--- title: "Gravity and Distributions" author: Jed Rembold date: January 23, 2025 slideNumber: true theme: tokyo-night-light highlightjs-theme: tokyo-night-light width: 1920 height: 1080 transition: slide --- ## Announcements - Homework 1 - Should have everything you need to do all of P1 and potentially start P2 - Second check-in will go out this weekend! Don't forget to fill it out! - Just a few of you still need to fill out the information gathering poll [here](https://docs.google.com/forms/d/e/1FAIpQLSdaUjeiwdLBnon0uamSm0lKLgxC88S24TsB70UcPs0_8yrzFQ/viewform?usp=sharing) ## Recap - Kepler found that all orbiting objects move in an ellipse (Kepler's 1st Law) - The central object exists at one of the focus points - Ellipse and orbit vocabulary - Using the `fitellipse` libraries ## Today's Plan - Kepler continued - Laws 2 and 3 - Gravity - Distributions - Histograms/Density plots - 2D variants # Return of Kepler ## Kepler's 2nd Law - A line joining a planet to the Sun sweeps out equal areas in equal times - Results in planets moving faster when close to the Sun and slower when far away \begin{tikzpicture}%%width=60% [every node/.style={font=\sf, color=Black}] \coordinate (sun) at (1.65,0); \draw[very thick] (0,0) ellipse (3cm and 2.5cm); \fill[opacity=0.5,Red] (sun) -- (-25:3cm and 2.5cm) arc (-25:25:3cm and 2.5cm); \fill[opacity=0.5,Red] (sun) -- (172:3cm and 2.5cm) arc (172:188:3cm and 2.5cm); \fill[inner color=yellow, outer color=orange] (sun) circle (2mm); \node[right] at (3,0) {Fastest}; \node[left] at (-3,0) {Slowest}; \end{tikzpicture} ## Kepler's 3rd Law :::incremental - The square of a planet's orbital period is proportional to the cube of its semi-major axis. - A planet's _period_ is the time it takes to complete an entire orbit - An orbit's semi-major axis we defined earlier to be half the major axis - Mathematically, this looks like: $$ p^2 \propto a^3 \qquad\text{or}\qquad \frac{p^2}{a^3} = \text{constant} $$ where $p$ is the period and $a$ is the semi-major axis - For objects orbiting the Sun, if you choose units of years and AU, then $$ p^2 \approx a^3 $$ - 1 _AU_ = 1 astronomic unit = average distance from Earth to Sun ($1.496\times10^{11}$ meter) ::: ## Projections - When observing, you rarely get a perfectly top-down view of an ellipse - Often, you are looking at some view projected from the side - This does not change the measurement of the period! - It may make measuring the semi-major axis more difficult though - Since many planets (and moons) have _mostly_ circular orbits, you can estimate the major axis by just taking the largest end-to-end distance that you observe # Newton's Gravity ## Why Gravity? :::::cols ::::col - Kepler told us that there **must** be these relationships, but he couldn't say _why_ - Newton found that, if the direction that something is moving changes, then it **must** have experienced some force - Newton's big connection (one of them) was determining that the necessary force turns out to be from gravity (at least for most astronomic objects) :::: ::::col \begin{tikzpicture}%%width=100% [scale=0.7] \fill[inner color=yellow, outer color=orange] (0,0) circle (1cm); \draw[dashed, -latex, thick] (3,0) arc (0:340:3cm); \foreach \a in {0,45,...,350}{ \draw[orange, -latex] (\a:2.7) -- (\a:1.3); } \node at (3,0) {\includegraphics[width=1cm]{world.png}}; \end{tikzpicture} :::: ::::: ## Gravity - Gravity is the universal attractor - Anything with mass attracts anything else with mass - Strength of force increases with the amount of mass involved - Strength of force decreases rapidly with distance between the masses \begin{tikzpicture}%%width=70% \coordinate (ball1) at (0,0); \coordinate (ball2) at (8,0); \coordinate (d) at ($(ball1)-(0,1.5)$); \draw[Green] (ball1) --+(0,-2); \draw[Cyan] (ball2) --+(0,-2); \fill[ball color=Green!50] (ball1) circle (1cm); \fill[ball color=Cyan!50] (ball2) circle (1cm); \node at ($(ball1)+(0,1.2)$) {$M_1$}; \node at ($(ball2)+(0,1.2)$) {$M_2$}; \draw[<->, thick] (d) -- node[midway,below] {d} (d-|ball2); \draw[-latex, ultra thick,Red] (ball1) -- +(4,0); \draw[-latex, ultra thick,Red] (ball2) -- +(-4,0); \node[Red] at (4,1) {\scalebox{1.5}{$F_g = G \frac{M_1 M_2}{d^2}$}}; \end{tikzpicture} ## Newton meets Kepler's 3rd :::{style='font-size:.9em'} - Kepler already had worked out $$ \frac{a^3}{p^2} = \text{same value for all planets orbiting Sun} $$ - Newton worked out, starting with the force, that two objects held in orbit by gravity would obey: $$ \frac{a^3}{p^2} = \frac{G(M_1 + M_2)}{4\pi^2} $$ where: - $M_1, M_2$ are the masses of the objects in _kilograms_ - $a$ is the average separation between the objects in _meters_ - $p$ is the orbital period in _seconds_ - $G$ is the gravitational constant ($6.67\times10^{-11}\,\tfrac{m^3}{kg\,s^2}$) ::: ## Some nicer units - Put in more convenient units, Newton's formulation of Kepler's 3rd boils down to: $$ \frac{a^3}{p^2} = (M_1 + M_2)_\odot $$ where - $M_1, M_2$ are the masses of objects in _solar masses_ (multiples of the Sun's mass) - $a$ is the average separation of the objects in _AU_ - $p$ is the orbital period in _years_ - For the Sun and most planets, $M_1 + M_2 \approx 1 M_\odot$ - If you can measure $a$ and $p$, then you can work out the mass of the objects! # Visualizing Distributions ## Distributions :::::cols ::::col - There is a **lot** of stuff in space! - To understand bulk properties of objects, we commonly need to resort to looking at various _distributions_ - Distributions can be comprised by looking at almost any variable of interest (or in some cases multiple variables) :::: ::::col ![](../images/distribution.png){width=100%} :::: ::::: ## Common usages: - EDA: exploring and bringing to light interesting properties or trends in the gathered data - By fitting mathematical expressions to the distribution, you now have a tool to use in further analysis - This is particularly common in simulations, where simulations pull from known distributions to create physically realistic situations - What type of mathematical distribution fits best can potentially tell you about possible inner workings that yielded that distribution - Classification: sometimes there are several distinct groupings in a distribution that can lead to various types of classification ## Histograms - A _histogram_ is a visual indication of how many times a certain variable appears - Y-axis represents counts, and the x-axis is divided into a number of bins - Data points that have a value appearing within a particular bin contribute to that bin's count - Commonly will want to try several binning methods or sizes ## Generating Histograms ::::::{.cols style='align-items: flex-start'} ::::col :::{.block name=Python} - Matplotlib can generate plots directly: ```python plt.hist( variable_list, bins=num_bins ) plt.show() ``` ::: :::: ::::col :::{.block name=R} - Just use the `hist` function: ```R hist( variable_list, breaks=num_bins ) ``` - Or, in ggplot ```R ggplot(data=df, aes(x=variable) ) + geom_histogram() ``` ::: :::: :::::: ## When Bins fail - The choice of bins can often lead to distributions that might feel pretty subjective - One option is to use one of the auto-binning algorithms provided by your histogram software (read documentation) - Another, which especially makes sense for continuous data, is to look at a Kernel Density Estimate, or KDE plot - Most common kernel is that of a Gaussian, but can be different - Essentially stacks multiple Gaussians for each point, and then sums them together - Results in a "smoother" looking histogram ## Generating KDE Plots ::::::{.cols style='align-items: flex-start'} ::::col :::{.block name=Python} - Pandas can generate plots directly: ```python df[column name].plot(kind='kde') ``` - Does require `scipy` under the hood, so will need it installed - Can also install Seaborn and use its variants of you like ```python import seaborn as sbn sbn.kdeplot(df[column name]) ``` ::: :::: ::::col :::{.block name=R} - Just use the `plot` and `density` functions: ```R plot(density(variable_list)) ``` - In ggplot: ```R ggplot(data=df, aes(x=column)) + geom_density(bw='bcv') ``` ::: :::: :::::: ## Activity! - [Here](../demos/satellites.csv) is a collection of satellite information for satellites orbiting the Earth - It has a lot of (some nonsense) columns - You will be interested in the column named `'Apogee (km)'` - This table originally included some satellites that are **really** distant from the Earth, but I've already filtered it a bit down to our region of interest - Create both histogram and KDE plots of the distribution of satellite apogee distances - Can you see the outer cluster formed from geosynchronous satellites? ## 2D Histograms and KDE Plots ::::::cols ::::col - Sometimes you have multiple variables that you want to visualize together as a distribution - There are 2D analogs of both histograms and KDE plots - One variable along each axis - Counts or density still determine color :::: ::::col ![](../images/distribution2d.png) :::: :::::: ## Multivariate Distribution Creation ::::::{.cols style='align-items: flex-start'} ::::col :::{.block name=Python} - Histogram in Matplotlib ```python plt.hist2d(xs, ys, bins=20) ``` - KDE plot easiest through Seaborn ```python sbn.kdeplot(x=xs, y=ys, fill=True) ``` ::: :::: ::::col :::{.block name=R} - Histogram through ggplot ```python ggplot(data=df, aes(x=xs, y=xs) ) + geom_bin_2d() ``` - KDE plot through ggplot ```R ggplot(data=df, aes(x=xs, y=xs) ) + geom_density_2d_filled() ``` ::: :::: ::::::