SN 2008am: A SUPER-LUMINOUS TYPE IIn SUPERNOVA

The host of SN 2008am is SDSS J122836.32+153449.6 that appears to be a faint, extended object in the Sloan Digital Sky Survey (SDSS), with an r' magnitude of ∼20 (see Figure 1). The position of the centroid of the host is α= 12^h28^m363 and δ= +15^d34^m50^s and its redshift is z= 0.2338 (Yuan et al. 2008a; Section 2.4). We estimate the luminosity distance of the SN to be 1130 Mpc assuming a ΛCDM cosmology with Ω_Λ= 0.73, Ω_M= 0.27, and H₀= 73 km s⁻¹ Mpc⁻¹. On that basis, the host has an absolute magnitude $M_{r^{\prime }} \sim$ −20 mag and, thus, is not a dwarf galaxy. The morphology of this galaxy is unknown, but the shape of its SDSS photometric SED and the optical spectrum showing narrow emission features of H, [N ii], [O ii], and [O iii] agree well with an SB1 galaxy template spectrum (Figure 9; Section 2.4). The metallicity of the host, estimated from the flux ratios of these narrow emission features, is sub-solar (Z∼ 0.4 Z_☉), similar to the hosts of other SLSNe (Neill et al. 2011; Stoll et al. 2010; Section 2.4).

Zoom In Zoom Out Reset image size

SN 2008am was discovered in unfiltered ROTSE-IIIb images on 2008 January 10.4 UT (MJD= 54475.4; Yuan et al. 2008a) when it had an unfiltered magnitude of 18.7. The position of the SN in the ROTSE images was determined to be α= 12^h28^m3625 and δ= +15^d34^m491, slightly offset from the center of its host that appears in the SDSS catalog (Figure 1). ROTSE-IIIb continued to gather unfiltered photometric data for ∼200 days after the discovery. The ROTSE-IIIb photometry is summarized in Table 1. The ROTSE data are shown in Figure 2 along with the detection limits over the course of the photometry. The ROTSE-IIIb photometry includes an early upper limit and data points during the rising phase of the supernova light curve that can be used to constrain the explosion date of the SN. To determine the explosion date, we converted the ROTSE magnitudes to luminosities assuming zero bolometric correction, and fit radioactive decay diffusion models (Section 4.1) to the resulting light curve. The fitting parameters are the effective diffusion time, the nickel mass, the explosion date, and a parameter that controls the gamma-ray leakage. The best-fit radioactive-decay diffusion model is shown in Figure 3. This fitting process yields an explosion date of MJD_expl= 54438.8 ± 1, approximately 18 days prior to the first real detection in the observed frame and about 14 days in the rest frame. We note that this model was employed only to determine the explosion date of the SN and may not account for the real physical situation in SN 2008am (see Section 4.1). The ROTSE-IIIb maximum occurred at MJD= 54480.4 (2008 January 15.0), about 34 rest-frame days after the explosion.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

At the distance of 1130 Mpc for the ROTSE unfiltered peak magnitude of 18.0 mag, the absolute ROTSE peak magnitude of SN 2008am is −22.3 mag (uncorrected for extinction). The ROTSE response curve peaks in the red and it is calibrated to the USNO-B1.0 system (Smith et al. 2003). There is always a slight offset from a true R magnitude due to the fact that the shape of a supernova SED is very different from the reference stars used by ROTSE, but, to a good approximation, this absolute peak magnitude is close to the real absolute R magnitude of the event. This establishes that SN 2008am is a super-luminous event; one of the most luminous supernovae ever discovered, placing the supernova in the family of SLSNe together with SN 1992ar (Clocchiatti et al. 2000), SN 1999as (Hatano et al. 2001), SN 2003ma (Rest et al. 2009), SN 2005ap, SN 2006gy, SN 2006tf, SN 2008es, SN 2008fz (Drake et al. 2010), SN 2008iy (Miller et al. 2010), SN 2007bi (Gal-Yam et al. 2009), SCP06F6 (Barbary et al. 2009), and recently discovered luminous PanSTARRS transients (SN 2009kf, Botticella et al. 2010; SN 2010gx, Pastorello et al. 2010).

SN 2008am was followed up with photometric observations ranging from the ultraviolet (UV) through the infrared (IR). The Peters Automated Infrared Imaging Telescope (PAIRITEL; Bloom et al. 2006) obtained J-, H-, and K_s-band photometry over a 25 day period (Table 2). The PAIRITEL J, H, and K_s fluxes are calibrated to the Two Micron All Sky Survey (2MASS) catalog (Skrutskie et al. 2006). The infrared light curves of SN 2008am over this phase appear to be flat and are probably heavily contaminated by the host. In the K_s-band, especially, the detections are most probably indicative of the host rather than the SN. Thus, we refer to them only as upper limits throughout this work. Although the infrared observations were obtained for only a small portion of the life of the SN, they can be used to better constrain the SED toward the infrared region for some phases. Contribution by dust IR radiation to the observed J, H, and K_s fluxes cannot be ruled out, but we have made no allowance for that process.

The Palomar 60 inch (P60) telescope obtained optical photometry in the bands g, r, i', and z' for a period of ∼330 days in the rest frame. Photometry on the P60 frames was performed using the aperture photometry routines in IRAF.⁹ The aperture radius was set to be 20 pixels (7.57 arcsec), and the background level was measured in an annulus with 30 pixels (11.36 arcsec) inner and 50 pixels (18.93 arcsec) outer radii, centered on the source. The P60 data were calibrated via five local tertiary standard stars having Sloan g', r', i', and z' magnitudes in the SDSS catalog (Table 3). The applied P60 filters were g and r (similar to the Thuan–Gunn filters; Thuan & Gunn 1976), and i' and z' that resemble the SDSS filters (Fukugita et al. 1996), although systematic differences exist between the P60 filters and their standard counterparts (Cenko et al. 2006). Due to the lack of standard fields observed simultaneously with the SN field, only approximate calibration of the P60 photometry was possible. As a first approximation, the SN magnitudes were tied to the g', r', i', and z' magnitudes of the local tertiary standards assuming only a zero-point shift and no color term. Table 4 details the results of the P60 photometry of SN 2008am. The error caused by the neglect of the color term was investigated by comparing the observed differential magnitudes and colors of the local standards with their cataloged magnitudes. Only the data from the g filter are affected by the lack of the color term, systematically at a level of ∼0.1 mag. No significant magnitude shifts were detected in any other filters. The resulting g', r', i', and z' AB magnitudes of SN 2008am were then transformed to fluxes adopting the filter parameters and flux zero points of Cenko et al. (2009). The lack of the color term caused less than 1σ error in the g' fluxes, where σ is the random error of the photometry. The SN fluxes were subsequently corrected for host contamination and interstellar reddening. The host correction was done by removing the host fluxes obtained from SDSS. For the interstellar reddening we used the interstellar absorption maps of Schlegel et al. (1998), giving E(B − V)_MW= 0.025 mag and the Milky Way reddening law parameterized by Fitzpatrick & Massa (2007) adopting R_V= 3.1. Throughout this work, we use only the Milky Way reddening correction to obtain the final photometry for SN 2008am. The 50–180 day slope of the P60 g, r, i' light curve is estimated to be 0.0065 ± 0.0006 mag day⁻¹. The z' light curve is somewhat flatter over the same period.

Swift photometry was obtained by the Ultraviolet/Optical Telescope (UVOT; Roming et al. 2005) covering the first ∼80 rest-frame days after maximum. The conversions between Swift magnitudes and fluxes were computed based on the calibration using the Pickles stellar templates (Poole et al. 2008) instead of the GRB templates included in the Swift CALDB. The result of the Swift photometry is detailed in Table 5. In the latest three epochs, the UVW1, UVM2, and UVW2 fluxes seem to level off. This may be due to the increasing contribution from the flux of the host galaxy relative to the decreasing SN flux at later epochs.

The rest-frame light curves of SN 2008am for all the available photometric bands are shown in Figure 4. The top left panel shows the PAIRITEL IR J and H light curves, the top right panel the Swift UVOT optical and UV light curves and the bottom panel the ROTSE unfiltered and P60 optical light curves. The data sets have been offset for clarity, with longer wavelength on top and shorter wavelength toward the bottom of each panel. The broad photometric wavelength coverage allows us to better constrain the SED of the SN.

Zoom In Zoom Out Reset image size

SN 2008am was observed with the Very Large Array at 8.46 GHz on 2008 February 19.37 UT, approximately 30 rest-frame days after optical maximum (Chandra & Soderberg 2008). No source was detected at the SN position above 120 μJy, which can be considered as a 3σ upper limit. Swift X-ray Telescope (XRT; Burrows et al. 2005) images of SN 2008am were obtained for the six epochs in parallel with the UVOT observations. No X-ray source was detected at the position of the SN, after co-adding all XRT observations. Assuming a power-law spectrum with spectral index of −2, the X-ray flux upper limit was estimated to be ∼10⁻¹³ erg cm⁻² s⁻¹ (or, equivalently ∼10⁴³ erg s⁻¹) at ∼50 rest-frame days since explosion.

The availability of multi-band simultaneous photometry for some epochs helps us construct SEDs for SN 2008am and thus study the evolution and basic properties of the event. Our criteria for selecting the photometric epochs for which we constructed the SEDs were two: close sampling in time and maximum possible wavelength coverage. Those criteria led to the choice of 10 epochs for which we constructed the photometric SEDs. For four of those epochs we had available UV+Optical+IR observations (hereafter UVOIR), for four only Optical (P60 data, hereafter OPT) and for two (08-01-30 and 08-02-25) Optical+IR (see Figure 4).

For the 08-02-25 epoch, we interpolated between the previous (08-02-16) and the next (08-03-14) epoch for which we had UV data. The error of this interpolated value was estimated using error propagation analysis for a linear fit. We note that the UV flux for the interpolated 08-02-25 epoch is uncertain since the UV light curve decline of emission line objects like SN 2008am is not well constrained at later times. Adding the interpolated UV data to the Optical+IR data gave us a total of five epochs of UVOIR data and one epoch (08-01-30) for OIR data that we will later use to estimate the pseudo-bolometric light curve (LC) of the event.

Although we see no sign of classic supernova photospheric P Cygni lines (see Sections 2.4 and 3), the blue continua of the spectra are consistent with thermal emission. This emission could arise in shocked CSM that is modestly optically thick (τ∼ 1) to absorption. To produce a quantitative diagnostic, we fit single-temperature blackbody curves to the set of the five UVOIR and one OIR photometric SEDs in order to determine effective blackbody temperatures in the rest frame. We included the reddening corrections discussed in Section 2.2. The blackbody curve fits were done in the rest frame of the SN. Bolometric luminosities were then derived from the integral of the flux from the corresponding blackbody at the adopted distance of the supernova. Effective blackbody radii were derived from the luminosity and blackbody temperature. We did not attempt to fit the OPT epochs with single-temperature blackbody curves as the uncertainties of the fit would be large, given the lack of UV data. We estimated lower limits for the bolometric luminosity in these four epochs by integrating the observed SED.

Figure 5 shows the photometric SEDs of SN 2008am for the six selected epochs together with the best-fit (lowest χ²) blackbody curve in each case. Figure 6 shows two examples of UVOIR SEDs and their blackbody curve fits and nearly contemporaneous spectral continua at +22 days and +33 days after maximum, respectively. Blackbody fits to the spectroscopic data were also done in the rest frame and included the reddening corrections of Section 2.2. The spectra of SN 2008am were scaled to the simultaneous photometry in each case, before any reddening corrections were applied. The temperatures derived from the spectral fits agree with those derived from the fits to the SEDs within the errors at these two epochs (Figure 6).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 6 summarizes the characteristics of the blackbody fits to the UVOIR and OIR SEDs of SN 2008am, the effective blackbody temperature, T_bb, the effective blackbody radius, R_bb, and the derived bolometric luminosity, L_bol. Figure 7 presents the evolution of T_bb (upper panel), R_bb (middle panel), and L_bol (lower panel). The filled triangles correspond to the fits to the UVOIR and OIR SEDs and the open circles to those of four early Hobby–Eberly Telescope (HET) and Keck spectra (the fifth, obtained on 2008 January 30.3 has a slightly anomalous slope that we attribute to observing conditions; Section 2.4). The single-temperature blackbody fits to the SEDs of SN 2008am show considerable uncertainty and scatter. The photometric points scatter around the best-fit blackbody curves and around the spectral continua. Differences between the spectral and photometric results at similar epochs may be attributed to the fact that the optical spectra do not accurately constrain the maximum of the blackbody curve and that we assumed a single blackbody to fit the SEDs and not more complex models. At the latest epochs, there could be other effects (for example, dust formation) that affect the final estimates. The NIR data are always in excess with respect to the fitted curves. That may be an indication that the single-temperature blackbody models do not accurately represent the emission properties of SN 2008am, but it is also possible that the NIR data are still somewhat contaminated by the host galaxy and only represent upper limits to the corresponding supernova flux. Another possibility is that the NIR excess is a sign of early warm dust emission (Meikle et al. 2007; Mattila et al. 2008; Kotak et al. 2009; Fox et al. 2010). The IR data are thus especially uncertain, but these data have relatively little weight in the blackbody fits. The blackbody temperatures and radii derived here are only indicative of the general conditions and their trends, and not to be treated as quantitatively precise nor as evidence that the emission is truly blackbody.

Zoom In Zoom Out Reset image size

The effective blackbody temperatures are in the range of 10,000–12,000 K, as presented in Table 6 and Figure 7, and are constant in time within the scatter. The pseudo-bolometric light curve implies a total radiated energy of about 10⁵¹ erg. Other estimates of the total radiated energy will be given in Section 4 based on several physical models. We will argue in Section 4.4 that the energy powering SN 2008am is most probably the interaction between the ejecta and the CSM and thus the estimates of T_bb, R_bb, and L_bol do not correspond to an expanding photosphere coincident with the ejecta of the SN. The ejecta of the SN and the CSM shocks may be hidden behind an optically thick CSM. In this case, the values of T_bb, R_bb, and L_bol refer to conditions in an optically thick CSM with τ∼ 1 representing the effective photosphere of the CSM.

We acquired a total of seven spectra of SN 2008am and its host galaxy spanning ∼2 yr after discovery. Four of them were taken with the HET Low Resolution Spectrograph (LRS; Hill et al. 1998) on 2008 January 29.3 UT, 2008 January 30.3, 2008 February 18.3, and 2008 February 25.3, corresponding to +11, +12, +27, and +33 rest-frame days after maximum, respectively. Two other spectra were obtained with the Keck-I Low Resolution Imaging Spectrograph (LRIS; Oke et al. 1995) on 2008 February 12.0 UT (+22 days after maximum in rest frame) and 2009 March 31.0 (+352 rest-frame days after maximum). Another Keck-LRIS spectrum of the host galaxy was taken on 2010 January 09.0 UT, +554 rest-frame days after maximum, when the transient had faded below the detection limit. All spectra were reduced in the standard way using IRAF. The instrumental resolution was ∼17 Å for the HET spectra and ∼6 Å for the Keck spectra.

We will refer to the spectra obtained within a month after maximum as "early-phase" and the +352 day Keck spectrum as "late-phase," respectively. All spectra were corrected for Milky Way reddening (as described in Section 2.2) and scaled to contemporaneous photometric data. The epochs of the early spectroscopic observations relative to the light curve are shown in Figure 3.

Figure 8 shows the spectral evolution of SN 2008am. Note that the first HET spectrum (obtained on 2008 January 29.3) is plotted for completeness, but omitted from further analysis because of the availability of the second spectrum (taken one day later, on January 30.3) that was obtained during better observing conditions giving better signal-to-noise ratio (S/N). The January 30.3 spectrum has a somewhat discrepant slope for reasons we have been unable to resolve, but the continuum slope of the spectra does not enter into our analysis except for perhaps a very small effect on line profiles. The 2008 January 29.3 spectrum was used in the blackbody fits (Section 2.3; Figure 7).

Zoom In Zoom Out Reset image size

The spectra were deredshifted by z= 0.2338 determined from the narrow emission features in the host spectrum. This value is very close to the one derived by Yuan et al. (2008a) based on the early-phase HET spectra.

The early-phase spectra show prominent features of H (Hα, Hβ, and Hγ). The He i 5876 Å line is detected in the February 12 Keck spectrum and with a smaller S/N in the February 25 HET spectrum. Na D may also contribute to this feature, but we were unable to make a definite identification.

The HET data had inferior seeing and lower spatial resolution of the spectrograph that made it impossible to fully resolve and separate the SN and the host. Given those instrumental differences between HET-LRS and Keck-LRIS we chose to analyze the spectra uncorrected for host extinction, so that we could treat all the data in a uniform way. The host probably contributes to some of the detected emission lines; [O ii] 3727 Å and [O iii] 5007 Å are undoubtedly present, and [O iii] 4959 Å can also be weakly detected in all HET spectra. Fortunately, the host contribution to the continuum should be much less, because the +554 day Keck spectrum, which is attributed to the host galaxy, shows mainly a flat, low continuum.

Due to better resolution and seeing, the host subtraction was possible for the early-phase and late-phase Keck spectra. The small visible spatial extension of the galaxy required a background region defined as close to the SN as possible to achieve good background subtraction. Most of the galactic forbidden emission lines have been removed successfully; however, this process resulted in a slight oversubtraction in the core of Hα for the early-phase Keck spectrum, forming an "absorption" dip on top of the line. That feature is due to the reduction process and should not be interpreted physically (see below). Although we show the host-subtracted early Keck spectrum for comparison, we note again that we do not use it for our analysis since host subtraction cannot be performed for the HET spectra given the different spatial resolution.

The late-phase Keck spectrum is also dominated by a broadened Hα line, but all other SN features have already faded below detectability. The shape of Hα strongly suggests that the transient was still visible at +352 days after maximum. A few narrow features at Hβ and around ∼5000 Å also appear that are probably due to contribution from the host.

The host spectrum, obtained with Keck-LRIS at +554 rest-frame days after maximum (Figure 9), shows the usual narrow emission features of galaxies with ongoing star formation. Beside Balmer lines, we identified [O ii] 3727 Å, [O iii] 4959, 5007 Å, [N ii] 6548, 6583 Å, and He i 5876 Å. There are also indications for the [S ii] 6716, 6731 Å features, but that region is heavily contaminated by tellurics, preventing a definite identification. All lines as well as the shape of the continuum (Figure 9; Section 2.1) can be very well matched by an SB1 galaxy template (Kinney et al. 1996). The metallicity of the host was estimated by computing the line flux ratios and the spectral indices N2 and O3N2 defined by Pettini & Pagel (2004). These resulted in an oxygen abundance of 12 + log(O/H) = 8.38 ±  0.15, significantly below the solar abundance value (∼8.7 ± 0.1). This oxygen abundance suggests a sub-solar metallicity for the host, about Z ∼ 0.4 Z_☉. There is growing evidence that SLSNe, such as SN 2008am, occur mostly in metal-deficient hosts (Neill et al. 2011; Stoll et al. 2010).

Zoom In Zoom Out Reset image size

None of the SN 2008am spectra show any sign of broad features characteristic of most SNe during the photospheric phase. There is no sign of P Cygni profiles. The Balmer lines in the early-phase spectra have FWHM ∼25 Å in the rest frame, which are usually referred to as "intermediate-width" lines (e.g., Stathakis & Sadler 1991). Such intermediate-width features are common characteristics of Type IIn SNe (e.g., Schlegel 1990). Based on these observed features, SN 2008am is certainly a member of the Type IIn subclass. The early-phase spectra are similar to those of SN 1988Z (Stathakis & Sadler 1991), SN 1995G (Pastorello et al. 2002), and the early spectra of SN 2006gy (Smith et al. 2010), all having blue continua with intermediate-width Balmer lines and without strong P Cygni profiles. In Figure 10, we plot the +22 day Keck spectrum together with the spectrum of SN 1998Z at +39 days after maximum light that had intermediate-width lines with no P Cygni components (Stathakis & Sadler 1991; Turatto et al. 1993; Aretxaga et al. 1999). A spectrum of SN 2006gy taken 19 rest-frame days after maximum with the HET by one of us (R.Q.) is also shown in Figure 10 for comparison. Although some differences in the fine details exist, we argue below that SN 2008am and SN 2006gy exhibited remarkable similarities both in their spectral appearance and evolution.

Zoom In Zoom Out Reset image size

Figure 11 shows the evolution of the emission lines that can be attributed to SN 2008am. For the early Keck spectrum, we give both the total flux profile (dashed red curve) and the line after host subtraction (solid red curve). The left panel presents Hα, the middle panel Hβ and the right panel the He i/Na D feature that is only weakly detected. Dotted vertical lines mark the rest wavelength of the features. In Table 7, we list the basic parameters derived for Hα and Hβ: the shift of the line center with respect to its rest-frame position (Δλ₀, expressed in km s⁻¹), the full width at half maximum (FWHM), integrated flux (F), and equivalent width (EW) based on fitting of Lorentzian profiles (more line profile models will be examined below).

Zoom In Zoom Out Reset image size

In the earliest spectra, the line centers of both Hα and Hβ are redshifted with respect to their rest-frame position (determined from the narrow emission features of the host, as mentioned above). The average redshift is 115 ± 40 km s⁻¹ for both Balmer lines, consistent with some multiple-scattering models (Chugai 2001; Section 3). On the other hand, in the late-phase Keck spectrum the cores of the Balmer lines are slightly blueshifted by −60 km s⁻¹. The Hα profile in this spectrum looks narrower close to the peak, suggesting the presence of an unresolved narrow component, similar to the late-time Hα of SN 2006gy (Smith et al. 2010). This slight blueshift with respect to the rest-frame position may be explained by rotation of the host but might also arise from the effects of multiple scattering (Section 3).

The early-phase spectra have rest-frame FWHM line widths of ∼22 Å and ∼23 Å for Hα and Hβ, respectively. Formally expressing these widths in terms of velocity (see Section 3 for a discussion on the effects of scattering), the FWHMs are ∼1000 km s⁻¹ and 1400 km s⁻¹ for Hα and Hβ, respectively. Hβ seems to be broader in terms of velocity than Hα. After removing the host flux, Hα appears broader in the early Keck spectrum than in the uncorrected spectrum or in any HET spectra, but a more detailed analysis showed that this is an artifact due to the depressed amplitude of the line core as a result of the host galaxy removal.

The HET spectra contain fluxes from both the SN and the host, thus the EWs and line fluxes are higher than those derived from the host-subtracted early-phase Keck spectrum. Comparing the numbers in Table 7 and correcting for the small host oversubtraction in the Keck spectrum, the host contribution to EWs and line fluxes in the HET spectra are estimated to be ∼17 Å and ∼5 × 10⁻¹⁶ erg s⁻¹ cm⁻², respectively.

The evolution of the line strengths and EWs for SN 2008am is very similar to that presented by Smith et al. (2010) for SN 2006gy and to several other Type IIn SNe. Shortly after maximum light the EW of Hα is ∼30–40 Å, rising to ∼300 Å at late phases. The Hα integrated line fluxes, on the other hand, tend to decline in time. SNe are quite heterogeneous regarding the evolution of this parameter according to Smith et al. (2010), but the majority show a similar decline to that of SN 2008am. The line fluxes of Hβ also show a declining trend toward later phases, but the EW of Hβ is roughly constant with time.

During the early phase, the F(Hα)/F(Hβ) ratio is ∼2.3 ± 0.6, which is close to the expected value in case B recombination (Osterbrock, 1989). SN 2006gy showed a very similar flux ratio close to maximum light (Smith et al. 2010). This ratio increases up to ∼17 during the late phases in SN 2008am. This is also in accord with the observations of other SNe. An even higher flux ratio (∼30) was observed for SN 2006gy (Smith et al. 2010) and SN 2006tf (Smith et al. 2008) as well as for the strongly interacting Type IIn SN 1988Z (Turatto et al. 1993).

The He i 5876 Å line clearly shows up in the early-phase Keck spectrum and is also weakly detected in the HET spectra, although the latter data are noisy. The rest-frame FWHM of this line is formally measured to be 1800 ± 600 km s⁻¹. This line width is typical for He i lines in other Type IIn events (e.g., SN 2005la, Pastorello et al. 2008). The total integrated flux in the early Keck spectrum is measured to be ∼5.5 × 10⁻¹⁶ erg s⁻¹ cm⁻² by fitting a Lorentzian profile. It is possible that Na D (λλ 5890, 5896) absorption contaminates this feature making it weaker. The F(Hα)/F(He i 5876) flux ratio is ∼3, probably reflecting the temperature/ionization conditions in the line-forming region. We do not detect the He ii 4686 Å line in any of our spectra.

Smith et al. (2010) delineated three phases for SN 2006gy. In the first phase, between 0 and 90 days after explosion (extending 20 days after maximum), when SN 2006gy was within a factor of two of peak light, Hα showed a nearly symmetric profile with no P Cygni features. Smith et al. associate this phase with conditions where there is a shock wave deep within a dense, opaque circumstellar shell, but the photosphere is in the outer, unshocked CSM. The Hα line is presumed to be excited by photoionization and to be intrinsically narrow, but broadened by multiple scattering on hot, free electrons (Fransson & Chevalier 1989; Chugai 2001). In the second phase, 90–150 days after explosion (20–80 days after maximum light), the Hα line in SN 2006gy broadened somewhat, and developed a distinct, narrow P Cygni component with a velocity of ∼200 km s⁻¹ and strong absorption in the blue wing extending to ∼4000 km s⁻¹. The red wing is nearly constant during this phase and indicates a line of intermediate width of ∼1800 km s⁻¹. Smith et al. attribute this phase to a condition where the photosphere has receded to beneath the forward shock so that radiation from the shocked matter can escape freely. Smith et al. (2010) proposed that the width of the red wing is determined by the Doppler shift of the expanding CDS that is presumed to form between the forward and reverse shocks in the collision of the SN ejecta with the dense CSM (Chugai 2001). The third phase in SN 2006gy is the very late phase, 150–240 days after explosion, when the Hα line becomes narrower. This is the phase when the CSM interaction is expected to decline in strength and the line-emitting region to become optically thin. SN 2008am and SN 2006gy had different rise times in their light curves, but their spectral evolutions show similarities if account is taken for the different timescales.

We first examine the possibility that the intermediate-width lines in the early phase of SN 2008am obtain their wings from multiple electron scattering, a mechanism favored by Smith et al. (2010) for SN 2006gy. Our first four spectra of SN 2008am were obtained when the SN was within a factor of two of maximum light, in keeping with Phase 1 of SN 2006gy by Smith et al. Our spectra can be fit well with a single Lorentzian profile (see below) consistent with multiple electron scattering, and they show no sign of broad features nor P Cygni lines of any width, similar to Phase 1. For these reasons, we believe a plausible case can be made that, despite the different timescales, SN 2008am could be a close cousin to SN 2006gy, and that our early-phase spectra are formed in the phase when the photosphere was in a dense CSM shell, but beyond the forward shock. If this is the case, SN 2008am could very well have then proceeded to Phase 2 defined by Smith et al. (2010), but we simply failed to obtain any spectra in this phase, more than 2 mag below maximum light. At very late phases, the two objects displayed substantially narrower Hα profiles and are again quite similar. Further support for this model may come from the fact that in the early phase both Hα and Hβ showed peaks redshifted by ∼100 km s⁻¹ from their rest-frame wavelength. The same effect was observed for SN 2006gy by Smith et al. (2010) and it is also predicted by the electron scattering model of Chugai (2001). Chugai (2001) assumed a velocity profile in which the velocity decreased with radius as might be caused by radiative acceleration. Fransson & Chevalier (1989) assumed a homologous velocity profile in their multiple scattering models and found profiles with enhanced red wings and a small blue shift of the peak. The details of the line profiles might thus contain information on the velocity profiles in the scattering regions of SN 2008am and related events, but we have not explored this in any depth.

The second possibility is that the intermediate-width lines arise from the post-shock motion of the shocked CSM in a CDS when the photosphere is interior to the forward shock (Fransson 1984; Chugai et al. 2004; Dessart et al. 2009). This would correspond to our early spectra being already in the Phase 2 of Smith et al. (2010). The principal argument against this interpretation is that we do not see any of the narrow or broad absorption manifested by SN 2006gy when it was in this phase, as interpreted by Smith et al. (2010). It could be that the structure of the CSM around SN 2008am is such that the matter interior to and beyond the shock is not sufficiently optically thick to create appreciable absorption. Another possibility is that these absorption features might have been observed had we had better S/N and/or better spectral resolution. We thus cannot rule out this possibility, but find it somewhat less likely than the broadening by multiple electron scattering.

Another mechanism that has been proposed to account for intermediate-width lines is the inward propagation of shocks into dense clumps of matter that have been engulfed by the SN shock (Chugai & Danziger 1994), rather than a single post-shock shell. In this picture, the SN shock sweeps past clumps in the progenitor wind, but the resulting high pressure of the shocked low-density CSM drives a shock into the dense clumps. This may broaden the lines via Doppler motion, and for appropriate choices of cloud sizes, densities, and other parameters one can produce Hα lines of suitable width and intensity. The problem with this model in the current context is that it is designed to have dense clumps separated by less dense, optically thin material. The latter should allow the SN ejecta to be observed directly. Since we see no sign of high-velocity SN features, we find the configuration with the enveloping CDS, as attributed to SN 2006gy by Smith et al. (2010), to be preferable to the clumpy model, where the dense clumps have a rather small filling factor.

In the context of the CSM interaction picture, we considered three different models to account for the line profiles of the early-phase spectra of SN 2008am: (1) thermal Doppler broadening producing Gaussian line shapes, (2) single Thompson scattering on free electrons giving exponential profiles (Laor 2006), and (3) multiple scattering on hot free electrons resulting in Lorentzian profiles (Chugai 2001; Smith et al. 2010). The first phenomenon is common in stellar atmospheres. The second one is proposed for the broad-line region in active galactic nuclei (AGNs)/quasi-stellar objects. The third one might pertain to the dense CSM environment around Type IIn SNe.

We fitted Gaussian, Lorentzian, and exponential profiles to the observed features. The results are listed in Table 8 and plotted in Figure 12. The Gaussian fits were inferior compared to the other two models in terms of conforming to profile shapes. Gaussian models resulted in FWHM = 1500 ± 300 km s⁻¹ for Hα and 1600 ± 500 km s⁻¹ for Hβ. If interpreted as a simple thermal Doppler broadening, the corresponding temperature would be ∼10⁸ K, which is obviously too high. On the other hand, if the line width is attributed to bulk kinematic motion, these velocities are too low to be directly related to the expected velocities of SN ejecta.

Zoom In Zoom Out Reset image size

The exponential fits are motivated by the possibility that the line-forming region might be a photoionized, but less dense, optically thin medium, similar to the environment of AGNs. In this case, the observed line profiles are due to Thompson scattering on hot free electrons. If the medium is less dense, hence optically thin to electron scattering, single scattering is an adequate description. Laor (2006) investigated such a region and derived the emergent line shape to be ∼exp(−Δv/σ), where Δv is the Doppler shift from the line center and σ is the velocity. The electron-scattering optical depth can be expressed as τ_e = exp(−σ/(1.1σ_e))^2.222 where σ_e ∼ (kT_e/m_e) is the velocity dispersion of free electrons. The fits to the observed line profiles resulted in σ = 800 ± 100 km s⁻¹, which gives τ_e ∼0.03 if the electron temperature T_e is assumed to be 11,000 K, close to the continuum effective temperature of SN 2008am. The resulting optical depth is much below unity, verifying the general assumption of this model; however, it should be noted that higher optical depths produce narrower lines in this model. Since we observe intermediate-width lines earlier and a narrower line later, the prediction from the Laor model would be the strong increase of electron optical depth, the opposite of what is expected in an expanding, diluting SN environment.

The fitting of Lorentzians produced the best fit to all line profiles. The resulting FWHMs are ∼1000 ± 300 km s⁻¹ for Hα and ∼1700 ± 500 km s⁻¹ for Hβ. The narrower Hα profile may be the consequence of host contribution to the HET spectra, which is certainly stronger in Hα. Indeed, the FWHM of the host-subtracted early-phase Keck Hα profile is much closer to the average FWHM of the Hβ line. Since the line-forming medium is assumed to be optically thick in this model, the line widths from the Lorentzian fitting cannot be simply related to the physical conditions characterizing the whole line-forming region. A higher density of free electrons should produce wider profiles and prevent the direct escape of the original narrow-width line. As a consequence, as the line-forming medium expands, the line broadening should decrease.

If multiple scattering is the correct interpretation, then the width of the lines we measure in the early-phase spectra of SN 2008am cannot be attributed to bulk kinematic motion. The width of the lines gives an upper limit on the motion of the line-forming region but yields no constraint on the velocity of deeper, optically thick regions. The profiles may give information on the velocity distribution in the scattering region, since the velocity profile can affect the resulting scattering-line profile if the bulk velocity exceeds the electron thermal velocity (Fransson & Chevalier 1989; Chugai 2001). For multiple scattering, the line wings are primarily a measure of the electron scattering optical depth of the line-emitting region. Following Smith et al. (2010), we estimate the Thompson scattering optical depth from the formula $U = (1 - e^{-\tau _T})/ \tau _T$, where U is the ratio of the narrow, unscattered Hα line flux to the total Hα luminosity. Since we were unable to fully resolve the narrow component in the +358 day Keck spectrum, we use a conservative estimate of U ⩽ 0.5. This results in a lower limit of τ_T ⩾ 2 for SN 2008am (Smith et al. obtained τ_T ∼ 15 for SN 2006gy), consistent with an optically thick scattering medium.

Although there is no evidence for a classic SN photosphere, the light curve of SN 2008am (Figure 3) is reminiscent of SNe powered by radioactive decay. The first model that we consider for SN 2008am is thus one of radioactive decay diffusion that was developed by Arnett (1980, 1982) and generalized by Valenti et al. (2008; see also supplementary information in Soderberg et al. 2008). In this model, the power source of the SN luminosity is the radioactive decay of nickel and cobalt, the energy of which diffuses out from the expanding envelope. This model was developed in the context of Type Ia SNe and is appropriate in the absence of an H recombination phase with a constant opacity in the photospheric phase (the effects of H recombination were considered by Arnett & Fu 1989 and Chatzopoulos et al. 2009). The light curve is given by the following formula (Valenti et al. 2008; Chatzopoulos et al. 2009):

$$\begin{eqnarray} L(t) &=& M_{\rm Ni}e^{-x^{2}}[(\epsilon _{\rm Ni}-\epsilon _{\rm Co}) \int _0^x2ze^{z^{2}-2zy}dz + \epsilon _{\rm Co}\nonumber\\ &&\times\int _0^x2ze^{z^{2}-2yz+2zs}dz](1-e^{-At^{-2}}), \end{eqnarray} \tag{ 1 }$$

where x = t/t_m, t_m is the effective diffusion time which is generally close to the rise time to maximum, y = t_m/2t_Ni with t_Ni= 8.8 days, s = t_m(t_Co − t_Ni)/2t_Cot_Ni with t_Co= 111.3 days, M_Ni is the initial nickel mass, and _Ni = 3.9 × 10¹⁰ erg s⁻¹ g⁻¹ and _Co = 6.8 × 10⁹ erg s⁻¹ g⁻¹ are the energy generation rates due to Ni and Co decay. The factor $(1-e^{-At^{-2}})$ accounts for the gamma-ray leakage, where large A means that practically all gamma rays are trapped. The gamma-ray optical depth of the ejecta is taken to be τ_γ = κ_γρR = At⁻², assuming spherical uniform density ejecta with radius R = vt and the Ni/Co confined in the center. This yields A = (3κ_γM_ej)/(4πv²) which is controlled by the gamma-ray opacity, κ_γ. The t⁻² scaling follows from homologous expansion, which is one of the basic assumptions of the simple analytic models that we adopt here. Thus, the main parameters of this model are the nickel mass, M_Ni, and the effective diffusion time t_m, which corresponds to an ejecta mass as given by the following equation for a constant density envelope:

$$\begin{equation} M_{{\rm ej}}(z)=\frac{3}{10} \frac{\beta c}{\kappa } v \frac{t_{m,{\rm ob}}^{2}}{(1+z)^{2}}, \end{equation} \tag{ 2 }$$

where β is an integration constant equal to about 13.8 (Arnett 1982; Valenti et al. 2008), κ is the mean optical opacity, v the mean photospheric expansion velocity, t_m,ob the observed rise time, and z the redshift of the SN. As described in Section 3.2, we have concluded that the observed emission lines are dominated by multiple electron scattering so that their width yields no direct information on the bulk kinematic expansion velocity. In the following, we will scale our results to a characteristic velocity of v= 1000 km s⁻¹ that is consistent with the limit set by line widths and might be compatible with velocities expected for shocks traversing dense circumstellar shells.

We provide a characteristic fit of the radioactive decay diffusion model to the ROTSE light curve of SN 2008am in order to estimate model parameters for this SN and to determine the applicability of this model. For the ROTSE light curve of SN 2008am the explosion date is held fixed, established in Section 2.2. Thus, we are left with three fitting parameters: M_Ni, t_m, and A. The gamma-ray leakage in this model is very small over the range of our observations. A decent fit is obtained for M_Ni= 19 M_☉ and t_m= 41 days. The 41 day effective diffusion time corresponds to an ejecta mass M_ej= 0.2 M_☉, using Equation (2) with the fiducial values κ= 0.4 cm² g⁻¹, appropriate for Thompson scattering in a pure H plasma, and v= 1000 km s⁻¹. If we use v= 2000 km s⁻¹ instead, which would be above the upper limit for the velocity implied from the observed FWHM of the emission lines in the early spectra of SN 2008am, and adopt a lower value for the optical opacity, κ= 0.05 cm² g⁻¹, characteristic of metal-rich ejecta, the ejecta mass can be scaled up to 3.2 M_☉. Even for this low opacity, the velocity would have to be v∼ 12,000 km s⁻¹ for the diffusion mass to exceed the required nickel mass, and we have no rationale to adopt such a high velocity.

The left panel of Figure 14 shows the radioactive decay diffusion model (red solid curve) fitted to the ROTSE light curve of SN 2008am (blue filled squares). The radioactive decay rate of cobalt for M_Ni= 19 M_☉ is shown as the red dashed curve in the same plot for comparison. The light blue data points correspond to the pseudo-bolometric light curve of SN 2008am derived by the SED fits. As we have shown above, the value of M_ej would be greater for lower κ and higher v, but it never becomes equal to or higher than the nickel mass for reasonable choices of these parameters given the short rise time of SN 2008am. This makes the radioactive decay diffusion model unphysical for this event. Although a large amount of radioactive nickel has been suggested for some SLSNe as a product of pair-instability (for example, 22 M_☉ for SN 2006gy; Smith et al. 2007 and 4.5 M_☉ for SN 2006tf; Smith et al. 2008), the inconsistency between the total ejected mass and the nickel mass in SN 2008am is quite remarkable. Integrating under the solid red curve of the left panel of Figure 14 yields a total radiated energy output E_rad ≃ 2.1 × 10⁵¹ erg. On the other hand, the kinetic energy of the explosion in this model, E_KE = (1/2)M_ejv²_mean is 0.6 × 10⁴⁸ erg where $v_{{\rm mean}} = \sqrt{3/5} v$ assuming v= 1000 km s⁻¹ and for κ= 0.4 cm² g⁻¹. Even within the uncertainties of these estimates, the kinetic and total radiated energy are also found to be inconsistent. Radioactive decay may contribute to the output energy, but the most significant contribution must come from other mechanisms. We discuss other models in the following sections.

Zoom In Zoom Out Reset image size

Here we consider a shell-shock diffusion model, similar to that suggested by Smith & McCray (2007) for SN 2006gy. In this type of model, the energy that powers the SN light curve is produced by the diffusion of shock-generated energy through an optically thick CSM shell of large initial radius. Smith & McCray (2007) considered the diffusion of shock-generated energy in a homologously expanding CSM shell in the case where the shock energy input is instantaneous and at the time of maximum light. This type of model does not account for the observed rise of the light curve. Smith & McCray (2007) adopted an L∝r² rise for their model of SN 2006gy based on the early portion of the diffusion models of Arnett (1982). They did not self-consistently consider the input necessary to drive such a rise. They found the diffusion time on the decline to be about the observed rise time for SN 2006gy, but we consider that a coincidence, since such a model fails to account for SN 2008am, as we show below. In order to account for the rising part of the light curve of SN 2008am, we consider a forward shock that propagates through the CSM envelope and deposits energy for a time t_sh = t_max and then shuts off. The effect of the reverse shock will be considered below. This model is somewhat similar to the "top hat" magnetar-input model that is considered in Kasen & Bildsten (2010) with the exception that we solve for the general case of large initial radius. The luminosity deposition function in our model has the form L_sh(t) = E_sh/t_sh for t < t_sh and L_sh = 0 otherwise, where E_sh is the total kinetic energy deposited by the shock in the CSM shell. This model formally assumes that the luminosity E_sh/t_sh is deposited in the center of homologously expanding matter. While not totally self-consistent, this model captures the essence of the shell-shock model on both the rise and decline. Using this energy deposition function and the first law of thermodynamics coupled with the diffusion approximation (as was originally done by Arnett 1980, but for a radioactive decay input), it can be shown that the light curve is given by the following expression:

$$\begin{equation} L(t) = \left\{\begin{array}{@{}ll} \dsty \frac{E_{{\rm sh}}}{t_{{\rm sh}}} [1-e^{-(t^{2}/2t_{d}^2+R_{0}t/v t_{d}^{2})}],& t < t_{{\rm sh}},\\ \ms \dsty \frac{E_{{\rm sh}}}{t_{{\rm sh}}} e^{-(t^{2}/2t_{d}^2+R_{0}t/v t_{d}^{2})} [e^{(t_{{\rm sh}}^{2}/2t_{d}^2+R_{0}t_{{\rm sh}}/v t_{d}^{2})}-1],& t > t_{{\rm sh}}, \end{array}\right. \end{equation} \tag{ 3 }$$

where t_d is the diffusion timescale, v is the characteristic bulk velocity of the CSM shell, and R₀ is the initial radius of the optically thick CSM shell around SN 2008am. We note that in the case where R₀ is small, the result for small initial radius is recovered (Kasen & Bildsten 2010).

To evaluate this model, the second part of Equation (3) (t>t_sh) is fitted to the ROTSE light curve decline of SN 2008am in order to determine E_sh, R₀, and t_d. Then using the best-fitting parameters, we plot the expected rise to maximum for the event within this class of model. We first consider the case for which the diffusion time on the decline is forced to be equal to the input time, which is also the rise time to maximum (t_max= 34 days). The best fit to the data in this case, is obtained for E_sh = 1.6 × 10⁵¹ erg and R₀ = 3.0 × 10¹³ cm if R₀ is constrained to be larger than the smallest possible progenitor radius (10¹¹ cm for Wolf–Rayet stars). Forcing t_d on the decline to be equal to the rise time of 34 days means the light curve falls below the indicated lower limits. The result of this failed "fit" is shown as the red dashed curve in the middle panel of Figure 14. We next consider the general case where t_d>t_sh, and we also fit t_d as a free parameter. The best fit is obtained for E_sh = 5.5 × 10⁵¹ erg, R₀ = 1.0 × 10¹⁴ cm, and t_d= 87 days. The result of this fit is shown as the red solid curve in the middle panel of Figure 14. The 87 day diffusion time corresponds to a CSM shell having a mass M_shell= 1.0 M_☉ (using Equation (2), again for v= 1000 km s⁻¹ and κ= 0.4 cm² g⁻¹). For the extreme values of v= 2000 km s⁻¹ and κ= 0.05 cm² g⁻¹ the shell mass is 16 M_☉. The radius of the shell at maximum light adopting a constant expansion velocity equal to 1000 km s⁻¹ is R_max = 3.7 × 10¹⁴ cm. These results are consistent with a relatively large optically thick CSM shell around SN 2008am. The total radiated energy within the context of this model is equal to 10⁵¹ erg. Note that the rise in this simple "top hat" model is concave upward rather than convex.

We conclude that the derived parameters for the CSM shell of SN 2008am based on a shell-shock diffusion model show that it may be less massive and smaller than the ones determined for SN 2006gy (M_shell= 10 M_☉, R_shell= 2.4 × 10¹⁵ cm; Smith & McCray 2007) and for SN 2006tf (M_shell= 18 M_☉, R_shell= 2.7 × 10¹⁵ cm, Smith et al. 2008 using the same type of model). It should be noted that the estimates for the properties of the shell of SN 2006tf are very uncertain due to the lack of data during the rising part of the light curve and thus the lack of an accurate explosion date. The larger mass derived for SN 2006gy is mainly determined by the larger rise time of 70 days versus 34 days for SN 2008am and by the larger velocity of 4000 km s⁻¹ versus the fiducial 1000 km s⁻¹ that we have adopted for SN 2008am.

An estimate of the optical depth of the CSM shell under the assumption that the derived shell radius is significantly larger than the radius of the progenitor and for constant density profile is given by

$$\begin{equation} \tau _{{\rm shell}}=\frac{3 \kappa M_{{\rm shell}}}{4 \pi R_{{\rm shell}}^{2}}. \end{equation} \tag{ 4 }$$

Assuming that electron scattering in a hydrogen plasma is the main source of opacity (κ= 0.4 cm² g⁻¹), v= 1000 km s⁻¹ (and taking for the CSM shell the derived parameters M_shell= 1.0 M_☉ and R_shell = R_max = 3.7 × 10¹⁴ cm) an estimate for the optical depth of the shell around SN 2008am at maximum light is τ_max∼ 1390 while for SN 2006tf it is τ_max∼ 480 (Smith et al. 2008) and for SN 2006gy it is τ_max∼ 330 (Smith et al. 2007). Thus, although less massive, the CSM shell of SN 2008am is very optically thick due to its small radius compared to the shells of those other SLSNe, which results in higher density. The values of the optical depth will vary for different choices of optical opacity. For the derived optical depth, we can estimate when the shell will become optically thin. Since τ ∝ R⁻², $R_{2} = R_{{\rm max}} \sqrt{\tau _{{\rm max}}/\tau _{2}}$ for a constant expansion velocity, where R₂ = R_max + vΔt. Combining those two equations yields $\Delta t = R_{{\rm max}} (\sqrt{\tau _{{\rm max}}/\tau _{2}}-1)/v$. Thus, we can estimate how long will it take for the CSM to become optically thin (τ₂ = 1) for τ_max= 3600 and R_max = 3.7 × 10¹⁴ cm. This yields Δt ≃ 4.3 yr for v= 1000 km s⁻¹.

An estimate for the mass-loss rate is given by $\dot{M} = M_{{\rm shell}} v_{w}/(t_{{\rm max}} v)$. For SN 2008am (M_shell= 1.0 M_☉), we find $\dot{M} = 0.9 \times 10^{-2} v_{w}$ M_☉ yr⁻¹ for v= 1000 km s⁻¹ and with v_w in units of km s⁻¹. For the range of wind velocities 10 km s⁻¹ <v_w< 1000 km s⁻¹ the inferred mass-loss rates range from 0.1 up to 10 M_☉ yr⁻¹. These extraordinary mass-loss rate estimates imply a very massive luminous blue variable (LBV)-type progenitor for SN 2008am, as was suggested for SN 2006gy (Smith et al. 2008). We note again that the choice of 1000 km s⁻¹ as the expansion velocity of the circumstellar shell, which was constrained by fitting Lorentzian profiles to the Balmer emission lines in the early spectra of SN 2008am (see Section 3.2), is just for illustration purposes.

We also investigated the possibility that SN 2008am is powered by the reverse shock that we expect to have formed due to an ejecta–CSM interaction. Chevalier & Fransson (2003) estimate the density of the swept up matter behind the reverse shock to be

$$\begin{equation} \rho _{{\rm rev}}=\frac{(-n-4)(-n-3)}{2} \rho _{\rm CSM}, \end{equation} \tag{ 5 }$$

where n is a constant that describes the density profile of the ejecta (ρ_ej ∝ Rⁿ). The density behind the reverse shock is higher than that behind the forward shock for n ⩽ −7. The luminosity from the reverse shock scales as L_rev ∝ t^{−3/(−n−2)}. The reverse shock may be adiabatic or radiative depending on the optical depth of the shell. We fit a power law of the form L = At^p to the decline of the observed ROTSE light curve in order to get an estimate of n and determine whether this model can account for SN 2008am. This procedure effectively assumes that the reverse shock dominates the forward shock and that the diffusion time in the reverse-shocked gas is small. The best-fit power-law model gives p = −1.1 ± 0.3 which corresponds to n = −4.7 ± 0.1. Since n> − 7 we conclude that the density behind the reverse shock may be smaller than that behind the forward shock and hence that the luminosity from the reverse shock may be lower than that of the forward shock.

A related model for the luminosity source for SN 2006gy (Ofek et al. 2007), SN 2002ic (Hamuy et al. 2003) and SN 2005gj (Aldering et al. 2006) is that of the interaction between the ejecta and the CSM in the case where the latter is dense, but optically thin. In such a model, a radiative shock forms due to the collision between the ejecta and the CSM (Chevalier & Fransson 1994). We consider the luminosity produced by the forward shock. The luminosity in this case given by (Ofek et al. 2007)

$$\begin{equation} L = 2 \pi \rho _{\rm CSM} R^{2} v_{{\rm sh}}^{3}, \end{equation} \tag{ 6 }$$

where ρ_CSM is the local CSM density, R = v_sht is the radius of the shock at time t and v_sh the velocity of the shock. Assuming that the shock enters the optically thin CSM envelope at a radius R₀ and that the density profile follows a power law of the form ρ = ρ₀(R/R₀)^m, where R = R₀ + v_sht and m is the slope of the density profile (for a constant velocity wind m = −2), we can rewrite Equation (6) as

$$\begin{equation} L = 2 \pi \rho _{0} R^{2} v_{{\rm sh}}^{3} [(R_{0}+v_{{\rm sh}}t)/R_{0}]^{m}. \end{equation} \tag{ 7 }$$

In our analysis, we assume that the shock velocity is constant for simplicity. More accurate solutions that take the change of the shock velocity into account can be found in Chevalier (1982) and Chevalier & Fransson (2003). If the CSM around the progenitor star is everywhere optically thin, R₀ represents the radius of the progenitor star.

An estimate for an upper limit of the mass-loss rate of the progenitor of SN 2008am in the context of an optically thin CSM comes from the X-ray flux upper limit measurement (see Section 2.2). According to Immler & Kuntz (2005), the X-ray luminosity from thermal Bremsstrahlung is given by

$$\begin{equation} L_{X}=\frac{4}{(\pi m)^{2}} \Lambda (T) \left(\frac{\dot{M}}{v_{w}}\right)^{2} (v_{{\rm sh}} t)^{-1}, \end{equation} \tag{ 8 }$$

where L_X is the X-ray luminosity, m_p the mean mass per particle (we adopt m_p = 2.1 × 10⁻²⁴ g for an H+He plasma), and Λ(T) the cooling function of a plasma that has temperature T. As in Immler & Kuntz (2005), we adopt Λ(T) = 3 × 10⁻²³ erg cm⁻³ s⁻¹ for T = 10⁷ K which is characteristic of the post-shock temperature. For the available XRT upper limit of L_X = 10⁴³ erg s⁻¹ corresponding to ∼50 rest-frame days after explosion and 16 days after maximum, we obtain an upper limit of $\dot{M} = 0.07$ (v_w/1000 km s⁻¹) M_☉ yr⁻¹. It should be noted that this model assumes a constant wind-density parameter $w = \dot{M}/v_{w}$.

Most of the radiation that is produced in models of the radiative forward shock is emitted at ultraviolet and X-ray wavelengths. We assume in the context of this model that the ROTSE light curve is a good proxy to the true bolometric light curve of SN 2008am as would be the case if a substantial fraction of UV/X-ray luminosity is absorbed and re-radiated by the shell that forms behind the forward shock. Then, the best fit of Equation (7) to the ROTSE light curve of SN 2008am would provide us with estimates of ρ₀, R₀, and m.

The optical depth of the CSM is

$$\begin{equation} \tau _{{\rm thin}} = \kappa \int _{R_{0}}^{\infty } \rho dR = -\frac{\kappa \rho _{0} R_{0}}{m+1}, \end{equation} \tag{ 9 }$$

where m < −1. In our fitting process, we demand that this optical depth beyond R₀ is less than one for κ= 0.4 cm² g⁻¹. Thus, we choose the best-fit model for which this condition is met in order to be self-consistent with the initial assumption of an optically thin CSM. We were unable to determine a satisfactory fit of this simple optically thin model to the ROTSE light curve of SN 2008am that was self-consistent, in terms of the optical depth. For all the best-fit models the optical depth was well above unity, and for models with τ_thin < 1 the derived values for the density and the radius were unphysical.

A purely optically thin model not only fails to account for the light curve of SN 2008am but also for the observed spectroscopic characteristics. Chevalier (1982) and Chevalier & Fransson (2003) predict that the bulk of the luminosity from optically thin forward shock emission is in the UV and X-ray region of the spectrum. It seems unlikely that the optically thin case would reproduce the spectra and spectral evolution of SN 2008am. In addition, this model being optically thin, it is incompatible with the failure to see SN photospheric features. We conclude that an optically thin ejecta–CSM interaction model alone does not provide a satisfactory explanation for SN 2008am.

Next, we discuss a hybrid model in which the CSM comprises both an optically thick and an optically thin region. This is somewhat similar to the model proposed by Chugai & Danziger (1994) in which the CSM contains two components: an optically thin rarefied wind and optically thick clumps or an optically thick disk around the progenitor. In this model, the light curve is described by the sum of Equations (3) and (7). We assume the whole structure follows a single power-law density profile ρ ∝ R^m so that the optical depth at a given point is

$$\begin{equation} \tau = \frac{-\kappa \rho _{0} R}{m+1} \left(\frac{R}{R_{0}}\right)^{m}. \end{equation} \tag{ 10 }$$

We take the point τ= 1 at radius R = R₁ to represent the boundary between the optically thick and optically thin regions. In this case, R₀ corresponds to the inner initial radius of the optically thick part of the CSM shell, which is also the radius of the progenitor star, where the density is ρ₀. The radius of unity optical depth is R₁ = [ − R^m₀(m + 1)/(κρ₀)]^1/(m+1), and we demand that this radius is larger than R₀ so that the CSM has both an optically thin and an optically thick part. This requires a slightly different set of fitting parameters: R₀, t_sh, t_d, m, ρ₀, and E_sh, and we again fix t_sh = t_max= 34 days. The best fit (plotted in the middle panel of Figure 14) is the same as that obtained in the purely optically thick shell case indicating that any contribution from optically thin emission is negligible. Therefore, for a hybrid model, the optically thick component (Equation (3)) dominates all phases of the observed ROTSE light curve of SN 2008am.

Kasen & Bildsten (2010) and Woosley (2010) proposed the idea that the light curves of some SLSNe may be powered by the spin-down of young magnetars. In such a model, the energy input by the magnetar is given by the dipole spin-down formula:

$$\begin{equation} L_{p}(t) = \frac{E_{p}}{t_{p}} \frac{l-1}{(1+t/t_{p})^{l}}, \end{equation} \tag{ 11 }$$

where E_p is the initial magnetar rotational energy, t_p is the characteristic time scale for spin-down that depends on the strength of the magnetic field and l= 2 for a magnetic dipole. For a fiducial moment of inertia, the initial period of the magnetar in units of 10 ms is given by P₁₀ = (2 × 10⁵⁰ erg s⁻¹/E_p)^0.5. The magnetic field of the magnetar can be estimated from P₁₀ and t_p as B₁₄ = (1.3P²₁₀/t_p,yr)^0.5, where B₁₄ is the magnetic field in units of 10¹⁴ G and t_p,yr is the characteristic time scale for spin-down in units of years.

Adopting Equation (11) as the energy deposition function (instead of the corresponding one for the radioactive decays of Nickel and Cobalt that leads to the Arnett 1980 solution) and using the first law of thermodynamics coupled with the diffusion approximation, it can be shown that the light curve of a supernova powered by a magnetar is given by the following expression:

$$\begin{equation} L(t)=\frac{E_{p}}{t_{p}} e^{-x^{2}/2} \int _{0}^{x}e^{z^{2}/2} \frac{z}{(1+yz)^{2}}dz, \end{equation} \tag{ 12 }$$

where x = t/t_d and y = t_d/t_p with t_d being the characteristic diffusion time. In this treatment, we assume that the initial radius of the progenitor star, R₀, is small. We fit Equation (12) to the ROTSE light curve of SN 2008am and we obtain E_p = 3.2 × 10⁵¹ erg, t_d= 64 days and t_p= 64 days. Note that in this model, neither t_p nor t_d corresponds to the time of maximum light. The fit of this model is shown as the red solid curve in the right panel of Figure 14. Using the fitted values of E_p and t_p, the implied initial period of the magnetar would be 2.5 ms and the magnetic field ≃0.7 × 10¹⁴ G. The total radiated energy implied by the magnetar model is ∼10⁵¹ erg. The values of P and B lie within the range predicted in Duncan & Thompson (1992) for magnetars assuming the field to arise in an α-Ω dynamo. The magnetar model does provide a reasonable fit to the data including the rise time (Figure 14, right panel). Although the magnetar model provides a decent fit to the light curve, it gives no natural explanation for the emission lines.