---
title: "Absorption and Doppler"
author: Jed Rembold
date: February 4, 2025
slideNumber: true
theme: tokyo-night-light
highlightjs-theme: tokyo-night-light
width: 1920
height: 1080
transition: slide
---


## Announcements
- I'm aiming to have HW1 feedback to you by the end of the week. They take me a while though
- Homework 2 is out!
  - Partner meetup time at the end of today
  - Should have enough to do problem 1 by the end of today and 2 and 3 by the end of Thursday
- We'll start looking at HR diagrams on Thursday

## Recap:
- We get two pieces of information when light enters our eyes: location and spectrum
- Light is an oscillating EM wave, with many possible wavelengths
  - Visible from 400nm to 720nm
- EM radiation comes from charges changing direction and speed
  - Hotter objects:
    - Produce more radiation
    - Produce "bluer" radiation
  - Planck's Law models this relation:
    $$ B(\lambda, T) = \frac{2hc^2}{\lambda^5}\frac{1}{\exp\left(\frac{hc}{\lambda k_B T}\right) - 1} $$

## Today's Plan
- Atomic spectral lines
  - Emission
  - Broadening
  - Absorption
- Doppler


## {data-background-image='../images/ch5_sun_spectrum.jpg'}

## Spectra and Composition
- Objects with thermal spectra are generally dense (wood, rocks, people, metals, stars, etc.)
- A perfect "blackbody" spectrum would _only_ give us information about the temperature
- Diffuse gases though can have more complex spectra, such that we can actually get information about chemical composition
	- Interestingly, this information also can help us identify some aspects about motion
	
	> - **Why is gas special?**


# Emission
## Life of an Electron
::::::cols
::::col
- Electrons orbit around the nucleus of an atom
- Only allowed in certain areas, which correspond to different _energy levels_
- The height of each energy level is determined by the atomic properties of the nucleus
::::

::::col

![](../images/energy_levels.svg)

::::
::::::

## Fingerprinting Light
- The discrete energy levels result in _line spectra_
- These spectra are unique for each atom and molecule, so they work as fingerprints!

\begin{tikzpicture}%%width=80%
[xscale=3, every node/.style={font=\sf}]
\pgfdeclarehorizontalshading{rainbow}{100bp}{
  color(0bp)=(violet);
  color(30bp)=(black!30!violet); 
  color(38bp)=(blue);
  color(42bp)=(black!20!cyan);
  color(46bp)=(green); 
  color(52bp)=(yellow);
  color(58bp)=(orange);
  color(65bp)=(red); 
  color(73bp)=(black!60!red);
  color(100bp)=(black!50!red)
}

\newcommand{\linespectra}[2]{
  \coordinate (init) at (#1);
  \fill[shading=rainbow, opacity=0.1] (init) rectangle +(4,1);
  \foreach \f in {#2}{
	\definecolor{col}{wave}{\f}
	%\fill[left color=transparent,right color=transparent,middle color=col] ($(init)+(\f/100-4,0)$) rectangle +(0.5mm,1cm);
	\fill[col, path fading=east] ($(init)+(\f/100-3.5+0.025,0)$) rectangle +(0.25mm,1cm);
	\fill[col, path fading=west] ($(init)+(\f/100-3.5+0.025,0)$) rectangle +(-0.25mm,1cm);
	%\node[white,anchor=west,font=\tiny, rotate=-90] at ($(init)+(\f/100-4+0.025,0)$) {\f nm};
  \draw[black,very thick] (init) rectangle +(4,1);
  }
}
  \linespectra{0,0}{656,486,434,410}
  \node[left] at (0,0.5) {Hydrogen};
  \linespectra{0,-1.5}{587,501,447}
  \node[left] at (0,-1) {Helium};
  \linespectra{0,-3}{670,610,548,460}
  \node[left] at (0,-2.5) {Lithium};
  \linespectra{0,-4.5}{645,615,470,467,464,459,441,407}
  \node[left] at (0,-4) {Oxygen};
\end{tikzpicture}

## On the Case
- Scattered around the room are ~4 different gas lamps, each with a different type of gas
- Given the descriptions below, can you identify which is corresponds to which gas?

  :::{style='font-size:.85em'}
  | Gas      | Description                                                            |
  |----------|------------------------------------------------------------------------|
  | Hydrogen | Two strong lines, one red, one greenish-blue, and a weaker blue-violet |
  | Nitrogen | Broad red and violet lines, strong green and yellow                    |
  | Mercury  | Strong yellow, green, and violet lines, numerous weaker red lines      |
  | Chlorine | Strong and evenly narrow lines at ROY-G-BIV                            |
  :::

# Broadening
## Seems Sus
- Stars (especially the surface) are made almost entirely of hydrogen
- So why do we get a (mostly) blackbody curve when looking at stars, and not sharp hydrogen emission peaks?


## Stars $\neq$ Gas Lamps
- Atoms in close proximity to one another influence each other's energy levels

:::::r-stack

:::{.fragment .current-visible}
\begin{tikzpicture}%%width=1000px
    \fill[ball color=Red] (0,0) circle (3pt);
    \foreach \r in {.5,1,...,2}{
      \draw[Black,dashed] (0,0) circle (\r cm);
    }
    \fill[ball color=cyan] (145:1) circle (2pt);
    \node[font=\sf] at (270:2.5) {Lonely atom in thin gas};
    \draw[Green,latex-] (2.4,0) -- +(1,0) node[align=center,right, font=\sf] {Nice, even\\energy levels};
\end{tikzpicture}
:::


:::fragment
\begin{tikzpicture}%%width=1200px
    \fill[ball color=Red] (0,0) circle (3pt);
    \foreach \r in {.5,1,...,2}{
      \draw[Black,dashed] (-\r/2,0) circle (\r cm);
    }
    \fill[ball color=cyan] ($(145:1)-(.5,0)$) circle (2pt);
    \fill[ball color=Red] (2.5,0) circle (3pt);
    \foreach \r in {.5,1,...,2}{
      \draw[Black,dashed] (2.5+\r/2,0) circle (\r cm);
    }
    \fill[ball color=cyan] ($(145:1)+(3,0)$) circle (2pt);
    \node[font=\sf] at (1.25,-2.5) {Crowded atoms in dense gas};
    \node[Green,font=\sf] (warp) at (1.25,2.5) {Warped energy levels!};
    \draw[-latex,Green] (warp.south) --+(240:1);
    \draw[-latex,Green] (warp.south)--+(300:1);
\end{tikzpicture}
:::
:::::


## Evolution of Spectra

:::::r-stack

:::{.fragment .current-visible}

\begin{tikzpicture}%%width=1200px
\pgfmathdeclarefunction{gauss}{3}{\pgfmathparse{#3*exp(-((x-#1)^2)/(2*#2^2))}}
\pgfdeclarehorizontalshading{rainbow}{100bp}{
  color(0bp)=(violet);
  color(30bp)=(black!30!violet); 
  color(38bp)=(blue);
  color(42bp)=(black!20!cyan);
  color(46bp)=(green); 
  color(52bp)=(yellow);
  color(58bp)=(orange);
  color(65bp)=(red); 
  color(73bp)=(black!60!red);
  color(100bp)=(black!50!red)
}
  \begin{axis}[samples=100, smooth,hide axis,width=10cm, height=6cm, xmin=-1]
    \addplot+[thick, Black,mark=none,shading=rainbow] {gauss(0,.1,1)+gauss(2,.1,1.3)+gauss(4,.1,.8)}\closedcycle;
  \end{axis}
  \node[font=\sf] at (4,-0.5) {Low Density Gas};
\end{tikzpicture}
:::

:::{.fragment .current-visible}

\begin{tikzpicture}%%width=1200px
\pgfmathdeclarefunction{gauss}{3}{\pgfmathparse{#3*exp(-((x-#1)^2)/(2*#2^2))}}
\pgfdeclarehorizontalshading{rainbow}{100bp}{
  color(0bp)=(violet);
  color(30bp)=(black!30!violet); 
  color(38bp)=(blue);
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  color(46bp)=(green); 
  color(52bp)=(yellow);
  color(58bp)=(orange);
  color(65bp)=(red); 
  color(73bp)=(black!60!red);
  color(100bp)=(black!50!red)
}
  \begin{axis}[samples=100, smooth,hide axis,width=10cm, height=6cm, xmin=-1]
      \addplot+[thick, Black,mark=none,shading=rainbow] {gauss(0,.3,1)+gauss(2,.3,1.3)+gauss(4,.3,.8)}\closedcycle;
  \end{axis}
  \node[font=\sf] at (4,-0.5) {Higher Density Gas};
\end{tikzpicture}
:::

:::{.fragment .current-visible}

\begin{tikzpicture}%%width=1200px
\pgfmathdeclarefunction{gauss}{3}{\pgfmathparse{#3*exp(-((x-#1)^2)/(2*#2^2))}}
\pgfdeclarehorizontalshading{rainbow}{100bp}{
  color(0bp)=(violet);
  color(30bp)=(black!30!violet); 
  color(38bp)=(blue);
  color(42bp)=(black!20!cyan);
  color(46bp)=(green); 
  color(52bp)=(yellow);
  color(58bp)=(orange);
  color(65bp)=(red); 
  color(73bp)=(black!60!red);
  color(100bp)=(black!50!red)
}
  \begin{axis}[samples=100, smooth,hide axis,width=10cm, height=6cm, xmin=-1]
      \addplot+[thick, Black,mark=none,shading=rainbow] {gauss(0,.7,1)+gauss(2,.7,1.3)+gauss(4,.7,.8)}\closedcycle;
  \end{axis}
  \node[font=\sf] at (4,-0.5) {Even Higher Density Gas};
\end{tikzpicture}
:::

:::{.fragment .current-visible}

\begin{tikzpicture}%%width=1200px
\pgfmathdeclarefunction{gauss}{3}{\pgfmathparse{#3*exp(-((x-#1)^2)/(2*#2^2))}}
\pgfdeclarehorizontalshading{rainbow}{100bp}{
  color(0bp)=(violet);
  color(30bp)=(black!30!violet); 
  color(38bp)=(blue);
  color(42bp)=(black!20!cyan);
  color(46bp)=(green); 
  color(52bp)=(yellow);
  color(58bp)=(orange);
  color(65bp)=(red); 
  color(73bp)=(black!60!red);
  color(100bp)=(black!50!red)
}
  \begin{axis}[samples=100, smooth,hide axis,width=10cm, height=6cm, xmin=-1]
      \addplot+[thick, Black,mark=none,shading=rainbow] {gauss(0,1.3,1)+gauss(2,1.3,1.3)+gauss(4,1.3,.8)}\closedcycle;
  \end{axis}
  \node[font=\sf] at (4,-0.5) {Really dense gas};
\end{tikzpicture}
:::
:::::

# Absorption
## Absorption Spectra and Goldilocks
- What if we have a cooler, diffuse gas, but with blackbody radiation shining on it?
- Some of the radiation will be absorbed
- Only the "just right" wavelengths corresponding to certain energy steps

\begin{tikzpicture}%%width=50%
  \coordinate (e) at (135:1);
  \coordinate (e2) at (125:2);
  \fill[ball color=Red] (0,0) circle (3pt);
  \draw[Black, dashed] (0,0) circle (1cm);
  \draw[Black, dashed] (0,0) circle (2cm);
  \fill[ball color=cyan] (e) circle (2pt);
  \draw[thick,-latex, Green!50!green, decorate, decoration={snake, post length=1mm}] ($(e)-(3,0)$) -- ($(e)-(1mm,0)$);
  \fill[ball color=cyan, opacity=0.5] (e2) circle (2pt);
  \draw[-latex, red] (e) to[bend right] (e2);
  \draw[thick, -latex, orange, decorate, decoration={snake, post length=2mm}] (-3,-1.5) --+ (6,0);
\end{tikzpicture}


## Sample Case: Hydrogen

\begin{tikzpicture}%%width=100%
[xscale=3, every node/.style={font=\sf}]
\pgfdeclarehorizontalshading{rainbow}{100bp}{
  color(0bp)=(violet);
  color(30bp)=(black!30!violet); 
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  color(46bp)=(green); 
  color(52bp)=(yellow);
  color(58bp)=(orange);
  color(65bp)=(red); 
  color(73bp)=(black!60!red);
  color(100bp)=(black!50!red)
}

\newcommand{\linespectra}[2]{
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  \foreach \f in {#2}{
	\definecolor{col}{wave}{\f}
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	\fill[col, path fading=east] ($(init)+(\f/100-3.5+0.025,0)$) rectangle +(0.25mm,1cm);
	\fill[col, path fading=west] ($(init)+(\f/100-3.5+0.025,0)$) rectangle +(-0.25mm,1cm);
	%\node[white,anchor=west,font=\tiny, rotate=-90] at ($(init)+(\f/100-4+0.025,0)$) {\f nm};
  \draw[black,very thick] (init) rectangle +(4,1);
  }
}
\newcommand{\absorbspectra}[2]{
  \coordinate (init) at (#1);
  \fill[shading=rainbow, opacity=0.8] (init) rectangle +(4,1);
  \draw[black,very thick] (init) rectangle +(4,1);
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	\fill[col, path fading=west] ($(init)+(\f/100-3.5+0.025,0)$) rectangle +(-0.25mm,1cm);
	%\node[white,anchor=west,font=\tiny, rotate=-90] at ($(init)+(\f/100-4+0.025,0)$) {\f nm};
  }
}

  \linespectra{0,0}{656,486,434,410}
  \node[above right] at (0,1) {Hydrogen Emission Spectra};
  \absorbspectra{0,-2}{656,486,434,410}
  \node[above right] at (0,-1) {Hydrogen Absorption Spectra};
\end{tikzpicture}

## That Spectra is Stellar!
::::::cols
::::col
- Hot gases in the surface regions of a star emit a blackbody spectrum, depending on their temperature
- Slightly cooler gases further out absorb some wavelengths, giving rise to absorption lines
::::

::::col
![](../images/ch5_sun_spectrum.jpg)
::::
::::::

## Viewing Absorption Differently

![](../images/solar_spectra.png)




## But wait, there's more!
<iframe width="1025" height="585" src="https://www.youtube.com/embed/EXxcwFfovkc" title="Train Horn Doppler Effect" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>


# Doppler
## The Doppler Effect
- The Doppler effect affects _all_ types of waves, including light!
- Approach waves get compressed (smaller wavelengths)
- Receding waves get stretched (larger wavelengths)

\begin{tikzpicture}%%width=70%
  \coordinate (s1) at (0,4);
  \foreach \r in {.2,.4,...,3}{
    \draw[Cyan] ($(s1)-(\r/1.5,0)$) circle (\r cm);
  }
  \draw[ultra thick, -latex,Green] (s1) --+(1.5,0);
  \fill[ball color=Red] (s1) circle (4pt);
  \coordinate (s2) at (10,4);
  \foreach \r in {.2,.4,...,3}{
    \draw[Cyan] ($(s2)-(\r/1.5,0)$) circle (\r cm);
  }
  \draw[ultra thick, -latex,Green] (s2) --+(1.5,0);
  \fill[ball color=Red] (s2) circle (4pt);
  
  \node at (4,0) {\includegraphics[width=0.75cm]{stickman.pdf}};
  \draw[Blue!50, thick, decorate, decoration={snake,segment length=1mm}] (3.7,.9) --+(145:3);
  \draw[Red!50, thick, decorate, decoration={snake,segment length=3mm,post length=0mm}] (4.3,.9) --+(35:3);
\end{tikzpicture}


## Putting Numbers to It
For our purposes:
$$ \frac{\lambda_{obs} - \lambda_{rest}}{\lambda_{rest}} = \frac{V}{c} $$
where:

- $\lambda_{obs}$ is the wavelength you, the observer, see
- $\lambda_{rest}$ is the normal wavelength you'd see if the object was not moving
- $V$ is the speed of the light source relative to you
  - Negative if coming towards you
  - Positive if moving away from you
- $c$ is the speed of light


## Astronomy Implications
- We can measure radial speeds of objects!
- Approaching objects are _blueshifted_
- Receding objects are _redshifted_


\begin{tikzpicture}%%width=100%
[xscale=2.5, every node/.style={font=\sf}]
\pgfdeclarehorizontalshading{rainbow}{100bp}{
  color(0bp)=(violet);
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  color(46bp)=(green); 
  color(52bp)=(yellow);
  color(58bp)=(orange);
  color(65bp)=(red); 
  color(73bp)=(black!60!red);
  color(100bp)=(black!50!red)
}

\newcommand{\absorbspectra}[2]{
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}
  \absorbspectra{0,0}{447,471,492,501,587,667}
  \node[anchor=east] at (0,.5) {Normal};
  \absorbspectra{0,-1.5}{457,481,502,511,597,677}
  \node[anchor=east] at (0,-1) {Redshifted};
  \absorbspectra{0,-3}{437,461,482,491,577,657}
  \node[anchor=east] at (0,-2.5) {Blueshifted};
\end{tikzpicture}


## Partners!
- I'm giving you a chance now to meet with your partner for HW2 to discuss schedules, strengths, weaknesses, and how you can best work together!
- Remember that there will be a check-in this weekend, so try to have at least something accomplished you can mention!

::::::{.cols style='align-items: flex-start'}
::::col
- Left rows (front to back)
	- Gabby & Maddie
    - Evan and Luna
    - Mamadou & Greg
    - Conor & Aurora
    - Evyn & Elliott
    - Sergio & Pearson
    - Oscar & Sage


::::

::::col
- Right rows (front to back)
	- Felicity & Tegan
    - Luca & Clay
    - Sawyer & Izzy
    - Salem & Jared
    - M & Sadie
    - Ema & Lucca

::::
::::::