Physics of Planets

Samenvattingen

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English below

Algemeen

Physics of Planets wordt gegeven door professor Tim Van Hoolst die een keer per week vanuit de Sterrenwacht afzakt naar onze universiteit. Het vak behandelt de structuur van planeten, met een focus op gravitatie (en getijden), en warmtegeleiding in planeten.

Tegen het einde van het semester geven studenten in groepjes van twee of drie personen een presentatie op 5 van de 20 punten. Het is hierbij de bedoeling een paper aan de cursus te linken.

Het examen zelf bestaat uit een deel theorie, gesloten boek, dat al dan niet nog mondeling wordt toegelicht. Nodige formules worden gegeven of kunnen worden gevraagd. De oefeningen zijn open boek.

Physics of Planets is taught by professor Tim Van Hoolst, who works at the Belgian Royal Observatory and drops by our university once a week. This course examines the interior structure of terrestrial and icy planets and satellites with most of the attention going to gravitational research, tides and heat transport.

In the first eight weeks, there is one lecture every week which usually include some exercises such as unfinished derivations. It is recommended that you already try these exercises to ask the professor questions about them if need be. After the Easter break, there are no more lectures. Instead, you form a team of usually 3 students and pick a topic by the end of the break (the prof provides some suggestions) in the form of a paper about planetary science. You are then expected to dig a bit deeper, (mathematically) work out some arguments and give presentation on your findings. This is graded for 5 out of the 20 points, meaning the exam only accounts for the remaining 15 points.

The exam is closed book, meaning you can't bring anything but a pen and a pocket calculator. The professor provides you with necessary equations and constants. Most questions will ask you to derive an equation and most of the derivations are rather short. There are also a handful of exercises, often using the equations you just derived, where you normally have to compute a quantity and compare it to observations/other estimates.

Exams

Exam 25 June 2021

Disclaimer

The exam usually takes 4 hours, in which you have to complete 5 or so questions. This year, the exam could only last 3 hours due to covid restrictions and was thus limited to 4 questions.

Questions

1) Derive an expression for the internal gravitational potential and field of a homogeneous ellipsoid with an equatorial flattening of zero. You are given an expression for the gravitational potential of a sphere as well as one for the radial distance to the surface of the ellipsoid in terms of the polar flattening.

2) Derive an expression for the temperature profile of a uniform sphere with radiative energy release. From this, find what the expected surface heat flux is and the central temperature. Use this to compute the central temperature of the Moon given its measured heat flux, radius, average density and radioactivity. What do you think about this central temperature?

3) Derive an approximation for how temperature evolves in a large body. You are given the equation for heat diffusion, as well as the definition of thermal diffusivity. Use this equation to find an approximate formula for the depth to which a significant thermal change has penetrated. Finally, use this to compute how deep thermal changes have occured in Earth.

4) Io is a very active object, proven by its great heat loss at the surface (which is given). For three different sources of heating, you must compute how great the expected surface heat flux is if all the produced energy went into surface heat flux and not e.g. melting. These are radiogenic heat production, tidal heating and cooling.

Examen Leuven 5 juni 2021

Info

Het examen was gesloten boek, 4 oefeningen in 3u (dit was in een coronajaar, volgende jaren is het waarschijnlijk iets anders).

Theorie

• - Leidt de relatie $L^2=\kappa t$ voor temperatuurschommelingen in een conductieve schil af. Je krijgt Fouriers law en de definitie van $\kappa$. Geef ook een schets bij je antwoord.

• - Laat zien waarom we Rayleighs number kunnen gebruiken om aan te tonen hoe effectief convectie in een omgeving is. Maak gebruik van twee tijdschalen: diffusie en convectie om je antwoord op te bouwen.

• - Leid de uitdrukking voor $\mathrm{\Delta }{\varphi }_{lm}$ door topografie in de Moho af. De uitdrukking voor $\mathrm{\Delta }\varphi$ is gegeven. Je moet beginnen van de algemene uitdrukking voor de potential: $\varphi (r)=-G\int \frac{\rho (r^{\prime })dr}{r-r^{\prime }}$. De uitdrukking voor $\frac{1}{r-r^{\prime }}$ is gegeven, alsook de addition theorem.

Oefening

• Oefening over Enceladus: berekenen een geschatte totale heat flow en energy loss, bereken hoe lang het duurt vooraleer Enceladus' oceaan compleet bevriest, bereken of radioactviteit de heat loss kan counteren, bereken of convectie mogelijk is in de surface ijslaag en of dit zou leiden tot meer of minder energieverlies. (elke constante die je nodig hebt is gegeven, alsook enkele formules)

Examen Leuven 24 juni 2011

Theorie

• Gegeven $Ra=\frac{\rho g\alpha (T1-T0)b^3}{\eta \kappa }$, thermal boundary layer $x=2.32\sqrt{\kappa t}$, $\frac{T-T1}{T0-T1}=1-erf(x/2\sqrt{\kappa t})$ en de definitie van de errorfunctie. Bepaal de tijd wanneer de boundary layer afbreekt. Bepaal ook de gemiddelde convectieve warmtestroom.

• Toon de definitie van $C_{lm}$ (Stokes coefficienten) aan in de zwaartekrachtpotentiaal. De definitie van $\mathrm{\Phi }$ is gegeven, net als het addition theorem, de afleiding voor $\frac{1}{|r-r^{\prime }|}$ en de lage-orde Legendre functies en de normalisatie en ortogonaliteitsregels uit de slides.

Oefeningen

• Estimate the tidal heating rate and radioactive energy production rate in the satellites Mimas and Enceladus of Saturn. Assume a constant radioactive energy production per unit mass and per unit time equal to that for the current Earth's silicate envelope (mantle + crust, to be determined from the date in the course notes). Mass of the silicate envelope of the Earth = 4.07*10^24 kg. Mass fraction of thes ilicate in Mimas and Enceladus is 18 wt% and 53 wt% respectively. For the tidal dissipation you may assume that the satellites are homogeneous. (Data on Mimas, Enceladus: M, R, a, e, n given for both satellites.) On which satellite would you expect geysers to occur?

• Demonstrate that $h_2=5/2$ and that $k_2=3/2$ for a homogeneous fluid (incompressible) planet from the definitions of the Love numbers. Start by deriving an expression for the additional gravitational potential ${\mathrm{\Phi }}^{\prime }(R)$ due to the tidal deformation of the planet in terms of the surface density associated with the radial tidal displacement H at the surface of the planet. Consider that the tidal potential, and therefore also the surface density, can be expressed as a Legendre polynomial of order 2. Note: we here follow the geodesy convention for the gravitational potential ( $F=\mathrm{\nabla }\mathrm{\Phi }$). Further use Bruns' formula, which states that the displacement of an equipotential surface due to an additional potential $\delta U$ (here: tidal potential + $\Phi'$) can be expressed as $H=\frac{\delta U(R)}{g(R)}$.