Puzzle #21 – HP3: The Guards of Azkaban

Azkaban Prison is found on a remote island in the North Sea. Its inmates are guarded by wraith-like beings called Dementors. They and the Ministry maintain a volatile accord: in exchange for their services, the Dementors use the prisoners as a source of sustenance. According to Professor Lupin:

Dementors are among the foulest creatures that walk this earth. They infest the darkest, filthiest places, they glory in decay and despair, they drain peace, hope and happiness out of the air around them … Get too near a Dementor and every good feeling, every happy memory, will be sucked out of you.

In a simple model of a Dementor’s influence on its victim, the rate of change of the victim’s happiness $H(t)$ is given by

$\displaystyle \frac{dH(t)}{dt}=-\frac{\alpha H(t)}{r^2 (t)}$

where $r(t)$ is the separation of the Dementor from its victim, and $\alpha$ is a constant which characterises the strength of the Dementor’s influence on its victim; the parameter $\alpha$ may vary from person to person, since some people (such as Harry) are more susceptible to Dementors’ presence than others.

Harry’s first encounter with a Dementor occurs on the Hogwarts Express. While travelling from King’s Cross to Hogwarts, the train is drawn to a halt by the guards of Azkaban. As the train is plunged into darkness, a Dementor opens the door of the compartment in which Harry is sitting, searching for the fugitive Sirius Black. It hovers on the threshold, a fixed distance $r$ from Harry.

Question 1: Find the dependence of Harry’s happiness on time, given that his initial happiness $H(0)=H_0$.

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By draining their happiness, a Dementor leaves its victim at the mercy of their worst memories. The despair induced by exposure to a Dementor is enough to cause victims to lose consciousness. For simplicity, assume Harry passes out when his happiness $H(t)$ reaches e-1 times its initial value (about 37%).

Question 2: Given that Harry loses consciousness after about ten seconds of exposure to the Dementor at a metre’s distance, evaluate the constant $\alpha$ for Harry (give your answer in units of m2s-1).

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In a climactic scene by the Black Lake, Harry is being closed in upon from all sides by a swarm of Dementors. Suppose there are $n$ Dementors forming a hemisphere around Harry, each of which approaches from a great distance, travelling directly towards him at constant speed $v$.

Question 3: Find Harry’s happiness as a function of time, using the initial condition $H(-\infty)=H_0$. Using your value of $\alpha$ for Harry, estimate how far away the Dementors are from him when he passes out, given that there are 10 or so Dementors converging and they fly at about 5 metres per second.

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Harry also encounters Dementors during a Quidditch match against Hufflepuff. Suppose Harry flies past a stationary Dementor, approaching from a great distance at speed $v$ and travelling along a straight line. Let the distance of closest approach to the Dementor be $b$.

Question 4: Find Harry’s final happiness ($H(\infty)$) having passed the Dementor, given the initial condition $H(-\infty)=H_0$. Assuming Harry passes within $\pi$ metres of the Dementor, how fast must Harry fly to avoid falling off his broom?

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Hint: for questions 3 and 4, the key is find an expression for $r(t)$ in terms of $t$. Your equations may be slightly simpler if the time $t=0$ is chosen to be that at which Harry and the Dementor(s) are closest. You might find this post useful.