Astrophysicists model the equilibrium temperature, T_{eq} (in kelvins, \text{K}), of a pla… — ACT

Astrophysicists model the equilibrium temperature, $T_{eq}$ (in kelvins, $\text{K}$ ), of a planet orbiting a star using the following equation:

T_{eq} = \left[ \frac{L(1 - a)}{16 \pi \sigma d^2} \right]^{1/4}

where $L$ is the star's luminosity, $a$ is the planet's albedo (the fraction of star radiation reflected by the planet), $d$ is the average distance from the star to the planet, and $\sigma$ is the Stefan-Boltzmann constant. Based on this model, match each proposed change in the physical parameters of the system (on the left) to its resulting effect on the equilibrium temperature $T_{eq}$ (on the right).

The distance $d$ from the star is multiplied by $4$ ( $4d$ ), with $L$ and $a$ held constant. $T_{eq}$ is multiplied by $\frac{1}{2}$ .
The star's luminosity $L$ is multiplied by $16$ ( $16L$ ), with $d$ and $a$ held constant. $T_{eq}$ is multiplied by $2$ .
The distance $d$ is multiplied by $2$ ( $2d$ ) and the luminosity $L$ is multiplied by $4$ ( $4L$ ), with $a$ held constant. $T_{eq}$ remains unchanged.
The term $(1 - a)$ is multiplied by $\frac{1}{81}$ , with $L$ and $d$ held constant. $T_{eq}$ is multiplied by $\frac{1}{3}$ .

Answer

Matching the parameter changes to their correct scaling factors: multiplying distance by 4 halves the temperature; multiplying luminosity by 16 doubles the temperature; multiplying distance by 2 and luminosity by 4 leaves temperature unchanged; and multiplying the albedo term by 1/81 scales temperature by 1/3.

Each relationship is correctly derived by applying the respective scaling factor to the variable and evaluating the term under the fourth root: multiplying the distance by 4 results in a factor of

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

; multiplying the luminosity by 16 results in a factor of

(16)^{1/4} = 2

; scaling both distance by 2 and luminosity by 4 scales the fraction by

4 / 2^2 = 1

, leaving the temperature unchanged; and scaling the albedo term

(1-a)

by 1/81 yields a factor of

(1/81)^{1/4} = 1/3

Step-by-Step Solution

Isolate the proportional relationship of each variable to

T_{eq}

by removing constants (

16

\pi

\sigma

T_{eq} \propto \left[ \frac{L(1-a)}{d^2} \right]^{1/4} = L^{1/4} (1-a)^{1/4} d^{-1/2}

This establishes how scaling each parameter mathematically impacts the overall temperature.

Determine the scale factor for the first scenario where distance

d

is multiplied by 4.

The distance term becomes

(4)^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2}

Because distance is squared and in the denominator under a fourth root, its scaling factor is

1/\sqrt{d}

Determine the scale factor for the second scenario where luminosity

L

is multiplied by 16.

The luminosity term becomes

(16)^{1/4} = 2

Luminosity is directly proportional under the fourth root, so scaling it by 16 doubles the final value.

Determine the scale factor for the third scenario where distance

d

is multiplied by 2 and luminosity

L

is multiplied by 4.

The joint factor is

(4)^{1/4} \times (2^2)^{-1/4} = 4^{1/4} \times 4^{-1/4} = 1

The scaling in the numerator (

4

) matches the scaling in the denominator (

2^2 = 4

), which cancels out completely.

Determine the scale factor for the fourth scenario where

(1-a)

is multiplied by

1/81

The albedo term scaling is

(\frac{1}{81})^{1/4} = \frac{1}{3}

The term

(1-a)

is directly proportional under the fourth root, so scaling it by

1/81

reduces temperature to

1/3

of its value.

Key Concept

Analyzing proportional scaling and fractional power relations in a multi-variable physical model.

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