The Spatial Scale of Star Formation

In this post I'll be doing #1 from this week's worksheet on star formation. The question itself is not that difficult, but its brief order of magnitude calculations give a good sense of the vast sizes involved in star formation. Here's the problem:

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:

a) If you let the size of your body represent the size of the star forming complex, how big would the forming stars be? Is there a biological structure in you that is roughly that size?

b) Within the Taurus complex there is roughly 3 x 10^4 solar masses 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 (11.34 g cm^-3) and air (0.0013 g cm^-3)

 Let's get some unit conversions first. we know that 1 parsec is about 200,000 AU, and 1 AU is about 100 solar radii. So, we have:

If we let the size of a star forming cloud to represent 1 body, we have:

Thus, the scaled size of a typical star is about 3 nanometers, which is about the diameter of a DNA helix.

Now, let's estimate the average density of the star-forming region. We will assume a spherical distribution of gas. The average density is given, then, by:

The mass of the cloud is about 3 x 10^4 solar masses:

The size of the cloud is 30 parsecs, so its radius is 15 parsecs:

Thus, the average density is:

The average density of a typical star, on the other hand, is:

The average densities are 21 orders of magnitude apart! Compared to the difference between lead and air (values given above), this difference is huge, highlighting just how sparsely distributed star-forming gas clouds are.