Where the X-Rays Are

Wow... It's been a while, hasn't it? Obviously, I'll accept some of the responsibility for that, but I also really want to blame school.  It's like they think I'm there to take classes and do homework or something...

Since it's been so long, you might be wondering what I've been up to. (Even if you aren't, I'm going to tell you anyway.  And besides, if you aren't interested, why are you reading this blog??) I decided to try something new (and stupid) this semester and take five classes -- Irish, Folklore Theory, Electromagnetism, and two astronomy research classes.  I don't want to burden you with a post that takes two hours to read, so if you want to read about one of the research projects, just jump on over to this beautiful page. The project is on hold for winter break, but it will pick back up.

Now, about my other research project.

This semester, I had the pleasure of working with Dr. Belinda Wilkes, the Director of the Chandra X-Ray Observatory.  One of the many projects she's working on right now is quantifying the orientation dependence of our Active Galactic Nuclei (AGN) observations. As a part of that, I was working with a single high-redshift quasar and trying to determine whether or not there was a galaxy cluster surrounding it.

(Professionally made) photo of the quasar I was studying this semester. 

One of my first questions was, "Why do we care if there's a galaxy cluster there?" Well, actually, I asked that question pretty far into the semester. At first, I was kind of busy asking myself things like "How do I open this file in ds9?" (but with a lot more expletives).   

But, when I got to the point where I wanted to know more about galaxy clusters, the whole project kind of wrapped itself together for me.  Galaxy clusters are really rare at high redshifts (z > 1) because they correspond to relatively young universes (about 4 Gyrs).  Such a young universe shouldn't have had enough time to let fully-formed clusters develop, but some still manage to exist.  Learning more about those young-universe galaxy clusters could lead to discoveries about cluster and galaxy formation.  But before we can learn about them, we have to find them, and quasars -- which tend to exist in really dense regions of space -- are good galaxy cluster homing beacons. 

To figure out if my quasar (I say "my," but I should make it very clear that I don't exactly own any objects in space...yet), 3C 270.1, is an effective homing beacon, I analyzed both radio and X-Ray data. 

EVLA Radio data (left) and Chandra X-Ray data (right)

If I had to wrap an entire semester of research into as few steps as possible, I guess I'd do it like this...

1. Define region
2. Extract counts
3. Fit Spectra

Huh, it seemed like a lot more work at the time....  Oh well, it was enough to keep me busy, and it was fun! 

Following those steps (more than once), I found the number of counts in the quasar's extended structure that we assumed to come from the possible galaxy cluster. I turned that count value into a luminosity and determined..... nothing conclusive. 

Basically, I spent 3 months trying to answer this simple yes-or-no question, and couldn't, but that's science for you.  And I really don't mind it  =)

Now that I'm on winter break, I'll have more time to actually write posts as often as I know I should. The next post I write will include a reunion with the best co-interns I ever could have asked for this past summer and my first AAS meeting! 

And, as a farewell gift, I give you this

It has absolutely nothing to do with my science (although, I'm sure someone, somewhere would be happy to work through all the physics of what she does), but she's a BAMF. And everyone deserves a little BAMF-ness in their day.  Enjoy! 

All I Wanna Do When I Wake Up in the Morning is See Your Eyes

Wow! Eleven weeks sure does go by fast!  It feels...maybe not like yesterday, but only last week that I walked into the NRAO building with no idea how awesome the future ahead would be.

But before I sing the praises of this summer and the other summer students (who I already miss like crazy), I'll talk about the final stages of my project.

Aren't we all so cute?! I can't wait to see them all again.

I think I was still struggling with Cloudy the last time I wrote a post (which was wayyy to long ago).  Well, there's good and bad news about that.  I usually like starting with bad news so I can end on a happy note, but doing that now just wouldn't make too much sense.

So, good news first:
I FIGURED IT OUT!  I understand Cloudy now (well, as much as anyone who isn't the creator can understand it without devoting their life to it).  It turned out that my biggest problem was that I was working with a model of Cloudy from 2013 and a Cloudy user manual from the 1990s... No wonder I was having so much trouble!

Now the bad news:
After I figured out how to use Cloudy and developed a rough understanding of what all of the output files meant, my mentor and I realized something--Cloudy was not going to be able to model out whole galaxy like we thought it would.  If you've been reading all of the posts about my summer (and if I've been putting the right stuff in my posts), you might remember that we were trying to constrain the physical properties of a nearby galaxy.  So we were trying to figure out its density, metallicity, temperature, etc. But galaxies aren't these big homogeneous creatures just floating about in space; they have different regions and those regions have different properties. In this project particular, we were concerned with the HII regions and the Photodissociation Regions (PDRs), or the ionized regions and the neutral regions, respectively.  When I finally got Cloudy working the way I wanted it to, we saw that it wasn't all that great at constraining the neutral regions of our galaxy, so we had to figure out another way to do that.  So it was a setback, but this is science! And in science, no result is still a result.

My battle with Cloudy over, and the different regions constrained, the next step in the project was to try and get some information out of the extra data we received about halfway through the summer.  We figured out its size, its mass, and how many stars it forms per year.  That took me to the end of the summer with my research! But I am still working on it, so I can continue to write posts about it and tell people how awesome radio astronomy is.

But the best part of this summer was that I wasn't limited to just sitting behind a desk, typing away at a computer.  I went to my first scientific conference--Transformational Science in the ALMA Era.  There were so many cool talks, some of which I actually kind of understood, and I got to meet some really cool radio astronomers from around the country.  I was even asked to be in charge of the official Twitter for the conference, which was a lot of fun!

All in all, this was an amazing summer. I learned a lot about radio astronomy, met some really cool people, and got a better idea of what I want to do when I graduate form college and become a "real person."  So, thank you NRAO for making this one of the best summers of my life!


And for those of you who were wondering what the title of this post meant, and for those who didn't pay any attention to the title but want to know now that I've brought it up...

At some point  this summer, I started listening to different 80's music playlists on Spotify and I kept going back to one song...

Today's the Day! The Day We Visit the VLA!

I know it's been a while since my last post, but that's the life of an aspiring scientist--sometimes the results come quickly and sometimes there's almost no progress, making it near impossible to write interesting blog posts about your research. And then there are those times when you don't make much progress in your own research because your job pays to fly you to one of the coolest astronomy-related sites in America!

This past week, I went with 11 other NRAO summer interns to visit the Karl Jansky Very Large Array in New Mexico.  The VLA is an interferometer of 27 radio antenna dishes that are used to take super high-resolution images of objects in our universe.  An interferometer is an array of telescopes that allows astronomers to combine the data coming from each individual dish.  Even if you don't know a lot about radio astronomy, that has to sound pretty freaking cool.

We got a pretty great tour of the facilities.  We started with the normal walking tour of the grounds, where we saw the Ron Bracewell Sundial, made from the pillars of an old telescope array.  Just a short walk away were the VLA's Whisper Dishes, a pair of parabolic dishes designed and placed so that two people standing near them could hear each other whispering from across the field.  A longer walk away, we were able to see the site's backup antenna being repaired in its hangar.  Going back the way we had come, we entered the operating building, the home of the VLA Control Room.  We talked with some of the telescope operators, but didn't get to do any operating (which was probably for the best because this is a multi-million dollar science project).  Our next and final stop was one of the dishes--dish 5 to be exact.  We climbed stairs and ladders until we made our way to the top, and it was such an amazing experience!

But don't worry! We didn't get flown out to New Mexico just to climb telescopes and buy really cool NRAO swag. 

Though I promise we bought plenty of swag

We had to do some science, too.  We had to choose between two projects-- one used the Very Long Baseline Array (VLBA) to image radio galaxies; the other, which I chose, used the VLA to try to identify a nearby dwarf galaxy.  We did this by mapping the neutral hydrogen (HI) in our target area, and depending on what we found, we would be able to determine whether or not the gas cloud was actually a galaxy.

On day one, after we chose our projects, they split us into groups and told us to reduce data.  No one in my group had ever used CASA to reduce data or worked with neutral hydrogen, but our supervisors did point us in the direction of an online CASA guide to follow. It was a slow, frustrating, and educational experience.  I'm far from an expert in CASA data reduction (I'm talking lightyear-scale far), but an image was made. We're not done analyzing yet, and we might have to go back and redo some of the calibration, but we're pretty sure our target is a galaxy.  We think that because on one edge of the target, the hydrogen is coming towards us, and on the other, it's moving away.  This indicates that the hydrogen is part of a larger, cohesive structure, and not just randomly floating out there in the universe. At least, not in this particular area.  

The morals of this post are:

1) If you ever get the chance to visit the VLA, take it in a heartbeat.

2) Reducing data is really hard when you've never done it before.

And, like always,

3) Science is cool!

Size Matters (Of Telescopes, Of Course)

It's really amazing how sometimes old memories are completely off base because you've built them up in your head so much and other times they're spot on.  Luckily for me, this past weekend's trip to NRAO's Green Bank site was even better than I remembered my high school trip being.

It started on Thursday morning.  Tierra (my roommate) and I had decided to go into work early so we could get some work done before we left Charlottesville.  Bad idea.  I didn't realize until I got to the office that I couldn't do anything, anyway, because I didn't have the values I needed to put into my code.  So instead of working, I walked around to the building's various vending machines and gathered snacks for the drive ahead of me.  You can never have too many snacks.

We packed in one suitcase, because we've reached that stage in our roommateship. 

We left Charlottesville a little before 9:00 and got lost almost immediately.  There was some unfortunate miscommunication between the student driving and the one navigating. But we made it to Green Bank in one piece, and just about 30 seconds behind the other van!

After eating a much-needed lunch, we got to tour the Green Bank Telescope, the 100-meter, 16 million-pound monstrosity that has the honor of being the largest movable structure in the world. It took three different elevator rides to travel the 450 feet to the top of the telescope, and it was both beautiful and terrifying.  Looking down on the 2-ish-acre surface of the GBT, I found myself trying to contain an unbelievably strong urge to jump down to the dish and slide from one edge to the other.  Of course I didn't do that, but trust me. It was a hard urge to resist.

See that on the right? That's the GBT. And it's huge. 

Later that night, we went to dinner with the Green Bank students and they invited us to a bonfire at their house.  It had been so long since I had been around fire! It brought back memories of camping, home (where I had a wood-burning stove), and going to school every day smelling like wood smoke. It was a great time, only enhanced by the fact that the Green Bank students have their own private playground.  The fun was short-lived, though, because we had to be in the telescope control room at 7:30 the next morning.

Fast forward through the 6 hours of sleep I got and we get to the actual observation.  We had two and a half hours of observing time on the GBT, which may not sound like much, but it's a crapload of time to give to a bunch of college students!  We are talking about one of the most highly sensitive single-dish telescopes in the country.  We mapped the neutral hydrogen in some nearby galaxies for an hour or so, then we improved the location measurement of a pulsar (basically, a collapsed supernova that spins really quickly).  I know, super casual, right? No, it was really, really cool.

That's about it for our Green Bank trip.  We all made it back to Charlottesville safely, the exhausted new owners of lots of NRAO swag.

Really quickly, I just want to say something about my own research.  We got some new data about our galaxy! My mentor was leading a discussion on galaxy formation, and when someone heard what galaxy we're working with, he told my mentor about some archival optical data.  What does this mean? It means pretty pictures!

Isn't it gorgeous?? I've been working with this galaxy for five weeks, and I had no idea what it looked like.  Now that I know it's basically a supermodel among galaxies, I'm even more excited about my work than I was before! 

WiaN? (What's in a Name)

I said before that I would write a post about acronyms in astronomy, and since I haven't made any breakthroughs in my project in the last week, here it is!

First, I need to make a distinction that I've always thought was pretty important, though others rarely share my opinion.  An acronym is an abbreviation made from the first letters of a series of words that can be pronounced as a new word.  An initialism  is an abbreviation that cannot be pronounced as a word.  Now we can start.

On any given day, I swear 10% of my conversations with people are made up of acronyms or initialisms.  NRAO, ALMA, NAASC, REU, VLA, GBT, FIR.... The list goes on and on, so since it would be impossible for me to write a reasonably-lengthed blog post about all of the abbreviations used in astronomy, I'm only going to talk about the ones that are really important to my internship and the ones I think are really cool.

Let's start big and work our way down.  NRAO.  When people ask me what I'm doing this summer, I usually start by telling them I'm doing some astronomy research.  Most people ask where, and I don't know if it's because I'm a lazy speaker or my quick response reflexes just haven't learned their lesson, but I always say "NRAO, in Virginia."  The other person will inevitably ask what NRAO is, and I'll tell them it's the National Radio Astronomy Observatory, and because they've already asked me three questions, they'll usually just smile and nod, even though they still have a slightly confused look on their face.  NRAO is a federally funded research group founded in the 50's whose focus is (surprise!) radio astronomy.  Oh, and it's a pretty freaking awesome organization.

ALMA, VLA, and GBT all stand for NRAO's different (arrays of) telescopes.  ALMA--which is an acronym, by the way--stands for the Atacama Large-Millimeter/submillimeter Array and is located in the Atacama Desert in Chile.  It's a giant international collaboration with the goal of getting super high-resolution radio data from space.  VLA--an initialism--stands for the Very Large Array (astronomers are really creative) and is located in Socorro, New Mexico.  The VLA is the super famous (I'm using this term very loosely) radio telescope array that can be seen in movies like Contact, Terminator Salvation, and 2010, to name a few.  The GBT is the Green Bank Telescope in Green Bank, West Virginia.  Its claim to fame is that it's the largest movable single-dish telescope in the world.

Now for some fun ones. One of the things that keeps me awake when I'm reading scientific papers is seeing all of the cool abbreviations. HERMES, GOALS, ZEUS, and SKA are just a few.  My favorite, though, is ALFALFA.  I saw this in a paper once and it immediately woke me up and kept me interested for the rest of the paper.  ALFALFA is the Arecibo Legacy Fast ALFA survey to find more pulsars.  (I know I just used an acronym in the description of an acronym, which is kind of copping out, but I couldn't find ALFA's definition to save my life.  If you know it, please let me know!)

Like I said, there's no way I could talk about all of the acronyms and initialisms, because I'm pretty sure that that blog post would take years to write, let alone read.  But now you (hopefully) know a little bit more than you did before.  And please feel free to comment back with questions about abbreviations I mentioned but didn't explain or ones I didn't even name that you want to know more about!  I'd love to help spread the knowledge.


Just FYI, I'm actually heading to do some observing with the GBT tomorrow, so you should definitely come back in a few days to read about those adventures!

It's All Coming Together!

I figured that it was probably time for me to talk about my research project again, and I actually have something to say today! So that's good timing.

Remember that code I was working on for all of last week?  Crushed it!! But I have to say that my greatest accomplishment of the week is finally putting all of the puzzle pieces together and seeing the big picture that is this project.  And I'm going to share that epiphany with you!

I'm going to start with another description of my project.  I'm determining the physical characteristics of a starburst galaxy--a galaxy that's undergone a lot of intense star formation--in the hopes that we'll be able to use it as a template for high-redshift galaxies.  The problem, though, is that galaxies aren't homogeneous creatures.  They vary in density, temperature, and composition throughout.   And that's great and all, but it's really annoying to model.

I'm particularly interested in the variances that come from two different regions of the galaxy--HII regions and PDRs.  HII regions are clumps of ionized hydrogen gas that occur when new stars are formed.  PDRs (Photodissociation Regions) are a little bit harder to explain.  Basically, they're regions in the inter-stellar medium (a fancy phrase for the space between stars) where UV light photons knock electrons off of dust grains.

Diagram of a PDR from Hollenbachs's Article on Dense PDRs

I intentionally made my explanation pretty light, so if you have questions, feel free to comment with them, and hopefully I'll be able to answer. 

Okay, now that you know that, on to my epiphany!  Until this week, I was kind of just going with the flow of my various assignments.  Reduce this data.  Identify these lines. Make these plots. And I did it all, faster than  and with more enjoyment than I expected, but I didn't really know why. But now I do, and here it is, step-by-step.  

1) Take this data, reduce it (this, I've learned, is a simple way of saying "take a massive glob of data and cut out all of the useless stuff to get down to the pretty information that will make you happy").  Why did I have to do this? I was looking for the spectral lines that I would then use to actually figure out the characteristics of the galaxy.  Pretty simple, actually.  

2) Take the newly reduced data and compare it to the models created by a scientist named Rubin.  Why?  Well, apparently, Rubin was modeling the HII regions with different parameters (density, temperature, composition, etc.) and recorded the intensities of different lines that for each individual model.  I compared my data to his to see which model of an HII region most accurately represented the data I had.  

3) Take the intensity values for each line and enter them into an online program that spits out contour maps of the PDRs.  I did this to model the PDRs...not too hard to understand in the grand scheme of things.

4) Use a program called CLOUDY (I will probably write a post in the near future about how much I hate this program) which is usually used to model nebulas.  It turns out that CLOUDY can also be used to model galaxies.  And since we don't know how much of each line's intensity comes from HII regions and how much comes from PDRs, we're using CLOUDY to (basically) figure out the how the galaxy is divided between the two.  

And that's it. Looking back now, I realize that it probably shouldn't have been to hard to see, but that's hindsight bias for you.  I guess the important bit is that I know why I did this stuff now.  

The Future's Gettin' Kinda Close

This past week has been pretty busy.  I've...

~Started working with entirely new data for my research project
~Produced two new codes
~Tried to write a proposal to get observing time on the VLBA (I'm definitely going to have to write a blog post about astronomers and their acronyms)
~Listened to a really hard-to-follow lecture on how to take data and turn it into an image
~Gone on a (short) tour of the tattoo shops in Charlottesville within bussing distance of my apartment (that has nothing to do with astronomy; it was just really fun)

But, most importantly, I've learned a lot about graduate school and the different paths that exist thereafter.

Let's start with last Friday.  A scientist at NRAO invited all of the summer students to his house for an informal discussion about grad school, career paths, networking, and all things future-related.  We were there for about five hours, but I learned so much (and I got to eat Indian food, which is always a positive in my book)!  We discussed GPAs, the GRE, how publishing papers works, the different career options after graduation, and a whole bunch of other stuff.  I left that night with two things on my brain:

1) I need to start meeting more people
2) OH MY GOD I HAVE TO GO TO GRAD SCHOOL AND FIGURE OUT WHAT I'M DOING WITH MY LIFE!!!

(Yes, the caps lock was necessary, because I was that frantic.)

So I took some deep breaths and emailed the TF from the astro classes I took last year.  He was a huge help and pretty much got rid of the crisis altogether.

Come Monday, I was ready for the next discussion about grad schools, and this time, I was excited.  Afterwards, trying to take care of that first thought, I asked someone if they could put me in contact with anyone who did astronomy education and public outreach.  They did, and this morning I met Tania Burchell, the wonderful woman who further opened my eyes to EPO.  She had so much great knowledge and advice to share, and is completely responsible for the creation of my new twitter page (@GoAstroMo because shameless plugs are great).

I guess I should probably say a little something about my research... I'm still coding.  That's about all I can say.  But I have found my new pump-up song.  Every time I hit a roadblock in my project, I listen to this, dance like an idiot, and continue typing away.

Coding Can Be...Fun!

Week 2 is done and it was even better than the first!  If I recall correctly, I was just starting my data analysis the last time I posted.  Here's what I've been up to since:

I finished that code I was working on at the end of the first week!  It took a couple days, but I finally figured out how to loop through the data I had reduced and add up the flux values for a range of selected wavelengths.  I was definitely making it more difficult than it had to be when I first tried to tackle the problem.  The hard part was actually identifying and calculating the different factors that contributed to the error in my intensity measurement.  But I did it!  It was my first solo-constructed script, and I was really damn proud.

Once I was done with that, my mentor had me start learning about modeling right away.  We're using these spectral lines to determine characteristics--temperature, density, metallicity, etc.--of a starburst galaxy, so one of the things I have to do is use the data that's been collected (by the Herschel telescope) to actually create a theoretical model of this galaxy.  My mentor told me to read a paper by Robert Rubin on modeling HII regions.  Most of the paper was filled with pages-long tables of data, which I was then asked to turn into a meaningful model with a code of my own creation.  And that's where it got really fun!

Simplified diagram of an HII Region from this University of California website

HII regions are large clouds of ionized hydrogen gas where a lot of star formation has just happened

Rubin's data came to me in 4-dimensional cubes with dimensions corresponding to metallicity, photon-number, temperature, and density.  My task was to turn it into a 5-dimensional "cube" with an extra axis representing the number of HII clouds in the galaxy model.  I had to do this because we don't know how many clouds the galaxy we're studying has, and creating a fifth dimension of data allows us to see which number of HII clouds most closely matches our own data.  After adding the extra dimension, I had to create chi square grids (which I now understand a lot better than I ever did when they tried to explain it in my introductory mechanics class) and calculate the probability that each model in Rubin's data actually matched our own.  So far, I've been working on this script for two days, and I'm still not done.  But every time I do something right, I can't help but do a little dance in my seat, and that's a great feeling!

On top of all that, I've been to two discussions about grad school and career paths in astronomy in the past 5 days.  I've barely been here for two weeks and they've already managed to convince me that I should pursue a Ph.D in astronomy.  But more on that in the next post, where I'll tell you all about my (new and very tentative) future plans.

Welcome to NRAO

I thought I was done with all of this blogging stuff for a while, but then my advisor suggested that I use this blog to document my summer internship.  It seemed like a pretty excellent idea, and his advice hasn't steered me wrong yet, so here I am. 

For those of you who don't know, I was granted the opportunity to work at the National radio Astronomy Observatory (NRAO) this summer.  When I boarded my train a little over a week ago, I literally only knew two things about the summer ahead:
1) My roommate's name was Tierra (there will almost definitely be more about her in future blogs).
2) My research project had something to do with a galaxy...somewhere in the universe.  

 

I was pretty terrified.  But I had already signed all of the official papers agreeing to show up, and I didn't want to find out what would happen if I didn't.  And I'm so glad I got on that train, because this last week has been amazing!

First, I'll give a quick description of the project I'm working on.  I'm using radio spectral lines to determine the physical characteristics of a galaxy, IRAS 08339+6517.  It's at a redshift of about 0.02, so it's far enough away that it looks like a point source in our images, but close enough that we can get some really good data from it.  Please, please, please ask me questions about the project!  Answering them will help me better understand what I'm doing. 

I started out last week with a bunch of data.  The program I was using, the Herschel Interactive Processing Environment (HIPE), provided the data in varying levels of reduction.  My first assignment was to go through the most processed level and see if I could find and identify the spectral lines.  It was actually pretty simple--look at the graphs and mark where there's a spike.  I found 6! 

My next assignment was to figure out how to reduce the data myself.  That was interesting.  It involved some coding, though since I was provided with a skeleton script, the bulk of my job was to add some values and do some debugging. 

Finally I got to do some analysis!  After 4 days of reducing data (which I understand is nowhere near as long as some people have to do it) to a useable form, I was asked to find the intensity of each spectral line.  I got to make my own script (with some help from my research mentor) and go through what I have identified as the typical stages of coding:

excitement
confusion
annoyance
hair-tearing anger
utter elation

And that was my first week!  I'm not completely sure what the next step is, but I'm here for 10 weeks, so I know I have a lot left to do. 

Comparison to Wasp-10b

This post is going to compare the parameters we get from analyzing these graphs

both graphs provided on Worksheet 11.1

with actual values from exoplanet.org.  

First, let's start with the impact parameter, b.  

$b=1-\delta^{1/2}\frac{T}{\tau}$

In the light curve above, 

$\delta$=0.028  

T = 102 minutes

$\tau\approx$18 minutes

Using these values and the equation above, I found b=0.99, which is a lot higher than the value given by exoplanet.org, b=0.3.  I believe the reason for this difference is a difference in data.  For example, in the light curve above, the length of the transit looks to be a little longer than an hour and a half, but on exoplanets.org, the transit is at least half an hour longer than that.  

Next, let's look at the ratio of the radii of the planet and star.  

$\delta=\frac{R_P^2}{R_*^2}$

$\frac{R_P}{R_*}=\delta^{1/2}$

For this system, we find that the ratio is equal to 0.17, so the planet is 0.17 times as big as the star.  

On worksheet 11.1, it says that the star is 0.8$R_\odot$.  With this information and the ratio we just found, we can tell that the planet is about 1.36 Jupiter radii big.  This is closer to the website's value of about 1.08 Jupiter radii.  The reason for this difference could be that the website lists the sun as being 0.7 solar radii big.  

Now we'll look at the ratio $a/R_*$.

On the worksheet, not taking into account the impact parameter, I used the equation 

$\frac{a}{R_*}=\frac{P}{\pi T}$

This equation gives us a ratio value of about 13.3, which is pretty close to the website's value 11.9.  The difference can be attributed to the fact that the website found a semimajor axis value dependent on the impact parameter.  

Finally we'll look at the densities of the star and planet.

We all know that density equals mass over volume.  But the fun thing about the mass of a star is that it can be related to semimajor axis and period of the system using Kepler's Third Law.

$M_*=\frac{4\pi^2a_P^3}{GP^2}$

$\rho=\frac{3\pi a_P^3}{R_*^3GP^2}=\left ( \frac{a_P}{R_*} \right )^3\frac{3}{GP^2}$

When we substitute provided values into this equation, we find that the density of the star is about 1.45 grams per cubed centimeter, which is pretty darn close to the website's value of 1.51 g/cm^3.  

WS 11.2, Problem 3: Goldilocks Zones

A Goldilocks Zone, or a "habitable zone," is a distance from a star where it is neither too hot nor too cold for a planet to have liquid water.  This is the criterion for life that most people have a greed on, but we're not sure that it's the only or most important factor in determining whether or not life can exist on a planet, so it's not the end all be all of life on other planets.

Now let's fi gure out what stars make the best targets for both the radial velocity technique and the transit method, assuming we are primarily interested in fi nding planets in their stars' habitable zones. For a planet in the habitable zone of a star of mass $M_*$...

a)  How does the Doppler amplitude, $K_{HZ}$, scale with stellar mass?

To solve this problem, let's first recall the equation for the Doppler amplitude that we found on a previous worksheet, generously dropping unnecessary constants.

$K\sim\frac{1}{M_*^{4/6}P^{2/6}}$

This isn't as obvious as it might seem, though, because an object's period depends on its mass, so we have to do some more algebra.

$P^{1/3}\sim\left ( \frac{a_P^{3/2}}{M_*^{1/2}} \right )^{1/3}=\frac{a_P^{3/6}}{M_*^{1/6}}$

Of course, the semimajor axis of a planet, $a_P$, is also dependent upon its star's mass.

$a_P^{3/6}\sim M_*^{3/6}$

When you combine all of these scaling factors, you find that

$K\sim\frac{1}{M_*^{4/6}M_*^{2/6}}=\frac{1}{M_*}$

b)  How does the transit depth, $\delta_{HZ}$, scale with stellar mass?

For this, we have to remember that the depth of the transit depends on the ratio of the radii of the star and planet.

$\delta\sim\frac{R_P^2}{R_*^2}$

We can use mass-radius scaling relations to simplify this

$\delta\sim\frac{1}{R_*^2}$

$M\sim R$

$\delta\sim\frac{1}{M_*^2}$

c)  How does the transit probability, Prob$_{HZ}$, depend on stellar mass?

We learned on worksheet 11.1 that the probability that a planet will pass in front of its star is

$\frac{R_*}{a_P}

Now we have to remember both the mass-radius scaling relation and the relationship between a star's mass and $a_P$.

$M\sim R$

$a_P=\frac{M_*a_*}{M_P}$

Then we can rewrite the probability as

$\frac{M_*M_P}{M_*a_*}$

So the probability of a transit happening doesn't really depend on the star's mass at all.


d)  How does the number of transits (orbits) per year depend on stellar mass?

This wording of this question is a little unclear, but it reminds me of part (c) from problem 2 on this same worksheet.

If the Sun were half as massive and the Earth had the same equilibrium temperature, how many days would our year contain?

To solve this, we set up a proportion between the period of a habitable zone planet and the period of our Earth under the conditions specified in the problem.

$\frac{P_{HZ}}{P_\oplus}\sim\frac{a_{HZ}^{3/2}M_\odot^{1/2}}{M_*^{1/2}a_\oplus^{3/2}}$

If we drop the constants, we get

$P_{HZ}\sim\frac{a_{HZ}^{3/2}}{M_*^{1/2}}$

Now we can substitute in the equation for the semimajor axis of the habitable planet and simplify to get

$P_{HZ}\sim M_*$

But the problem asks for the relationship between a star's mass and the number of orbits of a planet, which is inversely proportional to its period, so

Transits$\sim\frac{1}{M_*}$

e)  Based on this analysis, what are the best kinds of target stars for the search for habitable zone planets? What factors did we ignore in this analysis?

From parts (a) through (d), we can see that the Doppler amplitude and number of transits are both inversely proportional to stellar mass and the depth of the transit goes as the inverse square of the stellar mass.   We want all of these quantities to be as large as possible.  A high Doppler amplitude makes a transit more noticeable on an RV graph; a deep transit is more noticeable on a light curve; the more transits there are, the more opportunities we have to gather data.

Based on this, we can see that lower-mass stars make better targets for finding habitable-zone planets.


In doing this, we ignore the fact that lower-mass stars are harder to detect in general because their luminosities are so low.

WS 11.1, Problem 3: What's in a Light Curve?

This blog post is loosely modeled after problem 3 on worksheet 11.1.  I'm not going to answer the specific questions from the problem to generalize to all sets of data, but I am going to follow the path that the problem sets to discover what can be learned from a transit light curve.  This should be fun.

The problem gives us three pieces of information.

• The system's period is 3.1 days (not necessary since I'm not actually answering the parts of the problem, but now we know we can find a lot of the system's properties with its period).  
• This light curve

• And this radial velocity graph

The first part of the problem asks us to qualitatively describe the brightness distribution of the star.  We can tell this by looking at the bottom of the light curve.  The dots along the curve represent the amount of light in each image taken of the system.  The dots stay fairly close to the line, so the star's brightness is relatively evenly distributed.  But because the dip is more of a curve than a straight line, so we know that the center of the star is slightly more luminous than the edges.  This is to be expected due to the limb darkening effect, which basically just says that the center of the star will appear brighter.  (Yay for self-explanatory scientific terms!)

The second part asks us to find the ratio of the radii of the planet and star.  This is a little bit trickier than the first part because it involves actual math, but it's not that bad.  The theory behind the solution to this problem is that that the depth of the light curve depends on the ration of the areas of the two objects.  Actually solving the problem is a pretty logical process.  If the star is really big, the light curve will be pretty shallow.  In the same vein, the curve will be shallow if the planet is small (because it's not blocking a lot of the star's light).  So the depth of the curve goes as 

$\frac{R_P^2}{R_*^2}$

When the problem asks us to find the ratio of the objects' radii, we can find the average depth of the light curve and take its square root.  

The third part asks us to find the ratio of the planet's semimajor axis with the star's radius 

$\frac{a_P}{R_*}$

To solve this problem, we look at the duration of the transit.  If you believe that velocity is distance over time (which I do), then you can also believe that time is distance over velocity.  The distance that the planet has to travel to complete a transit is twice the star's radius, and it's velocity is $v_P$.  Lucky for us, the velocity of the planet is dependent upon its semimajor axis in the following way:

$v_P=\frac{2\pi a_P}{P}$

which means the ratio of the semimajor axis to the star's radius should be 

$\frac{a_P}{R_*}=\frac{P}{\pi t}$

where t is the duration of the transit.

It's important to realize that this equation is only true for equatorial transits (transits that happen along the equator of the star).  A more general equation would account for the transit time of a transit that travels along any chord on the star.

$T=T_{equatorial}(1-b^2)^{1/2}$

where b is called the impact parameter, which is basically the place on the star where the planet transits.  For equatorial transits, b is 0.  For transits where the planet just barely grazes the star--so it doesn't really cause any dip in the light curve at all--b is 1.  

$b=1-\delta^{1/2}\frac{T}{\tau}$

where $\tau$ is the time of egress--the time from the moment the planet first touches the star to the moment the last point of the planet crosses the first edge of the star--and delta is the depth of the transit.  

One of the parts of the question asks us to show that the ratio found above is related to density.  I started with the equation for density

$\rho_*=\frac{M_*}{4R_*^3}$

assuming that $\pi$ is 3.

But we can simplify this further by using the mass-radius scaling factor found in this post, which says that mass scales proportionally with radius.  

$\rho_*=\frac{1}{R_*^2}$

Because of this relationship, we can say that the planet's semimajor axis is related to the star's density in the following way:

$a_P=\frac{PR_*}{\pi t}=\frac{P}{\pi t sqrt{\rho_*}}$

If we know the radius of the planet (which we do, because we know the ratio of the radii of the two objects and the radius of the star), we can also find its density based on the 

$\rho_*=\frac{1}{R_*^2}$

equation.  

Here we have it.  Armed with three simple (I say simple, but in actuality, those graphs are not too easy to get) pieces of information, we can find the following properties

• ratio of the radii of the two objects in the system
• semimajor axis of the planet (which we can then use along with the mass ratios to find the semimajor axis of the star)
• the densities of both the star and the planet
• the impact parameter

Like I said at the beginning of this post, light curves are fun!

  

WS 10.2, Problem 2: Wobble, Baby, Wobble

a) Astronomers can detect planets orbiting other stars by detecting the motion of the star--its wobble|due to the planet's gravitational tug. Start with the relationship between $a_p$ and $a_*$  to find the relationship between speed of the planet and the star, $v_p$ and $v_*$ .

We started with the basic equation 

$a_*m_*=a_Pm_P$

We can also relate the semimajor axis and velocity in the following way

$v=\frac{2\pi a}{P}\Rightarrow a=\frac{vP}{2\pi}$

The first equation can now be rewritten and simplified as 

$\frac{v_*P}{2\pi}m_*=\frac{v_PP}{2\pi}m_P$

$v_*m_*=v_Pm_P$

b) Express the speed of the star, K, in terms of the orbital period P, the mass of the star $M_*$  and the mass of the planet.  Convert your units so that your expression is given as a speed in meters per second, with P measured in years, $M_*$ in solar masses and planet mass in Jupiter masses. 

Again let's start with the last equation from part (a)

$v_*m_*=v_Pm_P$

We can rearrange this to find the equation for the velocity of the star

$v_*= \frac{v_Pm_P}{m_*}$

We also know the following relationship between period and velocity

$v_P=\frac{2\pi a_P}{P}$

So now we can find K, the speed of the star

$K=v_*=\frac{m_P}{m_*}v_P=\frac{m_P}{m_*}\frac{2\pi a_P}{P}$

c) We can measure the velocity of a star along the line of sight using a technique similar to the way in which you measured the speed of the Sun's limb due to rotation. Specfi cally, we can measure the Doppler shift of stellar absorption lines to measure the velocity of the star in the radial direction, towards or away from the Earth, also known as the "radial velocity." What is the time variation of the line-of-sight velocity of the star as a planet orbits?

To someone who's grown up thinking that the solar system revolves around the sun, this problem might be a little bit conceptually difficult, because if the sun is truly the center of the solar system, it shouldn't move.  The point of this problem, however, is to show that the sun is not the exact center of the solar system (as I'm typing this, a bunch of dead white men are probably rolling in their graves because this problem debunks the work they devoted their lives to).  Instead, the sun wobbles--it moves away from the system's center of mass as the planet moves.  

If a planet is moving away from our viewpoint, the sun--assuming it only moves in our visual plane, and not up and down--is moving toward us.  If the planet is moving toward us, the sun has to compensate in its wobble and move away from the center of mass, or away from out viewpoint.  The sun's radial velocity would appear to be 0 (moving neither towards nor away from us) when the planet is at a point in its orbit when it's not moving towards or away from us, so when it's on the side of its orbital path.

d)  Sketch the velocity of a star orbited by a planet as a function of time. Denote the maximum velocity as K, and express the x-axis in terms of the number of periods, in intervals of P/4.

For this graph, we're starting at the point where the radial velocity is zero. The K is at it's peak hi and low at every other 1/4 of a period.  

e) What is the velocity amplitude, K, of an Earth-mass planet in a 1-year orbit around a Sun-like star?

This is another simple plug-and-chug problem where we take the equation from part (b) and substitute the sun's values where necessary.  

Thanks to WolframAlpha, I was able to find a K value of about 9.4 cm/s.  Earth's pretty freaking slow. 

WS 10.2, Problem 1: Center of Mass

In the first part of this problem, we were asked to draw a picture illustrating the relationship between the masses of a star and planet and their center of mass.

The relationship between the center of mass and and masses is

$x_{COM}=\frac{(-a_*m_*)+(a_Pm_P)}{m_*+m_P}$

If we set the center of mass to 0, like it is in the picture, the equation becomes 

$a_*m_*=a_Pm_P$

b) In a two-body orbital system the variable a is the mean semimajor axis, or the sum of the planet's and star's distances away from their mutual center of mass.  Label this on your diagram. Now derive the relationship between the total mass, , orbital period P and the mean semimajor axis a, starting with the Virial Theorem for a two-body orbit (assume circular orbits from here on).

So we start with the Virial Theorem like the problem told us to.  

$K=-\frac{1}{2}U$

$\frac{1}{2}(m_*v_*^2+m_Pv_P^2)=\frac{Gm_*m_P}{a}$

In order to conserve momentum, the periods of the star and planet have to be the same.  

$v_*=\frac{2\pi a_*}{P}$

$v_P=\frac{2\pi a_P}{P}$

The star is so much more massive than the planet, so it moves a lot more slowly.  When the two are compared, the velocity of the star is practically 0.  

The distance from the center of mass to the planet, $a_P$ is also a lot larger than the distance to the star, $a_*$.  Since $a=a_*+a_P$, $a=a_P$.  

We can now rewrite the Virial Theorem as 

$m_P\left ( \frac{4\pi^2a_P^2}{P^2} \right )=\frac{Gm_*m_P}{a_P}$

and we can rearrange it to say

$m_*=\frac{4\pi^2a_P^3}{GP^2}$

which is basically just Kepler's Third Law of Motion, so yay!

c) By how much is the Sun displaced from the Solar System's center of mass (a.k.a. the Solar System \barycenter") as a result of Jupiter's orbit? Express this displacement in a useful unit such as Solar radii.

This should be a pretty simple plug and chug problem.  

$a_\odot M_\odot=a_{Jup}M_{Jup}$

$a_\odot=\frac{a_{Jup}M_{Jup}}{M_\odot}$

Plugging in all of those values, we find that the sun is displaced $5\times 10^4$cm, or $7\times 10^{-7}$ solar radii.  

WS 10.1, Problem 2: White (I still don't know why the plural is spelled this way) Dwarfs

A white dwarf can be considered a gravitationally bound system of massive particles.

a) What is the relationship between the total kinetic energy of the electrons that are supplying
the pressure in a white dwarf, and the total gravitational energy of the WD?

This part is pretty much just a bunch of algebraic manipulation of the Virial Theorem.

$K=-\frac{1}{2}U$

The above equation, though, is only useful for relating the energies of one electron.  In order to find the relationship for the whole white dwarf, you have to multiply by the total number of electrons.  

$N_e\frac{1}{2}m_e v^2 = \frac{3}{10}\frac{GM^2}{R}$

but because of the relationship between momentum and velocity, 

$N_e\frac{1}{2}m_e \frac{p_e}{m_e}^2 = \frac{3}{10}\frac{GM^2}{R}$

$N_e\frac{p_e^2}{m_e} = \frac{3}{5}\frac{GM^2}{R}$

b)  Express the relationship between the kinetic energy of electrons and their number density n (Hint: what is the relationship between an object's kinetic energy and its momentum?)

So, we start with the last equation from part (a).  

$KE=N_e\frac{p_e^2}{m_e}$

In the problem, we're given the following information

• $\Delta p\Delta x>\frac{h}{4\pi}\Rightarrow \Delta p\sim \frac{1}{\Delta x}$

• $\Delta p\approx p$

• Volume\sim \Delta x^3$

If number density is number over volume, 

$n_e=\frac{N_e}{V}=\frac{N_e}{\Delta x^3}\sim \Delta x^{-3}$

We went through all of that so that we could find a relationship between the $p_e^2$ in the equation from part (a) and the number density.  

$p_e^2\sim n_e^{2/3}$

The total number of electrons, $N_e$, is equal to the number of protons, which is equal to 

$\frac{M_*}{m_p}$

After all of that, we can write 

$KE=\frac{M_*n_e^{2/3}}{m_p m_e}$

c) What is the relationship between $n_e$ and the mass M and radius R of a WD?

We kind of started solving this part in part (b).

$n_e=\frac{N_e}{V}$

We can get rid of a lot of constants in this equation to end up with 

$n_e\sim \frac{M}{R^3}$

d) Substitute back into your Virial energy statement, aggressively yet carefully drop constants, and relate the mass and radius of a WD.

First, let's recall the the equation from part (a).

$N_e\frac{p_e^2}{m_e} = \frac{3}{5}\frac{GM_*^2}{R_*}$

Based on all of the relationships we've found in parts (b) and (c), we can rewrite this equation as 

$\frac{M_*n_e^{2/3}}{m_p m_e}=\frac{3}{5}\frac{GM_*^2}{R_*}$

First, let's get rid of the constants we don't really need to have a basic understanding of the relationships between these quantities 

$M_*=\frac{M_*^2}{R_*}$

Finally, after simplifying, we find that the mass and radius of a White Dwarf are inversely proportional 

$M\sim R^{-1}$

WS 10.1, Problem 1: 'Til the Fusion Ends

We've talked about the birth of stars in molecular clouds. We also briefly discussed the main
sequence, on which stars are in hydrostatic equilibrium owing to energy generated by nuclear fusion in their cores. Now let's investigate what happens when a star like the Sun can no longer support itself via nuclear fusion.

a) At what rate is the Sun generating energy?

At first, we were confused by this question.  We knew it couldn't possibly be as simple as we were making it.  Then we realized we were being typical Harvard students and overthinking things, and that luminosity is given in ergs per second.  So, the sun is generating energy at a rate of

$L_\odot=4\times 10^{33}$ ergs/s

b) If fusion converts matter into energy with a 0.7% e fficiency, and if the Sun has 10% of its mass available for fusion (in the core only), how long does it take to use up its fuel supply? What is the general relationship between the mass of a star and its main-sequence lifetime?

The first step was to find the mass of the sun available for fusion.  

$(.1)(M_\odot)=(1\times 10^-1)(2\times 10^{33}g)= 2\times 10^{32}g$

The second step was to find the energy produces by fusion in the core using the equation for energy that I always knew, but never really knew when to use.  

$E=M_{fus} c^2=(2\times 10^{32}g)(3\times 10^{10}cm/s)$

But, the question states that the sun's fusion is only 0.7% efficient, so we have to multiply the energy by the efficiency.  

$E=M_{fus} c^2=(7\times 10^{-3})(2\times 10^{32}g)(3\times 10^{10}cm/s)=1.4\times 10^{51} ergs$

Finally, using dimensional analysis, we were able to find the final equation for the time.  

$t=\frac{E}{L_\odot}=4\times 10^{17}s\approx 1\times 10^{10} yrs$

d) The core will collapse until there is a force available to hold it up. One such force is supplied by degeneracy pressure. The pressure inside of a white dwarf star is provided by the motion of electrons. The electrons are in a tough situation: they can't occupy the exact same state, but there's not much room for them to coexist easily inside of a dense white dwarf. As a result, they must always be in motion to avoid other electrons (roughly speaking). This e ffect becomes important when the inter-particle spacing is of order the de Broglie wavelength $\lambda$, which is related to the momentum via 

$\lambda=frac{h}{v}$  

For a stellar core of a given temperature, which particles reach this critical density fi rst: electrons or protons?

The foundation of our solution to this problem was the equation for momentum:

$p=mv$

We had to compare the momentums of the electrons and photons in the star.  

$p_e=m_e v_e$

$p_p=m_p v_p=1800m_e v_p$ 

Because protons are just so much more massive than electrons, the momemtum of a proton is a lot bigger than the momentum of an electron.  Therefore, protons have a shorter de Broglie wavelength, so the protons reach their critical density first.   

e) If a typical white dwarf has roughly half the mass of the Sun and the radius of the Earth, what is the typical density of a white dwarf in grams per cubic centimeter? What is the volume of white dwarf material that weighs as much as a car?

This was pretty simple. 

$\rho_{WD}=\frac{m}{v}=1.5\times 10^6 g/cm^3$

We then figured that the average car is about 1 metric ton, so about $1\times 10^6$g.  

$v=\frac{m_{car}}{\rho_{WD}}=0.67$ cm^3