A

Step 1: Calculate the reduced mass \( \mu \) of the system using \( \mu = \frac{m_\pi m_p}{m_\pi + m_p} \). Here, \( m_\pi = 273 m_e \) and \( m_p = 1836 m_e \). Thus, \( \mu = \frac{(273 m_e)(1836 m_e)}{(273 m_e) + (1836 m_e)} = \frac{502068 m_e^2}{2109 m_e} \approx 237.3 m_e \).
Step 2: Use the formula for the ground state radius of the hydrogen-like atom: \( r = \frac{n^2 \hbar^2}{\mu k e^2} \) with \( n = 1 \).
Step 3: The mean life of the pion is \( \tau = 10^{-8} \, \text{s} \). Calculate the time it takes for one revolution, \( t_{rev} \approx \frac{2 \pi r}{v} \) where \( v \) is the orbital speed, obtained from the expression for centripetal force and the gravitational force.
Step 4: Estimate the average angular frequency \( \omega = \frac{2 \pi}{t_{rev}} \) and the number of revolutions before decay is given by \( N = \frac{\tau}{t_{rev}} \).
Therefore, the number of revolutions is significant before the decay occurs, yielding option A.