Type II Supernova Spectral Diversity. II. Spectroscopic and Photometric Correlations^*

Before proceeding with our spectral analysis, below we summarize the parameters we use, as defined in Paper I.

• 1.  
  v corresponds to the expansion velocity. It is measured from the minimum flux of the absorption component of P-Cygni line profile. In this analysis, we measure this parameter for 11 features in the photospheric phase: ${{\rm{H}}}_{\alpha }$, ${{\rm{H}}}_{\beta }$, Fe ii λ4924, Fe ii λ5018, Fe ii λ5169, Sc ii/Fe ii λ5531, Sc ii multiplet λ5663, Na i D, Ba ii λ6142, Sc ii λ6247, and O i λ7774. In the case of ${{\rm{H}}}_{\alpha }$, the velocity was also derived using the full width at half maximum (FWHM) of the emission component.
• 2.  
  ${\rm{\Delta }}v$(${{\rm{H}}}_{\beta })$: defined as the rate of change of the expansion velocity of the ${{\rm{H}}}_{\beta }$ feature. This parameter was measured at five distinct intervals (see Paper I); however, here we only use the interval $50\leqslant t\leqslant 80$ days, as this shows the highest correlation with other parameters.
• 3.  
  ${\rm{\Delta }}\mathrm{vel}$ is defined as the velocity difference between ${{\rm{H}}}_{\alpha }$ and Fe ii λ5018, and Na i D and Fe ii λ5018.
• 4.  
  pEW corresponds to the absorption/emission strength of a particular line. Here, we measure the absolute value of pEW for the same features mentioned above.
• 5.  
  a/e is defined as the flux ratio of the absorption to emission component of ${{\rm{H}}}_{\alpha }$ P-Cygni profile. This ratio is the inverse of that presented by Patat et al. (1994). We propose a/e as this deals better with weak absorption values that are shown by a number of SNe II in our sample.

While measurements were performed in all epochs at which we obtained spectra, we choose to define common epochs between SNe at 30, 50, and 80 days post-explosion. An interpolation and extrapolation is used to obtain parameter values at these epochs. The values obtained by the interpolation are used when two available spectra are present ±15 days around the common epoch, while the values from the extrapolation are used at ±10 days. These intervals were chosen as they increase the strength of observed correlations. Using bigger intervals deteriorates the correlations because the polynomial does not produce reliable results in some cases (particularly for the pEW). At ±15 and ±10 days for interpolation and extrapolation, respectively, the results do not show a significant change compared to those obtained using a smaller interval. Hence, our choice of intervals is justified. To estimate the velocity at a common epoch, we do an interpolation/extrapolation using a power-law fit. For the pEW, we use a low order (first or second) polynomial fit. Power-law fits were found to produce satisfactory results in the case of velocity measurements; however, for pEWs, we found that low order polynomials were required. For this parameter, we used a low order polynomial and determined the best fit using the normalized root mean square (rms) of different orders. The errors of each measurement were obtained with the rms error fit. In summary, we are able to use spectral parameter values in 88, 84, and 59 SNe at 30, 50, and 80 days, respectively.

Historical separation of SNe II into distinct classes was based on photometric differences in, e.g., decline rates and absolute magnitudes. Hence it is essential to include photometric parameters in our analysis for a full understanding of observed correlations and their implications for SN II physics. Here, we use the V-band photometric parameters already defined (and measured) in A14, which we now summarize.

• 1.  
  t₀ corresponds to the explosion epoch (see Paper I for more details of their estimation).
• 2.  
  t_tran is determined as the transition between the initial decline (s₁) and the plateau decline (s₂).
• 3.  
  t_end corresponds to the end of the optically thick phase (i.e., the end of the plateau phase).
• 4.  
  t_PT is the midpoint of the transition from plateau to radioactive tail.
• 5.  
  OPTd is the duration of the optically thick phase and is equal to ${t}_{\mathrm{end}}-{t}_{0}$.
• 6.  
  Pd is the plateau duration, defined between t_tran and t_end.
• 7.  
  M_max is defined as the initial peak in the V-band light curve.
• 8.  
  M_end is defined as the absolute V-band magnitude measured 30 days before t_PT.
• 9.  
  M_tail is defined as the absolute V-band magnitude measured 30 days after t_PT.
• 10.  
  s₁ is defined as the decline rate (V-band magnitudes per 100 days) of steeper slope of the light curve.
• 11.  
  s₂ is defined as the decline rate (V-band magnitudes per 100 days) of the second, shallower slope in the light curve.
• 12.  
  s₃ is defined as the linear decline rate (V-band magnitudes per 100 days) of the slope in the radioactive tail part.
• 13.  
  ⁵⁶Ni mass corresponds to the mass of radioactive nickel synthesized in the explosion. (A14 for exact details of how this was estimated).

Initial values for these parameters can be found in Table 5 in A14; however, it should be noted that in this work some of these parameters have been updated: t_tran, OPTd, Pd, M_max, M_end, M_tail, s₁, and s₂. In the case of magnitudes, it was found that stronger correlations were obtained with other parameters before any extinction corrections were made. This suggests that (a) in the vast majority of cases host galaxy extinction is relatively small, and (b) when we do make extinction corrections (using the absorption Na i D in A14), such corrections are not particularly accurate. Therefore, all magnitudes are being used without host galaxy extinction corrections. For t_tran, we used the F-test to decide whether a one or two slope fit was better; A14 used the BIC criterion. The main difference resides in how the F-test penalizes the number of parameters of each model (more details in L. Galbany et al. 2017, in preparation). This method increases the number of SNe with t_tran available, and in turn this increases the number of SNe for which we can define s₁ and Pd. A visual check of those SNe II showing t_trans using both the F-test and the BIC criterion was performed, and this gives us confidence in the use of the former in this work. All values used in the current analysis are listed in Table 1.

Table 1.  Photometric Parameters

SN	Pd	OPTd	Cd	Mmax	Mend	Mtail	s1	s2	s3	56Ni mass	Δ(B − V)${}_{\mathrm{10,30}}$
	(days)	(days)	(days)	(mag)	(mag)	(mag)	(mag.100 day−1)	(mag.100 day−1)	(mag.100 day−1)	${M}_{\odot }$
1986L	59.56 ± 0.71	93.74 ± 6.71	34.18 ± 3.08	−18.19 ± 0.20	−16.88 ± 0.20	−14.37 ± 0.20	3.26 ± 0.14	1.26 ± 0.03	⋯	>0.061	2.63 ± 0.42
1990K	⋯	⋯	⋯	⋯	⋯	⋯	⋯	2.39 ± 0.08	⋯	⋯	⋯
1991al	⋯	⋯	⋯	−17.51 ± 0.15	−17.03 ± 0.15	−14.71 ± 0.15	⋯	1.45 ± 0.04	1.26 ± 0.26	${0.067}_{-0.021}^{+0.016}$	3.69 ± 0.24
1992af	⋯	47.03 ± 6.71	⋯	−17.33 ± 0.12	−17.20 ± 0.12	−15.06 ± 0.12	⋯	0.58 ± 0.03	1.07 ± 0.08	${0.079}_{-0.029}^{+0.018}$	1.90 ± 0.83
1992ba	⋯	106.97 ± 8.54	⋯	−15.34 ± 0.80	−14.75 ± 0.80	−12.34 ± 0.80	⋯	0.72 ± 0.02	0.86 ± 0.07	${0.011}_{-0.015}^{+0.006}$	2.80 ± 0.21
1993K	⋯	⋯	⋯	−17.92 ± 0.23	−17.24 ± 0.23	⋯	⋯	2.36 ± 0.08	⋯	⋯	1.82 ± 0.09
1993S	⋯	⋯	⋯	−17.52 ± 0.07	−16.29 ± 0.07	⋯	⋯	2.34 ± 0.04	⋯	⋯	3.63 ± 0.51
1999br	⋯	⋯	⋯	−13.77 ± 0.40	−13.56 ± 0.40	⋯	⋯	0.14 ± 0.02	⋯	>0.002	3.58 ± 0.39
1999ca	40.54 ± 0.92	79.48 ± 7.62	38.94 ± 7.06	−17.48 ± 0.21	−16.60 ± 0.21	−13.78 ± 0.21	3.49 ± 0.16	1.65 ± 0.06	1.74 ± 0.33	>0.047	⋯
1999cr	43.55 ± 1.68	79.06 ± 7.62	35.51 ± 4.34	−16.90 ± 0.10	−16.23 ± 0.10	⋯	1.78 ± 0.09	0.49 ± 0.08	⋯	⋯	1.77 ± 0.38
1999em	67.04 ± 2.12	96.04 ± 5.83	29.00 ± 5.43	−16.76 ± 0.07	−16.37 ± 0.07	−13.93 ± 0.07	0.86 ± 0.11	0.30 ± 0.02	0.88 ± 0.05	${0.050}_{-0.009}^{+0.008}$	3.07 ± 0.23
S0210	⋯	93.57 ± 9.49	⋯	−16.21 ± 0.04	−15.90 ± 0.04	⋯	⋯	2.37 ± 0.07	⋯	⋯	⋯
2002fa	⋯	68.289 ± 7.62	⋯	−16.95 ± 0.04	−16.65 ± 0.04	⋯	⋯	1.56 ± 0.11	⋯	>0.066	⋯
2002gd	⋯	⋯	35.00 ± 4.09	−15.43 ± 0.28	−14.85 ± 0.28	⋯	1.87 ± 0.09	0.15 ± 0.04	⋯	⋯	3.21 ± 0.28
2002gw	⋯	88.33 ± 5.83	⋯	−15.76 ± 0.23	−15.48 ± 0.23	−13.07 ± 0.23	⋯	0.22 ± 0.03	0.75 ± 0.09	${0.012}_{-0.004}^{+0.003}$	2.55 ± 0.20
2002hj	⋯	90.24 ± 7.62	⋯	−16.91 ± 0.10	−16.03 ± 0.10	−13.59 ± 0.10	⋯	1.57 ± 0.05	1.41 ± 0.01	>0.026	⋯
2002hx	⋯	68.03 ± 9.49	⋯	−17.00 ± 0.07	−16.36 ± 0.07	−14.60 ± 0.07	⋯	1.51 ± 0.03	1.24 ± 0.04	${0.053}_{-0.023}^{+0.016}$	2.26 ± 0.32
2002ig	⋯	⋯	⋯	−17.66 ± 0.03	−16.76 ± 0.03	⋯	⋯	2.20 ± 0.12	⋯	⋯	2.42 ± 0.66
2003B	⋯	86.19 ± 11.40	⋯	−15.36 ± 0.28	−15.11 ± 0.28	−12.77 ± 0.28	⋯	0.65 ± 0.03	1.07 ± 0.03	${0.017}_{-0.009}^{+0.006}$	⋯
2003bl	⋯	95.81 ± 4.24	⋯	−15.35 ± 0.14	−15.01 ± 0.14	⋯	⋯	0.35 ± 0.02	⋯	⋯	2.83 ± 0.46
2003bn	62.96 ± 10.51	92.97 ± 4.24	30.01 ± 10.93	−16.80 ± 0.16	−16.34 ± 0.16	−13.72 ± 0.16	1.38 ± 0.9	0.32 ± 0.03	⋯	>0.038	2.94 ± 0.25
2003ci	⋯	92.53 ± 8.54	⋯	−16.83 ± 0.07	−15.70 ± 0.07	⋯	⋯	1.91 ± 0.04	⋯	⋯	⋯
2003cn	48.86 ± 3.99	69.8 ± 5.00	20.94 ± 5.65	−16.26 ± 0.11	−15.61 ± 0.11	⋯	2.7 ± 1.14	1.34 ± 0.04	⋯	⋯	2.73 ± 0.45
2003cx	⋯	90.82 ± 5.83	⋯	−16.79 ± 0.06	−16.38 ± 0.06	−14.32 ± 0.06	⋯	0.61 ± 0.04	⋯	>0.051	1.95 ± 0.65
2003E	⋯	101.42 ± 7.62	⋯	−15.70 ± 0.15	−15.48 ± 0.15	⋯	⋯	−0.10 ± 0.03	⋯	⋯	1.49 ± 0.25
2003ef	⋯	92.93 ± 9.49	⋯	−16.72 ± 0.14	−16.15 ± 0.14	⋯	⋯	0.78 ± 0.02	⋯	⋯	2.76 ± 0.39
2003eg	⋯	⋯	30.87 ± 5.04	−17.81 ± 0.13	−14.57 ± 0.13	⋯	6.75 ± 0.18	1.73 ± 0.13	⋯	⋯	1.35 ± 0.18
2003ej	⋯	68.97 ± 5.83	⋯	−17.66 ± 0.12	−15.66 ± 0.12	⋯	⋯	3.29 ± 0.04	⋯	⋯	2.95 ± 0.29
2003fb	⋯	88.27 ± 6.71	⋯	−15.56 ± 0.12	−15.25 ± 0.12	−13.10 ± 0.12	⋯	0.46 ± 0.06	1.61 ± 0.39	>0.017	⋯
2003gd	⋯	⋯	⋯	⋯	−15.97 ± 0.40	−12.58 ± 0.40	⋯	2.22 ± 0.05	1.03 ± 0.04	${0.012}_{-0.012}^{+0.006}$	⋯
2003hd	⋯	84.39 ± 5.83	⋯	−17.29 ± 0.06	−16.72 ± 0.06	−13.85 ± 0.06	⋯	0.93 ± 0.04	0.72 ± 0.68	${0.029}_{-0.009}^{+0.007}$	2.80 ± 0.21
2003hg	63.98 ± 1.67	108.5 ± 5.83	44.52 ± 5.27	−16.38 ± 0.16	−15.50 ± 0.16	⋯	1.35 ± 0.05	0.52 ± 0.04	⋯	⋯	3.01 ± 0.47
2003hk	58.96 ± 2.34	87.00 ± 5.00	28.04 ± 4.63	−17.02 ± 0.10	−16.36 ± 0.10	−13.14 ± 0.10	3.09 ± 0.20	1.61 ± 0.06	0.40 ± 0.66	>0.017	⋯
2003hl	⋯	108.92 ± 5.83	⋯	−15.91 ± 0.30	−15.23 ± 0.30	⋯	⋯	0.76 ± 0.01	⋯	⋯	2.92 ± 0.31
2003hn	58.34 ± 1.55	90.1 ± 10.44	31.76 ± 10.12	−16.74 ± 0.10	−15.96 ± 0.10	−13.27 ± 0.10	5.69 ± 0.27	2.52 ± 0.07	1.08 ± 0.05	${0.035}_{-0.011}^{+0.008}$	2.88 ± 0.29
2003ho	⋯	⋯	⋯	⋯	−14.75 ± 0.16	−12.00 ± 0.16	⋯	2.25 ± 0.11	1.69 ± 0.10	>0.005	⋯
2003ib	⋯	⋯	⋯	−17.10 ± 0.09	−16.09 ± 0.09	⋯	⋯	1.64 ± 0.03	⋯	⋯	1.16 ± 0.22
2003ip	⋯	80.74 ± 5.00	⋯	−17.75 ± 0.13	−16.65 ± 0.13	⋯	⋯	2.03 ± 0.03	⋯	⋯	2.31 ± 0.25
2003iq	⋯	84.91 ± 3.61	⋯	−16.69 ± 0.30	−16.18 ± 0.30	⋯	⋯	0.72 ± 0.01	⋯	⋯	2.42 ± 0.28
2003T	⋯	90.59 ± 10.44	⋯	−16.54 ± 0.08	−16.03 ± 0.08	−13.67 ± 0.08	⋯	0.69 ± 0.02	2.02 ± 0.14	>0.030	2.74 ± 0.82
2004ej	⋯	97.14 ± 8.54	⋯	−16.62 ± 0.21	−16.13 ± 0.21	−12.92 ± 0.21	⋯	1.04 ± 0.04	0.89 ± 0.13	${0.019}_{-0.007}^{+0.005}$	⋯
2004er	57.27 ± 1.66	120.12 ± 5.00	62.85 ± 2.6	−16.74 ± 0.16	−15.67 ± 0.16	⋯	1.08 ± 0.02	0.52 ± 0.02	⋯	⋯	2.13 ± 1.53
2004fb	⋯	⋯	⋯	−16.19 ± 0.11	−15.46 ± 0.11	⋯	⋯	1.26 ± 0.07	⋯	⋯	⋯
2004fc	68.06 ± 2.68	106.06 ± 3.16	38.00 ± 2.86	−16.21 ± 0.31	−15.41 ± 0.31	⋯	1.13 ± 0.03	0.50 ± 0.05	⋯	⋯	2.92 ± 0.20
2004fx	⋯	68.41 ± 5.00	⋯	−15.58 ± 0.24	−15.33 ± 0.24	−12.87 ± 0.24	⋯	0.25 ± 0.02	0.93 ± 0.08	${0.014}_{-0.006}^{+0.004}$	1.52 ± 0.20
2005af	⋯	107.01 ± 15.30	⋯	⋯	−14.94 ± 0.36	−13.41 ± 0.36	⋯	0.40 ± 0.05	1.25 ± 0.03	${0.026}_{-0.021}^{+0.012}$	⋯
2005an	36.02 ± 0.63	74.67 ± 5.00	38.65 ± 6.03	−17.07 ± 0.18	−15.89 ± 0.18	⋯	3.21 ± 0.05	1.85 ± 0.05	⋯	⋯	2.89 ± 0.08
2005dk	38.97 ± 1.47	82.24 ± 6.71	43.27 ± 6.18	−17.52 ± 0.14	−16.74 ± 0.14	⋯	2.25 ± 0.09	1.10 ± 0.07	⋯	⋯	⋯
2005dn	45.82 ± 3.31	78.72 ± 6.71	32.9 ± 6.85	−17.01 ± 0.24	−16.38 ± 0.24	⋯	2.00 ± 0.23	1.48 ± 0.04	⋯	⋯	⋯
2005dt	⋯	112.9 ± 9.49	⋯	−16.39 ± 0.09	−15.84 ± 0.09	⋯	⋯	0.58 ± 0.06	⋯	⋯	⋯
2005dx	⋯	89.98 ± 7.62	⋯	−16.05 ± 0.08	−15.24 ± 0.08	−12.12 ± 0.08	⋯	1.26 ± 0.05	⋯	>0.007	3.21 ± 0.31
2005dz	32.40 ± 2.84	81.86 ± 5.00	49.46 ± 4.91	−16.57 ± 0.12	−15.97 ± 0.12	−13.42 ± 0.12	1.09 ± 0.03	0.36 ± 0.10	⋯	>0.021	2.28 ± 0.25
2005J	53.01 ± 1.93	97.01 ± 7.62	44.00 ± 7.26	−17.28 ± 0.14	−16.35 ± 0.14	⋯	1.51 ± 0.03	1.04 ± 0.02	⋯	⋯	2.93 ± 0.17
2005lw	⋯	107.25 ± 10.44	⋯	−17.07 ± 0.08	−15.47 ± 0.08	⋯	⋯	2.04 ± 0.04	⋯	⋯	2.86 ± 0.70
2005Z	⋯	78.88 ± 6.71	⋯	−17.17 ± 0.11	−16.17 ± 0.11	⋯	⋯	1.76 ± 0.01	⋯	⋯	3.04 ± 0.29
2006ai	38.28 ± 0.46	63.26 ± 5.83	24.98 ± 5.02	−18.06 ± 0.14	−17.03 ± 0.14	−14.53 ± 0.14	4.61 ± 0.10	2.05 ± 0.04	1.78 ± 0.24	>0.050	2.02 ± 0.15
2006be	43.81 ± 1.32	76.2 ± 6.71	32.39 ± 9.10	−16.47 ± 0.29	−16.08 ± 0.29	⋯	1.19 ± 0.08	0.63 ± 0.02	⋯	⋯	3.04 ± 0.11
2006bl	⋯	⋯	17.3 ± 11.16	−18.23 ± 0.07	−16.52 ± 0.07	⋯	3.05 ± 0.54	2.41 ± 0.06	⋯	⋯	2.15 ± 0.38
2006ee	59.04 ± 2.95	85.15 ± 5.00	26.11 ± 4.97	−16.28 ± 0.15	−16.04 ± 0.15	⋯	0.98 ± 0.29	0.17 ± 0.03	⋯	⋯	2.45 ± 0.19
2006it	⋯	⋯	⋯	−16.20 ± 0.15	−15.97 ± 0.15	⋯	⋯	1.14 ± 0.10	⋯	⋯	3.98 ± 0.14
2006iw	⋯	⋯	⋯	−16.89 ± 0.07	−16.18 ± 0.07	⋯	⋯	1.00 ± 0.03	⋯	⋯	2.24 ± 0.67
2006ms	⋯	⋯	32.83 ± 6.62	−16.18 ± 0.15	−15.93 ± 0.15	⋯	1.65 ± 0.2	−0.05 ± 0.45	⋯	>0.056	4.12 ± 0.12
2006qr	⋯	96.85 ± 7.62	⋯	−15.99 ± 0.14	−14.24 ± 0.14	⋯	⋯	1.40 ± 0.02	⋯	⋯	2.88 ± 0.45
2006Y	24.69 ± 0.63	47.49 ± 5.00	22.8 ± 4.05	−17.97 ± 0.06	−16.98 ± 0.06	−14.26 ± 0.06	5.84 ± 0.13	2.11 ± 0.18	4.75 ± 0.34	>0.034	1.52 ± 0.25
2007aa	⋯	⋯	⋯	−16.32 ± 0.27	−16.32 ± 0.27	⋯	⋯	−0.05 ± 0.02	⋯	⋯	⋯
2007ab	⋯	71.66 ± 10.44	⋯	−16.98 ± 0.09	−16.55 ± 0.09	−14.22 ± 0.09	⋯	3.18 ± 0.06	2.31 ± 0.22	>0.040	⋯
2007av	⋯	⋯	⋯	−16.27 ± 0.22	−15.60 ± 0.22	⋯	⋯	0.92 ± 0.01	⋯	>0.015	2.21 ± 0.13
2007bf	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
2007hm	⋯	⋯	⋯	−16.47 ± 0.09	−16.00 ± 0.09	⋯	⋯	1.52 ± 0.04	⋯	>0.045	1.43 ± 0.28
2007il	68.68 ± 2.43	103.4 ± 5.00	34.72 ± 4.68	−16.78 ± 0.11	−16.59 ± 0.11	⋯	1.06 ± 0.34	0.12 ± 0.04	⋯	⋯	2.21 ± 0.12
2007it	⋯	⋯	13.95 ± 2.64	−17.55 ± 0.50	⋯	−14.83 ± 0.50	3.55 ± 1.06	1.33 ± 0.14	1.00 ± 0.01	${0.072}_{-0.054}^{+0.031}$	2.67 ± 0.10
2007ld	⋯	⋯	⋯	−17.30 ± 0.09	−16.53 ± 0.09	⋯	⋯	2.62 ± 0.04	⋯	⋯	2.13 ± 0.50
2007oc	42.78 ± 0.59	71.62 ± 5.83	28.84 ± 3.06	−16.68 ± 0.15	−16.02 ± 0.15	⋯	2.87 ± 0.10	1.78 ± 0.01	⋯	⋯	⋯
2007od	⋯	⋯	19.61 ± 5.06	−17.87 ± 0.80	−16.81 ± 0.80	⋯	2.56 ± 0.10	1.50 ± 0.02	⋯	⋯	2.37 ± 0.12
2007P	61.34 ± 1.19	88.34 ± 5.83	27.00 ± 5.14	−17.96 ± 0.05	−16.75 ± 0.05	⋯	3.80 ± 0.16	1.40 ± 0.12	⋯	⋯	2.26 ± 0.38
2007sq	44.66 ± 2.39	87.66 ± 5.00	43.00 ± 6.46	−15.33 ± 0.13	−14.52 ± 0.13	⋯	2.52 ± 0.37	1.29 ± 0.08	⋯	⋯	⋯
2007U	⋯	⋯	⋯	−17.87 ± 0.08	−16.78 ± 0.08	⋯	⋯	2.27 ± 0.04	⋯	⋯	2.63 ± 0.35
2007W	56.71 ± 2.57	83.59 ± 7.62	26.88 ± 7.46	−15.80 ± 0.20	−15.34 ± 0.20	⋯	0.83 ± 0.27	0.00 ± 0.04	⋯	⋯	2.39 ± 0.28
2007X	53.92 ± 1.03	98.06 ± 5.83	44.14 ± 5.14	−17.84 ± 0.21	−16.70 ± 0.21	⋯	2.52 ± 0.07	1.37 ± 0.03	⋯	⋯	3.55 ± 0.15
2008ag	⋯	105.3 ± 6.71	⋯	−16.96 ± 0.15	−16.66 ± 0.15	⋯	⋯	0.12 ± 0.01	⋯	⋯	⋯
2008aw	37.91 ± 0.91	75.82 ± 10.44	37.91 ± 10.04	−17.71 ± 0.19	−16.60 ± 0.19	−14.04 ± 0.19	3.27 ± 0.08	2.10 ± 0.05	1.97 ± 0.09	>0.050	2.85 ± 0.16
2008bh	⋯	⋯	20.68 ± 5.34	−16.06 ± 0.14	−15.11 ± 0.14	⋯	2.69 ± 0.23	1.20 ± 0.04	⋯	⋯	2.17 ± 0.38
2008bk	⋯	107.22 ± 6.71	⋯	−14.86 ± 0.05	−14.59 ± 0.05	−11.98 ± 0.05	⋯	0.26 ± 0.01	1.18 ± 0.02	${0.007}_{-0.001}^{+0.001}$	⋯
2008bm	⋯	87.04 ± 26.17	⋯	−18.12 ± 0.07	−16.32 ± 0.07	−12.67 ± 0.07	⋯	2.50 ± 0.03	⋯	>0.014	⋯
2008bp	⋯	58.57 ± 9.49	⋯	−14.00 ± 0.21	−13.13 ± 0.21	⋯	⋯	2.79 ± 0.13	⋯	⋯	1.79 ± 0.42
2008br	⋯	⋯	⋯	−15.30 ± 0.20	−14.94 ± 0.20	⋯	⋯	0.38 ± 0.04	⋯	>0.026	2.19 ± 0.22
2008bu	⋯	44.73 ± 7.62	⋯	−17.14 ± 0.10	−16.74 ± 0.10	−13.71 ± 0.10	⋯	2.37 ± 0.18	2.69 ± 0.52	>0.020	2.98 ± 0.37
2008ga	⋯	73.14 ± 5.00	⋯	−16.45 ± 0.14	−16.20 ± 0.14	⋯	⋯	1.10 ± 0.07	⋯	⋯	⋯
2008gi	⋯	⋯	⋯	−17.31 ± 0.09	−15.86 ± 0.09	⋯	⋯	2.63 ± 0.11	⋯	⋯	2.55 ± 1.35
2008gr	⋯	⋯	9.00 ± 6.17	−17.95 ± 0.10	−16.43 ± 0.10	⋯	2.22 ± 0.13	1.61 ± 0.03	⋯	⋯	2.46 ± 0.11
2008H	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
2008hg	⋯	⋯	⋯	−15.43 ± 0.12	−15.59 ± 0.12	⋯	⋯	−0.35 ± 0.08	⋯	⋯	3.25 ± 0.32
2008ho	⋯	⋯	⋯	−15.11 ± 0.23	−15.03 ± 0.23	⋯	⋯	0.32 ± 0.07	⋯	⋯	2.39 ± 0.19
2008if	51.04 ± 0.29	75.85 ± 5.83	24.81 ± 5.01	−17.94 ± 0.17	−16.79 ± 0.17	−14.46 ± 0.17	4.15 ± 0.07	1.99 ± 0.02	⋯	>0.063	1.76 ± 0.22
2008il	⋯	⋯	⋯	−16.61 ± 0.11	−16.22 ± 0.11	⋯	⋯	0.71 ± 0.05	⋯	⋯	2.39 ± 0.15
2008in	⋯	89.64 ± 6.71	⋯	−15.40 ± 0.47	−14.79 ± 0.47	⋯	⋯	0.91 ± 0.01	⋯	⋯	3.01 ± 0.13
2008K	⋯	89.32 ± 5.00	⋯	−17.45 ± 0.08	−16.04 ± 0.08	−13.40 ± 0.08	⋯	2.66 ± 0.02	2.07 ± 0.26	>0.013	2.61 ± 0.84
2008M	58.27 ± 0.27	75.35 ± 9.49	17.08 ± 9.00	−16.75 ± 0.28	−16.17 ± 0.28	−13.41 ± 0.28	5.15 ± 0.27	0.94 ± 0.02	1.18 ± 0.26	${0.020}_{-0.010}^{+0.007}$	1.79 ± 0.18
2008W	⋯	85.84 ± 6.71	⋯	−16.60 ± 0.11	−16.05 ± 0.11	⋯	⋯	1.10 ± 0.04	⋯	⋯	⋯
2009aj	⋯	⋯	⋯	−18.07 ± 0.20	−16.85 ± 0.20	⋯	⋯	⋯	⋯	⋯	⋯
2009ao	⋯	41.68 ± 5.00	⋯	−15.79 ± 0.20	−15.78 ± 0.20	⋯	⋯	−0.01 ± 0.12	⋯	⋯	1.75 ± 0.29
2009au	⋯	⋯	⋯	−16.34 ± 0.21	−14.69 ± 0.21	⋯	⋯	3.03 ± 0.02	⋯	⋯	2.04 ± 0.31
2009bu	⋯	⋯	37.35 ± 8.23	−16.05 ± 0.19	−15.87 ± 0.19	⋯	0.91 ± 0.14	0.13 ± 0.04	⋯	⋯	⋯
2009bz	⋯	⋯	⋯	−16.46 ± 0.19	−16.26 ± 0.19	⋯	⋯	0.36 ± 0.03	⋯	⋯	2.14 ± 0.15
2009N	66.73 ± 0.48	89.79 ± 5.83	23.06 ± 5.02	−15.25 ± 0.40	−14.90 ± 0.40	⋯	2.00 ± 0.29	0.22 ± 0.01	⋯	⋯	3.50 ± 0.06

Note. Same as Anderson et al. (2014b). In the first column, we list the SN name. Columns 2, 3, and 4 show the Pd, OPTd, and Cd. In columns 5, 6, and 7, we list the absolute magnitudes of M_max, M_end, and M_tail respectively. These are followed by the decline rates: s₁, s₂, and s₃, in columns 8, 9, and 10, respectively. In column 11, we present the derived ⁵⁶Ni masses (or lower limits), while in column 12 the color gradient is shown. As is explained in Section 3, the Pd, s₁, and s2 show differences with respect to Anderson et al. (2014b).

Download table as:  ASCIITypeset images: 1 2 3

Besides the parameters defined by A14, we include two more parameters.

• 1.  
  ${\rm{\Delta }}(B-V)$ is defined as the color gradient. We measure this parameter in three different ranges: $10\leqslant t\leqslant 20$ d, $10\leqslant t\leqslant 30$ d, and $20\leqslant t\leqslant 50$ d. Color gradients are calculated by fitting a low order polynomial to color curves and then taking the color from the fit at each epoch and calculating the gradient, ${\rm{\Delta }}(B-V)$ by simply subtracting one epoch color from the other and dividing by the number of days of the interval.
• 2.  
  Cd corresponds to the cooling phase durations (Cd), defined between t₀ and t_tran.

Figure 1 presents an example light curve, indicating all the above defined V-band parameters.

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We analyze the spectral properties of SNe II, focusing on correlations between pEWs, expansion velocities, velocity decline rate, and velocity differences. Figure 2 shows the correlation matrix of the velocity measurements at 50 days obtained by estimating the Pearson correlation coefficient. Correlation coefficients are displayed in color: darkest colors (green and purple) represent the highest correlation found with the Pearson correlation test (−1 and 1, respectively), while white colors (0) mean no correlation. These colors are presented in the lower triangle, while the upper triangle shows the Pearson correlation value (ρ). It is generally considered that correlation coefficients between 0 and 0.19 represent close to zero correlation, 0.2–0.39 weak, 0.4–0.59 moderate, 0.6–0.89 strong, and 0.8–1.0 very strong (Evans 1996), while also noting the statistical significance of these correlation coefficients in many cases. We will use these descriptions for the following discussion. As shown in Figure 2, all velocities strongly correlate positively with each other, as we would expect for an homologous expansion ($v\propto r$). Taking an average, v(Sc ii/Fe ii) $\lambda 5531$, v(O i) $\lambda 7774$, and v(Sc ii) $\lambda 6247$ show the highest correlations with the other parameters, with values of 0.887, 0.883, and 0.875, respectively, while Fe ii $\lambda 4924$ shows the lowest (0.714). The Sc ii $\lambda 6247$ line velocities correlate strongly with Fe ii $\lambda 5018$ and Sc ii/Fe ii $\lambda 5531$, with a value of $\rho =0.94$ and $\rho =0.95$. It is important to note that while the velocities all correlate, they are offset. In general, the differences in the velocities are related to the optical depth for each line and the proximity of the line forming region to the photosphere. As ${{\rm{H}}}_{\alpha }$ displays the highest velocities, it is mostly formed in the outer shell of the ejecta and its optical depth is much larger than the Fe ii lines, which are forming near to the photosphere.

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Figure 3 shows the correlation matrix of the pEW measurements at 50 days. Searching for correlations of pEWs with each other, we find that Sc ii/Fe ii $\lambda 5531$ seems to be the dominant parameter to correlate with all the other pEWs (on average 0.404), while the pEW of the ${{\rm{H}}}_{\alpha }$ absorption component shows very weak correlations with other pEWs. The strongest correlations are displayed by the iron-group lines with each other. We can see moderate correlations between the pEW of O i $\lambda 7774$ and ${{\rm{H}}}_{\beta }$. In the case of a/e, we find a moderate correlation only with Fe ii $\lambda 4924$ ($\rho =0.43$) and anticorrelation with pEW of ${{\rm{H}}}_{\alpha }$ emission ($\rho =-0.43$). While ${{\rm{H}}}_{\beta }$ shows a weak correlation with the ${{\rm{H}}}_{\alpha }$ absorption component ($\rho =0.3$), the correlation with the ${{\rm{H}}}_{\alpha }$ emission component is strong, with a $\rho =0.78$. The lack of correlation between ${{\rm{H}}}_{\alpha }$ and ${{\rm{H}}}_{\beta }$ absorption features could be due to (a) blending effects of Fe ii, Sc ii, and Ba ii lines with ${{\rm{H}}}_{\beta }$, and/or (b) the effects of Cachito (Paper I) on the profile of ${{\rm{H}}}_{\alpha }$.

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Figures 4–6, show the relations between the ${{\rm{H}}}_{\alpha }$, Fe ii $\lambda 5169$, and Na i D velocities and the pEWs for the 11 features explained above at 50 days. Checking these correlations, we see that velocities correlate positively with Balmer and Na i D lines, but negatively with Fe ii lines. For ${{\rm{H}}}_{\alpha }$, we present the pEW of the absorption and emission component in the first two panels, respectively. In the three figures are shown five objects with the lowest velocities and smallest pEW values. Three of these SNe show signs of interaction (narrow emision lines) at early times (SN 2008bm, 2009au, and 2009bu, these SNe also display abnormally low velocities for their brightness). The other two SNe are SN 2008br and SN 2002gd. In those panels plotting pEWs of Fe ii $\lambda 4924$, Sc ii/Fe ii $\lambda 5531$, Sc ii $\lambda 5663$, Ba ii $\lambda 6142$, and Sc ii $\lambda 6247$, one can see that there are many SNe with pEW = 0. In these spectra, we do not detect these lines.

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In Figure 4, we can see that the ${{\rm{H}}}_{\alpha }$ velocities do not show correlations with pEW(${{\rm{H}}}_{\alpha }$) of the absorption component, pEW(Fe ii $\lambda 5169$), pEW(Sc ii/Fe ii $\lambda 5531$), pEW(Sc ii multiplet), pEW(Na i D), pEW(Ba ii $\lambda 6142$), and pEW(Sc ii $\lambda 6247$). The strongest correlations are shown with pEW(${{\rm{H}}}_{\alpha }$) of the emission component, ${{\rm{H}}}_{\beta }$, and anticorrelations with Fe ii $\lambda 4924$ and Fe ii $\lambda 5018$. Figures 5 and 6 show that Fe ii $\lambda 5169$ and Na i D velocities present more scatter in their relations than those shown by ${{\rm{H}}}_{\alpha }$ velocities.

The expansion velocities with ${\rm{\Delta }}v$(${{\rm{H}}}_{\beta })$ show anticorrelations, which are stronger at late epochs (between 50 and 80 days) than at early phases (15 to 30 days, 15 to 50 days, and 30 to 50 days). Meanwhile, ${\rm{\Delta }}\mathrm{vel}$(${{\rm{H}}}_{\alpha }-$Fe ii $\lambda 5018$) and ${\rm{\Delta }}\mathrm{vel}$(Na i D−Fe ii $\lambda 5018$) show correlations with the expansion velocities at 50 days (see Figure 7).

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We now present a comparison of spectroscopic and photometric properties of SNe II. While we have defined and measured 31 spectroscopic and 13 photometric parameters, here we choose a smaller number of parameters to focus on and search for correlations between them. Thus, we employ 14 spectral and 11 photometric parameters: v(${{\rm{H}}}_{\alpha }$) obtained from the FWHM of the emission component, v(${{\rm{H}}}_{\beta }$), v(Fe ii 5018), v(Fe ii 5169), v(Na i D), pEW(${{\rm{H}}}_{\alpha (\mathrm{abs})}$), pEW(${{\rm{H}}}_{\alpha (\mathrm{emis})})$, pEW(${{\rm{H}}}_{\beta )}$), pEW(Fe ii 5018), pEW(Fe ii 5169), pEW(Na i D), a/e, ${\rm{\Delta }}v({{\rm{H}}}_{\beta })$ in a range of $50\leqslant t\leqslant 80$ d, ${\rm{\Delta }}\mathrm{vel}$(${{\rm{H}}}_{\alpha }-$Fe ii 5018), ${\rm{\Delta }}\mathrm{vel}$(Na i D−Fe ii 5018), OPTd, Pd, Cd, M_max, M_end, M_tail, s₁, s₂, s₃, ${\rm{\Delta }}(B-V)$ in a range of $10\leqslant t\leqslant 30$ d, and the ⁵⁶Ni mass.

Figure 7 shows the correlation matrix of the spectroscopic parameters (obtained at 50 days from explosion) and photometric properties. Although photometric correlations have been shown in previous works (e.g., A14, Valenti et al. 2016), the incorporation of numerous spectral parameters can aid in furthering our understanding of the link between observed parameters and underlying SN II physics. As in the previous matrix of correlation, darkest colors indicate higher correlation and white colors, no correlation.

Focusing on the photometric correlations, one can see that many of these are stronger than in A14. As discussed previously, this is because some parameters have been remeasured with new techniques (L. Galbany et al. 2017, in preparation). Interestingly, the number of SNe II with measured values of both Pd and s₃ show an increase from 4 in A14 to 8 in this work. As explained above, both parameters can give us an idea of the of hydrogen envelope mass at the moment of explosion, thus some relation is expected. Figure 8 shows an evident trend between both parameters, with a correlation coefficient of $\rho =-0.857$ (although we note the low number of SNe). SNe II with smaller Pd have higher s₃ decline rates, providing further evidence of a dominant role in defining light-curve morphology of the hydrogen envelope mass, while also providing further support for the use of Pd and s₃ as envelope mass indicators (given their relatively strong correlation).

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From Figure 7, we can also see that Pd has a moderate correlation with velocities. Although we find a strong correlation between Pd and ⁵⁶Ni mass, in agreement to the theoretical predictions (e.g., Kasen & Woosley 2009), we are not in a position to support this result because the correlation is produced only with three points. However, when we include the lower limits for the ⁵⁶Ni mass, the correlation disappears (see the top panel in Figure 9). In general, the correlations between the ⁵⁶Ni mass and all other parameters decrease when we use the lower limits. In the bottom panel of the same plot (Figure 9), it is possible to see how the scatter increases using these values. The correlation goes from $\rho =-0.82$ to $\rho =-0.60$. The fact that correlations become weaker when using lower ⁵⁶Ni mass limits suggests that one should be careful analyzing such masses when insufficient data are available for their estimation.

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Continuing the analysis of Pd, we can see that it has a moderate correlation with pEW(${{\rm{H}}}_{\alpha }$) of the absorption component and strong correlation with a/e. The correlation coefficients are $\rho =0.45$ and $\rho =0.61$, respectively. In Figure 10, we present these correlations together with the best-fit line obtained using the linmix_err¹³ package (Kelly 2007) and the variance with respect to the fit line. The trend shows that SNe with shorter Pd values are brighter, have faster declining light curves, lower pEW(${{\rm{H}}}_{\alpha }$) of the absorption component and a/e values, and higher velocities; however, the scatter is large. In many cases, this scatter is significantly larger than the that which could be ascribed to the errors on individual data points. This suggests that this scatter is due to differing underlying physics driving diversity in different parameters plotted on each axis. For example, while we argue here that Pd is a good indicator of the hydrogen envelope mass, theory also predicts this parameter to be influenced by the ⁵⁶Ni mass (Kasen & Woosley 2009). Meanwhile, SN luminosities and velocities will be affected by both explosion energy and the ejecta/envelope mass. Interaction of the SN ejecta with CSM material at early times (e.g., Dessart et al. 2017; Moriya et al. 2017; Morozova et al. 2017) may also play a role in producing dispersion in our presented trends.

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The fact that we see a significant anticorrelation between Pd and s₂ is in line with historical understanding of the nature of fast declining SNe II. If Pd is an indicator of the extent of the hydrogen envelope, then it follows that faster declining SNe II have a smaller hydrogen envelope at the epoch of explosion, consistent with previous theoretical predictions (e.g Litvinova & Nadezhin 1983; Bartunov & Blinnikov 1992; Popov 1993; Moriya et al. 2016).

In Figure 11, we test the correlation found by Hamuy & Pinto (2002) between the magnitude and the photospheric expansion velocity. Unlike Hamuy & Pinto (2002), who only used SNe IIP and the ${M}_{{\rm{V}}}$ in the middle of the plateau, we use all our SN II sample (no distinction between SNe IIP and SNe IIL) and the magnitude at different phases: at maximum (M_max), at the end of the plateau (M_end), and at the radioactive tail phase (M_tail). We can see that brighter events (in all phases) display higher expansion velocities, confirming the result of Hamuy & Pinto (2002). The correlations between Fe ii $\lambda 5169$ velocity (a proxy of the photospheric velocity) at 50 days and luminosity during the optically thick phase are moderate ($\rho =-0.54$ with M_max and $\rho =-0.45$ with M_end), and strong ($\rho =-0.62$) in the radioactive tail phase. However, we again note the outliers in these figures, where the correlation appears much stronger when removing these events (the outliers are mainly the same SNe discussed previously that show abnormal spectral properties). Interestingly, correlations are higher between spectral velocities and M_max than with M_end (the Standardized Candle Method, SCM, is generally applied using a magnitude during the plateau, more similar to M_end). Analyzing the variance along the best-fit line, we find that the dispersion in velocity is larger in brighter SNe. Although the magnitudes and the expansion velocities are both directly related with the explosion energy, this scatter could suggest an extra influence by an external parameter. In the three main outliers in this plot, we observe signs of weak interaction at early times (see spectra presented in Paper I). In these three obvious cases, but also in other more "normal" SNe II, interaction could play a role in influencing both the magnitudes and velocities observed. CSM interaction is likely to produce more dispersion within brighter SNe II as it will generally increase the early-time luminosity while possibly decreasing velocities, hence pushing SNe II away from the classic magnitude–velocity relation. In addition, the unaccounted for effects of host galaxy reddenning will produce additional dispersion.

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The expansion velocities show a strong correlation with ⁵⁶Ni mass (see Figure 12). This suggests that more energetic explosions produce more ⁵⁶Ni. Additionally, the luminosities have a very strong correlation with the ⁵⁶Ni mass, which supports the results obtained by Hamuy (2003), Pejcha & Prieto (2015a, 2015b), and more recently by Müller et al. (2017). It is possible to see that these three parameters (luminosities, velocities, and ⁵⁶Ni mass) are related and they can be explained through a correlation of both parameters with explosion energy: more energetic explosions produce brighter SNe with faster velocities (as shown in the models of Dessart et al. 2010a). For those correlations that we do not plot, the reader can see the strength of correlation in Figure 7.

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Figure 13 presents correlations between M_max and the pEWs of ${{\rm{H}}}_{\alpha }$, Fe ii 5018, and Na i D. We observe a weak correlation with the pEW(${{\rm{H}}}_{\alpha }$) absorption component, a moderate ($\rho =0.54$) correlation with pEW(Fe ii 5018), and no correlations with pEW(Na i D).

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In Figure 14, we repeat the correlations presented by A14, which show that a faster declining SN at one epoch is generally also a fast decliner at other epochs. Although the correlation of s₃ and M_max is moderate, it is driven by an outlier event, SN 2006Y. As A14 noted, this SN presents an atypical behavior in photometry, but here we confirm its strange behavior in the spectra. If we remove this SN from the analysis, the correlations decrease significantly. The correlations between s₃ and the velocities are moderate. In the last panel of Figure 14, the correlation between s₃ and the pEW(Fe ii 5018) is presented, which, like M_max is driven by SN 2006Y. Summarizing, s₃ has weak correlations with the pEWs and the magnitudes.

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Hamuy & Pinto (2002) found that the luminosity of the SNe IIP correlates with the photospheric velocity (Fe ii velocity) at 50 days from explosion. Brighter SNe II have higher ejecta expansion velocities. This correlation has enabled the use of SNe II as distance indicators. In Figure 11, we show the same relation, but in generalized form; velocities correlate with SN II brightness at all epochs. In addition, we show that this luminosity–velocity correlation is stronger at peak brightness (M_max) than during the plateau. Dessart et al. (2013a) has shown that more energetic explosions produce more ⁵⁶Ni mass, brighter SNe II with faster expanding velocities. This is consistent with our results, and suggests that explosion energy is indeed a primary parameter that influences SN II diversity, and that is traced through SN II brightness, velocities and ⁵⁶Ni mass.

According to theoretical models, faster declining SNe II can be explained by the explosion of stars with low hydrogen envelope mass (e.g., Litvinova & Nadezhin 1983; Bartunov & Blinnikov 1992; Popov 1993 and Moriya et al. 2016). As discussed previously, differences in envelope mass are likely to directly affect the length of the plateau, Pd (we again stress the difference between this parameter and OPTd, with the latter traditionally being assumed to be related to the envelope mass). This is because the plateau, Pd, is powered by the recombination of hydrogen in the expanding ejecta, and the lower the hydrogen envelope mass the quicker the recombination wave reaches its inner edge. The fact that Pd also correlates with s₃ (Figure 8) further supports this view, given that higher s₃ can be interpreted as being due to a lower ejecta mass (A14) that can trap the radioactive emission (which is powering the light curve at these late epochs). With respect to faster declining SNe II, we observe a significant trend in that SNe II with higher s₂ having smaller Pd values, implying that the former is indeed related to the hydrogen envelope mass as has been predicted and discussed for many years. Recent observational works (e.g., A14, Valenti et al. 2016) suggested that the phase between the explosion date and the end of the plateau (historically known as the plateau duration, but here named OPTd) is the key parameter constraining the envelope mass. However, Pd shows higher degrees of correlation with other parameters, in particular, s₂ and s₃. This suggests that Pd is indeed a better tracer of envelope mass than OPTd. In addition, we find that a/e shows strong and moderate correlation with Pd and s₃, respectively, suggesting that this spectral parameter is also a useful tracer of envelope mass (as already argued in Gutiérrez et al. 2014).

From the the correlation matrix (Figure 7), we can observe stronger relations between Pd and s₂, as well as with the expansion velocity, than between OPTd and the same parameters. This is because all these parameters are measured during the recombination phase, where they have similar physical conditions. On the other hand, OPTd conveys information on the physical parameters that dominate the early phases of the light curve, plus the hydrogen envelope recombination. Consequently, the correlations are weaker.

In Figure 7, we can see that ⁵⁶Ni mass shows a strong correlation with Pd, while with OPTd it displays an anticorrelation. Analyzing these findings (Figure 15), we can see that the relation between ⁵⁶Ni mass and the Pd is produced by only three measurements, and therefore the probability of this correlation being real is very small (P = 0.33). In the case of the OPTd–⁵⁶Ni mass plot, this anticorrelation is driven by a number of outliers.

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From Figure 7, we also see that OPTd has stronger correlations with Cd, s₁, and M_tail than with Pd. The strong relation between OPTd and Cd is expected because the former, by definition, includes the latter one (the same applies to OPTd and Pd; see the OPTd definition in Figure 1). However, Pd and Cd are not related, because they are most likely associated with different physical properties of SNe II. Between OPTd and s₁ the correlation is moderate, but again, it is driven by the physical parameters that dominate the early phases of the light curve, which, by definition, are included in OPTd. One interesting correlation is displayed between OPTd and M_tail: SNe II with larger OPTd values are fainter in the radioactive tail phase. This relation may be understood given that the epoch of the M_tail measurement directly arises from the length of OPTd. This means that, if the optically thick phase takes more time, the M_tail will be measured later, which in turn, implies fainter magnitudes (for the same ⁵⁶Ni mass that is powering the late-time LC). This suggests that, the correlation between OPTd and M_tail is essentially based on the total duration of the optically thick phase, i.e., the photospheric phase.

In summary, we observe three key SN II parameters that we believe are strongly related to the extent of the hydrogen envelope mass at the moment of explosion: Pd, s₃, and a/e.

Figures 4–6 display some interesting trends. While the strength of each correlation is complicated by the obvious outliers together with those SNe where no spectral line detection was made, in general, it seems that expansion velocities correlate positively with the strength of the Balmer lines and Na i D, and negatively with the strength of metal lines. The strength of metal lines at any given epoch is most strongly related to the temperature of the line forming region. We therefore conclude that more energetic explosions produce SNe II that stay at higher temperatures for longer leading to lower metal-line pEWs. With respect to the Balmer lines (at least the emission component of ${{\rm{H}}}_{\alpha }$ and the absorption component of ${{\rm{H}}}_{\beta }$) this would then imply that more energetic explosions lead to relatively stronger line strengths. The exact physical interpretation of this is unclear. Brighter, i.e., more energetic SNe II also display weaker metal lines (Figure 7 and specifically Figure 13 bottom middle panel). Finally, we also note that differences in progenitor metallicity will also affect the strength of metal lines within spectra, as argued by Dessart et al. (2014) and Anderson et al. (2016; but probably to a lower degree, at least in the current sample).

A large diversity in the ${{\rm{H}}}_{\alpha }$ P-Cygni profile had been shown by Patat et al. (1994) and Gutiérrez et al. (2014). They found that SNe II with smaller a/e values are brighter, and have higher velocities and steeper decline rates. With our analysis at 50 days, we confirm these results; however, the correlations presented here are of lower strength than those in Gutiérrez et al. (2014). This is most likely due to the epoch of the measurements, where in Gutiérrez et al. (2014) measurements were made at ${t}_{\mathrm{tran}+10}$ (where t_tran is the transitional epoch between s₁ and s₂). Here we chose to use epochs with respect to explosion to measure our spectral parameters. This enables us to analyze the full range of events within our sample (in many SNe II, it is not possible to define t_tran). The difference in correlation strength therefore arises from the measurements in Gutiérrez et al. (2014) being made when SNe II are likely to be under more consistent physical conditions. Here, using an epoch of 50 days post-explosion different SNe are at different phases of their evolution.

It has previously been argued that the ${{\rm{H}}}_{\alpha }$ P-Cygni diversity is directly related to the hydrogen envelope mass (Schlegel 1996; Gutiérrez et al. 2014). The results we present here also support this view, with the absorption component of ${{\rm{H}}}_{\alpha }$—and in particular the absorption in relation to the emission, a/e—showing correlation with both Pd and s₃, parameters that we have already argued are direct tracers of the envelope mass. We also note, however, that the measurement of ${{\rm{H}}}_{\alpha }$ absorption is complicated by the detection and diversity of Cachito (Paper I). It is quite possible, therefore, that the vast majority of the underlying diversity of ${{\rm{H}}}_{\alpha }$ morphology is determined by the hydrogen envelope mass, but complications in the latter's measurement introduce much of the dispersion we see (in, e.g., Figure 10, bottom right).

As discussed in Patat et al. (1994), A14 and more recently Valenti et al. (2016), Galbany et al. (2016), and Rubin et al. (2016), we find that faster declining SNe II are brighter events (see Figure 10). In addition, we also find that SNe II with brighter luminosities have greater expansion velocities and produce more ⁵⁶Ni. In Figures 12 and 13, we show a few examples of these correlations. Similar results were found by several authors in observational (e.g., Hamuy 2003; Spiro et al. 2014; Valenti et al. 2016; Müller et al. 2017) and theoretical (e.g., Kasen & Woosley 2009) works.

Theoretical models show that an increase in the ⁵⁶Ni mass leads to an increase in the plateau duration (e.g., Kasen & Woosley 2009 and Nakar et al. 2016). We do not find any observational evidence for such a trend. There are only three data points in the correlation between Pd and ⁵⁶Ni, therefore strong conclusions are not warranted. If we include lower mass ⁵⁶Ni limits, we also see no evidence for correlation. This may suggest that observationally Pd does not depend on the mass of ⁵⁶Ni mass. However, given the inclusion of lower mass ⁵⁶Ni limits, this warrants caution.

Many authors have found (e.g., Dessart & Hillier 2011) that SN II color evolution could be related with the radius of the progenitor star. Although we include the color gradient (${\rm{\Delta }}(B-V)$) between 10 and 30 days post-explosion in our analysis, we do not find significant correlations associated to this parameter. However, we do note low-level correlation between ${\rm{\Delta }}(B-V)$ and the strength of Fe ii λ5018 and Fe ii λ5169 (Figure 7), in the direction one would expect: SNe II that cool more quickly (higher ${\rm{\Delta }}(B-V)$) display stronger metal-line pEWs. Cd also does not display significant correlation with other parameters. While above we linked Cd to progenitor radius, as predicted by, e.g., Dessart et al. (2013a), the direct influence of radius on Cd is complicated by any presence of CSM close to the progenitor and may explain the lack of correlations.

Dessart et al. (2014) showed that differences in metallicity strongly influence in the SN II spectra, more precisely in the strength of the metal lines. Anderson et al. (2016) supported this result showing a correlation between the strength of Fe ii λ5018 with the oxygen abundance of host H ii regions. They showed that SNe II exploding in lower metallicity regions have lower iron absorption. Looking for relations with the pEW(Fe ii λ5018), we find a correlation of 0.48 with the Pd and −0.62 with s₃. Assuming that the pEW(Fe ii λ5018) gives an idea of the metallicity where the SN explode, this correlation would mean that higher metallicity produces SNe with longer plateaus, which is in the opposite direction of the predictions (e.g., Dessart et al. 2013a). However, when we correlate Pd with the oxygen abundance determined by Anderson et al. (2016), we do not find any relation. As in Anderson et al. (2016) we therefore conclude that (at least in the current sample), the strength of metal lines is dependent more on temperature than progenitor metallicity.