Supernovas be exploding

We are bounded in a nutshell of Infinite space: Blog Post #15, Worksheet # 5.1, Problem #4: Supernovas be exploding

4. Some stars explode as supernovae (SNe). In particular, Type Ia Supernovae come from exploding white dwarfs in binary systems. For now, it’s not important to know how this happens. It is, however, critical to learn the consequences of this mechanism, because they too are standard candles.

(a) As with the Cepheids, we can analyze the light curve of Type Ia SNe to standardize them. Below is a set of light curves. Examine them carefully, considering quantities like light curve shape, width/timescales, relative & absolute luminosity, &c. Find a rough relation obeyed by all the Supernovae Type Ia. Note: Although the supernova light curves have many features, try to relate just one of them to the peak magnitude. In other words, your relationship should have the form: \[M_{Max} = A* Feature + M_0 \] , where \(A\) and \(M_0\) are variables you should find.

(b) Describe how you would measure the distance to a Supernova Ia.

(c) Measure the distance to SN Cornell, whose light curve is shown below.

(a) Previously, we were looking at Cepheid variables, a type of star that allows astronomers to measure distances to other corners of the universe more easily than anything that could be done with a parallax measurement, but this is still not the farthest reaching measuring tool we have at our disposal. One of the farthest reaching tools in astronomy is the sight of type Ia supernovae. With these, exploding stars that outshine the entire galaxy when they occur, astronomers can measure the distances to incredibly deep space objects.

By analyzing the different characteristics of each light curve, there is a particular point that seems to repeat itself consistently: the period and Mean Absolute Magnitude of every star line up in the same place, the exact center of each plot, which means these could all be a set of data points for another type of graph that yields the relationship we could extrapolate to other Supernovae.

Checking every star and finding the exact mean magnitude by measuring the \(M_{Max} \) and \(M_{x_0 , y_0 } \) and getting their average \[\left(Half Max Magnitude = \frac {M_{Max} + M_{x_0 , y_0 }}{2}\right),\] and find the corresponding Period coordinate on these graphs, drawing lines on the graphs in the y value we just found and taking the corresponding Period value.

These values for the 4 Supernovae in the plots give us the following data:

SNe Name	Period	Max Absolute Magnitude
Harvard	44	-19.75
Princeton	42	-19.8
Yale	52	-19.25
Brown	48	-19.5

Which can now be used in the same analysis used previously when interpreting Cepheid variable data.

Creating a graph of the Supernovae values yields:

Which gives us an equation in the form of Ax + B that can be used for any number of situation to find the absolute magnitude of a star. Thus, the value of A is 0.057, while the value of \(M_0\) is -22.22, with a linear feature.

(b) Knowing a way to estimate the Max Absolute Magnitude by knowing the time when the Supernova will be at half its peak magnitude offers a very clear way for how its distance can be understood by simply knowing its light curve with apparent magnitude. By using the Magnitude-Distance equation used with the Cepheid problem, \[M_{absolute} = M_{apparent} – 5 \times (\log_{10} (Distance )-1 ) ,\] and solve for distance (in parsecs) when \[d = \left( 10^ {2 + 0.4 (M_{max apparent} - M_{max absolute})} \right)^(\frac{1}{2}\].

(c) Using the knowledge of an equation which yields the Max Absolute Magnitude of a Ia Supernova, we can apply the several equations to the light curve of the Cornell SNe. Measuring its apparent magnitude light curve it maxes out at 7.0, so the mid-max-magnitude here would be 8.5, as per our earlier equation. This magnitude then corresponds to a specific time, around 50 days, which can now be put into the equation. \[M_{Max} = A* Feature + M_0 ,\] \[M_{Max} = 0.057* t + -22.22, \] \[M_{Max} = 0.057* 50 + -22.22, \]
 \[M_{Max} = -19.37 .\]

With this number, we can plug it back into the distance formula: \[d = \left( 10^ {2 + 0.4 (M_{max apparent} - M_{max absolute})} \right)^(\frac{1}{2}, \]  \[d = \left( 10^ {2 + 0.4 (7 – (-19.37))} \right)^(\frac{1}{2} ,\] \[d = \left( 10^ {12.55} \right)^(\frac{1}{2} , \]\[d = 1879316.8 pc = 1.9 \times 10^3 kpc . \] The total distance to the Cornell Ia Supernova.