Problem Statement
Suppose that there is a pulsar with a period of 5 milliseconds (5 ms) and a period derivative of 0.1 ms/s at $t = 0$ seconds. Make the assumption that the product of $\dot{P}P$ remains constant at all times.
- Using your knowledge from calculus and differential equations, obtain an expression for the pulsar’s period, $P(t).$
- Using your expression from the first part, determine what $P(t = 11 \, \mathrm{seconds})$ is.
- Create a plot of $P(t)$ from $t = 0$ seconds to $t = 1000$ seconds using the Python programming language.
- Imagine a scenario where you have two clocks: a pulsar clock that keeps time by counting the radio pulses received from the pulsar, and a “perfect” clock where $\dot{P} = 0$ that is synchronized with the pulsar clock when both clocks read zero time. When the perfect clock reads that a time of $t = \frac{6}{5}\frac{P_0}{\dot{P_0}}$ has passed, what time does the pulsar clock display?
Solution
- To set up a differential equation, we make use of the fact that $\dot{P}P = \dot{P_0}P_0$.
$$
\begin{equation} \dot{P} = \frac{dP}{dt} = P_0\dot{P_0} \end{equation}
$$
$$
\begin{equation} P\frac{dP}{dt} = \dot{P_0}P_0 \end{equation}
$$
$$
\begin{equation} P dP = \dot{P_0}P_0 dt\end{equation}
$$
$$
\begin{equation} \frac{P^2}{2} = \dot{P_0}P_0t +C \end{equation}
$$
$$
\begin{equation} P(t) = (2\dot{P_0}P_0t + 2C)^{1/2} \end{equation}
$$
To determine the unknown $2C,$ we use the fact that $P(t = 0) = P_0 = \sqrt{2C}$ so $P_0^2 = 2C$
$$
\boxed{ P(t) = (2\dot{P_0}P_0t + P_0^2)^{1/2} = (1 \frac{\mathrm{ms}^2}{\mathrm{s}} t+ 25\,\mathrm{ms}^2)^{1/2} }
$$
- Using the expression from part 1, we should get that:
$$
(1 \frac{\mathrm{ms}^2}{\mathrm{s}}(11 \,\mathrm{s})+ 25\,\mathrm{ms}^2)^{1/2} = \sqrt{36\, \mathrm{ms}^2} = \boxed{6 \,\mathrm{ms}}
$$
- Making a plot of the graph from $t = 0$ seconds to $t = 1000$ seconds.