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In a couple of papers I've read that it's usual practice to fit early rise of supernova luminosity curve by $t^2$. Let me describe how I see it:

I take the early fast rising part of luminosity curve and then fit them to a curve:

$L = a\cdot\left( t - t_0 \right) ^ 2 + L_0,$

Where $t_0$ is the moment of explosion.

Is this approach right?

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I think you misunderstood.

$(y-k) = a (x-h)^2$ is a parabola with its apex at $(h,k)$. Therefore, the formula you mentioned above has $t_0$ as the time at peak, not at the explosion.

For the case of SN, we normally do not know when the explosion happened. Mostly, we have an image before SN detection that can set a constraint for the explosion happening in between the last non-detection to the first detection.

However, most analytical models require the explosion time as the reference, so the parabola fit is applied as an approximation.

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  • $\begingroup$ Okay, but what point should we take as the explosion time if not $h$? Left-side zero of the parabola? $\endgroup$ – Tajimura Nov 14 '18 at 1:43
  • $\begingroup$ Intuitively, the time at zero flux from interpolating the parabola. But, in some cases, this might go beyond the last observed non-detection. In this case, it depends on you. Some people used the last non-detection. Some used the mid point between the last non-detection and the first detection. Some used a light model fit to the data with varying the explosion date (e.g., MOSFiT). $\endgroup$ – Kornpob Bhirombhakdi Nov 14 '18 at 4:13

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