Suppose that, as a pulsar slows down, the quantity \((1 / P)(\mathrm{d} P / \mathrm{d} t)\) stays constant (say at a value of \(-b,\) where \(b\) is a positive number) (a) If the initial period \((t=0)\) of the pulsar is \(P_{0} .\) find an expression for \(P(t)\), the period as a function of time. (b) If the initial rotation energy is \(E_{0},\) find an expression for \(E(t)\), the energy as a function of time. (c) If the pulsar is formed with \(P_{0}=10^{-3} \mathrm{s}\), how long will it take to reach \(P=3\) s?

Differential equations play a key role in modeling the period decay of pulsars. In the problem, we are given a specific form: \(\frac{1}{P} \frac{\text{d}P}{\text{d} t} = -b\). This is a first-order linear differential equation, where the rate of change of the period, \(P\), depends on \(P\) itself and a constant, \(b\). By separating variables and integrating, we can solve this equation to find an expression for \(P(t)\). Such equations are common in various fields of physics and engineering, where they describe processes that change over time.

The rotational energy \(E\) of a pulsar is directly influenced by its period \(P\). When dealing with rotating objects like pulsars, the rotational energy is inversely proportional to the square of the period: \ E(t) \propto \frac{1}{P(t)^2}\. This means, as the period \(P(t)\) increases, the energy decreases. In the given problem, the initial energy \(E_0\) changes with time as \(E(t) = E_0 e^{2bt}\) because of the dependence on the exponential function derived from the differential equation solution, demonstrating how tightly linked period and energy are in astrophysical phenomena.

Pulsar astrophysics is a branch dedicated to studying rotating neutron stars that emit beams of electromagnetic radiation. Pulsars are known for their extremely regular pulsation periods. However, these periods can decay over time due to energy loss. By studying this deceleration and associated period decay, astrophysicists gain valuable insights into the pulsar's age and the mechanisms behind their energy emission. The given problem demonstrates how mathematical models and differential equations are used to predict the time evolution of a pulsar's period and rotational energy.

Time evolution of pulsars refers to how key properties like period and energy change over time. In our problem, the initial period decay model implies that \(P(t) = P_0 e^{-bt}\). It illustrates how the period increases logarithmically as time progresses due to the negative exponential factor. Similarly, the energy evolution \(E(t) = E_0 e^{2bt}\) provides a means to understand how energy emission varies. Using these equations, we can estimate how long it takes for a pulsar to reach a certain period. The problem demonstrates the practical application of these concepts, showing the time it takes for a pulsar to slow from an initial period of \(10^{-3} \text {s}\to \text {3 s}\).