Question

Deimos's Orbit. Deimos orbits Mars at a distance of 23,460 km from the center of the planet and has a period of 1.263 days. Assume that Deimos's orbit is circular. Calculate the mass of Mars. Express your answer in units of kg. (Hint: Use the circular orbit velocity formula ; remember to use units of meters, kilograms, and seconds.) Please round the answer to four significant digits.

Answer 1

Answer:

M = 5.882 10²³ kg

Explanation:

Let's use Newton's second law to analyze the satellite orbit around Mars.

         F = m a

force is universal attraction and acceleration is centripetal

          a = v²/ R

the modulus of velocity in a circular orbit is constant

         v= d/T

the distance of the cicule is

        d =2pi R

       a = 2pi R/T  

we substitute

          - G m M / R² = m ( $- \frac{4\pi^2 R^2 }{T^2 R}$)

         G M = $\frac{ 4\pi ^2 R^3 }{T^2 }$

         M = $\frac{4 \pi ^2 R^3 }{ G T^2 }$

the distance R is the distance from the center of the planet Mars to the center of the satellite Deimos

         R = 23460 km = 2.3460 10⁷ m

the period of the orbit is

         T = 1,263 days = 1,263 day (24 h / 1 day) (3600s / h)

          T = 1.0912 10⁵ s

let's calculate

          M = $\frac{4 \pi ^2 ( 2.3460 \ 10^7)^3 }{5.67 10^{-11} \ (1.0912 \ 10^5)^2 }$

          M = 509.73418 10²¹  /8.66640 10⁻¹

          M = 58.817 10²² kg

          M = 5.882 10²³ kg