A special kind of exoplanet

We are bounded in a nutshell of Infinite Space: Week 8: Reading #6: A special kind of exoplanet

Within the realm of extrasolar planets which have led to one of the greatest periods of extrasolar discovery in recent time, there are a couple of key planet types which we, as living beings who would be searching for a home soon enough (be it climate change or ice ages or the Sun running out of fuel at the cause) would find very interesting. These are the Super-Earths, planets found in the planetary search the Kepler Space Telescope was commissioned to do, which have the unique properties of being similar to Earth in some ways, but very different in others. They are some of the only planets we have strong reason to believe they have liquid water, while still being tremendously different to the Earth we’ve inhabited for the past couple million years. These planets are incredibly close to their star, so close that they usually have a steam-full, if not dominated, atmosphere, partially the reason we are able to confirm they do have water, but more on this in a second. Furthermore, the reason they are called Super-Earths is how they are incredibly massive, more massive than Earth but less so than gas giants like Saturn, as well as being made up of rock and gas. These Super Earths are among NASA and JPL’s best guesses for habitable planets, considering the new definitions of a habitable zone because of the existence of liquid water on these planets. The habitable zone considers a certain distance from the Sun at which liquid water can exist and be at a temperature we consider “livable”, although this definition is rapidly changing with the onset of these Super Earths and icy moons like Ganymede and Europa.

The means by which we are now able to detect exoplanets is described by Maoz in chapter 2.2.4 of Astrophysics in a Nutshell, explaining the process of taking the measurements of luminosities of a star to measure whether the brightness dips at any point and thus indicates an obstruction, and if this were sufficiently periodic and regular, there is a confirmation of the planet’s existence. This technique in particular, although there are other ones, allow us to measure the absorption and emission lines of the object blocking the star, thus allowing us to understand its composition and possible signs for compounds like water. This had led to discoveries like the Super Earths, and most recently an Earth-like planet near a Sun-like star, the planet being in the habitable zone. The NASA article reads:

July 23, 2015 -- NASA's Kepler mission has now confirmed the first near-Earth-size planet in the “habitable zone” around a sun very similar to our star. For more information about this latest discovery, visit: NASA’s Kepler Mission Discovers Bigger, Older Cousin to Earth. NASA's Kepler Space Telescope, astronomers have discovered the first Earth-size planet orbiting a star in the "habitable zone" -- the range of distance from a star where liquid water might pool on the surface of an orbiting planet. The discovery of Kepler-186f confirms that planets the size of Earth exist in the habitable zone of stars other than our sun. While planets have previously been found in the habitable zone, they are all at least 40 percent larger in size than Earth and understanding their makeup is challenging. Kepler-186f is more reminiscent of Earth. "The discovery of Kepler-186f is a significant step toward finding worlds like our planet Earth," said Paul Hertz, NASA's Astrophysics Division director at the agency's headquarters in Washington.

Now all that’s left is to visit these planets, someday.

 References: 

Maoz, D. (2007). Astrophysics in a Nutshell. Princeton: Princeton University Press.

http://www.nasa.gov/images/content/646920main_pia15622-43_946-710.jpg

http://ifa.hawaii.edu/~howard/ast241/exoplanet_properties.pdf

http://www.nasa.gov/ames/kepler/nasas-kepler-discovers-first-earth-size-planet-in-the-habitable-zone-of-another-star/

 http://www.nasa.gov/sites/default/files/kepler186f_artistconcept_2.jpg

Solving the Big Bang’s Issues

We are bounded in a nutshell of Infinite Space: Week 8: Free Form #10: Solving the Big Bang’s Issues

As we mentioned just a bit ago, the Big Bang theory has some clear problems, now it’s time to understand what these are. First off, there is the Horizon problem, which come from direct information received from the CMB. Here, astronomers measured points from the CMB in all different directions, and found the SAME, EXACT information, the same temperature, redshift, density population, number count, and beyond. This is confounding on account of the fact that for this type of uniformity to occur, all the points of the CMB need to be in contact with one another in the extremely early universe, in order for the information to be consistent. However, under the past Big Bang Model, this was impossible considering the size of the universe when it released the CMB is large enough for the light cones (the possibility of the light of each point at a point earlier in time) of the points measured to have met with each other and exchanged data.

The other clear problem with the Big Bang is the unlikelihood of our living in a Flat Universe, as opposed to a closed or open universe. As we saw in http://ay16-rodrigocordova.blogspot.com/2015/11/turns-out-euclid-was-right-about-math.html , there are several geometric configurations which could have described the ultimate fate of the universe, these being an open case (where the universe would expand endlessly, and quick enough that halos would not have formed in the early universe and thus galaxies would have never emerged), the closed universe (a continuous expansion and contraction of the universe, in a cyclic manner), and a flat case, which leaves the possibility of several types of expansion. We, according to many tests done on the CMB, live in a flat universe and thus also live in a precarious state of universal geometry which effectively allows for us to be now pondering these questions. How then, are we in such an unlikely state?

These two problems characterize the main issues we have observed in the universe (not based on theory and things we do not see, as is the case of the theorized but unperceived magnetic monopoles), the same problems astrophysicists and cosmologists worked to find a solution. Thanks to the work of Alan Guth, Andrei Linde, Paul Steinhardt, and Andy Albrecht, the theory of inflation was created. Inflation postulated that at a point after Planck Time (when gravity separated from the other Fundamental Forces, described by the only definition of the fundamental constants which make a time unit: ) \[ t_P = \sqrt{\frac{hG}{c^5}} = 5.6 \times 10^{-44} s\] there was a massive expansion of the order \(10^{40\sim 50}\) which straightened the universe into being flat (same way one would stretch out a piece of paper) and was early and quick enough to spread the initial information of the universe all across the CMB. This meant the universe went through an expansion from being \(10^{-28} m \) in length to being 0.73 meters in the span of \(10^{-36} s\), the most aggressive and rapid expansion in the history of the universe. 

References: 

Carroll, B. W., & Ostlie, D. A. (2007). An Introduction to Modern Astrophysics. San Francisco: Pearson: Addison Wesley.

Images from: http://www.esa.int/var/esa/storage/images/esa_multimedia/images/2013/03/planck_cmb/12583930-4-eng-GB/Planck_CMB.jpg

http://media.radiosai.org/journals/Vol_05/01JUL07/images/FeatureArticles/sifi/Fig-4.jpg

Ever noticed how The Big Bang Theory (the show) doesn’t actually talk about the Big Bang?

We are bounded in a nutshell of Infinite Space: Week 8: Free Form #9: Ever noticed how The Big Bang Theory (the show) doesn’t actually talk about the Big Bang?

The Big Bang theory presents our modern standard cosmological model. It includes in the description of a Universe which began in the form of a quantum state of such high energy and density that the Fundamental Forces of the Universe, Gravitational, Electromagnetic, and the Strong and Weak Nuclear Forces, had not separated. This also corresponds to the fact we still have no physical description for the universe before gravity had separated from the other three, a conundrum which continues to perplex astronomers and physicists alike, which could, in the future, have a solution when considering the theories on quantum gravity (but more on this later). Anyways, the Big Bang Theory describes how the universe expanded into a flat universe, first off a radiative driving force of expansion, followed by baryonic matter, and now approaching the epoch where the expansion is controlled by the vacuum energy, Dark Energy (queue deeper voice and lightning/thunder in the background) (more information on the equations which describe these expansion states can be found on: http://ay16-rodrigocordova.blogspot.com/2015/11/cosmology-101-part-2.html ). The universe expanded off initial gravitational and density perturbations/fluctuations to cause a complex web of baryonic gravitational systems, all held in line (and drawn together) by the only source of gravity strong enough to hold the forces in play, the mysterious Dark Matter.

However, just what caused the Big Bang, or rather, what are the driving mechanisms: my guess is as good as yours, with some initial condition we are able to construe. These are the first moments of the universe where quantum fluctuations determined the universe, meaning there were unpredictable changes and shifts in the nature of the pre-modern physics understandable era of the universe, all related to how the Universe began with a Singularity of Space-Time, and expanded from there on from the photonic and later radiative and other driving mechanisms. But, these definitions created a problem in the Big Bang, a perception of the early universe which did not line up with the observations of the Early Universe made on the Cosmic Microwave Background and other early sources. These are the Horizon, Magnetic Monopole, and Flatness Problems, predictions and observations made of the Early Universe and its effect in contemporary observations which contradict each other, or otherwise prove highly unlikely in the way they resulted nowadays (more on this in a bit). Regardless of these problems, and their proposed solutions, the Big Bang Theory represents the most robust theory to describe the beginning of the universe, adding onto decades of research to develop a model explain the expanding universe we continue to live in.  

                         (Disclaimer: Although the video represents the initial sound waves emanating form the Big Bang, the noise is quite annoying)

References: 

https://www.youtube.com/watch?v=WksTp06Go2c

Carroll, B. W., & Ostlie, D. A. (2007). An Introduction to Modern Astrophysics. San Francisco: Pearson: Addison Wesley.

It's a miracle we find anything in the Universe at all

We are bounded in a nutshell of Infinite Space: Week 8: Worksheet #14: Problem #1: It's a miracle we find anything in the Universe at all

1. Draw a planet passing in front of its star, with the star on the left and much larger than the planet on the right, with the observer far to the right of the planet. The planet’s semimajor axis is a.

(a) Show that the probability that a planet transits its star is \(R_\star / a\), assuming \(R_P \ll R_\star \ll a\). What types of planets are most likely to transit their stars?

(b) If 1% of Sun-like stars in the Galaxy have a Jupiter-sized planet in a 3-day orbit, what fraction of Sun-like stars have a transiting planet? How many stars would I need to monitor for transits if I want to detect 10 transiting planets?

a. From the images, we know the planet is transiting in front of the star from our perception of it. Furthermore, we can see that the area the planet could be in in order for us to properly see it is actually smaller than the area of the star, it being farther away, however we can disregard this because of the extreme distance this system is from the observer. Therefore, the chance of observing the planet in front of the star from our point of view is the small cross section of the star we are able to observe directly, and this entire path as a belt around the Sun indicating the possibility of the planet being in our line of sight at one point of its orbit, over the area of the sphere of radius a where the planet could conceivably be: \[ \frac{Area ~of~belt}{Surface ~area ~of ~the ~sphere}\] and now defining the areas of these surfaces: \[ \frac{\sigma_A}{A_{Sphere}} = \frac{2\pi a (2R_\star)}{4\pi a^2},\] which gives an elegant proportion the question asked for: \[ \frac{\sigma_A}{A_{Sphere}} = \frac{R_\star}{a}\]

b. Now assuming we want to find 10 planets, how many stars do we need to look at randomly? Starting off with the equation we just derived, we know the percentage is equivalent to the star and planet’s: \[R_\odot / a = \% ,\] and also using the simplified version of Kepler’s third law relating years to AU, so as to find the A from the information given: \[ P^2 = a^3 ,\] writing the time in years: \[ 3 days~\to \frac{3~days}{365 days/year} \approx \frac{1}{122} ~years,\] Now we plug this into the equation and find the semi-major axis: \[ \left(\frac{1}{122}\right)^2 = a^3,\] \[ a = 0.04 AU.\]  

Next, we plug these into the equation we found earlier, plugging in the radius of the Sun in cm and the semi-major axis which is now turned into the equivalent in cm: \[ 1 AU = 1.5 \times 10^{13} cm,\] \[\frac{R_\odot}{a} = \frac{7\times 10^{10} cm}{1.5 \times 10^{13} cm \cdot 4 \times 10^{-2}} ,\] \[\frac{R_\odot}{a} = \frac{7\times 10^{10} cm}{6 \times 10^{11} cm }\] \[ \frac{R_\odot}{a} = 1.1 \times 10^{-1} .\] This represents the percentage chance of the planet being in an orbit observable to us. Next we multiply this percentage by the percentage of stars with a mass/radius comparable to that of the Sun \(1\%\): \[ 1.1 \times 10^{-1} \cdot 0.01 = 0.0011 = 0.11 \% ,\]  which is the percentage of Sun-like stars with a visible transiting planet. Finally we now compute how many stars we need to observe to see the 10 planets multiplying 10 times the whole of stars divided by the percentage of Sun-like stars with orbiting planets: \[ 10 \cdot \frac{100}{0.1} = 10 \times 10^3 = 10^4 = 10,000, \] meaning you would need to look at a lot of stars to see the 10 transiting planets.

Turns out stars rotate within their solar system

We are bounded in a nutshell of Infinite Space: Week 8: Worksheet #13: Problem #1: Turns out stars rotate within their solar system 

1. (a) We often say that planets orbit stars. But planets and their stars actually orbit their mutual center of mass, which in general is given by \( x_{com} = \Sigma_i m_i x_i / \Sigma_i m_i \) . Set up the problem by drawing an x-axis with the star at \( x = -a_\star\) with mass \(M_\star \), and the planet at \(x = a_P\) and \(m_P\). Also, set \( x_{com} = 0\). How do \(a_P\) and \(a_\star\) depend on the masses of the star and planet?

(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: \( a = a_P + a_\star\). Label this on your diagram. Now derive the relationship between the total mass \( M_\star + m_P \approx M_\star\), 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).

(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. (Potentially useful numbers \(M_\odot \approx 1000 M_{Jup}\) and \( a_{Jup} \approx 5.2 AU\).)

a. Taking this problem as a simple center of mass problem, we can use the illustration below:

 and now we can better understand the original sum equation: \[ x_{com} = \frac{\sum_i m_i x_i }{\sum_i m_i} ,\] and identifying all the masses and their position relative to the center: \[ x_{com} = \frac{M_\star (a_\star) + m_P a_P}{M_\star + m_P},\] and setting \(x_{com}\) to 0:  \[ 0 = \frac{M_\star (- a_\star) + m_P a_P}{M_\star + m_P},\] we continue simplifying  \[ \frac{-m_P a_P}{M_\star + m_P} = \frac{-M_\star a_\star}{M_\star + m_P},\] and a simpler equality begins to emerge:  \[ m_P a_P = M_\star a_\star,\] meaning we find a direct correlation of mass of an entangled system between the objects’ masses and their semi-major axis of rotation: \[\frac{m_P}{M_\star} = \frac{a_\star}{a_P}\]

b. 

Using the virial theorem (similar to a problem directly involving the Orbital period earlier in the semester) we can find the correlation of mass to Period, and semi-major axis. Starting off with the base equation: \[ K = -\frac{1}{2} U ,\] we begin representing these energy values with the objects in question, mainly the gravitational potential energy and the kinetic energy (the movement of the star at the center of the system is negligible here):  \[ \frac{1}{2} m_P v^2 = -\frac{1}{2} \frac{-GM_\star m_P}{a_\star + a_P},\] and we take into account the definitions discussed by the problem: \[M_\star + m_P \approx M_\star,\] \[a_\star + a_P = a,\]  which allow the previous equation to simplify down to: \[ v^2 = \frac{GM_\star}{a},\] and now we define the velocity as a rotational system: \[ v = \frac{2\pi a}{P} ,\] and plug this in: \[ \frac{4\pi^2 a^2}{P^2} = \frac{GM_\star}{a},\] which is then placed simplified into a version of Kepler’s third law: \[ P = \left(\frac{4\pi^2 a^3}{GM_\star}\right)^{\frac{1}{3}} \]

c. Using the equations we just derived, we know the proportion is \[ \frac{m_P}{M_\star} = \frac{a_\star}{a_P}, \] and we substitute in the values for the mass of Jupiter and the axis of the orbit of Jupiter: \[ \frac{1/1000 M_\odot}{1 M_\odot} = \frac{a_\star}{5.2 AU},\] which solves to: \[\frac{5.2 AU}{1000} = a_\star\] \[a_\star = 5.2 \times 10^{-3} AU,\] and knowing a particular equivalence of: \[ 1 AU =    214.94 R_\odot,\] we can see how the Sun’s orbit is shifted by the mass of Jupiter by the following distance: \[ a_\star = 1.1 R_\odot.\]  

The Sun's Force Field

We are bounded in a nutshell of Infinite Space: Reading #5: The Sun's Force Field

From the reading of chapter 3.10 -3.12 of Astrophysics in a Nutshell, we begin to understand the different processes which lead up to the nuclear reactions which create the energy necessary to maintain the Sun. Furthermore, these equations lead us to understand the form and structure of the Sun. At the center of the Sun, we find the reactive core, the source of the Sun’s energy and the place where all the Nuclear Fusion takes place in order to power the Sun. This process of Nuclear Fusion works off quantum concepts like quantum tunneling, energy levels, and fundamental baryonic particles like neutrinos to combine Hydrogen atoms into other isotopes of the Hydrogen, to then be successfully combined into Helium atoms. This exchange and release of energy is what provides the Sun with energy, and begins radiating it outwards into the Radiative zone, followed by the outer Convection Zone, and then reaching the Photosphere were the photons are finally emitted after tens of thousands of years of traveling through the dense gases of the Sun. Above the initial atmosphere of the Photosphere, there exists the Chromosphere where the photons begin to travel outwards into space, and heat up this region more so than the Photosphere itself. Finally the Corona is the upper portion of the Atmosphere where the atmosphere is hottest and extends outward for several million miles.

However, the reach of the Sun’s electromagnetic influence does not end at the photons emitted to reach the objects rotating around it, rather the Sun does much more than we usually remember. Here we refer to the Heliosphere, the pressure exerted by the Sun and the Solar Wind into the reaches of the Solar System, going beyond the Kuiper belt to act as a protective sphere from the Interstellar Medium fraught with dangers and damaging sources of energy. The Heliosphere is continuously maintained by the outward pressure exerted by the Sun’s emissions and counteracted by the Interstellar Medium’s pressure as the Solar system moves across the Milky Way.

Thus far, we have had only one man-made object pass the Heliosphere: the Voyager 2 probe. Carrying on it the evidence of our existence and capabilities in the form of a phonograph disk, the Voyager missions were our first glimpses into the farthest parts of the Solar System, as well as the origin of the farthest image we have of the Earth:   

References: 

https://writescience.files.wordpress.com/2015/07/pale-blue-dot-wallpaper-1900x1200.jpg

https://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Sun_poster.svg/4000px-Sun_poster.svg.png

https://upload.wikimedia.org/wikipedia/commons/8/8e/NewHeliopause_558121.jpg

Carroll, B. W., & Ostlie, D. A. (2007). An Introduction to Modern Astrophysics. San Francisco: Pearson: Addison Wesley.

Maoz, D. (2007). Astrophysics in a Nutshell. Princeton: Princeton University Press.

Squiggle Math II: The Dawn of Differentiation

We are bounded in a nutshell of Infinite Space: Worksheet # 12, Problem #1: Squiggle Math II: The Dawn of Differentiation

1. In this worksheet we’re going to do some “order of magnitude differentiation”. Let’s start off with a simple example that is completely unrelated to stellar structure:

(a) The velocity of a particle is \(v = \alpha t^2\) where \(\alpha\) is a constant, and we want to find the scaling of position with time. First write down the equation in the form of a differential equation for x, the position. Next, we are going to say that \(dx \approx \Deltax \sim x \) and \(dt \approx \Delta t \sim t\) . In English: “\(dx\) is approximately the change in x, which scales as x.” Now it should be easy to show the scaling of x with t. What is the form of this scaling relationship?

(b) So you are probably saying to yourself, “This doesn’t feel right mathematically. How can you treat differential quantities with such disdain?!” But this is a simple differential equation, so you can actually integrate it. What do you get? How does it compare to your scaling relationship?

a. This system of mathematics fairly similar to the “squiggle math” we had worked with on a previous occasion, in the sense we are establishing the simple relationships and the way different aspects scale with one another in an equation. This is the case of a standard velocity equation with a constant: \[ v = \alpha t^2,\] and if we were to describe velocity in terms of the differentiated form of position, we see that:  \[ \frac{dx}{dt} = \alpha t^2 ,\] and as such becomes a description of the relationship once we “integrate” to find the scaling relationship of the true position, not the change in it: \[ \frac{x}{t} \sim \alpha t^2 ,\] therefore:  \[ x \sim \alpha t^3,\] and equivalently: \[ x \propto t^3,\] which is the scaling relationship we were looking for.  

b. However, we could to this calculation precisely, finding the correct proportionality and scaling factors of the equation we solved a minute ago: \[ \frac{dx}{dt} = \alpha t^2 ,\] we separate the variables: \[ dx = \alpha t^2 dt,\] and integrate using simple integration limits: \[ \int^x _0 dx = \int^t _ \alpha t^2 dt \] \[ x = \alpha \frac{1}{3} t^3 ,\] and now we see that the scaling relationship of the previous problem is perfectly sensible, and with the correct integration we may find the precise scaling factors which make the equation precise:  \[ x = \frac{a}{3} t^3 \] \[ x \propto t^3.\]