Q. 28 A new planet is discovered orbit... [FREE SOLUTION]

Chapter 13: Q. 28 (page 354)

A new planet is discovered orbiting the star Vega in a circular orbit. The planet takes $55$ earth years to complete one orbit around the star. Vega鈥檚 mass is $2.1$ times the sun鈥檚 mass. What is the radius of the planet鈥檚 orbit? Give your answer as a multiple of the radius of the earth鈥檚 orbit.

Short Answer

Expert verified

The radius of the planet's orbit is $4.34 脳 10^{5}$ times of the earth's orbit radius.

Step by step solution

Given information

Time period = $55 e a r t h y e a r s = 1.734 脳 10^{9} s,$ mass of the planet = $2.1 脳 1.989 脳 10^{30} k g = 4.177 脳 10^{30} k g .$

Calculation

According to Kepler's 3rd law, the time period is given by $T^{2} = \frac{4 蟺^{2} R^{3}}{G M} .$

Where, T is the time period, R is the required radius of the orbit, G is the universal gravitational constant and M is the required mass of Vega.

Now solving we get,

$T^{2} = \frac{4 蟺^{2} R^{3}}{G M} R^{3} = \frac{T^{2} 脳 G M}{4 蟺^{2}} R = {({(1.734 脳 10^{9} s)}^{2} 脳 \frac{(6.67 脳 10^{- 11} N m^{2} / k g^{2}) (4.176 脳 10^{30} k g)}{4 蟺^{2}})}^{1 / 3} R = {(2.121 脳 10^{37} m^{3})}^{1 / 3} R = 2.77 脳 10^{12} m$

The radius of the earth's orbit is $6.37 脳 10^{6} m .$

Radius of the planet is $4.34 脳 10^{5}$ times of the earth's orbit radius.

Final answer

The radius of the planet is $4.34 脳 10^{5}$ times of the earth's orbit radius.

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