Background

Type 1a supernovae act as standard candles in measuring astronomical distances because of the consistency in the nuclear reaction processes which lead this particular class of stellar explosion. This consistency lends a similarity to their absolute magnitudes, which are found to exhibit little variation from a value of -19.3 (~5 billion times brighter than the Sun). Although there is some variation, this can be corrected for by normalizing the peak luminosities of the supernovae to their light curves. Thus, type 1a supernovae may be used as standard candles with known precision distances using a prescribed distance modulus.

This distance modulus, $$\mu$$, is the difference between an object's apparent magnitude, $$m$$, which is the measure of the object's brightness as registered by an Earth-based observer, and the object's absolute magnitude, $$M$$, the measure of the object's intrinsic brightness:

(1) $$\mu = m - M$$

The distance modulus is also related to the object's distance from Earth, $$d$$, by

(2) $$\mu = 5\space log_{10}(d)-5$$

where $$d$$ is measured in parsecs.

Solving for $$d$$ gives

(3) $$d=10^{(\frac{\mu}{5}+1)}$$

Converting $$d$$ to megaparsecs by dividing (3) by $$10^6$$ gives

(4) $$d=10^{(\frac{\mu}{5}-5)}$$

The redshift, $$z$$, of an astronomical object is a measure of the amount that light coming from the object has been stretched before reaching an observer. This redshift is directly related to the relative velocity between the object and the origin of the redshift measurement. For very nearby objects, this relation is given by

(5) $$v_{nearby}=cz$$

where $$c$$ is the speed of light in $$km/s$$ (and thus $$v$$ is also in $$km/s$$, as $$z$$ is unitless).

For objects with a redshift greater than around 0.1, special relativity must be taking into consideration, and a new formula can be derived for $$z$$, using the Lorentz factor to adjust the observed frequency of the light, which in turn is used to calculate the relativistic redshift:

(6) $$z=\sqrt{\frac{c+v}{c-v}}-1$$

Thus the more generalized velocity-redshift relation, incorporating the relativistic correction, is

(7) $$v=c(1-\frac{2}{(z+1)^2+1})$$

So, measuring the redshift of type 1a supernovae gives information about the relative velocity between a given supernova and the Earth.

The Supernova Cosmology Project offers a dataset which contains measured distance moduli and redshifts of observed type 1a supernovae. These can be used, respectively, to calculate the distance to and velocity of each stellar explosion event. A comparison of these distances and velocities clearly shows that more distant supernovae are receding away from the Earth at higher speeds. This observed phenomenon is how astrophysicists came to realize that the expansion of the universe is indeed accelerating. Analysis of these supernovae events is therefore extremely relevant to forming accurate cosmological models, and fitting our models to this data can help inform our understanding of important parameters such as the densities of matter and dark energy in our universe.

Approximating the distance-velocity relationship

Let's start with a plot of the distance modulus, $$\mu$$, against the redshift, $$z$$. This gives a non-linear correlation, as shown here:

FIGURE 1
This relation clearly cannot be approximated by a linear function. Adding a simple least-squares fit in Fathom gives this atrocity:

FIGURE 2
So, a new approach must be taken to uncover the non-linear correlation between these two values. Because the function bears some semblance to an exponential, we can explore two options. First, we can apply a logarithm to our x-axis variable, $$z$$. Doing so with $$log_{10}$$ gives a very nice linearization:

FIGURE 3
We can then use a Search program in Python to approximate the slope, $$m$$, and intercept, $$b$$, for a best-fit line. We use a two-parameter Search and define our 'computeMeasureUsing' as such:
Note that 'math' must be imported in order to use the log function. Running the Search yields values of $$m=5.53$$ and $$b=43.99$$, giving a best-fit function of

(8) $$\mu=5.53log_{10}(z)+43.99$$

Plotting (8) on Fig. 3, we find a much better fit than the simple least-squares fit of the original function in Fig. 2.

FIGURE 4
We can then use (4), (7), and the best-fit function to derive a function for the velocity (in $$km\space s^{-1}$$) of a supernova event as a function of its distance away from the Earth (in $$Mpc$$):

(9) $$v=c(1-\frac{2}{(10^\frac{-3.798}{1.106}d^\frac{1}{1.106}+1)^2+1})$$

We can thus use this as a best-fit line for the distance-velocity plot:

FIGURE 5
Alternatively, we can deal with the non-linearity of Fig. 1 by using a Search program in Python to find an exponential approximation function (instead of logging the x-axis). In this case, we again use a two-parameter Search but instead define our 'computeMeasureUsing' by the following:
Running this Search gives $$m=44.06$$ and $$b=0.06$$, giving the best-fit function as

(9) $$\mu=44.06z^{0.06}$$

Plotting (9) on Fig. 3, we find a nice approximation:

FIGURE 6
Using (4), (7), and this new best-fit function, we find an approximation for a supernova's velocity (in $$km\space s^{-1}$$) as a function of its distance (in $$Mpc$$) as:

(9) $$v=c(1-\frac{2}{((\frac{25+5log(d)}{44.06})^{(\frac{50}{3})}+1)^2+1})$$

Adding this to a graph of the distance-velocity plot, we see that it provides a nice fit for the data:

FIGURE 7
Notice that the residuals are extremely similar in Fig.'s 5 and 7.

Reflections

The extreme similarity in the residuals between the two methods is not terribly surprising, given that logarithms and exponents are effectively inverse operations. I would be interested, however, in pursuing a more detailed analysis of the residuals in order to uncover which technique truly provided a more accurate approximation. I could imagine a Python program which could be created to compare these residuals to the calculated distance-velocity dataset.

It's also worth noting that I also performed velocity conversions such as the ones shown above which did not incorporate relativistic corrections. These graphs exhibited less compression along the velocity axis (as expected), and they were more linear due to the relativistic effects being obviously more pronounced for objects having a larger velocity (and thus at a further distance). The residuals for these graphs also exhibited much larger residuals.

FIGURE 8
Further, as a fun aside, it's neat to see that the Hubble constant can be estimated using the data points for supernovae which are measured to be at relatively nearby distances from the Earth. Hubble's law says that $$v=H_0d$$, where $$H_0$$ is the Hubble constant of our present time ($$H$$, often termed the Hubble parameter, is actually time-dependent and uses a scale factor parameter which depends upon the particular cosmological model being considered). Because we are seeing nearby astronomical structures as they were in the recent past, the effect of the universe's expansion is minimized and so we see a linear correlation between the distance and the velocity. I've added a movable line to a plot of relatively nearby supernovae, and $$H_0$$ can be seen to be estimated at 69.9, which is remarkably close the the accepted value of $$H_0=71\space km\space s^{-1}\space Mpc^{-1}$$.

FIGURE 9