A likely inverse-Compton emission from the Type IIb SN 2013df.

The inverse-Compton X-ray emission model for supernovae has been well established to explain the X-ray properties of many supernovae for over 30 years. However, no observational case has yet been found to connect the X-rays with the optical lights as they should be. Here, we report the discovery of a hard X-ray source that is associated with a Type II-b supernova. Simultaneous emission enhancements have been found in both the X-ray and optical light curves twenty days after the supernova explosion. While the enhanced X-rays are likely dominated by inverse-Compton scatterings of the supernova's lights from the Type II-b secondary peak, we propose a scenario of a high-speed supernova ejecta colliding with a low-density pre-supernova stellar wind that produces an optically thin and high-temperature electron gas for the Comptonization. The inferred stellar wind mass-loss rate is consistent with that of the supernova progenitor candidate as a yellow supergiant detected by the Hubble Space Telescope, providing an independent proof for the progenitor. This is also new evidence of the inverse-Compton emission during the early phase of a supernova.

Supernova (SN) explosions produce strong X-ray emissions by the interactions between the fast outward-moving ejecta materials and the almost stationary circumstellar medium (CSM) deposited by the progenitors’ stellar wind. The CSM would be heated up to T ≥ 109 K by the forward shock (often termed as circumstellar shock) in which free-free cooling and/or inverse-Compton (IC) cooling are crucial to radiate the thermal energy out in X-rays within the first months12. For electron temperatures of the shocked CSM, T ≥ 3 × 109, the effect of Comptonization on the X-ray emission becomes more important leading a domination over the free-free emission in 1–10 keV, even for a Compton-thin electron gas (i.e., )12. As the luminosity and the spectral slope of the Comptonized flux strongly depends on the optical depth that is totally determined by properties of the pre-supernova stellar wind, observing the post-supernova X-rays provides a unique way to trace the mass-loss history of the SN progenitor that gives constraint on the fundamental stellar properties.

Supernova (SN) 2013df, residing in the constellation Coma Berenices, was discovered by the Italian Supernovae Search Project (ISSP) on 7.87 June 2013 UT with an apparent magnitude of 14th. Optical spectrum from the 10-meter Keck II telescope at Keck Observatory showed that it is a Type IIb SN with weak hydrogen emission features observed in the early phase3. The host galaxy of SN 2013df, NGC 4414, was observed on 1999 April 29 by the Hubble Heritage Project (GO/DD 8400; PI: Keith Noll) with the Wide Field and Planetary Camera 2 (WFPC2) of the Hubble Space Telescope (HST) of which a three-band (F439W, F555W, and F814W) image mosaic was obtained in total fourteen WFPC2 pointings with a total exposure time of 5400 sec. By matching the WFPC2 images with a post-explosion 20 sec exposure taken by HST/WFC3 (F555W) on 2013 July 15, a yellow progenitor candidate within an astrometric accuracy of 5.5 mas was discovered4. Although the source was undetected in WFPC2/F439W band with a 3σ lower limit of 25.65 mag, they concluded that the candidate progenitor is likely a yellow supergiant with an effective temperature of Teff = 4250 ± 100 K, a bolometric luminosity of , and an effective radius of by the absolute magnitudes of V = −6.89 mag and I = −8.17 mag. The authors also compared the maximum luminosity at the Type IIb secondary peak to those of the SNe IIb 1993J, 2008ax, and 2011dh and estimated less than of 56Ni synthesized in the SN 2013df explosion.

In addition to the discovery of the yellow supergiant progenitor, another interesting scientific finding for SN 2013df is the early X-ray detection by the Swift X-ray Telescope (XRT)5. In spite of the limited resolution of Swift/XRT and the contamination from the diffuse emission of the host galaxy NGC 4414, the source is spectrally hard making it outstanding from the soft X-ray diffuse gas and showing the detection significance. Besides, early radio emissions were also detected by electronic Multi-Element Radio Linked Interferometer Network (eMERLIN; at 5 GHz6) and the Very Large Array (VLA; at various frequencies from 1.5 to 43.7 GHz7). While it has been suggested that the radio and X-ray emissions primarily originated from synchrotron and bremsstrahlung, respectively, we here present the X-ray spectrum and the multi-wavelengths light curve of SN 2013df and argue the origins of the X-rays to be dominated by the Compton scattering at the peak and the free-free emission in the rest of the time. In this work, the distance to NGC 4414 is assumed to be 16.6 Mpc8.

Results

X-ray/Optical Light curves

Double-peaked optical light curves are the most distinguished observational feature of Type IIb SNe. Shock wave heating of the progenitor’s hydrogen envelope is the main cause of the first optical peak. While the amount of the hydrogen envelope (i.e., a few of 9) is insufficient to support a sustainable thermal radiation, the decay of 56Ni synthesized in the explosion dominates the optical spectrum as the shock emission goes down, and produces the secondary optical peak10. In the case of SN 2013df, the secondary optical peak showed up at about day 20 since the SN discovery11. Surprisingly, a simultaneous X-ray brightening (~4σ significance; see Methods for details) is also detected by Swift/XRT (Fig. 1) with a photon index of and an X-ray luminosity of L1–7keV ≈ 3.2 × 1039 erg s−1 (1–7 keV; see Methods for details). From the Swift/UVOT v-band and X-ray light curves, the X-ray brightening reached the maximum as the secondary optical emission peaked. In addition to the synchronicity of the emission peaks, the rising and decaying rate of the X-ray flux is about the same as those of the optical lights. It is presumably more than a coincidence and the X-ray emission should be closely related with the SN photospheric radiation. This is reminiscent of Inverse-Compton scatterings of the SN lights by the shocked hot electron gas surrounding the SN as the source of the excess X-rays.

Figure 1

Multi-wavelength light curves of SN 2013df with day zero fixed at the SN discovery.

IC scattering as the origin of the X-ray peak

As mentioned in the introduction, IC is an important cooling process in the early phase of a SN, even if the shocked CSM electron gas (i.e., T ≥ 109 K) is Compton-thin (i.e., τ ~ 0.01; see Methods for a detailed explanation). In fact, the IC X-ray spectrum can be described by a single power-law1 and the spectral properties are closely related to the stellar wind (or the CSM) density of the progenitor, which can be represented by the ratio of the mass-loss rate (i.e., in units of ) to the velocity (i.e., w10 in units of 10 km s−1) of the pre-explosion wind. With the X-ray luminosity and the photon index obtained by Swift/XRT, we apply the inverse-Compton scattering model for SNe1212 to explain the peak X-ray emissions observed in SN 2013df and constrain the wind parameters of the progenitor in the following (see Methods for a detailed description).

As the photons of the secondary SN peak are the seeds of the IC X-rays, the temperature and bolometric luminosity of the SN are both important to affect the IC emission. Adopting the blackbody temperature of the secondary peak of T ≈ 6900 K11 and the absolute v-band magnitude of M = −16.80 mag (D = 16.6 Mpc; see the Methods section), we estimated the bolometric luminosity, hence the radius of the SN photosphere, which is about . With an assumption of a typical SN shock velocity V = 104 km s−1 (i.e., T = 1.36 × 109 K; see Methods) and the equations from eqs (2, 3, 4) for the optical depth, the photon index, and the IC luminosity, we computed the photon indices of the IC emission and the corresponding X-ray luminosities (i.e., 1–7 keV) with different wind parameters (a SN age of 20 days is assumed), which are shown with the best-fit XRT spectral values in Fig. 2. Comparing with the Swift/XRT data, it is clear that matches well with the best-fit results. The corresponding optical depth is τ ≈ 0.04, which fulfils the minimum criterion of τ > 0.01. In addition, we also used the UVOT color temperature of the secondary peak (i.e., T ≈ 5500 K; see the Methods section) for the same analyses and the result is similar to the previous one with a slightly larger (τ ≈ 0.05). Therefore, we conclude that the wind parameter of the progenitor is approximately .

Figure 2

Photon index versus logarithmic scale luminosity of the SN Comptonization model (red line; a SN photosphere of T = 6900 K is assumed) plotted with contours of the best-fit at 68%, 90%, and 95% confidence levels (grey shadows).

The upper two plots are based on Approach A while the lower two are based on Approach B (see the text for details).

A free-free emission as the “quiescent” component

We attribute the so-called “quiescent” off-peak X-ray emission component to be a free-free emission of the shocked hot gas as indicated by the extremely flat X-ray spectrum observed (i.e., Γ ≈ 1.0, equivalent to a zero spectral index; See Methods for details). According to Fransson et al.2, the free-free luminosity density is also a strong function of and a simplified version for V = 104 km s−1 around 1 keV can be written as

By adopting ζ = 0.86 (see Methods) and , the free-free X-ray luminosities (i.e., 1–7 keV and ΔLff = 6 keV) in the time interval of t = 10 d to 60 d are in the range of 6 × 1039 to 0.7 × 1039 erg s−1, which are in a good agreement of the Swift/XRT data.

Discussion

Chevalier13 suggested that such Compton scattering radiations should dominate the UV band, equivalent to an energy range around 10–100 eV, while emissions in the X-ray band (i.e. 0.1–10 keV) are mainly caused by free-free scatterings. However, the Comptonization flux could sometimes be compatible to the free-free radiations in soft X-ray band; for instance, the luminosity ratio of the Compton-to-free-free processes of SN 1993J (i.e., LComp/Lff) in ROSAT range (i.e., 0.1–2.4 keV) for τ ≈ 0.05 and T ≈ 109 K was computed to be 1.36 at day 72 and the ratio could increase to LComp/Lff = 341 with an electron temperature of T = 5 × 109 K. Although the ROSAT range is slightly different from the Swift/XRT range used here, the electron scattering optical depth as well as the shock temperature are indeed very similar to what inferred from the Swift/XRT peak emission spectrum. This ensures the possibility of the IC scattering scenario in such an optically-thin hot gas. Moreover, the inferred of SN 2013df is indeed very close to the one of SN 1993J inferred with the early X-ray data (i.e., 2). While the SN 1993J progenitor has been firmly characterized as an early K-type supergiant star14, the similarity between the pre-explosion stellar winds of the two SNe suggests that the SN 2013df progenitor was also a yellow supergiant, supporting the cool supergiant detected by the HST4 as the true progenitor.

Other Possible Mechanisms

Synchrotron radiation is another possible mechanism to produce non-thermal X-rays in SNe. However, the expected X-ray emission inferred from the observed synchrotron radio spectrum is at least an order of magnitude fainter than what we have seen in SN 2013df 7. We also considered an inverse Compton emission of the synchrotron radio photons as the origin of the observed X-rays, but in this case the X-ray-to-radio ratio would be too large (i.e., the radio luminosity at 5 GHz is L = 2.6 × 1036 erg s−1 7 and L/L > 2000) comparing with other well-studied SNe (e.g., LIC/Lsynch < 40 for the SN 1998bw15). Therefore, both the above two mechanisms are not applicable in the case of SN 2013df.

Alternatively, it has been suggested that the X-rays of SN 2013df are dominated by the bremsstrahlung thermal emissions from either the forward shock (Tfs ≥ 109 K) or the reverse shock (Trs ≥ 107 K)7. Although we agree that the free-free emissions significantly contribute the observed X-rays in the early phase (see the previous section), bremsstrahlung alone is insufficient to explain the correlation between the optical/X-ray light curves. While the X-ray photon index of SN 2013df is harder than it should be in a Comptonization dominant case (Γph ≈ 3; see Fig. 2), a hybrid of thermal bremsstrahlung and Inverse-Compton is a good way to understand the multi-wavelength data as we suggested.

Importance of the Inverse-Compton emission for SN studies

Inverse-Compton model has been well established to explain X-ray emissions of various SN events, for examples, the Type II-L SNe 1979C, 1980K, the famous Type II-b SN 1993J, and the Type II-P SN 2006bp121316. In fact, the IC scenario is also promising to explain the mysterious X-ray emissions of the special class GRB-SN associations (i.e., hydrogen-stripped Type Ic and possibly Ib supernovae detected well after the associated GRBs) and SN shock breakout events171819. While an IC X-ray brightening has been theoretically shown possible during the optical peaks of the SN2021, it has been a missing piece before for a long time. Although the recent multi-wavelength study of the Type Ia SN 2011fe (a.k.a., PTF 11kly) in M101 placed a marginal constraint on such a correlation by demonstrating a correlated X-ray/optical light curve profiles22, the X-rays of SN 2011fe are actually undetected (i.e., the X-ray profile was constructed based on three upper-limits) and the corresponding flux upper limits place a 3σ constraint on the progenitor mass loss rate of (note: it could be an interstellar stellar medium instead), which just ambiguously restricts the X-ray/optical relation. The X-ray brightening of SN 2013df shows a clear correlation with the secondary optical peak, which is an important example for the SN ejecta and CSM interaction model.

The IC X-ray component study could also benefit various researches on Type Ia. As stated earlier, Type Ia supernovae are also possible IC X-ray emission sources, although no significant X-ray detection has been made from any known Type Ia (except for the SN 2014J in which 56Co lines at energies of 847 and 1238 keV were detected by INTEGRAL23). If IC scattering is significant, a few percent (i.e., ~τ/2) of the SN optical radiations will be IC scattered and the optical measurement will deviate from the true intrinsic SN brightness. As Type Ia supernovae are well-known as standard candles to measure cosmological distances by tracing accurately the unique SN decline, the measured brightness deviation could be important to fine-tune the distance ladder. IC emissions are also crucial to distinguish the proposed origins of Type Ia SNe, which are single-degenerate (a main-sequence star or red giant companion) and double-degenerate scenarios (another white dwarf companion), by examining density of the surrounding medium.

On a conservative basis, the connection between the X-ray brightening of SN 2013df and the predicted IC X-ray emission may not be exclusively proven here. Nevertheless, this work certainly provides a new angle to the community on how simultaneous optical/X-ray data can be used to search for the IC emission of SNe. In particular, rapid Swift multi-wavelength observations of SNe have been shown to be useful to such a search. Hopefully, Swift will bring us more solid evidence of the IC emissions of SNe in the future.

Methods

Swift X-ray Telescope (XRT)

A series of Swift ToO observations for SN 2013df was triggered since 2013 June 13 (6 days after the optical discovery) with a total number of 26 observations until 2013 August 6. From the combined 49 ks X-ray Telescope (XRT) observation, a hard X-ray source was detected at α(J2000) = 12h26m29s.326, δ(J2000) = 31°13′37″.87 with an uncertainty of 3.6″ (radius, 90% confidence), which is less than 0.2″ offset from the optical position. No significant X-ray source was found at the same position in the previous XMM-Newton observations taken in 2005–2006 with a 3σ upper limit of 2.96 × 10−15 erg cm−2 s−1 or 9.8 × 1037 erg s−1 (0.5–4.5 keV) deduced by the XMM-Newton FLIX server.

We downloaded the Swift observations from the HEASARC archive and extracted XRT spectra and light curves using the HEAsoft (version of 6.11) built-in tasks, xrtgrblc and xrtgrblcspec. The source photons were selected using a circular region with a radius of 15″ while a background subtraction was done using a source-free annulus region around the host galaxy. Unless otherwise mentioned, the uncertainties are in 90% confidence level.

The Swift/XRT light curve (Fig. 1; at least 10 counts per bin with Poisson errors) shows that the X-ray emission lasted for about fifty-five days since the first Swift observation, during which the X-rays gradually rises from ~5 × 1039 erg s−1 at day 7 (with respect to the SN discovery here), reaches a peak luminosity of ~1 × 1040 erg s−1 at about day 18, and drops steadily over the last 40 days of observations (Fig. 1). For the spectral model fitting, the XRT spectrum was binned with at least 15 counts per bin and weighted with the Gehrels approximation24 (i.e., for a bin with N counts) for a better uncertainty estimation in cases of N < 20. We fitted the averaged “hard” XRT spectrum (i.e., 1–7 keV; to avoid possible contaminations from the soft X-ray diffuse emission of the host galaxy) with an absorbed power law model, of which the best-fit values are NH < 2.6 × 1022 cm−2 and ( with dof = 3). In addition, we tried a thermal bremsstrahlung model to fit the data, but the best-fit plasma temperature hit the boundary (i.e., 200 keV) to fail the fit. Thus, we fixed the temperature to kT = 100 keV (or T ~ 109 K, a typical temperature of a free-free X-ray emission for supernovae2), which cured the fit and gave NH < 1.6 × 1022 cm−2 and (dof = 4). Though a smaller value than that of the power-law model was obtained, we do not consider the improvement significant as the becomes 0.62 (dof = 4) if we fixed the photon index to Γph = 1. Instead, the simple power-law and the thermal bremsstrahlung models are both significant to describe the data and the former model fits the spectrum slightly better.

As the photoelectric absorption is much more important below 1 keV, we extended the lower energy range of the XRT spectrum down to 0.3 keV to have a better constraint on the column density, hence to improve the fits. We also set the minimum allowed column density to the Galactic value25 of NH = 1.6 × 1020 cm−2 to prevent an non-logical null absorption. For an absorbed power law, the best-fit parameters are improved to and NH < 2.4 × 1021 cm−2 ( with dof = 5; Fig. 3) yielding an unabsorbed luminosity of L0.3–7keV ≈ 6.8 × 1039 erg s−1. For an absorbed thermal bremsstrahlung (temperature fixed to 100 keV), the best-fit NH changes to NH < 2.9 × 1021 cm−2 ( with dof = 6) and the luminosity is L0.3–7keV ≈ 6.5 × 1039 erg s−1. Considering the poorly constrained NH, we also computed the “hard” X-ray luminosity L1–7keV ≈ 6.1 × 1039 erg s−1 (power-law) and L1–7keV ≈ 5.5 × 1039 erg s−1 (bremsstrahlung), with which the influence of the greatly uncertain NH on the investigation can be minimized.

Figure 3

Swift/XRT spectrum of SN 2013df with the best-fit power-law of and NH < 2.4 × 1021 cm−2 ( with dof = 5).

The X-ray brightening

As mentioned earlier, an X-ray brightening is seen between day 12 and 25. We found that the brightening light curve can be described by a Gaussian function on top of a flat light curve, A × exp(−(t − t)2/(2σ2)) + B, where the the best-fit values are A = (4.8 ± 0.8) × 10−3 cts s−1, t = 17.1 ± 0.7 d (time at the peak), σ = 5.3 ± 0.8 d, and B = (2.0 ± 0.2) × 10−3 cts s−1 (χ2 = 1.6; dof = 5). Alternatively, one may consider the apparent brightening as a statistical fluctuation. In this case, we have modelled the data with a flat light curve and the best-fit count rate is (2.9 ± 0.5) × 10−3 cts s−1 (χ2 = 18.6; dof = 8). Clearly, there is a significant improvement (i.e., Δχ2 = 17) when considering the brightening a real detection (i.e., the Gaussian model) rather than a fluctuation (i.e., the flat model). To find out whether this improvement is significant, we performed Monte Carlo simulations by generating 106 flat light curves and fitting them with both models. Among the 106 trials, 141208 of them did not converge in the Gaussian fit (i.e., a Gaussian is totally unnecessary in these cases), 858765 of them have an improvement less than Δχ2 = 17, and only 27 trials were improved by Δχ2 ≥ 17. This clearly shows that the X-ray brightening detection is not likely produced by chance (i.e., p-value ≈ 0.003% or statistical significance ~4σ).

To investigate spectral features of the X-ray brightening, we used the observations of IDs from 32862008 to 32862015 to extracted a stacked X-ray peak spectrum. Given that the photon statistic is not good (i.e., spectral source counts: 28), we adopted a binning factor of at least 7 counts per bin to produce a 4-bin X-ray spectrum. To elaborate the small binning factor applied, it is a strategy that sacrifices the signal-to-noise (S/N) ratios of single bins to exchange for a larger number of degree-of-freedom. Otherwise, there will be only 1 or 2 bin(s) in the spectrum and leave no freedom for fitting. In fact, low-count binning is common in X-ray astronomy for faint sources (e.g., the Chandra spectral analysis of A0620-0026). Additionally, we applied the W-statistic (CASH-statistic27 with Poisson distributed background) to deal with the fitting process, instead of the standard Chi-squared statistic (i.e., Gaussian distribution is assumed), which does not work properly for low-count spectra (see the XSPEC on-line manual for details; the difference between W-statistic and Chi-squared performances is shown in Fig. 2). This W-statistic (or CASH-statistic) has been widely used in spectral fitting with low-count X-ray spectra (e.g., the hyperluminous X-ray source candidate, CXO J122518.6 + 1445452829). It is therefore a valid statistical model for the cases of SN 2013df. Although our data have a low number of photon counts, the quality of the spectral fits can be reflected by the uncertainties of the estimated parameters given a proper statistic applied. We also note that spectral binning is not a necessary process for W-statistic fitting. However, binned spectra provide χ2 values of the best fits (not the fitting statistic though) that are good indicators of the goodness-of-fit (i.e., one cannot tell the goodness-of-fit from the W-statistic) to guide our data analysis.

With the Galactic hydrogen column density of NH = 1.6 × 1020 cm−2 assumed, the spectrum could be power-law distributed with a photon index of Γph = 1.4 ± 0.6 and the averaged luminosity of the X-ray peak is L0.3–7keV ≈ 8.1 × 1039 erg s−1 (or L1–7keV ≈ 6.6 × 1039 erg s−1). Comparing with the “quiescent” spectrum extracted from the off-peak observations (i.e., Ob. ID: 32862001–00032862005, 32862016–00032862027; Table 1), of which the best-fit values are Γph = 1.0 ± 0.4 and L0.3–7keV ≈ 4.9 × 1039 erg s−1 (or L1–7keV ≈ 4.4 × 1039 erg s−1), the peak X-ray emissions are spectrally softer, although insignificant. We also tried to subtract the peak spectrum by the “quiescent” one to probe for any additional emission component (i.e. IC emissions) leading to the X-ray enhancement as the observed peak. Two approaches have been used to eliminate the “quiescent” emission contribution, either by the observed spectrum itself (i.e., serving the “quiescent” spectrum as the “background” spectrum; namely approach A) or the best-fit power law model (i.e., including the fixed best-fit “quiescent” spectral model of Γph,fix = 1.0 and normfix = 1.4 × 10−5 photons keV−1 cm−1 s−1 at 1 keV, to the model of interest; approach B). For the approach A, the subtracted spectrum can be described with a photon index of and an X-ray luminosity of L0.3–7keV ≈ 4.3 × 1039 erg s−1 (or L1–7keV ≈ 3.2 × 1039 erg s−1). For the approach B (i.e., two power-law components of which the “quiescent” one has been entirely frozen), the best-fit photon index and X-ray luminosity are and L0.3–7keV ≈ 3.6 × 1039 erg s−1 (or L1–7keV ≈ 2.5 × 1039 erg s−1), respectively. While the fitting results of both methods are consistent with each other, the approach A generally performs better than the approach B in terms of the uncertainties of the best-fit parameters. Figure 2 shows the contour maps of the best-fit photon indices and luminosities, which fully revealed their uncertainties. Combining with the IC emission spectral model for SNe, the maps are indeed very useful to constrain the wind parameter of the SN progenitor (i.e., for a shock velocity of V = 104 km s−1).

Table 1

Swift/XRT Observation log of SN 2013df.

Date	Ob. ID	Exposure Time (s)	Note
5.94	32862001	2969.27	Low
8.36	32862002	1977.85	Low
9.95	32862003	139.86	Low
8.68	32862005	1940.39	Low
10.62	32862006	3348.88	Raising
10.62	32862007	489.50	Raising
11.69	32862008	497.00	Peak
11.69	32862009	2677.11	Peak
12.70	32862010	2310.00	Peak
15.23	32862011	1578.28	Peak
16.84	32862012	1523.35	Peak
18.57	32862013	1800.55	Peak
23.30	32862014	2170.14	Peak
24.92	32862015	1947.89	Peak
27.17	32862016	1573.29	Low
28.78	32862017	1997.83	Low
30.78	32862018	1840.50	Low
33.26	32862019	1353.55	Low
35.25	32862020	1221.18	Low
36.65	32862021	1558.31	Low
38.80	32862022	1156.25	Low
45.66	32862023	3638.54	Low
49.93	32862024	1086.33	Low
53.61	32862025	4060.58	Low
58.48	32862026	452.01	Low
59.62	32862027	3940.71	Low

Swift Ultraviolet and Optical Telescope (UVOT)

Swift/UVOT observed SN 2013df simultaneously with XRT using all six UVOT filters (central wavelengths are from 1928 to 5468 Å). We used a HEAsoft task uvotmaghist to compute the magnitude/flux density of each UVOT image using an aperture radius of 3″ (the optimum aperture size for UVOT images; see the Swift/UVOT on-line analysis thread for details) with an annulus source-free background around the SN. A prominent double-peaked light curve feature of Type IIb is clearly shown in v and b bands, but marginally seen from u to um2 bands and invisible in uw2 band (Fig. 1). The extinction corrected UVOT magnitudes of the first and second peaks are listed in Table 2 with A = 0.30 mag11. Based on the relation between B − V colors and color temperatures (T in Kevin) for blackbody radiations (i.e., B − V ≈ (7090/T) − 0.71), inferred temperatures of the peaks are T = 8500 ± 4000 K and T = 5500 ± 800 K for the first and second peaks, respectively. These temperatures are roughly consistent with those obtained by optical spectroscopic fittings in Morales-Garoffolo et al.11 (i.e., T = 7700 ± 500 K at day 9 and T = 6900 ± 500 K at day 22).

Table 2

Swift/UVOT magnitudes of SN 2013df.

	v	b	u	uvw1	uvm2	uvw2
1st Peak	14.47 (0.04)	14.59 (0.04)	13.67 (0.05)	14.09 (0.05)	14.31 (0.05)	14.77 (0.06)
2nd Peak	14.30 (0.06)	14.87 (0.06)	14.67 (0.07)	15.17 (0.06)	15.58 (0.07)	…

The IC emission spectral model for SNe

According to Fransson et al.2, the electron scattering optical depth behind the circumstellar shock is dependent of the shock velocity (V) and the stellar wind properties, which could be described as (note: it is a modified version for general uses as the original version is dedicated for the Type IIb SN 1993J, see Fransson et al.2 and references therein for details):

The is the key parameter for the stellar wind description, where is the mass-loss rate in units of and w10 is the wind velocity in units of 10 km s−1. There is another parameter ζ, defined as ζ = (1 + 2nHe/nH)/(1 + 4nHe/nH) for describing the chemical composition of the wind and nHe/nH = 0.1 is assumed throughout this analysis (i.e., ζ = 0.86, which has been widely used to explain the X-ray emission of SN 1993J2). In addition, the formula is constructed under an assumption of a steady stellar wind flow (i.e., ρ ∝ r−2), which is the simplest case of a circumstellar density profile. Fransson1 has shown that there is a fraction of ~(τ/2) of the supernova photospheric blackbody radiation up-scattered n times by the energetic shock electrons to form high-energy X-ray emissions, even though the hot shock gas is optical thin to electron scattering. For optical depths smaller than , each seed photon will at most be up-scattered only one time before it escapes the hot shock. In this case, the energies of the IC scattered photons will be doubled (i.e., ΔE/E ≈ 4kT/mc2 ≈ 1 for , with a typical shock temperature T ≈ 109 K) to possibly produce an excess UV tail at the shorter wavelength end of the blackbody spectrum, but certainly no X-rays in keV levels. For τ > 0.01, the number of electron scattering is large (i.e., n > 1) and the Compton cooling is therefore important. In this case, the Comptonized spectrum is a single power-law in X-rays (i.e., N(E) ∝ E−), with a photon index of

where α = kT/mc2 (note: the γ index is a spectral index (i.e., F(E) ∝ E−), instead of a photon index, in the original version1). The Compton luminosity above 13.6 eV could be described by a semi-empirical expression as

where R and Teff are the radius and the temperature of the photosphere, respectively, and T is the temperature of the shocked gas. According to Fransson et al.12, the shock temperature can be computed for a given shock velocity V by

The inferred shock temperature (T) is also the temperature of ions (Tion) in the shocked hot gas. As the equipartition time of the plasma is relatively short (i.e., about 1 day) in such a high density shocked region, the ion temperature is approximately the same as the electron temperature (i.e., T ≈ Tion)1.

Additional Information

How to cite this article: Li, K. L. and Kong, A. K. H. A likely inverse-Compton emission from the Type IIb SN 2013df. Sci. Rep. 6, 30638; doi: 10.1038/srep30638 (2016).