91Ó°ÊÓ

In order to answer this question, follow the step-by-step solution provided: Step 1: Calculate the time dilation factor, which is given by the formula \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\). In this case, \(v = 0.87c\). Step 2: Determine the time intervals according to Earth's clocks (t_Earth) using the equation: \(t_{Earth} = \gamma t_{astronaut}\), where \(t_{astronaut} = 12\) hours. Step 3: Calculate the observed frequency by Earth observers (f_Earth) using the Doppler effect formula for light: \(f_{Earth} = f_{astronaut}\frac{1}{1 + \frac{v}{c}}\). Step 4: Calculate the time intervals between received reports by Earth observers (t_received) by taking the inverse of the observed frequency: \(t_{received} = \frac{1}{f_{Earth}}\). The interval at which the reports are sent according to Earth clocks is \(t_{Earth}\), and the interval at which the reports are received by Earth observers is \(t_{received}\).