TAU Chapter 1 --- Scales of Space and Time

Increasing size scales, roughly a factor of a billion each step

One of the most important ideas to begin to understand is the immense differences in scales and size that we'll be investigating.  It's hard to get around the prejudice that limits our imaginations to things we've encountered on the Earth --- to most of us, "small" means something like a sand grain, perhaps a fraction of a millimeter (the smallest marks on a metric ruler or meterstick).  "Huge" might bring to mind a mountain or vast expanse of forest, maybe thousands of meters or many miles across (or tall).  This somewhat provincial attitude will be strongly challenged by the objects and distances we encounter even in our own Solar System, not to mention the unimaginable vastness of the Universe as a whole.  The entire Earth, as we'll see, is a tiny speck floating in a huge expanse; but even the smallest sand grain on the Earth is unfathomably gigantic when viewed from the perspective of its constituent atoms.  The astonishing promise is that all these fantastically different scales are all described by the same physical laws, which gives us some hope of understanding the Universe around us.

The above images represent a series of increasing size scales typical of objects we'll be studying.  Atoms begin our scale on the small end.  There are 92 different atoms, or building blocks, out of which all the “ordinary” matter in the Universe is made.  Each type, or element, has atoms of different sizes; a rough average size for an atom is about a ten-billionth of a meter.  Lining up a billion atoms will just about span the width of a couple of apples.  But wait, I’ve just invoked an enormous number that is quite beyond most of our imaginations already, so have I really told you anything at all?  Just how big is a billion?  It is a number of increasingly common usage, describing the economic costs of massive projects as well as the populations of the largest countries.  It is written as a 1 followed by 9 zeroes: 1,000,000,000.  As a shorthand, we write it also as  \( 10^9 \), indicating that it is 10 multiplied by itself 9 times.  If you don’t have some way of imagining a billion things, though, it’s difficult to make any sense out of the relative scales of thousands, millions, and billions.  These days, such a sense is needed to evaluate national-level programs.  Is a $10 billion project a lot more expensive than a $500 million one?  (Yes!  20 times the cost.)  If we can save $70 million from a proposal costing $7 billion, what percentage savings is that, really?  (Only about 1%!)  And so on.  

A thousand little boxes

First, let’s try to visualize a thousand things (1,000, or \(10^3\)):  The bottom of this cube is a square made of 10 rows of 10 boxes in each row, so there are 100 boxes on the bottom layer.  To make the cube, we stack 10 of these layers on top of each other, so you should convince yourself that there are a thousand boxes in the cube.

A million little boxes

Now let's step up to a million things (1,000,000, or \(10^6\)).  You should see, especially if you look at the full-resolution image, that each little box has itself been subdivided into a thousand still smaller little boxes.  There are now a thousand groups of a thousand things, which is a million.

Ok, now on to a billion!  Continuing the pattern, if you can somehow imagine each of the tiny boxes above subdivided into a thousand still tinier boxes, then there will be a billion little boxes in the big cube.  So a billion is a thousand groups of a million.  One fairly easy way to try to visualize a billion is to consider an ordinary meterstick.  The smallest marks are millimeters, so there are a thousand of them along the stick.  If you imagine a large box that is 1 meter square on the bottom by 1 meter high (a cubic meter), then there will be 1 billion millimeter-sized boxes contained inside.  Since a sand grain is perhaps a millimeter across, then a box of sand 3 feet high, 3 feet wide, and 3 feet deep would contain something like a billion grains.  If you were to lay out all the millimeter boxes next to each other in a line, it would stretch a distance of 1000 kilometers, or about 600 miles (from Oklahoma City to Denver, roughly) --- if atoms were just barely visible, like grains of sand, then everyday objects like apples would be the size of a few states across.  Of course, this works for counting anything, so it’s also interesting to try to imagine a million seconds, which is about 11 and a half days.  A billion seconds, though, is almost 32 years; you are likely to live somewhere between 2 and 3 billion seconds.  The Universe has been around about a billion human lifetimes, or about a million times longer than modern humans have existed.  It'll be useful to remember that these expansive scales exist not only for space, but for time.  We tend to think that a second is a short snippet of time, but there are processes that happen on fantastically short timescales.  Ordinary yellow light, for example, is ultimately the result of something vibrating 600 trillion (a thousand billion) times each second!  There are very important reactions we'll discuss later that occur only for a duration of \(10^{-18}\) seconds (a billionth of a billionth of a second).  In perspective, there have only been about \(10^{18}\) seconds since the Big Bang (half that, if you're being picky), so as many of these events could occur each second as there have been seconds since the beginning of the Universe!

The relative size of the Sun and the Earth --- the Sun is more than 100 times larger across, and over a million Earths would fill up the Sun's volume.

Now let’s look at the sizes of objects again in the first image.  The first step from an atom to the everyday scales represented by an apple is an increase in scale by a factor of a billion.  If we take the next step and line up a billion apples, then we're approaching the size of the Sun (see the above image!).  Really, the Sun is about 15 billion apples across --- truly gigantic compared to any objects humans have been accustomed to dealing with throughout our history.  It is remarkable that just in the last 100 years or so we have pretty well figured out the physics of atomic interactions a billion times smaller than we as well as the inner workings of the Sun and other stars ten billion times larger.  It’s a wonderful thing that the Universe seems to obey understandable laws on these wildly different scales.  We routinely study objects on even larger scales, though.  200 billion stars more or less like the Sun are gathered in our Milky Way galaxy, a truly immense structure that is about six billion times larger than the distance from the Earth to the Sun!  Just as roughly the size of an atom is to an apple, and that apple is to the Sun, our solar system is to our enveloping galaxy.  It is fascinating that we can view the gigantic Milky Way every clear night from Earth; our perspective in trying to fully comprehend this object is in some way similar to an atom's perspective of aggregate objects such as we.

A slice through our Universe showing the distribution
of galaxy clusters

But the Milky Way galaxy is not the largest structure in the Universe, although 100 years ago we were not aware of anything larger.  There are estimated to be around 150 billion galaxies in the observable Universe, which apparently spans the equivalent distance of about a million Milky Ways across and is arrayed in beautiful networks of galaxy clusters and superclusters, as shown here (each dot represents a galaxy cluster, with ours at the center of this map).  Only now, for the first time in human history, we are able to map the large-scale structure of the Universe.  Many previous cultures wondered and guessed at the nature of the Universe as a whole and what it might look like.  We are the first privileged generation to finally address and answer these fundamental questions addressing our cosmic context and place in the Universe.

TAU Introduction

Introduction

Our home as seen from the International Space Station

Roughly 2.5 million years ago, our ancestors began using tools to improve their lives and, in some (perhaps implicit) sense, also began learning about the world in which they were embedded.  Only very recently, with the aid of new “tools”, have we been able to peek out and learn about our nurturing blue home and its proper context within the vast unknown of the outside Universe.  

At least as important as any particular discovery we’ve made is the realization that the Universe is seemingly accessible --- by performing experiments and applying rational arguments, we can actually learn about its history and evolving nature.  Here, I'll try to outline the modern understanding of the workings of the Universe.

In my mind, there are at least 3 high-level goals one should pursue while reading something like this:

You should emerge on the other side with some appreciation of your cosmological context, your relative place in the Universe at large and the real relationships in scale between the cosmos as a whole and its constituent parts.  We tend to live our lives ignorant of the realities and relationships of nature on scales very much smaller than we and those much larger (and not just spatial scales, but timescales too; some things happen very much more quickly or slowly than is easily observed).  You will hopefully develop an appreciation for these frameworks that bind all objects and processes together into one intimately woven history and future.  To paraphrase Neil deGrasse Tyson, we are a small part of, and live in, the larger Universe; it's important to understand, though, that the Universe and the products of 14 billion years of cosmic evolution are also within us.  We are a consequence of what has come before and what continues to happen around us.

An understanding of the physical scales involved should also be paired with an understanding of the evolution of the Universe in time.  A crucial point is that, as a consequence of the natural laws we've discovered, the Universe must change over time.  It is not the same today as it was yesterday, and is certainly different from a few billion years ago.  If it really evolves, where’s the evidence?  Shouldn't these changes be observable somehow?  Yes!  Read on...

The engine that drives our understanding about how the Universe behaves is the process of science.  You will (hopefully) develop an appreciation of this human enterprise while keeping in mind not only the chains of logic and inference we use to make sense of the cosmos around us, but the assumptions we tacitly use to learn about the Universe.

For example, to do science, we typically make the following assumptions about the Universe at large:

The Universe does not operate via "Magic"

There is a consistent set of underlying rules and principles that governs the behavior of all objects or phenomena.  To investigate something scientifically means, in part, to believe that it can be explained as a natural consequence of either known laws or new principles that may extend the known ones.  

Humans can understand these rules

This is not as obvious as it sounds.  Naively, we might think that we can understand any set of rules of behavior, but let's be cautious.  You'll learn, for example, that we've largely solved the mystery of how the moon orbits the earth; it is, in fact, the same force that causes apples to fall out of trees.  We think we understand that.  But just try explaining that to a dog.  It's not just that they're not interested --- they're not.  More fundamentally, the dog's brain is made of the same stuff as ours is, as far as we know, but it seems safe to say that dogs are somehow limited in their capacity to understand the Universe compared to a human.  Perhaps the number of neurons and connections in our brains is sufficient to appreciate abstract concepts that might escape the dog.  This works all right for our current model of gravity, but you could imagine some natural process operating through a set of rules that are far too abstract or complex for us to understand.  There may be creatures elsewhere with fantastically more complex brains that can understand these new rules, but we might forever find them incomprehensible.  Perhaps quantum mechanics, the description of the rules followed by very small particles, is already fundamentally beyond our conceptual understanding.  In any case, all I'm saying is that just because we assume everything operates according to rules doesn't mean that we will be able to always understand all the rules that govern the Universe.  To do science, though, we assume that we can. 

The same rules apply everywhere

This breakthrough is commonly first attributed to Galileo and Newton.  It is tremendously important since it allows us to perform experiments here on Earth and then assume that the same rules and processes apply in the same way even in the most distant galaxy.

Now, it's possible that at least one, or all, of the above assumptions are not true.  We still don't know, for example, if consciousness has a purely mechanical or physical explanation.  To investigate it scientifically, we have to assume that it does.  Even if such a model for consciousness is developed, it's important to keep in mind the following caveat regarding scientific theories:

No scientific theory can ever be proven "true"

(Really, I like to say “model” rather than the overused “theory”).  They can certainly be proven false, but never true. The best we can ever say is that all the available evidence supports a given model.  Indeed, as we’ll see, most initial models that have ever been developed have yielded to modification and replacement over time.  New models are ever more precise and can better explain new observations gathered over longer times and with better technology.  We'll see many examples of this as we go along.  People often make mistakes on both sides; to say that some model is “just a theory” (evolution, gravity) is ignorantly dismissive of the evidence in favor of its wide adoption (a theory in science is much stronger than a “guess” in popular language. A reasonable “guess” in science is called a hypothesis.)  But to assume that a given theory is “true” in some ultimate sense is also naive. Any theory is subject to further testing and refinement, or perhaps replacement; however, new theories are still constrained to be consistent with everything we’ve observed so far.

Evolutionary history of mammals

Finally, I should draw a distinction between “fact” and “theory” since they are frequently confused in popular media.  Facts are observable outcomes of experiments.  Anywhere you go on the surface of the Earth, a dropped object will fall towards the center of the Earth.  You can even precisely measure how objects fall, at such-and-such a rate, and find that it is roughly constant no matter where you go on the surface of the Earth.  

Newton noted that the path of the Moon around the Earth also seems to obey the same general principle.  We imagine now that it “falls” towards the center of the Earth.  As we accumulate facts, then, we then may generate some abstract set of rules that explain these observations.  In this case, that would originally be Newton's gravitational theory.  It generalizes these facts into a unifying mathematical framework that is able to reproduce the observations and then further predicts the outcome of some new set of observations, like the motion of the planets around the Sun.  Later we might find that there are facts that contradict our theory, so it will need to be modified.  Einstein's modification of Newton's theory is just such an example, as we'll see.  So there are the “facts” of gravity and the “theory” of gravity that knits the known facts together in an explanatory model.  In another familiar case, there are the facts of evolution (the genetic code of organisms change from generation to generation, organisms compete with each other for limited resources in a changing environment and may pass on to their offspring any traits that give them an advantage), and the theory of evolution attempts to explain these facts using concepts such as genetic mutations and natural selection.

Writing an eBook

Part of the reason for this blog's existence is to support my writing of an introductory astronomy book for use by my students (and anyone else interested).  My intent is for it to be published in an eBook format, probably for iPads, and simultaneously available in roughly the same form as a series of posts on this blog and pdf for download.  The advantage to the eBook format would be portability and the ability to see the pertinent animations and play with the physical simulations that will be embedded as small widgets.  

It's long been my soapbox rant that in education we seriously underuse computing technology if we only employ it to show static pages.  Computers are so fast and powerful these days that, in my opinion, the best way to employ them is to harness their ability to interact with the reader by the use of animations and playful simulations.  So I suppose the logical step is to stop preaching about it and actually get down to fulfilling the rhetoric.  

The text, in some kind of order, will appear on the "Astronomy" page; this page will mostly be used as support in noting the "making of..." certain aspects as well as other bloggy topics of things I'm interested in outside the scope of the text.  I'd be very appreciative of any constructive criticism along the way to make things better or more understandable.

One of the daunting challenges in taking on a solo writing project (especially one in which the visual elements are so critically important) is creating the illustrations, both static and dynamic.  For the static pictures, I’ll be using primarily Blender, a free 3D modeling/rendering/animation program.  I’ve worked with it on and off for a while and am always amazed at the results one can achieve with a little tinkering.  One of the images for the introductory chapter is here, “Million”, illustrating visually something that is important to understand for the rest of the text —- the scale of commonly used numbers.  It’s hopefully clear here, after inspecting the image, that a million is a thousand groups of a thousand things.  It’s my opinion that an illustration has the greatest chance to be instructive if it’s aesthetically appealing; an image that people want to look at and draws their gaze has a much better chance of inspiring curiosity.  The banner image on this blog's home page is another example --- it's so easy to tell students that the Sun is about 100x bigger than the Earth, but it's much more impactful to show them in a way that is both accurate and perhaps memorable.

A Million Little Boxes