Okay, so I need to figure out how many times longer the orbital period of planet #2 is compared to planet #1. Let me start by understanding the problem.

Both stars have the same mass, similar to the Sun. The planets detected are both similar to Neptune's mass, but that might not be directly relevant here. The key data points are the radial velocity (RV) shifts for each planet. Planet #1 has a shift of 5 milliarcseconds (mas), and planet #2 has a shift of 7 mas. Both have circular orbits, so Kepler's laws should apply here.

Radial velocity is the speed at which the star moves towards or away from us due to the planet's orbit. The RV method measures this velocity, and the shift in wavelength (Doppler shift) is proportional to the velocity of the star. The formula for the RV semi-amplitude (K) is given by:

K = ( (G * (M_p) ) / ( (M_star + M_p)^(1/3) ) )^(1/3) * (1 / (2π)) )^(1/3) * (1 / sqrt(1 - e²)) ) * (1 / (P^(1/3)) )

But since both planets have circular orbits, e=0, so that term becomes 1. Also, the mass of the planet (M_p) is much smaller than the star's mass (M_star), so M_star + M_p ≈ M_star. Therefore, the formula simplifies to:

K ≈ (G * M_star)^(1/3) / (2π)^(1/3) * (M_p)^(1/3) / P^(1/3)

Wait, maybe I should look up the exact formula for K. Let me think. The RV semi-amplitude K is given by:

K = ( (2πG) / (P * c) )^(1/3) * (M_p * sin i) / ( (M_star + M_p)^(2/3) )

But since the orbital inclination i is not given, and if both planets are detected by the same method, perhaps we can assume that the inclination is the same, or maybe that the ratio doesn't depend on it. Alternatively, perhaps the formula simplifies when considering the same star and planet mass.

Wait, but the problem states that both stars have masses similar to the Sun, and the planets are similar to Neptune. But the RV shift is given in terms of wavelength shift, which is proportional to the velocity. The RV shift (Dλ/λ) is equal to v / c, where v is the velocity of the star, and c is the speed of light. So the ratio of the RV shifts would be the ratio of the velocities of the stars.

But the RV shift is given in milliarcseconds. Wait, no. Wait, the RV method measures the Doppler shift in wavelength. The formula for the Doppler shift (Δλ/λ) is equal to v/c, where v is the velocity of the star towards or away from us. So if planet #1 causes a shift of 5 mas, that's Δλ1/λ = v1/c = 5e-3 (since 1 mas is 1e-3 arcseconds, but wait, arcseconds are a unit of angle. Wait, maybe I'm confusing units here. Let me clarify.

The RV shift is measured as a change in wavelength (Δλ) relative to the rest wavelength (λ). The Doppler shift formula is:

Δλ/λ = v/c

But the problem states that the periodic wavelength shift is 5 milliarcseconds for planet #1 and 7 milliarcseconds for planet #2. Wait, but arcseconds are a unit of angle, not wavelength. Hmm, maybe there's a confusion here. Alternatively, perhaps the problem is using a different measure, like the semi-amplitude of the radial velocity curve, which is K. But the question mentions the periodic wavelength shift, which is the Doppler shift. So the shift in wavelength is proportional to the velocity.

But I'm getting a bit confused. Let me think again. The RV method measures the velocity of the star as a function of time. The semi-amplitude K of the RV curve is given by:

K = ( (G * (M_p) * sin i) / ( (M_star + M_p)^(2/3) ) )^(1/3) / (2π) )^(1/3) * (1 / sqrt(1 - e²)) )

But since the orbits are circular, e=0, so sqrt(1-e²)=1. Also, M_p << M_star, so M_star + M_p ≈ M_star. Therefore, the formula simplifies to:

K ≈ (G * M_star)^(1/3) * (M_p)^(1/3) / ( (2π)^(1/3) * P^(1/3) )

But K is in velocity units. However, the problem gives the RV shift as a periodic wavelength shift. So the shift in wavelength (Δλ) is related to the velocity (v) by Δλ/λ = v/c. Therefore, the RV semi-amplitude K is equal to Δλ/λ * c. Wait, but let me check the exact relationship.

The Doppler shift formula is Δλ = λ * v/c, so Δλ/λ = v/c. Therefore, the semi-amplitude K in velocity terms is K_v = (Δλ/λ) * c. So if we have the RV semi-amplitude K_v, then the wavelength shift is K_v / c. But in the problem, the shifts are given as 5 mas and 7 mas. Wait, but mas is milliarcseconds, which is a unit of angle. How does that relate to velocity?

Hmm, perhaps the problem is using a different approach. Let me think. The radial velocity semi-amplitude K is related to the orbital period P and the mass of the planet. The formula connecting K, the period P, the star's mass M, and the planet's mass m is:

K = ( (G * M) / (4π²) )^(1/3) * (m / (M + m))^(1/3) * (1 / P)^(1/3) )

But again, since m << M, this simplifies to:

K ≈ ( (G * M) / (4π²) )^(1/3) * (1 / P)^(1/3) )

But K is in velocity. However, the problem gives the RV shifts as wavelength shifts. So perhaps we need to relate the wavelength shift to K. Since Δλ/λ = v/c, and v is the velocity of the star, which is K. Therefore, Δλ/λ = K / c. Therefore, the semi-amplitude K is equal to (Δλ/λ) * c. So if the RV shift is given as 5 mas and 7 mas, then Δλ1/λ = 5e-3 (if mas is converted to decimal, but wait, no. Wait, arcseconds are units of angle. The problem says "periodic wavelength shift of a spectral line at a given wavelength". So perhaps the shift is given as a fractional change in wavelength, which is Δλ/λ = v/c. Therefore, if the RV semi-amplitude is K = v = Δλ/λ * c. So K (in velocity) is (Δλ/λ) * c. Therefore, the RV semi-amplitude K1 for planet #1 is (5e-3) * c, and K2 for planet #2 is (7e-3) * c. Wait, but 5 mas is 5e-3 arcseconds. Wait, but arcseconds are a unit of angle, not wavelength. Wait, perhaps the problem is using a different unit. Alternatively, maybe the problem is giving the semi-amplitude in terms of the wavelength shift, which is proportional to the velocity. So perhaps the ratio of the RV semi-amplitudes K1 and K2 is equal to the ratio of the velocity amplitudes. But how does that relate to the orbital periods?

But maybe I'm overcomplicating. Let's think in terms of Kepler's third law. The orbital period P is related to the semi-major axis a, the mass of the star M, and the mass of the planet m. The formula is:

P² = (4π²/G(M + m)) * a³

Again, since m << M, this simplifies to:

P² ≈ (4π²/(G M)) * a³

But the semi-amplitude of the RV K is given by:

K = (2π G)^(1/3) * (M_p sin i) / ( (M_star + M_p)^(2/3) * (1 - e²)^(1/2) )^(1/3) ) * (1/P)^(1/3)

Wait, maybe another approach. Let's recall that for a circular orbit, the RV semi-amplitude K is related to the orbital period P and the mass of the planet. The formula is:

K = ( (G * M_p) / (4π²) )^(1/3) * (1 / P)^(1/3) / ( (M_star + M_p)^(2/3) )

But since M_p << M_star, we can approximate M_star + M_p ≈ M_star. Therefore:

K ≈ (G * M_p / (4π²))^(1/3) * (1 / P)^(1/3) / (M_star^(2/3)) )

But wait, this seems a bit messy. Let me check the exact formula. The correct formula for the RV semi-amplitude is:

K = ( (2π G) / (P c³) )^(1/3) * (M_p sin i) / ( (M_star + M_p)^(2/3) ) )

But since M_p << M_star, this simplifies to:

K ≈ ( (2π G) / (P c³) )^(1/3) * (M_p / M_star^(2/3) )

But I'm getting a bit stuck here. Maybe a better approach is to use the relation between the RV semi-amplitude K and the orbital period P. Let's recall that for a star of mass M and a planet of mass m in a circular orbit, the RV semi-amplitude K is given by:

K = ( (G M) / (4π²) )^(1/3) * (m / (M + m))^(1/3) * (1 / P)^(1/3) )

But again, since m << M, this becomes:

K ≈ ( (G M) / (4π²) )^(1/3) * (1 / P)^(1/3) / (M^(2/3)) )

Wait, that simplifies to:

K ≈ (G / (4π²))^(1/3) * (1 / P)^(1/3) * (1 / M^(1/3) )

But that would give K proportional to (1/P)^(1/3). Therefore, K ∝ (1/P)^(1/3), so P ∝ (1/K)^3.

But the problem gives the RV shifts as 5 mas and 7 mas. Wait, but these are in terms of wavelength shift, not velocity. So perhaps we need to relate the wavelength shift to the velocity. The RV shift is Δλ/λ = v/c. Therefore, the velocity semi-amplitude K is K = (Δλ/λ) * c. So if the RV shift is 5 mas (which is 5e-3 arcseconds), but wait, arcseconds are units of angle. Wait, maybe the problem is using a different unit here. Alternatively, perhaps the RV shift is given as the fractional change in wavelength, which is v/c. So if the shift is 5 mas, that would be 5e-3 in terms of a fraction (since 1 mas is 1e-3 of an arcsecond, but arcseconds are a unit of angle, not wavelength). Wait, this is getting confusing. Let me clarify.

Wait, the problem says "periodic wavelength shift of a spectral line at a given wavelength". So the shift is a change in wavelength, not a fractional change. So Δλ = 5 mas (milliarcseconds), and similarly for 7 mas. But arcseconds are a unit of angle, not wavelength. So perhaps the problem is using a different measure, or perhaps it's a typo. Alternatively, maybe the problem is using a different unit, like micrometers or something, but that's not specified. Alternatively, perhaps the problem is using the fact that the RV shift (Δλ/λ) is proportional to the velocity of the star. So if the RV semi-amplitude is K, then Δλ/λ = K/c. Therefore, K = (Δλ/λ) * c. So if the RV shift is given as 5 mas and 7 mas, then the velocities are K1 = 5e-3 * c and K2 = 7e-3 * c. But this seems like a big assumption, and the problem might be using a different approach.

Alternatively, perhaps the problem is simplifying things by considering that the RV shift (Δλ) is proportional to the velocity, so the ratio of the RV shifts is the ratio of the velocities. Therefore, v1/v2 = Δλ1/Δλ2 = 5/7. But wait, the problem states that the shifts are 5 mas and 7 mas. If we take these as velocity shifts, then v1 = 5, v2 =7. But that would make the ratio v1/v2 =5/7. But then how does this relate to the orbital periods?

Wait, but Kepler's third law says that P² ∝ a³/(M), and for the same M, P² ∝ a³. But also, the RV semi-amplitude K is related to the orbital parameters. Let's recall that K is given by:

K = (2π G)^(1/3) * (M_p sin i) / ( (M_star + M_p)^(2/3) * (1 - e²)^(1/2) ) * (1/P)^(1/3)

But since M_p << M_star, and e=0, this simplifies to:

K ≈ (2π G / (P c³))^(1/3) * (M_p / M_star^(2/3)) )

But I'm not sure if that's the right formula. Let me check another source. The correct formula for the RV semi-amplitude is:

K = ( (G * M_p) / (4π² * (M_star + M_p)) )^(1/3) * (1 / P)^(1/3) * (1 / sqrt(1 - e²)) )

Again, since M_p << M_star and e=0, this becomes:

K ≈ (G / (4π²))^(1/3) * (M_p)^(1/3) * (1 / P)^(1/3) / (M_star^(2/3))

But this seems complicated. Maybe a better approach is to use the relation between K and the orbital period. Let's assume that the stars have the same mass, M, and the planets have the same mass, m. Then, the RV semi-amplitude for each planet would be:

K1 = ( (G M) / (4π² P1³) )^(1/3) * m^(1/3) * (1 / (1 + m/M)^(2/3) )

But since m << M, (1 + m/M)^(2/3) ≈ 1, so:

K1 ≈ ( (G M) / (4π² P1³) )^(1/3) * m^(1/3)

Similarly for K2:

K2 ≈ ( (G M) / (4π² P2³) )^(1/3) * m^(1/3)

Therefore, the ratio K1/K2 would be:

(K1/K2) = [ (P2/P1) ] / 1 = (P2/P1)^(1/3)

Wait, because:

K1/K2 = [ (G M / (4π² P1³) )^(1/3) * m^(1/3) ] / [ (G M / (4π² P2³) )^(1/3) * m^(1/3) ) ]

The G, M, and m terms cancel out, leaving (P2³/P1³)^(1/3) = P2/P1.

Wait, so K1/K2 = P2/P1.

Therefore, if K1/K2 = 5/7, then P2/P1 = 5/7. Wait, but that would mean planet 2's period is (5/7) times that of planet 1, which would make planet 2's period shorter. But the problem states that both planets have circular orbits, and we need to find how many times longer planet 2's period is compared to planet 1. If K1/K2 = 5/7, then P2 = (5/7) P1, which is shorter. But the question asks for how many times longer is planet 2's period. So if P2 = (5/7) P1, then planet 2's period is shorter, so the answer would be 5/7 ≈ 0.714, which isn't one of the options. Wait, but maybe I made a mistake in the ratio.

Wait, let's check again. If K1 is the RV shift for planet 1, and K2 for planet 2. If the RV shift is proportional to the velocity, and velocity is higher for a closer planet (since for circular orbits, v = sqrt(GM/a)), so closer planets orbit faster. But the problem states that the wavelength shifts are 5 mas and 7 mas. If the shift is larger, that suggests a higher velocity. So if planet 1 has a shift of 5 mas and planet 2 has 7 mas, then K1/K2 = 5/7. Then, according to the earlier relation, K1/K2 = P2/P1. Therefore, P2 = (5/7) P1. So planet 2's period is shorter. But the question asks how many times longer is planet 2's period than planet 1's. If P2 = (5/7) P1, then planet 2's period is shorter by a factor of 5/7. So the answer would be 5/7 ≈ 0.714. But the options are ~1.40, ~1.96, ~0.85, ~0.36. So 0.85 is close to 0.714, but not exact. Hmm.

Wait, but maybe I messed up the ratio. Let me re-examine. If K1/K2 = v1/v2, and from the RV formula, K is proportional to (M_p / (M_star + M_p)^(2/3)) * (1/P)^(1/3). Since M_p << M_star, this simplifies to K ∝ (1/P)^(1/3). Therefore, K1/K2 = (P2/P1)^(1/3). Therefore, (K1/K2)^3 = P2/P1. Therefore, if K1/K2 = 5/7, then (5/7)^3 = P2/P1. (5/7)^3 = 125/343 ≈ 0.364. So P2/P1 ≈ 0.364, meaning planet 2's period is about 0.36 times that of planet 1, so planet 2's period is shorter by a factor of ~0.36. Which would correspond to option D: ~0.36. But the question is asking how many times longer is planet 2's period than planet 1's. If planet 2's period is shorter, then it's 0.36 times as long, so 1 / 0.36 ≈ 2.78 times shorter. But the question is phrased as "how many times is the orbital period of planet #2 longer than that of planet #1?" So if planet 2's period is shorter, the answer would be less than 1, which matches option D. But the options are:

A. ~1.40

B. ~1.96

C. ~0.85

D. ~0.36

So according to this, the answer would be D. But let me check again.

Wait, perhaps I made a mistake in the relation between K and P. Let's go back to the formula. The RV semi-amplitude K is given by:

K = (2π G / (P c³))^(1/3) * (M_p sin i) / (M_star^(2/3)) )

Assuming edge-on orbit (sin i = 1) and M_p << M_star. Then, K ∝ (1/P)^(1/3). Therefore, K1/K2 = (P2/P1)^(1/3). Therefore, (K1/K2)^3 = P2/P1.

Given K1 = 5 mas, K2 =7 mas. Therefore, K1/K2 =5/7. Therefore, (5/7)^3 = P2/P1. (5/7)^3 = 125/343 ≈ 0.364. So P2/P1 ≈ 0.364. Therefore, planet 2's period is approximately 0.36 times that of planet 1. So the period of planet 2 is shorter by a factor of 0.36. Therefore, the question asks how many times longer is planet 2's period than planet 1's. Since it's shorter, the answer would be 0.36 times, which is option D.

But wait, the problem states "how many times is the orbital period of planet #2 longer than that of planet #1?" If planet 2's period is 0.36 times planet 1's, then it's shorter, so the answer is 0.36 times as long. Therefore, the answer is D. But let me check the options again. The options are:

A. ~1.40

B. ~1.96

C. ~0.85

D. ~0.36

So D is ~0.36, which matches the calculation. But wait, in the problem statement, the RV shifts are 5 mas and 7 mas. But if these are velocity shifts, then K1/K2 =5/7. Then (5/7)^3 =125/343≈0.364≈0.36. So the answer is D. But the options include 0.85, which is approximately (sqrt(5/7)) or something else. Wait, maybe I made a mistake in the cube. Let me check again.

If K ∝ (1/P)^(1/3), then K1/K2 = (P2/P1)^(1/3). Therefore, (K1/K2)^3 = P2/P1. So if K1=5, K2=7, then (5/7)^3 = (125)/(343) ≈0.364. So yes, P2/P1≈0.364. So planet 2's period is ~0.36 times that of planet 1. Therefore, the answer is D.

But wait, another approach: perhaps the problem is considering the wavelength shift as proportional to the velocity, but the formula for the RV shift (Δλ/λ) is equal to v/c. Therefore, the RV shift in terms of velocity is K = (Δλ/λ) * c. So K1 =5e-3 * c, K2=7e-3 *c. Then, since K ∝ (1/P)^(1/3), then K1/K2 = (P2/P1)^(1/3). Therefore, (5/7) = (P2/P1)^(1/3). Therefore, (P2/P1) = (5/7)^3 ≈0.364. So same result. Therefore, planet 2's period is ~0.36 times planet 1's, so the answer is D.

But wait, the problem states that the RV method detected both planets. RV method measures the Doppler shift in the star's spectrum. The period of the star's Doppler shift corresponds to the planet's orbital period. So the period measured from the RV is the same as the planet's orbital period. Therefore, the periods are P1 and P2 for the two planets. Therefore, the ratio P2/P1 is as calculated. So the answer is D. But let me check if there's any other factor I might have missed.

Wait, but the problem states that the stars have the same mass, but the planets have similar Neptune masses. But since the RV semi-amplitude K depends on the planet's mass, but in the formula, K is proportional to (M_p)^(1/3). However, the problem states that the planets have similar masses, so M_p1 ≈ M_p2. Therefore, the ratio K1/K2 would be determined by the inverse cube root of the period ratio. Therefore, the previous calculation holds. Therefore, the answer is D. ~0.36 times.

Wait, but the options have ~0.85 as C. Maybe I made a mistake in the cube. Let me check again. Wait, if K1/K2 =5/7, then (5/7)^3 =125/343≈0.364. But maybe the problem is considering the ratio the other way. If K1 is for planet 1, which has a shift of 5 mas, and K2 is for planet 2, shift of7 mas. So K1=5, K2=7. Then K1/K2=5/7. Therefore, (5/7)^3=0.364. So P2/P1=0.364. Therefore, planet 2's period is 0.36 times that of planet 1. So the answer is D. ~0.36.

But the options don't have ~0.36 as the answer, but the problem says "how many times is the orbital period of planet #2 longer than that of planet #1?" So if planet 2's period is 0.36 times that of planet 1, then it's shorter. So the answer is 0.36 times as long. Therefore, the correct answer is D. ~0.36.
</think>