Interesting physics problem about ice ball in Sun's orbit

Question

An ice ball flew into the Sun's orbit and began to melt. Find the time it takes for this ball to melt if the distance from it to the Sun is d and the initial radius of the ball is R0. Ice density ρ, specific heat of fusion λ. Assume that the outside of the ball melts first, and the liquid water immediately evaporates and does not absorb heat. The sun's radius is are R_S, and the temperature is T_S. Ignore thermal radiation from ice.

Answer 1

This is an interesting problem! To solve it, we can use principles of thermodynamics and heat transfer.

First, we need to calculate the mass of the ice ball. The mass can be calculated using the density and the initial volume of the ice ball:

m = (4/3)πR0^3ρ

Next, we need to calculate the heat required to melt the ice ball. The heat required can be calculated using the mass of the ice ball and the specific heat of fusion:

Q = mL

Next, we need to calculate the rate at which the ice ball absorbs heat from the sun. We can assume that the ice ball absorbs heat only from the sun-facing side, which is a hemisphere with radius d. The rate of heat transfer can be calculated using the Stefan-Boltzmann law, which relates the rate of heat transfer to the temperature difference and the surface area:

P = σA(TS^4 - Tice^4)

where σ is the Stefan-Boltzmann constant, A is the surface area of the hemisphere, TS is the temperature of the sun, and Tice is the temperature of the ice ball.

We can assume that the temperature of the ice ball is initially equal to its melting point, which is 0°C or 273 K. As the ice ball melts, its temperature will remain constant until all the ice has melted. Once all the ice has melted, the temperature of the water will start to increase.

We can calculate the time it takes for the ice ball to melt by dividing the heat required to melt the ice ball by the rate at which the ice ball absorbs heat from the sun:

t = Q/P

Substituting the expressions for Q and P, we get:

t = (4/3)πR0^3ρL / σπd^2(TS^4 - Tice^4)

Simplifying, we get:

t = (4/3)πR0^3ρL / σπd^2(TS^4 - Tice^4)

Simplifying, we get:

t = (4/3)R0^3ρL / σd^2(TS^4 - Tice^4)

Substituting the values for the constants, we get:

t = (4/3)R0^3ρL / (5.67×10^-8)(d^2)(TS^4 - 273^4)

This gives us the time it takes for the ice ball to melt in seconds.