An Empirical Fitting Method for Type Ia Supernova Light Curves. II. Estimating the First-light Time and Rise Time

Zheng & Filippenko (2017) proposed an empirical function that can fit the optical light curves of SNe Ia. It is a variant of the broken power-law function shown as

$$\begin{eqnarray}&&L=A^{\prime} {\left(\displaystyle \frac{t-{t}_{0}}{{t}_{b}}\right)}^{{\alpha }_{r}}{\left[1+{\left(\displaystyle \frac{t-{t}_{0}}{{t}_{b}}\right)}^{s{\alpha }_{d}}\right]}^{-2/s},\end{eqnarray} \tag{ 1 }$$

where $A^{\prime}$ is a scaling constant, t₀ is the first-light time, t_b is the break time, ${\alpha }_{r}$ and ${\alpha }_{d}$ are the two power-law indices before and after the break (respectively), and s is a transition parameter. This function⁴ is mathematically analytic, derived directly from the photospheric velocity evolution function with some reasonable assumptions. Zheng & Filippenko (2017) show that this function canfit the optical light curves of the prototypical Type Ia SN 2011fe, where they fixed t₀ during the fitting process because it is well estimated (see Nugent et al. 2011). In the following section, we expand this method to a set of nine extremely well-observed SNe Ia, estimating their values of t₀ (which means we treat t₀ as a free parameter during the fitting). We then apply the method to a larger sample of SNe Ia with well-observed optical light curves.

We adopt an IDL implementation of mpfit (Markwardt 2009)⁵ for all the fitting procedures. During our test fitting, we found that it is difficult to constrain t₀ if t₀, ${\alpha }_{r}$, and s are all set free, because these three parameters are related when estimating t₀. Hence, it is necessary to fix a few other parameters in order to get a good constraint on t₀. But before doing that, it is important to understand each parameter and statistically study them with at least a few SNe Ia.

Fortunately, there are nine SNe Ia that were discovered very early and monitored well thereafter, making them suitable for our purposes. Their first-light times are well estimated directly from the data; thus, for these SNe Ia, we could fix t₀ during the fitting and then study the properties of the other parameters in Equation (1). This well-observed SN Ia sample (see Table 1) includes SN 2009ig (Foley et al. 2012), SN 2011fe (Nugent et al. 2011), SN 2012cg (Silverman et al. 2012b), SN 2013dy (Zheng et al. 2013), iPTF13ebh (Hsiao et al. 2015), ASASSN-14lp (Shappee et al. 2016), and three SNe Ia discovered by the Kepler spacecraft (KSN-2012a, KSN-2011b, and KSN-2011c; Olling et al. 2015). Most of them are normal SNe Ia, except for the three Kepler SNe with unknown subtype, since we only have broadband light curves but no spectra, and iPTF13ebh which was categorized as a "transitional" event between normal SNe Ia and the fast-declining subluminous SN 1991bg-like objects owing to its relative largely ${\rm{\Delta }}{m}_{15}(B)$ (1.79; see Hsiao et al. 2015).

Table 1.  The Nine Extremely Well-observed Type Ia Supernovae

SN	Known t0(MJD)a	Filter	tb	${\alpha }_{r}$	${\alpha }_{d}$	s	${\chi }^{2}$/dof
ASASSN-14lp	56998.39	V	23.2 ± 0.7	1.48 ± 0.13	2.25 ± 0.10	1.38 ± 0.14	109.43/20 = 5.47
ASASSN-14lp		u	25.0 ± 0.8	2.81 ± 0.08	5.37 ± 0.18	0.36 ± 0.03	350.62/29 = 12.09
ASASSN-14lp		g	33.5 ± 1.3	1.88 ± 0.04	4.37 ± 0.17	0.44 ± 0.03	366.15/36 = 10.17
ASASSN-14lp		r	25.7 ± 1.8	1.65 ± 0.03	2.80 ± 0.24	0.86 ± 0.13	301.56/22 = 13.70
ASASSN-14lp		i	15.0 ± 0.4	1.73 ± 0.03	1.82 ± 0.06	1.58 ± 0.14	156.04/19 = 8.21
iPTF13ebh	56607.85	B	18.7 ± 0.2	1.91 ± 0.04	3.58 ± 0.08	1.24 ± 0.06	49.52/16 = 3.09
iPTF13ebh		V	20.8 ± 0.5	1.90 ± 0.03	2.93 ± 0.13	1.11 ± 0.10	100.95/16 = 6.30
iPTF13ebh		u	15.8 ± 0.3	2.53 ± 0.11	3.90 ± 0.16	0.94 ± 0.11	32.83/13 = 2.52
iPTF13ebh		g	21.7 ± 0.4	1.99 ± 0.03	4.07 ± 0.13	0.84 ± 0.05	176.37/16 = 11.02
iPTF13ebh		r	14.9 ± 0.1	1.80 ± 0.03	1.58 ± 0.02	3.44 ± 0.19	94.93/16 = 5.93
iPTF13ebh		i	12.8 ± 0.2	1.87 ± 0.05	1.59 ± 0.05	3.16 ± 0.39	58.09/11 = 5.28
2011fe	55796.687	B	27.0 ± 0.7	2.52 ± 0.16	4.75 ± 0.15	0.53 ± 0.03	315.26/129 = 2.44
2011fe		V	19.5 ± 0.1	2.18 ± 0.04	2.33 ± 0.02	1.40 ± 0.04	548.44/126 = 4.35
2011fe		R	20.7 ± 0.4	2.17 ± 0.03	2.70 ± 0.08	1.12 ± 0.07	444.17/107 = 4.15
2011fe		I	15.0 ± 0.3	2.28 ± 0.04	2.14 ± 0.06	1.69 ± 0.14	475.75/105 = 4.53
2011fe		g	24.0 ± 0.3	2.21 ± 0.03	3.21 ± 0.05	0.80 ± 0.03	2892.49/504 = 5.73
2009ig	55062.910	B	30.8 ± 3.6	1.66 ± 0.10	4.25 ± 0.61	0.56 ± 0.15	83.81/28 = 2.99
2009ig		V	18.7 ± 0.4	1.51 ± 0.11	1.65 ± 0.05	2.41 ± 0.24	153.90/28 = 5.49
2009ig		R	26.9 ± 3.7	1.68 ± 0.05	2.99 ± 0.47	0.74 ± 0.20	174.66/24 = 7.27
2009ig		I	20.0 ± 3.1	1.86 ± 0.10	2.53 ± 0.42	0.83 ± 0.29	43.85/22 = 1.99
2013dy	56482.986	B	23.4 ± 0.6	1.87 ± 0.07	3.09 ± 0.11	0.97 ± 0.11	35.30/42 = 0.84
2013dy		V	19.7 ± 0.3	1.77 ± 0.04	1.99 ± 0.04	1.70 ± 0.14	44.92/42 = 1.06
2013dy		R	25.1 ± 3.0	1.85 ± 0.06	2.97 ± 0.45	0.82 ± 0.23	43.97/26 = 1.69
2013dy		I	18.5 ± 1.8	1.85 ± 0.10	2.35 ± 0.32	1.15 ± 0.35	8.88/23 = 0.38
2013dy		U	22.7 ± 1.7	2.18 ± 0.17	3.82 ± 0.36	0.71 ± 0.18	15.51/42 = 0.36
2012cg	56063.950	B	28.1 ± 3.7	2.06 ± 0.10	4.17 ± 0.67	0.58 ± 0.17	11.44/28 = 0.40
2012cg		V	20.9 ± 0.8	1.76 ± 0.04	2.10 ± 0.11	1.63 ± 0.24	20.20/28 = 0.72
2012cg		R	22.5 ± 2.1	1.83 ± 0.07	2.54 ± 0.32	1.13 ± 0.30	37.72/21 = 1.79
2012cg		I	16.4 ± 0.9	1.85 ± 0.08	1.95 ± 0.16	1.66 ± 0.39	14.67/19 = 0.77
KSN-2011b	995.710b	broad	23.1 ± 0.3	2.38 ± 0.02	3.33 ± 0.06	0.71 ± 0.03	2185.09/57 = 38.33
KSN-2011c	1075.914b	broad	22.7 ± 0.4	1.77 ± 0.05	2.63 ± 0.12	1.97 ± 0.23	303.76/60 = 5.06
KSN-2012a	1328.466b	broad	22.0 ± 0.7	2.01 ± 0.04	3.22 ± 0.13	0.70 ± 0.06	1153.31/53 = 21.76

Notes.

^aReferences are given in Section 2.2, and data within 3.0 days after t₀ are not included in the fitting. ^bKJD = MJD−54832.5. Its broadband filter is plotted as the R-band.

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We first fit the light curves by fixing t₀ because it is already well estimated for each SN. We then apply the fitting independently to each filter, including the data points from the beginning of the observational campaign until the SN entered the phase dominated by cobalt decay, which usually happens around three weeks after peak brightness in the B and Vbands.

Since SNe Ia generally exhibit a shoulder in the R-band and a second peak in the I-band, we restrict the fits to earlier times in the redder bands (R, I) than those in the bluer bands (B, V). Specifically, for the R and Ibands, the data points are cut off before the shoulder appears, but still after the main peak. For a few cases, additional U-band data or g, r, or i filters were used. Data with all of these filters are treated independently. Additionally, the three Kepler SNe were observed with a broad filter that is quite different from any existing standard filter. However, since we fit all filtered data independently, we apply the same fitting to the three Kepler SNe. Only when plotting do we represent these objects with the R-band, since the Kepler response function covers the wavelength range 420–900 nm and peaks around 600 nm (see the Kepler website⁶ ).

The fitting procedure is very similar for each SN as well as for each filter; thus, we show only one case as an example. Figure 1 illustrates the fitting results for SN 2013dy, with comments similar to those for the fitting of SN 2011fe by Zheng & Filippenko (2017; see their Figure 2).

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As shown in Figure 1, using just one single function given by Equation (1), the fitting results are surprisingly good for the included data, which cover a long time range and nearly a factor of 100 in flux (∼5 mag) for all filters (U, B, V, R, clear, and I). In particular, for the V-band, although we include only the data up to day 45, the model still matches the data well up to nearly three months after the explosion (though this might be just a coincidence). The flux residuals are mostly within the $1\sigma$ measurement uncertainties.

Using the same procedure, we apply the fitting to the eight other well-observed SNe Ia by fixing t₀. In general, the fitting results are very similar to those of SN 2013dy shown in Figure 1, though they do show some indications of diversity at extremely early times. For a few cases in our sample, including SN 2012cg and SN 2013dy, we find for the first few days (within three days after first-light time) that the data show an excess compared to the fit model, possibly from other contributions such as thermal emission produced by the impact of the SN shock on a binary companion star (e.g., Kasen 2010; Cao et al. 2015; Marion et al. 2016). In principle, for our purposes it is more appropriate to exclude these data from the fitting; we therefore excluded data within 3.0 days after t₀ during the fitting. However, in our later analysis of the first-light time estimation (in Section 2.3), we find that this early-time excess insignificantly affects the estimate (the changes are much smaller than the fitting error itself), so in Section 2.3 we retain these data during the fitting process.

After fitting to all nine of the SNe given above, we plot the histogram distribution in Figure 2 for each parameter from Equation (1), including t_b, ${\alpha }_{r}$, ${\alpha }_{d}$, and s, but not t₀ (which was fixed during the fitting procedure) and $A^{\prime}$ (which is simply a scaling factor). Note that since some SNe lack observations in one or a few filters, not all filters have the complete set of nine data points representing the nine SNe; however, whichever filter was observed is fitted independently and shown in the figure. These fitting results are also listed in Table 1.

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As shown in Figure 2, it is clear that ${\alpha }_{r}$ (top right panel) is the most concentrated parameter, with a mean value of 1.90 ± 0.18—which is very consistent with the commonly known t² model for most SNe Ia or the tⁿ model (n varies from ∼1.5 to ∼3.0) studied by various groups (e.g., Conley et al. 2006; Ganeshalingam et al. 2011; Firth et al. 2015). The parameter with the next-smallest dispersion is the break time t_b (top left panel), with a mean value of 23.0 ± 4.0 days, typically a few days after the time of peak brightness. The other two parameters, s (the smoothing parameter) and ${\alpha }_{d}$, are much less concentrated.

In conclusion, to better estimate the first-light time t₀ (as shown in the next section), it is appropriate to keep ${\alpha }_{r}$ and t_b constant during the fitting procedure because for each filter these two parameters exhibit a small dispersion among different SNe. Conversely, it is best to keep s and ${\alpha }_{d}$ as free parameters, since they are quite diverse. By doing this, Equation (1) has fewer free parameters (only four, after fixing ${\alpha }_{r}$ and t_b), and we can better constrain t₀ as demonstrated below.

In reality, very few SNe Ia have been discovered extremely early like the nine SNe mentioned above. However, for SNe Ia that were discovered reasonably early (1–2 weeks after explosion), one may estimate the explosion time t₀ by fitting the light curve using Equation (1).

Before applying this method to a larger set we need to test its validity. Our set of nine well-observed SNe Ia provides a perfect sample for such a test, since all of them already have a well-determined t₀ (which we denote as "known t₀"). We can then compare the "known t₀ and the t₀ value estimated from fitting Equation (1) to study the errors from the fitting method.

First, in order to simulate the relatively late-time discovery for the majority of less well-observed SNe Ia, we intentionally exclude some of the very early-time data points for the above nine SNe. However, a criterion needs to be established for how late to start using the data. Here, we define the limit to be the time prior to peak brightness when the SN is 1 mag fainter than the peak magnitude, denoted by ${t}_{\mathrm{pm}-1}-0;$ it typically occurs around 1–2 weeks before peak brightness. The first data point must be at least 1 mag fainter than the peak magnitude. We adopt this limit for selecting SNe to be fitted with Equation (1) because we require relatively early-time observations to constrain t₀. If there are additional observations one day earlier than that, their times will be denoted by ${t}_{\mathrm{pm}-1}-1$, and so on. The discovery times of the above nine SNe are extremely early, in some cases reaching ${t}_{\mathrm{pm}-1}-7$.

Next, in order to simulate the different discovery times of most SNe, for each of the above well-observed SNe we gradually include data points earlier than ${t}_{\mathrm{pm}-1}-0$ until all data are included. Each step gives a corresponding ${t}_{\mathrm{pm}-1}X$ value (X varies from 0 to −7). We then use Equation (1) to fit t₀, and we compare the estimated value with the "known t₀." Again, we use SN 2013dy as an example to demonstrate this procedure, which is very similar for all of our SNe in each band. Figure 3 illustrates the case of SN 2013dy with ${t}_{\mathrm{pm}-1}-3$. Note the difference with Figure 1; now we only include data after ${t}_{\mathrm{pm}-1}-3$ in order to simulate a late discovery. The data before ${t}_{\mathrm{pm}-1}-3$ are shown as crosses, which means they are not included. As mentioned above, for all fittings, to estimate t₀ we fix ${\alpha }_{r}$ to be 1.90 and t_b to be 23.0.

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After applying the t₀ fit to each SN for each band, and for each case with different ${t}_{\mathrm{pm}-1}X$ (X varies from 0 to −7), we compare the estimated t₀ with the "known t₀." The result is shown in Figure 4, where the upper panel shows the cases for each SN and each filter with different ${t}_{\mathrm{pm}-1}X$. For a certain ${t}_{\mathrm{pm}-1}X$, if there is more than one filter observed for an SN (as shown in Table 1, except for the three Kepler SNe that were only observed with one broadband filter), a mean value of t₀ from all filters is also calculated. The bottom panel shows the histogram of the t₀ offsets compared with the "known t₀"; each filter is color-coded.

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Three important and clear conclusions can be drawn from Figure 4. First, the offset between the fitting t₀ and the "known t₀" becomes smaller when earlier data are included—namely, the smaller the X value in ${t}_{\mathrm{pm}-1}X$, the smaller the offset. But once X reaches −3, then even earlier data do not help much for t₀ estimation. Second, B interestingly has the smallest offset, and no matter when starting with ${t}_{\mathrm{pm}-1}X$, the offset is always within ±1.5 days; we estimate a 1σ systematic error of ±0.7 days by adopting only the B-band estimate of the t₀ offset. Third, the mean offset with multiple filters is similar to that of the B band; thus, for t₀ estimation, one should use either the mean value or the value estimated from B alone. Technically, it is not surprising to see why the B band gives the best estimate of t₀; in other bands, the peak is usually broader than that of B, making B better for estimating t₀. Because of this, in the following we only adopt the t₀ estimate from B, even if the SN was observed with multiple filters. We will also add a systematic error of ±0.7 days for the t₀ estimated from B-band fitting.

In addition, Figure 4 shows that the fixed value of ${\alpha }_{r}=1.90$ is appropriate, giving a mean t₀ offset of 0.1 ± 0.7 days for the B-band, very close to 0 as expected. Furthermore, we performed two other sets of fitting with the ${\alpha }_{r}$ value fixed to be 1.50 and 2.30 (respectively), and found t₀ offset by 1.2 ± 0.8 and −0.8 ± 0.7 for B, confirming that the fixed value of ${\alpha }_{r}=1.90$ is appropriate.

From the above analysis, we have shown that Equation (1) provides a practical way to estimate t₀ by fixing ${\alpha }_{r}$ (to be 1.90) and t_b (to be 23.0) during fitting, and using only B (or g) data. Such an estimate of t₀ gives an additional systematic error of ±0.7 days. We can now apply this method to a larger sample of objects that were typically discovered around ${t}_{\mathrm{pm}-1}-0$, or slightly earlier, but still with good-coverage observations after discovery.

Ganeshalingam et al. (2010) published a sample of 165 SNe Ia from the Lick Observatory Supernova Search (LOSS: Filippenko et al. 2001; Leaman et al. 2011) database. From this sample, we selected 44 SNe Ia that are suitable for our purpose. As above, all satisfy the criterion that the first observation is earlier than the time that the SN brightness reaches one magnitude fainter than the peak (${t}_{\mathrm{pm}-1}-0$). (Of course, even earlier observations are preferred.) This lower limit criterion allowed us to choose 44 of the 165 LOSS SNe Ia. In addition, 6 SNe Ia were selected from the Harvard-Smithsonian Center for Astrophysics Data Release 3 (CfA3; Hicken et al. 2009), alongside 6 SNe Ia from the Carnegie Supernova Project (CSP; Contreras et al. 2010), making the final sample 56 SNe Ia, as listed in Table 2.

Table 2.  The 56 Well-observed Type Ia Supernovae

SN	Subtype	z	${t}_{0,B}$	${t}_{0,B}$ erra	${t}_{p,B}$	${t}_{r,B}$	${\rm{\Delta }}{m}_{15}(B)$	Δ
From LOSS
1998dh	Ia-norm	0.0077	2451013.6	0.7	2451029.4	15.7	1.21	0.0015
1998dm	Ia-norm	0.0055	2451043.1	0.7	2451060.9	17.7	0.98	−0.2488
1999by	Ia-91bg	0.0027	2451296.1	0.7	2451308.5	12.4	1.95	1.5087
1999cp	Ia-norm	0.0103	2451346.3	0.7	2451363.8	17.3	1.01	−0.0837
1999dq	Ia-99aa	0.0137	2451417.8	0.8	2451436.4	18.4	0.87	−0.3112
1999gp	Ia-norm	0.0260	2451531.1	0.8	2451549.6	18.0	0.73	−0.3633
2000cx	Ia-pec	0.0070	2451738.0	0.7	2451752.3	14.2	0.94	0.2128
2000dn	Ia-norm	0.0308	2451807.7	1.4	2451824.5	16.3	1.03	−0.0058
2000dr	Ia-norm	0.0178	2451821.0	0.7	2451834.1	12.9	1.80	0.9319
2000fa	Ia-norm	0.0218	2451874.7	0.8	2451891.7	16.7	0.89	−0.1939
2001en	Ia-norm	0.0153	2452176.2	0.7	2452192.8	16.3	1.19	0.1095
2001ep	Ia-norm	0.0129	2452183.0	0.7	2452200.0	16.8	1.43	0.1458
2002bo	Ia-norm	0.0053	2452340.9	0.7	2452356.7	15.7	1.10	−0.0620
2002cr	Ia-norm	0.0103	2452391.9	0.7	2452408.7	16.6	1.21	0.0403
2002dj	Ia-norm	0.0104	2452435.0	0.7	2452450.6	15.4	1.06	0.0109
2002dl	Ia-pec	0.0152	2452439.2	0.7	2452452.5	13.0	1.85	1.0411
2002eb	Ia-norm	0.0265	2452475.8	0.7	2452494.5	18.2	0.91	−0.3094
2002er	Ia-norm	0.0090	2452508.4	0.7	2452524.5	15.9	1.24	0.1195
2002fk	Ia-norm	0.0070	2452530.0	0.7	2452548.0	17.9	1.01	−0.1163
2002 ha	Ia-norm	0.0132	2452566.2	0.7	2452581.3	14.9	1.31	0.3856
2002he	Ia-norm	0.0248	2452571.6	0.8	2452585.9	13.9	1.32	0.4099
2003cg	Ia-norm	0.0053	2452713.2	1.0	2452729.4	16.1	0.93	−0.0201
2003fa	Ia-99aa	0.0391	2452788.0	0.7	2452807.3	18.5	0.86	−0.3534
2003gn	Ia-norm	0.0333	2452837.8	0.9	2452852.7	14.4	1.26	0.2899
2003gt	Ia-norm	0.0150	2452845.0	0.7	2452862.0	16.7	1.00	−0.0191
2003W	Ia-norm	0.0211	2452664.0	0.8	2452678.9	14.6	1.04	−0.1525
2003Y	Ia-91bg	0.0173	2452665.9	0.8	2452676.6	10.5	1.79	1.5668
2004at	Ia-norm	0.0240	2453075.2	0.7	2453092.1	16.6	1.02	−0.1025
2004dt	Ia-norm	0.0185	2453223.3	0.7	2453240.3	16.7	1.11	−0.1102
2004ef	Ia-norm	0.0298	2453250.0	0.7	2453264.2	13.8	1.41	0.3373
2004eo	Ia-norm	0.0148	2453262.4	0.7	2453278.5	15.8	1.37	0.2798
2005cf	Ia-norm	0.0070	2453517.7	0.7	2453533.7	15.9	1.03	−0.1080
2005de	Ia-norm	0.0149	2453581.1	0.7	2453598.8	17.4	1.18	0.0495
2005ki	Ia-norm	0.0203	2453690.0	0.7	2453705.4	15.1	1.25	0.3400
2005M	Ia-91T	0.0230	2453384.7	0.8	2453405.8	20.5	0.86	−0.4287
2006cp	Ia-norm	0.0233	2453879.6	0.8	2453897.5	17.5	1.12	−0.1903
2006gr	Ia-norm	0.0335	2453993.5	1.2	2454012.4	18.3	0.92	−0.3245
2006le	Ia-norm	0.0172	2454030.8	0.8	2454047.7	16.6	0.86	−0.2751
2006X	Ia-norm	0.0064	2453770.4	0.8	2453786.5	16.0	1.28	−0.0228
2007af	Ia-norm	0.0062	2454157.5	0.7	2454174.4	16.8	1.18	−0.0064
2007le	Ia-norm	0.0067	2454383.7	0.8	2454398.8	15.0	0.98	−0.1349
2007qe	Ia-norm	0.0244	2454412.6	0.9	2454429.0	16.0	1.01	−0.2030
2008bf	Ia-norm	0.0251	2454537.6	1.3	2454554.7	16.7	0.93	−0.1849
2008ec	Ia-norm	0.0149	2454658.2	0.7	2454674.0	15.6	1.32	0.1941
From CfA3
2001V	Ia-norm	0.0162	51955.183	0.7	51972.446	17.0	0.67	−0.2967
2005hk	Iax	0.0118	53666.101	0.8	53684.385	18.1	1.65	−0.2331
2006ax	Ia-norm	0.0180	53808.486	0.9	53827.026	18.2	1.09	−0.1657
2006lf	Ia-norm	0.0130	54030.365	1.0	54045.385	14.8	1.32	0.3188
2007bd	Ia-norm	0.0319	54191.882	0.7	54205.783	13.5	1.03	0.2778
2007ci	Ia-norm	0.0194	54233.021	0.7	54246.246	13.0	1.91	0.8581
From CSP
2005kc	Ia-norm	0.0137	3679.7901	0.7	3697.4031	17.4	1.19	0.0055
2005ke	Ia-91bg	0.0045	3684.9211	0.7	3698.2680	13.3	1.77	1.6190
2007on	Ia-norm	0.0062	4404.8806	0.7	4419.7535	14.8	1.87	0.9558
2008bc	Ia-norm	0.0157	4532.0783	0.7	4549.1804	16.8	0.87	−0.2790
2008gp	Ia-norm	0.0328	4760.8915	0.7	4779.0086	17.5	1.05	−0.0576
2008hv	Ia-norm	0.0125	4801.8928	0.7	4816.5393	14.5	1.22	0.2761

Note.

^aInclude the 0.7-day systematic error estimated from the fitting method.

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We applied the fitting to the above 56 SNe Ia using Equation (1), only with B data (no g-band data were collected for these 56 SNe Ia), and fixing ${\alpha }_{r}$ to be 1.90 and t_b to be 23.0. The resulting t₀ estimates are listed in Table 2.

Once the first-light time t₀ is estimated, it is easy to calculate the rise time as long as the time of peak brightness can also be determined. Technically, it is much easier to estimate the peak time than the first-light time t₀. Although the above fitting method can also give the peak time, it is more accurate and straightforward to estimate the peak time directly by fitting the light curve around peak brightness with a low-order polynomial. The time of peak B-band brightness is given in Table 2, along with the decline rate ${\rm{\Delta }}{m}_{15}(B)$. The rise time is simply the time duration from first-light time to the time of peak brightness, corrected for time dilation through division by $1+z;$ see Table 2 for the final results.