Equations of a Parabola: Standard to Vertex Form and Back Again

November 27, 2011

Highlighting the Vertex Form of the equation for a parabola.

The equation for a parabola is usually written as:

Standard form:

$y = ax^2 + bx + c$

where a, b and c are constants. This is the form displayed in both the VPython Parabola and Excel parabola programs. However, to make the movement of the curve easier, the VPython program also uses the vertex form of the equation internally:

Vertex Form:

$y = a(x-h)^2 + k$

where the point (h, k) is the location of the vertex of the parabola. In the example above, h = 1 and k = 2.

To translate between the two forms of the equation, you have to rewrite them. Start by expanding the vertex form:

y = a(x – h)2 + k

becomes:

y = a(x – h)(x – h) + k

multiplied out to get:

y = a(x2 – 2hx + h2) + k

now distribute the a:

y = ax2 – 2ahx + ah2 + k

finally, group all the coefficients:

y = (a)x2 – (2ah)x + (ah2 + k)

This equation has the same form as y = ax2 + bx + c if:

Vertex to Standard Form:

a = a
b = -2ah
c = ah2+k

And we can rearrange these equations to go the other way, to find the vertex form from the standard form:

Standard to Vertex Form:

$a = a$

$h = \frac{\displaystyle -b}{\displaystyle 2a}$

$k = c - ah^2 = c - \frac{\displaystyle b^2}{\displaystyle 4a}$

Summary

In sum, you can write the standard equation for a parabola as:

Standard form:

And you can write the equation for the same parabola in vertex form as:

Vertex form:

UPDATE 1: This app will automatically convert from standard to vertex form (or back again).

UPDATE 2: Automatically generate and embed graphs using this parabolic grapher app.

Citing this post: Urbano, L., 2011. Equations of a Parabola: Standard to Vertex Form and Back Again, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Global Temperature Model: An Application of Conservation of Energy

August 19, 2011

Energy cannot be either created or destroyed, just changed from one form to another. That is one of the fundamental insights into the way the universe works. In physics it’s referred to as the Law of Conservation of Energy, and is the basic starting point for solving a lot of physical problems. One great example is calculating the average temperature of the Earth, based on the balance between the amount of energy it receives from the Sun, versus the amount of energy it radiates into space.

Anything with a temperature that’s not at absolute zero is giving off energy. You right now are radiating heat. Since temperature is a way of measuring the amount of energy in an object (it’s part of its internal energy), when you give off heat energy it lowers your body temperature. The equation that links the amount of radiation to the temperature is called the Stefan-Boltzman Law:

$E_R = s T^4$

where:
T = temperature (in Kelvin)
s = constant (5.67 x 10-8 W m-2 K-4)

Now if we know the surface area of the Earth (and assume the entire area is radiating energy), we can calculate how much energy is given off if we know the average global temperature (the radius of the Earth = 6371 km ). But the temperature is what we’re trying to find, so instead we’re going to have to figure out the amount of energy the Earth radiates. And for this, fortunately, we have the conservation of energy law.

Energy Balance for the Earth

Simply put, the amount of energy the Earth radiates has to be equal to the amount of energy gets from the Sun. If the Earth got more energy than it radiated the temperature would go up, if it got less the temperature would go down. Seen from space, the average temperature of the Earth from year to year stays about the same; global warming is actually a different issue.

So the energy radiated (ER) must be equal to the energy absorbed (EA) by the Earth.

$E_R = E_A$

Now we just have to figure out the amount of solar energy that’s absorbed.

The Sun delivers 1367 Watts of energy for every square meter it hits directly on the Earth (1367 W/m-2). Not all of it is absorbed though, but since the energy in solar radiation can’t just disappear, we can account for it simply:

• Some if the light energy just bounces off back into space. On average, the Earth reflects about 30% of the light. The term for the fraction reflected is albedo.
• What’s not reflected is absorbed.

So now, if we know how many square meters of sunlight hit the Earth, we can calculate the total energy absorbed by the Earth.

The solar energy absorbed (incoming minus reflected) equals the outgoing radiation.

With this information, some algebra, a little geometry (area of a circle and surface area of a sphere) and the ability to convert units (km to m and celcius to kelvin), a student in high-school physics should be able to calculate the Earth’s average temperature. Students who grow up in non-metric societies might want to convert their final answer into Fahrenheit so they and their peers can get a better feel for the numbers.

What they should find is that their result is much lower than that actual average surface temperature of the globe of 15 deg. Celcius. That’s because of how the atmosphere traps heat near the surface because of the greenhouse effect. However, if you look at the average global temperature at the top of the atmosphere, it should be very close to your result.

They also should be able to point out a lot of the flaws in the model above, but these all (hopefully) come from the assumptions we make to simplify the problem to make it tractable. Simplifications are what scientists do. This energy balance model is very basic, but it’s the place to start. In fact, these basic principles are at the core of energy balance models of the Earth’s climate system (Budyko, 1969 is an early example). The evolution of today’s more complex models come from the systematic refinement of each of our simplifications.

If students do all the algebra for this project first, and then plug in the numbers they should end up with an equation relating temperature to a number of things. This is essentially a model of the temperature of the Earth and what scientists would do with a model like this is change the parameters a bit to see what would happen in different scenarios.

Feedback

Global climate change might result in less snow in the polar latitudes, which would decrease the albedo of the earth by a few percent. How would that change the average global temperature?

Alternatively, there could be more snow due to increased evaporation from the oceans, which would mean an increase in albedo …

This would be a good chance to talk about systems and feedback since these two scenarios would result in different types of feedback, one positive and one negative (I’m not saying which is which).

Technology / Programming

Setting up an Excel spreadsheet with all the numbers in it would give practice with Excel, make it easier for the student to see the result of small changes, and even to graph changes. They could try varying albedo or the solar constant by 1% through 5% to see if changes are linear or not (though they should be able to tell this from the equation).

A small program could be written to simulate time. This is a steady-state model, but you could assume a certain percent change per year and see how that unfolds. This would probably be easier as an Excel spreadsheet, but the programming would be useful practice.

Of course this could also be the jumping off point for a lot of research into climate change, but that would be a much bigger project.

References

Yochanan Kushnir has a page/lecture that treats this type of zero-dimesional, energy balance model in a little more detail.

Citing this post: Urbano, L., 2011. Global Temperature Model: An Application of Conservation of Energy, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Beach Geomorphology on Deer Island

July 10, 2011

Figure 1: Beach profile on Deer Island spit.

The western end of Deer Island extends a white, sandy, artificial, spit that partially covers the first of a series of riprap breakwaters that protect the waterfront development of the city of Biloxi. Although we’d landed there to pick up garbage as part of our coastal science camp, the beautifully developed beach profile was worth a few minutes.

Figure 2: Narrow beach typical of the east-west trending shorelines that are not exposed to the direct force of the waves.

The spit curves just ever so slightly northward, so it feels more of the direct force of waves blown all the way along the length of Biloxi Bay. The combination of unvegetated sand and stronger waves makes the beach along the spit looks very different from the beaches that parallel the shore. While the parallel beaches on Deer Island are covered in grass almost to the water’s edge (Fig. 2), the spit has a much wider beach, with a nicely developed sandbar protecting a shallow, flat-bottomed, water-saturated trough behind it (Fig. 1).

While the white beaches are pretty (that’s why they imported this sand after all), there are a number of fascinating features in the trough.

Figure 3: On our Natchez Trace hike we found it quite easy to stick fingers into the red precipitate at the bottom of the stream.

The first, and most obvious question is, why the reddish-orange color in the fine grained sediment at the bottom of the trough? A microscope and a little geochemical analysis would be useful here, however, lacking this equipment, we can try drawing parallels with some of our experiences in the past. In fact, we should remember seeing the same color in some of the streambeds when we were hiking in Natchez Trace State Park in Tennessee (Fig. 3). My best guess at that time was that the red was from iron in the groundwater being oxidized when it reached the surface.

Figure 4: The rich black of decaying organic matter, sits just beneath the rusty-orange surface sediement.

Figure 5: Green, organic matter, freshly deposited at the edge of the trough. If it decays while saturated with water it will turn black. Note also the splay of white sand at the top of the picture.

This is probably not a bad guess for the red in the trough as well, since there is some fresh groundwater discharge from the shallow watertable on the island. However, I suspect that the story is a bit more complex, because the rich black color of the organic matter just beneath the surface (Fig. 4) suggest that the shallow water and surface sediment in the trough is lacking in oxygen. On the other hand, it’s not uncommon to have steep geochemical gradients in boundary environments like this one.

The physical and geochemical gradients extend horizontally as well as vertically. At the edges of the trough the organic matter just beneath the surface is green, not black (Fig. 5), because this is the color of the undecayed algae.

At the seaward side of the beach, the waves of Biloxi Bay lap against the sand bar. When the tide rises, and the wind picks up, these waves wash over the crest of the sand bar pushing water and sediment over the top into trough. When the sand washes evenly over the top it creates thin layers (possibly one layer with each high tide). If you cut into these layers you’ll see little the laminations in profile, which, because the layering is close to horizontal, look like the lines of topography on a map (Fig. 6). When the waves wash over small gaps in the sandbar the sediment it transports is deposited in a more concentrated area – these are called sand splays – that overlap and cover some of the fine-grained, orange sediment at the edge of the trough. These are both two of the small ways that the sand bar moves, slowly pushing inland.

Figure 6: Sand splay and laminations on the landward side of the sand bar. The laminations are created by even overwash of the sandbar, while the splay is the result of more concentrated flow.

Bioturbation

The features on the bottom of the trough are a quite interesting because of the observable effects of bioturbation (disturbance by organisms) (Figs. 7, 8 & 9).

Figure 7: In close-up, the holes of the crabs and the mixture of colors looks like an arid, volcanic landscape photographed from space.

Figure 8. Digging deep beneath the orange surface sediment, small crabs create mounds of white sand.

Figure 9. Footprints of predators. Paleontologists use features like these that are preserved in rocks to discover interpret what the relationships between organisms was like in the past.

Citing this post: Urbano, L., 2011. Beach Geomorphology on Deer Island, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Longshore Drift and Pufferfish

May 31, 2011

A groin strains to hold back the longshore drift. It is, as always, only partially successful.

It was about 1.5 kilometers from the Research Lab to the estuary where we spent our first morning sampling (overview of the trip is here).

Elevated beach house.

Walking along the beach to get there, we could see the beach houses to the right of us, across the narrow road of East Beach Drive, standing tall on columns to keep them above the reach of the storms. According to Stephanie, our guide, the storm surge from Hurricane Katrina in 2005, reached awfully close to the tops of the columns. The research lab, which is not elevated, lost an entire building to that hurricane. Indeed, much of the coast is still recovering from Katrina’s damage.

The white sandy beach, on the other hand, looked beautiful, which was a bit odd. After all, how did it survive the storm? Furthermore, when you think about it, this beach is located behind a string of barrier islands, which protect the coast from the full force of the waves coming out of the Gulf of Mexico, so how come there is enough wave energy to maintain a sandy beach. The relatively calm waters should allow finer grained sediment, like clay and silt, to settle out, and this area really should be a marsh. The answer, it seems, is that this is an artificial beach. Every few years, thousands of tons of sand are dumped along the coast to “replenish” the beachs.

Without beach replenishment the beaches would revert to salt marshes like this one.

This coastline really should be a tidal marsh, like the one we found when we got to the estuary. These Gulf-coast salt-marshes are fronted by a relatively short version of smooth cordgrass (spartina alterniflora), backed up by the taller, and more common black needlerush (Juncus roemerianus Scheele) .

Longshore Drift

Now, if this is a low-energy environment that allows silt and clay can settle out of the water column, where does the sand go so that it has to be replenished every so often? It is gradually moved along the coast by longshore drift.

Longshore drift moves sand along the coast in the direction of the wind. Image via the USGS.

Waves hit the beach at an angle. As they break, the turbulent swash pushes sand up the beach at the same angle as the movement of the waves. As the wave retreats, the backwash, drawn by gravity, pulls sand perpendicularly down towards the water. The net effect, is that sand gradually moves down the coastline with each swash and backwash of the waves.

Since dumping tons of sand is expensive, engineers try other things to prevent the sand from running off down the beach. Someone, a very long time ago, had the great idea to build a wall sticking out from the beach to impede the sand in its unwanted migration. This type of wall is called a groin (or sometimes a groyne in polite company), and it does stop the sand. In fact, the sand builds up on the upwind side of the groin. Unfortunately, it does not stop the longshore drift on the downwind side, and that results in the erosion of a bay on that side.

A groin impedes longshore drift. Note that the waves approach the beach at an oblique angle.

Pufferfish

Beaches are also great places to find random things washing up. We lucked upon an unusually large pufferfish (family: tetraodontidae). It was quite puffed up. It was also quite dead.

Pufferfish.

Pufferfish are famous for being extremely poisonous. According to the National Geographic page on pufferfish, their tetrodotoxin over a thousand times more poisonous than cyanide, and there is no known antidote.

Citing this post: Urbano, L., 2011. Longshore Drift and Pufferfish, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Global Atmospheric Circulation and Biomes

April 21, 2011

We’re studying biomes and I don’t know a better way to consider how they’re distributed around the world than by talking about the global atmospheric circulation system. After all, the primary determinants of a biome are the precipitation and temperature of an area.

Diagram showing global atmospheric circulation patterns.

It’s a fairly complicated diagram, but it’s fairly easy to reproduce if you remember a few fairly simple rules: hot air rises; the equator is hotter than the poles; and the Earth rotates out from under the atmosphere.

Hot air rises

Light from the Sun hits the equator directly but hits the poles at a glancing angle, so the equator is warmer than the poles. Warm air at the equator rises while cold air at the poles sinks.

The equator receives more direct radiation from the Sun. A ray of light from the Sun hits the ground at an angle near the poles so it’s spread out more. More radiation at the equator means the ground (or ocean) is warmer, so it warms up the air, which rises.

The warm air can’t rise forever, gravity puts a stop to that. If we did not have gravity the atmosphere would float off into space (and the universe would be a fundamentally different place). Instead, when the air reaches the upper atmosphere at the equator it diverges, heading either north or south toward the poles.

From all around the hemisphere the air converges on the poles. The air is cooling as it moves away from the equator, and when it gets to the pole it sinks to ground level and then makes the journey back to the equator. It’s a cycle, aka a circulation cell.

Hot air rising near the equator and sinking near the poles creates a cycle, a circulation cell, in each hemisphere. In this picture, the winds at ground level (dashed blue lines) would always be blowing from the poles toward the equator. This is what the world might be like without the coriolis effect.

Standing on the ground, the wind would always be blowing towards the equator from the poles. If you were in the northern hemisphere, in say Memphis, you would always be getting northerly winds.

The ITCZ and the Polar High

At the equator the rising air also takes with it water vapor that was evaporated from the oceans or from the land (evaporation and transpiration, which are together called evapotranspiration). The warm air cools as it moves up in the atmosphere and the water vapor forms clouds.

You get a lot of clouds and rainfall anywhere there is a lot of rising air.

Because air is coming together, converging, from north and south at the equator, and the equator is in the middle of the tropics, the zone where you get all this rising air is called the Inter-Tropical Convergence Zone or ITCZ for short (that’s an acronym by the way). The ITCZ is pretty easy to identify from space.

The line of clouds near the equator shows where air is converging at ground level and rising to create clouds. It's called the ITCZ. (Image via NOAA, which also hosts real-time images of the Tropical Atlantic that show the ITCZ very well).

All the rain from the ITCZ, and the warmth of the equator means that when you go looking for tropical rain forests, like the Amazon and the Congo, you’ll find them near the equator.

Locations of rainforests (more or less). Notice that in addition to the Congo and Amazon, Indonesia is pretty well forested too. All because of the ITCZ.

Now at the pole, the air is sinking downward from the upper atmosphere. Sinking air tends to be very dry, and places with sinking air also tend to be dry (it’s not a coincidence). So although the poles are covered with ice, they actually tend to get very little snowfall. What little snow they get tends to accumulate over tens, hundreds and thousands of years but the poles are deserts, arctic deserts, but deserts all the same.

The region of sinking air near the poles is called the polar high because of the high pressure generated by all that descending air.

We’ll complicate the picture of atmospheric circulation now, but the ITCZ and the polar high don’t change.

The Earth Rotates

The complication is the coriolis effect. You see, as the Earth rotates it kind-of drags the atmosphere with it. After all, the atmosphere isn’t nailed down. It’s got it’s own motion and intertia, and doesn’t necessarily want to rotate with the Earth.

Deflection of the wind, represented by a ball, because of the movement of the Earth beneath it. The ball here moves in a straight line but it appears to curve because the Earth is rotating out from under it. Click the image for a bigger, better version.

So a wind blowing from the North pole to the equator gets deflected to it’s right; the northerly wind becomes an easterly.

I could write an entire post about coriolis (and I will) but for now it shall suffice to say that the low-level wind from the pole gets deflected so much that it never reaches the equator. The high-level wind from the equator never reaches the pole, either. Instead of the one, single, circulation cell in each hemisphere, three develop, and you end up with the picture at the top of this post.

In this diagram, the convention is that it shows the circulation cells along the side of the globe, in profile, while the arrows within the circle of the globe show the wind directions on surface.

Note also that the winds in the region just north of the equator (where the label says “Tropical Air”, come from the northeast. These are the northeast trade winds that were vital to the transatlantic trade in the days of sailing ships. Know about them help a lot in the Triangular Trade game.

The Sub-Tropical High and the Sub-Polar Low

With three circulation cells you add the sub-tropical high, and the sub-polar low to the ITCZ and polar high as major features that affect the biomes.

Remember, rising air equals lots of rain, while descending air is dry.

So the sub-tropical high, with its descending air, makes for deserts. Since it’s in the sub-tropics these are hot deserts, the type you typically think about with sand-dunes, camels and dingos.

Sub-tropical deserts from around the world. They're located in the zones 30 degrees north and south of the equator at the sub-tropical high. Base map by Vzb83 via Wikimedia Commons.

The USGS also has a great map that names the major deserts.

Biomes

So if we now look at the map of biomes and climates from around the world we can see the pattern: tropical rainforests near the equator, deserts at 30 degrees north and south, temperate rainforests between 40 and 50 degrees latitude, and arctic deserts at the poles.

Map of biomes from around the world. The different biomes are closely related to the general atmospheric circulation model. (Image adapted from Sten Porse via Wikipedia)

Citing this post: Urbano, L., 2011. Global Atmospheric Circulation and Biomes, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Art and Science: Flow Paths

April 16, 2011

Butterfly.

I’ve been helping my wife model the fluid flow through her apparatus, and she has some really neat results from some experiments where two chemicals react and block off the regular, symmetrical flow.

The streamlines look a bit like butterfly wings to me, so I modified the image a little. The original flow paths through the circular apparatus are below. I’m not sure which image I like better.

Flow paths through a circular cell. Mineral precipitates (not shown) are blocking flow through the middle.

P.S. The other thing I learned from this little exercise is how to write Scalable Vector Graphics (svg) files (W3C has an excellent reference). With svg’s, like other vector graphics formats, no matter how big you blow them up you never loose resolution like you would do with a regular, rasterized image. Unfortunately, I still have to figure out how to include the svg files on this blog, so these png images will have to do for now.

Citing this post: Urbano, L., 2011. Art and Science: Flow Paths, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

How Microscopic Shells can tell us the History of the Earth’s Climate

April 10, 2011

Seeing the bigger picture.

Looking at the smear slides of Coon Creek Sediment Matrix got me thinking about just how important these little, microscopic shells have been for what we know about the Earth’s past climate. In fact, they provide the background knowledge that we have about the changes in climate that we’re seeing today.

Deep sea drilling vessel, JOIDES Resolution. Image via the National Science Foundation.

Back in the 1970’s the Deep Sea Drilling Project collected a lot of sediment cores from all around the world. The deeper you drill under the sea bed the older the sediments are, so micropaleontologists could look at how the organisms that lived in a certain area changed over time. Certain forams that could only live in warm oceans were found living far to the north. By combining all the information from all the sediment cores, they could construct paleo-geographic maps showing what the climate was like in the far past. It’s one of the reasons we know that the Jurassic climate was a lot warmer than today’s climate.

Then they invented mass spectrometers.

Mass specs can find the mass of individual atoms. Calcium carbonate has the chemical formula CaCO3. Water, as we should know by now, is H2O. They both have oxygen atoms, but not all oxygen atoms are equal; some are more equal. Actually, the mass of any atom is made up of the mass of the protons plus the mass of the neutrons in its nucleus. Now, by definition, any atom with eight protons is oxygen; however, while oxygen usually has eight neutrons, it sometimes has nine or even ten.

Your standard oxygen, with eight protons and eight neutrons has an atomic mass of sixteen, and is written as 16O or oxygen-16. Well, oxygen with ten neutrons is going to have a mass of eighteen (8p + 10n) and be called oxygen-18 (18O). These different versions of the same element are called isotopes.

Oxygen-18 has two more neutrons than the much more common oxygen-16. Note that both atoms have eight electrons, but their masses don't count because electrons are really small compared to the protons and neutrons which have about the same mass.

Water molecule with a molecular mass of 20.

What does this have to do with climate? Well a water molecule with two hydrogen atoms, each weighing one atomic mass unit, and one oxygen-16 atom will have a molecular mass of 18, while a water molecule with an oxygen-18 atom will have a mass of 20. When water evaporates from the oceans, the water with the lighter isotope will have an easier time going from liquid to a gas in the atmosphere.

So, during an ice-age for instance, lots of water evaporates from the oceans, falls on land as snow, and then gets trapped in the enormous glaciers that cover entire continents. Since the lighter water molecules evaporate easier from the oceans, they’re the ones that will end up falling as snow and being compressed into glacial ice. The water molecules left behind in the ocean will tend to have the heavier oxygen-18 isotopes. Since the forams use the ocean water as part of the process of creating their calcium carbonate shells, the oxygen from the water ends up in the carbonate (CO3) of the shells. Since the ocean water has extra oxygen-18s during an ice-age, then the shells will have extra oxygen-18 isotopes during an ice-age.

Ridge of ice from the continental glacier in Greenland. Glacial ice will have lighter isotopes than the oceans the water originally evaporated from.Image by Konrad Steffen from the U.S. Antarctic Survey.

Therefore, by measuring the amount of heavy oxygen-18 isotopes that are in a single shell, we can tell how large the glaciers were at the time that shell formed, and tell what the global climate was like.

Of course there are some interesting complexities to the story, but that’s the general idea of how the microscopic shells of long-dead plankton can tell us about the history of the Earth’s climate.

Citing this post: Urbano, L., 2011. How Microscopic Shells can tell us the History of the Earth’s Climate, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

We are Stardust: Supernovas and the Heavy Elements

March 31, 2011

Expanding globe of debris from the explosion of Tycho's Star. Tycho Brahe observed the star as it went supernova about 540 years ago. The red is the debris, the stardust, created by the explosion. Image from NASA.

We could have been talking about the nuclear meltdowns in Japan, but I’m not sure. Our conversations tend to wander. I remember trying to explain where the carbon atoms, that are so essential for life, came from. It’s been a while since we saw this topic, so I figured it wouldn’t hurt to go it over again. And then I found this wonderful image of the Tycho supernova from the Chandra space telescope. Supernovas are where the heaviest atoms are formed.

In the beginning … the big bang created just the smallest elements, hydrogen and helium. But even these tiny things have gravity, so they pull each other together until there’s so much stuff that the pressure at the center of the clump is enough to fuse hydrogen atoms together.

Now fusion is easy to confuse with chemical bonding that occurs around us every day. After all, the hydrogen in the atmosphere is usually in the form of H2, which is two hydrogen atoms bonding together by shared electrons.

With fusion, on the other hand, the single protons that make up the nuclei of the hydrogen atoms are pushed together to create a bigger atom, helium. I say pushed together, because it takes a lot of pressure to fuse atomic nuclei. And it also releases a lot of energy. Notice all that heat and radiation that comes from the Sun? All that energy was created by the fusion of hydrogen atoms; the smallest element, hydrogen, fuels the stars.

Fusion of two hydrogen atoms to create helium, compared the chemical bonding of hydrogen atoms to produce hydrogen gas (H2). The nutrons are left out for clarity.

The huge amounts of energy released by fusion makes fusion power one of the holy grails of nuclear energy research. If we were able to create and control self-sustaining fusion reactions, just like what happens in the Sun, we would have a source of tremendous energy. There is a lot of research in this area. Some people have figured out how to build fusion reactors in their basements, but these use a lot more energy than they produce so they’re not very useful as a power plant (Barth, 2010). The ITER reactor, currently being built in France, aims to be the first to produce more electricity than it uses.

Now back to the stars. Hydrogen atoms fuse to form helium, but it takes a lot more pressure to create larger atoms: carbon has six protons, nitrogen seven, and oxygen eight. These elements are essential for life (as we know it). The only time stellar forces are great enough to produce these are when stars explode; an exploding star is said to have gone nova. Bigger atoms, like iron (26 protons), gold (79 protons), and uranium (92 protons) need even greater forces, forces that only occur when the largest stars go supernova.

DNA. (from Wikipedia)

So if these elements are only produced in novae and supernovae, how did they get to Earth? How did they get into your DNA?

Well when stars explode, a lot of these newly formed elements are blasted off into space. It’s a sort of cosmic dust. We could even call it stardust. It’s matter, just like the hydrogen and helium from the big bang, only bigger, which means they have more mass, which means they have more gravity.

Formation of the solar system (model).

The gravity pulls the stardust together with the hydrogen and helium sill floating around in space (there’s a lot of it), to form new stars, and, now that there are the larger elements to create them, rocks, asteroids, and planets.

So, if you think about it, some stars needed to have been formed, lived their lives (which consists of fusing hydrogen atoms until they run out), and exploded to create the matter that makes up the planets in our solar system and the calcium in our bones, the sodium in our blood, and the carbon in our DNA.

Notes:

1. Lots of information about Tycho’s Star on SolStation.com.

Citing this post: Urbano, L., 2011. We are Stardust: Supernovas and the Heavy Elements, Retrieved January 22nd, 2018, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.