Kepler's Third Law is useful here. It states that for two objects orbiting the same celestial body the expressions
`T_1^2/R_1^3` and `T_2^2/R_2^3`
are the same, where `R_1` and `R_2` are the radiuses of orbits and `T_1` and `T_2` are the periods.
In our problem `T_1` is given and `R_2=2R_1` ("an asteroid orbits the Sun twice the orbital radius of the satellite"). Therefore,
It is approximately 622 days. This is the answer.
Johannes Kepler discovered his three laws of planet motion analyzing the results of his teacher's astronomical observations. The man who found the deep cause of these and other laws and found the relation between this `T^2/R^3` and the mass of a central object, was Isaac Newton.