Okay, so I need to figure out the ratio of the equilibrium temperatures between Planet_4 and Planet_2 in this exoplanet system. Let me start by recalling the basics. The problem states that the planets have circular orbits and are detected using the TTV method, which I think refers to Transit Timing Variations. But maybe that's just context here, and the main thing is that their orbital periods are in a ratio of 1:2:2.5:3.5:5. All planets have the same albedo. The question is about the equilibrium temperatures of Planet_4 and Planet_2. 

First, I remember that equilibrium temperature depends on the luminosity of the host star and the planet's distance from the star. The formula for equilibrium temperature is T = T_star * sqrt(R_star/(2a)) * (1 - albedo)^(1/4). Wait, actually, the formula I recall is T_planet = T_star * sqrt(R_star/(2a)) * (1 - A)^(1/4), where A is the albedo. Since the albedo is the same for all planets, the only factors affecting the temperature ratio between Planet_4 and Planet_2 would be the star's temperature, the star's radius, and their orbital distances. 

But the problem doesn't mention the star's properties. Hmm. Wait, maybe the star's luminosity is the same for all, which would mean that the star's T and R are the same for all planets. So the ratio of temperatures would depend only on the inverse square root of their orbital distances. Because the formula simplifies if the star is the same. Let me see.

The equilibrium temperature formula, when the star's luminosity is considered, is T_planet = T_eq = ( (L_star * (1 - A) ) / (16 * π^2 * σ * a^2) )^(1/4). Wait, maybe it's better to use the simplified version that relates the equilibrium temperature to the inverse of the square root of the semi-major axis. 

Yes, the equilibrium temperature is proportional to (L_star * (1 - A) )^(1/4) / a^(1/2). But since L_star is the same for all planets (same star), and A is the same, the ratio of temperatures between two planets would be the square root of the inverse of the ratio of their semi-major axes. Wait, let me get that right.

Let me write the formula again. The equilibrium temperature formula is:

T = ( ( L * (1 - A) ) / (16 * π^2 * σ * a^2) )^(1/4)

But since L = 4πR_star^2 σ T_star^4, substituting that gives:

T_planet = ( (4π R_star^2 T_star^4 (1 - A)) / (16 π^2 σ a^2) )^(1/4)

Simplifying inside the brackets:

4π / 16 π^2 = (1)/(4π)

So,

T_planet = ( (R_star^2 T_star^4 (1 - A)) / (4 π a^2) )^(1/4)

Then, raising to the 1/4 power:

T_planet = (R_star^(1/2) T_star) * (1 - A)^(1/4) ) / ( (4 π)^(1/4) a^(1/2) )

But since (4 π)^(1/4) is a constant, for the purpose of ratio between two planets, that constant cancels out. So the ratio of T_planet4 to T_planet2 would be:

(T_4 / T_2) = sqrt( (a_2 / a_4) )

Because the other terms (R_star, T_star, albedo) are same for all planets. Wait, is that correct?

Wait, let's check the exponents. Let's see:

The formula for T_planet is proportional to ( (R_star^2 T_star^4 (1 - A)) / a^2 )^(1/4 )

Which simplifies to (R_star^(1/2) T_star * (1 - A)^(1/4)) / (a^(1/2)) )

So T_planet is proportional to (R_star^(1/2) T_star (1 - A)^(1/4)) / a^(1/2)

Since R_star, T_star, and (1 - A) are the same for all planets, the ratio T_4 / T_2 would be (a_2 / a_4)^(-1/2) = sqrt(a_2 / a_4)

Wait, no. Wait, if T is proportional to 1/sqrt(a), then T_4 / T_2 = sqrt(a_2 / a_4). Because T is inversely proportional to sqrt(a). So higher a means lower T.

So the ratio would be sqrt(a2 / a4). But the problem gives the orbital periods in a ratio of 1:2:2.5:3.5:5. So Planet_1's period is 1, Planet_2 is 2, Planet_3 is 2.5, Planet_4 is 3.5, Planet_5 is 5.

Since orbital period P is related to semi-major axis a via Kepler's third law: P^2 ∝ a^3. So a ∝ P^(2/3).

Therefore, a_n = k * P_n^(2/3), where k is a constant.

So a_2 = k * (2)^(2/3)

a_4 = k * (3.5)^(2/3)

Therefore, the ratio a_2 / a_4 = [ (2)^(2/3) ] / [ (3.5)^(2/3) ] = (2/3.5)^(2/3) = (4/7)^(2/3)

So the ratio of temperatures T_4 / T_2 = sqrt( a_2 / a_4 ) = sqrt( (4/7)^(2/3) ) = (4/7)^(1/3)

Because sqrt(x^y) = x^(y/2). So here, x is (4/7), y is 2/3. So (4/7)^( (2/3)*(1/2) ) = (4/7)^(1/3)

So (4/7)^(1/3) is the cube root of 4/7. Let's compute that.

4/7 is approximately 0.5714. The cube root of 0.5714 is approximately 0.83, since 0.8^3 is 0.512 and 0.83^3 is about 0.5717. So 0.83^3 ≈ 0.5717, which is very close to 4/7 ≈ 0.5714. So the cube root of 4/7 is approximately 0.83. Therefore, the ratio T_4 / T_2 is approximately 0.83. But wait, let me check again.

Wait, the ratio T_4 / T_2 is sqrt(a_2 / a_4) = sqrt( (2/3.5)^(2/3) )? Wait, no. Wait, a_2 = (2)^(2/3) * k, a_4 = (3.5)^(2/3) *k. So a_2 / a_4 = (2/3.5)^(2/3) = (4/7)^(2/3). Then sqrt of that is (4/7)^(1/3). So yes, that's correct.

So 4/7 is approximately 0.5714. The cube root of 0.5714 is approximately 0.83. So the ratio T_4 / T_2 is ~0.83. So looking at the options, option C is ~0.83. So the answer should be C.

Wait, but let me make sure I didn't mix up the planets. The orbital periods are in the ratio 1:2:2.5:3.5:5. So Planet_1 has period 1, Planet_2 period 2, Planet_3 2.5, Planet_4 3.5, Planet_5 5. So Planet_4's period is 3.5, Planet_2's period is 2. So a_4 = (3.5)^(2/3) * k, a_2 = (2)^(2/3) *k. So a_2 is smaller than a_4. So Planet_2 is closer to the star than Planet_4. Since T is inversely proportional to sqrt(a), Planet_2 is hotter than Planet_4. So T_4 / T_2 would be less than 1. So the ratio is 0.83, which is option C. That makes sense.

Alternatively, maybe I confused the ratio. Let me check again. The ratio of the temperatures is T_4 / T_2. Since Planet_4 is farther away than Planet_2, its temperature should be lower. So the ratio should be less than 1. If the ratio is ~0.83, that matches. So the answer is C.
Therefore, the ratio of the equilibrium temperatures between Planet_4 and Planet_2 is approximately 0.83.
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