Fields · Question 8

Calculate Calculate the gravitational field strength at the surface of the planet.

1. 1

   g = GM/R² = 6.67 × 10⁻¹¹ × 6.4 × 10²³ / (3.4 × 10⁶)²

2. 2

   g = 3.7 N kg⁻¹

Determine Determine the escape speed from the surface of the planet.

1. 1

   escape requires the total energy to be zero: ½mv² = GMm/R

2. 2

   v = √(2GM/R) = √(2 × 6.67 × 10⁻¹¹ × 6.4 × 10²³ / 3.4 × 10⁶)

3. 3

   v = 5.0 × 10³ m s⁻¹

Explain Explain why gravitational potential is always negative, and state what a gravitational potential of −1.3 × 10⁷ J kg⁻¹ at the planet's surface means.

1. 1

   gravitational potential is defined as zero at infinity, and gravity is always attractive

2. 2

   so bringing a mass in from infinity releases energy, and its potential energy must be less than zero everywhere else

3. 3

   the value means that 1.3 × 10⁷ J of energy must be supplied to move each kilogram from the surface to infinity

Discuss The planet has almost no atmosphere, while the Earth retains a substantial one. Discuss how the escape speed helps to explain this difference.

1. 1

   the molecules of a gas have a distribution of speeds, with some far above the mean at any temperature

2. 2

   molecules whose speed exceeds the escape speed at the top of the atmosphere can leave the planet permanently

3. 3

   this planet's escape speed of 5.0 km s⁻¹ is less than half the Earth's 11.2 km s⁻¹, so a much larger fraction of its molecules exceed it

4. 4

   over geological time that fraction is lost, and the lightest gases go first — which is why any remaining atmosphere is thin and dominated by the heavier molecules