SN2019dge: A Helium-rich Ultra-stripped Envelope Supernova

SN2019dge was discovered by ZTF, which runs on the Palomar Oschin Schmidt 48-inch (P48) telescope (Dekany et al. 2020). The first real-time alert (Patterson et al. 2019) was generated on 2019 April 7 10:18:46 (JD  = 2458580.9297) for a g-band detection at 20.66 ± 0.34 mag and J2000 coordinates $\alpha ={17}^{{\rm{h}}}{36}^{{\rm{m}}}46\buildrel{\rm{s}}\over{.} 75$, $\delta =+{50}^{{\rm{d}}}{32}^{{\rm{m}}}52\buildrel{\rm{s}}\over{.} 2$. On April 8, a new alert was flagged by a science program filter on the GROWTH Marshal (Kasliwal et al. 2019), which is designed to look for fast-evolving transients (Ho et al. 2020a).

2.2.1. HST Observation

Hubble Space Telescope (HST) observations were obtained as part of our HST "Rolling Snapshots" pilot experiment (GO-15675). This new observational approach requires the PI to update a list of objects of interest each week before the schedule is built, giving the scheduler flexibility to choose a possible source for snapshots. Under this program, we obtained a NUV spectrum using the WFC3 G280 grism, a short (60 s) direct image of this field in the F300X filter to set the wavelength scale of the spectrum, as well as a longer exposure (200 s) in the F350LP filter. The image in the F350LP filter is shown in Figure 1. It has very similar throughput to the zeroth order of the G280 grism.

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SN2019dge resides in a compact galaxy SDSS J173646.73+503252.3. From our follow-up spectra (see Section 3.2), we measure a host redshift of z = 0.0213, corresponding to a luminosity distance of D_L = 93 Mpc. Figure 1 shows that there is a surface brightness peak to the northwest of SN2019dge (∼0.2 kpc away), which might trace a star-forming region. A thorough analysis of the host-galaxy properties is given in Appendix C.

2.2.2. Optical Photometry

Following Yao et al. (2019), we perform forced point-spread function (PSF) photometry on ZTF difference images generated with the ZTF real-time reduction and image subtraction pipeline (Masci et al. 2019). ZTF image subtraction is based on the Zackay et al. (2016) image subtraction method. The sky region of SN2019dge is covered by two ZTF fields with "fieldid" (i.e., ZTF field identifier) 763 and 1799. We exclude all data in field 1799 because the reference image was constructed using images obtained between 2018 May 25 and 2019 July 12, which is after the explosion of the transient. Although the ZTF name of this object (ZTF18abfcmjw) may indicate that the transient was discovered in 2018, this is due to an alert generated on 2018 July 7 from a candidate detection in negative subtraction (reference minus science) in field 763. We note that the seeing during that night was 42, larger than 99% of Palomar nights. The irregularly shaped PSF might cause over-subtraction around the galaxy nucleus in the difference imaging process.

Because field 763 was included in both the Northern Sky Survey with two epochs (one epoch each in g and r) per three nights and the extragalactic high-cadence survey with six epochs (three epochs each in g and r) per night (see Bellm et al. 2019a for the design of ZTF experiments), SN2019dge is visited multiple times every night. Therefore, single-night flux measurements in the same filter are binned (by taking the inverse variance-weighted average). This gives a pre-explosion r-band limit of 18.95 mag (5σ limit computed at the expected position of the transient) on April 4 10:36:34. We convert 5σ detections to AB magnitudes for further analysis.

Following the discovery of SN2019dge, we obtain follow-up photometry in ugriz with an optical imager (IO:O) on the Liverpool Telescope (LT; Steele et al. 2004). Digital image subtraction and photometry for LT imaging is performed using the Fremling Automated Pipeline (FPipe; Fremling et al. 2016). Fpipe performs calibration and host subtraction against Sloan Digital Sky Survey reference images and catalogs (SDSS, Alam et al. 2015).

LT and P48 photometry are shown in Figure 2. Absolute and apparent magnitudes are corrected for Galactic extinction $E(B-V)=0.022$ mag (Schlafly & Finkbeiner 2011). We assume R_V = 3.1, and adopt the reddening law from Cardelli et al. (1989). We do not correct for host-galaxy contamination given the absence of Na I D absorption in all spectra at the host redshift. To estimate the epoch of maximum light, we interpolate the g- and r-band photometry with three-order polynomial functions, as shown in the inset of Figure 2. The time window used in the fit is from MJD = 58581.2 to 58585.2. SN2019dge is found to peak at ${M}_{g,\mathrm{peak}}=-16.45\pm 0.03$ mag on MJD = 58583.19, and ${M}_{r,\mathrm{peak}}=-16.27\pm 0.02$ mag on MJD = 58583.39. Hereafter we use phase (Δt) to denote time with respect to the g-band maximum light epoch, MJD = 58583.2.

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We obtain one epoch of late-time imaging with the Wafer Scale Imager for Prime (WASP) mounted on the Palomar 200-inch telescope at Δt ≈ 85 d. The data are obtained in r band with a total exposure time of 900 s divided into dithered exposures of 300 s each. The data are reduced using standard techniques as described in De et al. (2020a). Image subtraction is performed using archival reference images from the Dark Energy Legacy Survey (Dey et al. 2019), using the method described in De et al. (2020a). The median 5σ limiting magnitude of the image is $r\approx 25$ mag. However, the depth at the transient location is limited by the noise from the bright host galaxy, and the transient is not detected to a 5σ limiting magnitude of r = 22.1 mag.

We also perform forced photometry on archival PTF/iPTF difference images spanning 2009 May 7–2016 June 13 (Law et al. 2009; Rau et al. 2009). No historical detection is found.

2.2.3. Swift Observation

2.2.4. Radio Follow-up

2.2.5. Spectroscopy

3.1.1. Peak Luminosity and Rise and Decline Timescales

The g- and r-band peak luminosity of SN2019dge (≈−16.3 mag) is around the lower limit of stripped envelope SNe (Drout et al. 2011; Taddia et al. 2018; Prentice et al. 2019), and akin to those of the Ca-rich gap transients, which occupy the luminosity "gap" between novae and SNe (peak absolute magnitude M_R ≈ −15.5 to −16.5 mag, Kasliwal et al. 2012).

To characterize the rise and decline timescales of SN2019dge, following Ho et al. (2020b), we calculate rise time (t_rise) defined by how long it takes the r-band light curve to rise from 0.75 mag below peak to peak, and decline time (t_decay) determined by how long it takes to decline from peak by 0.75 mag (corresponding to half of maximum flux). Because SN2019dge shows no evidence of hydrogen (Section 3.2) and exhibits a fast rise (Figure 2), we compare the t_rise, t_decay, and peak absolute magnitude between SN2019dge and two other groups of transients:

• 1.  
  Fast-evolving hydrogen-deficient transients that are fainter than normal SNe Ia (i.e., <−19 mag), including SN2002bj (Poznanski et al. 2010), SN2005ek (Drout et al. 2013), PTF09dav (Sullivan et al. 2011), SN2010X (Kasliwal et al. 2010), PTF10iuv (Kasliwal et al. 2012), iPTF14gqr (De et al. 2018c), iPTF16hgs (De et al. 2018b), SN2018kzr (McBrien et al. 2019), and SN2019bkc (Chen et al. 2020).
• 2.  
  "Fast-evolving luminous transients" (FELT, Rest et al. 2018) or "fast blue optical transients" (FBOT, Margutti et al. 2019). We select well-studied representative objects of this population, including KSN2015K (Rest et al. 2018), iPTF16asu (Whitesides et al. 2017), AT2018cow (Prentice et al. 2018; Perley et al. 2019), SN2018gep (Ho et al. 2019a), and ZTF18abvkwla (also known as the Koala; Ho et al. 2020b).

In Figure 3, peak magnitudes are given in (observer-frame) r band, except for KSN2015K, for which we only have observations in the Kepler "white" filter, and iPTF16asu, for which the rise was caught only in g band. We correct only for Galactic extinction to compute ${M}_{r,\mathrm{peak}}$ (assuming no host extinction). Note that iPTF14gqr and iPTF16hgs are SNe exhibiting double-peaked light curves. Because the rise to first peak was not captured, an upper limit of ${t}_{\mathrm{rise}}$ is calculated by taking the time difference between the first r-band detection and the latest prediscovery nondetection,²⁰ and absolute magnitude of the first r-band detection is considered to be a fainter limit of ${M}_{r,\mathrm{peak}}$ (plotted in the upper panel). In the lower panel, because observations of iPTF14gqr do not extend to 0.75 mag below its second peak, we present a lower limit of its t_decay.

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It is clear from the upper panel of Figure 3 that SN2019dge rose faster than normal Ca-rich events, such as PTF09dav and PTF10iuv. The t_rise of ≈2.0 days is similar to the population of FELTs/FBOTs, but SN2019dge is substantially fainter. In the subluminous regime, iPTF14gqr has t_rise comparable to SN2019dge, and its first peak has been postulated to be caused by the diffusion of shock-deposited energy out of an envelope around the progenitor star (De et al. 2018c).

The bottom panel of Figure 3 shows that ${t}_{\mathrm{decay}}$ of SN2019dge is longer than that for the most rapid-fading SNe Ibc, such as SN2005ek, SN2018kzr, and SN2019bkc. Its decay timescale is more similar to SN2002bj, SN2010X, the population of Ca-rich transients (PTF09dav, PTF10iuv, iPTF16hgs), and likely iPTF14gqr. It has been suggested that the latter group of events has a radioactivity-powered main peak with a low mass of nickel (${M}_{\mathrm{Ni}}\lesssim 0.1{M}_{\odot }$).

3.1.2. Bolometric Evolution

We construct the bolometric light curve at epochs where at least detections in two filters are available by fitting the spectral energy distribution (SED) with a blackbody function (see details of model fitting in Appendix B.1). We plot the physical evolution of SN2019dge with comparisons to iPTF14gqr and iPTF16hgs in Figure 4, in which we have adopted the explosion epoch of iPTF14gqr, iPTF16hgs, and SN2019dge estimated by De et al. (2018c, 2018b), and Section 4.1 of this paper, respectively. The bolometric luminosity of SN2019dge reaches $\sim 5\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ at ∼1.5 days after the explosion epoch. The subsequent decline displays an initial fast drop of $0.36\,\mathrm{mag}\,{\mathrm{days}}^{-1}$ at age 2–9 days, and transitions to a slower drop of $0.11\,\mathrm{mag}\,{\mathrm{days}}^{-1}$ at age 10–30 days.

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The bolometric temperature of SN2019dge reaches as high as ∼2.3 × 10⁴ K at age 1.5 days and rapidly falls afterwards. The maximum T_bb is much hotter than that observed in normal SNe Ibc (6000–10,000 K, Taddia et al. 2018). Its early light-curve evolution is slower than iPTF14gqr, but similar to iPTF16hgs and several other stripped envelope SNe displaying double-peaked light curves (e.g., see Figure 2 of Fremling et al. 2019). Their first peaks have been modeled by cooling emission from an extended envelope around the progenitor after the core-collapse SN (CCSN) shock breaks out (Modjaz et al. 2019). After ∼8 days past explosion, T_bb flattens to 6000 ± 1000 K, similar to the behavior of normal SNe Ibc at a much later phase (∼30 days after explosion, Taddia et al. 2018).

Assuming that the photospheric radius can be approximated by R_bb and linearly expands at early phase, we fit a linear function to the first few R_bb versus time measurements of SN2019dge, which gives $\approx 8000\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The radius then remains flat at $\sim 6.7\times {10}^{3}\,{R}_{\odot }$ during age 8–30 days, and even appears to slowly recede. The reason for this is that the temperature drops to a recombination temperature for helium and the opacity becomes small. As a result, the outer layers of SN ejecta become more transparent, and deeper regions of the ejecta are being probed (Piro & Morozova 2014).

3.1.3. Color Evolution

We compare the color curves of other fast transients to that of SN2019dge in Figure 5, in corresponding pairs of g − r (or B − R) and r − i (or R − I). For the double-peaked events iPTF14gqr and iPTF16hgs, "maximum" time corresponds to epoch of maximum light in the second peak.

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The early-time blue color of SN2019dge arises from the high-temperature peak. Among other events, SN2002bj, iPTF14gqr, and iPTF16hgs exhibit early colors as blue as SN2019dge. Subsequently, SN2019dge displays a color starting out blue and turning redder with time, consistent with a cooling process.

One unusual feature of SN2019dge is that at ∼6–9 days after maximum light, the g − r color becomes bluer by ≈0.2 mag, while after that the color continues to redden. We notice that iPTF14gqr exhibits a similar trend—around 4 days before the second peak, its g − r color stays flat before getting redder afterwards, while around 2 days before the second peak, its r − i color also turns bluer by ≈0.2 mag.

3.2.1. Early Spectral Evolution

The very early spectra at −1.1, −0.1, and +0.4 days show a blue continuum and strong galaxy emission lines from the underlying H II region (see Figure 6). In addition, these spectra also show prominent He I λ5876 and high-ionization He II λ4686 narrow emission lines. We computed the equivalent width (EW) of He II emission using the spectral line and continuum wavelength ranges given by Khazov et al. (2016). The EW is found to be −7.56 ± 1.07 Å, −2.66 ± 1.30 Å, and −3.77 ± 0.16 Å in the −1.1 days, −0.1 days, and +0.4 days spectra.

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In Table 2, we show the measured FWHM velocities of some emission lines by fitting a Gaussian to the line profile. Because the [S II] λλ6716, 6731 doublet is definitely from the host galaxy, their line widths serve as a practical measurement of instrumental line-broadening. As shown in column 2, FWHM velocities of the He II and He I emission lines are $\sim 550\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and $\sim 580\,\mathrm{km}\,{{\rm{s}}}^{-1}$, much broader than the resolution of $\approx 270\,\mathrm{km}\,{{\rm{s}}}^{-1}$, whereas Hα is not well resolved. Thus, we infer that the hydrogen emission is from the host galaxy, while the helium lines are from photoionized material exterior to the SN.

Table 2.  Rest-frame FWHM ($\mathrm{km}\,{{\rm{s}}}^{-1}$) of Narrow Emission Lines in the +0.4 days, +85.3 days, and +314.4 days LRIS Spectra

Transition	+0.4 days	+85.3 days	+314.4 days
He II λ4686	552 ± 36	⋯	⋯
He I λ5876	582 ± 86	272 ± 19	298 ± 24
He I λ6678	⋯	282 ± 28	339 ± 59
He I λ7065	⋯	230 ± 37	218 ± 26
Hα	291 ± 49	263 ± 13	280 ± 14
[S II] λ6716	285 ± 52	254 ± 14	274 ± 18
[S II] λ6731	263 ± 78	251 ± 14	264 ± 15
[O I] λ6300	⋯	263 ± 28	231 ± 24

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3.2.2. Photospheric Phase Spectral Evolution

Broad transient features are present in the +12.0 and +14.3 days spectra (Figure 7). These spectra are taken at the photospheric phase where emission comes from a photosphere receding (in mass coordinates) back through freely expanding SN ejecta. The HST spectrum contains little host-galaxy contamination due to its high angular resolution. Prominent galaxy emission lines in the DBSP spectrum are identified and plotted in light red to emphasize transient features. The existence of the P-Cygni He I λ5876 profile and nonexistence of hydrogen nominally classify SN2019dge as a Type Ib SN. We measure the velocity of the He I λ5876 line by fitting a parabola to the absorption minimum. The resulting fits give velocities of $\approx 6000\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and $5900\,\mathrm{km}\,{{\rm{s}}}^{-1}$ for the +12.0 days and +14.3 days spectra, respectively. This is lower than velocities of normal SNe Ib measured from the He I λ5876 absorption minimum ($\sim {10}^{4}\,\mathrm{km}\,{{\rm{s}}}^{-1}$, Liu et al. 2016), but higher than that in Type Ibn SNe ($\sim 3000\,\mathrm{km}\,{{\rm{s}}}^{-1}$, Hosseinzadeh et al. 2017).

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In Figure 8, we compare the photospheric phase optical spectra of SN2019dge with other helium-rich events. Note that the DBSP spectrum has host emission lines masked. SN2019dge is different from normal helium-rich stripped envelope SNe Ib/IIb or SNe Ibn in the sense that its P-Cygni absorption minimum in the He I λ5876 line is weaker. The feature at $\sim 5000\,\mathring{\rm A}$ is often attributed to He I λ5016 and Fe II triplet λλλ4924, 5018, and 5169 (Liu et al. 2016). The shape of this feature in SN2019dge is similar to those in normal SNe Ib/IIb at much later phases (∼20 days post maximum), indicating that the spectral evolution of SN2019dge is faster. The complex absorption profile at ∼4500 Å has been identified as a blend of Fe II, Mg II λ4481, and He I λ4472 (Hamuy et al. 2002). In the DBSP spectrum, we detected O I λ7774 and broad Ca II at ∼8500 Å (due to the λλ8498, 8542, and 8662 triplet) with clear P-Cygni profiles. Both are major features of stripped envelope SNe (Gal-Yam 2017).

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In Figure 9, we compare the HST NUV spectrum with spectra of other types of SNe. The UV part of SN2019dge is much weaker than a blackbody extrapolation of the optical spectra would predict. This has also been seen in normal thermonuclear and CCSNe, and interpreted as strong metal-line blanketing caused by iron-peak elements, particularly Fe II and Co II (Gal-Yam et al. 2008). SN2019dge bears a close resemblance to SN1993J between 2000 Å and 4000 Å. In Figure 9, we also marked the rest wavelength of Mg I λ2852 and Mg II λλ2796, 2803. The emission features at ∼2760 Å in SN2019dge and Gaia16apd are similar to the bump at ∼2730 Å in SN1993J, which was found to be a NLTE Mg II emission line (Jeffery et al. 1994). This resonance line is blueshifted from its rest wavelength, and is suggested to come from a circumstellar region that is distinctly separated from the SN photosphere in velocity and excitation conditions (Panagia et al. 1980; Fransson et al. 1984).

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3.2.3. Late-time Spectral Evolution

Figure 10 shows late-time spectra of SN2019dge obtained at +85.3, +143.1, +171.1, and +314.4 days. The general shape of the spectra is determined by the host galaxy, while possible SN features are marked by the dashed lines. The right panels (b), (c), and (d) highlight emission lines at wavelengths of He I, [O I], and [Ca II]. In panel (c) of Figure 10, the [O I] λλ6300, 6363 feature consists of two narrow emission peaks. These doublet transitions share the same upper level (${}^{3}{{\rm{P}}}_{\mathrm{1,2}}$–¹D₂). The observed intensity ratio R ≡ F(6300/6364) ∼ 3.1 agrees with the nebular condition, as one would expect in the optically thin regime (Leibundgut et al. 1991; Li & McCray 1992). In panel (d), we mark the position of the [Ca II] doublet in dashed lines, but only the λ7324 line is clearly detected. It presents a double-peaked profile with a peak separation of $\sim 400\,\mathrm{km}\,{{\rm{s}}}^{-1}$.

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From panel (a) of Figure 10, it is also clear that in the +85.3 days spectrum, the He I and [Ca II] lines have broader emission components with Lorentzian profiles at the bases of the narrow emission lines. These Lorentz-shaped components are not visible in the +314.4 days spectrum. Therefore, to further investigate the broader features, we subtract the +314.4 days' spectrum from the +85.3 days spectrum. The resulting subtraction (Figure 11) reveals intermediate-width (FWHM $\sim 2000\,\mathrm{km}\,{{\rm{s}}}^{-1}$) components of He I, [Ca II], and the Ca II IR triplet. It shares a close resemblance to some Type Ibn SNe, such as SN2011hw (Pastorello et al. 2015) and SN2015G (Shivvers et al. 2017). These intermediate-width features are too narrow to be explained by emission from SN ejecta. Instead, they are probably emitted by a cold, dense circumstellar medium (CSM) shell formed by radiative cooling from the post-shock material, as has been proposed to be the case in interacting Type IIn/Ibn SNe (Chugai & Danziger 1994; Smith 2017).

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Table 2 (column 3 and 4) gives the measured FWHM velocities of narrow emissions shown in panels (b), (c), and (d) of Figure 10. It can be seen that the measured FWHM of other emission lines are similar to the [S II] line width. Therefore, we conclude that the observed narrow emissions are not resolved.

Because of the low resolution of our LRIS spectra, we cannot directly rule out the possibility that the narrow lines are emanating from the host galaxy. However, there is evidence indicating that they are not merely a background contamination of an underlying H II region:

• 1.  
  In the +85.3 days spectrum, the He I and [Ca II] narrow emissions are on top of intermediate-width Lorentzian components characteristic of electron scattering (Huang & Chevalier 2018), which fades away in the +314.4 days' spectrum. However, the hydrogen Balmer lines do not have a broader base in any of our spectra.
• 2.  
  The flux intensities of He I, [O I], and [Ca II] lines decrease by a factor of approximately 2 from +85.3 days to +314.4 days, consistent with the temporal evolution from an emission mechanism connected to the aging supernova. As a comparison, the line strengths of the strongest emissions in normal ionized nebulae (Hα, [O III], [O II], [S II], etc.) do not follow this behavior.
• 3.  
  Although the He I and [O I] lines labeled in panel (a) of Figure 10 have been observed in H II regions (Peimbert et al. 2000, 2017), the doublet [Ca II] λλ7291, 7324 has not been detected in gaseous nebulae (Kingdon et al. 1995).

Taken together, we suggest that the narrow components ($\lesssim 270\,\mathrm{km}\,{{\rm{s}}}^{-1}$) of He I, [Ca II], [O I], and Ca II are also associated with the transient. Their widths might be consistent with the typical velocities of pre-shock CSM. The detection of these lines at >300 days after the SN explosion suggests that the circumstellar shell extends to $\gtrsim 2\times {10}^{16}\,\mathrm{cm}$ (∼1000 au) from the progenitor.²¹

Supernovae light curves are mainly powered by shock energy or radiative diffusion from a heating source. We first examine if the peak of SN2019dge is likely to be powered by the radioactive decay of ⁵⁶Ni→ ⁵⁶Co→ ⁵⁶Fe. With a peak luminosity of ${L}_{\mathrm{peak}}\approx 5\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and a rise time of ${t}_{\mathrm{peak}}\approx 2$–$4\,\mathrm{days}$, SN2019dge falls into the unshaded region of Kasen (2017, their Figure 1), where an unphysical condition of M_Ni > M_ej is required. Therefore, we rule out radioactivity as the power source for the fast rise of the light curve.

There have been clues for the early emission mechanism of SN2019dge,:

• 1.  
  The fast ${t}_{\mathrm{rise}}$ (Figure 3), initial high temperature (middle panel of Figure 4), blue color (Figure 5), and relatively fast color evolution of SN2019dge are reminiscent of shock cooling emission (SCE, Nakar & Piro 2014; Piro 2015; Piro et al. 2020).
• 2.  
  The color jump in g − r is observed 6–9 days after maximum (left panel of Figure 5). It is roughly at this phase that the change in bolometric luminosity decline rate transitions from 0.36 $\mathrm{mag}\,{\mathrm{days}}^{-1}$ to 0.11 $\mathrm{mag}\,{\mathrm{days}}^{-1}$ (upper panel of Figure 4). This supports the idea that the dominant power mechanisms before and after this transition are different.

Therefore, we model the early light curve as cooling emission from shock-heated extended material, which is located at the outer layers of the progenitor or outside of the progenitor. We use models presented by (Piro et al. 2020, hereafter P20) to constrain the mass, radius, and energy of the extended material (M_ext, R_ext, and E_ext, respectively). Compared with the SCE model introduced in Piro (2015), P20 does a better job of resolving the velocity gradient in the ejecta, and the velocity is higher near the surface of the material. Details of the model fitting to multiband observations are illustrated in Appendix B.2.

In Figure 12, the bolometric light curve measured in Section 3.1 is shown in black. We also show late-time r-band ν L_ν measurements in gray empty circles as a proxy of bolometric light-curve evolution. The dashed green line shows the best-fit model of ${M}_{\mathrm{ext}}={9.71}_{-0.27}^{+0.28}\times {10}^{-2}{M}_{\odot }$, ${R}_{\mathrm{ext}}={1.19}_{-0.05}^{+0.06}\times {10}^{13}$ cm (i.e., ${171.2}_{-7.7}^{+8.3}{R}_{\odot }$), ${E}_{\mathrm{ext}}=({5.30}_{-0.24}^{+0.25})\times {10}^{49}\,\mathrm{erg}$, and explosion epoch at phase ${t}_{\exp }=-2.91\pm 0.02$ days.

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Given the simple assumptions of the model, we expect the constraints on M_ext and ${R}_{\mathrm{ext}}$ to be only approximately accurate. We thus conclude that the early shock cooling emission was produced by an extended envelope with a mass of $\sim 0.1\,{M}_{\odot }$ located at a radius of ∼1.2 × 10¹³ cm (170 R_⊙).

Early-time low-velocity He II λ4686 emission (Section 3.2.1) has been detected in nearly 20 hydrogen-rich CCSNe and one hydrogen-poor SN, iPTF14gqr. This feature often fades away a few hours to a few days after the explosion (Yaron et al. 2017). The high ionization potential of this line requires high temperature or an ionizing flux, which might come from either shock breakout or CSM interaction (Gal-Yam et al. 2014; Smith et al. 2015). Because of the rapid decrease in T_bb at the three epochs of our early-time spectra and the similarity between SN2019dge and iPTF14gqr, we favor shock cooling emission as the origin of recombination helium lines. Therefore, we can use the luminosity of the He II λ4686 line to make an order-of-magnitude estimate on properties of the emission material, following the procedure given by Ofek et al. (2013b) and De et al. (2018c).

Assuming that the immediate CSM around the progenitor has a spherical wind-density profile of the form, ρ = K r⁻², where r is distance from the progenitor, $K\equiv \dot{M}/(4\pi {v}_{{\rm{w}}})$ is the wind-density parameter, v_w is the wind velocity, and $\dot{M}$ is the mass-loss rate. The integrated mass of the emitting material from r to r₁ is

$$\begin{eqnarray}&&{M}_{\mathrm{He}}={\int }_{r}^{{r}_{1}}4\pi {r}^{2}\rho (r){\rm{d}}r=4\pi K\beta r\end{eqnarray} \tag{ 1 }$$

where $\beta \equiv ({r}_{1}-r)/r$ is assumed to be of order unity.

We can relate the mass of the He II region to the He II λ4686 line luminosity using

$$\begin{eqnarray}&&{L}_{\lambda 4686}\approx \displaystyle \frac{{{An}}_{e}{M}_{\mathrm{He}}}{{m}_{\mathrm{He}}}.\end{eqnarray} \tag{ 2 }$$

Here

$$\begin{eqnarray}&&A=\displaystyle \frac{4\pi {j}_{\lambda 4686}}{{n}_{e}{n}_{{\mathrm{He}}^{++}}},\end{eqnarray} \tag{ 3 }$$

where j_λ4868 (in $\mathrm{erg}\,{\mathrm{cm}}^{-3}\,{{\rm{s}}}^{-1}\,{\mathrm{sr}}^{-1}$) is the emission coefficient for the λ4686 transition. m_He is mass of a helium nucleus, ${n}_{{\mathrm{He}}^{++}}$ is the number density of doubly ionized helium, and n_e is the number density of electrons.

Assuming a temperature of 10⁴ K, electron density of ${10}^{10}\,{\mathrm{cm}}^{-3}$, and Case B recombination, we get $A=1.32\,\times {10}^{-24}\,\mathrm{erg}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}$ (Storey & Hummer 1995). Using ${n}_{e}=2{n}_{{\mathrm{He}}^{++}}$ and the density profile, Equation (2) can be written as

$$\begin{eqnarray}&&{L}_{\lambda 4686}\approx \displaystyle \frac{8\pi A\beta }{{m}_{\mathrm{He}}^{2}}\displaystyle \frac{{K}^{2}}{r}.\end{eqnarray} \tag{ 4 }$$

The location of the emitting region can be constrained by requiring that the Thompson optical depth (τ) in the region must be small for the lines to escape. We require

$$\begin{eqnarray}&&\tau ={n}_{e}{\sigma }_{{\rm{T}}}{\int }_{r}^{{r}_{1}}{\rm{d}}r=\displaystyle \frac{2{\sigma }_{{\rm{T}}}K\beta }{{m}_{\mathrm{He}}r}\lesssim 1\end{eqnarray} \tag{ 5 }$$

Thus

$$\begin{eqnarray}&&{r}^{2}\gtrsim {\left(\displaystyle \frac{2{\sigma }_{{\rm{T}}}\beta }{{m}_{\mathrm{He}}}\right)}^{2}\displaystyle \frac{{L}_{\lambda 4686}{m}_{\mathrm{He}}^{2}r}{8\pi A\beta }\end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray}&&r\gtrsim {L}_{\lambda 4686}\displaystyle \frac{{\sigma }_{{\rm{T}}}^{2}\beta }{2\pi A}\end{eqnarray} \tag{ 6b }$$

The +0.4 days emission line flux is measured to be $F\,=(8.99\pm 0.71)\times {10}^{-16}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, corresponding to ${L}_{\lambda 4686}\,=9.0\times {10}^{38}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Hence, we get $r\gtrsim 4.8\times {10}^{13}\beta \,\mathrm{cm}$, $K\,\gtrsim 1.2\times {10}^{14}\,{\rm{g}}\,{\mathrm{cm}}^{-1}$, and ${M}_{\mathrm{He}}\gtrsim 3.7\times {10}^{-5}{\beta }^{2}\,{M}_{\odot }$. Adopting a wind velocity of ${v}_{{\rm{w}}}\approx 550\,\mathrm{km}\,{{\rm{s}}}^{-1}$ as measured from the He II FWHM, the mass-loss rate can be constrained to be $\dot{M}\gtrsim 1.1\times {10}^{-4}\,{M}_{\odot }{\mathrm{yr}}^{-1}$. Note that these estimates can be affected if the CSM cannot be well characterized by a spherically symmetric ρ(r)  ∝  r⁻² density profile, or if the emitting region was confined to a thin shell ($\beta \ll 1$). Our radio upper limits are not deep enough to place stringent constraints on the mass-loss parameters (see details in Appendix B.5).

After subtracting the SCE from the bolometric light curve, the remaining light curve has a peak luminosity of ${L}_{\mathrm{peak}}\approx 6\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and a rise time of t_peak ≈ 9 d. In the shaded region of Kasen (2017, Figure 1), this falls between the M_Ni = 0.1 M_ej and M_Ni = 0.01 M_ej lines, indicating that the remaining component can be powered by ⁵⁶Ni decay. Apart from this, the moderate ${t}_{\mathrm{decay}}$ of SN2019dge (bottom panel in Figure 3) is similar to that found in a few Ca-rich transients, and consistent with coming from radioactivity. Here we use two methods to estimate M_ej and M_Ni.

First of all, we use analytical models (Arnett 1982; Valenti et al. 2008; Wheeler et al. 2015) to constrain the nickel mass (M_Ni), a characteristic photon diffusion timescale (τ_m), and a characteristic γ-ray escape timescale (t₀). Details of the model fitting are given in Appendix B.3. The dotted blue line in Figure 12 shows the best-fit model of ${M}_{\mathrm{Ni}}={1.57}_{-0.03}^{+0.04}\times {10}^{-2}\ {M}_{\odot }$, ${\tau }_{{\rm{m}}}={6.53}_{-0.17}^{+0.18}$ days, and ${t}_{0}={23.86}_{-0.76}^{+0.78}$ days. Thus, using Equation (B6), the ejecta mass can be estimated to be

$$\begin{eqnarray}&&{M}_{\mathrm{ej}}=0.38\pm 0.02\left(\displaystyle \frac{{v}_{\mathrm{ej}}}{8000\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)\left(\displaystyle \frac{0.07\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}}{{\kappa }_{\mathrm{opt}}}\right)\,{M}_{\odot }\end{eqnarray} \tag{ 7 }$$

Here we adopt the mean opacity of SNe Ibc found by Taddia et al. (2018). In Section 3.2.2, the photospheric velocity of $\approx 6000\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is measured at phase Δt ∼ 12 days (i.e., ∼15 days post explosion). At that time, Figure 4 shows that R_bb stays roughly flat, indicating that a certain amount of ejecta in the outer layers should have a velocity greater than that measured from the He I absorption minimum. Therefore, we adopt the $\approx 8000\,\mathrm{km}\,{{\rm{s}}}^{-1}$ measured from early R_bb evolution (see Section 3.1.2) to be a more appropriate estimate for v_ej. The kinetic energy is then calculated to be

$$\begin{eqnarray}&&{E}_{\mathrm{kin}}=\displaystyle \frac{3}{10}{M}_{\mathrm{ej}}{v}_{\mathrm{ej}}^{2}=(1.44\pm 0.08)\times {10}^{50}\,\mathrm{erg}\end{eqnarray} \tag{ 8 }$$

Khatami & Kasen (2019, hereafter KK19) presented improved analytic relations (compared with the original Arnett 1982 model) between t_peak and L_peak. When t < 10 days, ${\varepsilon }_{\mathrm{Ni}}(t)\gg {\varepsilon }_{\mathrm{Co}}(t)$ (see Equations (B2) and (B3)), and hence we have an exponential heating function

$$\begin{eqnarray}&&{L}_{\mathrm{heat}}(t)={L}_{0}{e}^{-t/{\tau }_{\mathrm{Ni}}}\end{eqnarray} \tag{ 9 }$$

where ${L}_{0}={M}_{\mathrm{Ni}}\times {\epsilon }_{\mathrm{Ni}}$. In this case, KK19 (Equation (21)) shows that the relation between peak time and luminosity is

$$\begin{eqnarray}&&{L}_{\mathrm{peak}}=\displaystyle \frac{2{L}_{0}{\tau }_{\mathrm{Ni}}^{2}}{{\beta }^{2}{t}_{\mathrm{peak}}^{2}}\left[1-(1+\beta {t}_{\mathrm{peak}}/{\tau }_{\mathrm{Ni}}){e}^{-\beta {t}_{\mathrm{peak}}/{\tau }_{\mathrm{Ni}}}\right]\end{eqnarray} \tag{ 10 }$$

where β ∼ 4/3 gives a reasonable match to numerical simulations. With ${L}_{\mathrm{peak}}\approx 6\times {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and t_peak ≈ 9 days, we get an estimate of ${M}_{\mathrm{Ni}}\sim 0.017\,{M}_{\odot }$.

${M}_{\mathrm{ej}}$ can be estimated using Equation (23) of KK19:

$$\begin{eqnarray}&&\displaystyle \frac{{t}_{\mathrm{peak}}}{{t}_{{\rm{d}}}}=0.11\,\mathrm{ln}\left(1+\displaystyle \frac{9{\tau }_{\mathrm{Ni}}}{{t}_{{\rm{d}}}}\right)+0.36,\end{eqnarray} \tag{ 11 }$$

where t_d is the characteristic timescale without any numerical factors

$$\begin{eqnarray}&&{t}_{{\rm{d}}}={\left(\displaystyle \frac{{\kappa }_{\mathrm{opt}}{M}_{\mathrm{ej}}}{{v}_{\mathrm{ej}}c}\right)}^{1/2}.\end{eqnarray} \tag{ 12 }$$

We derive t_d ≈ 15.4 days, which implies

$$\begin{eqnarray}&&{M}_{\mathrm{ej}}\approx 0.33\left(\displaystyle \frac{{v}_{\mathrm{ej}}}{8000\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)\left(\displaystyle \frac{0.07\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}}{{\kappa }_{\mathrm{opt}}}\right)\,{M}_{\odot }\end{eqnarray} \tag{ 13 }$$

The kinetic energy of the ejecta is then ${E}_{\mathrm{kin}}\approx 1.3\times {10}^{50}\,\mathrm{erg}$.

In conclusion, the estimates derived from simplified model fitting and new analytic relations from KK19 are roughly the same. The ejecta mass (${M}_{\mathrm{ej}}\sim 0.3{M}_{\odot }$), the nickel mass (${M}_{\mathrm{Ni}}\sim 0.017{M}_{\odot }$), and the total kinetic energy (${E}_{\mathrm{kin}}\sim 1.3\,\times {10}^{50}\,\mathrm{erg}$) of SN2019dge are very small.

At early times, the cooling emission from shock-heated surrounding material of ${M}_{\mathrm{ext}}\sim 0.1\,{M}_{\odot }$ and ${R}_{\mathrm{ext}}\sim 1.2\,\times {10}^{13}\,\mathrm{cm}$ ($170\,{R}_{\odot }$) corroborates that the progenitor of SN2019dge is a star with an extended envelope. Indeed, stellar evolution models predict envelope radii of 10–200 R_⊙ for helium stars with zero-age helium core masses within 2.5–3.2 M_⊙ that have stripped all of the hydrogen-rich envelope (Woosley 2019; Laplace et al. 2020). Therefore, the early-time shock cooling light curve serves as strong evidence that SN2019dge is the explosion of a star with inflated radius (not a compact object).

The ⁵⁶Ni mass of $\sim 0.017\,{M}_{\odot }$ inferred from the radioactivity-powered decay is much greater than that produced in electron-capture SNe ($\sim {10}^{-3}\,{M}_{\odot }$, Moriya et al. 2014), whereas the ejecta velocity of $\approx 8000\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is larger than that expected in fallback SNe ($\sim 3000\,\mathrm{km}\,{{\rm{s}}}^{-1};$ Moriya et al. 2010). Therefore, we conclude that SN2019dge is associated with the class of Fe CCSNe.

As noted in the introduction, the majority of SNe Ibc, with M_ej in the range of 1–5 M_⊙, are believed to come from binary evolution. The small amount of ejecta mass seen in SN2019dge (M_ej ∼ 0.3 M_⊙) requires extreme stripping prior to the explosion in a binary system, which suggests an ultra-stripped progenitor (Tauris et al. 2013).

Compared with iPTF14gqr, in which the second peak of the light curve suggests ${M}_{\mathrm{ej}}\sim 0.2\,{M}_{\odot }$, SN2019dge has a higher ejecta mass. In particular, the helium-rich photospheric spectra indicate that SN2019dge has a greater amount of helium in the ejecta. He I emission lines are nonthermally excited by collisions with fast electrons, which result from Compton processes with γ-rays from ⁵⁶Ni decay (Dessart et al. 2012; Hachinger et al. 2012). On the other hand, the weak absorption strength in the He I P-Cygni profile (Figures 7 and 8) suggests that the helium envelope mass of SN2019dge is substantially lower than that in a canonical Type Ib SN (Fremling et al. 2018). While the stripping in SN2019dge is less extreme than for iPTF14gqr, the striking similarities between these two events indicate that they probably originate from similar channels.

The He II λ4686 flash-ionized emission comes from optically thin material located at ∼5 × 10¹³ cm (700 R_⊙). This is even larger than the expected orbital separation required for extreme stripping. Therefore, material at such a large radius might be ejected prior to the explosion, with a mass-loss timescale $t\sim 5\times {10}^{13}\,\mathrm{cm}/(500\,\mathrm{km}\,{{\rm{s}}}^{-1})\sim 10\,{\rm{d}}$. The inferred mass-loss rate of $\dot{M}\gtrsim {10}^{-4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ is much higher than that observed in Galactic Wolf–Rayet stars (Smith 2014). Additionally, the photospheric and late-time spectra of SN2019dge signify interaction with a helium-rich extended dense shell, which may also consist of gas originally ejected by the progenitor as a stellar wind or deposited by binary interaction. The high mass-loss rate and short ejection timescale can be achieved in the final stages of stellar evolution by several mechanisms: 1. a powerful outflow driven by super-Eddington wave energy deposition during the last few years before explosion (Quataert & Shiode 2012); 2. explosive mass ejection due to violent silicon flashes within a few weeks before the explosion of low-mass helium stars (Woosley 2019); 3. nonconservative case BB²² mass transfer in binary evolution of ultra-stripped stars (Tauris et al. 2015).

Here we discuss possible evolution paths of SN2019dge's progenitor.

We first consider the scenario in which SN2019dge comes from a progenitor more massive than $\sim 15\,{M}_{\odot }$ in a binary system that loses its mass in case B mass transfer. Yoon et al. (2010) showed that subsequent wind mass loss is weak at subsolar metallicity of Z ≈ 0.004 (similar to the Z ≈ 0.005 calculated in Appendix C), such that the final mass of the primary at the time of core-collapse will be higher than 3.8 M_⊙. This will lead to M_ej ≳  2.3 M_⊙, assuming that the explosion forms a neutron star of 1.5 ${M}_{\odot }$. This inferred ejecta mass is much higher than that observed (${M}_{\mathrm{ej}}\sim 0.3\,{M}_{\odot }$), so this scenario is not favored.

We next consider the possibility that the primary has an initial mass ${M}_{1}\lesssim 15\,{M}_{\odot }$. In many binary configurations, the primary will be stripped of its hydrogen envelope during case B mass transfer, which could be either stable or unstable. The fairly low-mass ($M\lesssim 4\,{M}_{\odot }$) primary helium star will expand after core helium depletion, followed by case BB mass transfer, which strips much of its helium envelope. Model 3 of Yoon et al. (2010) and a few of the models from Table 1 of Tauris et al. (2015) are examples of systems whose evolution may be similar to the progenitor of SN2019dge, whose initial mass as a helium star was likely M_He ∼ 3 M_⊙. Zapartas et al. (2017) performed population synthesis simulations, showing that for the pre-SN helium star to be stripped to a final mass $\lesssim 2\,{M}_{\odot }$, a relatively wide range of companion mass is possible. So, the nature of the companion star remains unclear, and both main-sequence stars and compact objects are possible. What is clear is that case BB mass transfer was stable in this system, because the post-common envelope orbital separation of unstable case BB mass transfer would be much smaller than the inferred radius of the progenitor of SN2019dge (Laplace et al. 2020).

The final mass of the helium envelope depends on the initial mass of the helium star and the orbital period of the compact binary. To reconcile with the ejecta mass observed in SN2019dge, we expect a small final envelope mass (≲0.3 M_⊙) but large enough for optical helium features to be observed in the SN explosion ($\gtrsim 0.06\,{M}_{\odot }$, Hachinger et al. 2012). This can be achieved in a system in which the progenitor is a helium star in a compact binary with ${P}_{\mathrm{orb}}\gtrsim 0.2$ days at the start of the helium burning phase (Tauris et al. 2015). However, the large inferred radius of the progenitor requires an orbital separation a  ≳  200 R_⊙, implying an orbital period ${P}_{\mathrm{orb}}\gtrsim 100\,{\rm{d}}$ at the time of explosion, depending on the companion mass. We note that the large inferred radius of the progenitor is near the maximum photospheric radius reached by low-mass M  ≲  3 M_⊙ helium stars (Kleiser et al. 2018a; Woosley 2019; Laplace et al. 2020), potentially challenging the assertion that case BB mass transfer has occurred. However, exploding stars without case BB mass loss retain significantly larger ejecta masses than our constraints for SN2019dge (Tauris et al. 2015; Woosley 2019), producing light curves that rise and fade too slowly (Kleiser et al. 2018a).

In addition to SN2019dge and iPTF14gqr, we search the literature for other subluminous fast-evolving hydrogen-poor SNe whose light curves can potentially be well fitted by an early-time SCE component and a radioactivity-powered second peak with small M_ej. We recover iPTFF16hgs (De et al. 2018b) and SN2018lqo (De et al. 2020a) as ultra-stripped SN candidates. Here we apply our modeling approach described in Sections 4.1 and 4.3 to iPTF16hgs and SN2018lqo to distill the physical parameters of these two events. We show the results in Table 3. The ejecta masses of iPTF16hgs and SN2018lqo are greater than that in SN2019dge and iPTF14gqr by a factor of ∼3, and fall inside the range of M_ej expected in explosions of a helium star orbiting a compact object, but are at the upper side of the boundaries (Tauris et al. 2015).

Table 3.  Model Parameters for Hydrogen-poor Subluminous Fast-evolving SNe where the Bolometric Light Curve Can Be Fitted with a Shock Cooling Powered Component and a Radioactivity-powered Component

Name	vej	${\tau }_{{\rm{m}},{\rm{A}}82}$	${M}_{\mathrm{ej},{\rm{A}}82}$	${M}_{\mathrm{Ni},{\rm{A}}82}$	${M}_{\mathrm{ej},\mathrm{KK}19}$	${M}_{\mathrm{Ni},\mathrm{KK}19}$	${R}_{\mathrm{ext}}$	${M}_{\mathrm{ext}}$
	($\mathrm{km}\,{{\rm{s}}}^{-1}$)	(d)	(M⊙)	(10−2 M⊙)	(M⊙)	(10−2 M⊙)	(1013 cm)	(10−2 M⊙)
iPTF14gqr	10,000	${4.63}_{-0.20}^{+0.19}$	${0.24}_{-0.02}^{+0.02}$	${8.14}_{-0.15}^{+0.14}$	0.24	8.57	${6.09}_{-3.18}^{+8.73}$	${2.59}_{-0.34}^{+0.46}$
iPTF16hgs	10,000	${12.32}_{-0.97}^{+099}$	${1.68}_{-0.25}^{+0.28}$	${2.51}_{-0.22}^{+0.20}$	0.83	2.30	${2.45}_{-1.80}^{+14.08}$	${9.27}_{-2.48}^{+3.40}$
SN2018lqo	8250	${9.71}_{-0.37}^{+0.52}$	${0.86}_{-0.06}^{+0.09}$	${2.75}_{-0.07}^{+0.09}$	0.63	3.08	${0.77}_{-0.51}^{+3.63}$	${7.30}_{-4.06}^{+8.14}$
SN2019dge	8000	${6.53}_{-0.17}^{+0.18}$	${0.38}_{-0.02}^{+0.02}$	${1.57}_{-0.03}^{+0.04}$	0.33	1.69	${1.19}_{-0.05}^{+0.06}$	${9.71}_{-0.27}^{+0.28}$

Note. Model parameters with the subscripts of "A82" and "KK19" refer to radioactivity-powered ligth curve models of Arnett (1982) (see Appendix B.3) and Khatami & Kasen (2019), respectively. Reference: iPTF14gqr (De et al. 2018c), iPTF16hgs (De et al. 2018b), SN2018lqo (De et al. 2020a), SN2019dge (this work).

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A full discussion of the progenitors of iPTF16hgs and SN2018lqo is beyond the scope of this paper. Here we refer to a recent study conducted by De et al. (2020a), which classified these two objects into the "green Ca-Ib" subclass in the Ca-rich SNe category. This class of objects is spectroscopically similar to SNe Ib at maximum light, and does not exhibit line-blanketed continua at ∼3500–5500 Å. De et al. (2020a) proposed that pure helium-shell detonations or deflagrations can explain their photometric and spectroscopic properties. Although it has been suggested that the existence of early first peak can distinguish ultra-stripped SNe from other Ca-rich transients arising from helium dotonation on the surface of white dwarfs (Nakaoka et al. 2020), the early-time peak might also be caused by radioactive decay from short-lived isotopes in the outermost ejecta (De et al. 2020a). For example, the double-peaked Ca-rich transient SN2018lqo occurs in an elliptical galaxy, which is not expected to be the host for ultra-stripped SNe.

Although we include only hydrogen-poor events in this comparison, we note that SN2019ehk, a Ca-rich transient that exhibits flash-ionized hydrogen in its early-time spectra (Jacobson-Galán et al. 2020) as well as possible hydrogen photospheric features (De et al. 2020a), has also been suggested to be an ultra-stripped CCSN (Nakaoka et al. 2020). If both iPTF16hgs and SN2019ehk are bona fide ultra-stripped SNe, there might exist a continuum of ejecta mass, from normal stripped envelope SNe ($1\,{M}_{\odot }\lesssim {M}_{\mathrm{ej}}\lesssim 5\,{M}_{\odot }$), to a higher degree of stripping in the progenitors of SN2019ehk and iPTF16hgs ($0.5\,{M}_{\odot }\lesssim {M}_{\mathrm{ej}}\lesssim 1\,{M}_{\odot }$), to a more extreme degree of stripping seen in SN2019dge (${M}_{\mathrm{ej}}\sim 0.3\,{M}_{\odot }$), to the most extreme stripping seen in iPTF14gqr (M_ej ∼ 0.2 M_⊙). The remaining amount of helium, the companion mass, and final orbital separation might become smaller along this sequence.

6.1.1. Using the BTS Sample

SN2019dge was followed up as a part of the ZTF Bright Transient Survey (BTS, Fremling et al. 2020), which aims to spectroscopically classify all extragalactic transients in ZTF brighter than 18.5 mag at peak. Because BTS reads from only the ZTF public alert stream (highlighted with a greater marker size in Figure 13), SN2019dge peaks between 18.5 and 19.0 mag in the BTS sample. Thanks to the relatively high spectroscopic completeness (≈89%) at the brightness limit of 19.0 mag, we can directly place constraints on the rates of 19dge-like ultra-stripped SNe using the BTS sample.

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SN2019dge peaked at an absolute magnitude of −16.44 mag in g band. At the BTS peak brightness limit of 19.0 mag, objects similar to SN2019dge would be detectable out to 123 Mpc. Thus, taking only the local 123 Mpc volume within redshift of z = 0.028, we compare the number of CCSNe brighter than 19.0 mag at peak that were found in the BTS experiment in its first 12 months of operations (between 2018 June 1 and 2019 June 1). In this time period, BTS classified a total of 116 CCSNe in this volume. As such, the detection of one object in this sample constrains the rate of ultra-stripped SNe to be ∼0.86% of the CCSNe rate brighter than M = −16.44 mag in this volume.

Taking the observed luminosity function of CCSNe in the local universe (Li et al. 2011b), we find that ≈50% of CCSNe are fainter than M = −16.44 mag. The luminosity function corrected rate of 19dge-like events is then ∼0.43% of the local CCSNe rate. The inferred rate is consistent with that estimated in Tauris et al. (2015), but higher than that inferred for iPTF14gqr-like events (Hijikawa et al. 2019). Adopting the CCSNe volumetric rate of $0.7\times {10}^{-4}\,{\mathrm{Mpc}}^{-3}\,{\mathrm{yr}}^{-1}$ (Li et al. 2011a), the volumetric rate of 19dge-like ultra-stripped SNe rate is ∼300 ${\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$. This rate estimation is only a lower limit, because the fast photometric evolution of objects similar to SN2019dge can be easily missed due to the slower 3 days' cadence of the ZTF public survey.

6.1.2. Using the CLU sample

We utilize simsurvey (Feindt et al. 2019), a python package designed for assessing the rates of transient discovery in surveys like ZTF. To simulate the expected yield of a specific type of transient given a volumetric rate, simsurvey requires three inputs: (1) A survey schedule. We use the actual ZTF observing history in g and r band between 2018 June 1 and 2019 June 1 in any of the public or collaboration surveys as the input survey plan. (2) A transient model. We construct a light-curve template of SN2019dge (see details in Appendix B.4). Using the template, we generate a TimeSeriesSource model in the sncosmo package (Barbary et al. 2016). (3) A function to sample the transient model parameters. Transients are injected out to a redshift of z = 0.044, because objects further out are not expected to peak brighter than 20.0 mag.

We examine the expected number of detected 19dge-like SNe for a range of input rates. For each input rate, we perform 300 simulations of the ZTF observing plan. To select transient candidates that would have passed the selection criteria of the BTS or CLU experiment and been flagged as an object with photometric properties consistent with being a 19dge-like ultra-stripped SN, we apply cuts on the simulated light curves as described below.

For the BTS filter, we use only public survey pointings, and reject SNe at low Galactic latitudes ($| b| \leqslant 7^\circ$) to be consistent with the BTS experiment (Fremling et al. 2020). In either the g- or r-band light curve, we identify peak light as the brightest detection in the simulated light curve, and require the following:

• 1.  
  peak magnitude $\lt 19.0$ mag
• 2.  
  within 4.1 days before peak, there must be at least one detection or one upper limit deeper than 1.5 mag below peak
• 3.  
  within 15 days after peak, there must be at least three detections, and the measured decline rate must be greater than $0.07\,\mathrm{mag}\,{\mathrm{days}}^{-1}$.

Criterion (ii) is set to require that the fast rise of the light curve can be recognized from the observation. This is essential because if we discover SN2019dge only at the radioactive tail, we will probably classify it as a low-velocity SN Ib. Criterion (iii) is made to ensure that the rapid decline of the light curve can be captured, such that the small ejecta mass can be inferred.

For the CLU filter, we use all ZTF pointings, and require that in either the g- or r-band light curve:

• 1.  
  peak magnitude < 20.0 mag
• 2.  
  the light curve must satisfy at least one of the following criteria: (1) within 4.1 days before peak, there must be at least one detection or one upper limit deeper than 1.5 mag below peak; (2) within 2.5 days before peak, there must be at least one detection deeper than 0.75 mag below peak
• 3.  
  same as criterion (iii) applied in the BTS filter.

We apply the above criteria to the actual observations of CCSNe in the BTS and CLU sample. We identify one other SN—ZTF18abwkrbl (SN2018gjx)—that passes our criteria. However, ZTF18abwkrbl is a SN IIb that clearly shows hydrogen in the spectra, and can therefore be excluded as an ultra-stripped SN candidate (Tauris et al. 2015).

In Figure 14, we show the number of transients that pass our selection criteria as a function of the input volumetric rate. The solid line and shaded region indicate the mean and 68% credible region of the 300 simulations. In the actual BTS experiment, there was only one detected ultra-stripped SN. Therefore, we consider the range of volumetric rate where one falls within the shaded red region as a constraint on the rate of 19dge-like ultra-stripped SNe. This gives R_19dge in the range of 1400–28,000 ${\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$.

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Using the fact that there were zero ultra-stripped SN detected in the actual CLU experiment, the gray shaded region in Figure 14 might suggest ${R}_{19\mathrm{dge}}\lesssim 4500\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$. However, this upper limit needs to be corrected for offset distribution and galaxy catalog incompleteness. First of all, as discussed in De et al. (2020a), the CLU experiment is restricted to transients coincident within 100'' of the host-galaxy nuclei. SN2019dge and iPTF14gqr are $0\buildrel{\prime\prime}\over{.} 5$ and 24'' from their host galaxies (all within 100''). Although a large sample of ultra-stripped SNe is needed to examine the host offset distribution of this class of objects, the fact that they arise from massive binary evolution suggest that the correction due to this factor should be small. Second, the incompleteness of the input galaxy catalog possibly leads to an underestimation of ultra-stripped SNe rate by a factor of 55%–80%, as indicated by the RCF. We adjust for such an incompleteness by increasing the upper limit from 4500 to 8200 ${\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$.

Combining results from the BTS and CLU experiments, we derive a 19dge-like ultra-stripped SNe rate of 1400–8200 ${\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$, corresponding to 2%–12% of CCSNe rate.

Given the low mass of ultra-stripped progenitors, we expect to see SCE from the inflated pre-explosion star, as has been clearly seen in the case of iPTF14gqr and SN2019dge in the fast early-time evolution and blue colors of the optical light curve. As is shown by Nakar & Piro (2014, Figure 2), rise time of the SCE is determined by mass of the extended material M_ext, while the peak luminosity is mainly modulated by R_ext. We demonstrate this dependence in Figure 15. We simulate SCE with the P20 model by varying M_ext and R_ext, and at the same time setting ${E}_{\mathrm{ext}}=5.30\times {10}^{49}\,\mathrm{erg}$ (the value found in SN2019dge).

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In the upper panel, ${t}_{\mathrm{rise}}$ is defined in the same way as in Section 3.1.1 (rise time from half-max to max). The rising part of the cooling light curve can be captured by a three-day, two-day, and one-day cadence optical survey at ${M}_{\mathrm{ext}}\gtrsim 0.14\,{M}_{\odot }$, $\gtrsim 0.07\,{M}_{\odot }$, and $\gtrsim 0.03\,{M}_{\odot }$, respectively. Transients with ${M}_{\mathrm{ext}}\gtrsim 0.15\,{M}_{\odot }$ will not pass our selection criteria in Section 6.2. In the bottom panel, we show the expected absolute luminosity at peak of the g-band cooling light curve. As is readily shown, for ultra-stripped progenitors with an extended radius $\lesssim {10}^{13}\,\mathrm{cm}$, a survey like ZTF is only sensitive to objects in the local universe (≲100–150 Mpc) for the subsequent evolution of the light curve to be well characterized. Taken together, we conclude that our estimation of the ultra-stripped SNe rate does not include ultra-stripped progenitors with M_ext ≳  0.15 M_⊙ or ${R}_{\mathrm{ext}}\lesssim {10}^{13}\,\mathrm{cm}$.

If the companion of the pre-explosion helium star is a low-mass main-sequence star, a white dwarf, or a black hole, SN2019dge will not be the progenitor of a double neutron star system, and thus the inferred ${R}_{19\mathrm{dge}}$ is not connected with R_DNS. Even in the case that the companion is a neutron star, if the forming DNS binary has orbital periods more than ∼1 days, it will not merge within the age of the universe (Tauris et al. 2015). Therefore, the above estimation of the ultra-stripped SNe rate should provide an upper limit to the local coalescence rate density (${R}_{\mathrm{DNS}}$) of double neutron stars not formed via dynamical capture in a globular cluster (East & Pretorius 2012; Andrews & Mandel 2019).

To model the multiband light curve with a blackbody function, we utilized the Monte Carlo Markov Chain (MCMC) simulations with emcee (Foreman-Mackey et al. 2013). We test the performance of three types of model priors for the blackbody radius (R_bb) and temperature (T_bb): (i) T_bb and R_bb are uniformly distributed in the range of [10³, 10⁷] K and [10, 10⁶] R_⊙, respectively (ii) the two parameters are logarithmically and uniformly distributed in the same ranges (iii) the two parameters follow Jeffreys prior (Jeffreys 1946) in the same ranges.

Within the ensemble, we use 100 walkers, each of which is run until convergence or 100,000 steps, whichever comes first. The test for convergence follows steps outlined in Yao et al. (2019) and Miller et al. (2020). We adopt the 68% credible region (i.e., 16th and 84th percentiles of posterior probability distributions) as the model uncertainties quoted in Table B1.

Table B1.  Physical Evolution of SN2019dge from Blackbody Fits

${\rm{\Delta }}t$	$L({10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1})$	R (${10}^{3}\,{R}_{\odot }$)	T (103 K)
−2.74	${2.98}_{-1.41}^{+578.84}$	${1.45}_{-1.14}^{+1.14}$	${14.21}_{-5.12}^{+101.35}$
−1.72	${44.63}_{-15.84}^{+43.85}$	${2.20}_{-0.46}^{+0.44}$	${22.75}_{-4.14}^{+7.60}$
−0.69	${26.96}_{-0.87}^{+0.92}$	${3.50}_{-0.07}^{+0.07}$	${15.90}_{-0.27}^{+0.28}$
0.24	${22.87}_{-0.61}^{+0.62}$	${4.46}_{-0.08}^{+0.08}$	${13.51}_{-0.21}^{+0.21}$
1.09	${17.50}_{-0.37}^{+0.40}$	${4.89}_{-0.10}^{+0.10}$	${12.07}_{-0.18}^{+0.19}$
3.26	${8.34}_{-0.20}^{+0.22}$	${7.26}_{-0.44}^{+0.46}$	${8.23}_{-0.28}^{+0.31}$
4.24	${7.29}_{-0.22}^{+0.26}$	${7.38}_{-0.83}^{+0.88}$	${7.88}_{-0.46}^{+0.54}$
5.25	${6.10}_{-0.15}^{+0.16}$	${8.77}_{-0.73}^{+0.78}$	${6.92}_{-0.30}^{+0.33}$
6.83	${6.18}_{-0.46}^{+0.63}$	${7.09}_{-0.92}^{+1.01}$	${7.72}_{-0.62}^{+0.75}$
7.98	${5.29}_{-0.21}^{+0.24}$	${8.00}_{-0.70}^{+0.75}$	${6.99}_{-0.35}^{+0.39}$
8.98	${5.49}_{-0.36}^{+0.44}$	${6.79}_{-0.85}^{+0.94}$	${7.66}_{-0.56}^{+0.65}$
10.05	${4.55}_{-0.27}^{+0.37}$	${7.42}_{-0.96}^{+1.03}$	${6.99}_{-0.52}^{+0.63}$
10.92	${4.49}_{-0.44}^{+0.64}$	${6.63}_{-1.06}^{+1.25}$	${7.37}_{-0.75}^{+0.93}$
11.89	${4.04}_{-0.21}^{+0.24}$	${7.42}_{-0.75}^{+0.83}$	${6.79}_{-0.40}^{+0.45}$
13.06	${3.34}_{-0.10}^{+0.10}$	${7.52}_{-0.66}^{+0.69}$	${6.43}_{-0.29}^{+0.32}$
14.05	${3.08}_{-0.09}^{+0.10}$	${7.05}_{-0.59}^{+0.63}$	${6.51}_{-0.29}^{+0.32}$
14.97	${2.82}_{-0.12}^{+0.15}$	${7.30}_{-1.06}^{+1.21}$	${6.25}_{-0.49}^{+0.57}$
16.09	${2.45}_{-0.09}^{+0.09}$	${6.17}_{-0.58}^{+0.59}$	${6.56}_{-0.29}^{+0.33}$
23.96	${1.02}_{-0.06}^{+0.07}$	${5.47}_{-1.12}^{+1.23}$	${5.59}_{-0.54}^{+0.74}$
27.00	${0.72}_{-0.08}^{+0.08}$	${4.08}_{-1.01}^{+1.26}$	${5.93}_{-0.70}^{+0.91}$
27.98	${0.78}_{-0.06}^{+0.07}$	${5.73}_{-1.29}^{+1.68}$	${5.10}_{-0.57}^{+0.68}$
29.23	${0.83}_{-0.09}^{+0.11}$	${4.76}_{-1.56}^{+2.47}$	${5.64}_{-0.93}^{+1.23}$
33.24	${1.09}_{-0.11}^{+0.16}$	${6.99}_{-1.74}^{+2.47}$	${5.02}_{-0.56}^{+0.65}$

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We examine the fitting results under different choices of priors in Figure B1, which shows the posterior distribution of T_bb using data obtained on April 7 (top panels) and April 9 (bottom panels). Many early stages of SN evolution feature extremely high temperatures. At an epoch when both UV and optical data are available (April 9), the posterior does not depend on the particular choice of prior, and the model parameter can thus be well constrained. However, at our first detection epoch, for which only optical data are available (April 7), the posterior strongly depends on the prior. For a linearly flat prior, high numbers receive a lot of "weight," making the "multi-peaks" shape posterior in the upper-left panel of Figure B1. Log prior and Jeffreys prior generally give similar results. In this study, we adopt results using log prior. However, we note that the choice of prior does not affect final estimates of maximum luminosity, or the model fits for shock cooling and ⁵⁶Ni decay.

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In Figure B2 we show the photometry interpolated onto common epochs, and fit to a blackbody function to derive the photospheric evolution. The resulting evolution in bolometric lumonosity, photospheric radius, and effective temperatures is listed in Table B1.

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We use the P20 analytical expression for the shape of the early-time light curve in terms of ${M}_{\mathrm{ext}}$, R_ext, and E_ext. Following P20, we adopt typical values of the density structure n ≈ 10 and δ ≈ 1.1

We fix $\kappa \approx 0.2\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ as appropriate for a hydrogen-deficient ionized gas, and assign wide flat priors for all model parameters, as summarized in Table B2. We include only observations up to Δt = 2 d in the fitting. We found that this particular choice of Δt − 2 days instead of 1 days or 3 days in general does not affect the final inference for the model parameters. Figure B3 shows the corner plot of logR_ext, $\mathrm{log}{M}_{\mathrm{ext}}$, t_exp, and ${E}_{\mathrm{ext},49}$.

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Table B2.  Shock Cooling Model Parameters θ and their Priors

θ	Description	Prior
$\mathrm{log}{R}_{\mathrm{ext}}$	log10 of extented material radius in cm	${ \mathcal U }(-5,25)$
$\mathrm{log}{M}_{\mathrm{ext}}$	log10 of extented material mass in ${M}_{\odot }$	${ \mathcal U }(-4,0)$
texp	explosion epoch in MJD relative to 58583.2	${ \mathcal U }(-8,-2.76)$
${E}_{\mathrm{ext},49}$	${E}_{\mathrm{ext}}$ divided by ${10}^{49}\,\mathrm{erg}$	${ \mathcal U }(0.1,100)$

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The maximum a posteriori model is visualized by solid lines in Figure B4 color-coded in different filters. Nevertheless, the early evolution of the light curve is well captured by this model.

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For ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe decay powered explosions, the energy deposition rate is

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{rad}}={\varepsilon }_{\mathrm{Ni},\gamma }(t)+{\varepsilon }_{\mathrm{Co},\gamma }(t)\end{eqnarray} \tag{ B1 }$$

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{Ni},\gamma }(t)={\epsilon }_{\mathrm{Ni}}{e}^{-t/{\tau }_{\mathrm{Ni}}}\end{eqnarray} \tag{ B2 }$$

$$\begin{eqnarray}&&{\varepsilon }_{\mathrm{Co},\gamma }(t)={\epsilon }_{\mathrm{Co}}\left({e}^{-t/{\tau }_{\mathrm{Co}}}-{e}^{-t/{\tau }_{\mathrm{Ni}}}\right)\end{eqnarray} \tag{ B3 }$$

where ${\epsilon }_{\mathrm{Ni}}=3.90\times {10}^{10}\,\mathrm{erg}\,{{\rm{g}}}^{-1}\,{{\rm{s}}}^{-1}$, ${\epsilon }_{\mathrm{Co}}=6.78\times {10}^{9}\,\mathrm{erg}\,{{\rm{g}}}^{-1}\,{{\rm{s}}}^{-1}$, ${\tau }_{\mathrm{Ni}}=8.8$ d, and τ_Co = 111.3 d are the decay lifetimes of ${}^{56}\mathrm{Ni}$ and ${}^{56}\mathrm{Co}$ (Nadyozhin 1994). The effective heating rate is modified by the probability of thermalization, and thus ${\varepsilon }_{\mathrm{heat}}\leqslant {\varepsilon }_{\mathrm{rad}}$.

The bolometric light curve can be generally divided into the photospheric phase and the nebular phase. The photospheric phase can be modeled using equations given in Valenti et al. (2008, Appendix A), with modifications given by Lyman et al. (2016, Equation (3)),

$$\begin{eqnarray}&&\begin{array}{l}{L}_{\mathrm{phot}}(t)={M}_{\mathrm{Ni}}{{\rm{e}}}^{-{x}^{2}}\\ \times \ \left[({\epsilon }_{\mathrm{Ni}}-{\epsilon }_{\mathrm{Co}}){\displaystyle \int }_{0}^{x}(2z{{\rm{e}}}^{-2{zy}+{z}^{2}}){\rm{d}}z\right.\\ \left.+{\epsilon }_{\mathrm{Co}}{\displaystyle \int }_{0}^{x}(2z{{\rm{e}}}^{-2{zy}+2{zs}+{z}^{2}}){\rm{d}}z\right]\end{array}\end{eqnarray} \tag{ B4 }$$

where $x=t/{\tau }_{{\rm{m}}}$, $y={\tau }_{{\rm{m}}}/(2{\tau }_{\mathrm{Ni}})$,

$$\begin{eqnarray}&&s=\displaystyle \frac{{\tau }_{{\rm{m}}}({\tau }_{\mathrm{Co}}-{\tau }_{\mathrm{Ni}})}{2{\tau }_{\mathrm{Co}}{\tau }_{\mathrm{Ni}}},\end{eqnarray} \tag{ B5 }$$

$$\begin{eqnarray}&&{\tau }_{{\rm{m}}}={\left(\displaystyle \frac{2{\kappa }_{\mathrm{opt}}{M}_{\mathrm{ej}}}{13.8{{cv}}_{\mathrm{phot}}}\right)}^{1/2}\end{eqnarray} \tag{ B6 }$$

In the nebular phase the SN ejecta becomes optically thin, such that the delay between the energy deposition from radioactivity and the optical radiation becomes shorter. The bolometric luminosity is then equal to the rate of energy deposition: L_neb(t) = Q(t). At any given time, the energy deposition rate Q(t) is Wheeler et al. (2015; Wygoda et al. 2019):

$$\begin{eqnarray}&&Q(t)\approx {Q}_{\gamma }(t)\left(1-{e}^{-{\left({t}_{0}/t\right)}^{2}}\right)\end{eqnarray} \tag{ B7 }$$

where ${Q}_{\gamma }(t)={M}_{\mathrm{Ni}}{\varepsilon }_{\mathrm{rad}}$ is the energy release rate of gamma-rays, and t₀ is the time at which the ejecta becomes optically thin to gamma-rays. Here the difference between the energy deposition rate of gamma-rays and positrons is neglected.

To fit the shock cooling subtracted bolometric light curve with a simple radioactive decay model, we do not divide the data into photospheric phase and nebular phase, but instead adopt the following formula for the whole light curve:

$$\begin{eqnarray}&&L(t)={L}_{\mathrm{phot}}(t)\left(1-{e}^{-{\left({t}_{0}/t\right)}^{2}}\right)\end{eqnarray} \tag{ B8 }$$

Priors of the model parameters are summarized in Table B3, and Figure B5 shows the corner plot of τ_m, logM_Ni, and t₀.

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Table B3.  ⁵⁶Ni Decay Model Parameters θ and Their Priors

θ	Description	Prior
${\tau }_{{\rm{m}}}$	characteristic photon diffusion time in day	${ \mathcal U }(1,20)$
$\mathrm{log}{M}_{\mathrm{Ni}}$	log10 of nickel mass in ${M}_{\odot }$	${ \mathcal U }(-4,0)$
t0	characteristic γ-ray escape time in day	${ \mathcal U }(20,100)$

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To construct a template for SN2019dge in the ZTF g and r filters, we model the observed light curve by a Gaussian process. We denote the measurements as $({\bf{x}},{\bf{y}})$, where ${\bf{x}}$ is $\mathrm{MJD}-58583.2$, and y is flux calculated as ${10}^{-0.4m}\times {10}^{8}$ (m is magnitude). We choose a kernel in the form of Matern covariance function (Rasmussen 2003, Equation (4.17):

$$\begin{eqnarray}&&{k}_{3/2}(x,{x}^{{\prime} })=A\left(1+\displaystyle \frac{\sqrt{3}r}{l}\right)\exp \left(-\displaystyle \frac{\sqrt{3}r}{l}\right)\end{eqnarray} \tag{ B9 }$$

where $r=| x-{x}^{{\prime} }|$. A and r in Equation (B9) are chosen to minimize the negative log likelihood function (see, e.g., Equation (2.43) of Rasmussen 2003).

We perform the fit from x = −10 d to x = +40 d, and the obtained templates are shown in Figure B6.

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Radio emission in SNe is produced by shock-accelerated electrons in the circumstellar material as they gyrate in the post-shock magnetic field when the shock freely expands. Should the CSM be formed by a pre-SN stellar wind, the radio synchrotron radiation can be used to probe the pre-explosion mass loss (Chevalier 1982). High-frequency (ν > 90 GHz) bright ($\nu {L}_{\nu }\gtrsim {10}^{40}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$) radio sources are often found to be associated with gamma-ray bursts (GRBs), tidal disruption events, and relativistic transients (see Figure 6 of Ho et al. 2019b). Among normal SNe Ibc, moderate submillimeter luminosity at $\sim 5\times {10}^{37}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ has been observed in SN1993J (Weiler et al. 2007) and SN2011dh (Horesh et al. 2013).

Our SMA observations constrain the submillimeter luminosity of SN2019dge to $\nu {L}_{\nu ,230\mathrm{GHz}}\lt 5.3\times {10}^{39}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and $\nu {L}_{\nu ,345\mathrm{GHz}}\lt 3.0\times {10}^{40}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. We place these upper limits in physical context using the synchrotron self-absorption model given by Chevalier (1998). The expected radio luminosities are computed at 230 and 345 GHz for two types of circumstellar environments—one with a wind density with the same parameterization as that adopted in Section 4.2, and the other with a constant-density environment (ρ = constant).

Adopting the explosion epoch found in Section 4.1, our SMA observations were obtained at 2.75 days after explosion. Given the early time of these observations, we consider constant shock velocities at 0.1–0.25 c, as found to be typical in SNe Ibc (Wellons et al. 2012). We assume an electron energy power-law index of p = 3, a volume-filling factor f = 0.5, and that the electrons and magnetic field in the post-shock region share constant fractions of the post-shock energy density, i.e., _e = _B = 0.1.

The expected radio luminosity predicted by the Chevalier model in the two environments at 230 GHz are shown in Figure B7 by the color maps, and the black contours indicate our 3σ limits. As can be seen, only small regions have expected luminosity higher than the 3σ limits (indicated by the hatched regions), and thus our observations are not deep enough to provide stringent constrains on the circumstellar properties. Compared with 230 GHz, the parameter space is even more poorly constrained at 350 GHz and are thus not shown.

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