# What's smaller: a neutrino, or a string from string theory [closed]

I've recently read an article that stated "If an atom were as big as the solar system, a neutrino would be the size of a golf ball". I watch the science channel, and on (I believe) the show How the Universe Works they mentioned that if a hydrogen atom was the size of the solar system a string would be the size of a tree (approx 50 ft tall). Could someone explain this to me, please?

## closed as unclear what you're asking by CuriousOne, Diracology, ACuriousMind♦, Gert, user36790 Jul 25 '16 at 4:46

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• You need to stop reading pop-science. This is total nonsense. Somebody probably took some cross sections and interpreted them in a mechanical ball collision model. That's not how quantum field theory works. – CuriousOne Jul 24 '16 at 20:07
• Part of the deal with string theory is that the known particles are assumed to be states of the strings. So the answer could be "the neutrino is a string". But as CuriousOne says what you read has a lot of assumed context behind it which means it doesn't mean what you think it means (and may very well not mean what the author of the article thought it meant). – dmckee Jul 24 '16 at 20:15

## 2 Answers

I will try to illustrate the sizes for you in my answer. However, with the examples you gave from the article and TV show, that is nonsense...they may be right with the size comparisons, but that is not an illustrative point. You kind of have to take that type of pop-science with a (large) grain of salt.

First, let's familiarize ourselves with units. We'll go with metric, because it isn't such a pain to convert. The diagram below shows units down to the Planck Scale with a few objects. Okay, now that we've got some idea, let's expand this chart out a bit. A honeybee is about 2 centimeters long. And it is about one hundredth the length of the average adult male. Now, this really isn't beyond our reckoning. It isn't very exotic for us; we've seen honeybees. Now, then, let's think about a dust mite. (Gross, I know; bear with me.) A dust mite is 50 times smaller than a honeybee. And, again, bear with me, but here's a picture of a dust mite. And these guys are less than $\frac{1}{2}$ of a millimeter in length, or (if you want to go in micrometers) it's about 400 micrometers in length. Now, let's go smaller. The diameter of a human hair (on average) is 100 micrometers. And here's a picture of a zoomed in human hair. And this, obviously, is about $\frac{1}{4}$ the length of a dust mite. Then, at about 6 to 8 micrometers in diameter is the human cell. This is so, so tiny. And a diagram of a cell is below. While we certainly can't see a cell, we can sort of see spider silk, which is about 3 to 8 micrometers in diameter. This is the limit of human perception. On the same sort of scale as the cell, mostly smaller (1 to 10 micrometers) are bacteria. Tons of bacteria. And I said this was the limit of human perception, but really, spider silk is the exception, not the rule here. And, to continue with the picture theme, below is a diagram of a bacteria cell. Let's get even smaller. Let's go to the scale of nanometers. Just to point out here, a nanometer is $\frac{1}{1000}$ of a micrometer. This is the scale of a virus, which can be anywhere from 20 to 400 nanometers. An "artist's rendering" of a virus is below. Okay, now to DNA. The double helix of a DNA molecule is about 2 nanometers wide. Just to point out again, how small we are, you can put 500 million DNA molecules side by side (remember, in width) to get a meter. This is tiny. And a diagram of DNA is shown below. Okay, now to introduce angstroms. An angstrom is not really a part of the conventional metric system, but 0.1 nanometers is an angstrom. Now, 2 angstroms is equal to the width of a water molecule. A diagram of a water molecule is shown below. Okay, now the diameter of a hydrogen atom is roughly one angstrom. This is the scale of the stuff we are made up of...and we aren't really near neutrinos, and we're definitely not near strings. I think this is helpful though, to get a sense of scale. Okay, here's a diagram of a hydrogen atom. Okay, now the nucleus of a hydrogen atom has the radius of $\frac{1}{10000}$ of an angstrom. This is so tiny it's ridiculous, really. And an electron has a radius of $3*10^{-5}$ angstroms. This is nuts. Now to get to the two things you give as examples. I couldn't really find a number to even approximate the diameter or radius of a neutrino, but the mass of a neutrino is $0.320 ± 0.081 eV/c^2$. This is so, so tiny, I can't even describe how small this is. To give you an idea, though, it is much smaller than an electron. And electrons are really tiny.

A string is on the order of the Planck Scale. Now would be a good time to reference that very first chart again. See where the size of an atom is? Now, scroll down a tad. See where the Planck Scale is? A string should be about $10^{-33}$ centimeters in length, or about a millionth of a billionth of a billionth of a billionth of a centimeter. And at this point, I can't even include a diagram of this stuff, because we've got no idea what this looks like. None, zilch, nada.

So to (finally) answer your question, a string from string theory is much, much much, much, much, much, much smaller than a neutrino.

Now, this may be mind-blowing (correction: it should be), but I hope this helps you get an idea of scale without resorting to nonsensical comparisons.

• You are fighting a tidal wave of Pop Sci but it is worth it. When the Higgs Boson was discovered, I was asked by lots of "ordinary folks" to tell them more about it. There is a huge interest in science projects that get media attention, it's just a shame how it's presented. Pictures of dust mites, that's okay, but please, never, ever pictures of spiders...........Lovely answer. – user108787 Jul 24 '16 at 23:12
• @count_to_10, I agree with your sentiments about spiders. I specifically picked a none-gross picture because I'm kind of bug adverse. =) Thanks! – heather Jul 24 '16 at 23:13
• This is simply wrong. Neutrinos have no known size. That is all that you can say about their extent. And the alleged "electron radius" here is a useful constant that is in no way related to the size of electrons which are known to show no structure down to less than $10^{-18}\,\mathrm{m}$. – dmckee Jul 25 '16 at 0:39

The mass of the neutrinos are estimated to some tenths of an $\mathrm{eV}$. The masses of atoms are mostly between 1 and 300 $\mathrm{GeV}$. Thus, considering the masses, this golf ball comparison isn't okay in my opinion. In my mind, comparing the mass of the Moon to the mass of the Solar System would be more realistic.

Their sizes can't be easily compared, because the elementary particles don't really have a size. Currently they are considered point-like particles with uncertain position. This uncertainty depends on their impulse (which depends on their speed and their mass) because of Heisenberg's uncertainty principle:

$$\Delta p \cdot \Delta r \ge \frac{\hbar}{2}$$

The masses of strings is a subject that belongs to professionals. I am not sure it is easy to even define their mass. Their sizes are in the order of the Planck length (around $1.6\cdot10^{-35} m$). Compare this to the size of the nuclei ($10^{-15} m$).

• That makes a little more sense, but it would give us a relativistic moon. At 1eV a neutrino is already moving near the speed of light. :-) – CuriousOne Jul 24 '16 at 23:55