# Week 10: The Final Distance

May 17, 2024

Hello, everyone! Welcome back to my blog! This will be the last blog post, so thank you for keeping up with my journey! Last week, I practiced my presentation and read up on how other scientists found the distance to other supernovae. This week, I’ll use the light curve of my supernova alongside the method described in a paper by Folatelli et al. to find the distance to my supernova.

**The Method****:**

The general idea is that since all type Ia supernovae “go supernova” with the same amount of mass, the absolute magnitude of all type Ia supernovae will be constant. This means that we can find the distance if we know the apparent magnitude of the supernova (or how bright it looks from Earth), and simply use an equation relating brightness to distance to obtain our final answer. Why this phenomenon works is easy to understand — if you take a lightbulb and walk away from it, you find that it gets dimmer. By knowing how bright it is to your eyes and how bright it actually is (say, to a person right next to the bulb), you can calculate the distance.

The absolute magnitude of the supernova is denoted as MB. I took this value from several databases, where it was calculated through alternative (but more imprecise) methods of finding distances (such as the Tully-Fisher approximation). I took MB to be -19.17 for the g-band.

We also need to know the distance modulus, which is defined as m-M, or the difference between the apparent and absolute magnitudes. My value of apparent magnitude was taken from my graph.

Here are the actual equations I ended up using:

µ = m – M = 5*log(d/10 parsecs)

µ_{sn}= m_{B} – M_{B} – b_{x} [△m_{15}(B) – 1.1]

△m_{15}(B) = m_{B}(15) – m_{B}(max)

What we want is µ_{sn }, or the distance modulus of our specific supernova. I took m_{B }(the peak apparent magnitude of my supernova) from my graph — it is around 11.35. Next, b_{x }(1.32) is the slope of the luminosity decline rate, which was calibrated using multiple supernovae (I took the value from Folatelli et. al). The term m_{B}(15) represents the apparent magnitude 15 days later, which I also took from my graph. My data was sporadic, and there was no picture of my supernova taken precisely 15 days after the day it was brightest, so I approximated the magnitude 15 days after the peak. I approximated it to be around 12.07.

My µ_{sn }was calculated to be 31.02. Solving for d — 31.02 = 5 * log(d/10pc), so d is around 15995580 parsecs, which can be rounded to 16 megaparsecs (MPC). The latest estimates from NASA’s extragalactic database for the distance to the supernova range from 14-17 MPC — these estimates were calculated through less precise methods (such as the Tully-Fisher approximation, which has around a 20% error bound).

So after a lot of data manipulation and a few errors, we’ve finally arrived at a distance value! What made this process exciting for me was how similar it was to real research conducted at universities and labs around the world. Thanks for following my journey!

I’ll be presenting my project on May 18th at the Senior Project Symposium (Fremont Marriott Silicon Valley), so if you can make it to the event, be sure to check out my project!

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