Thursday, January 04, 2007

A Googol of Particles?

A few weeks ago, some friends asked me whether or not there are a googol particles in the universe. I copped out at the time, pleading hunger, but it's an interesting question, so I found myself trying to answer it yesterday.

First off, a googol is defined as 10100, or a 1 followed by 100 zeros. It's a big number, which is why the Google people wanted to name their company after it; unfortunately one of their first buyers wasn't very good at spelling... I'm also going to define "in the universe" to mean "in the known, observable universe." Obviously, if the actual universe is infinitely large, then there will be more than a googol particles in it. Second is the question of what we mean by "particle." I'll start by assuming that particle means "atom," specifically hydrogen or helium atoms, since there are relatively few other atoms around. I'm also going to take "known universe" to mean "observable universe," and for both of these terms to refer to the entire observable universe at an age of 13.7 billion years old. This isn't actually how the universe is observed; because it takes time for light to travel to us, the farthest away we can see in the universe corresponds to when the universe was very young. Taking this into account would needlessly complicate the question and the calculations.

To get a feel for numbers of particles in big, massive things: how many particles are there in the Sun? The mass of the Sun is roughly 2 x 1057 times that of the mass of the proton, and since about one out of every four atoms in the Sun is a helium atom (which has the same mass as four protons), there are roughly 1057 particles in the Sun.

So how many particles are there in our galaxy, the Milky Way? The mass of the Milky Way is difficult to define, and we have to be careful to only talk about the baryonic mass for now—that is, the mass that is in the atoms we are trying to count. Let's say it is 1011 solar masses; the number of particles in the Milky Way is therefore 1011 x 1057 = 1068.

From here, we can estimate how many particles are in the known universe if we have a good estimate for how many galaxies are in the known universe. This is a tricky number to estimate, because we can't actually see all of these other galaxies. We are also assuming that the Milky Way is a typical galaxy of typical mass. The internet gives a lot of different numbers for the "total number of galaxies in the universe." NASA's "ask an astronomer" page claims 125 billion, which is the same as 1.25 x 1011; other sources give similar answers, so I'll use 1011. Under these assumptions, we calculate that there are on the order of 1079 "particles" (specifically, atoms) in the known universe.

Another way to do this calculation is to first figure out the number density of atoms for the universe, and then multiply by the total volume of the known universe to obtain a total number of atoms. I don't really feel like going into cosmology right now, but essentially, the density of the universe is inexorably linked with its geometry. We know that that universe is pretty damn flat, so we know what the density is fairly well. ("Flat" here means just what it sounds like; you can think of it as meaning that the three angles of a triangle have to add up to 180°.) This "critical" density is approximately 10-29 g/cm3. This is really really tiny: for comparison, the density of water is 1 g/cm3, the density of air at sea level is roughly 10-3 g/cm3, and the density of air in the best vacuum that can be made on earth is 10-20 g/cm3. Cosmologists tell us that only 4% of the universe is made up of atoms, so the density of atoms is more like 4 x 10-31 kg/m3. The mass of a typical atom is still about the mass of two protons, so this corresponds to about 1 x 10-7 atoms per cubic centimeter. (If you don't like fractional atoms, you can think of this as about 1 atom in every ten cubic meters instead.) The volume of the observable universe is determined by its radius. Even though the universe is 13.7 billion years old, its radius is not 13.7 billion lightyears; it's actually more like 93 billion lightyears, or about 9 x 1026 m. This gives us a volume of 3 x 1081 m3, and a total number of atoms in the observable universe of about 1077. I put a lot of assumptions and simplifications into these calculations, so it isn't too surprising that they give slightly different results. When I was doing the calculations yesterday, I was getting more like 1080 for both methods.

So we have determined that there are fewer than a googol atoms in the observable universe. This number won't increase by much if we expand the definition of "particle" to mean all electrons and quarks, the most fundamental particles of matter. But what if we include photons and neutrinos? The ground-up way to do this calculation is to start with a number density of photons (approximately 400 per cubic centimeter) and a number density of neutrinos (approximately 200 per cubic centimeter). Multiplying by the same volume as above, we now get a total of 2 x 1089 particles. There are therefore fewer than a googol known particles in the known universe. An easy check this number—as well as an alternate way of doing the calculation—involves using the known baryon-to-photon ratio of roughly 10-10. ("Baryon" being the word used in cosmology for "stuff that turns into atoms.") This is in fact approximately the ratio between the two total number of atoms and the total number of photons (and neutrinos) we calculated.

I said above that there this means that there are fewer than a googol known particles in the known universe. I have already mentioned that once we no longer restrict ourselves to the observable universe then there can clearly be more particles, but the distinction of known particles is important as well. I said earlier that about 4% of the universe by mass is made up of atomic-like particles; about 20% of the universe is made up of some other kind of mass that we don't really know what is, known as dark matter. A popular assumption is that the dark matter is some kind of as-yet unknown, un-detected particle—with some currently unkown mass. Using the calculation above, if there are going to be a googol dark matter particles in the observable universe, then the dark matter particle would need to be about 10-20 as massive as a proton—that's tiny—so stupidly small that it is in fact ruled out by the fact that we observe the universe to have structure. Such a small mass for the dark matter particle would lead to what is known as the "Hot Dark Matter" scenario; essentially, if the dark matter particle has very little mass, then regular matter won't get cold enough to condense and form nice things like stars and galaxies.

In conclusion, a googol is in fact a very very large number.

8 comments:

Stephen said...

I've seen numbers like 10^80th bandied about elsewhere.

One might consider the total energy of the Universe and estimate how many proton or photon equivelents this is. Then you could take into account dark matter and dark energy.

All in all, a fun exercise.

Stephen said...

I heard that during a supernova, some 10^57th neutrinos are released. Despite the fact that most of them pass through stars, planets, etc., without interaction with anything else, this pulse provides significant pressure, helping the star to actually explode, rather than fizzle.

One might expect that 10^57th particles would add alot to the total particle count in the Universe. But as mind-numbingly big a number as 10^57th is, it's hardly anything compared to 10^80th. It's this kind of comparison that helps one deal with really big numbers.

Understanding geologic time and astronomical distance is essential to having any idea how the Universe works.

For geologic time, one description shows modern sea shells - from freshly caught and eaten shell fish. Then, fossils from under a lava flow are dug up. The lava flow is known to be at least 100,000 years old. There are several layers, so that the fossils must be at least 1,000,000 years old. Yet, these fossils appear identical to the fresh ones. And yet, older fossils do show changes. And that, is geologic time.

Anonymous said...

"Even though the universe is 13.7 billion years old, its radius is not 13.7 billion lightyears; it's actually more like 93 billion lightyears."

I'm showing my ignorance here, but can you explain why this is the case? Thanks!

- Levi

Anonymous said...

Hello mollishka ,

I've just come across your article on particles, Googols and the Universe.

Have a read of these few words that I penned a year or so back.

THE SAND RECKONER
Isn't Google just about the most amazing research tool ever? For instance, with it you can discover that Google was going to be registered as Googol, but somebody got the spelling wrong. The founders of Google wanted a name that would relate to the immense amount of data that their website would have to store. So they thought of using Googol, the name for a very large number that was coined in 1920 by a nine-year-old nephew of an American mathematician.
A Googol is 1 followed by 100 zeros and looks like this:

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Or using scientific notation: 10100

A Googol is clearly huge, but does it exceed the number of stars in our boundless Universe? Well, the number of stars doesn't even come close! Astronomers have calculated that the number of fundamental particles (protons, neutrons and electrons) in the Universe is 'only' about 1080.
Now why would astronomers want to calculate that number? Well, probably because it can be done, and to show cleverness. Which is possibly why Archimedes did the same sort of calculation in about 250BC when he calculated the number of grains of sand that would fit into the then known Universe. Here are the opening sentences from his work: "The Sand Reckoner".






"Many people believe, King Gelon, that the grains of sand are infinite in multitude; and I mean by sand not only that which exists around Syracuse and the rest of Sicily, but also that which is found in every region, whether inhabited or uninhabited. Others think that although their number is not without limit, no number can ever be named which will be greater than the number of grains of sand. But I shall try to prove to you that among the numbers which I have named there are those which exceed the number of grains in a heap of sand the size not only of the earth, but even of the Universe"

His estimate was 1063 which has to be far, far fewer than the number that would fit into our known Universe. Let us show our cleverness by updating Archimedes's calculation for the first time in over 2000 years by using data obtained via Google.
Our Universe is estimated to have a diameter of about 160 billion light years, and as light travels at 300,000 km/s we can easily calculate its diameter in kilometres:

160 x 109 x 300,000 x 60 x 60 x 24 x 365.25 = 1.5 x 1024 km.

Multiplying this number by 1,000 gives us the diameter in metres, and by another 1000 gives us the diameter of our Universe in millimetres: 1.5 x 1030 mm.


Soil particles can be classified as having the following sizes:

Type of Mineral Particle Size Range
Sand 2.0 - 0.06 mm
Silt 0.06 - 0.002 mm
Clay less than 0.002 mm

If the above information was available to Archimedes it is possible that he would have chosen a particle of clay, rather than a grain of sand, to better demonstrate his cleverness. Let us do the same.
Clay 'comes in' sizes less than 0.002 mm so let's choose a size less than the 'average' size of 0.001 mm, say, a third of 0.002mm, i.e. 0.0007 mm.
We can now calculate the number of particles of clay that would fill the Universe. We don't need to calculate the respective volumes and divide one volume by the other. We can do what the Ancient Greeks would have done. We just divide the sizes and cube the result. Like this: (D/d) x (D/d) x (D/d).

So (1.5 x 1030)/0.0007gives us 2.14 x 1033, which cubed up gives us 9.8 x 1099. We can round up that answer because the Universe would have expanded slightly since you started to read this piece to arrive at the grand total of particles as 1 x 10100. Which is where we came in - it's a Googol!

This little journey, via History, Astronomy, Soil Mechanics and Mathematics, vividly demonstrates, better than any string of zeros, how huge a Googol really is. Archimedes, I'm sure, would have agreed.

Andrew Baxter
Amateur Astronomer & Professional Engineer
Oxfordshire England

Bark said...

Theoretical that is what this is, you have not included black holes and the matter streams they both engulf and emit (sub atomic particles) there is no way to calculate these figures you can only (for give the term) guess, unless you can fully map the universe factually and not theoretically.

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Jared said...

The calculation of the number of particles in the Universe must, of necessity, assume a uniform Universe. Such an assumption is challenged by the apparent explosion of space-time caused by "dark energy." The elasticity of space-time stretches the further away you go from our particular point in it. Geologic time stretches as one goes back, making calculations on the age of a sample suspect. Calculations on the radius of the Universe are also suspect, since space-time explodes away from us as we attempt to measure it. The best that can be said about the number of particles in the Universe is that they are "without number."

Agnius Vasiliauskas said...

@Stephen

There is infinite amount of numbers which are mind-numbingly bigger than googol. Take a look at googolplex or anything else between googol and infinity.

For the fun part -
How much images can be generated with dimensions 256x256 pixels ?
Answer is - (256^3)^(256^2) - in order approximatelly e ^ 1 000 000. This is truly an endless sea of picture "universe". When looking at this number we can now fully understand phrase that "artist is only limited by his/her imagination" ;)