# Akano's Blog

## Equation of the Day # 7: The Virial Theorem

Posted by Akano Toa of Electricity , in Math/Physics Dec 10 2013 · 58 views

This is one that I didn't really know much about until recently, so I thought I'd share it. Today's equation is known as the Virial theorem,

or, in component form,

The word "virial" comes from the Latin vis, which means "force" or "energy," and looking at the equation, it makes sense why it's called that. Here the big Σ means sum, the "k" index denotes the kth particle of a system of N particles, V is the potential energy function affecting the kth particle, T is the potential energy of all the particles in the system, and rk is the position of the kth particle. This essentially relates the kinetic energy of all the particles to the positions and forces exerted on each particle (since -grad V is the force when energy is conserved, which is an assumption we are making). The brackets 〈 〉 denote that we're taking an average, so 〈T〉 is the average kinetic energy, etc.

Now, you may be thinking, "okay, that's a cute equation, I guess, but I don't see how it's particularly useful." Okay, here's where the usefulness comes in. Let's say I want to know the mass of some distant galaxy, but I don't have a good galaxy-weighing device on hand. We know that the gravitational potential energy of an object is given by

where m is the mass of the star, M is the mass of the center of the galaxy, and r is the distance from the center of the galaxy.  Taking the distance r and multiplying by the gradient of the potential yields...the potential again, with a negative sign out front. So, for gravity,

Plugging this into the Virial theorem above and noting that 2T = mv^2 (where v is speed), we get that, for an object in the gravitational pull of an object of mass M,

Thus, we have at our disposal a way of measuring the mass of something like a galaxy by measuring only the speeds of stars and their distance away from the center. That's pretty incredible.

This actually is one of the ways scientists support the idea that there is dark matter in the universe; the Virial theorem gives an average of what speeds the stars in our galaxy should have based on their distance away from the center of the Milky Way, but what we actually observe is startlingly different. Thus, we can conclude that something is wrong with our knowledge of how gravity within a galaxy works. Based on this and other observations, the idea that there's extra stuff that can't be seen that adds to the gravitational force of a galaxy seems to be a reasonable idea.

In my research on diatomic hydrogen (H2), the Virial theorem is used in a different capacity. When figuring out the potential energy of an electron (or two) around the two positively charged protons, the virial has the Coulomb force term (which is just -V, just like gravity) and an additional term that pops up from assuming that the electrons are keeping the protons at equilibrium. I won't go too much into the physics, but the final product is

where E, T, and V are the total energy, kinetic energy, and potential energy of the electron(s), respectively, and R is the distance between the nuclei. This tells us something useful about the energy of the electrons; more specifically, it tells us about how the energy changes as you move the nuclei farther apart or closer together. In other words, since E = T + V,

which is very useful when constructing potential energy curves for hydrogen.

On a slightly related note, our lab's paper got published! Akano is now a for reals, published scientist! 8D

## I Caught Some Shinies!

Posted by Akano Toa of Electricity , in Pokémon Dec 01 2013 · 56 views

Over the past week I caught a shiny Remoraid (evolved it and gave it to Tekulo), shiny Octillery, and shiny Clauncher (traded from Tekulo's X version).

I have become a consecutive fishing master! I want to get a shiny Corsola next.

Also, I had a very enjoyable Thanksgiving with my family and our friends. Tomorrow I head back to my apartment, and then Monday is school as usual.

## Pokémon Y: Adventures on the GTS

Posted by Akano Toa of Electricity , in Video Games, Pokémon Nov 18 2013 · 52 views

So, I've been doing various things in the world of Pokémon recently. I obtained a Squirtle on Wonder Trade (that was awesome), a Froakie from Spain, various forms of Vivillon, a Riolu from Japan (リオル), and a Rotom (8D) from Japan (ロトム). I'm hoping to try the Masuda method to get a shiny Riolu, because shiny Lucario is awesome (and yellow).

Also, I caught a shiny Ditto in Pokémon Village. Luckiest chain ever (I definitely did not get a chain of 40...).

Posted by Akano Toa of Electricity , in Video Games, Life Nov 03 2013 · 73 views

Due to the combination of the servers being taken offline and me being in school, it's been a while since my last entry. I managed to find most of my entries from Google's cache (Aug 26 - now), so they've been posted as though I had posted them on their original entry days, but the comments have disappeared into the aether.

On a more jubilant note, I've been playing Pokémon Y and Ace Attorney: Dual Destinies recently (this is in no way like Tekulo's recent entry). I'm on the first part of case three of the latter, and I have to say that case two was very enjoyable. The twist was really good, and all the crazy weirdness of the people in that case now makes sense (it kind of reminds me of Turnabout Big Top from Justice for All).

In Pokémon Y, I have gotten through Victory Road and am now ready to face the Pokémon League:

Also, I think it's fantastic that I can customize my character with a yellow hoodie and well-matching fedora.

My party currently consists of Keaton (Delphox), Anubis (Lucario), Lapras, Sylveon, Snorlax, and my HM user (a linear combination of Machoke, Talonflame, Pansage in near equal probability). I absolutely love the fact that the Fennekin family is based on mages, and Sylveon is absolutely fantastic. Since evolving it, I don't think I've had it faint once.

## Visiting KK

Posted by Akano Toa of Electricity , in Life Oct 14 2013 · 29 views

So, it's currently my fall break, and I decided the best way to spend that time would be to visit my brother, KopakaKurahk. I arrived Saturday evening and enjoyed a fun day yesterday of meeting people at his grad school, checking out his lab, and enjoying some relaxation (also watching JonTron and MentalFloss and discussing physics and geology).

Good times.

EDIT: Also, apparently this is my 200th entry! This is a milestone for some reason!

Posted by Akano Toa of Electricity , in Video Games Oct 07 2013 · 13 views

So much awesome.

## My semester thus far

Posted by Akano Toa of Electricity , Oct 02 2013 · 9 views

Hi, all!

As you may have noticed, I've been kinda away from BZP recently due to the wonder that is me being in grad school. Last month began my third year into the foray of graduate studentdom, and since then I've been up to the following:
• Taking Thermodynamics/Statistical Mechanics (four lectures a week spread across Thursday and Friday)
• T.A.'ing three sections of introductory physics lab
• Doing research (My group recently submitted a paper which we have gotten back with editing suggestions. We're going to resubmit tomorrow and hopefully get published! I'm going to be a for-reals published scientist!)
• Buying a Wii U bundled with The Legend of Zelda: Wind Waker HD (Pictures of the awesome GamePad will follow. As of this writing, I have just beaten Gohdan)
• Finding out exactly what a true Hufflepuff is, anyways.
Pertaining to Wind Waker HD, they made many great improvements to the game. For one, Carlov will make as many figurines in one day as you give him, which is really fantastic. Also, you get the deluxe Picto Box much earlier (when you complete Lenzo's tasks for him), so you can get figurines much earlier. The Picto Box also holds 12 pictures now, and thanks to the Tingle Bottle, you can use pictographs sent to you from random people on Miiverse.

I'm very happy with the improvements made in the game. If only there weren't so much BLOOM (it's not that bad, though).

## My Afternoon

Posted by Akano Toa of Electricity , in Life Sep 14 2013 · 8 views

Sipping tea, grading intro labs and prelabs, and listening to the Golden Sun soundtrack while the cool air outside comes in.

## The Quantum World

Posted by Akano Toa of Electricity , in Math/Physics Sep 11 2013 · 9 views

Best Symphony of Science ever.

Also, how do I YouTube? Thanks, Xaeraz!

## First Intro Lab T.A.'ing of the semester

Posted by Akano Toa of Electricity , in Math/Physics Sep 10 2013 · 6 views

Akano Toa of Electricity

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Name: Akano
Real Name: Forever Shrouded in Mystery
Age: 23
Gender: Male
Likes: Science, Math, LEGO, Bionicle, Ponies, Comics, Yellow, Voice Acting
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Misses the 90's. A lot.