2016 D3 | Stretching Type Ia supernova light curves

Type Ia supernovae are important for measuring large extragalactic distances. Their brightening and dimming follow a characteristic light curve.

The light curves of all Type Ia supernovae can be fitted by the same model when scaled appropriately. First express them in the rest frame of the host galaxy by removing the cosmological time dilation: observed intervals $\Delta t_{\rm obs}$ are stretched by $(1+z)$, and the rest-frame interval is $\Delta t_{\rm gal}$.

In the rest frame, a supernova fades by two magnitudes from its peak in a time $\Delta t_0$. If the time axis is further scaled by a factor $s$ ($\Delta t_s = s\,\Delta t_{\rm gal}$) so that the scaled $\Delta t_0$ is the same for every supernova, the light curves have the same shape. It also turns out that $s$ depends linearly on the absolute magnitude at peak,

where $a$ and $b$ are constants. The magnitudes in the table are already corrected for interstellar and atmospheric extinction. $\Delta t_{\rm obs} = t - t_{\rm peak}$.

(D3.1) Compute $\Delta t_{\rm gal}$ for all three supernovae and plot the three rest-frame light curves. (15)

(D3.2) Take $s_2 = 1.00$ for SN2006IS. Calculate $s_1$ and $s_3$ for SN2006TD and SN2005LZ by finding $\Delta t_0$ for each. (5)

(D3.3) Compute the scaled times $\Delta t_s$ and plot the three light curves to check they now have an identical profile. (14)

(D3.4) Calculate $M_{\rm peak,1}$ (SN2006TD) and $M_{\rm peak,2}$ (SN2006IS), then $a$ and $b$. (6)

(D3.5) Calculate $M_{\rm peak,3}$ and the distance modulus $\mu_3$ of SN2005LZ. (4)

(D3.6) Use $\mu_3$ to estimate the Hubble constant $H_0$ and the characteristic age of the Universe $T_H$. (6)