Over on the Category Theory Community Server, John van de Wetering asked me how many times a typical solar neutrino oscillates on its flight from the Sun to the Earth. I didn’t know, and I thought it would be fun to estimate this.

So let’s do it! Let’s do a rough calculation, and worry about details later. For those too lazy to even jump to the end, here are the results:

• A neutrino takes about 500 seconds to travel from the Sun to the Earth.

• Because a typical solar neutrino moving is moving close to the speed of light, time dilation affects it dramatically, and the time of travel from the Sun to the Earth experienced by the neutrino is much less: very roughly, 1/6 of a millisecond.

• There are different kinds of oscillation. If we keep track only of its *slower* oscillations, a typical solar neutrino oscillates roughly once for each 1250 meters of its flight through space.

• As it travels from Sun to Earth, this typical neutrino does about 120 million oscillations.

Let’s start at the beginning.

The Sun emits a lot of electron neutrinos. Most are produced from a reaction where two protons collide and one turns into a neutron, emitting a positron and an electron neutrino. The proton and neutron then stick together forming a ‘deuteron’, but let’s not worry about that.

More importantly, the energy of the neutrinos produced from these so-called pp reaction is at most 400 keV. That means 400,000 eV, where an eV or ‘electron volt’ is the energy an electron picks up as it falls through a potential of one volt. If you look at this chart:

you’ll see *most* solar neutrinos have a somewhat lower energy. Let’s say 300 keV.

By comparison to the rest mass of a neutrino, this is huge. Nobody knows neutrino masses very accurately—as we’ll see, people know more about *differences of squares* of the three neutrino masses. But a very rough estimate for the rest mass of the lightest neutrino might be 0.1 eV/c^{2}. Here like particle physicists I’m measuring mass in units of energy divided by the speed of light squared. An eV, or electron volt, is the change in energy of an electron as it undergoes a one-volt change in potential.

This mass could be way off, say by a factor of 10 or more. But it’s good enough to show this: solar neutrinos are moving *very* close to the speed of light!

Remember, the energy of a moving particle, divided by its ‘mass energy’, the energy due to its mass, is

Our solar neutrino, using our very rough guess about its mass, has

It has an energy 3 million times its rest energy! That gives

or

or using a Taylor series trick

or if I didn’t push the wrong button on my calculator

This is ridiculously close to the speed of light.

It’s more useful to remember that our neutrino’s energy is roughly 3 million times what it would be at rest. And relativity says that due to time dilation, the passage of time experienced by this neutrino is slowed down by the same factor!

It takes 500 seconds for light to go from the Sun to the Earth. Our neutrino will take a tiny bit longer—the difference is not worth worrying about. But because of time dilation, the travel time ‘experienced by the neutrino’ will be

This figure is very rough, due to how poorly we know the neutrino’s mass, but it’s about a 1/6 of a millisecond.

Now let’s think about how the neutrino oscillates.

To keep things simple, let’s assume our electron neutrino gets out of the Sun without anything happening to it. What happens next?

There are three flavors of neutrino—and as it shoots through space, what started as an electron neutrino will ‘oscillate’ between all three flavors, like this:

Here black means electron neutrino, blue means muon neutrino and red means tau neutrino.

You’ll notice that both high-frequency and low-frequency oscillations are going on. This is because the three flavors of neutrino are nontrivial linear combinations of three ‘mass eigenstates’, each of which has a phase that oscillates at a different rate. Two of the mass eigenstates are very close in mass, and this small mass difference causes a small energy difference which causes the slower oscillation. The third mass eigenstate is farther away from the other two, so we also get a more rapid oscillation. As you can see, this is especially noticeable in how the neutrino flickers back and forth between being a muon and a tau neutrino.

But all this is a bit complicated, so let’s just focus on the slower oscillations. How many of *those* oscillations happen as our friend the neutrino wings its way from Sun to Earth?

To estimate this, let’s pretend there are only the two mass eigenstates that are very close in mass, and ignore the third. The two masses and are not actually known very accurately. What we know is

The reason we know these *differences in squares of mass* is actually by doing measurements of neutrino oscillations: these differences actually determine the frequency of the neutrino oscillations! Let’s see why.

If something has energy quantum mechanics says its phase will oscillate over time like this:

where is Planck’s constant and the minus sign is just an unfortunate convention. But all we detect is the absolute value of this, which is just 1: that doesn’t change. So to actually *see* oscillations we should think about something that can have two different energies and . Then we need to think about things like

or other linear combinations of these two functions. But their difference illustrates the point nicely: we have

and the absolute value of *this* changes with time! It’s

and the takeaway message here is that it oscillates at a frequency depending on the energy difference,

So, if we have two kinds of neutrino, it’s the *energy difference* of the two mass eigenstates that determines how fast a superposition of these two will oscillate. It’s very similar to how when two piano strings are oscillating at almost but not quite the same frequency, you’ll hear ‘beats’ as they go in and out of phase—and the frequency of these beats depends on the *difference* of their piano strings’ frequencies.

So energy differences are what we care about. But how is energy related to mass? In units where the speed of light is 1, special relativity tells us this:

where is mass and is momentum. One of the mind-blowing moments of my early physics education was watching someone do a Taylor expansion for *low* momenta and getting this:

It looks more impressive if you don’t set the speed of light equal to 1:

So we see that at low momenta the energy is Einstein’s famous plus the kinetic energy famous from classical mechanics before relativity!

But all this is useless for our solar neutrino, which is ‘ultra-relativistic’: it’s moving almost at the speed of light! Now is much bigger than not smaller, in units where So we should do a different Taylor expansion, where we treat as the small perturbation:

Cute, eh? Everything is backwards from what I learned in school: we just switch and

This shows us that if we have a neutrino with some large momentum and it’s a linear combination of two different mass eigenstates with masses and it’ll be a blend of two energies:

and

So, the energy difference is

and this is what determines the rate at which the neutrino oscillates.

If we stop working in units where we get

So, the frequency of oscillations is

This frequency says how the relative phase rotates around in radians per second. But it’s more useful to think about radians per distance traveled; let’s call that Since our neutrino is moving at almost the speed of light, to get this we just divide by

And because the neutrino is ultra-relativistic, its momentum almost obeys Here could be either or they’re so close the difference doesn’t matter here. So we get

This is why people doing experiments with neutrino oscillations measure *differences of squares* of neutrino masses, not neutrino masses.

For our solar neutrino we’re assuming

and remember

Plugging these in we get

Now it gets annoying, and this is where I usually make mistakes. We use

and get

It’s funny how multiplying and dividing all these large and tiny numbers leaves us with something at the human scale!

But actually my computation was sloppy at one point. I warned you! I think it’s actually off by a factor of two. Wikipedia says right answer is

and this gives

So, the neutrino oscillates at a rate of about one radian every 3200 meters! And to get the *wavelength* of the oscillation we need to multiply by So our solar neutrino makes a complete oscillation about once every 20 kilometers!

And the distance from the Earth to the Sun is 150 million kilometers. So, our neutrino oscillates about 7.5 million times on its trip here.

You should take all this with a grain of salt since I easily could have made some mistakes. If you find errors please let me know! I leave you with a puzzle:

**Puzzle.** Where does the missing factor of 2 come from?

I don’t think you need to know fancy physics to solve this. I think the mistake is visible in my calculations.