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We have given that, An astronaut travels to a star system $4.5ly$away at a speed of $0.90c.$

We have to find the time taken for the journey according to Mission Control on earth.

The astronaut's departure from Earth is the first event. The astronaut's arrival at the star system is the final event. The interval between these Earthly events is:

$$\begin{gathered} 鈭�t=\frac{4.5ly}{0.9c} \\ 鈭�t=\frac{4.5ly}{0.9lyly} \\ 鈭�t=5.0y \end{gathered}$$

We have given that, An astronaut travels to a star system $4.5ly$away at a speed of $0.90c.$

We have to find the time taken for the journey according to the astronaut.

Two occurrences occur at the same time for the astronaut and can be measured with only one clock. As a result,

$$\begin{gathered} 鈭�蟿=\sqrt{1-\frac{v^2}{c^2}}路鈭�t \\ 鈭�蟿=\sqrt{1-{\left(\frac{0.9c}{c}\right)}^2}路(5.0y) \\ 鈭�蟿=2.2y \end{gathered}$$

We have given that, An astronaut travels to a star system $4.5ly$away at a speed of $0.90c.$

We have to find the time elapses between the launch and the arrival of the first radio message from the astronaut saying that she has arrived.

The total elapsed time $T_{elapsed}$is:

$$\begin{gathered} T_{elapsed}=鈭�t+4.5y \\ {T}_{elapsed}=9.5y \end{gathered}$$