A) the earth year is 365.25 days long. How long would our year be if earth followed the same orbit, but was only the mass of Mars? Answer this in earth days.
B) How long would our year be if earth were the same mass, but had the orbital distance of mercury Answer this in earth days.
C) What do you conclude about the effect of mass of the body in orbit?
A) To find the length of our year if Earth had the mass of Mars, we need to understand the relation between the mass of a planet and its orbital period. According to Kepler's Third Law of Planetary Motion, the square of the orbital period is directly proportional to the cube of the semi-major axis of the orbit.
In this case, the semi-major axis of Earth's orbit remains the same (as stated in the question), while the mass changes to that of Mars. The mass of Mars is approximately 0.107 times the mass of Earth.
Using the proportional relation, we can calculate the new orbital period as follows:
(365.25^2) / (1^3) = (x^2) / (0.107^3)
Simplifying this equation gives:
x = √[(365.25^2) / (0.107^3)]
Evaluating this expression gives x ≈ 687.11 Earth days.
Therefore, if Earth had the mass of Mars, our year would be approximately 687.11 Earth days long.
B) Similarly, to determine the length of our year if Earth had the orbital distance of Mercury, we need to understand the relation between the orbital distance and the orbital period. According to Kepler's Third Law, the square of the orbital period is directly proportional to the cube of the semi-major axis.
The semi-major axis of Mercury's orbit is approximately 0.387 times the semi-major axis of Earth's orbit, and the mass of Earth remains the same.
Using the proportional relation, we can calculate the new orbital period as follows:
(365.25^2) / (1^3) = (x^2) / (0.387^3)
Simplifying this equation gives:
x = √[(365.25^2) / (0.387^3)]
Evaluating this expression gives x ≈ 88.81 Earth days.
Therefore, if Earth had the orbital distance of Mercury, our year would be approximately 88.81 Earth days long.
C) Based on the calculations from the previous two scenarios, we can conclude that the mass of a body in orbit has a direct impact on the length of its year. When the mass of a planet decreases, such as in the case when Earth has the mass of Mars, the year becomes longer. On the other hand, when the mass remains the same, but the distance from the Sun changes, such as in the case when Earth has the orbital distance of Mercury, the year becomes shorter.
Therefore, we can infer that the mass of a body in orbit affects the gravitational force acting upon it, resulting in different orbital periods and subsequently influencing the length of a year.