Day 33 Banneker Institute

This post will also be a special post, it will be about another Harvard graduate student, Luke Kelley. I was lucky enough to interview Luke about his work and the life of a graduate student. Luke also gave advise that will help future graduate students. Luke's main area of research is in massive black hole mergers. He studies what processes mediate this process, what mechanisms, and how long it takes for this process to occur.

Luke's love for Astrophysics began when he first watched Star Trek. His fascination for science took him to UC Santa Cruz where he did research in particle physics. He then change into Astrophysics where all aspects of physics are used, unlike in some Physics sub fields where only one or two aspects of Physics are used.

Graduate school differs a lot when compare to other level of education. In graduate school you suppose to be self motivated and independent. This is unlike other levels of education where you are guided trough the learning process. Luke advised undergraduate students to focus on getting research experience, and good recommendation letters that will speak about your research potential. These are the key elements that will help undergraduates get into graduate school.

Life in graduate school is also a lot different, in the sense that you choose when to work. Some work early in the morning and some late at night but the key is to not get sidetracked. Another important factor is finding a balance so you don't over work yourself.

Luke Kelley's websites and contact info:
https://www.cfa.harvard.edu/~lkelley
https://www.youtube.com/user/lzkelley

lkelley@cfa.harvard.edu

Two really interesting simulations:

https://www.youtube.com/watch?v=w7d-7Q0fGHE

https://www.youtube.com/watch?v=JVZEAA-0058

"There is no great genius without a mixture of madness."

-Aristotle

Day 32 Banneker Institute

Bayes' Theorem

Hola mundo,

Today I'll be explaining Bayes' Theorem and it's use in predicting events. The Bayes' Theorem uses a prior hypothesis based on a set of parameters and prior knowledge of the system that you are trying to predict, but you do not build a hypothesis of the model only of the result. The theorem uses a likelihood that event happening, giving that your parameters are true. The prior and the likelihood are completely independent of each other. Bayes' Theorem also uses a term named evidence (E) which in my case I am not using because it's a normalization term.

$$P(a|D) = \frac{{\Pi}(a)L(D|a)}{E}$$
$\Pi(a)$ is the prior where a is the hypothesis in the form of parameters a.

$L(D|a)$ is the likelihood where D is the data of the system in question.

I use this theorem to find the probability of getting heads after flipping a fair coin. I ran a few trials and plotted the results of the formula below.

$$P*H^{k}(1-H)^{n-k}$$
Where P is the prior, His the probability of getting heads, n is the number of trials, and k is the number of times we get heads. Below is a plot with the probability density distributions of my trials. As you can see the maximum probability of my trials are close to 0.5 which is the probability of getting heads using a fair coin if you flip the coin an infinite amount of times.

 

"Happiness is when what you think, what you say, and what you do are in harmony."

-Mahatma Gandhi

Day 31 Banneker Institute

This post is going to be different, I am not going to talk about my progress in my research, but instead about a Harvard Astronomy graduate student, Fernando Becerra. Fernando gave some advise for undergraduate students that want to go to graduate school. He spoke about the differences between graduate school and other levels of education. And he described how the lifestyle of a graduate student is like.

Fernando decided to study because he wants to understand how the physical world works. The way that he wants to do this is through the laws of Physics, and the language of Mathematics. Fernando is currently researching star formations, galaxy mergers ,and the first black holes in the universe. His work involves writing computer programs that will simulate the environments that these systems are born in. You can see some of his simulations in his website linked below.

Fernando described that biggest difference between graduate school and other levels of education is that you more independent. In High School and undergrad you mostly absorb knowledge, but graduate students generate knowledge. In graduate school is when people start to think and work like a researcher. Upcoming graduate students should have an open mind, and should know that research is not the same as the classroom. The skills needed to be a good researcher are different from the skills needed to get a good grade.

Upcoming graduate students need to able to explore to ideas and be flexible to change research projects at the beginning of their graduate careers. They need to know that academia is NOT THE ONLY PATH. Graduate school is the time to train yourself so you can obtain the skills necessary for your dream profession. In terms of the lifestyle of a graduate student, there is no one style. Everyone decides when to work and when not to work. Some people work early in the morning, and others late at night. The main thing to keep in mind is to be consistent.

Fernando's Website and Contact Information:
http://www.cfa.harvard.edu/~fbecerra
fbecerra@cfa.harvard.edu

"Even if you fall on your face, you're still moving forward."

-Victor Kiam

Day 30 Banneker Institute

I manage to fix the eccentricity calculation by changing the units to SI units. The values for the insolation of the planets is still a little strange, but I need to analyse this a little deeper.

"You must be the change you wish to see in the world."

-Mahatma Gandhi

Day 29 Banneker Institute

After calculating all of my parameters for all three planets, I noticed that the values for the insolation and the eccentricity are too large. These values don't make physical sense, but the mathematics seems to be correct. The only thing that I can do is to go back and double check my python code for any errors.

"A leader is best when people barely know he exists, when his work is done, his aim fulfilled, they will say: we did it ourselves."

-Lao Tzu

Day 28 Banneker Institute

I was able to calculate the insolation (S) and the eccentricity (e) of all three planets. The insolation of a planet is the amount of thermal energy that a planet obtains from its host star. The eccentricity is the shape of the orbit of an object as shown below.

$$e = \frac{|1 -  (\frac{\rho_{obs}}{\rho_{true}})^{2/3}|}{1 + (\frac{\rho_{obs}}{\rho_{true}})^{2/3}}$$

$$L = \frac{r_{*}^{2}*T_{*}^{4}}{T_{sun}^{4}}$$

$$S = \frac{\frac{L*a}{r_p}}{(\frac{r_{earth}}{a_{earth}})^{2}}$$

"Life is a series of natural and spontaneous changes. Don't resist them - that only creates sorrow. Let reality be reality. Let things flow naturally forward in whatever way they like."

-Lao Tzu

Day 26 Banneker Institute

I was not able to calculate the eccentricity today, but I was able to calculate the limb darkening components $u_1$ and $u_2$. I did this by using the formulas shown below. These formula can be found in my adviser's paper, http://arxiv.org/abs/1308.0009v2.
$$u_1 = 2\sqrt{q_1}q_2$$ $$u_2 = \sqrt{q_1}(1 - 2q_2)$$

Limb darkening is the optical effect that is cause due to the amount of light receive from an object dependent on the distance that you are from the centre of the object, as shown above.

"Work hard for what you want because it won't come to you without a fight. You have to be strong and courageous and know that you can do anything you put your mind to. If somebody puts you down or criticizes you, just keep on believing in yourself and turn it into something positive."

-Leah LaBelle

Day 24 Banneker Institute

Today I finish calculating the inclination of the planets, the temperature of the planets, the transit duration of the planets, and their respective $\frac{a}{R_*}$ values. The next step in the process is to compare this values to the values in the following published paper http://iopscience.iop.org/0004-637X/804/1/10/. After this is done I will calculate the eccentricity of the planets, finally!

"The only place success comes before work is in the dictionary."

-Vince Lombardi

Day 23 Banneker Institute

More complex mathematics was done today. I calculated the transit duration by using the following formula.$$t_{14} = \frac{P}{\pi}arsin(\frac{R_*}{a}\sqrt\frac{(1 + \frac{R_p}{R_*})^{2} - (\frac{a}{R_*}cosi)^{2}}{1 - cos^{2}i})$$
I also calculated the temperature of the planets by using the following formula.
$$T_p = T_*\sqrt\frac{R_*}{2a}$$

"It is difficult to understand the universe if you only study one planet."

-Miyamoto Musashi

Day 22 Banneker Institute

Today I manage to calculate $\frac{a}{R_*}$ which is the ratio of the semi major axis and the radius of the star. Because a >> $R_*$ this is a good approximation to obtain the distance beween the star and the planet. $$\frac{a}{R_*} = \frac{\rho_* G P_p^2}{3\pi}$$

I also calculated the inclination of each planet, by using the impact parameter b and $\frac{R_*}{a}$

$$i = \cos^{-1}(b\frac{R_*}{a})$$

"There is nothing outside of yourself that can ever enable you to get better, stronger, richer, quicker, or smarter. Everything is within. Everything exists. Seek nothing outside of yourself."

-Miyamoto Musashi

Day 21 Banneker Institute

I finish calculating the parameters with their uncertainty for all three planets. After using Multinest in Dr. Kipping's computer which runs a Markov Chain Monte Carlo(MCMC) program. Below are a few of the parameters that I obtained through this process. The numbers on the left are the parameter values, the ones in the middle are the lower bound uncertainty and the the last numbers are the upper bound uncertainty.

Radius ratio: planet over star

[0.035206658283169809, -0.0010722697357797295, 0.0018395676087651774]

Period of planet b

[10.05403551325956, -0.00024616330074600512, 0.00025023083783537459]

Mid-transit time of planet b

[2456813.4188600937, -0.0010220510885119438, 0.0010063056834042072]

"The world is a dangerous place to live; not because of the people who are evil, but because of the people who don't do anything about it."

-Albert Einstein

Day 20 Banneker Institute

I think this is a good time to review all of the progress that has been made in the last 4 weeks working on the planetary system K2-3. From the raw data to the separation of the planetary light curves, and the histograms of the most probable values for the parameters that describe this recently discover system. Below are a few examples of this plots.

"You can search throughout the entire universe for someone who is more deserving of your love and affection than you are yourself, and that person is not to be found anywhere. You yourself, as much as anybody in the entire universe deserve your love and affection."

-Buddha

Day 19 Banneker Institute

Today not a lot of work was done on the project. The focus was mostly on the physics behind the rest of the parameters that I will have to calculate. Most of it is cover in the following paper http://arxiv.org/pdf/astro-ph/0206228.pdf. Dr. Kipping and I were specially looking at equation 3, and how we could used it to calculate the orbit distance divided by the stellar radius.

"Research is what I'm doing when I don't know what I'm doing."

-Wernher von Braun

Day 18 of Banneker Institute

With the histograms of the parameters, I was able to calculate the median value of each parameter and their uncertainty. The uncertainty is obtained by calculating +/- one sigma away from the median point. This can be illustrated by the diagram below, where μ on the x axis represent the median point.

"True knowledge exists in knowing that you know nothing."
 
-Socrates