Static vs Dynamic Typing

Static vs Dynamic Typing

• Static Typing involves types known and checked at compile time, requiring explicit declaration of types.
• Static typing allows for early error detection.
• The compiler can optimize static typed code based on type information.
• Dynamic Typing features types checked at runtime, without explicit type declaration.
• Dynamic typing allows flexible and concise code.
• Errors in dynamic typing can occur during runtime.

Memory Management

• Under Static Typing memory allocation is determined at compile time.
• With static typing memory usage is fixed.
• Memory access is efficient in static typing.
• Poor memory management results in leaks or overflow in static typing.
• Memory allocation and deallocation occur during runtime in Dynamic Typing.
• Dynamic typing supports flexible memory usage with automatic garbage collection.
• Dynamic Typing may sometimes be less efficient due to runtime overhead.

Performance

• Static Typing code is generally faster during execution due to compile-time optimizations.
• There is less runtime overhead when doing static typing.
• Static typing is suitable for performance-critical applications.
• Dynamic Typing might be slower due to runtime type checking.
• Dynamic typing has more runtime overhead than static typing.
• Dynamic typing is more suitable for rapid prototyping and scripting.

Examples

• Code in C++, and Java are examples of Static Typing as shown below

int add(int a, int b) {
    return a + b;
}

int add(int a, int b) {
    return a + b;
}

• Code in Python, and JavaScript are examples of Dynamic Typing as shown below

def add(a, b):
    return a + b

function add(a, b) {
    return a + b;
}

Topic Subtitle

• Reaction rates for $aA + bB \rightarrow cC + dD$ can be expressed as disappearance of reactants: $-\frac{1}{a} \frac{d[A]}{dt} , -\frac{1}{b} \frac{d[B]}{dt}$
• Reaction rates for $aA + bB \rightarrow cC + dD$ can be expressed as appearance of products: $\frac{1}{c} \frac{d[C]}{dt} , \frac{1}{d} \frac{d[D]}{dt}$
• The overall reaction rate is expressed as $rate = -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$
• Reaction Rate is affected by concentration of reactants, temperature, surface area, catalysts, and pressure.
• Rate law relates reaction rate to the concentration of reactants, expressed as $rate = k[A]^m[B]^n$
• In the rate law expression: $k$ is the rate constant, $[A]$ and $[B]$ are the concentrations of reactants & $m$ and $n$ are the reaction orders with respect to A and B.
• The overall reaction order is the sum of individual orders: $m + n$
• Rate laws can be determined using experimental data like initial rates or the isolation method.
• The Method of Initial Rates measures the initial rate at varied initial concentrations.
• The Isolation Method keeps all reactants except one in excess to determine the order with respect to the changing reactant.
• Zero-Order Reactions have a rate independent of concentration, with a Rate Law: $rate = k$
• The Integrated Rate Law for zero-order reactions is: $[A]_t = -kt + [A]_0$
• Half-life for zero-order reactions is $t_{1/2} = \frac{[A]_0}{2k}$
• First-Order Reactions have a their rate directly proportional to concentration, and follows Rate Law: $rate = k[A]$
• A first-order reaction has an Integrated Rate Law of: $ln[A]_t = -kt + ln[A]_0$
• Half-life for a first-order reaction is $t_{1/2} = \frac{0.693}{k}$
• Second-Order Reactions have a rate proportional to the square of the concentration and follow Rate Law of: $rate = k[A]^2$
• The Integrated Rate Law for second-order reactions is: $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
• Half-life for a second-order reaction is: $t_{1/2} = \frac{1}{k[A]_0}$
• Elementary steps includes simple steps that represent the progress of a reaction at the molecular level.
• The Rate-Determining Step is the slowest step in a reaction mechanism that determines the overall rate of the reaction.
• Molecularity is the number of molecules colliding in an elementary step where unimolecular corresponds to one molecule
• Molecularity is the number of molecules colliding in an elementary step where bimolecular corresponds to two molecules
• Molecularity is the number of molecules colliding in an elementary step where termolecular corresponds to three molecules
• The Arrhenius Equation describes the temperature dependence of rate constants of $k = Ae^{-\frac{E_a}{RT}}$ where $k$ is the rate constant, $A$ is the pre-exponential factor (frequency factor), $E_a$ is the activation energy, $R$ is the ideal gas constant ($8.314 J/(mol \cdot K)$), and $T$ is the absolute temperature (in Kelvin).
• Activation energy can be determined using the two-point form: $ln(\frac{k_2}{k_1}) = \frac{E_a}{R}(\frac{1}{T_1} - \frac{1}{T_2})$
• Homogeneous Catalysts are in the same phase as the reactants.
• Heterogeneous Catalysts are in a different phase from the reactants.
• Enzymes are biological catalysts (proteins).
• Catalysts provide an alternative reaction pathway with a lower activation energy.
• Catalysts do not change the thermodynamics of the reaction

Lab 8: Data Visualization Objectives

• After completing this week’s lab, you should be able to:
• Create basic statistical plots (histograms, box plots, scatter plots) using matplotlib.
• Customize plot aesthetics to meet publication-quality standards.
• Generate multiple plots using matplotlib’s subplot functionality.

Instructions

• For this lab, we will be using a subset of the Integrated Postsecondary Education Data System (IPEDS) data set, focusing on a sample of 100 US-based colleges and universities.
• The data set contains the following variables:
• INSTNM: Institution name
• CITY: City
• STABBR: State abbreviation
• ZIP: Zip code
• CONTROL: Control of institution (1=Private, 2=Private nonprofit, 3=Public)
• LOCALE: Degree of urbanization (11=City, 12=Suburb, 13=Town, 21=Rural)
• ADM_RATE: Admission rate
• SAT_AVG: Average SAT score
• UGDS: Enrollment of undergraduate certificate/degree-seeking students
• TUITIONFEE_IN: In-state tuition and fees
• PCTPELL: Percentage of undergraduate certificate/degree-seeking students receiving Pell Grants
• COMPL_RP: Completion rate
• import pandas as pd
• import matplotlib.pyplot as plt

Task 1: Histograms

• Create a histogram of in-state tuition and fees (TUITIONFEE_IN).
• Customize the plot with a title, x-axis label, and y-axis label.
• Experiment with different bin sizes to find an appropriate representation of the data.

Task 2: Box Plots

• Generate box plots of the average SAT score (SAT_AVG) for each type of institution control (CONTROL). Add appropriate labels and a title to the plot.
• Discuss any differences you observe between the groups.

Task 3: Scatter Plots

• Produce a scatter plot of the percentage of students receiving Pell Grants (PCTPELL) versus the completion rate (COMPL_RP).
• Include axis labels and a title.
• Comment on any relationship you observe between these two variables.

Task 4: Subplots

• Create a figure with two subplots side by side.
• In the first subplot, display a histogram of undergraduate enrollment (UGDS).
• In the second subplot, show a box plot of the admission rate (ADM_RATE).
• Make sure to label each subplot appropriately.

Task 5: Aesthetics

• Improve the aesthetics of your plots by changing colors, adjusting transparency, and adding grid lines.
• Explain how these changes enhance the readability and visual appeal of your figures.

Task 6: Saving Plots

• Save one of your plots as a high-resolution image (e.g., PNG or PDF).
• Ensure that the saved image has clear labels and is suitable for inclusion in a report or presentation.

Submission

• Submit your Python script or Jupyter Notebook containing the code for each task, along with a brief written summary of your findings and observations.

Why VLE?

• Separation is a cornerstone of chemical engineering.
• Distillation is the most common separation technique, relying on VLE.

Phase Equilibrium

• Equilibrium is a system state showing no tendency to change.
• Phase Equilibrium is equilibrium between two or more phases.
• Vapor-Liquid Equilibrium (VLE) is equilibrium between liquid and vapor phases.

VLE Criteria

• Temperature, pressure, and chemical potential of each species are equal in all phases:
• $T^{\alpha} = T^{\beta} =... = T^{\pi}$
• $P^{\alpha} = P^{\beta} =... = P^{\pi}$
• $\mu_{i}^{\alpha} = \mu_{i}^{\beta} =... = \mu_{i}^{\pi}$
  • Where:
    • $\alpha, \beta, \pi$ represent different phases.
    • $\mu_{i}$ is the chemical potential of component i.

Chemical Potential

• $\mu_{i} = (\frac{\partial G}{\partial n_{i}}){T,P,n{j}}$ Where:
  • $G$ is the Gibbs free energy
  • $n_{i}$ is the number of moles of component i.
• This partial molar property is evaluated at constant temperature, pressure, and number of moles of all other components.

Simplified Criteria

• For VLE of a multicomponent mixture:
• $T^{l} = T^{v}$
• $P^{l} = P^{v}$
• $\hat{f}{i}^{l} = \hat{f}{i}^{v}$
• Where:
  • $\hat{f}_{i}^{l}$ is the fugacity of component i in the liquid phase
  • $\hat{f}_{i}^{v}$ is the fugacity of component i in the vapor phase

Pure Species VLE

• For a pure species:
• $T^{l} = T^{v} = T^{sat}$
• $P^{l} = P^{v} = P^{sat}$
• $f^{l} = f^{v}$
• Where:
• $T^{sat}$ is the saturation temperature
• $P^{sat}$ is the saturation pressure
• $f^{l}$ is the fugacity of the pure liquid
• $f^{v}$ is the fugacity of the pure vapor

Vapor Pressure

• The vapor pressure ($P^{sat}$) is the pressure at which the liquid and vapor phases of a pure substance coexist in equilibrium at a given temperature.

Antoine Equation

• $\ln P^{sat} = A - \frac{B}{T + C}$
  • Where:
    • $P^{sat}$ is the saturation pressure
    • $T$ is the temperature
    • $A, B, C$ are substance-specific parameters

Vapor Pressure Equations

• Table: Vapor Pressure Correlations | Equation | Parameters | Temperature Range | | :------------------------------- | :--------- | :---------------- | | $\ln P^{sat} = A - \frac{B}{T}$ | A, B | Narrow | | $\ln P^{sat} = A - \frac{B}{T+C}$ | A, B, C | Wide |

Clausius-Clapeyron Equation

• $\frac{d P^{sat}}{dT} = \frac{\Delta H_{vap}}{T \Delta V_{vap}}$
  • Where:
    • $\Delta H_{vap}$ is the enthalpy of vaporization
    • $\Delta V_{vap}$ is the change in volume upon vaporization

Simplified Clausius-Clapeyron Equation

• $\frac{d \ln P^{sat}}{dT} = \frac{\Delta H_{vap}}{RT^2}$
  • Assuming: Ideal gas behavior for the vapor phase Negligible liquid volume compared to vapor volume Constant $\Delta H_{vap}$

Integrating Simplified Equation

• $\ln P^{sat} = -\frac{\Delta H_{vap}}{RT} + C$
  • Where C is the integration constant.

Vectores

Suma de Vectores

Método Analítico

• Componentes rectangulares de un vector:

$\overrightarrow{A} = A_x\hat{i} + A_y\hat{j}$

$A_x = A\cos(\theta)$ $A_y = A\sin(\theta)$

$\overrightarrow{R} = \overrightarrow{A} + \overrightarrow{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$

$R_x = A_x + B_x$ $R_y = A_y + B_y$

$R = \sqrt{R_x^2 + R_y^2}$

$\tan(\theta) = \frac{R_y}{R_x}$

$\theta = \arctan(\frac{R_y}{R_x})$

• Dados los vectores:

$\overrightarrow{A} = 2\hat{i} + 2\hat{j}$ $\overrightarrow{B} = 4\hat{i} - 3\hat{j}$

• El vector resultante $\overrightarrow{R}$ es

$\overrightarrow{R} = (2+4)\hat{i} + (2-3)\hat{j}$ $\overrightarrow{R} = 6\hat{i} - 1\hat{j}$

$R = \sqrt{6^2 + (-1)^2} = \sqrt{37} \approx 6.08$

$\theta = \arctan(\frac{-1}{6}) = -9.46°$

Producto Escalar (Producto Punto)

$\overrightarrow{A} \cdot \overrightarrow{B} = |\overrightarrow{A}||\overrightarrow{B}|\cos(\theta)$

$\overrightarrow{A} \cdot \overrightarrow{B} = (A_x\hat{i} + A_y\hat{j}) \cdot (B_x\hat{i} + B_y\hat{j}) = A_xB_x + A_yB_y$

• Dados los vectores:

$\overrightarrow{A} = 2\hat{i} + 2\hat{j}$ $\overrightarrow{B} = 4\hat{i} - 3\hat{j}$

• El producto escalar $\overrightarrow{A} \cdot \overrightarrow{B}$ es

$\overrightarrow{A} \cdot \overrightarrow{B} = (2)(4) + (2)(-3) = 8 - 6 = 2$

Producto Vectorial (Producto Cruz)

$\overrightarrow{A} \times \overrightarrow{B} = |\overrightarrow{A}||\overrightarrow{B}|\sin(\theta)\hat{n}$

$\overrightarrow{A} \times \overrightarrow{B} = (A_x\hat{i} + A_y\hat{j} + A_z\hat{k}) \times (B_x\hat{i} + B_y\hat{j} + B_z\hat{k}) = (A_yB_z - A_zB_y)\hat{i} + (A_zB_x - A_xB_z)\hat{j} + (A_xB_y - A_yB_x)\hat{k}$

$\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$

• Dados los vectores:

$\overrightarrow{A} = 2\hat{i} + 2\hat{j} + 0\hat{k}$ $\overrightarrow{B} = 4\hat{i} - 3\hat{j} + 0\hat{k}$

• El producto vectorial $\overrightarrow{A} \times \overrightarrow{B}$ es

$\overrightarrow{A} \times \overrightarrow{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 2 & 0 \ 4 & -3 & 0 \end{vmatrix} = (0-0)\hat{i} + (0-0)\hat{j} + (-6-8)\hat{k} = -14\hat{k}$

Reguläre Ausdrücke

• Ein regulärer Ausdruck (Regex oder RegExp von engl. regular expression) ist in der theoretischen Informatik eine Zeichenkette, die einer bestimmten Menge von Zeichenketten entspricht.
• Mithilfe von regulären Ausdrücken konnen also Muster in Texten gefunden, ersetzt oder extrahiert werden.

Syntax

Beispiele

• Einige Beispiele für reguläre Ausdrücke und ihre Bedeutung:

Hallo - Findet die Zeichenkette "Hallo". \d+ - Findet eine oder mehrere Ziffern. [a-z]+ - Findet eine oder mehrere Kleinbuchstaben. [A-Z][a-z]* - Findet ein Wort, das mit einem Grossbuchstaben beginnt, gefolgt von null oder mehr Kleinbuchstaben. ^https?://.*.de$ - Findet eine URL, die mit "http://" oder "https://" beginnt und mit ".de" endet.

verwendung

• Reguläre Ausdrücke werden in vielen Programmiersprachen und Texteditoren unterstützt. In Python kann man beispielsweise das re-Modul verwenden, um mit regulären Ausdrücken zu arbeiten:

import re

text = "Hallo Welt! 123"

## Findet alle Zahlen im Text
zahlen = re.findall(r'\d+', text)
print(zahlen) # Ausgabe: ['123']

## Ersetzt alle Zahlen durch "XXX"
neuer_text = re.sub(r'\d+', 'XXX', text)
print(neuer_text) # Ausgabe: Hallo Welt! XXX

• Dieses Modul stellt Funktionen wie search(), match(), findall() und sub() zur Verfügung, um Muster in Zeichenketten zu suchen, zu finden und zu ersetzen.

Communication Model

• The general communication model's components: Source, Encoder, Channel, Decoder, Sink
• Source-> message
• Encoder -> codeword
• Channel -> noisy codeword
• Decoder -> message

Channel Definitions

• Discrete Memoryless Channel (DMC) input alphabet is $\mathcal{X}$
• Discrete Memoryless Channel (DMC) output alphabet is $\mathcal{Y}$
• Discrete Memoryless Channel (DMC) channel transition probabilities are $p(y|x)$, $x \in \mathcal{X}$, $y \in \mathcal{Y}$
• "Memoryless" means $\displaystyle p(y_n|x_1, \dots, x_n, y_1, \dots, y_{n-1}) = p(y_n|x_n)$
• Channel Use: Each time a symbol is sent through the channel.
• Index set ${1, 2, \dots, M}$.
• Encoding function $f: {1, 2, \dots, M} \rightarrow \mathcal{X}^n$, yielding codewords $x^n(1), x^n(2), \dots, x^n(M)$.
• Decoding function $g: \mathcal{Y}^n \rightarrow {1, 2, \dots, M}$.
• Rate of the code: $R = \frac{\log M}{n}$ bits per channel use.
• Conditional probability of error: $\lambda_i = P(g(Y^n) \neq i | X^n = x^n(i))$, for $i \in {1, 2, \dots, M}$.
• Maximal probability of error: $\lambda^{(n)} = \max_{i \in {1, 2, \dots, M}} \lambda_i$.
• Average probability of error: $P_e^{(n)} = \frac{1}{M} \sum_{i=1}^M \lambda_i = \frac{1}{M} \sum_{i=1}^M P(g(Y^n) \neq i | X^n = x^n(i))$.
• A rate $R$ is achievable if there exists a sequence of $(2^{nR}, n)$ codes such that $\lambda^{(n)} \rightarrow 0$ as $n \rightarrow \infty$.
• The channel capacity is the supremum of all achievable rates, i.e. $$C = \sup {R: R \text{ is achievable} }$$

Jointly Typical Sequences

• The set $A_{\epsilon}^{(n)}$ of jointly typical sequences $(X^n, Y^n)$ with respect to $p(x, y)$ is the set of $n$-sequences with empirical entropies close to the true entropies: $$A_{\epsilon}^{(n)} = \left{ (x^n, y^n) \in \mathcal{X}^n \times \mathcal{Y}^n : \begin{array}{l} \left|-\frac{1}{n} \log p(x^n) - H(X)\right| < \epsilon, \ \left|-\frac{1}{n} \log p(y^n) - H(Y)\right| < \epsilon, \ \left|-\frac{1}{n} \log p(x^n, y^n) - H(X, Y)\right| < \epsilon \end{array} \right}$$
  • Where:
    • $p(x^n) = \prod_{i=1}^n p(x_i)$
    • $p(y^n) = \prod_{i=1}^n p(y_i)$
    • $p(x^n, y^n) = \prod_{i=1}^n p(x_i, y_i)$ If $(X^n, Y^n)$ be sequences of length $n$ drawn i.i.d. according to $p(x^n, y^n) = \prod_{i=1}^n p(x_i, y_i)$, then
    • $P((X^n, Y^n) \in A_{\epsilon}^{(n)}) \rightarrow 1$ as $n \rightarrow \infty$
    • $|A_{\epsilon}^{(n)}| \leq 2^{n(H(X, Y) + \epsilon)}$ If $(\tilde{X}^n, \tilde{Y}^n) \sim p(x^n) p(y^n)$, then
    • $$P((\tilde{X}^n, \tilde{Y}^n) \in A_{\epsilon}^{(n)}) \leq 2^{-n(I(X; Y) - 3\epsilon)}$$

Channel Capacity Theorem

• The capacity of a discrete memoryless channel is given by $$C = \max_{p(x)} I(X; Y)$$ where $I(X; Y)$ is the mutual information between the input and output of the channel, and the maximum is taken over all possible input distributions $p(x)$.

Introduction

• Specific intensity $I_{\nu}$: energy flux per unit frequency per unit solid angle.

$d E=I_{\nu} d A \cos \theta d \Omega d \nu d t$

$I_{\nu}$: erg $\mathrm{cm}^{-2} \mathrm{~s}^{-1} \mathrm{~Hz}^{-1}$ ster $^{-1}$

• Isotropic radiation: radiation field is the same in all directions.
• Mean intensity: average of specific intensity over all directions

$J_{\nu}=\frac{1}{4 \pi} \int I_{\nu} d \Omega$

• Isotropic radiation: $J_{\nu}=I_{\nu}$
• Energy density:

$u_{\nu}=\frac{1}{c} \int I_{\nu} d \Omega$

• Isotropic radiation: $u_{\nu}=\frac{4 \pi}{c} I_{\nu}=\frac{4 \pi}{c} J_{\nu}$
• Flux: net energy crossing surface in direction $\vec{n}$

$F_{\nu}=\int I_{\nu} \cos \theta d \Omega$ Isotropic radiation: $F_{\nu}=0$

Opacity, Emission

• Opacity: absorption coefficient $\alpha_{\nu}$: fraction of intensity lost per unit length

$d I_{\nu}=-\alpha_{\nu} I_{\nu} d s$

$\alpha_{\nu}$: $\mathrm{cm}^{-1}$

• Optical depth:

$d \tau_{\nu}=\alpha_{\nu} d s$

$\tau_{\nu}=\int \alpha_{\nu} d s$

$I_{\nu}(s)=I_{\nu}(0) e^{-\tau_{\nu}}$

• Opaque: $\tau_{\nu} \gg 1$
• Transparent: $\tau_{\nu} \ll 1$
• Emission coefficient $j_{\nu}$: energy emitted per unit length per unit time per unit solid angle per unit frequency

$d I_{\nu}=j_{\nu} d s$

$j_{\nu}$: erg $\mathrm{cm}^{-3} \mathrm{~s}^{-1} \mathrm{~Hz}^{-1}$ ster $^{-1}$

• Source function:

$S_{\nu} \equiv \frac{j_{\nu}}{\alpha_{\nu}}$

• Radiative transfer equation:

$\frac{d I_{\nu}}{d s}=-\alpha_{\nu} I_{\nu}+j_{\nu}$

$\frac{d I_{\nu}}{d \tau_{\nu}}=-I_{\nu}+S_{\nu}$

Thermal Radiation

• Thermodynamic equilibrium (TE): system in complete equilibrium

$I_{\nu}=B_{\nu}(T)$

$B_{\nu}(T)$: Planck function

$B_{\nu}(T)=\frac{2 h \nu^{3}}{c^{2}} \frac{1}{e^{h \nu / k T}-1}$

$B_{\lambda}(T)=\frac{2 h c^{2}}{\lambda^{5}} \frac{1}{e^{h c / \lambda k T}-1}$

• Kirchhoff's Law (TE):

$j_{\nu}=\alpha_{\nu} B_{\nu}(T)$

$S_{\nu}=B_{\nu}(T)$

• Local thermodynamic equilibrium (LTE): matter in TE at a certain $T$, radiation not necessarily in TE

$S_{\nu}=B_{\nu}(T)$

• Radiative transfer equation (LTE):

$\frac{d I_{\nu}}{d \tau_{\nu}}=-I_{\nu}+B_{\nu}(T)$

• Solution to radiative transfer equation:

$I_{\nu}\left(\tau_{\nu}\right)=I_{\nu}(0) e^{-\tau_{\nu}}+\int_{0}^{\tau_{\nu}} e^{-\left(\tau_{\nu}-\tau_{\nu}^{\prime}\right)} B_{\nu}\left(T\left[\tau_{\nu}^{\prime}\right]\right) d \tau_{\nu}^{\prime}$

• Optically thick source: $\tau_{\nu} \gg 1$

$I_{\nu} \rightarrow B_{\nu}(T)$

• Optically thin source: $\tau_{\nu} \ll 1$

$I_{\nu}=I_{\nu}(0)+\left(B_{\nu}(T)-I_{\nu}(0)\right) \tau_{\nu}$

• If background source absent:

$I_{\nu}=B_{\nu}(T) \tau_{\nu}=j_{\nu} L$

• L: geometrical depth of the source

Blackbody Radiation

Planck function

• $B_{\nu}(T)=\frac{2 h \nu^{3}}{c^{2}} \frac{1}{e^{h \nu / k T}-1}$

$B_{\lambda}(T)=\frac{2 h c^{2}}{\lambda^{5}} \frac{1}{e^{h c / \lambda k T}-1}$

• Wien's displacement law: peak of $B_{\nu}(T)$

$\nu_{\max } \approx 5.88 \times 10^{10} \mathrm{~Hz} \mathrm{~K}^{-1} T$

$\lambda_{\max } \approx 0.29 \mathrm{~cm} \mathrm{~K} / T$

• Stefan-Boltzmann law: integrated Planck function

$F=\int B_{\nu}(T) d \Omega=\sigma T^{4}$

$\sigma=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}}=5.67 \times 10^{-5} \mathrm{erg} \mathrm{cm}^{-2} \mathrm{~s}^{-1} \mathrm{~K}^{-4}$

• Energy density:

$u=\frac{4 \pi}{c} \int B_{\nu}(T) d \nu=a T^{4}$

$a=\frac{4 \sigma}{c}=7.56 \times 10^{-15} \mathrm{erg} \mathrm{cm}^{-3} \mathrm{~K}^{-4}$

Brightness Temperature

• Brightness temperature $T_{b}$: temperature a blackbody would have to have to give the same intensity at frequency $\nu$

$I_{\nu}=B_{\nu}\left(T_{b}\right)$

• Rayleigh-Jeans limit: $h \nu \ll k T$

$B_{\nu}(T) \approx \frac{2 \nu^{2}}{c^{2}} k T$

$T_{b}=\frac{c^{2}}{2 k \nu^{2}} I_{\nu}$

• Optically thin source:

$I_{\nu}=B_{\nu}(T) \tau_{\nu}$

$T_{b}=T \tau_{\nu}$

Microscopic Processes

• Einstein coefficients:
  • $A_{21}$: spontaneous emission
  • $B_{12}$: absorption
  • $B_{21}$: stimulated emission

$n_{1} B_{12} \bar{J}=n_{2} A_{21}+n_{2} B_{21} \bar{J}$

• $\bar{J}$: average intensity at the transition frequency
• Einstein relations:

$g_{1} B_{12}=g_{2} B_{21}$

$A_{21}=\frac{2 h \nu^{3}}{c^{2}} B_{21}$

$g_{i}$: statistical weight of level $i$

• Thermal distribution:

$\frac{n_{2}}{n_{1}}=\frac{g_{2}}{g_{1}} e^{-h \nu / k T}$

• Absorption coefficient:

$\alpha_{\nu}=\frac{h \nu}{4 \pi} n_{1} B_{12} \phi(\nu)\left(1-e^{-h \nu / k T}\right)$

• $\phi(\nu)$: line profile function

Radiation from Moving Charges

• Larmor's formula: power radiated by a nonrelativistic accelerated charge

$P=\frac{2 q^{2} a^{2}}{3 c^{3}}$

• $q$: charge
• $a$: acceleration
• Relativistic Larmor's formula:

$P=\frac{2 q^{2}}{3 c^{3}} \gamma^{6}\left(a_{\perp}^{2}+\gamma^{2} a_{|}^{2}\right)$

$\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$: Lorentz factor

$a_{\perp}$: acceleration perpendicular to velocity

$a_{|}$: acceleration parallel to velocity

• Dipole approximation:

$\frac{d P}{d \Omega}=\frac{q^{2} a^{2}}{4 \pi c^{3}} \sin ^{2} \Theta$

• $\Theta$: angle between acceleration and direction of observation
• Thomson scattering: scattering of radiation by free electrons

$\frac{d \sigma}{d \Omega}=\frac{e^{4}}{m^{2} c^{4}} \frac{1}{2}\left(1+\cos ^{2} \Theta\right)=r_{0}^{2} \frac{1}{2}\left(1+\cos ^{2} \Theta\right)$

$r_{0}=\frac{e^{2}}{m c^{2}}=2.82 \times 10^{-13} \mathrm{~cm}$: classical electron radius

$\sigma_{T}=\int \frac{d \sigma}{d \Omega} d \Omega=\frac{8 \pi}{3} r_{0}^{2}=6.65 \times 10^{-25} \mathrm{~cm}^{2}$: Thomson cross section

Bremsstrahlung

• Bremsstrahlung: free-free emission
• Electron accelerated by ion.
• Thermal bremsstrahlung:

$\epsilon_{f f}^{\text {nonrel }}=\frac{16 \pi e^{6}}{3 \sqrt{3} m_{e}^{2} c^{3}}\left(\frac{2 \pi m_{e}}{3 k}\right)^{1 / 2} T^{-1 / 2} n_{e} n_{i} Z^{2} g_{f f}$

$\epsilon_{f f}^{\text {nonrel }}$: emission coefficient

$n_{e}$: electron density

$n_{i}$: ion density

$Z$: ion charge

$g_{f f}$: Gaunt factor $\sim 1$

• Optically thin thermal bremsstrahlung:

$I_{\nu}=j_{\nu} L$

$I_{\nu} \propto \nu^{