Worksheet 9 Problem 1

1. The spatial scale of star formation: The size of a modest star forming molecular cloud, like the Taurus region, is about 30 pc. The size of a typical star is, to an order of magnitude, the size of the Sun.

While this is a relatively simple problem, I think it is very important in helping us (attempt to) conceptualize the incredible vastness of space. As such I have included it in my blog posts for this week. 

Part of the Taurus Molecular Cloud

 (a) If you let the size of your body represent the size of the star forming complex, how big would the forming stars be? Can you come up with an analogy that would help a layperson understand this difference in scale? For example, if the cloud is the size of a human, then a star is the size of what? 

We can start by drawing the two different "regions" to allow us to set up a proportion of their size. The first region will be the Taurus region with a diameter of 30 parsecs and the second region will be a human who is 180 cm tall (~5'11"). 

We to find how big a star the size of the sun is relative to the Taurus region in terms of the human body. Thus we are looking for something on the human body that would be as large as one star. Could that be an eye? A hair? A cell? We know it's probably pretty small, but finding out just how small requires setting up a proportion and solving.

We now know that the diameter of one Sun would be approximately 3 nanometers if the Taurus region were the size of the human body. Clearly 3 nanometers is a very small number. Let's put it into some context. The diameter of a cell is \(3\:\times\:10^{-3}\:cm\) and the diameter of an HIV virus is \(10^{-5}\:cm\). The Sun would be even smaller than that! It would be around the size of globular protein \((4\:\times\:10^{-7}\:cm\). That's incredibly small! Our sun isn't even a speck of dust when compared to the Taurus region, it's beyond microscopic. Now consider that the Taurus region only one of thousands of star forming regions, each with thousands of relatively tiny suns, in our galaxy alone. The mass and volume of our Sun aren't even rounding errors in the overall mass and volume of the Galaxy, let alone the universe!

b)Within the Taurus complex there is roughly \(3\:\times\:10^{4}M_{sun}\) of gas. To order of magnitude, what is the average density of the region? What is the average density of a typical star (use the Sun as a model)? How many orders of magnitude difference is this? Consider the difference between lead \(\rho_{lead}\:=\: 11.34 g\: cm^{-3}) and air \(\rho_{air}\:=\: 0.0013 g\:cm^{-3}\ ). This is four orders of magnitude, which is a huge difference!

 Let's start off by quantifying all of the necessary values that we will need to solve this problem. Density is equal to mass divided by volume. Thus, in order to calculate the density of the Taurus Region and the Sun we need to know their respective masses and volumes which are as follows:

Now we can use these values to easily calculate our desired densities and then analyze them.

We see here that there is a huge discrepancy between the density of the Taurus region and the density of the Sun. The sun, even though it contains so much gas, is slightly more dense than water and has about one-tenth the density of lead. The density of the Taurus region is smaller by a factor of \(10^{22}\)! Considering that the 4 orders of magnitude difference between air and lead is pretty significant, this is astounding! But how is it that the Taurus Region has such a low density even though it's made up of hundreds of thousands of stars similar to our sun? The answer is that it is not of uniform density, and therefore not one giant 30 parsec star. The stars are separated by vast distances, leaving an unfathomable amount of empty space in the region. Just like in the previous part, the calculations for this problem were rather simple, but the results are incredibly significant. The Universe has an unimaginably large amount of empty space separating celestial bodies. This creates a vast expanse of, well, space. I find this ultra-low density to be fascinating and helpful in understanding cosmic distances.