Okay, so I need to find the ratio of the equilibrium temperatures between Planet1 and Planet2. Let me start by recalling what equilibrium temperature depends on. I remember that the equilibrium temperature of a planet is determined by the balance between the incoming solar radiation and the outgoing thermal radiation. The formula I think is Teff_planet = Teff_star * sqrt(R_star / (2a)) )^(1/2), but wait, maybe I need to think again. Oh right, the formula is Teff = ( (L_star * R_planet^2) / (16 * π^2 * σ * D^2) )^(1/4), but maybe there's a simpler way.

Alternatively, the equilibrium temperature can be approximated using the Stefan-Boltzmann law. The incoming flux from the star is L/(4πD²), where L is the star's luminosity and D is the distance from the star. The planet's outgoing flux is σT⁴. So setting them equal: L/(4πD²) = σT⁴. Solving for T gives T = [ L/(16πσD²) ]^(1/4). But the star's luminosity L is related to its radius and temperature via the Stefan-Boltzmann law: L = 4πR_star² σ Teff_star⁴. Substituting that into the previous equation, T_planet = [ (4πR_star² σ Teff_star⁴)/(16πσD²) ) ]^(1/4). Simplifying that, the 4π cancels with 16π, giving (R_star² Teff_star⁴)/(4 D²) )^(1/4). Taking the fourth root, that becomes (R_star / (2 D))^(1/2) * Teff_star. So T_planet = Teff_star * sqrt(R_star / (2 D)).

Wait, but in the question, we have planets with different masses and radii. But equilibrium temperature only depends on the star's properties and the planet's distance from the star. The problem says both planets are orbiting the same star, so D (distance) is the same for both? Wait, no, the problem doesn't mention the distance. Wait, but the transit method and radial velocity method are used. The transit method gives the size of the planet (from the dip in the star's brightness) and the orbital period. The radial velocity gives the mass of the planet and the orbital period. But since both planets have the same orbital period (because transit gives the period and same star), maybe the semi-major axis is the same? Wait, no, transit gives the period, but the semi-major axis depends on the mass of the planet. Wait, but if both planets have the same orbital period, then their semi-major axes would be different because their masses around the star are different. But the star's mass is given as 1.5 solar masses. So using Kepler's third law: (a³)/(T²) = (G(M_star + M_planet))/(4π²). Since the planet's mass is much smaller than the star's mass (Earth is 0.003 solar masses, so 5 Earth masses is negligible compared to 1.5 solar masses), we can approximate M_planet << M_star. So a³ ≈ (G M_star T²)/(4π²). So for each planet, a = [ (G M_star T²)/(4π²) ]^(1/3). Since both planets have the same T (same orbital period), their a would be the same. Wait, but the problem says "both planets are in circular orbits". So yes, same period implies same semi-major axis. Therefore, their distances from the star are the same. So D is the same for both. Therefore, the equilibrium temperature depends only on Teff_star and the ratio of R_star/(2D). Since D is the same for both, the ratio of equilibrium temperatures would be the same as the ratio of Teff_star multiplied by sqrt(R_star/(2D)) for each planet. Wait, but the star's Teff is given as 6300 K. So both planets are orbiting the same star at the same distance. Therefore, their equilibrium temperatures should be the same. But that can't be right because the options don't have 1.0. Wait, maybe I'm missing something here.

Wait, the question says "the star has a mass of 1.5 times that of the Sun, a radius 1.2 times that of the Sun, and an effective temperature (Teff) of 6300 K." So the star's Teff is 6300 K. The planets are orbiting this star. The equilibrium temperature formula is T_planet = Teff_star * sqrt(R_star/(2D)). But since both planets are orbiting the same star, D is the same. Therefore, T_planet is the same for both. But the options don't include 1.0. Wait, maybe I made a mistake here. Let me check the formula again.

Wait, maybe the formula is different. The equilibrium temperature is given by T = Teff_star * sqrt(R_star/(2D)) * (1 - A)^(1/4), where A is the albedo. But the problem states that both planets have the same albedo. So the albedo term cancels out when taking the ratio. Therefore, the ratio of equilibrium temperatures would be the same as the ratio of Teff_star multiplied by sqrt(R_star/(2D)) for each planet. But since both planets are orbiting the same star at the same distance (same D), the ratio would be 1.0. But the options don't have 1.0. Hmm, that's confusing. Maybe I need to re-examine the problem.

Wait, the problem says "the correct answer to this question: Two planets...". Wait, but the question is asking for the ratio of equilibrium temperatures between Planet1 and Planet2. But according to my previous reasoning, this ratio should be 1.0, which isn't an option. Therefore, I must be missing something here.

Wait, maybe the problem is not about equilibrium temperature but about something else. Let me read the question again carefully. Oh wait, the question says "the ratio of the equilibrium temperatures between Planet1 and Planet2". But according to the formula, if both planets are at the same distance from the star, their equilibrium temperatures would be the same. But the options don't include 1.0. So perhaps there's another factor involved here. Let me think again.

Wait, maybe the star's radius is given as 1.2 times the Sun's radius. The formula for equilibrium temperature is T_planet = Teff_star * sqrt(R_star/(2D)) * (1 - A)^(1/4). But if the star's radius is 1.2 R_sun, and the planets are at the same distance D from the star. So R_star is 1.2 R_sun. But the star's radius is not the same as the distance D. Wait, no, the star's radius is separate. The distance D is the orbital distance. So the formula still holds. So if D is the same for both planets, their equilibrium temperatures would be the same. But again, the options don't have 1.0. So maybe the problem is not using the equilibrium temperature formula but something else. Let me think.

Wait, the problem mentions forbidden line [OI] at 6300 Å and Doppler shifts. Wait, the forbidden line is a spectral line that is not emitted by the star but by the planet's atmosphere. The shift in the forbidden line is due to the planet's atmosphere absorbing the star's light and re-emitting it, causing a shift. The amount of shift depends on the planet's temperature. The maximum shift (0.03 Å and 0.04 Å) would be related to the planet's temperature. Wait, maybe the question is about the effective temperature of the planet's atmosphere rather than the equilibrium temperature. But the question specifically asks for the equilibrium temperatures. Hmm.

Alternatively, maybe the problem is not using the standard equilibrium temperature formula but considering other factors. Let me think again. The problem states that Planet1 induces a periodic shift of up to 0.03 Å in the forbidden line [OI] at 6300 Å, while Planet2 results in a periodic Doppler shift of up to 0.04 Å. The albedo is the same for both, so the depth of the absorption features (the shift) depends on the planet's temperature. The shift is proportional to the planet's temperature. Wait, the Doppler shift (or the shift in wavelength) is related to the planet's temperature. The maximum shift occurs when the planet is moving towards or away from us, causing the Doppler effect. The amount of shift (Δλ) is proportional to the planet's velocity, which in turn depends on the planet's orbital velocity. Orbital velocity of the planet is sqrt(G M_star / a), where a is the semi-major axis. But since both planets are orbiting the same star and have the same orbital period (because they are detected by transit, which gives the period), their semi-major axes are the same, as per Kepler's third law. Therefore, their orbital velocities are the same. But the problem mentions the Doppler shift of the forbidden line. Wait, the forbidden line is emitted by the planet's atmosphere, so the shift in the line is caused by the planet's temperature. The maximum shift (Δλ) is proportional to the planet's temperature. The formula for Doppler shift is Δλ/λ = v/c, where v is the planet's temperature-induced velocity in the atmosphere. But wait, the planet's temperature is related to the depth of the absorption line. The depth of the line depends on the planet's temperature. The maximum shift (Δλ) is proportional to the planet's temperature. So if Planet1 has a shift of 0.03 Å and Planet2 has 0.04 Å, then the ratio of their temperatures would be 0.03 / 0.04 = 0.75. But that would be a ratio of 0.75, which is not among the options. Alternatively, maybe the shift is proportional to the square root of the temperature. Wait, the Doppler shift (or the width of the line) is related to the planet's temperature. The thermal broadening of spectral lines is often approximated by the formula Δλ/λ ≈ (λ c) * sqrt( (k T) / (m v²) ) ), but I might be mixing things up. Alternatively, the shift in the line (Δλ) is proportional to the planet's temperature. Let me think. The Doppler shift caused by the planet's temperature could be related to the escape velocity or the thermal speed of particles in the atmosphere. The thermal speed is proportional to sqrt(T). The maximum shift would be proportional to the thermal speed, which is proportional to sqrt(T). Therefore, the ratio of the shifts would be sqrt(T1 / T2). If the shifts are 0.03 and 0.04, then sqrt(T1/T2) = 0.03/0.04 = 0.75, so T1/T2 = 0.75² = 0.5625. But that's not matching any options either. Hmm.

Alternatively, maybe the problem is referring to the Rossiter-McLaughlin effect or something else, but I'm not sure. Wait, the problem mentions both the transit and radial velocity methods. The transit method measures the planet's size (from the dip in brightness) and the orbital period. The radial velocity method measures the Doppler shift of the star's spectral lines due to the planet's gravitational pull. The radial velocity amplitude K is given by K = (v_p) * (M_p / (M_p + M_star))^(1/3), where v_p is the planet's orbital velocity. Since M_p is much smaller than M_star, K ≈ v_p * (M_p / M_star)^(1/3). But the problem gives the Doppler shift of the forbidden line as 0.04 Å and 0.03 Å. Wait, maybe the forbidden line shift is related to the Rossiter-McLaughlin effect, which is the Doppler shift due to the planet's orbit. But I'm not sure. Alternatively, the shift could be related to the planet's orbital speed. If Planet1 causes a shift of 0.03 Å and Planet2 causes 0.04 Å, then the ratio of their orbital speeds is 3:4. Since both planets have the same orbital period (same transit period), their orbital velocities would be v = 2πa / T. Since T is the same, v is proportional to a. But from Kepler's third law, a³ ∝ T² M_star. Since T is the same and M_star is the same, a is the same. Therefore, their orbital velocities are the same. Therefore, the shifts should be the same. But the problem says they are different. So perhaps this line of reasoning is incorrect. Maybe the forbidden line shift is a different effect.

Wait, maybe the forbidden line shift is due to the planet's temperature. The maximum forbidden line shift (Δλ) is proportional to the planet's temperature. So if Planet1 has a shift of 0.03 Å and Planet2 has 0.04 Å, then the ratio of their temperatures would be 0.03/0.04 = 0.75. But according to equilibrium temperature formula, T is proportional to sqrt(R_star/(2D)) * Teff_star. But if D is the same, then T is proportional to sqrt(R_star). Since Planet1's star has a radius of 1.2 R_sun, Planet2's star is also 1.2 R_sun (same star). Wait, both planets are around the same star, so R_star is the same. Therefore, their equilibrium temperatures would be the same, leading to a ratio of 1.0. But the options don't have that. Therefore, perhaps the problem is not about equilibrium temperature but about something else, like the depth of the forbidden line. Let me think again.

Alternatively, maybe the problem is referring to the effective temperature of the planet's atmosphere, which is different from the equilibrium temperature. The effective temperature of the planet's atmosphere can be higher due to greenhouse effects. But the problem states that both planets have the same albedo. Albedo is the fraction of light reflected, so it affects the incoming flux. The equilibrium temperature formula is T_eq = Teff_star * sqrt(R_star/(2D)) * (1 - A)^(1/4). Since both planets have the same albedo, the ratio of their equilibrium temperatures would be the same as the ratio from the star's Teff and the distance D. But since D is the same, the ratio is 1.0. Therefore, this can't be the case. So perhaps the problem is not using the equilibrium temperature formula but something else.

Wait, the problem says "the correct answer to this question: Two planets...". Wait, maybe the question is not about the equilibrium temperatures but about another parameter, like the depth of the forbidden line. The depth of the forbidden line is proportional to the planet's temperature. If Planet1 has a shift of 0.03 Å and Planet2 has 0.04 Å, then the ratio of their temperatures would be 0.03/0.04 = 0.75. But the options don't have 0.75. However, the problem asks for the ratio of equilibrium temperatures, not the ratio of the forbidden line shifts. So perhaps there's a misunderstanding here.

Alternatively, maybe the problem uses the formula for the change in wavelength (Δλ) due to the Doppler effect, which is Δλ = λ * (v/c), where v is the velocity. The maximum velocity of the planet in the orbit is v = 2πa / T. Since both planets have the same period T, their velocities are proportional to a. From Kepler's third law, a³ ∝ T² M_star. Since T and M_star are the same, a is the same. Therefore, v is the same. Therefore, Δλ is the same. But the problem states different Δλ values. Therefore, perhaps the formula is different. Maybe the forbidden line shift is related to the planet's temperature, and the temperature is related to the equilibrium temperature. But if the equilibrium temperatures are the same, the shifts should be the same, which they are not. Therefore, perhaps the problem is using a different approach.

Wait, maybe the problem is using the formula for the maximum forbidden line shift, which is proportional to the planet's temperature. The formula for the maximum shift is Δλ_max ≈ (λ / c) * v_p, where v_p is the planet's escape velocity. The escape velocity is sqrt(2 G M_p / R_p). But since M_p and R_p are different for each planet, their escape velocities would be different. Planet1 has a mass of 7 Earth masses and radius 1.7 Earth radii. Planet2 has 5 Earth masses and 1.3 Earth radii. Let's compute their escape velocities.

Escape velocity v_e = sqrt(2 G M_p / R_p). Let's compute the ratio of v_e for Planet1 and Planet2. Let M_p1 = 7 M_earth, R_p1 = 1.7 R_earth. M_p2 =5 M_earth, R_p2=1.3 R_earth. So v_e1 / v_e2 = sqrt( (7/5) * (1.3/1.7) ) ). Let's compute that: 7/5 = 1.4, 1.3/1.7 ≈ 0.7647. Multiply them: 1.4 * 0.7647 ≈ 1.0706. Square root is sqrt(1.0706) ≈ 1.0345. So the escape velocities are in a ratio of ~1.0345. If the forbidden line shift is proportional to the escape velocity, then the ratio of shifts would be ~1.0345. But the problem states shifts of 0.03 and 0.04, which is a ratio of 0.75. So this doesn't add up. Therefore, perhaps the shift is not directly proportional to the escape velocity.

Alternatively, the shift could be related to the planet's thermal velocity. The thermal speed is proportional to sqrt(T), and the maximum shift would be proportional to that. So if the equilibrium temperatures are the same, the shifts should be the same. But they are different. Therefore, perhaps the problem is not about equilibrium temperatures but about something else.

Alternatively, maybe the problem is using the fact that the forbidden line shift is proportional to the planet's radius. But that doesn't make sense. Alternatively, maybe the problem is using the fact that the shift is proportional to the planet's mass. But the shifts are 0.03 and 0.04, which are not proportional to the masses (7 vs 5). So that doesn't fit either.

Wait, the problem states that both planets have the same albedo. The albedo affects the incoming flux, which affects the equilibrium temperature. But the equilibrium temperature formula is T_eq = Teff_star * sqrt(R_star/(2D)) * (1 - A)^(1/4). Since albedo is the same, the ratio of T_eq would be the same as the ratio of Teff_star * sqrt(R_star/(2D)). But both planets have the same star and distance, so the ratio is 1.0. Therefore, this suggests that the answer should be 1.0, but that's not an option. Therefore, perhaps the problem is not asking about equilibrium temperature but something else.

Wait, the problem mentions that both planets are detected using both transit and RV methods. The transit method gives the size of the planet (radius), and the RV method gives the mass. The question is about the ratio of equilibrium temperatures. But according to the formula, if both planets are at the same distance from the star and have the same albedo, their equilibrium temperatures are the same. But the options don't have 1.0. Therefore, perhaps the problem is using a different approach. Maybe the problem is assuming that the star's Teff is 6300 K, and each planet's effective temperature is being calculated based on their own properties. Let me try that.

The effective temperature of a planet can be calculated using the formula T_planet = T_star * sqrt(R_star / (2D)) * (1 - A)^(1/4). But again, if D is the same, the ratio is 1.0. But perhaps the star's Teff is different from the planet's Teff? Wait, no, the star's Teff is given as 6300 K. The planet's Teff would be lower. Wait, no, the planet's Teff is the effective temperature of the planet's atmosphere. The equilibrium temperature is the effective temperature of the planet. So I'm back to the same problem.

Alternatively, maybe the problem is using the formula for the maximum forbidden line shift, which is proportional to the planet's temperature. Let's consider that. If the forbidden line shift is proportional to the planet's temperature, then the ratio of the temperatures would be the ratio of the shifts. Planet1's shift is 0.03, Planet2's is 0.04. So ratio is 0.03/0.04 = 0.75. But the options don't have 0.75. However, the options are 1.30, 0.98, 1.05, 0.53. So perhaps the ratio is inverted? 0.04/0.03 ≈ 1.33, which is close to option A (1.30). But why would the ratio be 4/3? Maybe there's another factor involved. Let's think again.

The forbidden line shift is proportional to the planet's temperature. Let's say Δλ = k * T_planet, where k is a constant. Then T1/T2 = Δλ1 / Δλ2 = 0.03 / 0.04 = 0.75. But if the problem is asking for T2/T1, then it's 4/3 ≈ 1.33, which is close to option A (1.30). However, the problem states "the ratio of the equilibrium temperatures between Planet1 and Planet2". If the ratio is T1/T2 = 0.75, then the ratio of Planet2 to Planet1 is 1.33. But the options don't have 1.33. The closest is 1.30 (option A). However, the problem states that the forbidden line shift for Planet1 is 0.03 and for Planet2 is 0.04. So if we take the ratio as T1/T2 = 0.03/0.04 = 0.75, then the ratio of equilibrium temperatures is 0.75, but that's not an option. Alternatively, maybe the problem is considering the inverse, so 0.04/0.03 = 1.33. But the question asks for the ratio between Planet1 and Planet2, so T1/T2 = 0.75. But since 0.75 isn't an option, perhaps this approach is incorrect.

Alternatively, maybe the forbidden line shift is not directly proportional to the temperature but to the square root, due to the broadening mechanism. The Doppler shift (or velocity) is related to the thermal speed of particles in the atmosphere, which is proportional to sqrt(T). Therefore, if the shift is proportional to sqrt(T), then the ratio of shifts would be sqrt(T1/T2). So if Δλ1/Δλ2 = sqrt(T1/T2). Then (Δλ1/Δλ2)^2 = T1/T2. So T1/T2 = (0.03/0.04)^2 = (0.75)^2 = 0.5625. Which is approximately 0.56, and the closest option is D (0.53). But this is speculative. However, the problem states that both planets have the same albedo. Albedo affects the equilibrium temperature, but if the equilibrium temperature is the same, then the ratio would be 1.0. But the forbidden line shifts are different, which complicates things.

Alternatively, perhaps the problem is using the fact that the forbidden line shift is proportional to the planet's radius. But the shifts are 0.03 and 0.04, which don't correspond to the radii given (1.7 and 1.3 Earth radii). So that doesn't fit. Alternatively, maybe the shift is proportional to the mass. But the shifts are 0.03 and 0.04, which don't match the masses (7 and 5). So that's not it either.

Given that the problem seems to tie the forbidden line shift to the planet's temperature, but the options don't match the straightforward ratio, perhaps there's another factor. Let's consider the formula for the maximum forbidden line shift. The maximum Doppler shift (Δλ/λ) is given by (v/c), where v is the velocity of the planet in the orbit. Since both planets have the same orbital period (same transit period), their orbital velocities are the same. Therefore, the shifts should be the same. But the problem states different shifts. Therefore, perhaps the problem is using a different approach.

Wait, maybe the problem is considering the Rossiter-McLaughlin effect, where the shift is due to the star's rotation. But the star's rotation period isn't given. Alternatively, the problem might be referring to the star's rotation causing a shift, but since the star's rotation period isn't given, this seems unlikely.

Alternatively, maybe the problem is using the fact that the forbidden line shift is proportional to the planet's mass. But again, the shifts are 0.03 and 0.04, which don't align with the masses (7 and 5). Therefore, this approach seems invalid.

Alternatively, perhaps the problem is using the fact that the star's radius affects the escape velocity of the planet. But the star's radius is 1.2 times that of the Sun, and the planet's radius is given. But I'm not sure how that ties in.

Alternatively, maybe the problem is using the fact that the forbidden line shift is proportional to the square root of the planet's mass. But 7^0.5 ≈ 2.64, 5^0.5 ≈ 2.24. The ratio is 2.64/2.24 ≈ 1.18, which is close to option B (0.98) but inverted. However, this is speculative.

Alternatively, perhaps the problem is using the fact that the forbidden line shift is proportional to the planet's density. Planet1 has mass 7 and radius 1.7, density is 7/(1.7^3) ≈ 7/4.913 ≈ 1.42. Planet2 has mass 5 and radius 1.3, density 5/2.197 ≈ 2.28. The ratio is 1.42/2.28 ≈ 0.62. The inverse would be 1.61. But this doesn't match the options.

Alternatively, maybe the problem is using the fact that the forbidden line shift is proportional to the planet's radius. But the shifts are 0.03 and 0.04, and the radii are 1.7 and 1.3. The ratio is 1.7/1.3 ≈ 1.30. Which is option A. But why would the radius affect the forbidden line shift? The radius would affect the planet's size, but the forbidden line shift is due to the temperature of the atmosphere, not the size. Unless larger planets have atmospheres with different escape velocities, but escape velocity depends on mass and radius.

Alternatively, perhaps the problem is considering that the forbidden line shift is proportional to the planet's escape velocity. As calculated earlier, the escape velocities are in a ratio of ~1.0345. But the shifts are 0.03 and 0.04, which is a ratio of 0.75. This doesn't align. Therefore, perhaps the problem is using a different approach.

Wait, perhaps the problem is considering the fact that the forbidden line shift is proportional to the planet's orbital velocity. Since both planets have the same orbital period, their orbital velocities are the same. Therefore, the shifts should be the same. But the problem states they are different. Therefore, perhaps this line of reasoning is incorrect.

Alternatively, perhaps the problem is using the fact that the forbidden line shift is proportional to the planet's orbital speed, but since the problem states different shifts, this seems contradictory. Therefore, perhaps the problem is not about the forbidden line shifts but about something else.

Given that I'm stuck here, perhaps I should consider another approach. Let's think about the equilibrium temperature formula again. If the equilibrium temperature of Planet1 is T1 and Planet2 is T2, then the ratio T1/T2 = [ (R_star / (2D))^(1/2) * (1 - A)^(1/4) ] for Planet1 divided by the same term for Planet2. Since both planets are around the same star and distance, and same albedo, the ratio is 1.0. Therefore, the answer should be 1.0. But since that's not an option, perhaps the problem is considering another factor.

Wait, the problem mentions that the forbidden line shift is 0.03 Å for Planet1 and 0.04 Å for Planet2. If the forbidden line shift is proportional to the planet's effective temperature, then the ratio of temperatures is 0.03/0.04 = 0.75. But considering the albedo, which affects the incoming flux, but albedo is the same. Wait, the equilibrium temperature formula is T_eq = Teff_star * sqrt(R_star/(2D)) * (1 - A)^(1/4). Since both planets have the same albedo, the ratio of T_eq is the same as the ratio of Teff_star * sqrt(R_star/(2D)). But both planets are around the same star and same distance, so the ratio is 1.0. Therefore, the ratio of the forbidden line shifts must be related to something else. Perhaps the forbidden line shift is proportional to the planet's temperature, which in turn is the equilibrium temperature. Therefore, the ratio of forbidden line shifts should be the same as the ratio of equilibrium temperatures. But in the problem, the ratio is 0.03/0.04 = 0.75, which would imply the ratio of equilibrium temperatures is 0.75. But since 0.75 isn't an option, perhaps the problem is considering the inverse ratio, 4/3 ≈ 1.33, which is close to option A (1.30). However, the problem asks for the ratio of Planet1 to Planet2. If the forbidden line shift is proportional to the temperature, then T1/T2 = 0.75, which is not an option. If the problem is considering the ratio of Planet2 to Planet1, then 4/3 ≈ 1.33, which is close to 1.30. But the question says "the ratio of the equilibrium temperatures between Planet1 and Planet2", which is T1/T2. So 0.75 isn't an option. Therefore, perhaps this approach is incorrect.

Alternatively, perhaps the forbidden line shift is proportional to the square root of the temperature. Then T1/T2 = (Δλ1/Δλ2)^2 = (0.03/0.04)^2 = 0.5625, which is close to option D (0.53). But the problem states that the forbidden line shift for Planet1 is 0.03 and Planet2 is 0.04. If the shift is proportional to sqrt(T), then the ratio of T1/T2 = (Δλ1/Δλ2)^2 = (0.03/0.04)^2 = 0.5625. But this would make the ratio 0.56, which is close to option D (0.53). However, this is speculative and the problem doesn't state that the shift is proportional to sqrt(T). Therefore, this might not be the correct approach.

Alternatively, perhaps the forbidden line shift is proportional to the planet's mass. But Planet1 has mass 7 and Planet2 has 5. The ratio is 7/5 = 1.4. But the shifts are 0.03 and 0.04, which is a ratio of 0.75. This doesn't align. Alternatively, if the shift is proportional to the planet's radius, the ratio is 1.7/1.3 ≈ 1.30, which is option A. However, the forbidden line shift is not directly proportional to the radius; it's related to the temperature. Therefore, this approach is invalid.

Given that I'm stuck and the time I've spent, perhaps I should consider that the problem is not about equilibrium temperature but about something else, and the answer is 1.0, but since that's not an option, perhaps the correct answer is based on the star's temperature and planet's radius. Wait, the star's Teff is 6300 K. The planets' equilibrium temperatures would be lower. Let's compute the equilibrium temperature for each planet.

First, calculate the equilibrium temperature formula:

T_planet = Teff_star * sqrt(R_star / (2D)) * (1 - A)^(1/4)

But since both planets are at the same distance D, and same albedo (A), the ratio of their equilibrium temperatures would be 1.0. Therefore, the answer should be 1.0. But since that's not an option, perhaps the problem is considering the star's Teff and the planet's radius to compute the planet's temperature. However, the formula for equilibrium temperature doesn't directly involve the planet's radius. Wait, maybe the problem is using the formula for the maximum forbidden line shift, which is related to the planet's temperature. Let's assume that the shift is proportional to the planet's temperature. Then, T1/T2 = 0.03/0.04 = 0.75. But since the options don't have this, perhaps the answer is 0.75, but since it's not an option, maybe the closest is option D (0.53). But this is a guess.

Alternatively, perhaps the problem is considering the ratio of the squares. Then (0.03/0.04)^2 = 0.5625, which is close to option D (0.53). However, this is speculative.

Alternatively, perhaps the problem is using the fact that the forbidden line shift is proportional to the planet's density. Planet1's density is 7/(1.7)^3 ≈ 7/4.913 ≈ 1.42. Planet2's density is 5/(1.3)^3 ≈ 5/2.197 ≈ 2.28. The ratio is 1.42/2.28 ≈ 0.62. The inverse is 1.61. But this isn't close to the options.

Alternatively, perhaps the problem is using the fact that the forbidden line shift is proportional to the planet's radius. Then the ratio is 1.7/1.3 ≈ 1.30, which is option A. But why would the radius affect the shift? The radius affects the gravity, but the shift is due to the temperature. Unless the radius affects the escape velocity, but escape velocity is sqrt(2 G M / R), which for Planet1: sqrt(2 G (7 M_earth)/(1.7 R_earth)) = sqrt( (7/1.7) * (2 G M_earth / R_earth) ) = sqrt(4.117) * sqrt(escape velocity of Earth). The escape velocity of Earth is ~11.2 km/s. So Planet1's escape velocity is ~4.117^(1/2) * 11.2 ≈ 2.03 * 11.2 ≈ 22.8 km/s. Similarly, Planet2's escape velocity is sqrt(5/1.3) * 11.2 ≈ sqrt(3.846) * 11.2 ≈ 1.96 * 11.2 ≈ 21.9 km/s. The ratio is ~22.8/21.9 ≈ 1.037. If the forbidden line shift is proportional to escape velocity, then the ratio is ~1.037. But the problem states shifts of 0.03 and 0.04, which would imply a ratio of 0.75. Therefore, this approach is invalid.

Given that I'm stuck, perhaps the correct answer is 1.0, but since it's not an option, maybe the problem is considering the star's Teff and the planet's radius to compute the planet's temperature. Let's try that.

The equilibrium temperature formula is T_planet = Teff_star * sqrt(R_star/(2D)) * (1 - A)^(1/4). Since D is same and A same, the ratio is 1.0. Therefore, the answer should be 1.0. But since the options don't have this, perhaps the answer is D (0.53), but I'm not sure.

Alternatively, perhaps the problem is using the formula for the maximum forbidden line shift, which is proportional to the square root of the planet's temperature. Let's say Δλ ~ sqrt(T). Then, T1/T2 = (Δλ1/Δλ2)^2 = (0.03/0.04)^2 = 0.5625. Close to option D (0.53). However, this is speculative.

Alternatively, perhaps the problem is using the fact that the forbidden line shift is proportional to the planet's mass. Then, T1/T2 = (mass1/mass2)^(1/2) = sqrt(7/5) ≈ 1.18. But the shifts are 0.03 and 0.04, which is a ratio of 0.75. So this doesn't align.

Alternatively, perhaps the problem is using the fact that the forbidden line shift is proportional to the planet's radius. Then, T1/T2 = (radius1/radius2) = 1.7/1.3 ≈ 1.30. Which is option A. But why would the radius affect the shift? The radius affects the escape velocity, which affects the thermal broadening, but the shift is due to thermal motion. However, if the problem is oversimplifying and assuming that the shift is proportional to the radius, then option A would be the answer. However, this seems incorrect.

Alternatively, perhaps the problem is using the formula for the maximum forbidden line shift, which is proportional to the square root of the planet's mass. Then, T1/T2 = sqrt(7/5) ≈ 1.18. But the shifts are 0.03 and 0.04, ratio 0.75. This doesn't align.

Alternatively, perhaps the problem is using the formula for the maximum forbidden line shift, which is proportional to the square root of the planet's radius. Then, T1/T2 = sqrt(1.7/1.3) ≈ sqrt(1.307) ≈ 1.14. But the shifts are 0.03 and 0.04, ratio 0.75. Doesn't align.

Given all these possibilities, I'm inclined to think that the correct approach is to realize that the equilibrium temperatures of both planets are the same, and thus the ratio is 1.0. However, since 1.0 is not an option, perhaps the problem has a typo or I'm misunderstanding something. Alternatively, perhaps the problem is considering the ratio of the star's Teff to the planet's Teff. But since the star's Teff is 6300 K, and the planets' Teff are lower, but the ratio would be the same for both planets. Therefore, the ratio is 1.0.

But since the options don't have 1.0, perhaps the answer is D (0.53), but I'm not confident. Alternatively, perhaps the problem is considering the ratio of the forbidden line shifts as the inverse of the ratio of the equilibrium temperatures. If the forbidden line shift is proportional to the temperature, then T1/T2 = 0.03/0.04 = 0.75. Therefore, the ratio of Planet1 to Planet2 is 0.75. But since 0.75 isn't an option, perhaps the closest is option D (0.53), but that's not close. Alternatively, maybe the problem is considering the inverse, 4/3 ~1.33, which is option A. But again, the problem says "ratio of Planet1 to Planet2", which would be ~0.75. But since that's not an option, perhaps the problem is incorrect.

Alternatively, perhaps the problem is considering the ratio of the masses. Planet1 has mass 7, Planet2 has 5. The ratio is 7/5 = 1.4. But the shifts are 0.03 and 0.04, which is 0.75. Therefore, this doesn't align.

Alternatively, perhaps the problem is using the formula for the maximum forbidden line shift, which is proportional to the square root of the temperature. Therefore, T1/T2 = (0.03/0.04)^2 = 0.5625. Close to option D (0.53). Therefore, perhaps the answer is D.

But given the time I've spent and the uncertainty, I think the most plausible answer, even though it's speculative, is that the ratio is approximately 1.0, but since that's not an option, the closest is D (0.53). However, I'm not confident. Alternatively, if the problem is considering the inverse ratio, which would be ~1.33, which is option A. But the problem asks for Planet1 to Planet2, so if the shift ratio is 0.75, but the answer is 1.33, which is the inverse, perhaps that's the intended answer. However, the problem says "ratio of the equilibrium temperatures between Planet1 and Planet2", which is T1/T2. Therefore, the answer should be 1.0, but since it's not an option, perhaps the problem is wrong.

Alternatively, perhaps the problem is considering the star's radius. The star's radius is 1.2 R_sun. The planets' radii are 1.7 and 1.3 Earth radii. The ratio of the star's radius to planet's radius would be 1.2 R_sun / 1.7 R_earth ≈ 1.2 * (695,700 km) / (6,371 km) ≈ 1.2 * 109.2 ≈ 131. So, 131. But this is unrelated to the forbidden line shifts.

Alternatively, perhaps the problem is using the formula for the Rossiter-McLaughlin effect, where the shift is proportional to the planet's orbital velocity. Since both planets have the same period, their orbital velocities are the same. Therefore, the shifts should be the same. But the problem states different shifts. Therefore, perhaps this line of reasoning is invalid.

Given all that, I think the most logical conclusion is that the ratio of equilibrium temperatures is 1.0, but since that's not an option, perhaps the intended answer is D (0.53), but I'm not sure. Alternatively, if the problem is considering the inverse ratio, 4/3 ≈ 1.33, which is option A. But the problem asks for Planet1 to Planet2, which would be ~0.75. Therefore, perhaps the intended answer is D (0.53), but I can't confirm. Alternatively, perhaps the problem is considering the ratio of the forbidden line shifts as the inverse of the temperature ratio. If the forbidden line shift is proportional to temperature, then T1/T2 = 0.03/0.04 = 0.75. Therefore, the answer is 0.75, but since that's not an option, the closest is D (0.53). However, this is speculative.

Alternatively, perhaps the problem is considering the ratio of the forbidden line shifts as the inverse of the temperature ratio. If the shift is proportional to the square root of the temperature, then T1/T2 = (0.03/0.04)^2 = 0.5625, which is close to option D (0.53). Therefore, the answer might be D. But this is speculative.

Given the time I've spent and the uncertainty, I think I'll go with option D (0.53), assuming that the problem is considering the ratio of the forbidden line shifts as the inverse of the temperature ratio, and the shift is proportional to the square root of the temperature.
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