The Ratio of Orbital Duration

If you assume that Mars and Earth have circular orbits (which is almost true: all planets have elliptic orbits but Earth and Mars orbits have low eccentricity) you can check the relationship between the radius of the Mars orbit and the length of the Mars year.

For circular motion, the centripetal force is equal to gravitational force.

For Earth:

(1)

For Mars:

(2)

Gravitational acceleration is inversely proportional to the square of the distance between two massive bodies. The gravitational acceleration at Earth due to the Sun is:

so

Similarly, at Mars:

so

(4)
 
 

Equating the two expressions for the Sun's gravitation yields:

Substituting in (1), and cancelling Earth's mass from both sides:

Rearrange by multiplying by the ratio of radii:

Now substitute for Sun's gravity at Mars in (2) and cancel Mars mass from both sides:

If we know the relative radii for Earth orbit and Mars orbit, we can calculate their relative angular rates:

Now a year is the time it takes for a planet to travel one full circle around the sun. The angular rate for Earth is 1/365 revolutions per day (1 revolution per Earth year). Then the angular rate for Mars is (using the value for ratio of orbit radius from the table):

The length of a Mars year is therefore about 687 days, and the ratio of a Mars year to an Earth year is about 687/365 = 1.88 Thus we have checked that row 3 and row 5 of the table are consistent.