How much hair?

Thomas Sperling discusses some furry Fermi problems

The first Fermi problem I tackled was in an introductory astrophysics lecture. One of the questions on the first problem sheet was: Are there more mosquitoes in the world than stars in our galaxy? Questions like these are intriguing, easy to understand, and have a definite answer. However, with only limited information available, we need to use a creative way of guesstimating to answer them. In principle a mighty (but hypothetical) being like Maxwell’s demon or the flying spaghetti monster would be able to count all mosquitoes and stars at one instance in time (or, as physicists might put it, in one specific frame of reference), but this brute force approach might not be necessary to answer the question: educated guesses can do the job just as well.

Total Eclipse of the Hair-t. Bonnie Tyler would be proud.

The legendary Italian physicist Enrico Fermi was the master of these questions. He was a key player in the Manhattan project to build the first nuclear weapon—you might spot him in the background of Christopher Nolan’s Oppenheimer, which unfortunately doesn’t show how he estimated how much energy the bomb would release. During the first detonation of the bomb, the Trinity test on 16 July 1945, Fermi threw a piece of crumpled paper in the air and observed how far it was blown away by the shock waves released in the explosion. He came up with an instant estimate of the energy released by the bomb, which was later confirmed after the analysis of all the data obtained during the test. Fermi solved a Fermi problem in the most elegant way!

Proposing and solving Fermi questions is an essential skill not only for physicists. Companies use questions like How many golf balls would fit in a car? or How many leaves are on a tree? in job interviews to test applicants’ creativity and capability for abstraction.

For me, Fermi problems lurk around every corner and I think they make life more fun… like the other day when I was waiting at the hairdressers for a new hair cut. This is usually the time when you can think about the universe, and stumble across deep questions of major importance for humankind. Questions like: How quickly is my hair growing and how much hair do I have? As an astrophysicist, I wanted to think about them on a cosmic scale. Next time you’re struggling for small talk at the hairdressers, maybe you can ask them…

Are there more hairs or stars in the universe?

If we want to make an educated estimate, it’s often best to start with simple observations in our daily life. Looking in the mirror, my head is roughly a sphere, with diameter 16cm. About half of this sphere is covered with hair, so the total hair-ea on my head is $A_\text{head} = \frac12 \times 4\pi \times \frac{(16\,\text{cm})^2}{4} \approx 400\,\text{cm}^2$ or, to use traditional German units, $5.6 \times 10^{-5}$ standard football fields.

Hairy Earps: About half of the surface hair-ea of Mary’s head is covered in hair. Image: James Boyes, CC BY 2.0

If all the hairs were squashed up on top of each other, how many could fit on a head? Each hair is between 50 and $120\,\mu\text{m}$ wide—we’ll call it $80\,\mu\text{m}$—so the average cross section of a single hair is $\overline{A}_\text{hair} = \pi \times \frac{(80\,\mu\text{m})^2}{4} = 5000\,\mu\text{m}^2.$ This is about $7\times 10^{-13}$ standard football fields. So if all the hairs were tightly packed, we’d get an estimate of $A_\text{head}/\overline{A}_\text{hair} \approx \text{8,000,000}$ hairs on our heads. However, scalp hair is not densely packed—between two single hairs there is a distance. I had another look in the mirror, and observed that the average distance between two hairs is on the order of 1mm. That means between one hair and the next one, about 1mm $/$80$\mu$m $\approx$ 13 hairs could fit.

Rubeus Hair-grid: it’s helpful to assume that hairs grow in a grid pattern with 1mm gaps between hairs. That’s a gap of around 13 hair-widths, so only 1 out of 196 `hair spaces’ are filled.

There’s only one hair spot occupied in each square of $14\times 14$ hair diameters, so we should divide our estimate by 196. This means that there are about to get $\text{8,000,000} / 196 \approx \text{40,000}$ hairs on each person’s head, at a density of roughly 100 hairs per $\text{cm}^{2}$. In reality, hairs don’t grow in a grid pattern, so the actual number is a bit higher: the average person has 100,000 hairs on their head. We’re out by a factor of 2: pretty good going for a guesstimate!

We just need two more facts to be able to answer our question. Firstly, how many stars are there? In the observable universe we believe there are $10^{11}$ galaxies, with $10^{11}$ stars in each galaxy (on average)—so there are about $10^{22}$ stars in total. Secondly, there are about $8\times 10^{9}$ humans in our solar system. We’ve just worked out that each one has about $10^5$ hairs on their heads, adding up to $8\times 10^{14}$ hairs, which we’ll round up to $10^{15}$. Thus we have more hairs in our galaxy than stars but there are more stars in the observable universe than hairs!

How long would it take for a single hair to grow to the moon?

All this talk of stars is lovely, but I wanted to focus on something a bit closer to home. I wondered: how long would it take for a single hair to grow long enough to reach the moon?

Hair-ston, we have a problem. Image: Gregory H Revera, CC BY-SA 3.0

Let’s start with the speed of hair growth. As a child I learned that my scalp hair grows at a rate of about $v_\text{hair} = 1\,\text{cm}$ per month (As a grown-up I can check Wikipedia: the range is more or less $v_\text{hair} = 0.6$–$3.35\,\text{cm}/\text{month}$.). This seems like a natural unit to use for hair growth, but unfortunately it is not a SI unit, nor does it relate to football fields. As an astronomer, I’d rather work in more familiar units: $v_\text{hair} = 3.86\times 10^{-12}\,\text{km}/\text{s}.$ In order to have a better feeling of how fast this is, we can compare the speed of hair growth with the speed of light, $c_\text{light} = \text{299,792.458}\,\text{km}/\text{s}$. It turns out that photons travel through space by a factor of $c_\text{light}/v_\text{hair}\approx 10^{17}$ times faster than hair grows!

The distance between the Earth and the moon is $D_\text{moon} =\text{384,400}\,\text{km},$ which is about one light-second. Since hair grows $10^{17}$ times more slowly than light moves, our total growth time is $t=D_\text{moon}/v_\text{hair}\approx 10^{17}\,\text{s}$. In human-readable units this time is about $3\times 10^{9}\,\text{yrs}$—that’s pretty close to the age of Earth, or $4.5\times 10^{9}\,\text{yrs}$. Sending humans to the moon via spaceships based on human hair growth is thus not a very smart idea—not to mention the amount we’d be spending on shampoo.

Ah, but what if we all teamed up? Since we have about $8\times 10^{9}$ contributing people on the planet, the global human scalp hair production rate (or GHSHPR for short) is about $8\times 10^{9} \times \text{100,000} \times v_\text{hair} \approx 3000\,\text{km}/\text{s}$. This means that, if we all pitched in, we could grow enough hair to reach the moon in about two minutes!

On the other hand, with 8 billion people and $100\,000$ hairs each, sticking all those hairs together into one mega-hair might be a problem. We’d need $8 \times 10^{14}$ pieces of tape, and even if we could stick ten hairs per second, it would take us $8\times 10^{14}\times 0.1\,\text{s}$, or about two and a half million years, to glue it all together.

Total Eclipse of the Hair-t. Bonnie Tyler would be proud.

What if we wanted to cover the Earth in hair?

Let’s admit it—one big downside of our planet is that it’s not very fluffy. But we can make it fluffy (like the Tribbles in Star Trek) by covering it with hair!

The Earth’s radius is $R_\text{Earth} = 6371\,\text{km}$, so its surface area is $4 \pi \times (6371\,\text{km})^2 \approx \text{40,000,000}\,\text{km}^2$. To make the Earth satisfyingly hairy, we’d need $4\pi R^2_\text{Earth} /\overline{A}_\text{hair} \approx 10^{22}$ hairs. This is about a tenth of a mole of hair, but it’s several orders of magnitude more than the hair we’ve got available. Even if we all teamed up, it would take absolutely ages to grow enough hair. According to the American Academy of Dermatologists, most people lose about 50–100 hairs per day, and since we seem to have the same number of hairs from one day to the next, let’s assume we’re growing 100 hairs a day each. At 8 billion people on Earth, that’s $8 \times 10^{11}$ new hairs every day. It would take us $10^{10}$ days—or 27,000 years—to reach our goal. This is probably the main reason why our proposed plan of creating the fluffiest planet in the universe will never get funding. On the other hand, we do have enough hair to turn Manchester into Mane-chester. If anyone wants to get in on my grant application, please do let me know.

Thomas is a young, curious postdoc at the Thüringer Landessternwarte Tautenburg in Germany. He hunts protostellar microjets in our universe and likes to put google eyes on stuff. He hopes that the Hubble tension will be resolved soon, so that he can sleep better at night.

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