Because Of Science

Tuesday, December 20, 2011

Note: This post does not have so much to do with astronomy as it does with my feelings about astronomy and how my feelings have evolved.  It's going to read very much like a personal blog post.  If this does not interest you, stop reading here.

I talked to John Huchra today.  It's probably fairly obvious that this was a rather one-sided conversation, but such is life.  I wasn't going to mention him on this blog, but I think it's more appropriate now that it is no longer an ongoing class assignment.  And I can't write an astronomy blog without talking about him, seeing as he's the reason I'm studying the subject.

I'm told I first met Dr. Huchra at a party when I was an 8th grader.  We met by chance.  He was the married to the friend of a friend of my father's.  I don't remember this meeting at all.  The first time I remember meeting him, I was asking him for a job of sorts.  My high school had an institution called project week in which we all took a week off in January to do something exciting in a field that interested us.

So in January 2008 I found myself shadowing Dr. Huchra and doing some busy work that I now doubt was actually of any relevance to his research.  It's funny; I don't actually remember much of the science part so much as sitting down and having coffee with him.  I don't even like coffee.  I have an exclusive long term relationship with tea.  I only drank the coffee because he bought it for me.  He handed me a mug, we walked down to the lobby and filled up with some sludge that claimed to be from Starbucks.  I tried to pay and he told me I could get it next time.  I can't even remember what we talked about.  That's my most vivid memory of the professor.  It's also my greatest regret—the one I haven't been able to get out of my head since last October—I never bought him coffee.

When I arrived at Caltech, I was absolutely certain I wanted to do whatever it was that he did.  Never mind that I had no idea what he actually did.  He was my mentor and I was determined to make him proud.  I'm still not entirely sure what he did.  It's a bit beyond my current knowledge base.  Anyway, this is all a bit beside the point.  I talked to John Huchra today.  I went through my SURF last summer feeling guilty that I was working in his building when he was gone.  A building that I, as an undergrad, had no right to.  I talked about that too.  It's why I've been avoiding visiting him.  I may have a SURF at the CfA again this year, though, and I had to deal with it some time.

So we talked.  And I realised that over the last term I no longer want to be him.  Impress him, yes.  Live up to his expectations, yes.  But be him, no.  If I go into his field, it will be entirely a matter of chance.  I don't know what I want to do with my life anymore.  But I do know that one day I'm going to be a real astronomer working on whatever field I find interesting.  And I have a right to work at the CfA.

An Interview With Jan Vrtilek

Wednesday, December 14, 2011

As people reading this may know, we had an assignment a while back to interview an astronomer.  As we all know, astronomers are busy people.  I got one interview back, but Dr. Jan Vrtilek at the CfA was kind enough to fill out an interview when he had time.  I first met Dr. Vrtilek several years back as the father of a high school friend.  He has since transformed from "my friend's dad" to being a colleague and I had the pleasure of talking with him several times during my SURF at the CfA this past summer.  His support and advice were valuable resources.

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

My short answer would be that there's really no difference at this point in history, and that astronomy/astrophysics is really just a branch of physics (which is why a strong background in physics is likely the optimal preparation for an astronomy/astrophysics career). But a fuller accounting may be a bit more complicated: because astronomy/astrophysics is largely not an experimental science in the usual sense (with exceptions for some areas such as laboratory astrophysics) there is an ongoing history of empirical/statistical/classification studies that can run ahead of a well-developed physical understanding; historically, such work has been associated more with the "astronomy" than the "astrophysics" aspect of the science, and some of these historical associations are still reflected in the subject areas of papers that have shown up in, say, the Astronomical Journal rather than the Astrophysical Journal. But these are small effects, and not something that I think is much worth worrying about.

What are your primary areas of research as an astronomer/astrophysicist? How did you get interested in them?
I work principally in the are of galaxy groups and clusters, with emphasis on the mechanisms of AGN feedback on intragroup and intracluster gas. The decision to work in this area was essentially 'opportunistic': when I joined the Chandra project a couple of years before launch, I had just returned to the CfA from a couple of years working at NASA Headquarters in D.C., and there was an opportunity to move from my earlier area (molecular astrophysics) into a new, potentially very exciting area related to what Chandra would do (and indeed, it turned out that Chandra has made quite remarkable contributions in this area, so it turned out very well).

How did you get into astronomy/astrophysics? What did you study as an undergrad? Where did you go to graduate school and why?
I was a physics and math double major as an undergrad at Wisconsin/Madison. As my undergraduate time was drawing to completion, I know that I would need a specialization, and thought about what area would hold long-term appeal for me. Given my liking for the two astronomy courses that I'd taken, my experience with a quite unusual summer program in high school (the Summer Science program, that has been running for a half century, and is now operated as a nonprofit by its alumni: see, and the general excitement and promise of the field, the choice wasn't difficult. I applied to a half-dozen graduate schools, and five accepted me. These were days when travel was expensive (at least for me) and there was no internet; I chose Harvard based on rather general considerations and with far less information than a modern student would consider acceptable.

How has your career played out? Is it what you expected? What is the typical career arc of an astronomer/astrophysicist?
In a general sense, it's about what I expected. I never had a single conception of what my future must be ("university professor or bust"), but wanted to be active in forefront work that I found exciting, and my career has done that quite well. Although I did my thesis in active galaxies, the most exciting offer had me going to postdocs and then staff positions at Columbia, Goddard, and the CfA in molecular astrophysics and infrared astronomy. At that point, and with finances running low, I had an interesting offer from NASA Headquarters to join the Visiting Senior Scientist program (alas, it no longer exists) and become a visiting administrator for NASA's infrared and radio astrophysics programs. I had the great good fortune of landing in a team of terrific mentors and colleagues who are still now my close friends, and was able, among many duties, to work as program scientist for COBE, which I think of as one of NASA's greatest missions. I'm not sure that I'd want to do this work forever, but for two and a half years it was a tough but invaluable experience that taught me much about the management and funding of science, but also brought me into touch with a cross-section of the most interesting people and projects of that time. Just as I was finishing up, the Chandra project, a couple years before launch, was staffing its science center, and I was very pleased to get an offer to return to the CfA, in a new life as X-ray astronomer. This also gave me the chance to select a new research area, and I chose X-ray study of galaxy groups and clusters (as described in 'areas of research' above). 
Now all of this is hardly "typical", and I suppose that if it teaches a lesson it's that while planning and goals for your career path are necessary and important, what actually happens depends on the unpredictable fluctuations of opportunity. Flexibility is a virtue.

How have your goals evolved over the course of your career, if they have at all?
My goals have always been to "do something interesting" in the field, and I've never been committed a priori to some particular, narrowly-focused area. As a result of this approach, as well as of the practical consequences of managing a two-career family, I've had several switches of subfield. (The plus to this is a thrilling ride through a broad swath of astronomy; the downside is never having that confident settled feeling, but instead the sense of perpetually racing to catch up in the latest area.)

If you hadn't gone into academia, what would you be doing with your education in astronomy/astrophysics?
Given my long association with NASA projects, I suspect that I would be likely to be working in the aerospace industry.

What is the best part of being an astronomer/astrophysicist? The worst?
I'll identify the two most positive aspects for me: 
(1) The intellectual excitement and challenge. I hardly have to expand much on this for you -- you're already in the middle of it. For me it has aspects of using physics I like in really cool situations on the grandest scale, a sense of contribution, however modest, to an enduring and worthwhile enterprise, and the pleasure of great variety. 
(2) The company of my colleagues. I have been very fortunate in my colleagues and friends. I try to be surrounded by people who are smarter than I am (I strategy I strongly encourage!), and who are also people of decency, humor, and integrity with whom I work on a basis of enduring trust and friendship. This matters immeasurably to the quality of life. 
As to the worst, there are the inevitable exhaustion, frustration, and uncertainty. But you are not likely to escape occasional encounters with these in any challenging profession, and there is no line of work that I can think of these days that will offer you a sure path to peace and security!

What can aspiring astronomers/astrophysicists do to make things easier for themselves? i.e., what do you wish you'd known as an undergrad?
I can't offer much more here than the obvious: get a really solid physics and math background. Work with a strong sense of strategy. Find good advisors and listen to them. Find talented, reliable, congenial collaborators. 
There is maybe one point at which particular care is warranted. When you're going for your PhD, your choice of advisor and research group is critical. This is a spot where problems can arise that cost you much time and frustration. You'll want to assess very carefully the project (for how workable it is and how it will place you for your future plans), the advisor (for success with earlier students), and the entire team (for the "spirit" and effectiveness of the group).

What has been the most difficult stage of your career so far? What have been some notable inspirations along the way?
I suppose the very high level of uncertainty at various points, especially early on. There was a whole series of moves at 2 or 3 year intervals, which means being in job-search mode about half the time. It came with the uncomfortable realization that there's a fair amount of luck involved; I've gone through searches with only one offer at the end, and that makes the role of chance all too clear. I can't say too much about the importance of a really positive and supportive personal life to sustain you through the difficult spots, and to help you enjoy the smooth stretches!

Any final thoughts for the undergraduate astronomy student?
Not at the moment. You have a good set of questions! If there's anything you'd like me to clarify or expand on, please just let me know.

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!

Stellar Properties From Afar (Problem 1)

Wednesday, October 26, 2011

Considering the angular diameter of the sun and the astronomical unit, we can estimate the radius of the sun, the AU in solar diameters, and the mass of the sun using Kepler's 3rd law.

Applying basic trigonometric identities and taking the astronomical unit a to be the distances from us to the closest point of the sun to us (i.e., the centre of the circle we see from earth), we can see that:

Multiplying through by a we get a value for the radius of the sun.  It is clear from here that if we divide a by twice the solar radius we can easily determine the answer to the second part of our question.  Finally, we have Kepler's 3rd law:

Where P is the period of the earth and G = 6.7 x 10-8 dyne-cm2/g2.  From here we can solve for the mass of the sun.

Solving the first equation using a = 1.5 x 1013 cm we get the radius of the sun equal to 6.545 x 1010 cm which is very close to the actual value of 6.955 x 10107 s.  Dividing, we get 1 AU = 114.6 solar diameters.  Then, solving for the mass of the sun in Kepler's 3rd law with P = 3.154 x 107 s, we have the mass of the sun equal to 2.007 x 1033 g which is a surprisingly accurate number.

Surface Temperature of Planets

Friday, October 21, 2011

by Eric S. Mukherjee, Nathan Baskin, and I forget who else (sorry).

In this problem we consider the how the temperature of the sun affects the temperature of the earth.  This is possible to estimate by assuming both the sun and the earth to behave like perfect blackbodies.

Assuming the Earth has constant surface temperature and that it behaves like a blackbody, we can estimate the surface temperature using the energy emitted by the sun.  We also assume the sun to be a perfect blackbody.  Under these assumptions we can find the surface temperature of the Earth by knowing the temperature of the sun, the radius of the sun, the mass of the sun, the mass of the earth*, and the radius of the earth.

We start with the equation for flux at the surface of a blackbody (σ is the Stefan-Boltzmann constant):

From this we derive the luminosity of the sun by multiplying through by the surface area:

Then the flux of the sun at the surface of the earth is (where a is the astronomical unit):

If we consider the area of the earth through which the flux passes, it is the circle of area π R2.  Multiplying through by this quantity we get the power input to the earth from the sun.  We then realise that this is necessarily equal to the power output of the earth due to energy conservation which, at the surface of the earth, is equal to σT4π R2.   Thus we have an equation of the form:

Using this equation with R= 695,500 km and T= 5778 K, we get T⊕ = 279 K.

This temperature that we calculate is around 5.5°C which sounds reasonable for an earth without accounting for atmospheric greenhouse effects and allowing for the temperature at the poles.  The true average temperature of the earth is around 16°C but that is measured with the warming effect of the atmosphere.  The sun is not a perfect blackbody which also contributes to the difference between our calculation and the true value.

I'd like to thank the entire Ay 20 class and teachers for collective brainpower due to the fact that I can't remember who exactly worked on this problem and I'm sure we drew from the knowledge of many people in the room.  I'd also like to thank the superior computational power of Wolfram Alpha for bringing to my attention that there exponents matter when calculating ratios and that the temperature of the earth is most definitely not 1270 K.

*Note: It has been brought to my attention by Professor Johnson that the masses of the earth and sun do not actually factor into this calculation at all unless we need them to derive some of our other known constants.

A slight belated correction on AGN

As my readers may remember, a few weeks back I posted about the properties of AGN and how they affect their host galaxies.  One of these ways I listed was star formation rate.  Actually, a bit less than two weeks ago an article was reprinted from UCSD by ScienceDaily that AGN do not stop star formation as previously thought.

Prior research showed a correlation between the presence of AGN and the lack of star formation in galaxies.  This new study claims that this was a function of observational bias.  Older, more massive galaxies are easier to detect, and are also the ones with decreased star formation rate.  This study finds AGN in all types of galaxies including those in which stars are still being formed.

Read the full article here.

Eric Answers

Wednesday, October 19, 2011

Hey all, as you may know I've been collecting astronomy questions from people.  Now I'm going to answer a few of them.

Jessica and choirqueer asked: "Astronomy vs. astrology vs. astrophysics.  What is the difference?"

I've combined the two questions for ease of answering.  Technically, astronomy has more to do with the qualitative or observational study of all objects not contained in the Earth's atmosphere.  Astrophysics is part of astronomy, but is focused on the applications of physics to astronomy and understanding why things are the way they are through physics.  The title of this blog comes from something an old friend of mine used to say despite the fact that he was, in fact, an astrophysicist.
Astrology is completely different in today's world although in antiquity astrology was astronomy.  Astrology is a belief that astronomical phenomena affect our lives as humans on earth and is widely regarded as unscientific.  I personally don't believe constellations and planetary motion have any effect on our lives as constellations are patterns that humans have assigned to stars which aren't even necessarily close together and planets are predictably governed by physics, but belief is very personal and I am not one to tell people they are wrong.

 LilyForest asked: "How noisy is the sun, assuming we could actually here it?"
I actually attended a colloquium at the CfA that dealt with this over the summer.  It was a fascinating topic.  Here's a video from ESA talking about the vibrational modes of the sun way more eloquently than I possibly could.
Basically, the sun produces "noise" due to its surface vibration.  However, this noise is generally not in the human audible range and also cannot reach us on earth.  Here is a clip from Stanford of the audio from 3 modes: Solar Sounds

 NastyNate (Nathaniel) asked: "How many total planets have been discovered and recorded in the universe?"
This is actually a question for my professor, who studies planets outside of our solar system.  However, since this is my blog and not his, according to this website which seems like a credible source run by a Paris Observatory scientist gives the current number a 694 planet candidates found outside of our solar system as of today.

Greg asked: "Given the universe is expanding at an accelerating rate, will the rate of expansion eventually pass the speed of light?
The expansion of the universe is a very tricky subject.  Currently, the recession velocity of galaxies due to the expansion, which is proportion to the speed of light multiplied by the redshift, can be greater than the speed of light for redshifts greater than 1.  However, this is greatly dependant on the coordinate system and reference frame.  Since we can argue that galaxies are moving apart due to expansion of the universe, the short answer is yes: the expansion for distant objects is even currently greater than the speed of light.  The coordinates in which these are moving faster than c, are not the same coordinates used in relativity so this doesn't really contradict relativity.  Presumably this is a result of the odd behaviour of comoving coordinates which are explained best in Ned Wright's tutorial.

Jogirl asked: "Why is Pluto not considered a planet anymore?  How can it be a planet one day and not the next?"
Pluto no longer fits the criteria for a planet.  According to the International Astronomical Union, the current criteria for a planet are the following:
  1. It is in orbit around the sun
  2. It has sufficient mass to have taken on a spherical shape due to self gravity
  3.  It has cleared the neighbourhood of its orbit.
Looking at 1 and 2, Pluto may be a planet.  However, it does not fill the third requirement.  Pluto has very little mass in comparison to the combined mass of numerous objects in its orbit.  In comparison, the Earth is by far the most massive object in its orbit.  Basically, Pluto failed to gravitationally bind or expel the other similarly small objects in its immediate neighbourhood and is therefore one of many similar objects in the area rather than The One Large Thing in its orbit, if that makes any sense.
One answer I can give you for your second question is that science progresses by falsification.  This means that the rules in science are constantly changing and things are being redefined in order to comply with new rules.  As we learn more about the universe we realise that some things we thought before are not the case.  For instance, we now know that the earth orbits the sun.  It's not that one day the sun orbited the earth and then it changed, but that science changed and our theory was then modified to better fit the new model.

Measuring the Astronomical Unit

Tuesday, October 18, 2011

I  apologise in advance for the hurried write up.

We claim that we can measure the astronomical unit using nothing but the following image of Mercury transiting the sun as viewed from a satellite in near earth polar orbit.  From this we can also determine the semimajor axis of Mercury's orbit and the orbital period of the satellite.

Methods and Conclusions
In order to solve this problem, we have to notice the sinusoidal motion of Mercury against the backdrop of the sun.  This is an effect of the parallax using the a distance roughly the diameter of the Earth as a baseline.  We also notice that there is a parallax effect on the sun that has been corrected in the image.  As a result, the angle we measure by comparing the amplitude of the sine wave to the radius of the sun is the angle α between the centre of the sun and the apparent position of mercury against the sun.  We then define β as half the angle from one position of the satellite at the pole to the other pole through the centre of the sun (see figure) and θ/2 as the parallactic angle of mercury compared to the backdrop of the stars.

It is easy to see from this diagram that π = α + β + π - θ/2 or θ/2 = α + β.  Using small angle approximations for tangent, we get β = R/a and θ/2 = R/Δa with R the radius of the earth, a being the astronomical unit and Δa defined as the distance from the Earth to Mercury.

We also know the orbital period of Mercury which is 87 days and the orbital period of the Earth which is 365 days.  Dividing, we get:
Then we use Kepler's 3rd Law we get:
Which gives us Δa = 0.62a.

Therefore, we have:

Drawing a circle on a printout of the satellite image that coincides with the border of the sun, we get the amplitude of the wave to be 1mm and the diameter of the disc to be 218mm.  Using the ratio of these numbers and the fact that the angular diameter of the sun is 0.5 degrees, we get α = 0.00004 radians.  We use R = 6378km and then get a = 9.71e12 cm which is roughly 0.65 of an actual astronomical unit.

We see that the semimajor axis of Mercury is a-Δa = 0.38a.  We also have the orbital period of the satellite using Kepler's 3rd Law:  .  We get P = 1:24:17 hours as the orbital period of TRACE the satellite (using the actual mass of the earth in our calculation).

Our result for the astronomical unit was off by a bit under 50% of the actual value.  This error most likely came from the imprecise measurement of the diameter of the sun on the page and the amplitude of the sine wave across the sun.  This was done by hand with no compass, which left a lot of room for error.

Thank you to Professor Johnson for providing the orbital period of Mercury, Jackie for pointing out the parallax of the sun, and Daniel for measuring the diameter of the circle.

Tau Ceti From Palomar

Sunday, October 9, 2011

I wrote up a mix of 2 and 3 from the work sheet on LST. Hope you don't mind

Palomar is an important observatory for the Caltech astrophysics community. Because of this, we seek to constrain the visibility of the star Tau Ceti from the Palomar Observatory over the course of the year. We determine first whether its declination allows it to be seen and then when it can be observed. We also examine the elevation from month to month. We find that Tau Ceti is observable from Palomar during certain times of the year.

Tau Ceti is a star located in the constellation Cetus at RA = 1:44 (25.0167 degrees) and Dec = -15.9375 degrees in the southern hemisphere of the celestial sphere. The Palomar Observatory is located at 33.358 degrees north and 116.864 degrees west. It is important to understand how and where Tau Ceti can be observed from Palomar in order to study the star. To do this, we determine the elevation of Tau Ceti when it is on the meridian passing through Palomar as well as the times that it is observable optically.


First we determine what the maximum elevation of Tau Ceti is from Palomar. The elevation of a star is the angle it forms with a line connecting the centre of the earth to the observer's local horizon. We calculate this using the equation:

Where a is elevation, delta is declination, phi is the local latitude, and H is the hour angle (the difference in LST from the meridian at the time of viewing). When the star is on the meridian, it is clear that H = 0. We then get:

Since one local sidereal day is a 360 degree revolution of the earth, or the time it takes for the meridian to re-align with the object in question, it is obvious that this is the elevation any time Tau Ceti is on the meridian and also that this is the maximum elevation of Tau Ceti as observed from Palomar.

Next we notice that if we set a = 0, or say the star is at the horizon, we can find the hour angle the time the star rises. This is given by:

Dividing this by 15, we get H = 6.72 hours, which we round down to 6.5 to account for the fact that stars at the horizon are invisible to the ground based observer.

Plotting the elevation as a function of time for the 20th of each month of the year, we get the following graph. We choose the 20th of each month for convenience's sake as the vernal equinox is on 20 March.

This shows, as expected from considering the problem, that at LST = 1:44, or when Tau Ceti is aligned with the meridian, the elevation is always 40.71. The graph also shows the approximate times at which Tau Ceti is aligned with the meridian in UT for planning of observations. Now plotting the time (UT) of Tau Ceti's maximum elevation against the passing months, we get the following graph:

As expected, the change in time is a linear relationship to the time of year. The error bars of this graph mark the total time that Tau Ceti is above the horizon. Regions plotted in red are the times before and after sunrise and sunset, respectively. These are the times that Tau Ceti is actually visible from Palomar because interference from the sun is absent. Predictably, this range is from July to January, with maximum visibility in the October-November region where it is dark the entire time that Tau Ceti is above the horizon. This agrees with observation data that indicates Cetus is best visible in late autumn.

We thank Professor Johnson and Jackie for providing us with this problem to solve. Also, Wikipedia for the image describing horizontal coordinates and Professor Harold Geller for his notes on calculating the altitude of a star. We also thank Microsoft Excel for cooperating long enough to produce these graphs.

Clarifications on AGN

A few days back, Jackie asked:

I'm interested to hear about the ways we can study AGN! I would also be interested in seeing a diagram of one explained. That's a question some of the grad students here have gotten on their qual exam ("Diagram an AGN.") and I still feel a little blurry about the details.

What does it mean that AGN influence the color of their host galaxies?
Now that the horror that is Phys 12, which is by the way the sophomore physics course for astrophysics and physics students is over for the week, I am free to answer.

What Does An AGN Look Like?
To properly understand the parts of an AGN we must consider the two types. The currently accepted "unified" structure of an AGN looks this (personal communication with Dr. Andy Goulding, origin unknown):

The Black Hole and Accretion Disk:
I'd assume you know what a black hole is, but let's talk about the accretion disk. According to NASA, an accretion disk is a flat sheet of gas and dust that surrounds a black hole. The presence of an accretion disk indicates that there is material being incorporated into the black hole and that the black hole is accumulating mass.

The Jets:
It's important to note that jets do not occur in all of AGN, but when they do occur they are characterised by a beam of particles ejected in opposite directions. These are observed in the radio wavelength.

The Broad Line and Narrow Line Regions:
The Broad Line Region (BLR) of an AGN is the region of gas clouds immediately surrounding the black hole. It is characterised by relatively broad emission lines, hence its name. The Narrow Line Region (NLR) of an AGN is an outer layer of gas clouds in the area of the jets that produce strong "forbidden lines" which are not present in denser gases.

The Dusty Torus:
This is simply an obscuring torus around the accretion disk. Current research indicates that it is probably made up of gas and dust, but the distribution of gas and dust in the torus are not yet known (Antonucci, 1993).

Now let's put it all together. The two types refer to the inclination of the AGN with respect to us (the observer).

Type 1 AGN:
A type 1 AGN refers to an AGN in which the BLR is visible. To visualise this situation, a type 1 looks a bit like a donut with an accretion disk at the centre. i.e., a type 1 AGN is face-on as opposed to edge-on. Viewing a type 1 AGN will show a clear view of the BLR as well as the accretion disk and black hole.

Type 2 AGN:
A true type 2 AGN is a bit less straightforward. We call an AGN type 2 when the BLR/accretion disk/black hole area is obscured by the dusty torus. This leaves us only the NLR visible for observation. There are some AGN that are neither entirely edge-on or face-on, since the inclination of galaxies does not change in intervals of 90°. However, a true type 2 AGN is completely edge-on.

AGN and Galaxy Colour
I have to admit, when I wrote that AGNs affect galaxy colour, I did so because my SURF mentor told me to rather than because I truly understood. I still can't claim to fully understand. However, I've spent some time reading through the relevant paper and found an interesting diagram to share (Hickox et al., 2009):

This diagram shows (a) the colour of host galaxies plotted against absolute magnitude with yellow circles, green stars, and red squares showing radio, X-ray, and infrared AGN, respectively and (b) the colour distribution in each of these wavelengths. Notice how in both the X-ray and infrared spectra, the distributions for AGN are shifted to the left in what is called the "green valley". To summarise, AGN do not influence the colour of their host galaxies beyond what is normally seen, but they may influence the proportion of a particular colour in the X-ray and infrared spectra.

CCAT vs. Keck: Angular Resolution and Telescope Size

Saturday, October 8, 2011

by: Eric S. Mukherjee, Nathan Baskin

This is not the actual write up I was going to do for this week. I just thought this problem was a cool look at the many factors that go into designing a telescope and decided to write it up anyway. The other one will follow.

It's very easy to assume that a bigger telescope is always the more precise one. The question is, is this assumption always true? In this post, we examine an order of magnitude solution of problem 3 of the problem set on optics to determine whether this is the case.

The angular resolution of a telescope is the smallest angular separation at which the telescope can resolve two distinct objects. Any angle smaller than this registers as one larger object when viewed by the telescope in question. Caltech is building CCAT, a 25m telescope which observes in the 350-850 micron wavelength. We would like to know how the angular resolution compares to that of Keck, a 10m telescope observing in the near-infrared J-band at 1.25 microns.

We want to find angular resolution, which is given by the small angle approximation:

We then convert all of our known quantities into the same units. In this case, meters. As we can see, the angular resolution has a direct dependence on the wavelength, so we need only convert the minimum wavelength for CCAT:

Immediately we can see that these differ by two orders of magnitude while the diameters, 25 m and 10 m, respectively, are on the same order of magnitude. We can ignore the factor of 1.22 since it is shared by the two equations and is close to 1 and find that

We find that despite the greater diameter of CCAT, there is a difference of roughly a factor of 100 between the angular resolutions of the two telescopes. This was immediately obvious from the differences between the two wavelengths, since the diameters of the two telescopes have the same order of magnitude. Solving the equation for angular resolution we get precisely:


We find in the end that in order to determine a telescope's power we must refer to the wavelength at which it is observing as well as the diameter and that in order to have an equivalent angular resolution at the desired wavelength, CCAT would have to be a 2500 m telescope.