A team of astrophysicists modeled the equilibrium surface temperature, T (in Kelvin, \text… — ACT

A team of astrophysicists modeled the equilibrium surface temperature, $T$ (in Kelvin, $\text{K}$ ), of airless rocky planets orbiting a distant star. According to the model, the temperature is predicted by the following equation:

T = T_0 (1 - a)^{1/4} D^{-1/2}

where:
- $T_0$ is a star-specific constant equal to $400\text{ K}$ .
- $a$ is the planet's albedo (reflectivity), ranging from $0.0$ to $1.0$ .
- $D$ is the planet's distance from the star in astronomical units ( $\text{AU}$ ).

Based on this model, arrange the four planets (W, X, Y, and Z) shown in the diagram in order of their predicted equilibrium surface temperature, from lowest to highest.

1Planet W ( $a = 0.9375$ , $D = 4.0\text{ AU}$ )
2Planet X ( $a = 0.9375$ , $D = 1.0\text{ AU}$ )
3Planet Y ( $a = 0.0$ , $D = 1.0\text{ AU}$ )
4Planet Z ( $a = 0.0$ , $D = 0.25\text{ AU}$ )

Answer

The correct order of the planets from lowest to highest predicted equilibrium surface temperature is Planet W, Planet X, Planet Y, and Planet Z.

Substituting the specific physical parameters into the model equation

T = 400(1-a)^{1/4}D^{-1/2}

yields the exact temperatures:

100\text{ K}

for Planet W,

200\text{ K}

for Planet X,

400\text{ K}

for Planet Y, and

800\text{ K}

for Planet Z, demonstrating an ascending sequence from W to Z.

Step-by-Step Solution

Identify the parameters of each planet and analyze the model equation:

T = T_0 (1 - a)^{1/4} D^{-1/2}

The constant

T_0 = 400\text{ K}

. Planet W has

a = 0.9375, D = 4.0\text{ AU}

. Planet X has

a = 0.9375, D = 1.0\text{ AU}

. Planet Y has

a = 0.0, D = 1.0\text{ AU}

. Planet Z has

a = 0.0, D = 0.25\text{ AU}

Listing the parameters clearly helps set up the mathematical calculations for comparison.

Calculate the predicted equilibrium temperature for Planet W and Planet X.

For Planet W,

T = 400(1 - 0.9375)^{1/4}(4.0)^{-1/2} = 400(0.0625)^{1/4}(0.5) = 400(0.5)(0.5) = 100\text{ K}

. For Planet X,

T = 400(1 - 0.9375)^{1/4}(1.0)^{-1/2} = 400(0.0625)^{1/4}(1.0) = 400(0.5)(1.0) = 200\text{ K}

Evaluating the fractional power

(1/16)^{1/4} = 1/2

and distance roots allows us to determine the temperatures for high-albedo planets.

Calculate the predicted equilibrium temperature for Planet Y and Planet Z.

For Planet Y,

T = 400(1 - 0.0)^{1/4}(1.0)^{-1/2} = 400(1.0)(1.0) = 400\text{ K}

. For Planet Z,

T = 400(1 - 0.0)^{1/4}(0.25)^{-1/2} = 400(1.0)(2.0) = 800\text{ K}

Evaluating the model for the zero-albedo planets establishes their temperatures.

Compare the calculated temperatures to order the planets from lowest to highest.

Planet W (

100\text{ K}

) < Planet X (

200\text{ K}

) < Planet Y (

400\text{ K}

) < Planet Z (

800\text{ K}

). The order is Planet W, Planet X, Planet Y, Planet Z.

Arranging the numerical values in ascending order directly determines the correct sequence.

Key Concept

Applying mathematical models with fractional and negative exponents to predict and compare astronomical states.

Estimated Time:2m 0s

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