### Issue 114 / November - December 2016

### Archimedes Universe

### Adeel M. Khan

Staring upwards at the vast night sky, which is riddled with stars, have you ever wondered just how many stars are up there? And more fundamentally, just how many *things *â€“ i.e., particles â€“ are up there? For people curious about humanity’s place in the universe, these questions have been asked by ancient thinkers and contemporary scientists, and have even been answered, at least as much as contemporary instruments allow us to answer them. Â

In a famous story about Archimedes, the ancient Greek mathematician sets out to calculate the hypothetical number of grains of sand that would fully occupy the known universe. Archimedes wished to know just how large our universe is, as well as how many particles might fill its vastness. Archimedes’ bold attempt at answering this question pioneered two important contributions: first, a numerical system to denote extremely large quantities (which led to modern-day *exponents*) and second, a guess at how voluminous the universe is and how many particles it might contain. The answer that Archimedes arrived at is that 10^{63} grains of sand would fill the known universe, which is a 1 followed by 63 zeroes.[1]

This number 10^{63} is as large in the physical world as it is deceptively compact in its written form. Ultimately however, Archimedes did not have the advantage of modern-day telescopes to verify his answer. Furthermore, he could not answer the natural and more scientifically-challenging follow-up question to his thought experiment: *just how many particles make up this universe*? In fact, the true number of particles composing the known universe is believed by modern day astronomers and astrophysicists to be over a quadrillion times the value of Archimedes’ number.

But how did scientists arrive at their number? That is, how do we count the particles in the universe? A basic version of the calculation is actually straightforward and is presented below.

First, before attempting to answer how many particles exist in the known universe we should define *particles *and *known universe*. There are several ways to characterize a particle, but popularly employed definitions are atoms, nuclei, or electrons. For the sake of this discussion’s simplicity, we shall use atoms, since they are readily understood. Therefore our task is to count the number of atoms that exist in the known universe.

As for the term *known universe*, we mean the observable universe as described in the standard model of cosmology that astronomers and astrophysicists use today. Based on calculations on the observed “drift velocities” of galaxies (how fast galaxies move away from one another), this standard model posits the universe to have a radius of 4.6 x 10^{10} light-years (i.e., 46 billion light-years or 3 Ã— 10^{23} miles), an age of 13.7 billion years, and a finite â€“ thus, countable â€“ number of galaxies, stars, and other celestial entities.

**How many stars are there?**

Just as cells are the basic units of an organism, the basic unit of the universe is the star. Thus to count the number of atoms in the universe, we first count the stars. We begin this calculation with the largest discrete entities in the universe â€“ galaxies â€“ which are essentially large clusters of stars. Galaxies come in various shapes and sizes. Our own Milky Way is a spiral-shaped galaxy and is slightly on the larger side, having a diameter of 110,000 light-years. In contrast, the Sagittarius Dwarf Elliptical Galaxy is only about 10,000 light-years in diameter and has a more ellipsoid shape. But whatever their features, how many galaxies are there in total?

By analyzing partitioned images from the Hubble Ultra Deep Field telescope, astronomers were able to create photographs of deep space, mapping our collective view of space from Earth. In these photographs, there are myriads upon myriads of galaxies. All of them are visible as discrete entities which can be computer-counted. With this method, scientists estimate the number of galaxies in the universe to be between 100 billion and 1 trillion. Thus, this number is on the order of 10^{11}.[2]

The next question is how many stars does an average galaxy contain? Quite a few, it turns out. An analysis of the light spectrum of galaxy images from the Hubble telescope, with extrapolation from data stemming from our own Milky Way, suggests that on average a given galaxy contains hundreds of billions of stars, which is again 10^{11}.[2] Combining these numbers, we calculate that there are approximately 10^{11} x 10^{11}Â = 10^{22} stars (10 sextillion) in the known universe.[2]

These 10^{22} stars differ greatly in their mass. For example, Eta Carinae, a star 8,000 light years away from Earth, is 100 times as massive as our sun. And the smallest known star, AB Doradus C, is found in the constellation Dorado and has a stellar mass of only 9% that of the sun. As an average, however, most stars are roughly as massive as our sun, which is 2 x 10^{30} kg. For comparison’s sake, the mass of the Earth is 6 x 10^{24} kg, putting an average star’s mass as approximately a million times more than Earth’s mass. Multiplying 10^{22} stars by 2 x 10^{30} kg per star gives us a remarkable 2 x 10^{52} kg as the total star mass in the universe.

**What makes up stars?**

Now we must ask how many atoms does an average kilogram of star mass contain? To answer that, we have to delve into the composition of a star. Stars vary in their chemical content and contain elements ranging from the smallest one known, hydrogen, all the way up to iron. As an interesting byproduct of the big bang, much of the known matter created was hydrogen, since it is the simplest element â€“ it is essentially one proton with one electron circling it. As a result, when stars formed from the available matter in the universe, they contained mostly hydrogen. As scientists analyzed the light spectrum of various stars, it was noted that star matter is surprisingly consistent in its average composition: it is roughly 78% Hydrogen and 27% Helium by mass.[3] The remaining 2% is made of trace elements such as Oxygen, Calcium, Silicon, and so on.

For the purposes of this simplified calculation, this 2% can be ignored. Then, by using the atomic masses of hydrogen (1.01 grams per mole) and helium (4.00 grams per mole) readily available from any periodic table of the elements, we can calculate that 1 kilogram of star matter is roughly 710 grams of elemental hydrogen and 270 grams of helium. Using basic knowledge of chemistry, we know that a gram of hydrogen is roughly 10^{24} atoms and a gram of helium is roughly 10^{23} atoms. Thus, we can calculate that there are approximately 4.6 x 1026 atoms in a kilogram of star matter.

**Finishing the calculation**

Now we can arrive at our answer to how many particles exist in the known universe: if there are 2 x 10^{52} kilograms of star matter in the universe, and 4.6 x 1026 atoms in each kilogram of this star matter, we yield (2 x 10^{52}) x (4.6 x 1026) = 9.2 x 10^{78}. So after rounding, *we can declare there are approximately 10 ^{79} atoms in the universe.*

This number is astoundingly large, yet looks deceivingly small. 10^{79} is over a million trillion trillion trillion trillion trillion trillions. For comparison, the estimated number of hairs on your head is 105, the estimated number of people on this earth is 109, and the number of atoms in a typical human body is 1027.[4] If we count the atoms of each person alive and add them together we still only have 1036 atoms â€“ that is to say, 1036 atoms make up the whole of humanity. If even humankind grew by a thousand-fold, meaning instead of 7 billion people alive, there were 7 *trillion*, the number of atoms making up the world’s population would still be only 1039.

Thus 10^{79} is an enormous number â€“ far larger than the grains of sand Archimedes proposed. It should also be noted that this is technically an underestimate of the true number of atoms in the universe since we did not include trace elements, mass from extraterrestrial planets, or mass from extrastellar phenomena such as the interstellar medium, intergalactic medium, black holes, brown dwarves, and other items of outer space. Also, this number deals only with visible matter, thus excluding dark matter (and dark energy), which to date remain very poorly understood concepts. Finally, we should also note that our estimate of 10^{79} atoms is not the only one. Various other calculating methods have produced other estimations of the number of particles in the universe, ranging from 1072 to 1087.

Regardless of what the precise count of atoms in the universe might be â€“ whether more or less than 10^{79} Â â€“ it is undoubtedly, as Archimedes noted, an awe-inspiring number. It makes the 1028 atoms in each of our bodies feel utterly humbled â€“ but also connected to the grand whole since we are a part of that awe-inspiring number. When glancing up at the star-riddled sky, realize that the vastness of this universe can be described as 10^{79} atoms organized into such things as stars and asteroids, people and plants, cars and computers. We humans are just a fraction of this astronomical quantity.

**References**

- Bradshaw, Gillian.
*The Sand-Reckoner*. New York: Forge, 2000. - Van Dokkum PG, Conroy C. A substantial population of low-mass stars in luminous elliptical galaxies. Nature, 2010; 468:940-942. Doi: 10.1038/nature09578
- Irwin, Judith A.
*Astrophysics: Decoding the Cosmos*. West Sussex: John Wiley & Sons, 2007. - Brandreth, Gyles.
*Your Vital Statistics: The Ultimate Book About the Average Human Being*. New York: Lyle Stuart, 1986.