Famed astronomer Carl Sagan once suggested in his book “Cosmos” that the number of stars exceeds the total number of grains of sand on all the beaches on Earth. This poetically descriptive statement has been widely circulated in the astronomical community. Standing on a beach, looking out, the grains of sand seem endless, too numerous to count. If such a quantity were to be magnified to encompass all the grains of sand on all the beaches of the Earth, then that number would be astronomical. According to Sagan, the number of stars in the universe far exceeds this amount, which is certainly incredible.
When attempting to estimate such unimaginable quantities, we encounter a problem: How accurate is this claim in reality? Perhaps the somewhat displeasing answer is that it is not exact, because the result largely depends on several assumptions, some of which are difficult to define clearly.
Such grand and quantifiable doubts are categorized as “Fermi problems,” named after the famous physicist Enrico Fermi. A Fermi problem is a rough estimation method, famous for applying simple methods to approach an accurate answer when dealing with complex issues that are nearly impossible to calculate. Astronomers usually aim for an accuracy within an order of magnitude of 10, which means the calculated value is considered acceptable if it is within one-tenth to ten times the range of the real answer. This practice is what astronomers describe as the concept of “orders of magnitude.” In such estimates, a two or threefold error in order of magnitude is not overly exacting, as long as it is close within an order of magnitude.
Having understood the requirements for precision of the answer, we might as well continue to explore how many stars twinkle in the sky. For example, our Milky Way galaxy is composed of several hundred billion stars. Being within the Milky Way, our observation of the external universe is mostly obstructed by gas and dust. Due to the vast range of star brightness, it is difficult to determine the exact number, but it is conservatively estimated that there are two hundred billion stars in the Milky Way.
Next, by multiplying the number of galaxies in the observable universe, we can get the total number of stars. In 2016, a group of astronomers researching the number of galaxies in the universe published their findings, roughly estimating that there are two trillion galaxies in the universe. If we simply multiply two hundred billion stars by two trillion galaxies, can we get the answer? At this point, patience is needed because the truth is more complex. In fact, when making this estimate, astronomers also considered galaxies whose total mass of stars exceeds a million times the mass of the Sun.
Compared to massive galaxies like the Milky Way, there exist some small galaxies whose mass is only one-twentieth millionth that of the Milky Way. This means we cannot simply use the Milky Way as the representative of all galaxies.
Excitingly, just as with the stars’ case, the number of these low-mass galaxies might far exceed that of the large-mass galaxies, so their numerous count may compensate for the fewer number of stars they contain. If we estimate based on each galaxy containing about a million times the mass of the sun, this estimate is already accurate enough—there is no need to fret over small discrepancies in the numbers.
However, there is still a problem: not every galaxy contains exactly one million stars. The mass of the Sun is unusually large compared to most stars; in fact, most stars are red dwarfs that are much smaller than the Sun. Among all stars, only about 10% have a mass equivalent to or greater than that of the Sun. This suggests that, in the universe, an average solar mass corresponds to roughly 10 stars. Therefore, we can estimate that each galaxy has, on average, about 10 million stars.
Based on the above calculation, we can conclude that the total number of stars in the universe is 10 million x 2 trillion = 20 quintillion, which is 200 quintillion or 2 x 1019 stars. Obviously, the universe is not lacking in stars. But how does this immense number compare with the amount of sand grains on Earth? Now it’s time for some more practical calculations.
Let’s estimate the total number of sand grains on all the beaches on Earth. The most direct estimation method is: first, determine the total volume of sand on these beaches (measured in cubic meters), and then multiply by the number of sand grains per cubic meter. This calculation is not complicated. As for the number of grains per cubic meter, it depends on the size of the sand grains, which typically range from 0.1 millimeter to 2 millimeters. For simplicity, let’s assume an average of 1 millimeter. This means one cubic meter contains 1000 x 1000 x 1000 = 1 billion sand grains. This number is truly astonishing. Therefore, with only a few hundred cubic meters of sand—an amount roughly the volume of a standard room—the number of sand grains might match the number of stars in the Milky Way.
To get the total volume of all the beach sand, one possible method is to consider a simplified beach model: assume a beach extends 50 meters from the seashore to inland and has a depth of 10 meters. With this, we only need to estimate the total length of all the beaches combined to find out their total volume.
Overall Length of Coastlines and Sand Beach Coverage
When people evaluate the total length of coastlines, a surprising discovery emerges—the total length of coastlines of all continents is about 2.5 million kilometers. Of these, approximately 30% consist of sandy beaches. Excluding the part surrounding Antarctica, we get a more realistic figure: the total length of sandy beaches is about 750,000 kilometers, which is 750 million meters.
Estimate of Total Volume of Sand
To estimate the total volume of sand, we can take this approach: a beach with a width of 50 meters, a depth of 10 meters, and a length of 750 million meters. According to this calculation, the total volume of sand is approximately 375 billion cubic meters. Assuming there are 1 billion grains of sand per cubic meter, the total number of sand grains would be an astonishing 37.5 quintillion grains.
Comparison of the Number of Stars and Sand Grains
According to a rough estimate, there are 4 sextillion grains of sand, which converts to 4 x 1020 grains. Interestingly, this number is about 20 times the number of stars in the known universe. The closeness of the two figures, contrary to expectations, challenges a common astronomical metaphor. However, as this is just a rough estimate, many assumptions could significantly affect the final number. For instance, varying sizes of sand grains could lead to a marked change in the number of grains. An extreme example is smaller sand grains, which could increase the count by a thousandfold, and even if there are deviations in the estimated length or depth of beaches, the number of sand grains would still be thousands of times that of stars. Although the number of stars in galaxies might be greater than our estimates, they are unlikely to pose a threat to the sheer abundance of sand grains.
Grains of Sand from Other Sources
It is worth mentioning that the calculations here only include the sand on the beaches. If we were to consider the sand from the seabed and the deserts on land, the amount of sand would significantly increase. Taking just the Sahara Desert as an example, the quantity of its sand grains might be hundreds of times more than the total amount of sand on all the beaches of the world.
The Challenge to Intuition and the Value of Mathematics
Such findings are undoubtedly surprising and illustrate the charm of what are known as Fermi problems: a simple calculation can provide an approximate answer, thus verifying the accuracy of intuition, or at least understanding how far it deviates from intuition. Intuition can often be wrong, especially when it involves dealing with such large numbers. Our brains are not particularly adept in this area and can easily be led astray. This demonstrates the importance of mathematics and science, which provide powerful tools to verify and calibrate our intuition.
[ad_2]
