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| 1 | +% minilecture04.tex: Tutorial 4 mini-lecture: Surfaces & Interiors |
| 2 | +% Planetary Systems, Kapteyn Institute, University of Groningen |
| 3 | +% A 5-10 minute recap, presented by the TAs at the start of Tutorial 4, |
| 4 | +% of the Lecture 7-8 concepts that Worksheet 4 trains. |
| 5 | + |
| 6 | +\documentclass[aspectratio=169, 11pt]{beamer} |
| 7 | + |
| 8 | +% ── Theme and macros (shared with the lecture decks) ──────── |
| 9 | +\usepackage{../../slides/common/beamerthemeIPS} |
| 10 | +\input{../../slides/common/macros} |
| 11 | +\graphicspath{{figures/}{../../slides/lecture08/figures/}{../../slides/lecture07/figures/}{../../slides/common/}} |
| 12 | + |
| 13 | +% ── Metadata ──────────────────────────────────────────────── |
| 14 | +\title[T4 mini-lecture: Surfaces \& Interiors]{Tutorial 4 Mini-Lecture:\\Surfaces \& Interiors: Craters, Pressure, \& Seismology} |
| 15 | +\subtitle{The concepts behind Worksheet 4 \textbullet\ Lectures 7--8} |
| 16 | +\author{} |
| 17 | +\institute{Kapteyn Astronomical Institute\\University of Groningen} |
| 18 | +\date{2026} |
| 19 | +\titlebgcredit{Earth's interior cross-section. Credit: NASA/JPL-Caltech} |
| 20 | +\titlebgopacity{0.55} |
| 21 | + |
| 22 | +% ════════════════════════════════════════════════════════════ |
| 23 | +\begin{document} |
| 24 | + |
| 25 | +% ── Title slide ───────────────────────────────────────────── |
| 26 | +\begin{frame}[plain,noframenumbering] |
| 27 | + \titlepage |
| 28 | +\end{frame} |
| 29 | + |
| 30 | +% ── Roadmap ───────────────────────────────────────────────── |
| 31 | +\begin{frame}{What Worksheet 4 trains} |
| 32 | + Five problems, all built on Lectures 7--8. The physics you need, per problem: |
| 33 | + \vskip4pt\relax |
| 34 | + \begin{enumerate} |
| 35 | + \item \textbf{Impact energetics on Mars}: impactor mass and energy, gravity-regime crater scaling, the simple-to-complex transition. |
| 36 | + \item \textbf{Reading ages from craters}: the crater chronology $N(T)$, its linear and exponential branches, the printed chronology figure. |
| 37 | + \item \textbf{Moment of inertia and the inside of Mercury}: the two-layer model, layer densities, what $C/MR^2$ measures. |
| 38 | + \item \textbf{Central pressure}: hydrostatic equilibrium, the uniform-density pressure profile, why it is a lower bound. |
| 39 | + \item \textbf{Seismology}: shadow zones, what liquid layers do to S and P waves, the Adams-Williamson assumptions. |
| 40 | + \end{enumerate} |
| 41 | + \vskip2pt\relax |
| 42 | + \keyresult{Every problem has the shape and level of one exam question. All parts are analytical; no computer is needed.} |
| 43 | +\end{frame} |
| 44 | + |
| 45 | +% ── P1 ────────────────────────────────────────────────────── |
| 46 | +\begin{frame}{Impact energetics on Mars} |
| 47 | + \small |
| 48 | + \begin{columns}[T] |
| 49 | + \column{0.52\textwidth} |
| 50 | + The gravity-regime scaling of Lecture 7: |
| 51 | + \[ |
| 52 | + \boxed{D \approx \left(\frac{E_k}{\rho_t\,g}\right)^{1/4}} |
| 53 | + \qquad D \propto L^{3/4} |
| 54 | + \] |
| 55 | + \begin{itemize} |
| 56 | + \item $E_k = \tfrac{1}{2}mv^2$ with $m = \tfrac{4}{3}\pi r^3 \rho$: the whole chain runs in SI units. |
| 57 | + \item The fourth root compresses hard: check how many orders of magnitude in energy correspond to one order of magnitude in diameter. |
| 58 | + \end{itemize} |
| 59 | + |
| 60 | + \column{0.44\textwidth} |
| 61 | + \begin{block}{Simple or complex?} |
| 62 | + The transition diameter scales as $D_t \propto 1/g$: stronger gravity collapses smaller cavities. Locate the martian transition from the lunar one, then classify your crater against it. |
| 63 | + \end{block} |
| 64 | + \end{columns} |
| 65 | + \vskip2pt\relax |
| 66 | + \keyresult{One scaling law connects the impactor you cannot see to the crater you can measure.} |
| 67 | +\end{frame} |
| 68 | + |
| 69 | +% ── P2 ────────────────────────────────────────────────────── |
| 70 | +\begin{frame}{Reading ages from craters} |
| 71 | + \small |
| 72 | + \begin{columns}[T] |
| 73 | + \column{0.52\textwidth} |
| 74 | + The lunar chronology of Lecture 7: |
| 75 | + \[ |
| 76 | + \boxed{N(\geq 1\,\mathrm{km}) = a\left(e^{\lambda T} - 1\right) + b\,T} |
| 77 | + \] |
| 78 | + \begin{itemize} |
| 79 | + \item Young surfaces lie on the linear branch: steady flux, $N \approx bT$, age is a division. |
| 80 | + \item Old surfaces lie on the exponential branch of the early bombardment: $N$ grows exponentially with $T$. |
| 81 | + \end{itemize} |
| 82 | + |
| 83 | + \column{0.44\textwidth} |
| 84 | + \begin{block}{Iterate, do not invert} |
| 85 | + The full $N(T)$ has no closed-form inverse. On the steep branch, neglect the small term, solve, then correct once with the term you dropped. On the shallow branch, check the dropped term is small instead. |
| 86 | + \end{block} |
| 87 | + \end{columns} |
| 88 | + \vskip2pt\relax |
| 89 | + \keyresult{The same measurement error means different things on different branches: check which branch you are on before you trust a crater age.} |
| 90 | +\end{frame} |
| 91 | + |
| 92 | +% ── P3 ────────────────────────────────────────────────────── |
| 93 | +\begin{frame}{Moment of inertia and the inside of Mercury} |
| 94 | + \small |
| 95 | + \begin{columns}[T] |
| 96 | + \column{0.52\textwidth} |
| 97 | + The two-layer model of Lecture 8: |
| 98 | + \[ |
| 99 | + \boxed{\frac{C}{MR^2} = \frac{2}{5}\,\frac{(f-1)\,x^5 + 1}{(f-1)\,x^3 + 1}} |
| 100 | + \quad f = \frac{\rho_c}{\rho_m} |
| 101 | + \] |
| 102 | + \begin{itemize} |
| 103 | + \item A uniform sphere gives $0.400$; every planet measures below it, and the deficit measures central condensation. |
| 104 | + \item Mass balance fixes the layer densities: core mass in core volume, the rest in the mantle shell. |
| 105 | + \end{itemize} |
| 106 | + |
| 107 | + \column{0.44\textwidth} |
| 108 | + \begin{block}{Small cores hide well} |
| 109 | + A core of radius fraction $x$ fills only $x^3$ of the volume. Before judging any claim from $C/MR^2$, compute how far a plausible core can move the number at all. |
| 110 | + \end{block} |
| 111 | + \end{columns} |
| 112 | + \vskip2pt\relax |
| 113 | + \keyresult{$C/MR^2$ is measured from spin and gravity without drilling a hole; it is the single most informative number about a planetary interior.} |
| 114 | +\end{frame} |
| 115 | + |
| 116 | +% ── P4 ────────────────────────────────────────────────────── |
| 117 | +\begin{frame}{Central pressure} |
| 118 | + \small |
| 119 | + \begin{columns}[T] |
| 120 | + \column{0.52\textwidth} |
| 121 | + Hydrostatic equilibrium at uniform density: |
| 122 | + \[ |
| 123 | + \boxed{P_c = \frac{2\pi}{3}\,G\bar\rho^2 R^2 = \frac{3GM^2}{8\pi R^4}} |
| 124 | + \] |
| 125 | + \begin{itemize} |
| 126 | + \item Derive it once: integrate $\dd P/\dd r = -Gm(r)\rho/r^2$ from the centre to the surface. |
| 127 | + \item The two forms must agree; use the second as a cross-check on the first. |
| 128 | + \end{itemize} |
| 129 | + |
| 130 | + \column{0.44\textwidth} |
| 131 | + \begin{block}{What changes if density is not uniform?} |
| 132 | + Compare the enclosed-mass term at fixed $r$ for a centrally concentrated profile against a uniform one. State only which term grows, not the direction of the final inequality. |
| 133 | + \end{block} |
| 134 | + \end{columns} |
| 135 | + \vskip2pt\relax |
| 136 | + \keyresult{The central pressure of a uniform body depends only on its mass and radius; the strong $M^2/R^4$ scaling separates the planets by orders of magnitude.} |
| 137 | +\end{frame} |
| 138 | + |
| 139 | +% ── P5 ────────────────────────────────────────────────────── |
| 140 | +\begin{frame}{Seismology: listening to interiors} |
| 141 | + \small |
| 142 | + \begin{columns}[T] |
| 143 | + \column{0.52\textwidth} |
| 144 | + The wave speeds of Lecture 8: |
| 145 | + \[ |
| 146 | + \boxed{v_S = \sqrt{\frac{\mu}{\rho}} |
| 147 | + \qquad |
| 148 | + v_P = \sqrt{\frac{K + \tfrac{4}{3}\mu}{\rho}}} |
| 149 | + \] |
| 150 | + \begin{itemize} |
| 151 | + \item A liquid has no shear modulus: $\mu = 0$ stops S waves entirely. |
| 152 | + \item P waves pass through liquids, but a speed drop at a boundary bends the rays and opens a shadow ring. |
| 153 | + \end{itemize} |
| 154 | + |
| 155 | + \column{0.44\textwidth} |
| 156 | + \begin{block}{Two shadows, two mechanisms} |
| 157 | + One shadow comes from absorption, the other from refraction. For each observation, state which mechanism makes it and which interior property it therefore proves. |
| 158 | + \end{block} |
| 159 | + \end{columns} |
| 160 | + \vskip2pt\relax |
| 161 | + \keyresult{What is missing from a seismogram tells as much as what arrives: shadow zones map liquid layers on any planet with a seismometer.} |
| 162 | +\end{frame} |
| 163 | + |
| 164 | +% ── Closer ────────────────────────────────────────────────── |
| 165 | +\begin{frame}{Working the worksheet} |
| 166 | + \begin{itemize} |
| 167 | + \item \textbf{Show your algebra.} The exam asks for the steps, not only the number. |
| 168 | + \item Carry \textbf{4 significant figures} through intermediate steps; round at the end. |
| 169 | + \item Use the \textbf{checkpoints} ("show that \ldots") to verify your work and to keep moving if a step resists. |
| 170 | + \item Qualitative parts want \textbf{2 to 3 sentences} of physics, not an essay. |
| 171 | + \item Problem 2 reads the chronology figure printed on the sheet; Problem 5 is entirely qualitative and trains the exam skill of arguing from mechanisms. |
| 172 | + \end{itemize} |
| 173 | + \vskip8pt\relax |
| 174 | + \keyresult{Worksheet and full solutions: \texttt{ips.formingworlds.space/worksheets.html}. We discuss questions, not presentations of the solutions.} |
| 175 | +\end{frame} |
| 176 | + |
| 177 | +\end{document} |
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