Answer
$\beta = 0.999 999 50 $
Work Step by Step
First we'll use the time dilation equation and solve for Beta;
$\Delta t=\Delta t_0 / \sqrt{1-\beta^2 }$
$\sqrt{1-\beta^2 } \Delta t=\Delta t_0 $
$\sqrt{1-\beta^2 } =\Delta t_0 / \Delta t $
$1-\beta^2 =(\Delta t_0 / \Delta t)^2 $
$\beta =\sqrt{1-(\Delta t_0 / \Delta t)^2} $
Our journey will take 1 year, ignoring the time it would take to turn around. Also, we would like earth to be 1000 years older. Therefore;
$\beta =\sqrt{1-(1year / 1000years)^2} $
$\beta = 0.999 999 50 $
