Two-Body Orbits: Where's The Centre of Mass?

Sunday, November 20, 2011

Second Authors: Nathan, Lauren

Consider a problem of a planet orbiting a star.  It's easy to see from Newtonian gravitation that they exert a force on one another.  But then, how, according to Kepler's Third Law can we say the planet orbits the star alone?  The star cannot remain fixed while feeling a force.  In order for Newton's Laws to hold, we must say that they both orbit a mutual centre of mass.  Using conservation of momentum, we can determine how far each body truly is from this centre of mass.

In order to balance forces, we notice that the planet and the star must be at opposite ends of their orbits at all times as seen in the following picture (not to scale).  It follows from this that they have the same orbital period and the same angular velocity.
We know that linear momentum is equal around the centre of mass, such that:

Dividing through, we get the relationship

Rearranging and using the mean semimajor axis, a, we can see that

We have shown here that star does indeed orbit the centre of mass, just as the planet does.  However, looking at these equations carefully, we find that except for very massive planets, the semimajor axis of the star's orbit is roughly zero and the semimajor axis of the planet's orbit is roughly equal to the mean semimajor axis.  As a result, we find that we can in fact use the assumptions implicit in Kepler's Law.

The Death of a Star

Second Authors: Nathan, Lauren

We know how stars are formed to a certain extent and that while they are on the main sequence, they are supported by hydrostatic equilibrium. However, when a star moves off the main sequence and can no longer support itself, what happens? We assume that at this point, the core of the sun has converted all of its mass to energy and is now undergoing gravitational collapse.

We know that the Sun generates energy throughout its lifetime at a rate of:


Assuming that the sun uses up the entire mass of its core (10% of a solar mass) as it undergoes fusion, and converts energy with 0.7% efficiency, we can determine the total energy it produces in its lifetime with


Dividing this number by the rate of energy production, we can determine the time it takes for the Sun to use all of its mass available for fusion. This time is


Now we know the core will collapse, but it won't collapse indefinitely.  We find that the core collapses to the point that the interparticle spacing is on the order of the De Broglie wavelength.  Since it is easy to see that electrons have greater momentum compared to protons of equal energy, electrons are the first to reach this critical density.  We can calculate this using the equations:

It is easy to tell that we have one molecule per cubic lambda.  So we have:

The actual value is 8 times this, for reasons I can't remember, but Nathan tells me Professor Johnson said the factor of 8 was okay to include in our calculations.  So multiplying this by the mass of a hydrogen atom and using T = the temperature of the sun's core, we get density

Which is more than twice the current maximum density of the sun's core.

We find that by converting 0.7% of the mass of the sun's core into energy, the sun's lifetime is roughly 10 billion years, which agrees with what scientists have predicted.  The density of the core after collapse will also be far greater than the current density of the sun's core, which is reasonable or it would not be able to support the sun post-collapse.

An Interview With an Astrophysicist: The Postdoc

Friday, November 18, 2011

As some readers may know, this past summer I had the privilege of being able to work for Dr. Andy Goulding at the Smithsonian Astrophysical Observatory.  Dr. Goulding is a first year Smithsonian Research Fellow in the High Energy Astrophysics division.  He did his PhD work in AGN activity at the University of Durham, UK in the fall of 2010 before moving to Boston as a postdoc.  Working with him was a wonderful educational experience that I'd happily do over if I had the chance.  He gave me a look into the life of a researcher and how much research differs from course work.

We've vaguely kept in touch since September and he generously agreed to answer a few questions on his career for me.  Of course, given the title of this blog, the first question was obvious.  He took a lot of time in answering these questions, and I definitely learned some new things.  For example, I had no idea that there was a difference between a postdoctoral research associate and a fellow.

What is the difference between an astronomer and an astrophysicist at this point in time?  Which, if you have a preference, are you?

From a professional point of view, this is really semantics. People have degrees/PhDs in astronomy and/or astrophysics - it depends on institution. However, it is more likely that someone who is an amateur (non-PhD) is considered to be an astronomer. Classically, an astrophysicist attempts to understand and interpret the astronomical observations through application of physics. My PhD is in astrophysics, so in the strictest sense, I am an astrophysicist.

What are your primary areas of research as an astronomer/astrophysicist?  How did you get interested in them?
Black hole growth and galaxy evolution. Black hole physics was considered a "popular science" in the 1990s, so when I was younger it was very easy to pick up a book in the local store, and this naturally progressed from a hobby into a profession.

How did you get into astronomy/astrophysics?  What did you study as an undergrad?  Where did you go to graduate school and why?
As an undergrad I studied Theoretical Physics - this involved particle theory, supersymmetry, advanced quantum theory and general relativity - this had little to do with astronomy, so it does not necessarily follow that your undergrad major must be your graduate major. Despite offers from several graduate schools in the UK, I decided to stay at my undergrad institution, Durham University, as it has a fantastic worldwide reputation as well as the largest astrophysics/cosmology department in the UK.

What precisely is a postdoctoral fellowship?  How does it fit in to a career in astronomy/astrophysics?
Once a predoc has completed their PhD, they will generally look for a Post-doctoral position at a different institution - these come in 2 flavors: (1) research associate and (2) fellowship. A research associate position is when a group/faculty has money available to employ a post-doc to carry out their specific research and help pre-doc students within the research group. A fellowship is generally a highly sort-after monetary prize which may or may not be linked with a specific institution and allows the holder to carry out the research of their choice - this is often predicated by the research proposal which is submitted in order to win the fellowship.

How has your career played out?  Is it what you expected?  What is the typical career arc of an astronomer/astrophysicist?
At this stage, I am not really in a position to answer this question. I completed my PhD in late 2010, and moved to Harvard shortly after to begin my fellowship which I had been accepted for earlier in the year. As I have only been here for a little over 12 months, it is not really possible to answer this question. 'Typically', an astrophysicist will expect to go from grad student (3-6 years) -> post-doc associate (2-3 years) -> [post-doc associate (2-3 years) ->] post-doc fellow (2-3 years) -> [post-doc fellow (2-3 years) ->] associate professor (3-10 years) -> tenured professor (indefinite). N.b., I added the other post-doc positions as some people prefer to stay as post-docs for longer periods of time to help with their publication records to move to the next stage and/or keep their teaching duties lower.

How have your goals evolved over the course of your career, if they have at all?
As I said above, my career is still becoming established. As I skipped the post-doc associate stage, and gained a fellowship on my first position, my current goal (for next year) will be to win a second prize fellowship to further expand my publication record.

If you hadn't gone into academia, what would you be doing with your education in astronomy/astrophysics?
My Masters degree is in theoretical physics, many people with this degree become statistical analysts and/or military defense specialists.

What is the best part of being an astronomer/astrophysicist?  The worst?
This is quite an interesting question as I'm pretty sure that you will get a different answer from every person. Being an astrophysicist is all about 'puzzle solving', we all have some intrinsic desire to answer the questions which are the most difficult to answer, so when we answer them, or move a step closer to answering them, this is the 'best part of being an astronomer'. However, it can be worst, you can find yourself working on one project for 6 months and then finding out that you have gone down completely the wrong avenue, and you have to start over. Of course, there are certainly perks too - for example, we travel to very exotic places (a lot), telescopes ares not generally in well-populated areas so you get to travel to places like Hawaii, the Canary Islands, Australia, the Atacama desert (Chile).

What can aspiring astronomers/astrophysicists do to make things easier for themselves?  i.e., what do you wish you'd known as an undergrad?
Of course, get very high grades and after that, you need something on your cv that will get you noticed (e.g., a summer studentship in a department)

What has been the most difficult stage of your career so far?  What have been some notable inspirations along the way?
At this point in time, astrophysics is struggling for funding from the government due to certain projects costing significantly more than was originally budgeted for, as such further funding money which would be relatively easy to propose for 5 years ago is not forth-coming. Hence, much more time must be put into proposing for the next year's projects, and this slows down the current research. This is not necessarily 'difficult' but it is certainly frustrating.

Any final thoughts for the undergraduate astronomy student?
Astronomy as an undergraduate student and astronomy research are nothing alike (as you have already seen, Eric).

Star Formation: Timescale and Stability

Sunday, November 13, 2011

Star formation is governed by the collapse of a cloud of particles into a gravitationally bound sphere which we call a star.  The radius of the could at which this occurs is called the Jeans Length, where the gravitational force of the cloud overcomes the thermal energy causing it to expand.  Here we examine the time scale of such a collapse and also calculate the Jeans Length.

In order to determine the time it takes for this collapse to occur in terms of the mass and size of the cloud, we consider a cloud of mass M and a test particle a distance away from it.  We assume the cloud has a mass given by

where r is the length of the major axis for an elliptical orbit of eccentricity 1.  By assuming such a geometry for the free fall, we can initially approximate the orbit to a straight line with a  mass M at one end and our test particle at the other.  Since this is a free fall, we can also approximate the time tff to be half the orbital period we get from Kepler's 3rd law (a = 1/2 r)
Substituting our mass formula into this equation, we get
The implicit assumptions are that we can even call this half an orbit, as an eccentricity 1 orbit is parabolic and therefore not periodic, and that we can approximate this orbit to a straight line.  Now in order to find the Jeans Length, we equate this to the dynamical time, or the time it takes a sound wave to cross this distance.  Let's define this as
Equating the two, we get the radius at which the cloud will undergo gravitational collapse
For an isothermal gas of constant density, this length signifies the minimum radius at which it will continue to be a gas and not collapse into a much denser formation.  This is the Jeans Length to an order of magnitude.  The actual formula for the Jeans Length is

We have hear calculated the free fall time for star formation as well as the radius at which the gravitational force between interstellar dust particles takes over.  It is important to note that since the density is radius dependent, the Jeans Length is not constant for all star forming clouds, but varies even with the change of radius due to collapse and we have
If we consider a cloud that starts out at the Jeans Length for its particular conditions, by the time it reaches half this radius the Jeans Length has decreased by a factor of √8.  As a result, the initial Jeans Length may actually govern how far the cloud will collapse for a given mass and radius.

Becoming An Astronomer

Saturday, November 12, 2011

We were recently, or not so recently—I'm very good at procrastinating—assigned a multi-part blogging to find out what it truly means to be an astronomer.  I realise as a sophomore astrophysics major that I still don't understand the specifics of what being an astronomer entails or means to me.  To quote my friend Alexa,

I just want to be one. So much. “Space” is, if you think about it, everything but Earth. When we study it, we’re pausing our narcissistic tendencies for just a moment. We’re not everything; we’re part of everything. Ignoring that is shameful.
She stated in the best way possible what attracts me to astronomy, but that still doesn't mean I know what astronomy is.  Right now I just think of astronomy as some nebulous loosely defined field of Things I Would Like To Do Because They Are Amazing, but that's not an acceptable answer to the question.  So without further delay I shall attempt to synthesise my thoughts on the topic.

The point of being a professional astronomer, in my experience, is to contribute understanding of what the universe is, how it is structured, how it came to be, and what its future might hold.  Most likely this is because the first astronomer I ever met was a professor of cosmology.  I've come to accept that he's probably the reason my main research interest tends to observational cosmology.  Of course, this is a very broad and relatively unhelpful answer to the astronomer question.  Sure, that's the intention, but how do we get there?

I think it's safe to assume that the journey to becoming a professional astronomer begins as an undergrad, or if you're very lucky, as a high school student.  I think mine was a bit of both, as I did get the opportunity as a junior to do some busy work for he of blog title fame.  But that was a week long, and although it was some exposure I doubt it's how careers in astronomy start.  Careers in astronomy, at least for a Caltech student, probably start with a SURF fellowship.  I know SURF was my first real look at what an astronomer does.  I sat alone in an office 8-9 hours a day, writing code in a language I'd never seen before to analyse data I didn't understand, and I had fun doing it.  I think that enjoyment that's what sets the astronomer apart from the average person.

Then the natural course of things is to go to graduate school.  This is where you decide What You Want To Do With Your Life.  As far as I know, you don't have to decide right away.  Unless you're in the UK in which case you need to know what you want to study before you've learnt anything about it.  At least, that is what my SURF mentor who is English tells me.  Being a grad student requires doing semi-independent research under the guidance of a faculty member who works on a similar topic.  You'll probably start to hate your field at some point during this process, but hopefully you'll get over it soon.  Next is the postdoctoral fellow.  I have no idea what a postdoc does.  Don't tell my SURF mentor that because he is one.

I believe your career options then become a) professor at an academic institution, b) research scientist some place like the SAO, or c) finance.  I'm sure there are more options, I'm just uneducated in that side of things.  The first two options strike me as pretty similar except the professor track astronomer will probably have to teach at some point or another.  This part is where you get to move on to independent research in topics that interest you.  You might find out something that only you know about, and that's a rewarding experience.  Although any work in astronomy is rewarding if it's what's truly exciting and inspirational to you.

I have given my impression of what it takes to become an astronomer.  So once again, we're back to the question of what does it mean to me to be an astronomer?  It's going to take a lot of work.  Astronomy, as it turns out, is hard.  But the work will be worthwhile because I'll be learning how the universe works, or how we think the universe works.  Maybe I'll end up amending some of that knowledge.  Who knows?  Being an astronomer means getting excited about the mysteries of space and our tiny place in it.  It means realising how small we really are in the grand scheme of things, accepting that, and moving on to understand why.  Most of all, it means that when your friend starts talking about M83 and means the band, this is all you can see.

Is There Life On Maaaars?

Sunday, November 6, 2011

I've been having an uncharacteristic moment of curiosity lately, and that curiosity is about life outside Earth.  Usually I don't care.  I'm much more of a "let's explore and discover the physical laws of the universe" kind of guy.  But today, it's all about life out there, and why not?  Some pretty interesting things have happened in the last week.

1. ESA's Mars500 Simulation Ended
So I have to admit, I knew nothing about this project until I read the article today.  Doesn't prevent me from thinking it's amazing.  In short, a crew of 6 was stuck together in an in-lab "spacecraft" for 17 months, performing the tasks necessary for a real mission to Mars including "entering" orbit and "landing" on Mars.  Conditions were controlled exactly as if they were actually travelling and they completed experiments on the problems brought about by long space missions.  Maybe this will open up opportunities for an actual space mission to Mars after studying the physiological and psychological effects of longterm isolation.  Very cool.  Here is a compiled video diary of their time during the simulation:

2. A New Way to Look for Aliens
Avi Loeb and Edwin Turner of the Harvard-Smithsonian Center for Astrophysics and Princeton University, respectively have suggested a new way to look for extraterrestrial intelligence: doing it the same way we find civilisation on earth.  They intend to look for the lights from their cities.  These two operate on the assumption that life evolves in the light of the nearest star and that any intelligent life forms would have learned to make light and extend their days.  They would have to find a way to filter out the light from the star.  They suggest that one method of doing this is to look for bright areas in a dark phase of the planet's orbit (think of the dark side of the moon).  Unfortunately, this method would require far more powerful telescopes than we now have, but it's definitely a start.

3. Organic Molecule "Sweet Spots"
This isn't technically, astrophysics, however I think it still has a place in a post about life outside Earth.  Astrobiologists at Rensselaer (one of the reasons I didn't apply there was I couldn't spell it on the first try) have discovered areas of higher methanol concentration surrounding some, but not all, newly formed stars.  Methanol is apparently one of the precursors to more complex organic molecules which may give rise to life.  They call this a "sweet spot" of physical conditions that allow these organic molecules to form.  Even more interestingly, from studying concentrations in comets, they have determined that our solar system is painfully average in the methanol department.  In other words, we're not all that special and life still managed to appear on earth.  The implication here is there may be other solar systems out there with greater methanol concentrations that lend themselves more easily to the appearance of life than our own!


Hydrostatic Equilibrium and the Sun

We would like to know how the sun is being "supported".  We assume that this mechanism is hydrostatic equilibrium, but to be sure we work through the derivation.

We know that the sun is somehow being prevented from gravitational contraction.  Our theory is that it is supported by hydrostatic equilibrium, which means that the internal pressure provides an opposing support force.  We calculate the gravitational force on a mass shell, the pressure required to balance it, and then derive the force equation for hydrostatic equilibrium.

Methods and Results
We first assume the Sun to be a spherical gas cloud with density ρ(r).  We consider a differential mass shell of this sphere with radius r.  We recall that the volume of a sphere is 4/3πr3 and that the differential volume is its derivative. Then we get a differential mass dM:

We know the equation for universal gravitation:
Here we let M be the total mass enclosed by the mass shell and m be the differential mass element.  As a result, we get the differential gravitational force to be:
We know that pressure is equal to force divided by area.  So we can say:
Now dividing by dr on both sides of the equation we arrive at the equation of hydrostatic equilibrium:

We have derived from simple physical laws that the equation for hydrostatic equilibrium is a plausible explanation for the way the sun is supported.  A quick search shows that we are indeed correct.  Hooray!