Okay, let's tackle this problem step by step. So, the question is about comparing the luminosities of two stars, Star_1 and Star_2. The given information is a bit detailed, so I need to parse it carefully.

First, let's list out all the given data:

1. Star_1 has a radius 1.5 times larger than Star_2. So, R1 = 1.5 * R2.
2. The mass of Star_1 is also 1.5 times that of Star_2. So, M1 = 1.5 * M2.
3. The wavelengths at which the stars appeared brightest are the same. That probably relates to their temperatures since the peak wavelength of a black body is inversely proportional to temperature (Wien's Law: λ_max = b/T, where b is Wien's displacement constant).
4. Radial velocities: Star_1 has a radial velocity of 0 km/s, and Star_2 has 700 km/s. Radial velocity relates to the Doppler effect, which can indicate the star's motion towards or away from us. However, since the problem mentions that they radiate as black bodies, maybe the Doppler shift isn't directly affecting their luminosity calculation. But I need to think about how radial velocity factors into this.
5. They want to find the factor by which the luminosity of Star_1 is greater than Star_2. The options are around 2.2 to 2.35.

Wait, but the question mentions that the stars radiate as black bodies. So their luminosity should be calculated using the Stefan-Boltzmann law: L = 4πR²σT⁴, where R is radius, T is temperature, and σ is the Stefan-Boltzmann constant.

So, to find L1/L2, we need to compute (R1/R2)² * (T1/T2)^4.

We know R1/R2 is 1.5. But we need to find T1/T2. The problem states that the stars have the same peak wavelength, which, according to Wien's Law, implies that T1/T2 = λ2/λ1. But since λ1 = λ2 (same wavelength), that would mean T1 = T2. Wait, that can't be right because then their luminosities would depend only on radius. But the problem says their luminosities are different, so maybe there's another factor.

Wait, hold on. Wien's Law states that λ_max * T = constant. If both stars have the same λ_max, then their temperatures must be the same. So T1 = T2. Then, according to the Stefan-Boltzmann law, L1/L2 = (R1/R2)^2 * (T1/T2)^4 = (1.5)^2 * (1)^4 = 2.25. But that's one of the answer choices, option B. However, the options also include C (2.23) and others. Wait, but why is there a discrepancy here?

Alternatively, maybe I'm missing something. Let's consider the radial velocities. The problem states that the astronomers determined the wavelengths at which the stars appeared brightest. But if the stars are moving towards or away from us, that would cause a Doppler shift. However, the observed wavelength would be different from the emitted wavelength. But the problem says the stars' own peak wavelengths are the same. Wait, no. If the stars are moving, that affects the observed wavelengths. But the problem states that through spectroscopy, they found that the stars appeared brightest at the same wavelength. Hmm, maybe that's the emitted wavelength, not the observed. Or perhaps the observed wavelengths are the same, which would imply that the Doppler shifts are such that the stars' emitted peaks are the same when accounting for their velocities.

Wait, let's clarify. The problem says: "the wavelengths at which the stars appeared brightest, and found that they are the same." So the observed wavelengths are the same. That would mean that the stars' light is shifted due to their velocities, but their intrinsic peak wavelengths (their temperatures) would be different. Because if a star is moving away (positive radial velocity), its light is redshifted, so the observed wavelength is longer. If a star is moving towards us (negative radial velocity), its light is blueshifted, observed wavelength is shorter.

But the problem states that the observed wavelengths are the same. So for Star_1 and Star_2, their observed λ_max are the same. Therefore, their intrinsic λ_max (their actual peak wavelengths) must be different. Because Star_1 is moving at 0 km/s, so its observed λ_max is the same as its intrinsic λ_max. Star_2 is moving at 700 km/s, but since velocity is positive (assumed to be moving away), its observed λ_max is longer than its intrinsic λ_max.

Wait, but the problem says the observed wavelengths are the same. So for Star_2, which is moving away (velocity positive), its observed λ_obs = λ_emit * (1 + v/c), where v is the radial velocity. So if Star_2 has v = 700 km/s, which is 700,000 m/s. The speed of light c is approximately 3e8 m/s. So v/c = 700,000 / 3e8 ≈ 0.002333. So the observed wavelength is λ_obs = λ_emit * (1 + 0.002333) ≈ 1.002333 * λ_emit.

But the problem states that the observed λ_max for both stars is the same. Therefore, Star_1's intrinsic λ_max (since it's at rest) must be λ_obs / 1.002333, and Star_2's intrinsic λ_max is λ_obs. Wait, no. Wait, for Star_1, since it's at rest, its observed λ_max is the same as its intrinsic λ_max. For Star_2, which is moving away, its observed λ_max is λ_emit * (1 + v/c) = λ_intrinsic * (1 + v/c). Since the observed λ_max for both is the same, then:

λ_obs = λ1_intrinsic = λ2_intrinsic * (1 + v2/c)

So, λ1_intrinsic = λ2_intrinsic * (1 + v2/c)

But from Wien's Law, λ_max * T = constant. So for each star, T = b / λ_max. Therefore, T1 = b / λ1_intrinsic, T2 = b / λ2_intrinsic.

From above, λ1_intrinsic = λ2_intrinsic * (1 + v2/c). Therefore, T1 = (b) / [λ2_intrinsic * (1 + v2/c)] = T2 / (1 + v2/c).

So T1 = T2 / (1 + v2/c). Therefore, T1/T2 = 1 / (1 + v2/c). Since v2 is 700 km/s, which is 7e5 m/s. c is 3e8 m/s. So v2/c = 7e5 / 3e8 = 7/(3e3) ≈ 0.002333. Therefore, T1/T2 ≈ 1 / 1.002333 ≈ 0.99767.

Wait, but this would mean that T1 is slightly less than T2. However, earlier reasoning suggested that if the observed wavelengths are the same, then the intrinsic temperature of Star_1 is lower, making its luminosity lower. But the problem states that L1 is greater than L2. Hmm, but according to the Stefan-Boltzmann law, if T1 is lower, then L1 would be (1.5)^2 * (0.99767)^4 ≈ 2.25 * (0.9908) ≈ 2.23. Which is close to option C (2.23). But let's check the exact calculation.

Alternatively, maybe I made a mistake in the Doppler shift reasoning. Let's re-examine.

The observed wavelength λ_obs is related to the emitted wavelength λ_emit by the Doppler effect. For a star moving away (positive radial velocity), the observed wavelength is longer: λ_obs = λ_emit * (1 + v/c). Therefore, if both stars have the same observed λ_max, then:

λ_obs = λ1_emit = λ2_emit * (1 + v2/c)

So λ1_emit = λ2_emit * (1 + v2/c)

But since λ_max is inversely proportional to temperature, T1 = T0 / λ1_emit, T2 = T0 / λ2_emit, where T0 is Wien's constant. Therefore, T1 = T2 * (1 + v2/c). Because λ1_emit = λ2_emit * (1 + v2/c), so T1 = T2 / (1 + v2/c).

Wait, no. Let's correct this. If λ_max is inversely proportional to temperature, then T ∝ 1/λ_max. So if λ1_emit = λ2_emit * (1 + v2/c), then T1 = T2 / (1 + v2/c). Therefore, T1/T2 = 1 / (1 + v2/c).

Wait, but v2 is 700 km/s, which is 700,000 m/s. c is approximately 3e8 m/s. So v2/c = 700,000 / 3e8 ≈ 0.002333. So 1 + v2/c ≈ 1.002333. Therefore, T1/T2 = 1 / 1.002333 ≈ 0.99767. So T1 is about 0.99767 times T2. Therefore, the temperature ratio is approximately 0.99767.

Now, the luminosity ratio L1/L2 = (R1/R2)^2 * (T1/T2)^4. We have R1/R2 = 1.5, T1/T2 ≈ 0.99767. So:

(1.5)^2 = 2.25

(0.99767)^4 ≈ (0.99767)^2 * (0.99767)^2. Let's compute:

First, 0.99767^2 = approx 0.99535. Then squared again: 0.99535^2 ≈ 0.9907. So (0.99767)^4 ≈ 0.9907.

Therefore, L1/L2 ≈ 2.25 * 0.9907 ≈ 2.229. Which is approximately 2.23. So the answer would be option C (~2.23). But let's check if this is accurate.

Alternatively, maybe I should use exact values. Let's compute v2/c exactly. 700,000 m/s divided by 3e8 m/s is 700,000 / 300,000,000 = 7/(300) = 0.023333333... Wait, no. Wait, 700,000 is 7e5, and 3e8 is 3e8. So 7e5 / 3e8 = 7/(3e3) = 0.002333333... So 1 + v2/c = 1.002333333...

So T1/T2 = 1 / 1.002333333... Let's compute this exactly. 1 / 1.002333333 = approximately 0.997673. Let's compute 1.002333333... as a fraction. 0.002333333... is 7/(3000). So 1.002333333... = 3007/3000. Therefore, T1/T2 = 1 / (3007/3000) = 3000/3007 ≈ 0.997033. Wait, let me check:

Wait, 7/(3000) = 0.002333333...

So 1 + 7/(3000) = 3007/3000. Therefore, 1 / (3007/3000) = 3000/3007 ≈ 0.997033. So T1/T2 ≈ 0.997033.

Therefore, (T1/T2)^4 = (0.997033)^4. Let's compute this step by step.

First, compute 0.997033^2:

0.997033 * 0.997033 ≈ (1 - 0.002967)^2 ≈ 1 - 2*0.002967 + (0.002967)^2 ≈ 1 - 0.005934 + 0.0000088 ≈ 0.9940748.

Then, square that result to get the fourth power:

0.9940748^2 ≈ (1 - 0.0059252)^2 ≈ 1 - 2*0.0059252 + (0.0059252)^2 ≈ 1 - 0.0118504 + 0.0000351 ≈ 0.9881847.

So (T1/T2)^4 ≈ 0.9881847.

Therefore, L1/L2 = (1.5)^2 * 0.9881847 ≈ 2.25 * 0.9881847 ≈ 2.2234. Which is approximately 2.223, so about 2.22. The closest option is C (2.23). However, the options given are A. ~2.32, B. ~2.25, C. ~2.23, D. ~2.35. So 2.223 is closest to 2.23, which is option C.

But wait, maybe I made a mistake in the calculation. Let me verify the exact value of (3000/3007)^4.

Alternatively, maybe using logarithms would be more precise. Let's compute ln(3000/3007) = ln(1 - 7/3000) ≈ -7/3000 - (7/3000)^2/2 - ... using the Taylor series for ln(1-x) ≈ -x - x²/2 - x³/3 - ... for small x.

Here, x = 7/3000 ≈ 0.0023333.

So ln(3000/3007) = ln(1 - 0.0023333) ≈ -0.0023333 - (0.0023333)^2 / 2 ≈ -0.0023333 - 0.000002694 ≈ -0.00233599.

Then, ln((3000/3007)^4) = 4 * ln(3000/3007) ≈ 4 * (-0.00233599) ≈ -0.00934396.

Exponentiating this gives e^(-0.00934396) ≈ 1 - 0.00934396 + (0.00934396)^2/2 ≈ 1 - 0.009344 + 0.000043 ≈ 0.990699. Wait, that's different from the previous calculation. Hmm, maybe my previous approximation was too rough.

Alternatively, using a calculator approach, but since I can't use a calculator, perhaps I need to check another way.

Alternatively, note that (3000/3007)^4 = [1 - 7/3000]^4 ≈ 1 - 4*(7/3000) + 6*(7/3000)^2 - 4*(7/3000)^3 + (7/3000)^4. Let's compute this:

First term: 1

Second term: -4*(7/3000) = -28/3000 ≈ -0.0093333

Third term: 6*(49/9,000,000) = 294/9,000,000 ≈ 0.0000326667

Fourth term: -4*(343/27,000,000,000) ≈ -1372/27,000,000,000 ≈ -0.0000000508

Fifth term: 2401/81,000,000,000,000 ≈ negligible.

Adding these up:

1 - 0.0093333 + 0.0000326667 - 0.0000000508 ≈ 0.9906993.

So approximately 0.9907. Which matches the earlier calculation. Therefore, (T1/T2)^4 ≈ 0.9907. Then L1/L2 = 2.25 * 0.9907 ≈ 2.229, which is approximately 2.23. So the answer should be C.

But wait, the problem states that the stars radiate as black bodies, so their luminosity is determined by their surface area and temperature. But the Doppler effect here affects the observed wavelength, which relates to their intrinsic temperature. So the key here is that the observed peak wavelength is the same, which implies that the intrinsic peak wavelengths are different due to their velocities. Therefore, the intrinsic temperatures are different, leading to a lower T1 than T2, hence lower luminosity than if they were at rest. However, the problem states that the luminosity of Star_1 is greater than Star_2. Wait, but according to our calculation, L1/L2 ≈ 2.23, which is greater. But according to our reasoning, T1 is less than T2, but R1 is larger. So the ratio of (1.5)^2 * (0.997)^4 ≈ 2.25 * 0.9907 ≈ 2.229. So L1 is greater. Therefore, the answer is C.

But let me check again. The problem states that the observed wavelengths are the same. So for Star_2, which is moving away, the observed λ is longer than the emitted λ. Therefore, the intrinsic λ_emit for Star_2 is λ_obs / (1 + v2/c). Since Star_1 is at rest, its intrinsic λ_emit is λ_obs. Therefore, λ1_intrinsic = λ_obs, λ2_intrinsic = λ_obs / (1 + v2/c). Therefore, T1 = b / λ1_intrinsic = b / λ_obs. T2 = b / λ2_intrinsic = b / [λ_obs / (1 + v2/c)] = b * (1 + v2/c) / λ_obs = T1 * (1 + v2/c). Therefore, T2 = T1 * (1 + 700/3e8) ≈ T1 * 1.002333. Therefore, T2 is slightly higher than T1. So T1/T2 ≈ 1 / 1.002333 ≈ 0.99767. Therefore, the temperature ratio is less than 1. So L1/L2 = (1.5)^2 * (0.99767)^4 ≈ 2.25 * 0.9907 ≈ 2.229. So again, L1 is about 2.23 times L2. Therefore, the answer is C.

But wait, the problem states that the mass of Star_1 is 1.5 times that of Star_2. Does mass affect luminosity here? Wait, the problem says they are interested in the luminosities of the stars, and they computed the luminosity using the radial velocities. Wait, radial velocities are used in determining the Doppler shift in the wavelengths, which affects the observed brightness. But the actual luminosity calculation is based on the intrinsic properties (radius and temperature). So the mass might not directly affect the luminosity calculation here, except through the Doppler shift. But the problem states that the stars radiate as black bodies, so their luminosity is determined by R and T. The radial velocities are used to determine the Doppler shift, which affects the observed wavelength, and hence the temperature. Therefore, the mass is given but not directly used in the luminosity calculation. So the mass is perhaps a distractor in the problem.

Therefore, the answer should be C (~2.23). However, the options given are A. ~2.32, B. ~2.25, C. ~2.23, D. ~2.35. So the correct answer is C.
</think>