Hordes Are Made of People, Too

A few months ago, I was casually scrolling through my Facebook newsfeed when I saw a post by Dr. Chanda Prescod-Weinstein about people protesting the construction of a telescope in Hawai'i.  I didn't think much of it at the time, but kept scrolling and eventually went back to watching some terrible sitcom on Netflix. Some time after that, my Facebook and Twitter feeds exploded with stories about that same telescope. Suddenly, I couldn't look at any form of social media without hearing about Mauna Kea or the Thirty Meter Telescope (TMT). I started paying attention then (after all, when Khal Drogo sends a message, you better listen), and now I have lots of feelings about the subject.

 

For those of you who aren't friends with a lot of socially aware astronomers on Facebook and didn't get that same barrage of TMT news, here's what's going on:  Astronomers want to build the TMT on top of Mauna Kea in Hawai'i.  Mauna Kea is a sacred mountain. There are many people in Hawai'i (and elsewhere) who would rather the astronomers build their very large, very destructive telescope somewhere else.  

I started actively seeking out articles about this conflict, written from both perspectives. I found lots of astronomers who support the construction TMT and I found lots of astronomers who oppose it.  You know what else I found lots of? Language that lumps entire groups of people into one single being. It's "the protesters" this, or "the Protectors" that (if the author knows what's up), or "horde of native Hawaiians attacking" (if the author really doesn't know what's up).  And that's not cool!

I'm not really here to to talk about what I think of the TMT. That's a topic for another blog post. But regardless of what side of this argument you're on, it's important for everyone to realize that this movement is made up of individuals, each with their own backgrounds and motivations for being involved. I recently had the opportunity to spend some time on Mauna Kea and meet some of the people those articles are about. I'd like you to meet them, too. (Anonymously, of course, because ain't nobody got time to rudely give away people's identities without their permission.) 

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S was the first person I met when I approached the Protector tent, shy and awkward and totally unsure of how to initiate conversation.  Within 30 seconds of meeting me, S gave me a hug and offered me a donut. He won me over then and there. He won me over again when he told me that he "used to be just a normal guy" until a few months ago, when he heard about the TMT and all of the consequences its construction would have. He quit his job and bought a one-way ticket to Hawai'i island, and he's been there ever since. 

The first time I saw P, he was answering questions about his plans to repopulate the native plant life on Mauna Kea. When I asked him about it later, he talked about the different kinds of plants he wanted to bring back and their various purposes, some of them medicinal. He heard I was an astronomy student and got so excited, not angry like some of the articles led me to expect.  When I hugged him goodbye, he was on his way out to the garden to plant some more herbs I had never heard of. 

T is a farmer who lives close to the mountain. She and her partner practice a specific kind of farming that uses the natural waterflow that comes off of Mauna Kea. In between telling me about the damage the TMT would do to one of the largest aquifers on the island, T offered me chili, fruit, and a local tea that she had brought up the mountain for lunch.  She told me my name sounded like the Hawaiian word for "to dream" and we bonded over a love of science. We touched noses before she left. 

L has been on top of the mountain almost every day since people started occupying the mountain in response the first attempts to break ground for the TMT.  He's a teacher, and he's fluent in Hawaiian.  When I told him I was trying to learn a little bit of the language, he sat down with me for 20 minutes to go over some basic vocabulary. l asked him why he was there. He said he felt a responsibility to his family, to his aunt who signed the anti-annexation petition back in 1897.He talked about the power of love to fix all troubles, and I kind of felt like I had been transported back to the 60's, but I was into it.

K is a college student just like me, studying environmental science. She knows people who have dropped out of school to become involved in the movement.  Like most of the others, she offered me food, but it was my first day on the mountain and I was too shy to take it. She asked me why I was there, and when I told her, she led me around and introduced me to people, making sure I knew I was welcome. 

J and R are the sweetest married couple. He's been retired for over 10 years and she's excited to retire next year. They've only been off the island once. J heard that I was an astronomy student there to ask questions for my senior thesis and offered to give me a tour of the land. She showed me plants that looked like swords sticking out of the ground and bushes that smelled like fish when I rubbed them. When R got back from wherever he had been, she asked if I wanted to go hiking with them. We hiked to the top of a sizable hill, and along the way, they told me about their children, life-long friendships, marathons they had run together.  They promised me a place to stay if I ever found myself in Hawai'i again and I made them promise to look me up if they ever found themselves in Boston. 

These are just a few of the people I had the pleasure of meeting, but I think you get the picture. 

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When we talk about groups that form around controversial matters, like the Protectors or Black Lives Matter, we have this tendency to remove their humanity.  We talk about them as if they're a Borg cube, sharing thoughts, plans, and motivations. I don't know which is the chicken and which is the egg here, but when we do this, it allows a few things to happen:

1. It's easier to perform (and write about) inhumane acts that are direct consequences of racially charged colonization if you don't focus on the victims as humans.
2. It makes it so that the actions of one person, no matter how far removed they are from the group's agenda, represent the entire movement. 
3. It justifies condemning the group for being "disorganized."  Neither the Mauna movement nor BLM claim to have centralized leadership, yet we expect them to act as if they do because of the way we present them in the media. 

It's time to stop this and get to know the individuals within the horde. Maybe then we would see our actions, past and present, in a different light. 

Faces Like Mine

I spent so much time this summer discussing racism, sexism, and any other (legitimate) "ism" you can think of.  I sat in discussion circles where we learned vocabulary, talked about historical events and motivations that lead to the conditions we have now, and brainstormed on how to enact real and positive changes in our society.  In all of those discussions, the one thing that never failed to be mentioned was representation in the media.

I'd now like to direct your attention to this video:

 When discussing race, there are a few questions that are bound to come up.

• Why are there so few people of color in STEM fields? 
• Why are crime rates among Blacks and Latin@s so high? 
• If white people can realize the American Dream and make successes out of nothing, why can't POCs? 

There are so many historical, political, and economic answers to those questions.  But this blog post is going to focus on one: representation of POCs in the media.

I've heard several times (mostly from older white men) that women and POCs aren't very present in STEM fields because they just aren't interested in STEM.   I'm sorry, what?  You mean to tell me, old white dude, that entire demographics of people are significantly less interested in science and math than you are? I gueeeeeeess that could be it.

OR, could it be that we (the women and POCs) are taught practically from birth that we don't belong in those fields?  My mother is a hardcore feminist, but even she gave me dolls to play with as a young child instead of trucks or toy rocket ships.  I watched Star Trek and Contact and Star Gate and countless other space-themed things growing up.  And you know what I learned from them?  I learned that, unless I was lucky enough to be the ONE black person in a cast of about 100, I didn't really belong in the field. All of that worked together to make it so that I didn't actually believe I could make it as an astronomer until last summer. LAST SUMMER!! That's absurd.

I've talked to people who believe that Black and Latin@ communities have higher crime rates than white communities because there's something about those racial groups that makes the people in them morally inferior to whites.  Now, I feel the need to make it abundantly clear that I'm not condoning (most) crimes.  But (ignoring the fact that Black and Latin@ people are more likely than whites to be persecuted for identical crimes, source here) perhaps there's a reason more Black and Latin@ people feel it's necessary to turn to a life of crime? Could that reason maybe be the fact that most roles in movies and on TV for Black and Latin@ actors revolve around crime?  I think it might be.

And finally, I've literally had people say to my face that white people can work their way out of poverty because they work hard and Black and Latin@ people are lazy.  I don't know about you, but there's just something about characterizing entire races of people like that that doesn't sit right with me. Instead, I wonder if it might have something to do with the fact that most AMerican Dream movies are about white people.  Maybe, if POCs saw people who look like them succeeding just as often as they see whites succeeding, they would start to see it as a possibility.  Because I guarantee that there little Black and Brown children in this country who don't even think success is something they should dare dream about.


This might be the point in the blog post where you say, "But Moiya, what about all those great Black and Brown athletes and singers? Aren't they good role models?" 

Yeah, sure.  Of course there are a few bad apples **cough cough** Chris Brown **cough cough** but for the most part, Black and Brown performers let young Black and Brown people know that they could make it big one day.  There's just one problem:  They're only performers.  I'm not saying that to belittle those professions, but what about the little Black boy who wants to grow up to be a lawyer, or the young Latina girl who wants to be a marine biologist?  Who are their role models? They don't have any, and that's truly heartbreaking. 

I'm not saying that all of the prejudices in the world would disappear if Hollywood and other major film and television institutions suddenly decided to cast POCs, but it sure would make a hell of a difference

Summer Reflections

Another summer internship season has come to a close.  My projects have wrapped up, I've given my presentations and handed in my papers, and I've said my goodbyes. Tho only thing I haven't done -- because the last two weeks have been such a whirlwind of science, presentations, and moving -- is reflect on what this summer has meant to me, so that's what I'm going to do in this blog post.

Science
In case you haven't devoted most of your attention to following my blog and remembering everything I've ever written (and I guess I can't really blame you too much if that's the case), I spent the summer researching the exoplanet system Kepler-186.  I did other things too, mostly regarding galaxies, but this project has a more concrete result, so I'll focus on it.

Kepler-186 is about 500 lightyears (\(\approx\)150pc) away.  The system is made of 5 planets orbiting an M-dwarf star (about half the size and mass of our own sun, but less than 10% the brightness of our sun).  The 5th planet, K-186f, is famous in the exoplanet community.  Hell, it's even famous outside of the astronomer community, as evidenced by the fact that it has its own Wikipedia page.  Its fame comes from the fact that it's practically Earth-sized (\(1.06 R_{Earth}\), according to my calculations) and is just the right distance away from its star that it could hold liquid water. 

There are some downsides to this system being so famous.  Mostly, it just means that any work I do regarding this system won't be new, but that's okay. I wasn't really looking to do completely original research this summer.  I wanted to do research that could form a scientific basis for the book I want to write as my senior thesis, and that's exactly what I did.

The big question I had to ask about K-186f was, "Is it habitable?"  Well, here's my answer in picture form:

 

Obviously, we don't have the technology yet to go and visit this planet, so we can't answer this question with certainty.  All we can do is run the planet through "habitability tests" and see if it could pass. The plot above shows that it (probably) passes the first few tests we put it through. 

The first is the Planet Test.  The dashed line at the bottom indicates the radius of the smallest exoplanet we've ever found (just slightly larger than the moon).  All of the planets are above that line. Yay.

The second is the Composition Test.  There are roughly two types of planets: rocky and gaseous.  Since it's pretty difficult for is to stand on gas, a planet has to be rocky for it to be habitable. The dashed line at the top of the plot shows the radius at which planets tend to stop being rocky, according to Leslie Rogers and her collaborators. All of the planets are below that line. Another yay.

The third and traditionally most important test is the Liquid Water Test. The grey box shows the range of distances from the host star where the temperature on the planet could be just right to hold liquid water.  You can see that K-186f spends its entire orbit (the width of the points represents the range of distances from the star that the planet experiences throughout its orbit, because none of the orbits are perfectly circular) in this Goldilocks Zone. A third yay!  

There are still other tests that we need to put K-186f through, but I'm happy with three yays for now.

Feelings 
I'll admit that I was a little bit jaded at the beginning of the summer. I knew I was going to do an exoplanet project but was convinced that I hated exoplanet research.  I was just as convinced that no group of people would ever be able to compare to the friends I made at NRAO last summer.  And, if I'm being honest with myself and you, I wasn't too happy about being part of a program meant exclusively for Black and Brown students.  It felt like I was cheating or receiving special treatment or admitting that I wasn't good enough to compete with the white kids.  (None of those things were true, of course, but I can't control the thoughts that run that deep in my head.)  It wasn't long before all that jade faded away.

I haven't forgotten about galaxies.  I still think they're really, really cool.  But I now realize that I shouldn't completely write off exoplanets as the most boring area of astronomy research.  I actually think they're, dare I say it, fun to study.

The NRAO kids from last summer are still some of my best friends -- both in and out of the astronomy community -- but I got pretty damn close to the Banneker kids, too.  And as much as I love my NRAO friends, I didn't need them as much as I needed my Banneker friends.  That was actually the best part of this summer. Being around so many talented Black and Brown people and forming such close relationships with all of them was literally a life changing experience.  It alleviated some long-held biases of mine, ended my life-long habit of not feeling "black enough" to spend time with other Black people, and opened my eyes to so so so many issues that exist, hidden, in our society.

I know this blog post wasn't the most well-written.  I had a lot of thoughts and just needed a way to get them out of my head.  The gist of this post, though, is that I needed this summer more than I ever could have realized.

Anti-Racism MUST EQUAL Anti-Sexism

I love Key and Peele.  Their sketches occupy at least 5 of top 10 spots on my list of favorite internet videos.  But lately, watching their videos has made me increasingly uncomfortable.  I'd like to take this moment to simultaneously thank the Banneker Institute and curse it (okay, not really that second one) for opening my eyes to the things in the world that I'm now starting to identify as problematic.

The show just premiered its fifth season. If each season has a dozen or so episodes and each episode has 4 or 5 sketches, that means there are about 200 K&P videos floating around the internet. That's a lot of videos to comb through and analyze for potentially (most likely unintentionally) offensive content. To spare you all that, I'll just talk about one(ish) in this blog post.

That one is Negrotown.

 

It's funny, right? And it's a catchy song with lots of cool dance moves and flashy outfits, all of which are things I really appreciate. It even provides a really nice satirical commentary on the current state of affairs in America.  

If you doubt that any of the things alluded to in this video are true, I encourage you

1. Watch this TED Talk on the the injustice of our so-called "justice system" 
2. Look up names like Sandra Bland, Eric Garner, Tanisha Anderson, and Tamir Rice.  Honestly the list of names could go on and on; these are just a few of the well-known ones. 
3. Read books like Claude Steele's Whistling Vivaldi or Jean Halley's Seeing White.
4. Or, if all of that is too much, literally just spend 5 minutes watching the news or scrolling headline stories on the internet, because this shit is everywhere. 

If you already believe that the things K&P mentioned in the video are indeed real, then you're ready for this next bit.

Go back to the video and start watching at \(\approx\)2:45.  This video is about the issues that Black people in America face, right?  Then why is it that the only complaint the Black women of Negrotown have is against other women and about their relationships with men??  I'm pretty sure Black women have bigger problems than "white women [taking black men] away." 

Maybe I shouldn't be making assumptions about the concerns of an entire demographic, but I'm definitely more worried about the fact that the median net worth of single Black women is less than 100 dollars, but the median for single white women is more than $40,000 (source here).  And about the fact that the infant mortality rate among Black women is more than twice that among white women (source here).  And about the fact that Black women earn 89 cents for every dollar that Black men make, and only 64 cents for every dollar a white man makes (source here). And so many other things.

I recognize that this video does not embody the entire Black rights movement. Nor is it demonstrative of the tone of every K&P sketch (some of them are incredibly feminist).  But it makes a nice litmus test for the mood of the movement.  Just as the problems solved in Negrotown are mostly those faced by Black men, or maybe I should say just as so few of the problems solved in Negrotwon are those faced by Black women, too few of the problems addressed by Black rights movements are aimed towards improving the lives of Black women.

I've participated in a few Black Lives Matter rallies and marches. And in each one of those few, we chanted one or two Black women's names, but the list of Black men's names went on and on.  Is this because only a couple Black women are being unjustly killed? Hell no!  It's because, though there might be a significant intersectional feminist movement, there isn't really an emphasis in most Black rights groups on intersection anti-racism. 

We should change that. 

Ian Czekala and Starfish

This blog post is a collaborative effort with fellow Banneker student, Justin Myles, who demonstrates brilliance in all aspects of his life, even though he goes to Yale.

Last week, our advisor John Johnson assigned us the task of finding a graduate student at the Center for Astrophysics and talking with them about their research.  His reason for giving us this assignment was many-fold.

1. It provides us with a (mandatory) opportunity to get to know one of the grad students in the department.
2. It gives us a chance to learn about a topic of research that might not necessarily be super close to our own.
3. The grad student we choose to “interview” gets free publicity.

It’s a win-win-win.  

That night, Justin and I both went home and scrolled through the list of CfA graduate students.  We both found Ian Czekala, and didn’t realize our overlapping intentions until the next morning, when we decided to do this project together.  (Who said Yalies and Harvard students couldn’t work together, huh?)

Ian, who had his first research experience as a summer student in primarily studies young stars and circumstellar disks. One of his recent achievements is developing a package called Starfish which fits an entire spectrum. This is a novel approach to spectroscopy, which is often limited to a narrow range of wavelengths (despite a wide range of data being collected) and a small number of species (e.g. Fe and Na). Starfish is written in Python, available on Github, and utilizes statistical methods -- all subjects which we’ve been learning about in class in the mornings. So we were both interested in learning more about Starfish.

Where other spectral fitting packages focus on fitting the spectral line itself, Starfish focuses on minimizing the residuals (or the difference) between the observed spectrum and the model while accounting for the covariance introduced by systematic discrepancies in the models.

In the above plot, the synthetic spectrum is shown in red and the data are in blue. The residuals are in black. Zooming in to the gray region, we can see a region in the residuals in which the noise is clearly not simply Poisson noise:

An autocorrelation reveals that there is significant correlation on scales roughly the size of a spectral line:

In each row of this plot, the left plot is the covariance matrix, which illustrates the covariance of adjacent pixels and the right plot shows the residuals of the synthetic spectrum fit to the data. It is computationally expensive to interpolate spectra, which is why the following method is useful: by identifying a region of relatively large residuals, and scaling the covariance values to be larger, the residuals decrease. This is shown by the progression from the first to the third row.

In particular, by adding first a global, then a local kernel to the covariance matrix, random draws influenced by the covariance matrix accurately predict the residual noise and in this way model the residual noise.

At this point, you might be wondering how Starfish could be used by the wider astronomical community.  Well, stellar astronomers aren’t the only people who deal with spectra.  Spectroscopy is a tool used in every sub-field of astronomy.  

Let’s say, for example, that you are an astronomer who studies the formation and evolution of galaxies.  (Though there are definitely some people who claim that studying galaxies is nothing more than studying large groups of stars at once.)  You’re working with several spectral lines from a single galaxy, trying to use them to determine the galaxy’s physical characteristics. How do you do that?

The short answer is: make a bunch of model galaxies and compare the line fluxes from those models to the actual line fluxes you observed.

That takes So. Much. Time. Modeling a galaxy is hard work for a computer.  Modeling a few thousand slightly different galaxies? Starfish could take off some of the strain by first identifying the model spectra that best match the observed lines. Using a flexible likelihood function like that advocated by Czekala et al. '14 would deliver realistic parameter estimates and uncertainties, while also potentially identifying any particular lines that are treated incorrectly by the models.

Prior Expectations and Bayes' Theorem

I am the most naive and gullible person I know.  I mean, I'm the reason they have those announcements in airports telling you not to accept any bags from strangers.  That's an extreme case, but there are more realistic applications of my gullibility, too.  Coin flipping, for example.

Coin flipping is my go-to method of settling disputes.  But that method is only fair if the coin is fair, meaning it's just as likely to return Heads as it is to return Tails.  That's not always the case, and this blog post is going to tell you why.

First, I'll start with a simple description of Bayesian Statistics.  As far as I can tell, this is exactly like the statistics we've been doing, but instead of letting the data dominate every step in the fitting process, we have things called "priors."

Priors are just expectations you have about what the parameters you're trying to find will be.

Let's say, for example, that someone hands you a bowl of ice cream, but doesn't tell you what it is.  You have a set of expectations -- or priors -- about what that ice cream will taste and feel like based on your experience (in science, that experience can be your own or it can be knowledge gleaned from already-done studies).  You expect that ice cream to be sweet and cold, right?  Well, plot twist! It's bacon ice cream.

Now that you've taken that bite and recovered from the initial shock, will you expect the next bite to be sweet? No, you know it's going to taste like bacon.  This is because you've adapted your expectations -- or modified your priors -- to match the data you observed when you took that bite.

In math language, this process looks like this:

$p(a|D) = \frac{\pi (A)\emph{L}(D|a)}{E}$

where \(\pi (A)\) is your prior expectation for parameter A (the flavor of the ice cream) and \(\emph{L}(D|A)\) is the likelihood of observing your Data D if A were true (what's the likelihood that you're eating ice cream if it tastes like bacon).  E is just a scaling factor used to represent the evidence you gather as you conduct more research. 

We wanted to test this out to gain a better understanding. We didn't have any ice cream readily available (and I wouldn't really want to eat bacon flavored ice cream, anyway), so we flipped coins instead.  In this case, the question we were trying to answer is: Is this coin fair?

We used an Oregon quarter from 2005 (with the custom state design, not the standard eagle) to conduct this test.  We flipped it 20 times and recorded how many times it returned Heads. 

We started with a prior of .25, meaning the coin should return Heads about once every four flips.  This was actually cheating a little, for the sake of learning.  We actually expected the coin to return Heads about half the time, but we wanted to see how the prior can be overtaken by the observed data. 

I don't know if it was a self-fulfilling prophecy type thing or if the coin was actually unfair (I'm leaning slightly toward the latter), but the coin returned Heads 5 out of 20 times. We plugged those numbers into our equations for likelihood and then plugged those numbers into the probability equation above, and voila! 

Everything converges around .25, which tells us that the probability of that 2005 Oregon quarter returning Heads really is 1 in 4.  I guess I know now which coin I'm using if I ever want to settle a dispute with someone.  

But, we weren't done.  Our data just happened to serendipitously match our priors.  That's great for science, I guess, but it's not all that great for learning.  So I re-did the analysis with the exact same numbers, but with a different prior, this time telling my computer that I expect the coin to be fair.  This is what I got:

The lines all appear to have the same amplitude because they've all been normalizes to have an amp of 1.  In reality, the lines corresponding to higher Ns would have smaller amplitudes because they represent smaller likelihoods.

Here, you can see that the prior is 0.5 and the first line is close to that (the green line corresponds to the point in the trial where we had flipped the coin 5 times and it had returned Heads twice).  But the others are far away from being fair.  They are closer to .25, which is the true probability of the coin returning Heads.  

I really liked this exercise.  It provided me with empirical evidence that there's a flaw in the way I view the world.  If something as basic as a quarter can lie so egregiously to me, I should really try to dial back that gullibility. 

"Fixing" Stereotype Threat

I was always a huge proponent of standardized tests.  The SAT?  That was my jam.  The ACT?  Loved it.  At least, I did until I took the physics GRE for the first time and completely tanked it.  Looking back at how well I had done on standardized tests before, I couldn't reconcile the astonishing difference in my performance.  I didn't understand what had changed. 

Then I read Whistling Vivaldi and, as my advisor John Johnson likes to say, I saw the Matrix.  I understood what was happening: stereotype threat.

Stereotype threat is a phenomenon that causes people to under-perform in situations where they're faced with a task that's associated with a negative stereotype about a group that person is a part of.  That sentence may not have made the most sense, so I'll give some examples. 

• Some time ago, people got it into their heads that women aren't good at math and science. Now, when women are put in a situation where they need to do math or science, the pressure is on them to perform well so as not to confirm that negative stereotype. 
• Everyone's heard the joke that white people aren't as good at sports as black people.  Because of that, white people feel extra (unconscious) pressure when they're competing in sports.

There are a lot of really awful things wrapped up in this phenomenon.  The first, which brings in the concept of intersectionality, is that no one person belongs to just one group, and the intersections of their social groups mean they experience different threats at different times. 

• Asian women simultaneously deal with the stereotype that women are bad at math and that Asian people are good at it.  Studies have been done to show that performance in cases like this depend on which stereotype -- or which facet of their identities -- the women were reminded of before they took the test. 
• Black women have to deal with two negative stereotypes surrounding their ability to do math and science.  Those two negatives don't cancel each other out (like they do in math, which I know despite the fact that I'm a black woman); they add up. 

The second is that stereotype threat has the worst effect on people who care about how they perform.  It's not enough to just be part of a negatively stereotyped group. You have to really want to do well. This means that the people suffering the most from this phenomenon are the ones who are already disadvantaged, but who have the drive and the passion to overcome the obstacles in their way. 

It's a pretty shitty thing for our brains to do to us, but not as shitty as it is for humans to let the culture that established those negative stereotypes in the first place to continue to exist.

Like I said, I didn't do well on the physics GRE.  I'm not saying the sole reason for that was because I was dealing with the effects of stereotype threat.  I could have studied more, and it definitely would have helped if I had actually taken the physics classes that teach the material the test covers.  But if you read Whistling Vivaldi, you will see that stereotype threat really does have a significant effect on test scores.

I'm planning on taking the GRE again in the fall, and I wanted to know if my awareness of stereotype threat would alleviate some of its effects.  Whistling Vivaldi didn't have any chapters on that.  I talked with my advisor about my questions and he pointed me in the direction of value affirmation.

Studies have shown  that asking people to list and talk about the things they value most about themselves can actually combat the effects of stereotype threat!  More than that, students who were explicitly reminded of a negative stereotype that applied to them (so, students who would be harmed the most by stereotype threat) scored just as high as the students in the group who faced no negative stereotype at all. 

Of course, now I'm left to wonder if my knowing about the positive effects of value affirmation will undo those effects altogether, but that's another blog post for another time.

I'm still struggling to let go of the idea that standardized tests are good tools for weeding out students.  Hell, I'm struggling to let go of the idea that we need to "weed out" students.  But at least I know that these are ideas I should consider letting go.  Now, if only we could get all of the old white men upholding the institution of The Standardized Test to fail one, maybe they'd see the Matrix, too.

Dark Side of a Planet

This is a long one, folks, but hold in there. I think it's one of my best yet. 

Since I declared Astrophysics as my major a year and a half ago, my friends seem to have delegated it to one of my primary defining characteristics.  "Do you know Moiya?"  "Oh yeah, she studies astrophysics, right?"  I'm okay with this.  One, there are weeks when I spend more time at the Center for Astrophysics than I do in my own room, so I can't deny that my life pretty much revolves around the subject.  Two, I absolutely love talking to people about astronomy.

My friends know this, so they ask me a lot of questions about the universe.  Sometimes, their questions are very specific: "What do they mean when they say the universe is expanding?"  But most of the time, their questions aren't really questions at all, but requests to hear something, anything, about astronomy.  Because of this, I've developed a sort of astro fact kitty that I keep in my back pocket at all times.  One of my favorite facts to pull out is: The moon is tidally locked with the Earth, which means that the time it takes the moon to rotate is exactly the same as the time it takes to revolve around Earth, so that we always see the same side of it. 

"Tidal locking of the Moon with the Earth" by Stigmatella aurantiaca - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Tidal_locking_of_the_Moon_with_the_Earth.gif#/media/File:Tidal_locking_of_the_Moon_with_the_Earth.gif

I love this fact because it's quick, relatively simple, and I spent a week last year learning about the science and math that explains it.  But I never thought much about the broader applications of this knowledge.  In my mind, the moon was tidally locked with the Earth and that was it.  But my mind was wrong, because all sorts of things get tidally locked, including planets!

One question you might have at this point (especially if you didn't follow the link above) is:  How does tidal locking work? 

If you want a really in-depth answer, I'm here to tell you that Google is your friend.  If a more qualitative answer will suffice, Googling isn't necessary.

Say you have two massive bodies in space, A and B, where A is much more massive than B.  The gravitational force from A literally changes the shape of B, forcing it to elongate, or bulge, along the axis that points towards A.  So, instead of looking like a basketball, planet B now looks more like a rugby ball.

       

Before B is tidally locked to A (B's rotation speed does not yet match its orbital speed), that bulge travels around B.  Depending on the relationship between the rotational and orbital periods (which one is longer than the other), the bulge will lag behind the planet in its orbit or point in front of it.  This asymmetrical bulge creates all kinds of messy forces, which act on the system until the bulge faces planet A, thus tidally locking B to A.


Cool. Now you (hopefully) have a better understanding of how tidal locking works.  But that's not really the point of this blog post. The point is to explore what it would be like to live on a planet that was tidally locked to its star. (Just the scientific implications, because, as a student of mythology, I could go on an on about the mythological and cultural implications of having one side of a planet in perpetual light and the other in perpetual darkness.)

I'm going to try to do some math later, so let's make some assumptions about this tidally locked system.

1. The planet is about 1/10th the size of Earth:  \(R_P = 6.4\times10^6 m\)
2. The planet's atmosphere is made of mostly oxygen.
3. The star around which that planet orbits is roughly the same size and temperature as our Sun:  \(R_{sun} = 7\times10^8 m\) , \(T_{sun} = 5800K\)

The way I understand it, there are two possible outcomes.

In the first, the planet rests right on the edge of the space where oxygen freezes:

$T_P^4 = \frac{T_*^4R_*^2}{4a^2} \Rightarrow a = \sqrt{\frac{T_*^4R_*^2}{4T_P^4}}$

This was found by setting the flux received by the planet from the star equal to the flux emitted by the planet. 

A quick Google search told me that oxygen freezes at \(\approx 50K\), so that's the temperature we'll use to find the distance, a, from the planet to its star. 

\(a_P \approx 10^{12} m\)

For reference, this is about 10 times farther than the Earth is from the Sun.

 Because it's just on the edge of oxygen's freezing point, the side of the planet that faces the star is almost frozen, and the side of the planet that faces away from it is frozen solid. Eventually, the ice would encroach upon the "warm" side and the entire atmosphere would become ice.  That would be the end of any life that could have possibly lived under such extreme conditions.

The second outcome is, at least for me, more exciting.  In this case, I want the planet to be close enough to its star that the whole planet would be warm enough to sustain life.  

When I was originally thinking about this, I toggled back and forth between thinking such a thing was possible and thinking the dark side would freeze no matter what.  I had memories that backed up both theories. I remembered reading about a deep, deep crevice on earth that was colder than the coldest places on the moon because it never received any sunlight, and I remembered spending summer afternoons sitting in shady spots that never really felt cooler than the sunny spots a few feet away.  Eventually, the memories of lazy summer days won, but I needed to back that intuition up with science.

To do this, I needed to revisit my old friend (when I say "friend," I really mean "bane of my existence") from a class on partial differential equations, the Heat Equation.  This equation, intuitively enough, can be used to describe the distribution of heat in an area over time.  Sounds pretty perfect for what I'm doing.  I recognize that this might be giving some people flashbacks to traumatic physics class experiences, so I won't treat this as a rigorous physics problem, but will instead do some mostly qualitative assessments.

We can project the sphere of a planet onto 2 dimensions because we're concerned with the distribution of heat over its surface.

 

This projection becomes an oval, and it's fairly simple to solve the heat equation over a plane.  

$T_t = c^2\left ( \frac{\delta^2 T}{\delta x^2} + \frac{\delta^2 T}{\delta y^2}\right )$

where c is determined by the initial conditions, which are just how much flux the bright side of the planet is receiving.  Now that I have all of the equations, all I need to do is set my boundary conditions (an acceptable range of temperatures, say \(275K < T_P <320K\) that can sustain life) and I can solve for the right set of initial conditions.  Yay!  

Okay, now we know how the math behind creating a habitable, tidally locked planet would work.  But how would that manifest itself physically? In other words, what must the physical characteristics of this planet be in order to maintain a reasonably uniform temperature? 

• The planet has to have a lot of liquid (maybe water).  
• Those of you who use water to heat your home know that it's a really efficient way of transporting and re-radiating heat.
• There has to be some way to trap the heat in. 
• This could be like our Greenhouse Gas Effect, which uses Carbon Dioxide and other gases to trap the heat within our atmosphere. 
• If there's a Greenhouse Gas Effect, it means there has to be something producing that much greenhouse gas, which likely points to life existing on the planet! 
• There has to be some internal heat source.
• This could be in the form of volcanoes, which means there's plate tectonic activity on this planet, which some believe is essential for forming and sustaining life.  

I don't quite know what the implications are of all this yet; I just thought it was fun and cool to think about. Do with this what you will  :)  

If I'm a Bitch, You're a Bitch

Warning: there is some (more) profane language ahead.  

Last Friday, Prof. Brittney Cooper, professor of Women & Gender Studies and African American Studies at Rutgers University, came to talk to the Banneker Institute.  She was asked to talk about intersectional feminism, and though the things she talked about weren't necessarily new to me, it did remind me of a discussion I had with some of the other Banneker students the week before.

First, for those not hip with the social justice lingo, I'll define intersectional feminism.

"Intersectionality" is a phrase coined by black women like Kimberle Crenshaw and Audre Lorde in the second half of the 20th century.  It is a concept that refers to the fact that different people experience different types of oppression based on their various traditionally-discriminated-against identities.  For example, a White Gay Man and a Straight Black Woman each experience their own complicated forms of oppression.

"Intersectional feminism" (as I understand it) is the idea that, in order to be a feminist -- someone who seeks to establish equal rights for men and women -- one cannot ignore intersectionality.  Being a feminist means gaining equality for all women -- black, latina, gay, trans* -- so to be a feminist is to be an advocator for all of these marginalized groups.

Okay, definition time is (probably) over.  Now let's talk about that discussion I had with the other students.

I don't remember how we ended up talking about this, but, during our lunch break, we were talking about what words are appropriate to say and when and who is allowed to say them.  More specifically, we were talking about the use of the word "bitch." 

One of the male students announced that most women don't mind when gay men call them a bitch.  Let's unpack this and talk about what's wrong with that statement.

First, I'd like to point out that it was a man who decided to speak for the women in the room about how women feel when they're called a certain name.  Men, it is never okay to tell a woman how she should feel about something.

Second, what is the difference between a gay man calling a woman a bitch and a straight man calling a woman a bitch?  Absolutely nothing.  This statement was made based on a widespread idea that gay men and women share a special bond, that one (women) fully embraces the other (gay men) as one of their own.  Where did this idea come from? Well, it likely came from the media.  Shows like Will & Grace and Sex & the City and movies like Mean Girls and Clueless all show the "gay guy best friend" dynamic.  This trope has invaded our culture so much so that even I grew up wondering when I would find my GGBF.

But what do all of those shows and movies have in common? Oh yeah, everyone's white.  I would absolutely LOVE it if anyone could tell me about a well-known show or movie where the GGBF trope was used with two black characters, But I don't think you'll be able to find many.  Do you know why? Because you can have a white gay guy on a TV show, and you can have a straight white woman.  (Hell, you can even have a white lesbian, but she would serve a totally different purpose than the GGBF.)  But you can't have a black gay dude or a black lesbian or a black woman without turning them into caricatures, because that would just be too much otherness.

The point of that rant is that we've been conditioned to think that white gay men and white women inherently go together and that they share equal social footing.  The GGBF can call a woman a bitch and it's okay, because he's just one of the girls.  No.  A gay man is not the same thing as a woman.  Saying so just reduces a gay man to his stereotypical femininity and reduces the woman to her interest in men.

My response in the moment, because saying all of that would have taken too long, was "My reaction to being called a 'bitch' depends first and foremost on the person's intention and then on my relationship with that person."

Thinking we were done with the topic, I turned back to my soup.  But I was wrong.  That same student told the room that he would never react well to someone calling him a bitch.

I tried, readers, I really, really tried not to say anything, but I couldn't let it go.  I had to ask him why.

His response: "Because a bitch is what you call a female dog, and I'm not a dog."

I didn't quite believe that this was the whole reason, so I asked which was worse, being called a bitch or being called a dick?  I don't remember how he answered, but this is an important place to stop and unpack the situation.

As a man, no regardless of the circumstances, he would be offended if someone called him a bitch.  Could it be because the word is so deeply associated with women?  Would he have the same strong reaction if someone called him a whore, which is also almost exclusively used to refer to women?  What about any of the other tens of words that are used as derogatory terms for women, as opposed to the handful of male-specific phrases?

All of this points toward one thing that I wish we had spent more time discussing on Friday with Professor Cooper: the Patriarchy.

I recognize that this is kind of a buzzword.  It's been thrown around so much in the past few years that it's started to lose its meaning for some people, and for others, it's become a joke to use when talking about man-hating, bra-burning feminists.  But it needed to be overused, because the Patriarchy over-exists.

How do you know it's there?  You can tell because the only time women and men are put in the same group is when that man is gay, and therefore considered by many to be "less of a man."  You can see it in the fact that the number of derogatory slang terms for women is literally orders of magnitude higher than the number of exclusively negative slang terms for men.  I see it every time I or one of my friends get catcalled on the street.  It's there every time one of my male colleagues/peers thinks he needs to explain simple concepts to me.  It's in so many places that I would run out of allotted characters in this blog post if I tried to name them all.

I don't know how to take down the Patriarchy any more than I know how to end racism.  But I know that the first step is getting everyone to recognize that it exists.  Maybe blog posts like this are the answer.  Or maybe it's funny, culturally relevant videos.

Whatever it is, I hope we find it soon, because I'm pretty damn tired of this. 

You Are Here

I grew up without TV or siblings, so I spent most of my time as a child playing in the woods and reading Trixie Belden books (it's like Nancy Drew, but she's way more tomboy-ish and the mysteries are way more interesting).  This means that I've read a lot of books that most people have never heard of, which is pretty cool.  But it also means that I am seriously behind on my Disney/Pixar game.

Snow White, Cars, The Little Mermaid, Monsters University, Sleeping Beauty, A Bug's Life.  Haven't seen any of them.  One of the other Banneker students, Ana Colon, learned my secret and made it her goal to educate me in the ways of Pixar films. Her lessons started two nights ago with Toy Story.

Toy Story is a great movie. (I can say that now, because I've actually seen it.)  But one thing bothered me.  Where do they live???  Do you know that scene where Andy's mom takes him to Pizza Planet? The one where Buzz and Woody get lost, which sets up the whole conflict of the movie?  Well, if you don't know what I'm talking about, I really can't judge you because two days ago I was right where you are now. But here's the scene I'm talking about:

 

I saw that scene two days ago and, being the astronomy nerd that I am, I couldn't stop wondering where the movie is set.  They never mention it in the movie, or at least I didn't notice any clues pointing toward the setting. (Neither did the super observant people who write buzzfeed articles about the things no one ever notices in movies.)  But LOOK AT ALL THOSE STARS!!!  

I figured I could use those stars (and other clues from the movie) to pinpoint a location for this movie that I'm told is a classic for my generation. 

My plan was to try and use common sense to find a place in the U.S. (the first of my basic assumptions) where you could see so many stars.  The very fact that you can see so many stars means that the movie is set in a place very far away from a major city.  I grew up in the middle of nowhere in Pennsylvania, and I couldn't even see that many stars, just because Pittsburgh was two hours away. 

Light pollution map of the U.S. from darksitefinder.com

What else can the stars tell us? 

Well, we know it's summer, because we never see Andy go to school.  We also know that days are longer in the summer, so stars come out later at night.  Let's make our second assumption and say that Andy's mother wouldn't take him and his sister to dinner any later than 7:30 PM.  This means that the time stamp in the picture above would be around 8 or 8:30.  In the summer, the sky is not dark enough at 8:30 to see that many stars.  

UNLESS you live in Arizona, where Daylight Savings Time doesn't exist.  Right now, I'm sitting in Cambridge, Massachusetts where it is \(\approx\)11:30 PM Eastern Time.  In Utah, it is currently \(\approx\)9:30 Mountain Standard Time and it's dark enough to see stars.  In Arizona, it is currently \(\approx\)8:30 Mountain Standard Time, and even though it's "earlier" there than it is in Utah, it's dark enough to see stars. 

Based on this (late-night) logic, I would be willing to bet that a) Toy Story is set in Arizona or b) Andy's mom has him and his sister on a messed up eating schedule. 

Also, you can't see any distant mountains in any outdoor scene in the movie, which tells us Andy and his family live far from any tall mountain ranges.  

All of this together tells me (and maybe tells you) that Toy Story 1 takes place in South or Southwestern Arizona.  

 

So there you have it!  With just a little bit of common sense and a picture of some stars, I was able to answer a question that's been bothering me for the last two days.  This is the power of astronomy.  

P.S.  To check myself, I also tried running the picture through this really awesome website called astrometry.net , but it didn't return any matches.  So now I "know" where the movie was set AND I know that the Toy Story animators just drew random points of light when they made this scene.  Yay science!!

Nemo me impune lacessit

So far this summer, this blog has been so serious!  If I didn't know anything about myself besides what I read in this blog, I would think I was really boring.  So this post is a story of something utterly ridiculous that happened to me last week.

A few of the other Banneker students and I were watching Game of Thrones (one of them had never watched it before!) when we heard a strange noise coming through the window. After a few minutes, we realized it was someone playing the bagpipes!  Like, seriously, who expects to hear someone casually playing the bagpipes on a Wednesday night? Not us, so we got pretty excited about it.

The sound stopped, and we still couldn't figure out where it was coming from, so we went back to watching GoT. Fifteen minutes later, it started up again.  This time, we were determined to find the source of the music. 

Ana, one of the other Banneker students, and I ran downstairs to the courtyard (I was in such a hurry to find this bagpiper that I didn't even put on shoes), where we saw another student yelling up at a window.  Breathless, Ana looked at him and screamed "Are you looking for the bagpipes, too?!?!"  He just stared at us, so we ran away. 

We walked around our dorm, trying to figure out where the bagpiper could possibly be hiding, and then we heard it again. We ran back to the courtyard and there he was.  Just walking around.  Playing the bagpipes like that's something everyone does on a Wednesday evening in a college dorm courtyard.  It was the same guy Ana had yelled at minutes earlier.  He looked at us and said "You found him."

Obviously, Ana and I needed a way to document this adventure, so she asked if we could take a selfie with him.  He looked confused, but said yes.

We left him, telling him that he should definitely keep playing.  I'm pretty sure he thought we were insane, but, I mean, we weren't the ones playing bagpipes for everyone in the dorm to hear.

He was playing again today.  We didn't run down to see him, but we sat and listened to his song. 

I still have no idea what it's supposed to be.  

You Shall Not Pass!

This post isn't going to be scientific. It's not going to be about some revelation I had, or about my new-found interest in racial social justice. It's basically just going to be me whining about things.

I'm blocked. I have three or four different projects going on right now and I can't seem to move forward on any of them.

I tried to read up on radiative transfer modeling, but just ended up staring at my computer screen for an hour before I fell asleep.  I tried to work on some coding, but couldn't get my fingers to type anything. I tried to start work on my thesis and my mind was a total blank; no words would come to it.  It's like there's a little Gandalf inside my brain stopping me from moving anywhere.  I'm surprised I've been able to type anything in this blog post.

I guess all I can do is wait, because trying to force myself to produce results has already given me enough of a headache.

On the bright side, all of the sitting around I've done today gave me the chance to remember that I asked a riddle a few weeks ago, but never provided the answer like I promised I would. If any of you managed to solve it, congratulations!  I spent so long trying to figure it out before it drove me crazy and I had to ask my friend for the answer. If you didn't get it, don't feel bad. It's a doozy.  Anyway, here's the solution.  I know I asked about cubes, but cue balls work just as well.

Tips for Wannabe Allies

Last night, I had my first experience calling someone out for making a racist comment.  I figured this was the perfect time to write my first social justist-motivated blog post.

First, I should explain what happened.

I'm at Harvard this summer to do astronomy research, which means the people I spend most of my time with are self-proclaimed astronomy nerds, so much so that we all wanted to hang out around a telescope at 1:00 AM on a Friday night.  We also all happen to identify as people of color. 

When we got to the door that leads to the telescope, a man in the next room immediately assumed we didn't have access (despite the fact that we were able to open the door) and told us we had to leave. We bypassed that situation and made it to the telescope, but some people wanted to talk about it more.

One member of the group said he thought that the man had assumed we didn't have access because most of us presented as people of color, and POCs aren't expected to belong in places like Harvard astronomical observatories. He also said he thought that some people in the group had automatically accepted this man as an authority figure because he presented as a white male.  Another member of the group (let's call him Shaun) said that was ridiculous. He had accepted the man as an authority figure because he spoke so confidently. 

There was some back and forth between the two, so I decided to jump in and try to end it. I told Shaun that

1.  The man was able to speak so confidently because he was a white man and white men are made to feel comfortable in most predominantly white spaces.
2. With knowledge of the white supremacy culture we live in today, it's safe to assume that many interactions like this one are racially motivated.
3. Assuming that it was racially motivated, his reaction was a result of being brought up in a culture that assumes POCs don't belong in academic settings, as was Shaun's acceptance of this man's word as law. 

I've been reading up on this subject lately, thanks to the reading assignments we have as part of the Banneker Institute.  I've read that when people get called out for their actions, they often get defensive or angry, or they try to detour the conversation and distract the person who called them out.  But it's one thing to read about these tactics and another thing entirely to see them in action.

Shaun immediately jumped to defend himself and say he wasn't a racist.  In fact, according to him, he couldn't be a racist because he was a minority, too, and by saying that the man questioned us because we were POCs, we were being reverse racist.

That was when I lost most -if not all- of my respect for Shaun. But I still felt like I needed to turn this into a teachable moment (I'm actually awful at letting things go and allowing moments to pass without making them worse).  It's now been about 24 hours since the incident, so I don't remember exactly what was said, and even if I did I wouldn't transcribe it all here, but there was an argument that went on way longer than it needed to. 

By the end of the night, some people's feelings were hurt, at least one relationship was irreparably damaged, and everyone was frustrated.  All of this because one person couldn't distinguish between being told that his actions were borne of a racist culture and being called a racist.

So, I'd like to give the following advice to anyone who reads this post:

1. If someone says that something you've said or done was offensive (whether it was racist, sexist, homophobic, etc.), recognize that they are not calling you a bad person. Instead of jumping to defend yourself, simply apologize, learn from your mistake, and move on.
2. If you are in a position of power in a setting, or you pass for someone who traditionally holds a position of power, and someone from a marginalized group says they think they experienced prejudice, do not tell them they are wrong. If you don't understand the situation, ask them to explain it (but don't be upset if they say no, because it is not their responsibility to educate you). Otherwise, express sympathy, ask what you can do to make the situation better, or shut the hell up. 
3. Do not use the term "reverse racism."  Ever. Like, seriously, that's just a dumb move.

If we lived in a perfectly logical world, I don't think these guidelines would be too hard to follow, but apparently we live in one where these are easier said than done.  But we can't expect it to get easier to do if we don't say it a lot, so let's spread them around and educate the Shauns of the world. 

Data Fitting

Oh hai there, folks. This one's going to be super heavy on the statistics.  But this blog is supposed to be a record of what I do this summer, and I've spent at least 1/3 of my time this week learning and practicing statistics with a coding twist.  By the end of this blog post, hopefully you'll know more about statistics, too (or, at least a kind of stats that astronomers use a lot).

First, I'll say that for some background on the material covered in this post, you should check out this one.  It talks about sigma and normal distributions, which have been really integral to the work I've been doing this past week.

Now let's get to the new stuff.  I said it had a coding twist, which meant that I had to write Python (learn more about Python here) functions that took in a few input variables and returned useful values and plots. In this post, I'm going to describe those functions and maybe even a little bit of the frustration I felt while I was writing them.

The first function I wrote is used to generate data sets. It does this by returning the dependent values of a polynomial function of any degree. To make this work, the user has to input an array of independent variables (or x values) and the coefficients to each term in the function and they have the option of including a sigma value.  That all sounds kind of abstract, even to me, so let's use a concrete example.

Let's say you want to make fake data that follows the function \(y = x^2+4x-3\), and because you know from the "Sigmas" post that no measurement is perfect, you want to add some "noise" or uncertainty to that data.  Let's say that each measurement could be as much as 2 off from the "true" value.

To do this, the user would input an array of x values: 0, 2, 4, 6, 8, 10
and the coefficients for each term (in order of increasing degree): -3, 4, 1
and the sigma value: 2
and the function produces these y values: -2.380, 6.168, 28.457, 56.444, 94.166, 134.499

Yay! But now we have to actually use that data, which is what the other functions are for.

The second function  is used to calculate how well modeled data matches observed data. In statistics, this is known as the likelihood.  The function takes as input the data (either real or the set you generated using the first function), the array of x values you used, and coefficient values like you used before.

The function works by generating a list of \(\mu\)s using the x values and the coefficients. Those \(\mu\)s are then used in the following equation:

$ln(L) = -\frac{1}{2}\sum ln\left ( 2\pi\sigma_i^2 \right )-\frac{1}{2}\sum\left ( \frac{D_i-\mu_i}{\sigma_i} \right )^2$

Because there are sums, this function returns a single value that basically just tells you how close your model is to your observation. That value us used in the next function.

The third function is my favorite.  It was one of those things that wouldn't work for the loooooongest time, but once I figured it out and looked at it, it was almost embarrassingly simple. It takes as input the "observed" data you have, the x values, and arrays of values that you want to test for each coefficient.  For example, with the values above, I would test out arrays from [-4,-2], [3,5], and [0,2].  This way, it's guaranteed that when the function tests out all these different values, it will test out the true ones.

All it does is create a grid or cube where the axes are the arrays of coefficients you want to test and calculate the likelihood at each point.  By the end, if you make a contour plot of the final grid, you can literally see which coefficients have the highest likelihood of matching your observed data. If you put those coefficients through the first function, you end up with a line that claims to be the best fit to your observed data. It's so cool!  

But, with more than 3 coefficients (higher degree polynomials), you can probably imagine that the grid or cube will get really big and calculating the likelihood that many times would cost a lot of computer time.  Lucky for us, there's another, simpler, more mathematical way to find the best fit to the data.

 

 Thank you, Adventure Time, for expressing my exact feelings about times like this so well. 

The fourth function  is just a lot of matrix math. This isn't a math blog, so I'm not going to write it all out for you, but I'll give you the gist. 

The second term in the long equation above?  The bit after the summation symbol is Chi Squared (\(\chi ^2\)).  Chi Squared is really important to statisticians, but the most important thing I learned about it this week is that we want it to be minimized.  How do you minimize things in math? You take the derivative and you set it equal to 0.  So that's what we did.  We differentiated \(\chi ^2\) (with respect to the coefficients we're trying to find), translated the summation terms into matrices, and set it equal to 0. 

The function returns the best-fit values for the coefficients, just like the last one.  But this function can do polynomials of any degree and it's so much faster than the last. 

So, that was my week.  Well, that and a little bit of research, but I'll save those stories for another blog post.