The weight of the astronaut is given by [tex]W=mg[/tex] where m=59.1 kg is his mass and [tex]g=9.81~m/s^2[/tex] is the gravitational acceleration on Earth.
To solve the problem, we must find the value of g on the new planet. g is given by [tex]g= \frac{GM}{r^2} [/tex] where G is the gravitational constant, M the mass of the planet and r its radius. The mass of the planet can be written as [tex]M=dV[/tex] where d is the density and V the volume. We can assume that the planet is a sphere, therefore the volume is proportional to [tex]r^3[/tex]: [tex]V= \frac{4}{3}\pi r^3 [/tex] and we can write the mass as [tex]M= \frac{4}{3} \pi d r^3[/tex] and then, g becomes [tex]g= \frac{GM}{r^2}= \frac{4}{3} \frac{G \pi d r^3}{r^2}= \frac{4}{3} G \pi d r [/tex] So, in the end g is proportional to the radius of the planet, r (because the density of the new planet d is the same as the Earth's one. If the radius of the new planet is twice the Earth's radius, g will be twice the value of g on Earth: [tex]g_{new}=2g=2\cdot9.81~m/s^2=19.62~m/s^2[/tex] And since the mass of the astronaut is always the same, the weight on the new planet will be twice the weight on Earth: [tex]W_{new}=mg_{new}=2mg=1159~N[/tex]
