Earth Size

Earth Size
Relative Size of the Sun and Earth

Monday, January 27, 2014

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.