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A new planet has been discovered in the Solar System, and it is traveling in a circular orbit around the Sun. The planet has a
semimajor axis of 48.1 au.
New Planet
What is the period of the new planet's orbit?

Respuesta :

oladayoademosu

The period of the new planet is 1.053 × 10¹⁰ s

Gravitational force on planet

The gravitational force on the planet, F equals the centripetal force on the planet, F'

F = F'

GMm/R² = mRω² where

  • G = universal gravitational constant = 6.67 × 10⁻¹¹ Nm²/kg²,
  • M = mass of sun = 1.989 × 10³⁰ kg,
  • m = mass of planet,
  • R = radius of planet = 48.1 au = 48.1 × 1.496 × 10¹¹ m = 7.1958 × 10¹² m,
  • ω = angular speed of planet = 2π/T where
  • T = period of planet

So, GMm/R² = mRω²

GMm/R² = mR(2π/T)²

GM/R³ = (2π)²/T²

Period of the planet

Making T subject of the formula, we have

T = 2π√(R³/GM)

Substituting the values of the variables into the equation, we have

T = 2π√(R³/GM)

T = 2π√((7.1958 × 10¹² m)³/[6.67 × 10⁻¹¹ Nm²/kg² × 1.989 × 10³⁰ kg])

T = 2π√(372.5952 × 10³⁶ m³/13.26663 × 10¹⁹ Nm²/kg)

T = 2π√(28.085 × 10¹⁷ Nm/kg)

T = 2π√(2.8085 × 10¹⁸ Nm/kg)

T = 2π × 1.676 × 10⁹ s

T = 10.53 × 10⁹ s

T = 1.053 × 10¹⁰ s

The period of the new planet is 1.053 × 10¹⁰ s

Learn more about period of planet here:

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