Kepler's third law establishes a relationship between an orbiting body's orbital period, P (in days), and its semimajor axis, A (in AU), described by the equation P^2 = kA^3, where k is a constant for all bodies orbiting the same star. Mars has a semimajor axis of 1.52 AU and an orbital period of approximately 687 days. To find the value of the constant k, use these values and solve the equation: 687^2 = k * 1.52^3 k = (687^2) / (1.52^3) Now calculate the value of k: k ≈ 191,649 / 3.5 k ≈ 54,757.14 Thus, the value of the constant k is approximately 54,757.14.

To find the value of the constant \( k \) using Kepler's third law, we need to use the equation \( P^2 = kA^3 \), where \( P \) is the orbital period and \( A \) is the semimajor axis of the orbit.

For Mars, we are given: \( P = 687 \) days, \( A = 1.52 \) AU.

We can plug in these values to solve for \( k \):

\( P^2 = kA^3 \) \( (687)^2 = k(1.52)^3 \)

Calculating \( P^2 \): \( 687^2 = 471969 \)

Calculating \( A^3 \): \( 1.52^3 = 1.52 \times 1.52 \times 1.52 \approx 3.51 \)

Now, substituting the calculated values back into the equation for \( k \): \( 471969 = k \cdot 3.51 \)

To solve for \( k \), divide both sides by 3.51: \( k = \frac{471969}{3.51} \approx 134465 \) (rounded to the closest whole number, since constants typically don't need extreme precision)

Therefore, the value of the constant \( k \) for Mars's orbit is approximately 134465 (when \( P \) is in days and \( A \) is in AU).