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]
