--- title: "Peak Star Types" author: Jed Rembold date: February 6, 2025 slideNumber: true theme: tokyo-night-light highlightjs-theme: tokyo-night-light width: 1920 height: 1080 transition: fade --- ## Announcements - I'm aiming to get HW1 feedback to you this weekend - Be working on HW2! - Don't forget the check-in form this weekend! Make sure you have touched base with your partner and have a plan! ## Recap - Atomic energy levels result in emission or absorption spectra - Exact levels depend on chemical composition - Hot diffuse gases emit, colder gases absorb - Movement towards or away from us shifts our perceived wavelengths - Redshifted (longer wavelength) going away from us - Blueshifted (shorter wavelengths) coming toward us ## Today's Plan - Peak Data Reduction - Luminosity - Stellar Distances - Star Types - HR Diagrams # Finding and Measuring Absorption Peaks ## Maximum Redshift - Stars are considered "hypervelocity stars" if they have radial speeds of around 500 km/s or higher - Take a moment to compute how much of a shift this would cause in the $H_\alpha$ line usually at 656.464 nm. . . . $$\Delta\lambda = \frac{500\times10^3}{3\times10^8}\times 656.464 = 1.094$$ - The shift would be only a single nanometer! - **Finding** the peaks isn't difficult, as they are largely where we expect - **Measuring** the slight differences is far more tricky ## Gaussians - Measure peak placement precisely involves fitting a model to the peak - One of the most common is that of a Gaussian: $$ g(x) = H + A\exp\left(\frac{-(x-x_0)^2}{2\sigma^2}\right) $$ where - $H$ is the offset or shift of the baseline - $A$ is the peak height above the baseline (can be negative) - $x_0$ is the $x$ at which the peak is centered - $\sigma$ is the spread of the peak ## The Continuum - Peaks often show up on areas of the blackbody spectrum that are heavily sloped - Fitting them well requires "flattening", or normalizing out this background _continuum_ - Most common approach is to fit a line or polynomial to just the background (not the peak!) and then divide the entire background by this signal - If done well, you should get a clean peak sitting on a level surface, ready for fitting - For velocities or identifying chemical composition, determining $x_0$ is the most important, as that is the wavelength the peak is centered at ## Example Using Solar Spectra - To illustrate this approach, let's investigate the $H_\alpha$ line in the solar spectra given [here](../demos/sun_spectra.csv) - Want to: - Establish there is a peak approximately where we expect it - Normalize the background - Fit a gaussian to determine peak location ## Velocity Computing ::::::cols ::::col - You can technically compute a velocity for each peak individually - The spectrum will generally have _many_ peaks - Don't just average them! Fit a line to them: $$\lambda_{obs} = \left(\frac{V}{c} + 1\right)\lambda_{rest} $$ :::: ::::col ![](../images/doppler_graph.svg) :::: :::::: . . . # Luminosity ## So what can we observe? ::::::{.cols style='align-items: flex-start'} ::::col - Light from the stars tells us: - Their location in the sky - Their overall brightness - Their intensity at different wavelengths (their spectrum) :::: ::::col - From these, we can determine: - Surface temperature - Radial motion - Distance (sometimes) - Size (in a fashion) - Power output or Luminosity - Mass (sometimes) :::: :::::: ## Luminosity ::::::cols ::::{.col style='font-size:.8em; flex-grow:1.5'} - We measure the apparent brightness $B$ of an object here at Earth (area under spectra) - Like ripples around a dropped rock though, brightness falls off with distance: - Unlike pond ripples, the waves spread out radially, so the energy gets spread over a sphere - Thus the luminosity is: $$ L = 4\pi d^2 \times B $$ where $d$ is the distance - The range of possible stellar luminosities is huge - $L_{sun} = L_\odot = 4 \times 10^{26}$ W - Dimmest at around $0.000001L_\odot$ - Brightest around $100000L_\odot$ :::: ::::col ![Energy at the source is spread over ever larger spheres](../images/ch12_LightFalloff.png){width=90%} :::: :::::: # Accounting for Distance ## Stellar Distances - Initially, coming from parallax - Shifts of the foreground relative to the background when the viewpoint changes - Parallax effects are larger for closer objects, and stars are **far** away - Need as large a baseline as possible: observing during 6 month intervals to be on opposite sides of the Sun - Parallax effects from stars are still **tiny**: generally less than an arcsecond - A _parsec_ is the distance that corresponds to a parallax angle of 1 arcsecond - Equivalent to 3.26 light-years, or 3.26$\times$ the distance light travels in a year - Measuring the parallax angle $p$ in arcseconds gives the distance in parsecs $d$: $$ d_{pc} = \frac{1}{p_{asec}} $$ ## Absolute Magnitude - Astronomers will also use _absolute magnitude_ as a proxy for luminosity - A star's absolute magnitude (commonly denoted $M$) is the magnitude it would seem to have if it was 10 parsecs away - Still requires knowing the distance to the star to compute: $$ m - M = 5\log_{10}\left(\frac{d_{pc}}{10}\right) $$ where $$ \begin{aligned} M &= \text{ absolute magnitude} \\ m &= \text{ apparent magnitude} \\ d_{pc} &= \text{ distance in pc}\\ \end{aligned} $$ # Classifying Stars ## Star Types ::::::cols ::::col - Stars were originally classified by the strength of their Hydrogen lines - The strongest were classified type A, all the way down to type O, which showed virtually no hydrogen lines :::: ::::col ![Original Star Types](../images/ch12_StarTypes.png){width=75%} :::: :::::: ## Scrambling the System - As more spectra were observed, the H lines were proving to be less reliable in predicting similar properties - Enter Annie Cannon - Hired as one of the Harvard Computers - Classified some 350,000 stars (yikes!) - Drastically simplied the system and eliminated many classes, focusing mainly on the Balmer line transitions - Once the relationship between spectra lines and temperature was understood, the letters were reordered to match the temperature trend \begin{tikzpicture}%%width=90% [every node/.style={font=\sf}] \pgfdeclarehorizontalshading{startype}{100bp}{ color(0bp)=(black!50!red); color(25bp)=(black!50!red); color(28bp)=(red); color(32bp)=(orange); color(36bp)=(yellow); color(40bp)=(white); color(50bp)=(cyan!50); color(60bp)=(cyan!50!blue); color(70bp)=(blue); color(75bp)=(blue!50!violet!70!black); color(100bp)=(blue!50!violet!50!black) } \shade[rounded corners,shading=startype, shading angle=180] (0,0) rectangle +(10,.5); \node at (1, 0.7) {O}; \node at (4, 0.7) {B}; \node at (5.5,0.7) {A}; \node at (6.8,0.7) {F}; \node at (7.5,0.7) {G}; \node at (8.5,0.7) {K}; \node at (9.4,0.7) {M}; \end{tikzpicture} ## HR Diagrams ::::::cols :::::col ![](../images/hr1.gif){width=70%} ::::: :::::col ![](../images/HRDiagram.png){width=70% .fragment} ::::: :::::: ## HR Diagram {data-background-color="var(--gray2)"} ![](../images/standalone/HR_diagram.svg) # HR Trends ## Star Sizes - Given certain names, you can perhaps guess how stellar size varies in an HR diagram - But why? - Recall that total brightness over some interval of wavelength is measured in watts per square meter - This would be the area under a spectra curve - This is why brightness drops off as it travels away from the star to us - This also means though that the total energy emitted from the surface of the star will depend on the star's size! - The area under the curve depends (heavily) on the temperature $$ L = 4\pi R^2_s \times \sigma T^4 $$ ## Size Trends ::::::cols ::::col - The total energy output of the star thus depends on both its size (radius) and its temperature - Cooler stars need to be much larger to have the same luminosity output! - Hot stars can be smaller :::: ::::col ![HR Diagram Size Dependence](../images/HR_sizes.svg) :::: :::::: ## Mass and HR Diagrams :::incremental - What about patterns in the mass of stars on the HR diagram? - Globally, there is no obvious trend - There do appear to be trends within the subgroups though: - Main sequence stars decrease in mass from upper left to lower right - White dwarfs are generally fairly low in mass - Giants and supergiants can vary wildly - Mass determines many of the equilibrium points in stars, so no clear trend is interesting! :::