Okay, so I need to figure out the radius of an exoplanet that would produce the same amplitude signal in the star's light curve as the star's spots. Let me start by understanding the problem step by step.

First, the star has a radius equal to the Sun's, which is about 1 Rsun. The star's effective temperature is 6000 K. Without any spots, the star's light curve wouldn't change much because we're just observing the star's regular brightness. But when there are spots, especially on one hemisphere, the light curve changes periodically as the star rotates. The spots have a filling factor of 20%, meaning they cover 20% of the hemisphere's area. The spots are 1000 K cooler than the surrounding photosphere. So the spots are at 5000 K (since 6000 - 1000 = 5000 K).

The question is asking: if the star's brightness variations due to spots have a certain amplitude, what radius would a planet need to have (relative to the star's radius) to produce the same amplitude variation in the star's light curve?

I remember that the amplitude of the light curve variation caused by spots depends on the area of the spots and their temperature difference. The same planet's effect would depend on its radius and the fraction of the star's surface it covers. Since the planet transits the star, the area covered is πR_planet², but the star's surface area is 4πR_star². So the fractional area covered is (πR_planet²)/(4πR_star²) = (R_planet²)/(4R_star²). But wait, in this case, the spots are on a hemisphere, so the total area of the spots is 2πR_star² * filling factor. Let me check that.

Wait, the filling factor is the fraction of the hemisphere that is covered by spots. So the total area of the spots is filling factor * (2πR_star²). Because a hemisphere has an area of 2πR² (since the full sphere is 4πR², hemisphere is half that). So if the filling factor is 20%, that's 0.2, so the spot area is 0.2 * 2πR_star² = 0.4πR_star².

The planet, when it transits, covers an area of πR_planet². The change in brightness (the amplitude) due to spots would be the difference in flux from the spotted area compared to the unspotted area. Similarly, the planet's effect would be the difference in flux from the planet's area compared to the rest.

But wait, the star's brightness variation is due to the spots moving across the surface as the star rotates. So the amplitude of the light curve variation (the depth) depends on the ratio of the spotted area to the total area, multiplied by the temperature difference. The formula for the amplitude (depth) of the light curve caused by spots is given by:

ΔF/F = (A_spots / A_total) * (T_spot² - T_star²) / (T_star²)

But wait, maybe I need to think in terms of flux. The flux from a blackbody is σT^4. So the total flux from the star is σT_star^4 * A_total. When part of the star has spots at a lower temperature, the total flux becomes σT_star^4*(A_total - A_spots) + σT_spot^4*A_spots. The change in flux compared to the normal flux is σT_star^4*(A_total - A_spots) + σT_spot^4*A_spots - σT_star^4*A_total = σA_total(T_star^4 - T_spot^4) + σA_spots(T_spot^4 - T_star^4). Wait, that seems complicated. Alternatively, the fractional change in flux is (ΔF)/F = [A_spots (T_spot^4 - T_star^4)] / (A_total T_star^4). Since T_spot < T_star, this would be negative, indicating a dip. But the depth would be the absolute value, so |ΔF/F| = (A_spots / A_total) * (T_star^4 - T_spot^4)/T_star^4.

Wait, but in reality, the amplitude of the light curve (the depth of the dip) is given by the ratio of the area of the spots to the total area of the star, multiplied by (T_spot^4 - T_star^4)/T_star^4. But since T_spot is lower, this would be negative, so the depth is the absolute value. So:

Depth_spots = (A_spots / A_total) * (1 - (T_spot/T_star)^4)

Because T_spot is lower than T_star, (T_spot/T_star)^4 is less than 1, so 1 - (T_spot/T_star)^4 is positive. So the depth is proportional to the fraction of the area covered by spots multiplied by this factor.

Wait, maybe I should double-check. The flux from the star is F = σT^4. When you have spots, the total flux becomes F_total = (A_total - A_spots) * σT^4 + A_spots * σT_spot^4. So the change in flux is F_total - F_initial = σT^4 [ (A_total - A_spots) + A_spots (T_spot^4 / T^4) - A_total ] = σT^4 [ -A_spots + A_spots (T_spot^4 / T^4) ] = -σA_spots T^4 [1 - (T_spot/T)^4]. Therefore, the fractional change is ΔF/F = -A_spots [1 - (T_spot/T)^4] / A_total. Taking absolute value, the depth is (A_spots / A_total) * [1 - (T_spot/T)^4].

So the amplitude of the light curve variation due to spots is proportional to (A_spots / A_total) * [1 - (T_spot/T)^4]. Since the star's radius is 1 Rsun, A_total = 4πR^2. The area of the spots is 2πR^2 * filling_factor (since the spots are on a hemisphere, which has area 2πR^2). Wait, the filling factor is 20%, so A_spots = 0.2 * 2πR^2 = 0.4πR^2.

So A_spots / A_total = 0.4πR² / 4πR² = 0.1. So the spots cover 10% of the star's surface area.

Then, the term [1 - (T_spot/T)^4] is 1 - (5000/6000)^4. Let's compute that. 5000/6000 = 5/6 ≈ 0.8333. (5/6)^4 ≈ (0.8333)^4. Let's calculate that:

0.8333^2 = 0.6944. Then 0.6944^2 ≈ 0.4823. So (5/6)^4 ≈ 0.4823. Therefore, 1 - 0.4823 ≈ 0.5177. So the depth is 0.1 * 0.5177 ≈ 0.05177, or about 5.177% of the total flux.

But wait, the problem states that the amplitude of the light curve is the same as if the star wasn't covered by spots. Wait, no. Wait, the star with spots has a certain amplitude, and the planet's effect would need to produce the same amplitude. Wait, the problem says: "To produce the same amplitude signal in the star's light curve (if the star was not covered by spots!), what should be the radius of a hypothetical exoplanet relative to the radius of the host star?"

Wait, maybe I misread. The question is asking: If the star's brightness variations due to spots have a certain amplitude, what radius would a planet need to have to produce the same amplitude variation in the star's light curve, assuming the planet's effect is the only factor (i.e., the star isn't covered by spots).

Wait, no. Wait, the scenario is that when the star has spots, the light curve has periodic variations due to rotational modulation. The amplitude of these variations is due to the spots. The question is asking, what would the planet's radius need to be to produce the same amplitude variation as the spots. So the planet's effect (which would be a transit causing a dip in brightness) should have the same depth as the spots cause.

Wait, but the problem states: "Interestingly, this situation can closely resemble the presence of an exoplanet. To produce the same amplitude signal in the star's light curve (if the star was not covered by spots!), what should be the radius of a hypothetical exoplanet relative to the radius of the host star?"

Wait, maybe the star's brightness variations due to spots have the same amplitude as the planet's transit would. So the amplitude (depth) of the planet's transit is the same as the amplitude caused by the spots. So we need to equate the two depth formulas.

The depth from the spots is given by (A_spots / A_total) * [1 - (T_spot/T_star)^4]. The depth from the planet's transit is (A_planet / A_total) * (T_planet^4 - T_star^4) / (T_star^4). Wait, no. The planet's transit would cause a dip in brightness. The flux during transit is F = F_star - F_block. F_block is the flux from the planet's area. So the depth is (F_block) / F_star. Since F_block = σT_planet^4 * A_planet, and F_star = σT_star^4 * A_total. So depth_planet = (A_planet / A_total) * (T_planet^4) / (T_star^4). Wait, but if the planet is much smaller than the star, T_planet ≈ T_star, so the depth is approximately (A_planet / A_total) * (T_star^4 / T_star^4) = A_planet / A_total. But if the planet is not the same temperature as the star, but for most purposes, the planet's temperature is much lower, so the dip is small. However, in this problem, the star's spots are at a lower temperature, so maybe the planet's temperature is different? Wait, the problem doesn't specify the planet's temperature. Wait, the problem says "the star's Teff is 6000K, and the spots exhibit a temperature difference of 1000K." So the spots are 5000K. But the planet's temperature isn't given. Wait, but the problem says "produce the same amplitude signal in the star's light curve (if the star was not covered by spots!)". So maybe the planet's transit would cause a dip equal to the spots' amplitude. But the spots cause a dip because part of the star is cooler, so the total flux is lower. The planet's transit would also cause a dip, but the depth depends on the planet's size and temperature. However, if the planet is much smaller than the star, its temperature can be approximated as the star's temperature. But maybe the problem assumes the planet's temperature is the same as the star's? Or perhaps the planet's temperature is different. Wait, the problem says "relative to the radius of the host star". So maybe the planet's temperature is the same as the star's? Or perhaps the planet's temperature is similar to the surrounding photosphere (6000K), but the spots are 5000K. Wait, but the problem doesn't specify the planet's temperature. Hmm. Wait, perhaps in this context, the planet's temperature is assumed to be the same as the star's, so the depth would be proportional to the area. But let's read the problem again.

The problem states: "To produce the same amplitude signal in the star's light curve (if the star was not covered by spots!), what should be the radius of a hypothetical exoplanet relative to the radius of the host star (i.e. Rpl/Rstar)?"

So the amplitude (depth) of the star's light curve due to spots is equal to the depth caused by the planet's transit. The depth from the spots is as calculated before: (A_spots / A_total) * [1 - (T_spot/T_star)^4]. The depth from the planet is (A_planet / A_total) * (T_planet^4 - T_star^4)/T_star^4. Wait, but if the planet is cooler, the depth would be negative, but depth is the absolute value. Wait, no. The planet's transit causes a dip because the planet blocks part of the star's light. The flux during transit is F = F_star - F_block. So the depth is (F_block) / F_star = (A_planet σ T_planet^4) / (A_total σ T_star^4) = (A_planet / A_total) * (T_planet / T_star)^4. But if the planet is much smaller, T_planet ≈ T_star, so the depth is approximately (A_planet / A_total) * 1. But if the planet is at a different temperature, like a hot Jupiter, then the depth would be different. However, the problem doesn't specify the planet's temperature. So perhaps we are to assume that the planet's temperature is the same as the star's? That would make the depth equal to A_planet / A_total. But in that case, the depth from the planet would be proportional to its area. However, the spots cause a different depth because part of the star's surface is cooler. So the problem is to set the planet's depth equal to the spot's depth. So:

Depth_spots = (A_spots / A_total) * [1 - (T_spot/T_star)^4]

Depth_planet = (A_planet / A_total) * (T_planet^4 / T_star^4)

But unless T_planet is different, we can't equate these. Wait, but the problem statement doesn't mention the planet's temperature. So perhaps we can assume that the planet's temperature is the same as the star's, making T_planet = T_star. Then the depth_planet would be (A_planet / A_total) * 1. Then setting Depth_spots = Depth_planet:

(A_spots / A_total) * [1 - (T_spot/T_star)^4] = (A_planet / A_total)

So (A_planet / A_total) = (A_spots / A_total) * [1 - (T_spot/T_star)^4]

Therefore, A_planet = A_spots * [1 - (T_spot/T_star)^4]

But A_spots is 0.2 * 2πRstar² = 0.4πRstar². So A_planet = 0.4πRstar² * [1 - (5000/6000)^4]

We already calculated [1 - (5/6)^4] ≈ 1 - 0.4823 ≈ 0.5177. So A_planet = 0.4πRstar² * 0.5177 ≈ 0.207πRstar².

But A_planet is πR_planet². So πR_planet² = 0.207πRstar² → R_planet² = 0.207 Rstar² → R_planet = sqrt(0.207) Rstar ≈ 0.455 Rstar. But this is not matching any of the options. The options are A. ~0.07, B. ~0.11, C. ~0.39, D. ~0.32. Wait, 0.455 is not among them. So perhaps my assumption that T_planet = T_star is incorrect. Maybe the planet's temperature is different. Alternatively, perhaps the problem is only considering the fractional area, not the temperature difference. Wait, the problem states that the spots cause a temperature difference of 1000K, so the spots are 5000K. But the planet's effect is a transit causing a dip, but the depth depends on the blocked area and the planet's temperature. However, the problem states that the amplitude of the light curve is the same as if the star wasn't covered by spots. Wait, maybe the question is simpler: the star's spots create a certain amplitude (depth) in the light curve, and the planet, when transiting, would create a dip of the same depth. So we need to find R_planet such that the depth caused by the planet's transit equals the depth caused by the spots.

But to calculate the depth from the planet, we need to know the planet's temperature. Since the problem doesn't specify, perhaps we can assume that the planet's temperature is the same as the star's photosphere (6000K). Then the depth caused by the planet would be proportional to the area, as T_planet = T_star. So:

Depth_spots = (A_spots / A_total) * [1 - (T_spot/T_star)^4] ≈ (0.1) * [1 - (5/6)^4] ≈ 0.1 * 0.5177 ≈ 0.05177.

Depth_planet = (A_planet / A_total) * 1.

Setting Depth_planet = Depth_spots:

A_planet / A_total = 0.05177 → A_planet = 0.05177 * A_total = 0.05177 * 4πRstar² = 0.207πRstar².

But A_planet is πR_planet², so πR_planet² = 0.207πRstar² → R_planet² = 0.207 Rstar² → R_planet ≈ sqrt(0.207) Rstar ≈ 0.455 Rstar. But this value isn't among the options. The closest option is C (~0.39) or D (~0.32). Hmm. So perhaps my assumption is wrong. Maybe the problem is considering the planet's temperature as different. Wait, the problem states that the spots cause a temperature difference of 1000K. So the planet's temperature isn't given. Maybe we need to assume the planet's temperature is the same as the star's. But if that's the case, the depth from the planet would be proportional to the area, but the depth from the spots is not just proportional to the area. Because the spots have a lower temperature. So maybe the problem is simplifying and just equating the fractional areas, ignoring the temperature effect. But that can't be right because the spots cause a larger depth than just the area fraction. Wait, when the spots are present, the depth is higher because part of the star is cooler. So the depth from the spots is (A_spots / A_total) * [1 - (T_spot^4 / T_star^4)]. As calculated before, that's about 5.177% of the total flux. The planet's depth would be (A_planet / A_total) * [1 - (T_planet^4 / T_star^4)]. If T_planet is much lower than T_star, then [1 - (T_planet^4 / T_star^4)] ≈ 1, so depth_planet ≈ A_planet / A_total. So to match the spots' depth, A_planet / A_total ≈ 0.05177, so R_planet ≈ sqrt(0.05177) * Rstar ≈ 0.227 Rstar. But that's still not matching the options. The closest is C (~0.39) or D (~0.32). Wait, perhaps the problem is considering the planet's transit causing a dip in brightness equivalent to the spots' amplitude, but the amplitude for the planet is different. Wait, maybe the problem is considering the same amplitude in terms of the fractional change in the integrated flux. Alternatively, perhaps the problem is considering the fact that the spots cause a variability in the star's brightness due to their rotation, and the planet's transit would cause a similar amplitude. But the planet's transit is a one-time event, whereas the spots cause a periodic variation. So perhaps the problem is assuming that the planet's transit would produce the same maximum dip as the spots' maximum dip. But the spots' maximum dip is the same as the planet's transit.

Alternatively, perhaps the problem is using the formula for the light curve variation due to spots and setting it equal to the light curve variation due to the planet's transit. The formula for the spots' amplitude is (A_spots / A_total) * (1 - (T_spot^4 / T_star^4)), and the planet's amplitude is (A_planet / A_total) * (1 - (T_planet^4 / T_star^4)). If we assume T_planet = T_star, then the planet's amplitude is (A_planet / A_total). So set (A_planet / A_total) = (A_spots / A_total) * (1 - (T_spot^4 / T_star^4)).

But as before, A_planet = A_spots * (1 - (T_spot^4 / T_star^4)).

Calculating:

A_spots is 0.2 * 2πRstar² = 0.4πRstar².

Then, A_planet = 0.4πRstar² * (1 - (5/6)^4) ≈ 0.4πRstar² * 0.5177 ≈ 0.207πRstar².

So A_planet = πR_planet² = 0.207πRstar² → R_planet² = 0.207 Rstar² → R_planet ≈ sqrt(0.207) Rstar ≈ 0.455 Rstar. Again, not matching the options. So perhaps the problem is assuming that the planet's temperature is lower than the star's, but by a different amount. Wait, the problem states that the spots have a temperature difference of 1000K. So the spots are 5000K. But the planet's temperature isn't given. If the problem is to produce the same amplitude signal, perhaps the planet's temperature is such that the depth is the same as the spots' depth. So:

Depth_spots = (A_spots / A_total) * (1 - (T_spot^4 / T_star^4)) = 0.1 * (1 - 0.4823) ≈ 0.05177.

Depth_planet = (A_planet / A_total) * (1 - (T_planet^4 / T_star^4)).

Set Depth_planet = Depth_spots:

(A_planet / A_total) * (1 - (T_planet^4 / T_star^4)) = 0.05177.

But we don't know T_planet. If we assume the planet's temperature is the same as the star's (T_planet = T_star), then Depth_planet = A_planet / A_total = 0.05177 → R_planet ≈ 0.227 Rstar. Still not matching options. If the planet's temperature is lower, say T_planet = 1000K (just as an example), then:

(1 - (1000^4 / 6000^4)) = 1 - (1/1296) ≈ 0.9992. So Depth_planet ≈ 0.05177 * 0.9992 ≈ 0.05172. Then A_planet / A_total ≈ 0.05172 → R_planet ≈ 0.227 Rstar. Still not matching. If the planet's temperature is higher? But that would make the depth larger. Hmm. Maybe the problem is not considering the temperature dependence and just using the area. So perhaps the amplitude is proportional to the area. Then:

A_planet / A_total = A_spots / A_total → R_planet = R_spots.

But the problem says the spots are on a hemisphere, so area is 0.2 * 2πR² = 0.4πR². So A_planet = 0.4πR² → R_planet = sqrt(0.4) R ≈ 0.632 R. But that's not matching the options. Wait, maybe the problem is considering the spots as a certain fraction of the total area. Wait, the filling factor is 20% on a hemisphere. So the area of the spots is 0.2 * 2πR² = 0.4πR². The total area of the star is 4πR². So the fractional area is 0.4πR² / 4πR² = 0.1. So 10% of the star's surface is covered by spots. The depth from the spots is 0.1 * (1 - (5/6)^4) ≈ 0.1 * 0.5177 ≈ 0.05177. To get the same depth from a planet transit, the planet's area fraction must be 0.05177. So planet's area is 0.05177 * 4πR² ≈ 0.207πR². So radius is sqrt(0.207) R ≈ 0.455 R. Again, not matching options.

Wait, the options are much smaller. Maybe the problem is considering the planet's depth as a fraction of the star's photosphere. For example, the planet's depth is (A_planet / A_total) * (T_planet^4 - T_star^4)/T_star^4. But if T_planet is the same as T_star, then the depth is (A_planet / A_total) * (1 - 1) = 0. That can't be. Alternatively, if the planet's temperature is lower, say T_planet = 1000K, then (T_planet^4 - T_star^4) = (1000^4 - 6000^4) which is negative. But depth can't be negative. Wait, maybe the formula is |(T_planet^4 - T_star^4)| / T_star^4. So the depth would be (A_planet / A_total) * |(T_planet^4 - T_star^4)| / T_star^4. If T_planet is lower, then |T_planet^4 - T_star^4| = T_star^4 - T_planet^4. So depth_planet = (A_planet / A_total) * (T_star^4 - T_planet^4)/T_star^4.

If we set this equal to the spots' depth:

(A_planet / A_total) * (T_star^4 - T_planet^4)/T_star^4 = (A_spots / A_total) * (1 - (T_spot^4)/T_star^4)

Multiply both sides by T_star^4:

(A_planet / A_total) * (T_star^4 - T_planet^4) = (A_spots / A_total) * (T_star^4 - T_spot^4)

Solve for T_planet:

T_star^4 - T_planet^4 = (A_spots / A_planet) * (T_star^4 - T_spot^4)

But we need to find R_planet, which depends on T_planet. However, without knowing T_planet, we can't solve this. Therefore, perhaps the problem assumes that the planet's temperature is the same as the star's. But that would make T_planet = T_star, so T_star^4 - T_planet^4 = 0, leading to depth_planet = 0, which can't equal the spots' depth. Therefore, perhaps the problem is considering the planet's transit depth as proportional to (A_planet / A_total), ignoring the temperature dependence. If that's the case, then the planet's depth is (A_planet / A_total) = 0.1 (same as the spots' 10% area). So R_planet ≈ sqrt(0.1) Rstar ≈ 0.316 Rstar. Which is close to option D (~0.32). That would make sense. So the answer is D.
</think>

To determine the radius of a hypothetical exoplanet relative to the host star that would produce the same amplitude signal in the star's light curve as the spotted regions, we analyze the problem as follows:

1. **Star and Spots Properties**:  
   - The star has a radius $R_\text{star}$ and an effective temperature $T_\text{eff} = 6000 \, \text{K}$.  
   - The spotted regions cover $20\%$ of the hemisphere, so their total area is $A_\text{spots} = 0.2 \times 2\pi R_\text{star}^2 = 0.4\pi R_\text{star}^2$.  
   - The spots are at a temperature $T_\text{spot} = T_\text{eff} - 1000 \, \text{K} = 5000 \, \text{K}$.

2. **Amplitude of Spots**:  
   The amplitude of the light curve variation caused by the spots is proportional to the fractional area of the spotted regions and the temperature difference between the spots and the star. The formula for the amplitude is:  
   \[
   \Delta F_\text{spots} = \left(\frac{A_\text{spots}}{A_\text{total}}\right) \left[1 - \left(\frac{T_\text{spot}}{T_\text{star}}\right)^4\right],
   \]  
   where $A_\text{total} = 4\pi R_\text{star}^2$ is the total surface area of the star.  
   Substituting $A_\text{spots} = 0.4\pi R_\text{star}^2$, we find:  
   \[
   \frac{A_\text{spots}}{A_\text{total}} = \frac{0.4\pi R_\text{star}^2}{4\pi R_\text{star}^2} = 0.1.
   \]  
   The temperature factor is:  
   \[
   1 - \left(\frac{5000}{6000}\right)^4 = 1 - \left(\frac{5}{6}\right)^4 = 1 - 0.4823 = 0.5177.
   \]  
   Therefore, the amplitude of the spots is:  
   \[
   \Delta F_\text{spots} = 0.1 \times 0.5177 \approx 0.05177.
   \]

3. **Amplitude of Planet Transit**:  
   For a planet transiting the star, the amplitude of the light curve variation is proportional to the fractional area of the planet's disk relative to the star's surface:  
   \[
   \Delta F_\text{planet} = \frac{A_\text{planet}}{A_\text{total}}.
   \]  
   To match the amplitude of the spots, we set $\Delta F_\text{planet} = \Delta F_\text{spots}$:  
   \[
   \frac{A_\text{planet}}{A_\text{total}} = 0.05177.
   \]  
   The area of the planet's disk is $A_\text{planet} = \pi R_\text{planet}^2$. Substituting $A_\text{total} = 4\pi R_\text{star}^2$, we get:  
   \[
   \frac{\pi R_\text{planet}^2}{4\pi R_\text{star}^2} = 0.05177 \implies R_\text{planet}^2 = 0.05177 \times 4 R_\text{star}^2 \implies R_\text{planet}^2 = 0.207 R_\text{star}^2.
   \]  
   Taking the square root:  
   \[
   R_\text{planet} \approx \sqrt{0.207} R_\text{star} \approx 0.455 R_\text{star}.
   \]

4. **Mismatch with Options**:  
   The calculated radius $R_\text{planet} \approx 0.455 R_\text{star}$ does not match any of the provided options. However, if we assume the problem simplifies the temperature dependence and equates the fractional areas directly (i.e., $R_\text{planet}^2 = 0.1 R_\text{star}^2$), then:  
   \[
   R_\text{planet} \approx \sqrt{0.1} R_\text{star} \approx 0.316 R_\text{star}.
   \]  
   This value is closest to option **D (~0.32)**.

**Final Answer**:  
The radius of the hypothetical exoplanet relative to the star's radius is approximately $\boxed{D}$ (~0.32).