Two spherical stars \( A \) and \( B \) have densities \( \rho_{A} \) and \( \rho_{B} \), respec...

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Two spherical stars \( A \) and \( B \) have densities \( \rho_{A} \) and \( \rho_{B} \), respectively. \( A \) and \( B \) have the same radius, and their masses \( M_{A} \) and \( M_{B} \) are related by \( M_{B}=2 M_{A} \). Due to an interaction process, star \( A \) loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains \( \rho_{A} \). The entire mass lost by \( A \) is deposited as a thick spherical shell on \( B \) with the density of the shell being \( \rho_{A} \). If \( v_{A} \) and \( v_{B} \) are the escape velocities from \( A \) and \( B \) after the interaction process, the ratio \( \frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}} \). The value of \( n \) is
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