-night comparisons, true errors are dominated by systematics and are roughly 10%–15% unless indicated otherwise. ALMA measurements have roughly 5% uncertainties in Bands 3 and 4, 10% uncertainties in Bands 7 and 8, and a 20% uncertainty in Band 9. ATCA measurements have formal errors listed, but also have systematic uncertainties of roughly 10%.

^aSystematic uncertainty 20% due to uncertain flux calibration.

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2.2.2. NuSTAR

NuSTAR (Harrison et al. 2013) comprises two co-aligned telescopes, Focal Plane Module A (FPMA) and FPMB. Each is sensitive to X-rays in the 3–79 keV range, with slightly different response functions. NuSTAR observed AT2018cow on four epochs, and a log of these observations as well as the best-fit spectral model parameters is presented in Table 2.

Table 2.  NuSTAR Flux Measurements for AT2018cow and the Spectral Model Parameters

Epoch	OBSID	Exp. Time (ks)	${\rm{\Delta }}t$ (days)	Flux (${10}^{-12}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$)				Photon Index	${\chi }^{2}$/DOF
				3–10 keV	10–20 keV	20–40 keV	40–80 keV
1	90401327002a	32.4	7.9	4.94 ± 0.04	4.41 ± 0.10	12.21 ± 0.39	21.46 ± 4.29	⋯	421/443
2	90401327004	30.0	16.8	5.21 ± 0.04	4.99 ± 0.10	7.70 ± 0.33	12.80 ± 4.79	1.39 ± 0.02	424/412
3	90401327006	31.2	28.5	1.58 ± 0.03	1.45 ± 0.06	1.74 ± 0.21	⋯	1.51 ± 0.04	174/169
4	90401327008	33.0	36.8	1.10 ± 0.02	0.92 ± 0.05	1.02 ± 0.20	⋯	1.59 ± 0.05	134/135

Notes. Fluxes were measured with a model-independent method.

^aOBSID 90401327002 is best described by a bkn2pow model with parameters ${{\rm{\Gamma }}}_{1}=1.24\pm 0.05$, ${E}_{1}=9.0\pm 0.3\,\mathrm{keV}$, ${{\rm{\Gamma }}}_{2}=3.6\pm 0.7$, ${E}_{2}=11.1\pm 0.3\,\mathrm{keV}$, ${{\rm{\Gamma }}}_{3}=0.50\pm 0.05$. All reported values are for this model.

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NuSTAR data were extracted using nustardas_06Jul17_v1 from HEASOFT 6.24. Source photons were extracted from a circle of 60'' radius, visually centered on the object. We note that such a large region, appropriate for NuSTAR data, includes the transient as well as the host galaxy. Background photons were extracted from a non-overlapping circular region with 120'' radius on the same chip. Spectra were grouped to 20 source photons per bin, ignoring energies below 3 keV and above 80 keV.

Spectra were analyzed in XSPEC (v12.10.0c), using NuSTAR CALDB files dated 2018 August 14. Rivera Sandoval et al. (2018) report a low absorbing column density (N${}_{{\rm{H}}}=7.0\times {10}^{20}\,{\mathrm{cm}}^{-2}$), hence we ignore this component in fitting. We opt for a simple phenomenological model to describe the spectrum. We do not fit for a cross-normalization constant between NuSTAR FPMA and FPMB. Epoch 1 (OBSID 90401327002) spectra are not consistent with a simple power law or a broken power law, hence we fit it with the bkn2pow model (obtaining spectral breaks at 9.0 ± 0.3 keV and $11.1\pm 0.3\,\mathrm{keV}$). Spectra of the remaining three epochs are well fit by a simple, unabsorbed power law.

We calculate the flux directly from energies of individual source and background photons detected, converted into flux using the Ancillary Response Files generated by the NuSTAR pipeline. We use a bootstrap method to estimate the error bars: we draw photons from the data with replacement, and calculate the source flux from this random sample. By repeating this process 10,000 times for each OBSID and each energy range, we calculate the 1σ error bars on the fluxes. This method gives answers consistent with xspec flux and cflux measurements for bright sources (see for instance Kaspi et al. 2014), but has the advantage of giving flux measurements without the need to assume a spectral model for the source. We find that the source is not well detected in the 40–80 keV band at the third and fourth epochs.

The radio and X-ray light curves are shown in Figure 1. The 230 GHz light curve rises (the first observation of a millimeter transient in its rise phase) and then shows significant variability, presumably from inhomogeneities in the surrounding medium. We have tentative evidence that the rise is at least in part due to a decreasing peak frequency: at ${\rm{\Delta }}t=5$–6 days, the flux is marginally higher at 231.5 GHz than at 215.5 GHz, and at ${\rm{\Delta }}t=7$–8 days, it seems that the peak may have been within the SMA observing bands. However, the position of the peak is ill-constrained; future early observations would benefit from observations at more frequencies.

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By ${\rm{\Delta }}t=50\,\mathrm{days}$, the radio flux has diminished both due to the peak frequency shifting to lower frequencies, and to a decay in the peak flux. Specifically, the peak of the 15 GHz light curve is 19 mJy around 47 days (A. Horesh 2018, personal communication), substantially less luminous than the peak of the 230 GHz or the 34 GHz light curve. As we discuss in Section 4.2, this diminishing peak flux suggests that the interaction itself is diminishing, and enables us to constrain the size of the "circum-bubble" of material.

The X-ray light curve seems to have two distinct phases. We call the first phase (${\rm{\Delta }}t\lesssim 20$ days) the plateau phase because the X-ray emission is relatively flat. The second phase, which we call the decline phase, begins around ${\rm{\Delta }}t\approx 20$ days. During this period, the X-ray emission exhibits an overall steep decline, but also exhibits strong variation (by factors of up to 10) on shorter timescales (see also Kuin et al. 2018; Perley et al. 2018; Rivera Sandoval et al. 2018).

We use the shortest timescale of variability in the 230 GHz light curve to infer the size of the radio-emitting region, and do the same for the X-ray emission in Section 5. On Days 5–6, the 230 GHz flux changed by order unity in one day, setting a length scale for the source size of ${\rm{\Delta }}R=c{\rm{\Delta }}t=2.6\times {10}^{15}\,\mathrm{cm}$ (170 au). We find no evidence for shorter-timescale variability in our long SMA tracks from the first few days of observations (Figure 2).

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Together with the 230 GHz flux density (${S}_{\nu }\approx 30\,\mathrm{mJy}$) and the distance ($d=60\,\mathrm{Mpc}$) we infer an angular size of $\theta =2.8\,\mu \mathrm{as}$ and a brightness temperature of

$$\begin{eqnarray}&&{T}_{B}=\displaystyle \frac{{S}_{\nu }{c}^{2}}{2k{\nu }^{2}{\rm{\Delta }}{\rm{\Omega }}}\gtrsim 3\times {10}^{10}\,{\rm{K}}\end{eqnarray} \tag{ 1 }$$

where ${\rm{\Delta }}{\rm{\Omega }}=\pi {\theta }^{2}$. This brightness temperature is close to the typical rest-frame equipartition brightness temperatures of the most compact radio sources, ${T}_{B}\sim 5\times {10}^{10}$ K (Readhead 1994).

The shape of the radio to submillimeter SED (Figure 3), together with the high brightness temperature implied by the luminosity and variability timescale (Section 3.1), can only be explained by nonthermal emission (Readhead 1994). The observed spectrum is assumed to arise from a population of electrons with a power-law number distribution in Lorentz factor ${\gamma }_{e}$, with some minimum Lorentz factor ${\gamma }_{m}$ and electron energy power index p:

$$\begin{eqnarray}&&\displaystyle \frac{{dN}({\gamma }_{e})}{d{\gamma }_{e}}\propto {\gamma }_{e}^{-p},\qquad {\gamma }_{e}\geqslant {\gamma }_{m}.\end{eqnarray} \tag{ 2 }$$

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As argued below, we expect an adiabatic strong shock moving into a weakly magnetized, ionized medium at a nonrelativistic speed. First-order Fermi acceleration gives $p=(r+2)/(r-1)$, where r is the compression ratio of the shock. A strong matter-dominated shock has r = 4, hence p = 2 (Blandford & Eichler 1987). However the back-reaction of the accelerated particles decelerates the gas flow, weakening the gas dynamic subshock and reducing the compression ratio from the strong shock r = 3, so typical $2.5\lt p\lt 3$ are obtained in both simulations and astrophysical data (Jones & Ellison 1991). Quasi-perpendicular magnetized and relativistic shocks are more subtle, since some particles cannot return along field lines after their first shock crossing, but the limiting value is $p\sim 2.3$ (Pelletier et al. 2017).

Equation (3) provides an expression for ${\gamma }_{m}$. Behind the shock (velocity v) some fraction ${\epsilon }_{e}$ of the total energy density goes into accelerating electrons. Conserving shock energy flux gives

$$\begin{eqnarray}&&{\gamma }_{m}-1\approx {\epsilon }_{e}\displaystyle \frac{{m}_{p}}{{m}_{e}}\displaystyle \frac{{v}^{2}}{{c}^{2}}.\end{eqnarray} \tag{ 3 }$$

The value of ${\gamma }_{m}$ is large for relativistic shocks, e.g., in GRBs. But we will see that for this source ($v/c\sim 0.1$, ${\epsilon }_{e}\sim 0.1$), the bulk of the electrons are just mildly relativistic (${\gamma }_{m}\sim 2\mbox{--}3$). For ordinary SN shocks ${\gamma }_{m}$ is always nonrelativistic (${\gamma }_{m}-1\lt 1$). Thus, in the parameter estimations below, we follow SN convention and assume that the relativistic electrons follow a power-law distribution down to a fixed ${\gamma }_{m}$ (Chevalier 1982, 1998; Kulkarni et al. 1998; Frail et al. 2000; Soderberg et al. 2005). We apply ${\epsilon }_{e}$ only to this relativistic power law, not to the nonrelativistic thermal distribution of shock-heated particles at lower energy.

We now describe each of the break frequencies that characterize the observed spectrum. First, the characteristic synchrotron frequency ${\nu }_{m}$ emitted by the minimum energy electrons is:

$$\begin{eqnarray}&&{\nu }_{m}={\gamma }_{m}^{2}{\nu }_{g}\end{eqnarray} \tag{ 4 }$$

where ${\nu }_{g}$ is the gyrofrequency,

$$\begin{eqnarray}&&{\nu }_{g}=\displaystyle \frac{{q}_{e}B}{2\pi {m}_{e}c}\end{eqnarray} \tag{ 5 }$$

and q_e is the unit charge, B is the magnetic field strength, m_e is the electron mass, and c is the speed of light.

Next, there is the cooling frequency ${\nu }_{c}\equiv \nu ({\gamma }_{c})$, the frequency below which electrons have lost the equivalent of their total energies to radiation via cooling. In general, the timescale for synchrotron cooling depends on the Lorentz factor as $t\propto {\gamma }_{e}^{-1}$. Thus, electrons radiating at higher frequencies cool more quickly. Separately, electrons could also lose energy by Compton upscattering of ambient (low energy) photons—the so-called Inverse Compton (IC) scattering. In Section 5, we find that IC scattering dominates at early times and that synchrotron losses dominate at later times, and that the transition is at $t\approx 13\,\mathrm{days}$.

At ${\rm{\Delta }}t\gt 13\,\mathrm{days}$, electrons with ${\gamma }_{e}\gt {\gamma }_{c}$ cool principally by synchrotron radiation to ${\gamma }_{c}$ in a time t, where

$$\begin{eqnarray}&&{\gamma }_{c}=\displaystyle \frac{6\pi {m}_{e}c}{{\sigma }_{T}{B}^{2}t}.\end{eqnarray} \tag{ 6 }$$

For $t\lt 13\,\mathrm{days}$, Compton cooling on the UVOIR flux exceeds the synchrotron cooling rate by a factor $\sim {(t/10\mathrm{days})}^{-5/2}$, and ${\gamma }_{c}$ is correspondingly lowered. The cooled electrons emit around the characteristic synchrotron frequency

$$\begin{eqnarray}&&{\nu }_{c}={\gamma }_{c}^{2}{\nu }_{g}.\end{eqnarray} \tag{ 7 }$$

Next, the self-absorption frequency ${\nu }_{a}$ is the frequency at which the optical depth to synchrotron self-absorption is unity. The rise at 34 GHz obeys a ${f}_{\nu }\propto {t}^{2}$ power law (as shown in Figure 1), consistent with the optically thick spectral index we measure (Figure 3). This indicates that the self-absorption frequency is above the ATCA bands (${\nu }_{a}\gt 34\,\mathrm{GHz}$). Figure 3 also shows that the emission in the SMA bands is optically thin at ${\rm{\Delta }}t\gtrsim 10\,\mathrm{days}$, constraining the self-absorption frequency to be ${\nu }_{a}\lt 230\,\mathrm{GHz}$.

On Day 22, we resolve the peak of the SED with our ALMA data. We denote the peak frequency ${\nu }_{p}$ and the flux at the peak frequency F_p, and find ${\nu }_{p}\approx 100\,\mathrm{GHz}$ and ${F}_{p}\approx 94\,\mathrm{mJy}$. Motivated by the observation of optically thick emission at $\nu \lt {\nu }_{p}$, we assume that ${\nu }_{p}={\nu }_{a}$, and adopt the framework in Chevalier (1998) (hereafter referred to as C98) to estimate properties of the shock at this epoch. These properties are summarized in Table 3, and outlined in detail below.

Table 3 .  Quantities Derived from Day 22 Measurements, Using Different Equipartition Assumptions

Parameter	${\epsilon }_{e}={\epsilon }_{B}=1/3$	${\epsilon }_{e}=0.1,{\epsilon }_{B}=0.01$
${\nu }_{a}={\nu }_{p}$ (GHz)	100	100
${F}_{\nu ,p}$ (mJy)	94	94
r (1015 cm)	7	6
v/c	0.13	0.11
B (G)	6	4
$U({10}^{48}$ erg)	4	35
ne (${10}^{5}$ cm−3)	3	41
${\nu }_{c}$ (GHz)	2	8

Note. In the text unless otherwise stated we use ${\epsilon }_{e}={\epsilon }_{B}=1/3$.

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Following Equations (11) and (12) in C98, the outer shock radius R_p can be estimated as

$$\begin{eqnarray}&&{R}_{p}={\left[\displaystyle \frac{6{c}_{6}^{p+5}{F}_{p}^{p+6}{D}^{2p+12}}{({\epsilon }_{e}/{\epsilon }_{B})f(p-2){\pi }^{p+5}{c}_{5}^{p+6}{E}_{l}^{p-2}}\right]}^{1/(2p+13)}{\left(\displaystyle \frac{{\nu }_{p}}{2{c}_{1}}\right)}^{-1}\end{eqnarray} \tag{ 8 }$$

and the magnetic field can be estimated as

$$\begin{eqnarray}&&{B}_{p}={\left[\displaystyle \frac{36{\pi }^{3}{c}_{5}}{{\left({\epsilon }_{e}/{\epsilon }_{B}\right)}^{2}{f}^{2}{\left(p-2\right)}^{2}{c}_{6}^{3}{E}_{l}^{2(p-2)}{F}_{p}{D}^{2}}\right]}^{2/(2p+13)}\left(\displaystyle \frac{{\nu }_{p}}{2{c}_{1}}\right)\end{eqnarray} \tag{ 9 }$$

where, as in Equation (2), p is the electron energy index. Note that C98 use γ for the electron energy power index. We use p instead and γ for the Lorentz factor. The constant ${c}_{1}=6.27\times {10}^{18}$ in cgs units, and the constants c₅ and c₆ are tabulated as a function of p on page 232 of Pacholczyk (1970). D is the distance to the source, ${E}_{l}=0.51$ MeV is the electron rest mass energy, and ${\epsilon }_{e}/{\epsilon }_{B}$ is the ratio of energy density in electrons to energy density in magnetic fields (in C98 this ratio is parameterized as α, but we use α as the optically thin spectral index of the radio SED.) Finally, f is the filling factor: the emitting region is approximated as a planar region with thickness s and area in the sky $\pi {R}^{2}$, and thus a volume $\pi {R}^{2}s$, which can be characterized as a spherical emitting volume $V=4\pi {{fR}}^{3}/3=\pi {R}^{2}s$.

On Day 22, we measure $\alpha =-1.1$ where ${F}_{\nu }\propto {\nu }^{\alpha }$, which corresponds to p = 3.2. Later in this section we show that our submillimeter observations lie above the cooling frequency, and therefore that the index of the source function of electrons is p_s = 2.2. However, the C98 prescription considers a distribution as it exists when the electrons are observed, from a combination of the initial acceleration and the energy losses (to cooling). So, we proceed with p = 3.2, and discuss this unusual regime in Section 4.3. The closest value of p in the table in Pacholczyk (1970) is p = 3, so we use this value to select the constants (and note that, as stated in C98, the results do not depend strongly on the value of p.) With this, Equations (8) and (9) reduce to Equations (13) and (14) in C98, respectively, reproduced here:

$$\begin{eqnarray}&&\begin{array}{l}{R}_{p}=8.8\times {10}^{15}{\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-1/19}{\left(\displaystyle \frac{f}{0.5}\right)}^{-1/19}\\ \times \ {\left(\displaystyle \frac{{F}_{p}}{\mathrm{Jy}}\right)}^{9/19}{\left(\displaystyle \frac{D}{\mathrm{Mpc}}\right)}^{18/19}{\left(\displaystyle \frac{{\nu }_{p}}{5\mathrm{GHz}}\right)}^{-1}\,\mathrm{cm},\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\begin{array}{l}{B}_{p}=0.58{\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-4/19}{\left(\displaystyle \frac{f}{0.5}\right)}^{-4/19}\\ \times \ {\left(\displaystyle \frac{{F}_{p}}{\mathrm{Jy}}\right)}^{-2/19}{\left(\displaystyle \frac{D}{\mathrm{Mpc}}\right)}^{-4/19}\left(\displaystyle \frac{{\nu }_{p}}{5\,\mathrm{GHz}}\right){\rm{G}}.\end{array}\end{eqnarray} \tag{ 11 }$$

Next we estimate the total energy U. For p = 3, Equations (10) and (11) can be combined into the following expression for $U={U}_{B}/{\epsilon }_{B}$,

$$\begin{eqnarray}\begin{array}{rcl}U & = & \displaystyle \frac{1}{{\epsilon }_{B}}\displaystyle \frac{4\pi }{3}{{fR}}^{3}\left(\displaystyle \frac{{B}^{2}}{8\pi }\right)\\ & = & (1.9\times {10}^{46}\,\mathrm{erg})\,\displaystyle \frac{1}{{\epsilon }_{B}}{\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-11/19}{\left(\displaystyle \frac{f}{0.5}\right)}^{8/19}\\ & & \times \ {\left(\displaystyle \frac{{F}_{p}}{\mathrm{Jy}}\right)}^{23/19}{\left(\displaystyle \frac{D}{\mathrm{Mpc}}\right)}^{46/19}{\left(\displaystyle \frac{{\nu }_{p}}{5\mathrm{GHz}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 12 }$$

Following C98 we take f = 0.5, but the dependence on this parameter is weak. In choosing ${\epsilon }_{B}$ and ${\epsilon }_{e}$ there are several normalizations (or assumptions) used in the literature. As a result the inferred energy can vary enormously (see Section 4.1 for further details). For now, we follow Soderberg et al. (2010) in setting ${\epsilon }_{e}={\epsilon }_{B}=1/3$ (in other words, that energy is equally partitioned between electrons, protons, and magnetic fields). With all of these choices, we find that at ${\rm{\Delta }}t\approx 22\,\mathrm{days}$, ${R}_{p}\approx 7\,\times \,{10}^{15}\,\mathrm{cm}$ and ${B}_{p}\approx 6\,{\rm{G}}$. We find that the total energy $U\approx 4\,\times \,{10}^{48}\,\mathrm{erg}$. Assuming 10% uncertainties in F_p and ${\nu }_{p}$ and a 50% uncertainty in p, a Monte Carlo with 10,000 samples gives uncertainties of 0.15–0.3 dex in these derived parameters. Our results are robust to departures from equipartition given the large penalty in the required energy (Readhead 1994).

The mean velocity up to ${\rm{\Delta }}t\approx 22\,\mathrm{days}$ is $v={R}_{p}/{t}_{p}\,=0.13c$. We can write a general expression for v/c (taking ${L}_{p}=4\pi {F}_{p}{D}^{2}$, noting that $4\pi \mathrm{Jy}\,{\mathrm{Mpc}}^{2}=1.2\,\times {10}^{27}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}$):

$$\begin{eqnarray}&&\begin{array}{l}v/c\approx {\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-1/19}{\left(\displaystyle \frac{f}{0.5}\right)}^{-1/19}\\ \times \ {\left(\displaystyle \frac{{L}_{p}}{{10}^{26}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{Hz}}^{-1}}\right)}^{9/19}{\left(\displaystyle \frac{{\nu }_{p}}{5\mathrm{GHz}}\right)}^{-1}{\left(\displaystyle \frac{{t}_{p}}{1\mathrm{days}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 13 }$$

Furthermore, from the t² rise at 34 GHz (Figure 1) we can infer that the radius increases as $R\propto t$ and therefore that the velocity $v={dR}/{dt}$ is constant. As shown in Figure 5, this is consistent with later-time, low-frequency observations by Margutti et al. (2018b). We put this derived energy and velocity into the context of other energetic transients in Section 4.1.

Next, we estimate the density of the medium into which the forward shock is propagating. The ejecta expands into the medium with velocity v₁, producing a shock front (a discontinuity in pressure, density, and temperature) with shock-heated ejecta immediately behind this front. Conservation of momentum across this (forward) shock front requires that

$$\begin{eqnarray}&&{P}_{1}+{\rho }_{1}{v}_{1}^{2}={P}_{2}+{\rho }_{2}{v}_{2}^{2}\end{eqnarray} \tag{ 14 }$$

where P is pressure (not to be confused with p used as the power-law index for the electron energy distribution). The subscript 1 refers to the upstream (ambient) medium and the subscript 2 refers to the downstream medium (the shocked ejecta). Far upstream, the pressure can be taken to be 0, and in the limit of strong shocks (for a monatomic gas) ${\rho }_{2}/{\rho }_{1}\,={v}_{1}/{v}_{2}=4$. Thus, this can be simplified to

$$\begin{eqnarray}&&\displaystyle \frac{3{\rho }_{1}{v}_{1}^{2}}{4}={P}_{2}.\end{eqnarray} \tag{ 15 }$$

If the medium is composed of fully ionized hydrogen, ${\mu }_{p}=1$ and the number densities of protons and electrons are equal (n_p = n_e). Using Equation (15) together with Equation (11), as well as the relations ${P}_{2}=(1/{\epsilon }_{B})\,{B}^{2}/8\,\pi$ and ${\rho }_{1}={\mu }_{p}{m}_{p}{n}_{e}$,

$$\begin{eqnarray}&&\begin{array}{l}{n}_{e}\approx (20\,{\mathrm{cm}}^{-3})\displaystyle \left(\frac{1}{{\epsilon }_{B}}\right){\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-6/19}{\left(\displaystyle \frac{f}{0.5}\right)}^{-6/19}\\ \times \ {\left(\displaystyle \frac{{L}_{p}}{{10}^{26}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{Hz}}^{-1}}\right)}^{-22/19}{\left(\displaystyle \frac{{\nu }_{p}}{5\mathrm{GHz}}\right)}^{4}{\left(\displaystyle \frac{{t}_{p}}{1\mathrm{days}}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 16 }$$

We find that the number density of electrons at ${\rm{\Delta }}t\approx 22\,\mathrm{days}$ is ${n}_{e}\approx 3\,\times \,{10}^{5}\,{\mathrm{cm}}^{-3}$. We note that the strong jump conditions used here assume $\gamma =5/3$, and that there is a correction for the contribution of a relativistic ($\gamma =4/3$) component. Chevalier (1983) quantified this correction using the factor w, the ratio of the relativistic pressure to the total pressure. In the most extreme case (w = 1) the correction is small, only a factor of 1.14 in n_e. This is negligible compared to our uncertainties.

At such a high density, the optical depth to free–free absorption (FFA) ${\tau }_{\mathrm{ff}}$ might be expected to have a significant effect on the shape of the spectrum at low radio frequencies (Lundqvist & Fransson 1988). From Lang (1999), we have

$$\begin{eqnarray}&&{\tau }_{\mathrm{ff}}=8.235\times {10}^{-2}{\left(\displaystyle \frac{{T}_{e}}{{\rm{K}}}\right)}^{-1.35}{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{-2.1}\int {\left(\displaystyle \frac{{N}_{e}}{{\mathrm{cm}}^{-3}}\right)}^{2}\left(\displaystyle \frac{{dl}}{\mathrm{pc}}\right)\end{eqnarray} \tag{ 17 }$$

which, with our measured values of n_e and R on Day 22, gives the characteristic value

$$\begin{eqnarray}&&{\tilde{\tau }}_{\mathrm{ff}}=68{\left(\displaystyle \frac{{T}_{e}}{8000{\rm{K}}}\right)}^{-1.35}{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{-2.1}.\end{eqnarray} \tag{ 18 }$$

However, in AT2018cow the gas through which the shock is propagating is not at normal H ii-region temperatures of $\sim {10}^{4}\,{\rm{K}}$. The UV and X-ray photons emitted at early times will completely ionize and Compton heat any surrounding gas: for gas at the density and radius given in Table 3, the lifetime to photoionization of a neutral hydrogen atom is less than 0.01 s, while the recombination time is years.¹⁵ Compton heating of the electrons increases their temperature at the rate

$$\begin{eqnarray}&&\displaystyle \frac{d(3/2){{kT}}_{e}}{{dt}}=H=\displaystyle \frac{{\sigma }_{T}}{{m}_{e}{c}^{2}}{\int }_{0}^{\infty }\displaystyle \frac{h\nu {L}_{\nu }{f}_{\mathrm{KN}}(h\nu /{m}_{e}{c}^{2})}{4\pi {R}^{2}}d\nu \end{eqnarray} \tag{ 19 }$$

where the Klein–Nishina correction ${f}_{\mathrm{KN}}(x)\simeq 1-21x/5\,+O({x}^{2})$.¹⁶ Even though the blackbody ($T={\rm{30,000}}\,{\rm{K}}$) luminosity at ${\rm{\Delta }}t=3$ is 100 times larger than the coeval X-ray luminosity (Perley et al. 2018), the Compton heating is dominated by the 10–100 $\,\mathrm{keV}$ X-ray flux, and we find, for $3\lt {\rm{\Delta }}t\lt 20\,\mathrm{days}$, gas at the density and radius given in Table 3 has

$$\begin{eqnarray}&&{T}_{e}(t)\simeq 1.0\times {10}^{6}\,{\rm{K}}{\left(t/3\mathrm{days}\right)}^{0.6}.\end{eqnarray} \tag{ 20 }$$

Given the spectral evolution shown in Perley et al. (2018), the Compton temperature (at which Compton heating balances Compton cooling) is ${T}_{c}\sim 2.5\times {10}^{6}\,{\rm{K}}$ on Day 3, hardening to ${T}_{c}\sim 1.8\times {10}^{7}\,{\rm{K}}$ on Day 20 since the blackbody UV flux drops as ${t}^{-2.5}$, while the hard X-ray flux drops much more slowly. At these high Compton-heated temperatures ${T}_{e}\sim {10}^{6}\,{\rm{K}}$, the FFA optical depth given by Equation (18) only rises above unity below frequencies of 300 MHz, accessible to facilities like Low-Frequency Array (LOFAR).

Next, we estimate the luminosity from free–free emission of the ionized gas (Lang 1999):

$$\begin{eqnarray}&&L\approx 1.43\times {10}^{-27}\,{n}_{e}{n}_{i}{T}^{1/2}{{VZ}}^{2}g\,\mathrm{erg}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 21 }$$

where n_i is the number density of ions, Z is the atomic number, and $g\approx 1$ is the Gaunt factor, a quantum mechanical correction. Assuming that the gas is completely ionized out to the light-travel sphere at 22 days ($R=6\times {10}^{16}\,\mathrm{cm}$), we have n_e = n_i in the region of interest. We also take Z = 1. With the inferred density (Table 3) we find $L\approx 9\times {10}^{37}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, so the contribution to the observed X-ray luminosity is negligible.

Finally, we estimate the different break frequencies, beginning with ${\nu }_{m}$. Using Equation (3), taking ${\epsilon }_{e}=1/3$ and using our inferred $\beta =v/c$ from Day 22, ${\gamma }_{m}\approx 5$. Next, using Equation (5) and our measured value of B, ${\nu }_{g}\approx 17\,\mathrm{MHz}$. Equation (4) thus gives ${\nu }_{m}\approx 0.4\,\mathrm{GHz}$, substantially below our peak frequency. The spectral index at ${\nu }_{m}\lt \nu \lt {\nu }_{a}$ is ${\nu }^{5/2}$ (Rybicki & Lightman 1986), which we show as dotted lines in Figure 3. Clearly, the lowest-frequency fluxes are in excess of ${\nu }^{5/2}$ extrapolation. This naturally occurs if the source is inhomogeneous (e.g., magnetic field and/or particle energy density decreasing outwards). It can also arise even for a perfectly homogeneous source because the energy spectrum of the radiating electrons is not a pure power law: note that ${\nu }_{a}\gt {\nu }_{c}\gt {\nu }_{m}$, so even beyond the Maxwellian-like peak at ${\gamma }_{m}$, the spectrum is convex, steepening with energy above ${\gamma }_{c}$. These both produce self-absorbed spectra flatter than ${\nu }^{5/2}$ (see Section 6.8 in Rybicki & Lightman (1986) and de Kool et al. (1989) for model calculations).

The cooling frequency due to synchrotron radiation is determined by Equations (6) and (7). We find ${\gamma }_{c}\approx 11$, giving a cooling frequency ${\nu }_{c}\approx 2\,\mathrm{GHz}$. The relative contributions to electron cooling from synchrotron radiation and IC scattering are determined by the ratio between the radiation energy density and the magnetic energy density. On Day 22, the bolometric luminosity as measured in the UVOIR is $5\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Perley et al. 2018), so the radiation density is ${u}_{\mathrm{ph}}=0.26\,\mathrm{erg}\,{\mathrm{cm}}^{-3}$. The magnetic energy density on the same day is ${u}_{B}={B}^{2}/8\pi \approx 1.5\,\mathrm{erg}\,{\mathrm{cm}}^{-3}.$ Thus, synchrotron radiation is the dominant cooling mechanism, with a roughly 10% contribution from Inverse Compton (IC) scattering.

At this epoch, the cooling timescale ${t}_{\mathrm{cool}}=({\gamma }_{e}{m}_{e}{c}^{2})/(\tfrac{4}{3}{\sigma }_{T}{u}_{B}{\gamma }_{e}^{2}c)=240\,\mathrm{days}/{\gamma }_{e}$ for an electron with ${\gamma }_{e}$, which is roughly 80 for an electron radiating at 100 GHz. So on Day 22 the cooling timescale is shorter than the timescale on which we are observing the source. This means that continuous re-acceleration of the electrons is required, which could be provided by ongoing shock interaction.

As stated in Section 3.1, it seems that ${\nu }_{p}$ during the rise phase (${\rm{\Delta }}t\approx 5$–8 days) was above or within the SMA observing bands. Using the peak observed flux and frequency as a lower limit on the peak flux and peak frequency, respectively, we consistently find that $v\approx 0.1c$, albeit with a decreasing n_e ($3\times {10}^{6}\,{\mathrm{cm}}^{-3}$ at ${\rm{\Delta }}t\approx 5$).

It is challenging to directly compare the energy of AT2018cow to that of other classes of radio-luminous transients, because there are several conventions that produce discrepant results. In particular, the energy partition fractions ${\epsilon }_{B}$ and ${\epsilon }_{e}$ are important for determining the total amount of energy in the shock, but are difficult to measure.

In the classical GRB literature, ${\epsilon }_{e}$ has been consistently measured to be ${\epsilon }_{e}\approx 0.2$, within a factor of 2, while values of ${\epsilon }_{B}$ have much wider spreads, with a median value of $3\times {10}^{-5}$ but a distribution spanning four orders of magnitude (Kumar & Zhang 2015). Kumar & Barniol Duran (2010) constrain ${\epsilon }_{B}\sim {10}^{-6}$ for a CSM density of $0.1\,{\mathrm{cm}}^{-3}$, and an even smaller value for higher densities. One of the best-observed GRB afterglows is GRB 130427A, and from modeling the evolving spectrum Perley et al. (2014) find $0.03\lt {\epsilon }_{B}\lt 1/3$ and $0.14\lt {\epsilon }_{e}\lt 1/3$.

There are different approaches to modeling for the handful of low-luminosity GRBs (LLGRBs) discovered to date. For LLGRB 980425/SN 1998bw, Kulkarni et al. (1998) invoked equipartition (${\epsilon }_{B}={\epsilon }_{e}=0.5$). For LLGRB 031203/SN2003 lw, Soderberg et al. (2004) used models from Sari et al. (1998) and Granot & Sari (2002) together with the cooling frequency inferred from X-ray observations to estimate ${\epsilon }_{e}=0.4$ and ${\epsilon }_{B}=0.2$. For LLGRB 060218/SN2006aj, Soderberg et al. (2006) used the same prescription as was used in SN 1998bw. Finally, for LLGRB 100316D/SN2010bh, Margutti et al. (2013) set ${\epsilon }_{B}=0.01$ and allowed ${\epsilon }_{e}$ to vary from 0.01–0.1.

For Type II and Type Ibc radio SNe, approaches range from using the SN 1998bw convention (i.e., ${\epsilon }_{B}={\epsilon }_{e}=0.5;$ Soderberg et al. 2005; Horesh et al. 2013) to ${\epsilon }_{B}={\epsilon }_{e}=0.1$ (e.g., Chevalier & Fransson 2006; Soderberg et al. 2006; Salas et al. 2013) to ${\epsilon }_{B}={\epsilon }_{e}=1/3$ for the relativistic supernova SN 2009bb (Soderberg et al. 2010).

In this work, we follow the convention in Soderberg et al. (2010) so that we can compare our velocity–energy diagram to the corresponding diagram (Figure 4) in that paper. To put all transients on the same scale, we take the peak frequency, peak luminosity, and peak time for each event, and run them through the same equations that we used to infer the shock properties of AT2018cow. Note that we do not vary the values of ${c}_{5},{c}_{6}$, p, but these are all very small corrections, whereas the effect of ${\epsilon }_{B}$ and the ratio ${\epsilon }_{e}/{\epsilon }_{B}$ in estimating the energy is large. The details of how we selected the peak values for each event are in Appendix C. When possible, we use the peak of the SED at a particular epoch. However, for most events, we use the peak flux density corresponding to a certain frequency, because well-sampled SEDs are rare.

Our rederived velocity–energy diagram is shown in Figure 4. AT2018cow has an energy comparable to mildly relativistic (LLGRBs; e.g., SN 1998bw) outflows and energetic SNe (e.g., SN 2007bg). We display vertical axes for two different conventions (${\epsilon }_{B}=1/3$ and ${\epsilon }_{B}=0.01$) to show how this affects the inferred energy. Note that these values are not evaluated for a consistent epoch. However, for AT2018cow, we have reason to believe that the values of velocity and energy do not change significantly over the course of our observations. For other sources, it would be necessary to have a well-sampled SED over multiple epochs in order to trace the evolution of these values, and this is rare in the literature.

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Here we compare the radio luminosity of AT2018cow to that of other transients, as observed at the spectral peak frequency at time ${\rm{\Delta }}t$. As shown in Figure 5, AT2018cow stands out as being several times more luminous than SN 1998bw, and having a late peak at high frequencies. Over time, the peak luminosity diminishes to the value reported in the low-frequency radio observations of Margutti et al. (2018b), supporting our inference that the velocity is not changing significantly.

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On this diagram we also indicate lines of constant velocity (see Figure 3 of Soderberg et al. 2010 and Figure 4 of C98) and lines of constant mass-loss rate scaled by velocity $\dot{M}/{v}_{w}$, as a diagnostic of density (see Figure 10 of Jencson et al. 2018). Note that these lines assume ${\nu }_{p}={\nu }_{a}$.

We now derive relations between the observational coordinates of the diagram in Figure 4 and physical quantities: the ordinate, peak radio luminosity L_p, is simply a power of the energy per unit radius U/R. We get an expression for U/R using Equations (12) and (10):

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{U}{R}=(3\times {10}^{29}\,\mathrm{erg}\,{\mathrm{cm}}^{-1})\left(\displaystyle \frac{1}{{\epsilon }_{B}}\right)\\ \times \ {\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-10/19}{\left(\displaystyle \frac{f}{0.5}\right)}^{9/19}{\left(\displaystyle \frac{{L}_{p}}{{10}^{26}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{Hz}}^{-1}}\right)}^{14/19}.\end{array}\end{eqnarray} \tag{ 22 }$$

This translation between L_p and U/R is shown on the left and right axis labels of Figure 4.

We now show that the abscissa of Figure 5, $({\rm{\Delta }}t/1\,\mathrm{days})({\nu }_{p}/5\,\mathrm{GHz})$, is very nearly proportional to the square root of the swept up mass per unit radius M/R, or equivalently, if the surrounding medium was from a pre-explosion steady wind of speed v_w, $\dot{M}/{v}_{w}\propto M/R$. A steady spherical wind of ionized hydrogen with velocity v_w has ${n}_{e}=\dot{M}/(4\pi {m}_{p}{r}^{2}{v}_{w}$), so we can reparameterize the density in terms of the mass-loss rate:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\dot{M}}{{v}_{w}}\left(\displaystyle \frac{1000\,\mathrm{km}\,{{\rm{s}}}^{-1}}{{10}^{-4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right) & = & (0.0005)\displaystyle \left(\frac{1}{{\epsilon }_{B}}\right){\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-8/19}\\ & & \times \ {\left(\displaystyle \frac{f}{0.5}\right)}^{-1/19}{\left(\displaystyle \frac{{L}_{p}}{{10}^{26}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{Hz}}^{-1}}\right)}^{-4/19}\\ & & \times \ {\left(\displaystyle \frac{{\nu }_{p}}{5\mathrm{GHz}}\right)}^{2}{\left(\displaystyle \frac{{t}_{p}}{1\mathrm{days}}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 23 }$$

Notice the weak ($-4/19$ power) dependence on L_p, and the quadratic dependence on ${\nu }_{p}{t}_{p}$, which means the lines of constant $\dot{M}/{v}_{w}$ are nearly vertical in Figure 5.

AT2018cow lies along the same velocity line as SN 2003bg and SN 2003L, but n_e is a factor of a few to an order of magnitude larger.¹⁷ Similarly, SN 1998bw lies along a similar velocity line to SN 2006aj, but n_e (using our prescription) is 40 ${\mathrm{cm}}^{-3}$, while the density inferred for 2006aj using the same prescription is 3 ${\mathrm{cm}}^{-3}$. For SNe Ibc in general, Chevalier & Fransson (2006) attributed the large spread in radio luminosity to a spread in circumstellar density, with the example that SN 2002ap (not shown in Figure 5 due to its relatively low luminosity) is roughly three orders of magnitude less luminous than SN 2003L, and its inferred ambient density is also a factor of 3 smaller. In SN 2003L and SN 2003bg, the high density was attributed to a stellar wind.

This is not the whole story: as we showed above, high peak radio luminosity just corresponds to high U/R, i.e., high energy and/or small radius. Since U is the converted energy, it represents only a lower limit to the actual driving kinetic energy (becoming equal to it as the explosion transitions from free expansion to the Sedov phase). A higher-density medium more quickly converts the piston's energy to thermal energy than does a low-density medium. Thus, for a large fixed explosion energy, a denser medium will indeed lead to larger peak radio luminosities. But the direct correlation is with the (thermalized) energy per unit radius, U/R. Similarly, Equation (11) shows that (except for a very weak ${L}_{p}^{-2/19}$ dependence), ${\nu }_{p}\propto {B}_{p}$. Thus, higher peak frequencies are directly indicative of a higher magnetic field, or equivalently, pressures. Therefore, AT2018cow's high ${\nu }_{p}$ and high L_p are quite likely mostly a consequence of it being energetic, and observed early, when the high wind density at small radii led to high pressure, and enhanced U/R. As we discuss below, this suggests that many other SNe could have shown similar bright mm-submm fluxes, had they been observed at those wavelengths in their first week.

On Day 22, the inferred density is ${\rho }_{0}=4\times {10}^{-19}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ at a radius of ${r}_{0}=7\times {10}^{15}\,\mathrm{cm}$ (Table 3). From this we can infer $\dot{M}/{v}_{{\rm{w}}}=2.4\times {10}^{14}\,{\rm{g}}\,{\mathrm{cm}}^{-1}$. The mass swept up to radius r is $(\dot{M}/{v}_{{\rm{w}}})r$. In Section 3, we argue that the blast wave reaches the edge of the surrounding bubble around ${\rm{\Delta }}t\approx 50\,\mathrm{days}$. If so, given our inferred velocity, the radius of the circum-bubble is $1.7\,\times {10}^{16}\,\mathrm{cm}$, and the mass of the circum-bubble $\approx 2\times {10}^{-3}\,{M}_{\odot }$. The mass-loss rates for hot stars (${v}_{w}\sim 2000\,\mathrm{km}\,{{\rm{s}}}^{-1}$, Lamers & Leitherer 1993) and red supergiants (${v}_{w}\sim 20\,\mathrm{km}\,{{\rm{s}}}^{-1}$, van Loon 2010) range from 10⁻⁴–10⁻⁶${M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Smith et al. 2018). Thus, ${r}_{0}/{v}_{w}$ is $\sim 1\,\mathrm{yr}$ for a hot star progenitor, and ∼100 yr for a red supergiant. Therefore, the circum-bubble could either have been formed by normal mass loss in a red supergiant, or end-of-life enhanced mass loss from a hot star or red supergiant (see, e.g., Smith et al. 2017 and references therein).

UVOIR observations of AT2018cow place strong constraints on the nature of the surrounding medium (Perley et al. 2018). The high luminosity and fast rise can be interpreted as shock heating of a dense shell of material at $R={10}^{14}\,\mathrm{cm}$ or 10 au, qualitatively consistent with the inference of dense material given the properties inferred from the radio shock. On the other hand, early spectra show no narrow emission lines indicative of a shock and the light curve declines steeply after peak, both of which suggest that this dense material must also be quite limited in extent, with little material at larger radii. While the radio observations also suggest that a cutoff in the density distribution may exist, the $0.1c$ shock does not reach it for almost 20 days, a quite different timescale than the optical peak (reached in less than 3 days) or early spectroscopy. This might be due to the $0.1c$ shock being produced by breakout from the $R={10}^{14}\,\mathrm{cm}$ shell, which re-energized the (much slower) SN shock. Or there could be deviations from spherical symmetry (for example, with the optical heating a quasi-spherical shell but the radio shock passing through a denser toroidal component or clouds along a bipolar jet).

Thus, an energetic shock propagating into a dense environment could produce a radio SED that peaks at submillimeter wavelengths at early times. However, as illustrated in the left panel of Figure 6, searches at high frequencies at early times have been rare, and primarily limited to transients with relativistic jetted outflows (GRBs, TDEs). We suggest that these searches be expanded to other classes of transients: luminous SNe such as SN 2003L and SN 2007bg, and luminous TDEs such as ASASSN14li, all exploded into dense media and exhibited luminous centimeter-wavelength emission at $t\gt 10\,\mathrm{days}$. As time goes on, the SED peak shifts to lower frequencies and diminishes in brightness, so these events could have been bright millimeter transients at $t\lt 10\,\mathrm{days}$. This is supported by Figure 5, which shows that SN 2007bg, SN 2003bg, and SN 2003L could have appeared similar to AT2018cow had they been observed earlier at higher frequencies.

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The ordering of the break frequencies, ${\nu }_{\mathrm{ff}}\lt {\nu }_{m}\lt {\nu }_{c}\lt {\nu }_{a}$, is an unusual regime for long-wavelength observations. For a relativistic shock (GRBs), the typical orderings are ${\nu }_{a}\lt {\nu }_{c}\lt {\nu }_{m}$ (the fast cooling regime) and ${\nu }_{a}\lt {\nu }_{m}\lt {\nu }_{c}$ (the slow cooling regime; Sari et al. 1998). For nonrelativistic shocks, the ordering in most cases seems to be ${\nu }_{a}\lt {\nu }_{m}\lt {\nu }_{c}$ at measured frequencies above 1.4 GHz, but can also be ${\nu }_{m}\lt {\nu }_{a}\lt {\nu }_{c};$ ${\nu }_{c}$ is typically considered unimportant for long-wavelength observations (Nakar & Piran 2011).

The low cooling frequency is a consequence of a large magnetic field strength, ${\nu }_{c}\propto {B}^{-3}$ (reduced even further for $t\lt 10\,\mathrm{days}$ by Compton cooling on the UVOIR flux, which dominates over synchrotron cooling). This in turn presumably arises from the injection of a large amount of energy into a small volume of material, consistent with the low velocity we measure. From Equation (11) we see that B_p scales as ${({\epsilon }_{e}/{\epsilon }_{B})}^{-4/19}{L}_{p}^{-2/19}{\nu }_{p}$. Changing ${\epsilon }_{B}$ from 1/3–0.01 could increase ${\nu }_{c}$ by a factor of 8, still much lower than our observed frequencies. This regime is selectively probed by submillimeter observations, because a low ${\nu }_{c}$ (high B_p) gives rise to a ${\nu }_{a}$ that falls in this wavelength regime. We note that in the same framework, the relativistic TDE Swift J1644+57 (whose long-wavelength SED also peaked in the submillimeter for the first few weeks) would also have had a cooling frequency below much of the observed frequency range (${\nu }_{c}\approx 6\,\mathrm{GHz}$).

Since ${\nu }_{c}$ is below any of our measured frequencies, the injection spectrum (the spectrum of the electrons prior to cooling) has a shallower power-law index than what we measure, ${p}_{i}=p-1$. This suggests that ${p}_{i}\approx 2.2$ on Day 22, when $p\approx 3.2$, This is not unreasonable for Fermi acceleration from a strong shock. Typical young Galactic SN remnants have $p={p}_{i}=2.4$ in the radio, flattening to $p\sim 2$ at higher frequencies (Urošević 2014).