Standardising type Ia supernovae

Adapted from IOAA 2016 (Bhubaneswar), data-analysis problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 20 October 2026.

A type Ia supernova brightens and then fades along a light curve whose shape is nearly the same for all of them, which makes them useful for measuring the distances of faraway galaxies. To compare their light curves, we first turn every observed interval of time $\mathrm{\Delta }{t}_{\text{obs}}$ into the interval $\mathrm{\Delta }{t}_{\text{gal}}$ that passed in the host galaxy, by dividing it by $1+z$, where $z$ is the redshift, because the expansion of the Universe stretches the observed intervals by that factor. The light curve fades by 2 magnitudes from its peak in a time $\mathrm{\Delta }{t}_0$ after the peak. Stretching the time of each supernova by a factor $s$, so that $\mathrm{\Delta }{t}_{\text{s}}=s\mathrm{\Delta }{t}_{\text{gal}}$ and the stretched $\mathrm{\Delta }{t}_0$ is the same for all of them, makes all the light curves the same shape. The factor $s$ turns out to depend linearly on the absolute magnitude $M_{\text{peak}}$ of the supernova at its peak, as $s=a+b{M}_{\text{peak}}$, where $a$ and $b$ are constants.

The table gives three supernovae, with the distance moduli $\mu$ of the first two, their speeds of recession $cz$, and their apparent magnitudes $m$ at times $\mathrm{\Delta }{t}_{\text{obs}}$ from their peaks. The magnitudes have already been corrected for extinction in our Galaxy and in the air.

(a) Find $\mathrm{\Delta }{t}_{\text{gal}}$ for every observation and plot the three light curves in the frames of their galaxies. (15 points)

(b) Taking the factor $s_2$ of SN 2006IS to be 1.00, find the factors $s_1$ of SN 2006TD and $s_3$ of SN 2005LZ from their values of $\mathrm{\Delta }{t}_0$. (5 points)

(c) Find $\mathrm{\Delta }{t}_{\text{s}}$ for every observation and plot the three stretched light curves, to show that they now have the same shape. (14 points)

(d) Find the absolute magnitudes at peak of SN 2006TD and SN 2006IS, and from them the constants $a$ and $b$. (6 points)

(e) Find the absolute magnitude at peak and the distance modulus of SN 2005LZ. (4 points)

(f) From the distance modulus of SN 2005LZ, estimate the Hubble constant $H_0$ and the characteristic age of the Universe $1/H_0$. (6 points)

AnswerThe stretch factors are $s_1=1.322$ and $s_3=1.163$, the constants are $a=7.678$ and $b=0.343$ per magnitude, and SN 2005LZ has $M_{\text{peak},3}=-18.99$ and ${\mu }_3=36.57$, which gives $H_0=58.5$ km/s per Mpc and $1/H_0=16.7$ billion years. The official solution reads $\mathrm{\Delta }{t}_0$ as 16.4, 22.0 and 18.8 days from smooth curves drawn by hand, within the day that it accepted either way. It then divides 22.2 days instead of 22.0 by the others, a slip that changes $s_1$, $s_3$, $a$ and $b$ but hardly changes $M_{\text{peak},3}$, and it finds the same $H_0$ of 58.5 km/s per Mpc. It also accepted an age of two thirds of $1/H_0$, the age of a flat universe that holds only matter.