Cosmic Scales

Our universe is 14 billion years old. With $100\ billion$ galaxies in observable universe. And on average of $100\ billion$ stars in a galaxy. Spanning about $43\ billion\ \text{light-years}$ around us. The scales we deal with in our day-to-day life, are much smaller in comparison. How do we appreciate the cosmic scale?

If I did my job rightI am new to these numbers and the approach. If you find mistakes, please do let me know., reading this article, you would have gained a few analogies to work with cosmic numbers. Among other things, you would find that if the $14\ billion\ year$ history of universe is squished into $72\ years$, most of human advancement happened in last minute. That if we squish a galaxy into the volume of a human being, about million stars would fit in $1\ \text{centimeter}$ cube. But the star themselves would be smaller than the size of an atom. i.e, most of the space is empty. And that at this scaling, next galaxy would be $25\ meters$ away.

Experiencing these scales is out of question. But we can build some familiarity and anthropomorphized sense for these scales. Two activities that I have found useful are analogyFor the analogies to work, we have map the numbers to human range. I haven't considered error margins in the calculations. And haven't spent enough effort to be precise. I think it won't be an issue. and playing with the numbers. To get familiar with numbers, you would have to play with them, yourself. Analogies are easier to share and build upon. This article plays with a few cosmic numbers and shares analogies around them, that I have found interesting.

We will get help from a fictitious Celestial Being, referred to as CB in the article, for some of our analogies. And give itTo avoid gender pronoun argument about the celestial being, let's use it to refer to the being. I assure you, it doesn't mind. some supernatural, but limited powers. First power it has is to be able to create universes, as an experiment. Something akin to petri-dish experiments, but at a much grander scale. We will look at CB's perspective on one such experiment (our universe) and our place in it.

We start with time. It's been 14 billion years, since the Big BangWhat happened before then, is CB's concern. when CB started the experiment.

Time

Here is CB's another supernatural power. For it, time seems to be passing much faster. CB started the experiment 72 CB Years ago, with a Big Bang. 1 CB year is equivalent to $\frac{14\ billion\ years}{72}$ 194 million human years . A CB year has days, hours, minutes, seconds, same as an Earth for humans1 year is not special from universe perspective. Neither is 1 second. Year is rotation of earth around sun. And second is a convention (or standard?). I use same notion of year and second for CB, to make things more palatable for our mind.. In other words, 1 CB second is about Since 72 CB Years is approximately equal to 14 billion human years. 1 CB second is about $\frac{14 \times 10^9}{72\ \times \ \text{seconds in a CB year}}$ or about 6 human years.6 years for humans. We get a year older, and for CB not even a second has passed. In rest of the section, assume all time-scales are CB's perspective, unless explicitly mentioned as human time-scale.

Average human life of 72 years, in CB's time would be Average human life expectancy is 72 years. 72 human years, would be about $\frac{72}{\text{human years in 1 CB second}}$ $\frac{72}{6}$ or 12 CB seconds.12 seconds. So, for CB, here is how timeline of our history would feel like:

• It's been 72 years since experiment started (with a Big Bang). Or the Universe is 72 years old.
• Earth appeared roughly 4.5 billion years ago.
  $\frac{4.5 \times 10^9}{\text{years per CB second}}$ or 22 years 5 months in CB's timeline, since experiment start.23 year after Big Bang. Ignoring rest of the universe and focusing on earth.
  • First life appeared 3.7 billion years ago. 53 years after Big Bang or 19 years ago.
  • Dinosaurs lived and went extinct in Dinosaurs lived somewhere between 245 and 66 million years ago. $\frac{14 \times 10^9 -\ 246 \times 10^6}{\text{1 CB seconds in human years}}$ - $\frac{66 \times 10^6}{\text{human years per CB second}}$
    1 year 2 month - 4 monthsyear 69.
  • When did human genus come into picture? Around year Our ancestors appeared around 2 million years algo. 70 (just 2 years ago from CB's perspective).
  • Our species, Homo Sapiens, only emerged around Homo Sapiens emerged around 300,000 to 200,000 human years ago. Remember, 1 CB second is about 6 human years. So, Homo Sapiens emerged $\frac{300,000}{6}\ seconds$ ago. Or, about 14 CB hours.14 hours ago.
  • Most of written human history, around 4000 human years, would be about 11 minutes for CB.
  • In last About 200 human years.30 seconds , humans learned that the solar system is part of a galaxy with 100 billion stars. And their galaxy is also nothing special but another in 100 billion or so galaxies. And are contemplating the origins of the experiment and much more.

It's fascinating to think how much has happened in last minute, from CB's perspective. Be it the understanding of what is within reach (nature), what is beyond/before/after (universe), what can be (AI, large societies, maths, space exploration) and much more. Human beings have expanded their understanding exponentially, in as many topics.

But, for CB to observe this, it would have to find Earth first. The planet where humans are. A planet, in a solar system, in a galaxy, in the universe.

Counting Stars

We will start by first getting a sense of the number of stars in the universe. Earth is part of a solar system with other planets and one star (Sun). Sun is one among 100 billion other stars in the Mikly Way galaxy. Milky Way itself is one among 100 billion galaxies in the universeThat we know these numbers is in itself interesting. So, in total, 100 billion * 100 billion stars in the universe. Building intuition around this number is going to be difficult. But let's play with these numbers.

First let's get some familiarity with billion.

Few billions

Human population around 9 billion right now. Let’s give each person 1 meter and make them stand one beside another, in a line. How long would that line be? That line could circle the earth $\text{population} \times \text{1 meter} \over \text{earth circumference}$ or $9 \times 10^9 \over 40 \times 10^6$ 200 times . If you could walk (without sleep) at 5 km/hour you can cover the circumference in 8000, or about a year (with few short breaks). What would be your age, 1 billion seconds after you are born? Answer: About 31 years, 6 months. Assuming 72 years as average age, most of us live about 2.27 billion seconds. Shake hands with one person for 1s, how long will it take to shake hand with all the humanity. That would be about 253 years. Definitely not something we have intuitive sense of. But, gives us a reference point to play with the number of stars in the universe.

Back to stars

Let's give CB another super-power, with limitation. It sees galaxies as we see our bodyHere is another fascinating scale. Our body houses around 30 trillion cells.. As an example, Milky Way galaxy filling a persons body. I will call this galaxy bodies. One galaxy is one galaxy body. Observable universe is $100\ billion$ galaxies. Or about Rounding human population to 10 billion. 100 billion galaxies is 10 times that.$10$ times the human population. In total there would be about 10 times human population galaxy bodies100 billion galaxies, round human population to 10 billion. That means about 10 times as many galaxies as humans.

Now consider all the stars in one galaxy body. That’s 100 billion stars packed one besides other, in our body. Imaging a small centimeter cube on your fingertip, that would be around One way to calculate this is first get volume of human body, in $cm^3$ and divide the number of stars in a galaxy with it. First, volume of human body. One way is to calculate it with density and mass of human body. Let's say average mass is 75 kg and density close to $1000\ kg/m^3$. So volume comes out to be $\frac{75}{1000} m^3$ or $0.075 m^3$. Converting to centimeters $7.5*{10}^4 cm^3$. Dividing number of stars in a galaxy with the volume, $\dfrac{100 * 10^9}{7.5 * 10^4}$ or $1.3 * 10^6$. A million stars in a centimeter cube.1 million stars. We are ignoring distance between the stars. We will look at distances in next section.

So, Earth would be somewhere in the Milky Way galaxy body. Let's say at the tip of a finger. CB doesn't know where exactly to look, so it will have to look one star at a time (given no waves or communication). Let's hope for the fictitious entities' sake, there are more planets with lifeSee: https://inference-review.com/article/a-lonely-universe.

But if only earth has life, assuming the celestial being can scan 1 star a CB second, a galaxy body would take it around 1 second per star. 100 billion stars. So, 100 billion seconds. That is about 3000 years.$3000\ CB years$ . That's 1 galaxy. Even if we assume that CB would find Sun after groing through half of the galaxy bodies, it will still take ${50\ billion\ stars} * {3000\ CB\ years}$$150,000\ \text{billion CB years}$ to find Sun. The being would have to be immortal, with abundant patience. Hope the human race survives that long.

But that's not the only concern. Stars are not packed together, like cells in the body. There is a lot of space between them.

Space

Within the galaxy body, if we scale the stars in proportion to the space available to them, the size of the stars would be about $10^{-9}\ cm$. That is smaller than an atom. I.e. most of the space inside galaxy body is empty. Yet, there are a million stars in it. On the other hand, if CB looks for neighboring galaxy bodies, they would be around $25\ m$ away. At the scale of stars, space is mostly empty. But at the scale of galaxies, space is quite packed. And all of the observable universe would have a width of Observable universe is $96 \times 10^9\ light-years$ wide. We are mapping $100k\ light-year$ wide galaxies to about $1 meter$ . So, universe would be $\frac{10^{11}}{10^7}\ meters$ wide. Or about $1000\ km$ wide. $1000\ km$. Let's work this out step by step.

Moon is about $400,000\ km$ away from Earth. That's big enough to fit in Earth diameter is $12,742\ km$. Diving distance by Earth's diameter, $400,000 \over 12,742$, or about $30$ Earths.30 Earths. Sun is $150 \times 10^6\ km$ away from Earth. Proxima Centauri, Sun's nearest neighboring star, is $4 \times 10^{13} \ km$ away. It's going to be difficult to deal with distances in $km$.

Instead, astronomical distances are mostly measured in light-yearsThere is also AU and cool-sounding parsec. You should check them out.. Let's start with speed of light. Light travels at about $300,000\ km/s$, or $3 \times 10^8\ m/s$. Speed of light is the fastest speed that can be achieved in our universeWith consequences like special relativity. Another fascinating topic to explore. Cars do about $100\ km/hour$ or $0.03\ km/s$. Fastest speed a man-made object has achieved is about $700,000\ km/hour$, or $200\ km/s$. It's a probe in space. And it's only $0.065\%$ of the speed of light.

What does speed of light have to do with measuring space? Light from moon takes 1.3 seconds to reach earth, covering $400,000\ km$. Light takes about 8 minutes to reach from Sun to earth, covering $150 \times 10^6\ km$. Light-year is the distance light travels in a year31 million seconds in a year. So, light travels $31 \times 10^6 \times 3 \times 10^5$ or about $10^{13}\ km$ in a year.. Light takes $4.2\ years$ to reach from Proxima Centauri to Sun, covering $4 \times 10^{13} \ km$. I.e. Proxima Centauri is $4.2\ \text{light-years}$ away from Sun. Using $\text{light-years}$ makes cosmic distances easier to deal with. After all, light from the stars and galaxies, is how we know what we know about the universe. Now we are ready to look at cosmic distances.

Let's assume stars, on average, are the size of Sun. About $10^6\ km$ in diameter. And separated from closest star by let's say 5 light-years. For these cases, the distance between stars is about $\frac{5\ \text{light-year in km}}{10^6}$ or $\frac{5 \times 10^{13}}{10^6}$. About $10^7$ $10^7$ times the size of the stars. If stars were a meter in diameter, their next neighbor would be about $10,000\ km$ away. For comparison, earth diameter is $12,742\ km$.

Milky Way galaxy is $100k\ \text{light-years}$ in diameter. And about 2.5 million light-years from Andromeda Galaxy. Again, I will assume this is somewhat representative of rest of the galaxies on average. The distance between galaxies is about $2.5 \times 10^6 \over 10^5$$25$ times the size of galaxies. If galaxies were a meter in diameter, their next neighbor would be 25 meters away. Galaxies are quite close to each other.

Now consider observable universe. It's 93 billion light-years wide. It is related to how old the universe is. The theory is, universe as we know it, started with a big bang, and has been expanding ever since. And we are seeing edges of that universe, with what light could travel in that time. So, universe might be much bigger than whats we can observe.

We figured in last section that a centimeter wide cube in our fictional galaxy body would have about 1 million stars in a galaxy body. Or on an edge of cube, there would be 100 stars, one after the other. Each star has $0.01\ cm$ to it. If we scale the star so that $0.01\ cm$ represents the distance between stars, the size of the star itself would be $10^{-9}\ cm$ in diameter. That is smaller than size of an atom. In other words the 1 cm cube is mostly empty, but still there are 1million stars in it. On the other hand, like we saw in previous paragraph. If the galaxy is 1 meter in diameter, neighboring galaxy would be 25 meters away.

Imagine CB trying to find a planet around a sun (which is barely the size of an atom) in some galaxy body, among 100 billion galaxy bodies. CB would need a lot more super-powers than what we have given it.

Before we part ways

There is much more to explore. Galaxies are moving. Sometimes through each other. Imagine galaxy bodies moving through each other, while distoring each others shape because of gravity. There are planets, stars in different sizes (neutron stars, red giants), black-holes, Nebulae, galaxy clusters and much much more. Worth reading something like Welcome to the Universe: An Astrophysical Tour, to know more.

My intent with this post, was to share some examples of scaling to human range. I find playing with numbers helps in getting familiar with them. Hopefully, this write-up will nudge you towards exploring on your own, if you haven't already.

But I can't stop without pointing out the wonder of our existence. Universe itself is fascinating, on how it came to be, what it is, what it's going to be. The basic elements it's made of, lead to planets and sun. Life happened on at least one of these planets. With all its variety and process, life (or evolution) is fascinating in its own right. Somehow this process created intelligence which can observe and grow new super-powers (technology for both observation and affect) and gain some understanding of both universe and the process of life itself. This is not even scratching the surface of what we have achieved as a species.

Further explorations

Two broad areas that this article covers is playing with numbers and scale of universe.

Playing with numbers

For playing with numbers, here are a few books that you might find interesting.

• Chapter 6. On Number Numbness - Metamagical Themas: Questing for the Essence of Mind and Pattern - Douglas R Hofstadter
• Innumeracy - John Allen Paulos
• Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers - Brian W. Kernighan
• The Machinery of Life - David S. Goodsell
  Cell biology is another fascinating area with scales difficult to experience. The Machinery of Life uses analogy and visualizations to introduce it.

Scale of universe

• Welcome to the Universe: An Astrophysical Tour - Neil deGrasse Tyson, Michael A. Strauss, J. Richard Gott
• Cosmic Distance Ladder - videos
• Powers of Ten™ (1977) - video
  
• Expansion of the universe, comoving coordinates
• How big is the solar system
• How the Universe is Way Bigger Than You Think

Acknowledgement

Thanks to Anantha Kumaran, Lalit Patel, Rohit Shinde, Saneef and others for their feedback. Much of the blog is influenced by books, videos and writeups by others on the net. Infact, there might be no original idea in this article. Unfortunately, it's difficult to keep track of all the sources.