Question

A comet has an orbit size of 3.5 AU & its orbit time is 6.54 years. T² = 42.875 years. What does R³ equal?

Answer 1

Kepler's third law allows us to find the answer for the orbital radius of the comet's orbit and the relationship of the magnitudes are:

• Orbital radio is a = 3.49 AU

• The ratio of the orbit to the cube to the period squared gives the Kepler constant.

Kepler studied the motion of bodies in the solar system creating mathematical relationships that describe their motion, Kepler's third law which is a direct application of Newton's second law to the orbit of the planets around the sun is

         T² = K a³

 

Where T is the period of the orbit, a is the semi-major axis of the ellipse and K is a constant called Kepler's constant given by

        K = $\frac{4 \pi ^2 }{GM_s}$  = 3 10⁻¹⁹ s²/m³

Where G is the constant of gravitation universes, M_s the mass of the Sun

In this case they indicate the size of the orbit tabulated a = 3.5 AU and the period T = 6.54 years, let's reduce the two magnitudes to the international system of measurements (SI)

         T = 6.54 years $( \frac{365 d}{1yr} ) ( \frac{24h}{1 d} ) ( \frac{3600s}{1 h} )$ = 2.062 10⁸ s

Let's calculate the radius of the orbit

          a = $\sqrt[3]{\frac{ T^2 }{K} }$  

          a = $\sqrt[3]{\frac{(2.062 \ 10^8 )^2 }{3 \ 10^{-19}} }$

          a = $\sqrt[3]{0.1417 \ 10^{36}}$  

          a = 0.5214 10¹² m

Let's reduce to astronomical units, which corresponds to the distance from the Sun to the earth

         a = 0.5214 10¹²  m ( $\frac{1AU}{1.496 \ 10^{11}}$ )

         a = 3.49 AU

Let's observe that it is practically equal to the tabulated value.

The ratio of the orbit to the cube to the period squared gives the Kepler constant, which is practically invariant for all bodies in the solar system.

In conclusion using Kepler's third law we can find the radius of the comet's orbit and the relationship of the magnitudes are:        

• Orbital radio is a = 3.49 AU

• The ratio of the orbit to the cube to the period squared gives the Kepler constant

Learn more here:  brainly.com/question/13825807

Answer 2

Answer:

42.875 AU is the answer.

Explanation: