Destroying Planets & Stuff

We've all grown up on Star Wars. One of the most vivid scenes of the movie that we all never forget is when the Death Star destroyed Alderaan. It immediately made a deep impression on our psyches. It bespoke the power of the Galactic Empire and cowed even us- the audience looking upon an illusion, to think twice before opposing the Emperor.

The spectacle also probably broke the movie's illusion somewhat- because we all asked ourselves whether such a thing as this were possible in the real world? Is it scientifically possible to destroy a planet like the Death Star destroyed Alderaan?

In a word: yes.

And the method is actually fairly simple as well. For something to be capable of destroying a planet (or star, or other astronomical object that is a sphere with a roughly average density), it needs to generate enough energy (in joules) to overcome that object's gravitational binding energy. That is, the energy by which a planet's gravity binds its mass together as a single object. Equal or exceed that, and the planet's mass is scattered far enough so that its gravity will not pull it back together again. This means that the debris will equal or exceed a planet's escape velocity (more on escape velocity in another post).

The equation for figuring out gravitational binding energy is as follows:

The gravitational binding energy of a spherical astronomical object (like a planet or star) is U. G is the gravitational constant (6.67x10^-11), M is the mass of the object, and r is its radius in meters (keep that in mind, because if you put its radius in kilometers, as you may find preferable when dealing with objects this big, you will get nonsensical results).

Now let's use the Earth as an example of putting this equation in action.

The Earth's mass, as you may know, is 5.97219×10^24 kilograms. Its radius in meters is 6,378,100.

Now with my favored calculator, this is very easily solved:

5.97219×10^24 = 5972190000000000000000000
6.67x10^-11 = 0.0000000000667

3 x 0.0000000000667 x 5972190000000000000000000^2 / (6378100 x 5) = 223796346390292093256612470798513.66394380771703171790972.

There is your gravitational binding energy in joules. Obviously this is a large and very abstract number that is difficult for most people to make sense of. For this reason, I generally prefer to break the joules down into TNT equivalent, which is far easier to grasp.

Using a simple energy converter, you will find that the answer in terms of TNT equivalent is 5.3488610514e+22 tons of TNT. That's 53.488 sextillion tons, which you can verbalize using this numbers to words converter. In metric prefixes, the number sextillion is denoted by the word "zetta." So the gravitational binding energy of Earth is 53.488 zettatons of TNT.

This is obviously, again, a staggering amount of energy (though not as much as the kinetic energy of the planet's orbit). For some perspective, the Tsar Bomba, the most destructive weapon ever constructed by man, at around 50-60 megatons (the word "mega" denotes the number million in metrix prefixes), was a quadrillion times less powerful.

This equation isn't perfect, of course. It assumes both uniform density (which astronomical objects do not possess), and a perfect sphere (which again, the Earth and other bodies are not), but this is a good estimate of the Earth's gravitational binding energy and is more than usable in a scientific context.

And if you're lazy and don't want to use the equation, you can try the SD.net Planetary Parameter Calculator, and look for the "Death Star yield LL." It will be second from bottom on the right. By inputting the parameters of your object to the left, the calculator will tell you the gravitational binding energy of your object, though the surface gravity requirement might be a hassle if you're looking for an answer for an exotic star, for instance.

So, summing up...

1. To destroy a planet, star, or other large spherical object in space, you must overcome its gravitational binding energy.
2. To solve for this you can do the equation given above.
3. This equation is only a (good) approximation most of the time however, because it assumes perfect spheres and uniform density, which most astronomical objects do not have.

As a final note, you can use the kinetic energy equation to get a more accurate measurement of the power of a planet destroyer like the Death Star. By taking the velocity of a planet's mass scattering beyond its original radius, you can find the energy behind the attack. This will often be far higher in entertainment media than the bare minimum gravitational binding energy. Such was the case with the Death Star. Planet Busters Death Star Destroyers Star Wars Binding Energy Astronomy

The Evolution of the Universe

Because our lives are so short, our tendency is to think of the universe as stable, unchanging, and eternal. This was codified into our species' most important doctrines for thousands of years, and even Einstein believed in this so fervently that he made the biggest blunder of his career- the Cosmological Constant, which appeared to be in direct defiance of his own magnum opus- the Theory of Relativity.

These theories of a static universe were exploded over the course of the 20th century, and in its closing years it was discovered that not only is the universe expanding, but accelerating in its expansion- driven by a mysterious force that has not yet been identified and only given the name of Dark Energy.

The universe, it turns out, is actually a very chaotic, dynamic, and unstable place. It merely appears stable in our human eyes because the time spans that these changes take place in are so long- but from the point of view of a black hole for instance, they are in fact quite rapid. Such is also the case with biological evolution. As Carl Sagan stated in episode eight of Cosmos: "from the point of view of a star, life on Earth evolved very rapidly."

So, where did the universe come from and where is it going? The influential book The Five Ages of the Universe by Fred Adams and Gregory P Laughlin goes over the question in detail. Unfortunately, an admirer of the beauty of the cosmos will not like what it is in store for. The ugly truth is that the 2nd Law of Thermodynamics essentially signs the universe's death warrant. Chaos must increase over time. Order turns to disorder, and energy in the universe must decrease.

The road map of the cosmos is marked below:

1st Age: The Primordial Era

This age began with the Big Bang and lasted until the first stars began to form. It is believed that the inflation of the universe began immediately after the Big Bang, instantly expanding it exponentially in size. The earliest thing we can see- the Cosmic Microwave Background Radiation, which took place around 380,000 years after the Big Bang, is a product of this period. The universe became transparent and matter, which at this point consisted only of hydrogen, helium, and a tiny amount of lithium, began to coalesce. In the centers of dense clouds of hydrogen, thermonuclear fusion began to take place.

The Cosmic Microwave Background, a remnant of the Primordial Era

2nd Age: The Stelliferous Era

The thermonuclear reactions at the end of the Primordial Era signified the creation of the first stars, and the blackness of the universe was set aglow with light. The "star bearing" age had begun. As you might expect, this is the era in which we live- the golden age of the universe, awash with light, color, and, we are given reason to expect, life.

The first generation of stars was typically very massive- hot blue stars that burned through their fuel in only a few million or tens of millions of years. Though this sounds like an impossibly long time, from the point of view of a star, it is but a brief instant.

These first generations of stars fused hydrogen into helium, and then helium into carbon and oxygen, and then those into even heavier elements. The short-lived blue giant stars exploded these heavier elements back out into space in spectacular supernovae, creating new stars and systems like our own sun and solar system, and it was through these heavier elements that the building blocks of life, created in the stars, eventually coalesced into life itself. In this era, the remains of dead stars are the building blocks of new ones. Stars are almost literally like phoenixes rising from the ashes.

Eventually however, there will be no hydrogen left to support new stars. Each generation of stars is slightly less massive than its predecessors, and the free clouds will eventually be used up. With no hydrogen left, the existing stars will blink out and not be replaced- starting first with the massive blue stars, and then moving down the line to mid-sized stars like the sun. The universe will be far dimmer, with only faint red dwarves still shining in the latter stages of this era. At this same time, it is also thought that the continuing expansion of the universe will push all galaxies not gravitationally bound to each other so far away that no information, traveling at the speed of light, will ever be exchanged between them again. The only visible things will be objects in each individual galaxy- in whatever form they may take. It is estimated that in about 100 trillion years, the last red dwarves will have finally perished, and the age of stars will be over.

If this sounds depressing to you, be prepared, for the end of the ride is far from here.

The Hubble Ultra Deep Field. By the end of the Stelliferous Era, all this will be gone.

3rd Age: The Degenerate Era

This age gets its name from the dominant forms of matter in the universe that will succeed the time of the stars- the various stellar remnants. Brown dwarves (which are stars that never shined), white dwarves, neutron stars, and black holes will be all that remain.

It isn't totally dark in the universe- not yet at least. There is still some radiation coming from the dying stellar remnants, and the occasional supernova may yet happen when collisions amongst white dwarves, for example, occur. But it is a mere shadow of the glories of the Stelliferous Era. The degenerate matter will continue to cool until it no longer shines and eventually, gravitational interactions will scatter the remnants into the aforementioned collisions, outside of the galaxy, or into black holes.

By the end of this era, in 10^40 years, only black holes are still left.

It is hypothesized by some that protons will decay in the Degenerate Era, which will dissolve matter as we know it and thus shorten the time span involved. It is not yet clear whether or not this will be the case, but whatever that case may be, the coming age makes even this one look good.

White dwarves colliding. In the Degenerate Era, stellar remnants like these are the dominant forms of matter.

4th Age: The Black Hole Era

Welcome to the far, far, far future, where the only objects of note remaining in the universe are black holes. No light is shining and all is dark.

But not forever. Not even black holes are exempt from the 2nd Law of Thermodynamics, and they too will evaporate in the form of Hawking Radiation. As their mass is continually lost, the black holes themselves will begin to emit light. In the last seconds of their lives, they will shine and explode colossally, lighting up the darkness of the universe for the first time in what can almost justly be called forever.

But this truly is the last hurrah- the last trick in the bag of the universe. Once the last black hole goes in about 10^99 years, there will truly be nothing left to see.

Artist's conception of a black hole. Even these evaporate very slowly.

5th Age: The Dark Era

Welcome to the true Dark Ages, where no light will shine- forever. All that remains now is a scattered smattering of sub-atomic particles- electrons, photons, and other particulate matter. Some annihilation events may occur when an electron and positron encounter each other (and essentially cancel each other out of existence) but these will make a snail's pace look like the speed of light. The future of this era seems to be all remaining matter either annihilating or moving so far away that it will never interact with each other.

Either way, it is not a bright future, and an immortal observer would look back and wonder at what had been.

Assesment:

Is there any way out of this death trap that the 2nd Law of Thermodynamics seems to be spelling out for the universe? The answer in most circumstances seems to be "no." Some theories of quantum mechanics predict or suggest that it might be possible to start a new universe, but that's a bit beyond my capability at this point.

This model also makes a number of assumptions- chief among them that the expansion of the universe will continue to occur without interruption. All observational data suggests this will be the case, but who is to say what crazy thing might be discovered in the future?

Beyond anything, the postulated fate of the universe makes me more self-reflective and appreciative of the time in which I'm alive. We are the legacy of the universe itself, privileged with experiencing the glories of its golden age. I think it is an insult to the cosmos that created us to not revel in the time that we're here and be productive members of the cosmic order. We should want to live the good life- and in some ways this brings us full circle, back to Aristotle who was so influential in the model of a static, unchanging universe. The ancient philosopher goes into detail about living the Eudaimon or desirable life- and the universe gives us the prerequisite to do that. It didn't always give us that, and it won't always. Scarcity then, makes the Eudaimon life even more desirable. And so, we must value it, treasure it, and put in the work to make it happen.

Aristotle, by Francesco Hayez (1811)

Five Ages of the Universe Cosmology Stars

The Most Famous Equation

Why is the sun so powerful? Why do nuclear weapons create such a big boom? The answer can be broken down to mass-energy equivalence, as expressed by the most famous equation in all of science.

Contrary to popular belief Einstein was not the first one to ever propose a relationship between mass and energy, but he is the one that created the formula, which is a natural result from his theories of relativity.

An examination of the wonderfully simple equation reveals the answer as to why atomic physics deals with such powerful levels of energy.

"c" stands for the speed of light- 299,792,458 meters per second. The equation then squares this number and multiplies that by the mass of the object. There is your answer in joules. As you can already guess, the number of joules (and thus the amount of energy measured) is staggering. Put simply, it's the most efficient release of energy known to science. Whereas the breaking of chemical bonds releases some energy, it pales in comparison to any nuclear process where a mass-energy dynamic is present. Let's try an example:

Say that one kilogram of matter is converted entirely into energy.

1 x 299,792,458^2 = 89875517873681764 joules.

That is equivalent to 21.480 megatons of TNT- a yield nearing one of the most powerful thermonuclear devices ever tested.

The Castle Yankee device, which converted less than one kilogram

And that's only one kilogram, one!

Mass-energy equivalence is serious stuff, friends.

What makes it all go is the loss of mass in a system. This lost mass then gets converted entirely into energy. When a nuclear weapon goes boom, the spectacularly destructive results that you see is due to a certain loss of mass in the original system. The same forces are at play inside the cores of the stars.

The sun for instance, loses 4 million metric tons per second, converting some of that lost mass into energy. The slightly less massive sun then radiates that energy at about 3.846 x 10^26 watts per second, or around 91.921 petatons of TNT per second (that's 91.921 quadrillion tons), over a trillion times more powerful than the Tsar Bomba. More massive stars do this at even higher rates and produce even more energy.

That's mass-energy equivalence in a nutshell.

Starting Simple: Kinetic Energy

We might as well start off this blog with a simple post about kinetic energy. Firstly, let's define it: kinetic energy is the energy (the capacity for doing work on something) of an object that is imparted by its motion. The reason why a fist hurts you when you get punched is due to the kinetic energy of the fist. The kinetic energy of a speeding car is the reason why it kills you when you get hit by it. This sounds simple, but is more pervasive and subtle than you may think. Heat (or lack thereof) can also be seen to be a form of kinetic energy. How? The thing that distinguishes something that is hot from something that is cold is the speed at which its constituent matter is moving. This is kinetic energy.

A-Rod hurt by the kinetic energy of the ball

So how is this thing called kinetic energy solved?

In classical, Newtonian mechanics, the kinetic energy of an object in joules (the measurement unit of energy) can be found by multiplying half of its mass times its velocity squared, as illustrated below:

KE = .5mv^2

So we can solve for the kinetic energy of an object in this way. Let's do something fun- something really big. How about the kinetic energy of the Earth itself in its orbit around the Sun?

How energetic is this planet?

The first order of business is to find the mass of your object (in this case the Earth) in kilograms. Kilograms is bolded because if you don't measure your object in kilograms, your calculation will be nonsensical. In the age of Wikipedia, this task which may have been a tall order in the past is today pretty easy.

The Earth's mass is listed as being 5.97219×10²⁴ kilograms, via a NASA source. Don't be intimidated by the 10^24 label. Any advanced calculator will be able to give you the answer (and most of your equations in school likely won't have numbers anywhere near as large). If you're feeling very masochistic, you can just write out the number 1 followed by 24 zeros after the number 9.

This calculator (which I'll do a post on later) can give you an easy answer: 5972190000000000000000000 (also do take note that you can just leave it as written above and work with it in that calculator).

Next, you need to find the velocity of your object in meters per second. This is important, especially for those of us living in the United States where our typical measurement of speed or velocity is in miles per hour. If you do not put your velocity in meters per second, your number will again be nonsensical.

Don't do this

A quick check back on Wikipedia reveals that the Earth orbits its parent star at a velocity of 29.78 kilometers per second on average. Simply multiplying this number by 1,000 gets your velocity in meters per second at 29,780.

Now that we have our components (5972190000000000000000000 kilograms, 29,780 meters per second), we can find the kinetic energy of this object in joules:

5972190000000000000000000 x .5 = 2986095000000000000000000

2986095000000000000000000 x 29,780^2 = 2648213572998000000000000000000000.

That is the kinetic energy of the Earth in joules. Needless to say, this is obviously a staggering amount of energy, far more than enough to destroy the planet itself (more on that fun stuff in a future post).

Now I'll let you cheat and use this kinetic energy calculator that I've been keen on using. It lets you solve for mass, kinetic energy, or velocity. It's pretty neat (do however make sure to put the whole mass into its proper component, not half of it also, no commas).

A quick check reveals that the answer it gives you is essentially the same number as doing it the long-hand way.

HOWEVER...

there is a snag. This equation and thus this answer is technically incorrect (or at least, was derived in an incorrect way).

No, this answer being technically incorrect (or incorrectly derived) isn't something even a scientist is typically going to have to worry about, but in other cases, it will show itself.

The problem arises due to the revelations of that most famous of all scientists, Albert Einstein. His experiments in the early 20th century proved that the Newtonian view of gravity, mass, and matter was incorrect. By revealing the true nature of space-time, Newton was eclipsed. This upset a lot of people.

Trollin' like a mo fo

The true, proper equation for kinetic energy can be found on this page.

Confused yet? Don't worry, I am too! You probably won't need to worry about this unless you're taking university-level physics (hence, not me), but I'll go into detail regarding why this way is technically correct (because it's at least an interesting caveat and will make you more knowledgeable about the universe).

While Newtonian mechanics works well in explaining the properties (in this case, kinetic energy) of everyday objects moving at normal speeds (yes, this includes things like stars and planets in their orbits, go figure), it cannot explain the properties of very small objects (like subatomic particles) or very big objects (like black holes). It also cannot measure for objects moving at velocities approaching the speed of light where the effects predicted by the Theory of Relativity start to become noticeable to an observer (things like time dilation and mass increase). Generally, this starts to happen at around 10% of the speed of light (29,979,245.8 m/s).

Going back to our example of the Earth in its natural orbiting speed, you can take a look at stardestroyer.net's fantastic relativity calculator (yes, things like this fortunately do exist to simplify Relativistic kinetic energy should you wind up needing to use it). Inputting the variables above, we find that the Newtonian and Relativistic answers in joules for Earth's kinetic energy are identical:

2.648E+33 (2648000000000000000000000000000000) joules, essentially the same answer arrived at multiple times above.

But how about if the Earth was orbiting at 50% of the speed of light (149896229 m/s), and somehow we were all still alive to ponder it?

The Newtonian kinetic energy is: 6.709E+40 joules.

The Relativistic kinetic energy is: 8.301E+40 joules.

A noticeable and important difference is revealed. Though it might seem small (a differential of 1.237 times), it is significant, and it will increase exponentially the closer you get to the speed of light.

So at the end of the day...

What have we learned?

1. Kinetic energy is the energy of an object based around its motion.
2. You can solve for an object's kinetic energy by taking half of its mass, and multiplying it by the square of its velocity (.5mv^2).
3. This equation however is only a very good approximation in most cases, and cannot explain for objects of very small or very large mass, or for objects traveling at a velocity nearing the speed of light (starting at around 10%).
4. Breathe a sigh of relief that for 99% of you, relativistic kinetic energy won't be anything you need to worry about.
5. Einstein trolled the entire scientific establishment with his findings.

Kinetic Energy Albert Einstein Isaac Newton Classical Relativistic Mechanics