Starstruck: November 2011

Friday, November 18, 2011

And again...

http://inspirehep.net/record/928153/files/arXiv:1109.4897.pdf?version=2

I'm still skeptical....

What does it mean to be a professional astronomer?

When I think of being a professional astronomer, I can’t help but keep thinking about how amazingly awesome it would be to go observing! After our class trip to Palomar, I want nothing more than to have a project where I can go observing. Which, of course, requires that I think up a project and get telescope time. But after that…… J

The process to become a professional astronomer, as far as I know, is basically constrained to academia. Professors and staff scientists of universities/ organizations such as CfA are the only people who can actually participate in this process, as it helps tremendously to be associated with a telescope-owning institution to study the sky. In this vein, to become a professional astronomer, one would need to follow the path of being an undergraduate -> grad student -> postdoc -> assistant professor -> professor. Each step of this transition needs something new. To become a grad student, I need to do research as an undergrad to learn research skills, and get good grades and test scores so I can get into a grad school. To become a postdoc, I would need to win a fellowship. I am unclear if this is the only way to become a postdoc. I’m also unsure how one goes about becoming a professor – I assume that if you are a really good postdoc, then you can apply for jobs and if you’re lucky, you’ll get one?

Also, your job can limit the science you can do. As a undergrad, I cannot apply for telescope time, unless someone qualified submits the proposal for me and I am listed only as a coauthor. Something I have learned in that last few months is how extremely important knowing people is. Who you know is almost more important than what you know. Working with people (or, in my case, learning from people) is one of the most important ways to grow as scientist. No matter how many papers on adaptive optics I read, I could never have gotten the same emotional fascination with them nor rough understanding of them that I did from listening to people explain it.

There also seems to be a lot of luck in becoming a professor – if the system you are studying happens to contain some crazy new structure, it could boost your career hugely. However, this also requires you have enough skill to recognize an opportunity when you see it. I have met a few postdocs who are doing such interesting projects, and being so successful with them, that I can’t believe they aren’t professors yet. This shows how high the standards are in the field. I think that rather than it always being the researcher who is the BEST who become a professor, there is an element of luck to it. There is not a definite measurement of being the “best” researcher, after all.

Like any field, I think the most important things to becoming a professional astronomer are connections and luck, with a necessary amount of skill to take advantage of the luck you have. More skill can offset a lack of luck, but luck cannot offset a lack of skill.

Sunday, November 6, 2011

The Formation of Stars

Introduction

Here we consider the time scale of star formation and stability. We do this by considering the time it would take a cloud of particles to collapse into a star, and in this process derive an approximation for the Jean’s Length.

This is the system we must consider. The grey cloud is all the dust orbiting the point mass. The test particle is at the far right edge of the red line. The red line is the semimajor axis of the ellipse. The test particle must travel along the red line to reach the center. The time it takes to travel this distance is the free fall time.

Solution

a. Consider a test particle in an e=1 “orbit” around a point mass.

An orbit with an eccentricity of 1 would be a straight line. That is, it would mean that that the planet would orbit back and forth straight across the point mass:

If the test particle starts at the edge of the orbit - that is, r away from the point mass, then the distance it will have to fall is equal to r (the semimajor axis of the whole clous - see image above!)

Kepler’s third law, from memory, is: $\bg_black dM = p(r)4\pi r^2dr$

The expression for mass of a sphere, in terms of its density, is: $M = \frac{4\pi r^{3}\bar{p}}{3}$

Combining these two expressions, we can solve for the free fall time by considering the period. Since the period, p, is the time it would take the test particle to “cross” over the point mass twice and return to its initial position, one half of the period will equal the free-fall time. Also, we can substitute 2a for r (see the diagram above).

$\frac{a^{3}4\pi ^{2}}{Gp^{2}} = \frac{4\pi r^{3}\bar{p}}{3}$

Solve for p (period. Note that this is different from the symbol for density, which is p-bar):

$\sqrt{\frac{3\pi a^{3}}{G r^{3}\bar{p}}} = {p}$

We know the relationships between t_ffand p, as well as r and a:

$\frac{p}{2} = T_{ff}$ $2a = r$

$\sqrt{\frac{3\pi a^{3}}{G (2a)^{3}\bar{p}}} = {\frac{p}{2}}$

$\sqrt{\frac{3\pi a^{3}}{8G a^{3}\bar{p}}} = {p}$

$\frac{1}{2}\sqrt{\frac{3\pi}{8G\bar{p}}} = T_{ff}$

$\sqrt{\frac{3\pi}{32G\bar{p}}} = T_{ff}$

This expression describes the free fall time of a particle on the edge of the cloud, and this the time it would take the cloud of dust to collapse into a star. We are assuming that the test particle experiences exactly one half a period of motion.

b. Determine the dynamical time for a sound wave to cross this same distance.

Defining the speed of sound as c_s, we can define the time it would take a sound wave to cross the same distance as a test particle as;

$t_{s} = \frac{2a}{c_{s}}$

We can solve for the length variable – that is, a.

$t_{s} = t_{ff}$

$\frac{2a}{c_{s}} = \sqrt{\frac{3 \pi}{32G\bar{p}}}$

$a = \frac{c_{s}}2{}\sqrt{\frac{3 \pi}{32G\bar{p}}}$

Which is equivalent to:

$R_{j} = \sqrt{\frac{3 \pi c_{s}^{2}}{128G\bar{p}}}$

This length signifies the distance at which the forces of gravity and pressure are at an equilibrium. The gas cloud will neither expand nor contract at this size.

c. What is the significance of Jean’s Length?

Jean’s Length shows the distance at which a cloud of dust would not be stable, and thus susceptible to collapse. The force of gravity attempts to pull the dust into the center, while the force of pressure tries to push it back out. If a sound wave takes longer to cross the distance than the actual particle of dust, then the force of pressure won’t be strong enough to prevent the dust cloud from collapsing. So, if a dust cloud is larger than Jean’s Length, that means that it cannot sustain itself, and will collapse into its center, resulting in stars!

d. Consider a collapsing cloud with a radius equal to Jean’s Length: when the cloud reaches a radius equal to one half of its initial radius, by what fractional amount had R_Jchanged?

To see the ratio of the Jeans’ Length for a cloud of equal mass but different radii, we must consider how the density changes from one size to another. So, first, we can find the densities of the two clouds, assuming a constant mass.

$\bar{p_{o}}=\frac{M}{\frac{4}{3} \pi R_{o}^{3}}$ $\bar{p_{J}}=\frac{M}{\frac{4}{3} \pi R_{J}^{3}}$

So:

$\bar{p_{J}}\frac{4}{3} \pi R_{J}^{3} = \bar{p_{o}}\frac{4}{3} \pi R_{o}^{3}$

$\bar{p_{J}}R_{J}^{3} = \bar{p_{o}} R_{o}^{3}$

We can plug in 0.5R_O and see how R_Jchanges. As R_o shrinks, the density p must grow by a balancing amount.

$X\bar{p_{o}}(0.5R_{o})^{3} = \bar{p_{o}} R_{o}^{3}$

$X\bar{p_{o}}0.125R_{o}^{3} = \bar{p_{o}} R_{o}^{3}$

$(8)\bar{p_{o}}0.125R_{o}^{3} = \bar{p_{o}} R_{o}^{3}$

So the density increases by a factor of 8 if the radius decreases by a factor of 2.

Thus,

$R_{J} =\sqrt{\frac{\pi c_{s}^{2}}{G \bar{p}}}$

If R becomes 0.5R_o then the density becomes 8p:

$R_{J} =\sqrt{\frac{\pi c_{s}^{2}}{G \bar{8p}}}$

$R_{J} =\frac{1}{^{\sqrt{8}}}\sqrt{\frac{\pi c_{s}^{2}}{G \bar{p}}}$

So, R_jdecreases by a factor of 0.35. This is fractional change of about 65%; the new value of R_J will be around 65% of the old one.

Conclusion

Jeans’ Length, the point of balance between the forces of pressure and gravity, can help you determine the behavior of a cloud of dust. By knowing the radius and mass, or by knowing just the density, you can calculate whether the cloud will expand or contract. If the cloud contracts, then it will likely become a star. Decreasing the radius of a cloud by 50% decreasing the Jeans’ Length by 35%, so the new length is 65% of the old one.

Acknowledgements

Thanks to Iryna and Monica! We had a lot of fun with this problem :)

Look what I found!!

So, as I attempt to do the write-up for the most recent worksheet, I instead find myself procrastinating by reading Astrobites... and look what I found!!

http://astrobites.com/2011/08/15/qa-with-john-johnson-what-should-you-do-in-graduate-school/

:) There are lots of other articles about grad school, both about getting in and what to do once you have gotten in, but I thought you guys you especially like this one!! For obvious reasons...

Here is one comment that he made regarding classes in graduate school, one that I think applies to us undergrads at Caltech: Professor Johnson explains that it was a bad thing that "I was just concentrating on getting the current week’s problem sets finished so I could get to the next one and then finish the course with a good grade." This is something I have been thinking about a lot the last few months, mostly inspired by the way Ay20 is run. I think I lot of people get through Caltech this way, but we don't have to!

Astrobites is actually pretty cool! There are a lot of contributors, resulting in very diverse reading! .... and many hours worth of procrastination... I started my write-up at three.... oops....