A newly discovered planet has a surface temperature of 330 K and a density of 6 g/cm3. What minimum radius does the planet need to have in order to hold water in its atmosphere indefinitely? (in km)
To determine the minimum radius a planet needs to hold water in its atmosphere indefinitely, we need to consider the planet's escape velocity.
The escape velocity is the minimum velocity an object needs to escape the gravitational pull of a planet. In the context of water, we can use the escape velocity to determine if water vapor molecules will have enough kinetic energy to escape the planet's gravitational pull into space.
Here's the step-by-step process to calculate the minimum radius of the planet:
Step 1: Calculate the escape velocity of the planet using the formula:
Escape Velocity = √(2 * G * M / R)
Where:
G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
M is the mass of the planet (in kg)
R is the radius of the planet (in meters)
Step 2: Convert the given density of the planet to the mass using the formula:
Mass = Density * Volume
Where:
Density is the density of the planet (in g/cm^3)
Volume is the volume of the planet (in cm^3)
Step 3: Convert the volume of the planet to the radius using the formula:
Volume = (4/3) * π * R^3
Where:
π is a mathematical constant (approximately 3.14159)
Step 4: Substitute the mass of the planet into the escape velocity formula calculated in Step 1 and solve for R.
Step 5: Convert the minimum radius from meters to kilometers.
Let's calculate the minimum radius of the planet:
Step 1: Calculate the escape velocity
- Let's assume the mass of the planet, M, is equal to the mass of Earth (5.972 × 10^24 kg).
- Gravitational constant: G = 6.67430 × 10^-11 m^3 kg^-1 s^-2
- Radius: R (unknown)
Escape Velocity = √(2 * G * M / R)
Step 2: Calculate the mass of the planet
- Density: 6 g/cm^3 (given)
- Volume = (4/3) * π * R^3
Mass = Density * Volume
Step 3: Solve for the radius
Volume = (4/3) * π * R^3
Step 4: Substitute the mass into the escape velocity formula
Escape Velocity = √(2 * G * M / R)
Step 5: Convert the minimum radius to kilometers.
By following these steps, we will be able to calculate the minimum radius of the planet in order to hold water in its atmosphere indefinitely.