Question

In: Physics

Calculate the temperature of a planet. A star has a surface temperature of 4000K and a...

Calculate the temperature of a planet. A star has a surface temperature of 4000K and a radius of R = 8 × 108 m It has a rocky, airless planet orbiting it at a distance of 1.0×1011 meters. The planet has a radius of 5.0 × 106 m. We will estimate 1 for all objects. (a) What is the total power output of the star?

(b) What is the power incident on each square meter at distance to the planet (i.e. on an imaginary sphere).

(c) How much power is absorbed by the planet? (Assume the planet is black. Use the fact that the planet looks like a disk.)

(d) In equilibrium, Pin = Pout. How much power does the planet radiate per square meter of surface area of the planet?

(e) What is the temperature of the planet? (Should humans try to live there?)

Expert Solution

what does this statement mean - "We will estimate 1 for all objects'' ????E

(a) Area of star = 4*pi*r²

Area = 4 *pi*8e8²

Area = 8.042e18 m²

The flux is given as

W = T⁴

where = 5.6704e-8 ( stefan boltzmann constant)

W = 5.6704e-8 (4000)⁴

W = 1.45e7 W/m²

Therefore, total power output is

P = W* Area

P = 1.1674e26 W

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(b) Area radiated by star power

A = 4 *pi*1e11²

A = 1.257e23 m²

Therefore,

power incident on each square meter = 1.1674e26 W / 1.257e23 m²

power incident on each square meter = 928.98 W/m²

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(c) area of cross section of planet

A = pi*r²

A = pi*5e6²

A = 7.854e13 m²

Therefore, power abosrbed = 928.98 W/m² * 7.854e13 m²

power abosrbed = 7.296e16 W

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(d) P_in = P_out = 7.296e16 W

P_out = T⁴A

T⁴ = P_out / A

T⁴ = 7.296e16 / 5.6704e-8 * 4 *pi *(5000000)²

T⁴ = 4.0956e9 K

T = 252.97 K

genius_generous answered 1 year ago

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