Final answer:
The orbital period of a planet orbiting a star with 10 solar masses at an orbital distance of 3.4 AU is approximately 1.98 years, as derived using Kepler's third law adjusted for stellar mass.
Step-by-step explanation:
The question asks us to calculate the orbital period of a planet that orbits a star with 10 solar masses at a distance of 3.4 AU (astronomical units) from that star. According to an extended version of Kepler's third law, which also includes the mass of the star, the orbital period P in years is given by the equation:
P² = a³ / M,
where a is the semimajor axis of the orbit (3.4 AU in this case) and M is the mass of the star compared to the mass of the Sun. Since this star has 10 solar masses, the equation becomes:
P² = (3.4³) / 10.
Calculating (3.4 AU)³ gives 39.304, and dividing this number by 10 gives 3.9304. Taking the square root of 3.9304 gives an approximate orbital period P of 1.9825 years.
Therefore, the planet would have an orbital period of approximately 1.98 years around the star with 10 solar masses at an orbital distance of 3.4 AU.