thehighastronomer-blog

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Planck Units

So you can get really serious about units.  Like, people can (and have) obsessed about what *exactly* do we mean by a meter, or a kilogram, or a second.  All of the units we use on a day to day basis are, by necessity, anthropocentric.  But if the defining feature (literally, a foot!), of our unit systems is based upon the scale of the world some clever apes find convenient, that introduces a significant bias into our attempts to understand the universe.

In an attempt to remove this bias, the Planck Units were introduced to represent a first-principles approach to unit systems.  It’s wrong to say they were designed to put a limit on how far our theories can describe.  Instead, the fact that our current laws stop working at the Planck Scale is an artifact of the Plank scale being built around the apparatus of our current laws, and the extremes allowed therein.

So below is my attempt to document these units, how they convert to common units, and what, if anything they represent.  As I’ll get into later, this whole process can be seen as an exercise in dimensional analysis, strictly mathematical, but I believe, and I make the case here, that they have a real meaning based around a first principles approach to units.  I have also decided to forego scientific notation when presenting these numbers, as the digits in their full glory help convey the extreme scales involved.

Planck Length

0.000 000 000 000 000 000 000 000 000 000 000 016 162 meters

If you took the smallest object for which a notion like diameter actually makes sense, an electron, and enlarged it so that one edge rested on the Earth, and the other edge touched the nearest star, Proxima Centauri, the Planck Length (Hereafter Lengthp) would only be the thickness of 5 sheets of paper.  The physical interpretation of Lengthp is a little squirrley.  One misconception is the oversimplification that it represents the atomic unit of space time, a distance below which there is no meaningful distance.  This is only partially right.  It is instead connected to the idea of information theory.  Information is necessarily quantized, as you can only reduce it to an abstraction ultimately expressible in binary, which are then divisible no further.  Lengthp (or more appropriately Areap) is the change in size of a singularity when it grows by one bit of information.  Thus, if any length smaller than Lengthp could be observed, created, or divided then information would be quantized further than binary, which it cannot.  Phew.

To measure anything Lengthp long would require an amount of energy so large in a single photon that it would collapse into a singularity, which would be unable to report anything about the object you’re trying to observe at that scale, because the singularity would “eat” the photon. (Yes, yes, yes black holes/singularities don’t properly *eat* anything, but om nom nom nom #highastronomermunchies).

Plank Time

0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 053 911 seconds

So if the smallest distance that makes sense is Lengthp, thus the shortest time that makes sense is how long it takes the fastest thing, light, to travel one Lengthp.  This time is the Planck Time (Timep).  To call it an instant seems an understatement

Imagine all of the Carbon in all of the life, plant and animal, on Earth.  If you erased each of those atoms, one every Timep, all of the Carbon atoms would be gone in a little over one one hundredth of a second.  And as much as I love science, I gotta let my math nerd shine here. That number of atoms is gigantic in practical terms, but pure math has rather efficient ways to write and describe it. For example, the Planck Time is between 1/37! And 1/38!, which is the probability of taking 38 numbered cards, shuffling them and placing them down in descending (or any) order.  It's the number of unique ways to arrange 38 objects.  It's also well within the bounds of what kinds of numbers graphing calculators can deal with without rounding.  Its less than the number of legal chess positions by a factor of a thousand.  But in scientific terms, as a measurement, it is a vanishingly short period of time.

Planck Temperature

141,680,800,000,000,000,000,000,000,000,000 Kelvin

The Big Bang Theory (no, not the goddamn show) implies that the universe originated in a singularity that cooled over time as it expanded.  Thus, the hottest meaningful temperature would occur the shortest meaningful amount of time (which is itself the time it takes the fastest thing to travel the shortest meaningful distance) after the beginning of cosmological expansion and cooling.  The temperature of the universe at this point in history is the Tempp.  Because if it were hotter than the Tempp, that would imply a time earlier than the Timep, which would imply either a speed faster than light, or a distance shorter than the Lengthp, which would imply that binary information isn’t quantized.

Have I mentioned that Planck was a fucking genius?  Oh yeah.  I have.

So how large is this temperature?  I love me some metric, but the one part of the Imperial system I find more relatable and easy to use is the Fahrenheit scale, so lets turn

141,680,800,000,000,000,000,000,000,000,000 Kelvin

into

255,000,000,000,000,000,000,000,000,000,000 Degrees Fahrenheit

Okay, now that we’re basking in the glow of the only redeeming unit in the entire Imperial system (okay, maybe besides the Hogshead), lets try to get a handle on how ridiculously hot this is.

The hottest thing mankind has ever made is the quark-gluon plasma in the proton beam at The Large Hadron Collider, which clocks in at

1,000,000,000,000 Degrees Fahrenheit

This is so far below the Planck Temperature that if you squared the LHC temp you’d still be off by a factor of 100,000,000.

If you had a thermometer, spanning from -40F (or Celsius ha!) to 120F, that spans 10 cm (I like to think I am the first person to deliberately mix Fahrenheit and meters) and tried to extend the scale on the thermometer to read Tempp,  The thermometer would have to extend by a distance greater than 180 times the diameter of the observable universe in order to register Tempp.  It would require so much Mercury (cause we measure it old school!) that you could make 967 planets the size of Mercury out of Mercury.

Planck Energy

1,956,100,000 Joules

Earlier we touched on the limits of the energy of a photon would need to measure a Lengthp, and we call that energy Energyp.  Measured in electron volts (I explained that unit here) it tips the scales at 1,220,800,000,000,000,000 eV, a number of eV that would make an experimental particle physicist dive for the fainting couches.  And indeed, a single photon with that much energy would collapse into a singularity, which would gobble itself up, and thus no photon can exceed that energy.  But in macroscopic units, it’s about the energy of gasoline in a car’s gas tank or a bit more energy than is contained in a single lightning strike.  Michael Phelps consumed this much caloric energy in 47 days training for the Olympics.  (Editorial note: How the hell does Wolfram Alpha know that?)

Planck Momentum

6.52485 kg m / s

Momentum, in macroscopic terms, is mass times velocity and represents the propensity (or lack thereof) of something to move when a force is exerted.  Since we are talking about a world of very tiny (or very large) things, this definition of momentum doesn’t help much.  Instead, we can look to photons, which both have momentum and travel at the most extreme allowable speed, the speed of light.  So thus the Planck Momentum (Momentump) is the momentum of one photon of energy equivalent to the Massp (which itself is the largest energy a photon can attain before going singularity).  

And that single photon has a lot of momentum (the most momentum a photon can have).  But on a human scale, its not a heck of a lot.  It’s about the momentum imparted on a wall if you walk into it at a decent pace.  A baseball thrown at 100mph carries about this much momentum.  

Planck Charge

0.000 000 000 000 000 001 875 546 Coulombs

Take the smallest sphere that makes sense, Lengthp, and pack the most energy into that be fit in without becoming a singularity, the Energyp, and measure the electric potential energy, and it will be one Chargep

This turns out to be about the electric charge of 12 electrons, or equivalently the positive charge of completely ionized Magnesium.  Neat.  

Planck Force

121,029,500,000,000,000,000,000,000,000,000,000,000,000,000 Newtons

Planck Force, in addition to being the name of Max Planck’s team of physicist superheros, represents the greatest amount of force possible, given the units we have established so far.  This is gonna be the last unit I cover, for now, and it does a great job of explaining why Planck units are so cool.

Planck units can be an exercise in dimensional analysis, whereby you move from one definition to another, seeking to cancel units until you get what you wanted, with no higher physics principles guiding your hand.  And while you can certainly start like that, I have made the case that these units have an implicit meaning behind them.  When researching Planck Force, I started with the dimensional analysis, namely that force is the change in momentum, so

Okay, so lets try to put some meaning on this.  Where in nature do we describe things with force?  Gravity.  So let’s take the largest point-masses that make sense, Planck Masses, and place them one Planck Distance apart.

Huh.  Neat.  I was excited to get the same result, but I thought that it would be odd if only gravity worked, so I dusted off Coulomb’s law for electrostatic repulsion.

I know it would require more explanation if this *weren’t* the case, but I still find it kind of spooky that, as long as you are talking in Planck units, all interpretations of force are equivalent.  Let me explain what this kind of perspective shift does to understanding.  A question one might ask before Planck Units is “why is electromagnetism so much stronger than gravity?”  This is an excellent question that has no obvious answer in the structure of the laws themselves.  But when speaking in Planck terms, you see that the difference in magnitude between the electromagnetic and gravitational forces is exactly proportional to the difference in magnitude between Massp and the Chargep, the meaning of both of which are explainable from first principles.  The physics of gravity and electromagnetism aren’t explained by Planck units, but they allow us to see deeper connective tissue between ideas that we might not otherwise have seen using anthropocentric units.

Alright, there’s plenty more Planck units that I wanted to get to, but I will leave that to another post.

[thehighastronomer-blog]

CNO Cycle

Today I want to talk about fusion.  Stellar nucleosynthesis, to be specific.  There are tons of ways that elements fuse in the cores of stars, but the two most common both turn Hydrogen (free protons, if you like) into Helium and energy.  There’s the proton-proton chain, a pathway I happen to really like.  Seriously.  I have it tattooed on my ankle.  Here.

That process represents the primary way stars create energy up until around 1.3 solar masses.  For example, approximately 92% of our sun’s energy comes from the proton-proton chain.  It initiates at around 4 million kelvin.  Crank the heat up to 15 million kelvin and you get the CNO cycle I am going to explain in a bit.   What I like about the CNO cycle is that it is catalytic:  that is, there is a ring of various nuclear reactions that involve heavier elements but does not consume or produce those elements.

First some basic concepts we need to understand:

Mass-energy equivalence:  Einstein gave us possibly the most famous equation ever:  E=MC^2.  It establishes that mass and energy are directly proportional, and can, under various conditions, be converted interchangeably.  It might be helpful to think of energy and mass, from this point forward, as different units of the same thing.

The Electron Volt:  Instead of using kilograms or joules, it is convenient in nuclear physics to use the megaelectron-volt, or MeV.  Technically it is the amount of energy required to move one million electrons through an electric field of 1 volt.  For a sense of scale, the whole proton-proton chain above has an output of 26.73 MeV.  A single nucleon (proton or neutron) has a mass/energy of around 931.5 MeV.  

Mass Defect and Fusion:  Get two atoms together and they fuse.  The mass before and the mass after the reaction will be different, and this amount is called the mass defect.  This mass/energy will manifest as kinetic/heat energy.

Positron Decay:  A nucleus is a constant battle between the attractive strong force and the repulsive electromagnetic force.  The protons really, really, really want to push away from each other.  But the more nucleons, protons and neutrons, you have the more cohesive the nucleus is.  If you have too many protons unbalanced by a number of neutrons, something has to give.  And sometimes what happens is the proton will transmute into a neutron.  Because charge and spin must be conserved, the positive charge will fly off as a positron (anti-electron) and the spin value will fly off as an electron neutrino.  

Notations for the equations:  I have written the elements somewhat unconventionally with the proton and neutron counts on the right side, as opposed to the total nucleon count on the upper left.  I find that this helps to show how the math “adds up.”  Black text is a catalyst, Blue is a reactant, light red is a product, dark red is kinetic energy.

And here we go!

So there are six reactions that all produce energy, and it doesn’t matter which one we start on, it is a closed loop.

Take C-12 and fuse it with Hydrogen, and you get N-13, a gamma ray, and energy.

That N-13, see how it has more protons than neutrons?  That’s unstable and with a given half-life it will decay via positron emission into C-13 thus.

Now that the neutrons are more abundant in the nucleus, it will happily fuse with another proton, giving N-14 a gamma ray, and energy.

The N-14 is still open to fusion, so it takes another proton in.

And now we are in the same problem as before.  Too many protons.  So the O-13 undergoes positron emission to N-15.

This N-15 will now fuse with another proton, and the O-16 will then fracture into C-12 and He-4.  And it begins again.

So 4 protons walk in, and a helium atom, some gamma rays, some positrons, neutrinos, and 26.7 MeV pop out.  

[thehighastronomer-blog]

Pigeon poop (and one of the most important pieces of data in cosmology.)

Before I get into this post I need to make a point.  Close your eyes.  Picture every image of the big bang you’ve ever seen in modern media.  I am fairly certain you just saw a bright point of light, then everything flying out of it, into the void.

This is wrong. Terribly, horribly wrong and it leads to a bunch of questions/misconceptions.

What did the big bang expand into?  Nothing. The big bang created the universe, which was itself, which expanded.  There is no way to be “outside” looking in.

Where did the big bang start, like can we point to a place in the sky where everything came from? No. The big bang happened everywhere, all at once.  Again, space ITSELF was created and expanded.  There’s even a clear cut observational explanation for this: we can fairly accurately, using spectral line red shift, figure out the radial velocity (how fast towards or away from us something is flying).  If there WERE a center of the universe, things on the far side would be flying away from us faster than things near us.  But that’s not what the data shows.  Velocity is proportional to distance in all directions. This means either we are the center of the universe and completely static, or that the space between everything on large scales is expanding and everything, including us, is flying away from everything else.

The last part of that image that is wrong is that there was nothing, then there was *stuff* almost immediately afterwards, flying apart.  And this is somewhat reinforced by media on theoretical physics, because really short timescales (10^-36 to 10^-43 seconds) our current understanding of physics breaks down and the “Buy my book to read about my grand theory of everything” authors come out of the woodwork (Looking at you Kaku, Greene, Smolin, et al).  Its an enigma, but the science is largely unsettled.

I want to share a part of the science that IS fairly settled, that goes against the image you have from above, and has an impact on nearly every part of modern cosmology. And it started with pigeon poop.

So, to set the stage, it is the middle of the 1960s.  Big bang cosmology is relatively well accepted, but vital information about the theory (including the value of the expansion rate of the universe, the Hubble Constant) is very much in the air.  Microwave astronomy was nascent, and one of the early tools used in the field was something called a Hogg Horn.  One of these receivers was installed by Bell Labs in Holmdel, New Jersey to help NASA detect passive radio waves from communications satellites.  Our protagonists, Arno Penzias and Robert Wilson, were working at Bell with this instrument, and their challenge was to try to pick out this remote, reflected microwave signal from the background. They managed to subtract out every ambient source they could find, from RADAR and radio broadcasts to the thermal radiation of the horn itself (they tried cooling it with liquid hydrogen to 5K). And no matter what they did there was this signal (similar to TV static) that could not be stamped out.

New hypothesis: Pigeons were roosting near the horn, so their droppings must be causing the signal! So they grabbed some mops, and spent a weekend scrubbing it clean.  When the anomaly persisted, they were convinced that they had eliminated all possible sources of error, and therefore the evidence had to be real.  It had to be extraterrestrial in origin, because they had eliminated all local sources.  It had to be extragalactic because it could be detected at all times, from all directions.  Simultaneously, a team of astrophysicists around the corner, in Princeton, were looking for evidence just like this pervasive microwave signal.

At this point I will swing into the modern narrative of that epoch in the universe’s development, as it largely agrees with the work of those researchers.  They got in touch with Penzias and Wilson, and the data matched their predictions to a high degree.  Its notable that both Penzias and Wilson won the Nobel in 1978 for this work, not the researchers (Dicke et al).

In the beginning, the four forces were unified.  Temperatures, pressures, and densities were so high that the format of the equations we use to describe them become symmetrical.  There is no practical difference between gravity and electromagnetism.  Strong and weak nuclear forces.  As the universe expanded in that first 10^-43 seconds enough for gravity to make an early break for the exit, becoming an independent force.  The three forces of the quantum standard model remained unified until later (10^-32 seconds when strong force splits off).  Inflation happens, and inflation is another post in the writing.  By about 10^-12 seconds the weak and EM forces separate.  The energy density is still stupendously high, but this is about as far as the LHC can probe with it’s collisions.  By about 1 second the quark/gluon plasma has cooled enough that things we recognize as hadrons (protons and neutrons) can exist individually. The universe was a plasma, with electrons refusing to get cozy with the hadrons, and critically, photons could not travel far without interacting with something.

The next 300,000ish years were spent in much the same highly ionized state.  The universe, everywhere, was opaque.  If you could somehow travel back in time and park your lawn chair and watch the festivities, besides being cooked to a cinder, you wouldn’t be able to see further than the material immediately in front of you.  The photons were going through Brownian Motion (or as computer scientists call it, the drunkard’s walk) where they travel a short distance, hit something, are absorbed and re-emitted in a random direction.  This happens in the middle of stars in the modern day, and it is why a photon, from creation to emission from the star, can take tens of millions of years.

So at 300,000 years the universe was getting bigger and cooler (the concepts of the ideal gas law are appropriate here), and the electrons decided it was time to settle into a nice hydrogen or helium atom in the suburbs.  The photons were allowed to travel in free space. And they did so, in every direction, all at once, from every point in space.  And this critically hinges on the understanding from above that the big bang was a phenomenon EVERYWHERE.

So what’s the connection to our pigeon-poop horn detecting microwaves?  Doppler shift.  Remember, Hubble established a linear relationship between the distance between distance from us and radial velocity.  Far things are going away from us faster than near things.  And the primary way to measure radial velocity, as I mentioned before, is with Doppler red shift.  If we have light that was emitted very far away (and a long time ago), in the intervening time the space it propagates through will have expanded, and this stretches the light from higher wavelength infra red rays to microwaves. Just at the frequency that Penzias and Wilson were picking up in their horn.

So how best to picture the CMB:  It is an opaque curtain around the entire sky, in every direction, beyond which we can never detect any light (because it was still coupled in the plasma).  As the universe ages, we will still see it, but we will see the particular photons emitted from further away than the points that emitted the photons we see now.

What can it tell us? Seemingly everything.

Hubble was right.  It is a very, very strong line of evidence for the big bang hypothesis.  Steady state universe theories would struggle to explain the sudden release of photons, in every direction, of fairly uniform energy.

Planck was right.   At the turn of the 1900’s people could not seem to match the predictions of the relationship between light frequency and intensity and increasingly precise measurements.  Their error, that Planck started a revolution by overturning, was that light must be emitted in discreet packets, called quanta.  His black body equation matched quite nicely to the data of the day.  At this point, Planck doesn’t need the CMB to chalk up a win, but it IS notable that the frequency vs intensity graph for the CMB matches his theory so precisely, the error bars look more like points than a range of error.  That graph is a beautiful graph.

We can find the age of the universe.  We know that electrons and protons decouple at a minimum temperature of 3000K.  The black body curve of the CMB today is just a copy of the 3000K curve, slid down the EM chart with redshift.  Since Hubble was right about the universe itself expanding, we can infer the distance traveled by the CMB to be around 13.8 billion light years.  Since it was light making that journey, the release happened about 13.8 billion years ago.

We can find our relative motion.  You are on a planet that’s spinning on its axis, orbiting around the sun, moving within the disk of the galaxy AND rotating around the center of the galaxy, and that galaxy is moving within our subsection of the galaxy cluster.  So how do we keep ourselves straight with regard to motion in the universe? The CMB, naturally.  Remember, it is a nearly uniform spectrum covering the entire sky in every direction.  So if we are heading more towards, say, the constellation Leo (which we are, by the way) then the CMB on that side will have a slightly lower redshift than the opposite side of the sky.  You just need to look for two poles in the CMB data where redshift rises to a peak on one side and falls to a low on the other.  

There are some things that it can’t answer, but can inform.  For example, why is there matter? This seems like an obvious question (and again, in honesty deserves its own post) but the laws of physics show absolutely no bias towards matter vs antimatter, yet observationally we can only see matter.  The fact that matter exists is a problem.  And the CMB can put constraints on the type of matter that existed in the early universe, from which we can infer what that opaque soup was like.

The CMB can tell us how the largest scale structures came to be, the superclusters, filaments, and voids. The areas of the CMB that are slightly warmer than average generally correspond to where the “stuff” is, and the cool spots correspond the large voids.  So the structure of the universe 13 billion years later is evidenced by tiny fluctuations in that image.

So that’s what the CMB is. The universe’s first picture, from when it was a baby.  It shows us where we came from, and what our potential future is.

[thehighastronomer-blog]

How I just spent my last half hour.

[thehighastronomer-blog]

High Energy Cosmic Rays

Last time, when I talked about momentum, I opened like 20 tabs from the Oh My God Particle about cool stuff I thought needed sharing.

What is a cosmic ray?

Basically anything that isnt light that is flying fast enough at us to not originate locally.  Hence cosmic.  The highest energy of them have been observed since the sixties, but serious questions were asked about their origin and mechanics after the Oh My God! Particle in 1991.

What kind of matter are they?

They’re typically charged, baryonic matter.  But it’s really, really important to take Einstein’s matter and energy equivalence to heart here.  As you’ll see in the next section, the fact that it was once a proton or iron nucleus doesn’t matter because it is now a beast almost entirely made of... energy... and any attempt to interact with it to determine it’s progenitor particle would end... poorly.

How energetic?

It is often said that the OMG particle has the momentum of a 100mph baseball, and that is useful in conveying one aspect of how energetic this particle was.  But because we don’t have a sense of subatomic scales, we can’t fully extend the analogy.  

Gamma Gamma Hey!

Special relativity has a lot of interesting and counter-intuitive predictions for behavior of fast things.  From the outside, their length of the traveler decreases.  Time, as experienced by the traveler, slows.  The mass of the traveler increases.  These are called Lorentz Transformations, and their mathematics, once derived are relatively simple.  The important equation common to all of these transforms is the definition of the Lorentz Factor (lowercase gamma).

( \gamma = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} )

Where v is the velocity of the traveler and c is the speed of light in a vacuum.  When you have any kind of dilation, it will be of this magnitude.  Essentially, gamma tells you how relativistic shit is gonna be.  The fastest thing we have hurled into space, Deep Horizons, has a factor of 1.000 000 001 469.  At fifty percent the speed of light, you’re are 1.155 times shorter, slower, and more massive to an outside observer.  The largest scientific machine (and probably any kind of machine), and it can get individual protons up to 

99.9999991% the speed of light, and that corresponds to a Lorentz Factor of 7,460.

So.  That OMG particle?  It had a Lorentz factor of 319,438,000,000.  So whatever it was, the overwhelming majority of it was energy, in a ratio of 

319,438,000,000 to 1.  

Where do they come from?

holds flashlight in front of face

NOBODY KNOWS!!

Of course I am just kidding.  Just like anything with such a relatively short observational history, Astronomers have a pretty good idea on what mechanisms can propel matter like this.  (If you’re thinking of rail guns you’re not far off).  The major differences amongst researchers is on what physical process is generating the conditions necessary.   I will primarily focus on the neutron star theory, but anywhere you have obscenely high magnetic fields and a source of charged material, you will get cosmic rays.

I just wrote a paragraph and a half on neutron stars, so thats probably going to be the next post.  When a big star dies a fiery death there is sometimes a neutron star that remains.  Dense beyond description and on the order of tens of kilometers in diameter.  Large stars spin, but not a lot (our sun rotates once every 26 days or so).  But when something spinning shrinks, the angular momentum is conserved and, like an ice skater pulling in their arms, they rotation speed increases.  So our neutron star will be spinning 10,000 times per second, or more.  Magnetohydrodynamics is several times more complicated than the name implies.  “Spinny thing makes magnet” should be sufficient for our purposes.  A magnet in the order of 10^8-10^11 tesla (on Earth the magnetic field is 10^-5 T, give or take).  Any iron that escaped in the supernova cloud from the neutron star’s explosion gets swept up by the field and ejected out of the poles to one day slam into an atmosphere like ours.

A competing model has them ejected, as part of the jets coming out of active galactic nuclei, with a similar magnetic field rail-gun type event.

Finally, there’s a very interesting idea that dark matter, when it enters the ergosphere of a supermassive black hole, will fragment and via a Penrose process (I’ll do a post on that too) ejects protons at phenomenally high energy.  This model is very much in the minority, but its neat nonetheless.

Still a problem remains

The cosmic rays, by their very nature, should not be able to travel vast (Beyond our local galaxy cluster) distances without being slowed by the cosmic microwave background.  There is an upper limit on kinetic energy, called the GZK-limit, above which it ceases to move freely through the universe without losing some momentum to... for lack of a better term, drag.  This would seem to rule out extremely distant sources from the early universe.

[thehighastronomer-blog]

The momentum of a butterflies wings.

Heres a journey through my ADD mind.

Some telescopes, like the Fermi space telescope, see faint objects by collecting a photon here or there, building up an image over long exposures.  Fermi, for example, can make a detection of between 5 and 10 photons from a single source. So I just pictured telescopes, floating out there in space, giant shiny mitts gently catching tiny little packets of light.  And my brain started searching for an analogy of what something with that *little* momentum would be like, or conversely, what something from our macroscopic world would consider to be low momentum.  How would it translate into the world of a photon?

So I imagined the lightest thing I could generating a push, the flapping of a butterflies wings.  I am not a true Fermi solver, I admit, I like to google some tidbits, but I make assumptions when it gets too complicated.  So the average monarch butterfly has an average wingspan of 10cm and, on average weighs 0.1g.  Assuming that a single half wing can flap only at most its full wingspan, each flap swings a wing (simplifying assmption: 1/5 of the body resides in the wing) weighing .0001kg 0.1m up and down on average 8 times a second, which means the one flap happens in 1/16 = 0.0625s.  The flap has a velocity of (0.1m)/(0.0625s) = 1.6m/s and a momentum of p = (1.6m/s)(0.0001kg) = 1.6x10^-4 kgm/s.

That doesn’t seem like a lot of momentum.  Its just a wing.  But what if it were a photon.  The product of a photons wavelength and momentum is the Plank constant, so I have L*1.6x10^-4 = 6.626x10^-34, and thus L = 4.14x10^-30 per meter.  On the order of 10^23 eV.  

Is that a powerful photon or a weak one?

Well, its more gamma ray than any gamma rays that nature produces every day.  For example the most energetic thing astronomers have found, ever, was a detection of a cosmic ray (since dubbed the Oh My God! particle).  This was a proton that screamed into our atmosphere (pretty sure it was screaming.  (sub parenthesis: just like muon showers, such a screaming particle would be living in a hell world, perceiving that time has essentially stopped.  I feel like Rick and Morty need to do an episode on this)) and the proton had the equivalent momentum as a baseball thrown at 100 mph, about 10^20 eV.

*record scratch*

Isn’t our butterfly flap photon 10^23 eV?  Yes.  But, unlike our photon, the Oh My God! particle (baseball) has mass.  The photon born of a butterfly flap has its entire momentum wrapped up in the energy of the photon itself.