By the looks of the title, we'll be talking about a couple of seemingly incongruent topics today. Between the amazing three day weekend (which took us by surprise), the national holiday that may be the fifth fundamental force, and the recent detection of gravitational waves, there seems to be quite a bit that has transpired since we last published a post.

Let us start by apologizing for the delay. We are constantly trying to improve our website and as such have recently began reformatting some of the equations to make them a lot more user friendly and not reliant on third party sources. What this means for you, is more reliable service. But in addition to improvements that we have begun undertaking, we are also striving to revive our blog and start talking about issues that you care about. So if there's anything that you would like us to talk about, mention, or explain please feel free to either comment below or send us a line at

We would like to inform our visitors - from countries outside the United States or those who tend not to get involved in the current events - that this Sunday is Valentine's Day. A holiday mainly associated, and marketed, for love and companionship also brings mixed feelings for others who may not have a companion. For those of you fortunate enough to be in a relationship this Valentine's Day, strive to spend time with your partner. Spending time is far more important than spending money - it really is the thought that counts. So, if you're both into science and would like to do some problems together we will be here to help all weekend :D. For those who are single, don't fret. This is the PERFECT time to go out an meet a wonderful individual. The obvious eventual goal would be to use EQNS together of course, but we wouldn't blame you if you choose to wait until the second date to mention us.

On to more concrete issues. WE HAVE FINALLY OBSERVED GRAVITATIONAL WAVES. This is actually a huge relief because we have spend an enormous amount of money (over $1 billion) in our quest to find these waves. Let us take a step back and briefly talk about what gravitational waves are. Every force in nature has a corresponding quantum particle. Electromagnetic force for example has a corresponding photon. But recall that there's a wave-particle duality that exists. So while light can be thought of as a stream of photons constantly hitting your retina, they can also be thought of as waves. And much like waves, they propagate through space and cancel each other out. The same is true for the gravitational force. Instead of photons, however, the gravitational force is complemented with gravitons. Unfortunately, until just now, we had never detected a graviton. Gravitational waves are to gravitons what electromagnetic radiation is to photons. In other words, they're a different interpretation / representation of the same thing. For the purposes of this explanation, we're going to talk about gravitational waves as waves and not particles.

To detect a gravitational wave, scientists have to detect an expansion of space-time. Take a second to digest that statement. There are several questions that may arise as you are doing so. First, how does space-time change? Isn't it what makes up everything? Doesn't that mean that I took will be affected by it? And second, how does one go about detecting it? In response to the first questions, space-time is constantly changing. Anything that has gravity distorts space time. In fact that's the reason that objects orbit around each other. When you think of gravity from this perspective it no longer seems arbitrary. These gravitational waves essentially report a change in space-time. The larger the change, the more prominent the waves. The change that led to these waves being detected was a collision between two black holes over a billion light years away. If the distance doesn't shock you, the fact that we are only now detecting a collision that occurred a billion years ago may. Why are the two numbers the same? Because it took light a billion years to get to us!

So how do they detect these waves? Large structures are built to detect changes in length. Because gravitational waves both contract and expand space-time. If you send light going in one direction and get one number and send it going in another direction and get another number (for the length of two objects that were initially the same length) then you have just detected a gravitational wave! Unfortunately for this to work, the structures must be massive (miles long) and one has to factor out all of the noise. Additionally even trucks passing by can skew the results... So here's hoping that this isn't another neutrino-faster-than-light incident and that we have actually detected something that we once though was impossible to detect!

Here's to detecting more gravitational waves! Oh, and Happy Valentine's Day!!

Be sure to stay tuned to more blog posts and to check out some interesting physics equations on EQNS.

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You're walking down the street, stumble on a rock, and fall head first into an open manhole. Unfortunately for you, this manhole leads to a tunnel that goes straight through the Earth. As you begin your descent into the dark abyss you figure that you probably have quite some time before you emerge on the other side. But how long do you have? When should you get ready to brace yourself?

This is where EQNS comes in! Let's calculate just how much time you'll have.

But before we can delve any deeper we have to make a few


  • The Earth is a sphere of radius 6.38 E+006 meters
  • The Earth has a constant density of 5,510 kg m^-3
  • Temperature doesn't vary (for your own safety)


Let's start by determining the gravitational force of the Earth:
Let m_1 be defined as the mass of sphere between you and the center of the Earth, m_2 be defined as your mass, and r be defined as the radius of the Earth. Because m_1 changes as you fall deeper into the Earth we should rewrite the above formula in terms of constants. Let's rewrite it in terms of density. Recall that:
Plug equation [3] into equation [2]:
Solve for m:
Plug equation [5] into equation [1]:
Notice that F is proportional to r, or in other words the force increases linearly with depth. This is analogous to the force exerted by a spring!
The period of oscillation by a mass on a spring can be described by the following equation:
Since we don't want to return to where we are originally, we actually care about half the period. So, let's divide T  by 2 and plug in :
Notice that your mass cancels out:
Finally, plug in the values as solve:
So you have a whole 42 minutes - plenty of time! Take out your phone and start playing around on it. And don't worry about "dropping it". It'll fall at the same rate you do - so you can't even lose it!

If you want to go into more detail on anything we discussed here, please be sure to read Wired's article on this same subject matter.

Be sure to stay tuned to more blog posts and to check out some interesting physics equations on EQNS.

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Much like a needy ex-girlfriend, Pluto has resurfaced once more - and this time more beautiful than ever before. New Horizons recently flew by Pluto, and over 5 hours later we were finally graced with Pluto's intricate contours and features. 

Over 85 years ago, when Pluto was first discovered, we knew very little about this dwarf-planet - whose status, in recent years, has changed more than Kim Kardashian's relationship status. Back in 1930, this is all we had to go off of: 
Photographic plates depicting the discovery of Pluto (
Nearly 70 years later, we were able to get an image of Pluto that was larger than a handful of pixels across. Although this image was still extremely blurry by today's standards, it served as a good first order approximation of Pluto's topography. The observations are listed in the small squares below, the larger images depict rendered computer models that are based on the pixeleted observations in the small squares.  
1996 Observations and renderings of Pluto (
Following these observations, Pluto stayed out of the public eye for nearly a decade while scientists were developing a probe that would give us a better understanding of this far away world. Google Trends actually serves as a good barometer for the popularity of Pluto, following 2004. Below is a graph of depicting the popularity of the search term "Pluto" normalized by the total number of searches made during that point in time. 
Popularity of the keyword "Pluto" over time. Letters depict news headlines that featured the keyword "Pluto" (
The graph depicts many interesting points in time - at least from Pluto's perspective. Here, we will only discuss the three highest peaks. We'll start with peak "I". 

Peak "I"

As many of you may recall, there were many very upset school children in 2006. They were so upset, in fact, that they ended up sending letters to everyone from NASA to Neil deGrasse Tyson. The controversy, if it can even be called that, lay in the fact that we never had a formal definition for a planet and as such have been "winging it" when it came to determining what was a planet and was a asteroid, comet, or Kuiper Belt Object.

By 2006, several other planet-like objects, like Ceres and Eris, had been discovered and astronomers realized that a formal definition was required. During one IAU, or International Astronomical Union, meeting in 2006 a formal definition was established. Unfortunately, this definition demoted Pluto from the status of "planet" to the status of "dwarf planet".

The new definition of a planet is:

"A planet is a celestial body that 
     (a) is in orbit around the Sun,
     (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and
     (c) has cleared the neighbourhood around its orbit. "

Unfortunately, Pluto did not conform to the third requirement and therefore lost its status as a "planet". A formal definition for a "dwarf planet" was also established during the same meeting:

"A 'dwarf planet' is a celestial body that
     (a) is in orbit around the Sun,
     (b) has sufficient mass for self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape,
     (c) has not cleared the neighbourhood around its orbit, and
     (d) is not a satellite. "

This made it incontrovertible that Pluto was no longer a planet, although even in the incontrovertible is often argued.

Peak "J"

Moving backwards in time, we can see that a few months before Pluto was demoted, it was abnormally popular. The reason for this actually ties in quite nicely with the recent observations of Pluto. In January of 2006, NASA launched the New Horizons probe to gain a better understanding of Pluto. The same New Horizons that sent us close-up images of Pluto just a few days ago. 

Peak "A"

The final peak obviously corresponds to the recent images sent by New Horizons. After reading through all of this, potentially boring, history I'll reward you with the beautiful images that were recently received. Enjoy!

Be sure to stay tuned to more blog posts and to check out some interesting astronomy equations on EQNS.

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Magnitudes have long been linked to earthquakes, so it was much to everyone's surprise when magnitudes begun being linked to stars as well! 

...Actually both earthquake intensities and stellar brightnesses have long been measured in magnitudes. Magnitudes are a great ways to take seemingly large, complicated, and convoluted data and convert it to something that a layperson can understand. In the case of earthquakes, the Richter magnitude scale converts seismological readings into a simple one or two digit number that clearly conveys the severity of the earthquake. Because high levels of precision aren't particularly important when discussing the intensity of an earthquake, the Richter magnitude scale employs logarithms to truncate all of the excess information.
Omega Centauri Wide-Field View (Courtesy of the ESA)
Stellar brightness is also conveniently measured in magnitudes. These magnitudes are much friendlier than the long luminosity values that we would otherwise use. For example, the sun has a luminosity of 384,600,000,000,000,000,000,000,000 Watts and an apparent magnitude of -26.7. Now, which value would you rather remember and use?

There are two types of magnitudes that you should be aware of:

Apparent magnitude: the magnitude of the object measured from Earth.
Absolute magnitude: the magnitude of the object measured at a distance of 10 parsecs away from it.

Magnitudes are not exclusive to stars, and instead can be extended to anything from Mercury to the Andromeda Galaxy. 

A brief historical note on magnitudes: they have existed long before the Richter scale. Magnitudes originated early on during the Greek and Roman Empires. They were used by everyone from Hipparchus to Ptolemy, but were only standardized in the mid-19th century. The reason that the magnitude scale seems arbitrarily centered was because it involves a lot of historical artifacts.

But let's stop dwelling on the past. How does one actually calculate the magnitude of a star? As it turns out, it's quite simple! A stars apparent magnitude can be computed in one of two ways:  
Be sure to learn more about magnitudes here!

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Aside from being one of the best television dramas of all time, Breaking Bad was also a great example of popular media bringing science to the forefront of entertainment. Breaking Bad, which follows the downward spiral of a brilliant chemistry high school teacher as he learns that he has cancer, reminded the public that chemistry is, for lack of a better word, awesome! But the show did more than just inspire youth to study chemistry for its explosive properties, it also made permanently implanted the word Heisenberg into everyone's minds. 

The name was inspired by one of the earliest quantum physicists, Werner Heisenberg. Heisenberg made many contribution to the field of physics, mainly in the branch of quantum mechanics. Heisenberg is probably best known for the Heisenberg's Uncertainty Principle:

This, seemingly simple, relation has made generations of physics students question everything they had come to believe was fact. This relation states that two variables, the uncertainty in a particle's position and the uncertainty in a particle's momentum are related by a constant. If we choose to peer deeper, we realize that for this relation to hold true the uncertainty in the position of the particle must be inversely proportional to the the uncertainty in the momentum of the particle.

Translating this observation to English, we see that the more certain we are in where precisely a particle is, the less certain we are in how fast it is moving. Taking this to the extreme, if we are nearly certain of exactly where the particle is, we are nearly infinitely uncertain as to its momentum. 

And this is just one of a plethora of various things that hold true in quantum mechanics, but that we are not familiar with seeing on a macroscopic scale. If you'd like to learn more about this principle and quantum mechanics in general, we suggest watching some lectures on Coursera and reading about it some more on Wikipedia. Although we would like to give you fair warning, quantum mechanics is not for the faint of heart. While the quantum mechanical conclusions are riveting, the path to said conclusions is often long and laborious, requiring advanced math.

To see other equations like the Heisenberg's Uncertainty Principle, check out College Modern Physics.

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PictureHeinsenberg vs. The Real Heisenberg (Courtesy of


National Popcorn Day was this past Monday, and many Americans took this opportunity to consume some delicious salted butter on their favorite edible vehicle: popcorn! But popcorn is more than just one of the most delicious and popular American snacks, it also serves as a good analogy for events that are constantly occurring in the cosmos

Imagine that a massive - at least 8 times more massive than our sun - star is represented by a corn kernel. When this kernel is placed in the microwave and heated it heats up and eventually pops. All the heat that is imparted on the kernel through the microwave's microwaves (that's why it's called a microwave after all) is released in three ways: light, sound, and heat. Unfortunately most of us can't witness the first of these three methods of energy release because the popped kernel does not glow in the visible part of the electromagnetic spectrum - unless you begin burning it of course. But heating the kernel does cause it to glow brighter in the infrared part of the electromagnetic spectrum.

The release of energy through sound and heat, on the other hand, are a lot more evident. It's precisely the sound, or popping, that reassures us that our popcorn is being cooked. But in addition to acting as a great cooking timer, the popping is also an effective way for the, now, popped kernels to lose energy. Finally, the popcorn loses the rest of its excess energy through the dissipation of heat to the air or surrounding medium.

But what happened when the kernels were heated and why did they pop? As it turns out, kernels aren't as dry as they appear. In fact, they have quite a bit of water inside their shells, which causes the kernel to vibrate when heated. Eventually the built up pressure becomes large enough such that it is able to overcome the shell's structure and it causes the kernels to pop.

So, what does this have to do with science? And specifically with the cosmos? Well, let's go back to our kernel - star analogy. Much like a kernel that's being heated in the microwave, a star also experiences extreme outward pressures. In a star, this pressure is due to nuclear fusion - the combination of at least two atomic nuclei into a new, and typically more massive, nucleus. Much like the kernel however, the star's outward pressure is fighting against a force pushing inwards: the gravitational force. The interplay between these two forces causes the star to significantly vary in size during its lifetime.

Unfortunately this is where the analogy breaks down. Unlike with kernels that pop when the internal pressure overcomes their shell, stars cause supernovae (Type II Supernovae in particular) after their internal pressure disappears and the gravitational force inwards wins out. This quick stellar (pun intended) collapse, causes a violent explosion, which we refer to as a Type II Supernova. Supernovae are more than exciting explosions, however. They are also the only way by which heavier elements (i.e. anything heavier than iron) form - naturally that is.

But the analogy doesn't completely break down! Whether you're talking about kernels or massive stars, one thing remains true: the popped product (popcorn / byproduct of a supernova) is a lot more beautiful, and delicious, than what you started out with.

To learn more about astrophysics, check out the college stellar astrophysics webpage here!

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PictureArtist's depiction of a black hole.
In light of recent attention brought to astronomy though movies such as Interstellar and Theory of Everything, we, here at EQNS, have decided to give a brief introduction to the mechanics and evolution of black holes.

Stellar evolution results in the formation of one of three things: white dwarfs, neutron stars, or black holes. Typically, the larger the star is during its life, the more mass it'll leave behind once it dies. Stars that fall within the Chandrasekhar limit - of roughly 1.4 times the mass of the sun - end their lives as white dwarfs, with their mass being held up through electron degeneracy pressure. Stars that are slightly larger than the Chandrasekhar limit - between 1.4 and 3 times the mass of the sun - end their lives as neutron stars, whose mass is held up through quantum degeneracy pressure. Stars any larger than roughly 3 times the mass of the sun are too massive to be held up by either electron degeneracy or quantum degeneracy pressure and collapse into a singularity that we refer to as a black hole. 

Type of Compact Star

White Dwarf
Neutron Star
Black Hole

Approximate Mass Requirement

Less than 1.4 times the Mass of Sun
Between 1.4 and 3 times the Mass of the Sun
Over 3 times the Mass of the Sun
Unfortunately, there are currently more questions than answers when it comes to black holes. Because of their mass and density, any light or mass that strays too close to a black hole becomes lost forever. Much like the Earth's gravitational field that keeps us from floating away into space and ensures that the moon continues to orbit the Earth, the gravitational field of a black hole swallows up everything including light. The Schwarzschild radius is defined to be the distance beyond which light and/or matter may escape the pull of a black hole. Anything closer than the Schwarzschild radius, will end up being devoured by the black hole.

The Schwarzschild radius is directly related to the mass of the black hole. The more massive the black hole, the larger the Schwarzschild radius. The equation for the Schwarzschild radius is given below:
Learn more about the Schwarzschild radius and other stellar astrophysics principles by checking out the equations on the college stellar astrophysics page here!
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Studying shouldn't be difficult, neither should navigating EQNS. In an attempt to mitigate the process even further, we recently recorded a YouTube video that gives a step by step guide to accessing the formula you're looking for on EQNS. 

Check it out here:
The days of aimlessly scouring the internet for equations that may have been covered in class, and are definitely needed for the homework or the exam, are over. On 01/12/15, EQNS Beta Version was released. A year in the making, EQNS completely transforms the art of problem solving. The one-step resource for solving problem sets and studying for exams, EQNS eliminates the time students spend trying to find the right tools (i.e. equations), and instead directs their focus on learning the material.

Science is intricately beautiful and profound, and has an extremely high barrier of entry. But this should no longer be the case, everyone deserves to understand the underlying physical and scientific principles that govern this world. And we believe that this can only occur with an improvement to the way we teach science - one equation at a time. 

If you know anyone who is currently studying astronomy, chemistry, geology, or physics in high school or college please share EQNS with them.