Okay, let's tackle this problem step by step. So, we have two binary star systems, system_1 and system_2. Both exhibit eclipses with different periods: system_1 has a period of 2 years, and system_2 has a period of 1 year. The question is asking by what factor system_1 is more massive than system_2, considering the mass is the sum of the two stars in each system.

First, I need to recall the basics of binary star systems and how their orbital periods relate to their masses. I remember that in a binary system, the two stars orbit a common center of mass. The period of their orbit (the time between eclipses) is determined by their combined mass and the distance between them. The more massive the system, the longer the orbital period, assuming the separation is the same. But wait, the separation might not be the same for both systems. Hmm, but the problem doesn't mention any differences in separation, so maybe we can assume they have the same orbital separation? Or perhaps the periods are the key here since they are given.

Wait, no, the periods are given as the time between two consecutive primary eclipses. For a binary system, the orbital period is the time it takes for both stars to orbit their common center of mass once. So the period here is the orbital period. So system_1 has an orbital period of 2 years, and system_2 has 1 year. But the masses of the stars would affect the orbital period through Kepler's third law. Let me recall that Kepler's third law states that the square of the orbital period is proportional to the cube of the semi-major axis divided by the sum of the masses. The formula is P² = (4π²/G(M₁ + M₂)) * a³. Since the semi-major axis (a) is the same for both systems? Wait, no, the problem doesn't say that. It just says the periods are 2 and 1 years. So if the periods are different, and assuming the same separation (which isn't stated), the masses would affect the period. But since the periods are different, maybe the separation is different as well. Hmm, this is getting complicated.

Wait, but the problem is asking about the mass ratio. The question is about the sum of the masses. So maybe there's another approach. Let me think. The problem also mentions radial velocity (RV) variations. The amplitudes of the RV sinusoidal variations are given for each system. For each system, the two stars have different amplitudes. System_1 has 10 km/s and 5 km/s, system_2 has 15 and 10 km/s.

I remember that in binary systems, the radial velocity semi-amplitude (K) is related to the mass function. The formula for the mass function is (M₂³ sin³ i) / ( (M₁ + M₂)² ), where i is the inclination angle. But since the system is eclipsing, the inclination is close to 90 degrees, so sin i ≈ 1. For an eclipsing binary, the inclination is likely close to 90 degrees, so we can approximate sin i as 1. Therefore, the mass function simplifies to (M₂³) / ( (M₁ + M₂)² ).

But wait, the RV semi-amplitude K is related to the mass of the companion. The formula for K is (2π G / c³) * (M₂ sin i) / ( (1 + M₂/M₁)² + (M₂/M₁) )^(1/2) ). Wait, maybe that's more complicated. Alternatively, in the case of a circular orbit (which is likely), the RV semi-amplitude K is given by K = (2π G / (P c)) * (M₂ sin i) / (M₁ + M₂)^(1/3) ), but I might be mixing up different forms.

Wait, maybe a better approach is to use the fact that the RV semi-amplitude K is inversely proportional to the sum of the masses. Let me think again. The RV amplitude depends on the mass of the companion. For each star in the binary, the RV semi-amplitude is proportional to the mass of the other star divided by the total mass. So K₁ (for star 1) is proportional to M₂ / (M₁ + M₂), and K₂ (for star 2) is proportional to M₁ / (M₁ + M₂). Since the total mass M = M₁ + M₂, then K₁ = (M₂ / M) * (something), and K₂ = (M₁ / M) * (something). But the "something" here relates to the orbital period and the separation. However, since both systems have different periods, the "something" would differ. Therefore, perhaps the ratio of the RV amplitudes can give us the mass ratio.

Wait, let's formalize this. Let's denote for system 1:

K₁₁ = 10 km/s (for star 1)
K₁₂ = 5 km/s (for star 2)

Similarly, for system 2:

K₂₁ = 15 km/s (for star 1)
K₂₂ = 10 km/s (for star 2)

In a binary system, the ratio of the RV amplitudes is K₁/K₂ = M₂/M₁. Because K₁ is proportional to M₂/(M₁ + M₂) and K₂ is proportional to M₁/(M₁ + M₂). So K₁/K₂ = M₂/M₁.

So for each system, we can calculate the mass ratio M₂/M₁ as the ratio of the smaller RV amplitude to the larger one. Wait, but wait. Let's check. Let's say K₁ is the RV amplitude for star 1, and K₂ for star 2. Then K₁/K₂ = (M₂ sin i)/(M₁ sin i) * (M₁ + M₂)/ (something). Wait, maybe I need to rederive this.

The formula for the RV semi-amplitude is K = (2π G / (P c)) * (M₂ sin i) / ( (M₁ + M₂) )^(1/3) * (1 / sqrt(1 - e²)) ), but assuming a circular orbit (e=0), this simplifies. However, another approach is to use the relation between the masses and the velocities. The center of mass is at a point where M₁ a₁ = M₂ a₂, where a₁ and a₂ are the semi-major axes of the two stars. Since the total separation is a = a₁ + a₂, then a₁ = (M₂/(M₁ + M₂)) a, and a₂ = (M₁/(M₁ + M₂)) a.

The RV semi-amplitude K is given by K = (2π G / (P c)) * (M₂ sin i) / (M₁ + M₂)^(1/3) * (1 / sqrt(1 - e²)) ). For circular orbits, sqrt(1 - e²) = 1. So K is proportional to (M₂ sin i) / (M₁ + M₂)^(1/3) * (1/P) ). Wait, maybe it's better to use the formula in terms of the mass ratio.

Alternatively, another approach is to note that the ratio of the RV amplitudes K₁/K₂ = M₂/M₁. Because K₁ = (M₂ / (M₁ + M₂)) * (something), and K₂ = (M₁ / (M₁ + M₂)) * (something). Therefore, K₁/K₂ = M₂/M₁. So the mass ratio M₂/M₁ is K₁/K₂.

Wait, let's test this with an example. Suppose M₁ = 1, M₂ = 1. Then K₁ = K₂, so ratio is 1. If M₂ is larger, say M₂ = 2, then K₂ would be larger than K₁. Wait, no. Wait, K₁ is the RV amplitude for star 1, which is (M₂ sin i) / (M₁ + M₂) multiplied by some other factors. Let me think again.

Alternatively, the RV semi-amplitude for each star is given by K₁ = (2π G / (P c)) * (M₂ sin i) / ( (M₁ + M₂) )^(1/3) * (1 / sqrt(1 - e²)) ), but perhaps it's more straightforward to consider that the sum of the velocities is constant. Wait, no. The two stars orbit the center of mass with velocities inversely proportional to their masses. So if M₁ is larger, then a₁ is smaller, so the velocity of star 1 would be smaller. Therefore, the RV semi-amplitude K₁ (for star 1) is proportional to M₂/(M₁ + M₂), and K₂ (for star 2) is proportional to M₁/(M₁ + M₂). Therefore, K₁/K₂ = M₂/M₁. So the mass ratio M₂/M₁ is equal to the ratio of the RV amplitudes of the two stars.

Therefore, for system_1, the two RV amplitudes are 10 km/s and 5 km/s. Let's assume K₁ (for star 1) is 10 km/s and K₂ (for star 2) is 5 km/s. Then M₂/M₁ = K₁/K₂ = 10/5 = 2. So the mass ratio M₂/M₁ is 2. Therefore, M₁ + M₂ = M₁ (1 + 2) = 3 M₁. So the total mass is 3 M₁.

Similarly, for system_2, the RV amplitudes are 15 km/s and 10 km/s. Let's say K₁ (for star 1) is 15 km/s and K₂ (for star 2) is 10 km/s. Then M₂/M₁ = 15/10 = 1.5. Therefore, M₂ = 1.5 M₁ for system_2. The total mass is M₁ + M₂ = M₁ (1 + 1.5) = 2.5 M₁.

Now, we need to find the factor by which the total mass of system_1 is more than that of system_2. So (M₁_system1 + M₂_system1) / (M₁_system2 + M₂_system2) = (3 M₁_system1) / (2.5 M₁_system2). But we need to relate the M₁'s of the two systems. Wait, but we don't have any information about the actual velocities or the periods in terms of the masses. Wait, the periods are given as 2 years and 1 year. Kepler's third law states that P² = (4π²/G(M₁ + M₂)) * a³. But since the separation a is different for each system, unless they have the same separation, the periods would be different. However, since the periods are given, and the periods are inversely related to the cube root of the total mass (if a is the same), but if a is different, then the relationship is more complex.

Wait, but the problem states that the periods are the time between two consecutive primary eclipses. For a circular orbit, the period is the same as the orbital period. Therefore, the period is given as 2 years for system_1 and 1 year for system_2. Kepler's third law is P² = (4π²/G(M₁ + M₂)) * a³. But since a is different for each system, we can't directly compare the masses based on the periods alone. However, if we can relate the separations of the two systems, perhaps through other information.

But we also have the RV amplitudes. The RV amplitude depends on the mass of the companion and the orbital period. The formula for the RV semi-amplitude is K = (2π G / (P c)) * (M₂ sin i) / (M₁ + M₂)^(1/3) * (1 / sqrt(1 - e²)). Since the orbits are circular, e=0, so sqrt(1 - e²)=1. Therefore, K = (2π G / (P c)) * (M₂ sin i) / (M₁ + M₂)^(1/3). But since sin i is approximately 1 for an eclipsing system, we can ignore that factor. Therefore, K ∝ (M₂) / (M_total)^(1/3) * (1/P).

But we have two systems with different P and different K. Let's write this formula for each system.

For system_1:
K₁ = (2π G / (P₁ c)) * (M₂₁) / (M_total₁)^(1/3)
Similarly, for system_2:
K₂ = (2π G / (P₂ c)) * (M₂₂) / (M_total₂)^(1/3)

But since we have the ratios of K, P, and M_total, maybe we can take the ratio of the two equations to eliminate some variables.

Let's take system_1 and system_2:

K₁ / K₂ = [ (2π G / (P₁ c)) * (M₂₁) / (M_total₁)^(1/3) ] / [ (2π G / (P₂ c)) * (M₂₂) / (M_total₂)^(1/3) ) ]

Simplifying, the 2π G / c terms cancel out, so:

K₁ / K₂ = (P₂ / P₁) * (M₂₁ / M₂₂) * (M_total₂ / M_total₁)^(1/3)

We can plug in the known values:

K₁₁ = 10 km/s, K₁₂ = 5 km/s. Wait, but for each system, we need to use the total K for each star. Wait, no. Wait, K₁ in the formula above refers to the RV semi-amplitude of the entire system. Wait, no. Wait, I might be confusing the RV semi-amplitudes of the individual stars with the system's total. Let me clarify.

Wait, the formula K = (2π G / (P c)) * (M₂ sin i) / (M₁ + M₂)^(1/3) * (1 / sqrt(1 - e²)) ) is for the semi-amplitude of the star's velocity. But in the problem, we are given the RV semi-amplitudes for both stars in each system. So for system_1, the two RV semi-amplitudes are 10 and 5 km/s. Similarly for system_2, 15 and 10 km/s.

But according to the earlier reasoning, the ratio of the RV semi-amplitudes for the two stars in a system is equal to the mass ratio of the two stars. So for system_1, M₂₁ / M₁₁ = 10/5 = 2. For system_2, M₂₂ / M₁₂ = 15/10 = 1.5.

So we have M₂₁ = 2 M₁₁ and M₂₂ = 1.5 M₁₂.

The total mass of each system is M_total₁ = M₁₁ + M₂₁ = M₁₁ + 2 M₁₁ = 3 M₁₁.

Similarly, M_total₂ = M₁₂ + M₂₂ = M₁₂ + 1.5 M₁₂ = 2.5 M₁₂.

Now, we need to relate M₁₁ and M₁₂. To do this, let's use the RV amplitudes and the periods.

For system_1:

The RV semi-amplitude for star 1 is K₁₁ = 10 km/s = (2π G / (P₁ c)) * (M₂₁ sin i) / (M_total₁)^(1/3)

Similarly, for star 2, K₁₂ = 5 km/s = (2π G / (P₁ c)) * (M₁₁ sin i) / (M_total₁)^(1/3)

But since sin i ≈ 1, we can write:

10 = (2π G / (2 c)) * (M₂₁) / (3 M₁₁)^(1/3)

5 = (2π G / (2 c)) * (M₁₁) / (3 M₁₁)^(1/3)

But wait, let's compute the ratio of K₁₁ and K₁₂ for system_1.

K₁₁ / K₁₂ = [ (2π G / (2 c)) * (M₂₁) / (3 M₁₁)^(1/3) ) ] / [ (2π G / (2 c)) * (M₁₁) / (3 M₁₁)^(1/3) ) ] = M₂₁ / M₁₁ = 2, which matches the earlier ratio. So that checks out.

But perhaps using the actual values and the periods can help us relate the masses. Let's try that.

First, for system_1:

K₁₁ = 10 km/s = (2π G / (P₁ c)) * (M₂₁) / (M_total₁)^(1/3)

We have P₁ = 2 years.

Similarly, for system_2:

K₂₁ = 15 km/s = (2π G / (P₂ c)) * (M₂₂) / (M_total₂)^(1/3)

P₂ = 1 year.

But we can write two equations for each system and then take their ratio.

For system_1:

10 = (2π G / (2 c)) * (2 M₁₁) / (3 M₁₁)^(1/3)

Similarly, for system_2:

15 = (2π G / (1 c)) * (1.5 M₁₂) / (2.5 M₁₂)^(1/3)

Let me simplify these equations.

Starting with system_1:

10 = (2π G / (2 c)) * (2 M₁₁) / (3 M₁₁)^(1/3)

Simplify numerator: 2 M₁₁ cancels with denominator:

= (2π G / (2 c)) * 2 M₁₁ / (3 M₁₁)^(1/3)

= (2π G / (2 c)) * 2 / (3)^(1/3) * M₁₁^(1 - 1/3)

= (2π G / (2 c)) * 2 / 3^(1/3) * M₁₁^(2/3)

Simplify constants:

= (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3)

Similarly, for system_2:

15 = (2π G / (1 c)) * (1.5 M₁₂) / (2.5 M₁₂)^(1/3)

Simplify numerator: 1.5 M₁₂ / (2.5 M₁₂)^(1/3) = (1.5 / 2.5) * M₁₂^(1 - 1/3) = (3/5) * M₁₂^(2/3)

So:

15 = (2π G / c) * (3/5) * M₁₂^(2/3)

Now, let's write both equations:

For system_1:

10 = (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3)   ...(1)

For system_2:

15 = (2π G / c) * (3/5) * M₁₂^(2/3)      ...(2)

Let me divide equation (1) by equation (2) to eliminate some variables.

(10) / (15) = [ (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3) ] / [ (2π G / c) * (3/5) * M₁₂^(2/3) ) ]

Simplify:

2/3 = [ (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3) ] / [ (2π G / c) * (3/5) * M₁₂^(2/3) ) ]

Cancel out π G / c:

2/3 = [ 2 / 3^(1/3) * M₁₁^(2/3) ] / [ 2 * (3/5) * M₁₂^(2/3) ]

Simplify numerator and denominator:

Numerator: 2 / 3^(1/3) * M₁₁^(2/3)

Denominator: 2 * (3/5) * M₁₂^(2/3) = (6/5) * M₁₂^(2/3)

So:

2/3 = [ (2 / 3^(1/3)) * M₁₁^(2/3) ) ] / [ (6/5) * M₁₂^(2/3) ) ]

Multiply both sides by denominator:

2/3 * (6/5) * M₁₂^(2/3) = (2 / 3^(1/3)) * M₁₁^(2/3)

Simplify left side:

(2/3)*(6/5) = (12/15) = 4/5

So:

4/5 * M₁₂^(2/3) = (2 / 3^(1/3)) * M₁₁^(2/3)

Divide both sides by 2:

2/5 * M₁₂^(2/3) = (1 / 3^(1/3)) * M₁₁^(2/3)

Take both sides to the power of 3/2 to solve for M₁₁ and M₁₂.

[ (2/5) * M₁₂^(2/3) ]^(3/2) = [ (1 / 3^(1/3)) * M₁₁^(2/3) ]^(3/2)

Simplify exponents:

(2/5)^(3/2) * M₁₂^( (2/3)*(3/2) ) = (1 / 3^(1/3))^(3/2) * M₁₁^( (2/3)*(3/2) )

Which simplifies to:

(2/5)^(3/2) * M₁₂ = (1 / 3^(1/2)) * M₁₁

Because (2/5)^(3/2) = (2^3)/(5^3)^(1/2) = 8/(5√5)

Similarly, (1 / 3^(1/3))^(3/2) = 1 / 3^(1/2) = 1/√3

So:

(8)/(5√5) * M₁₂ = (1/√3) * M₁₁

Multiply both sides by 5√5:

8 M₁₂ = (5√5 / √3) M₁₁

So:

M₁₁ = (8 * √3) / (5√5) * M₁₂

Simplify √3 / √5 = √(3/5), so:

M₁₁ = (8 / 5) * √(3/5) * M₁₂

Compute √(3/5) ≈ 0.7746

So:

M₁₁ ≈ (8/5) * 0.7746 * M₁₂ ≈ 1.6 * 0.7746 * M₁₂ ≈ 1.239 * M₁₂

So M₁₁ ≈ 1.239 M₁₂

Now, the total mass for each system:

M_total₁ = 3 M₁₁ ≈ 3 * 1.239 M₁₂ ≈ 3.717 M₁₂

M_total₂ = 2.5 M₁₂

So the ratio M_total₁ / M_total₂ ≈ 3.717 / 2.5 ≈ 1.487

Which is approximately 1.49. Looking at the options, the closest is D. ~0.7, but wait, 1.49 is close to 1.5. Wait, the options given are A. ~0.4, B. ~1.2, C. ~0.6, D. ~0.7. Wait, 1.49 is not among the options. Did I make a mistake here?

Wait, let's check the calculations again. Let's go back step by step.

We had:

From system_1:

10 = (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3)   ...(1)

From system_2:

15 = (2π G / c) * (3/5) * M₁₂^(2/3)      ...(2)

Dividing (1) by (2):

10/15 = [ (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3) ] / [ (2π G / c) * (3/5) * M₁₂^(2/3) ]

Simplify:

2/3 = [ (2 / 3^(1/3)) * M₁₁^(2/3) ) ] / [ (6/5) * M₁₂^(2/3) ) ]

Wait, denominator is (2 * (3/5)) = 6/5?

Wait, 2π G / c divided by (2π G / c) is 1, but there's a 2 in the numerator and 1 in the denominator. Wait, let's re-express the denominators correctly.

Wait, system_1 equation:

10 = (2π G / (2 c)) * (2 M₁₁) / (3 M₁₁)^(1/3)

Which is (π G / c) * (2 M₁₁) / (3^(1/3) M₁₁^(1/3))

= (π G / c) * 2 / 3^(1/3) * M₁₁^(1 - 1/3) = (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3)

System_2 equation:

15 = (2π G / (1 c)) * (1.5 M₁₂) / (2.5 M₁₂)^(1/3)

= (2π G / c) * (3/5 M₁₂) / ( (5/2) M₁₂ )^(1/3 )

= (2π G / c) * (3/5) * M₁₂ / ( (5/2)^(1/3) M₁₂^(1/3) )

= (2π G / c) * (3/5) / (5/2)^(1/3) * M₁₂^(1 - 1/3) 

= (2π G / c) * (3/5) / (5/2)^(1/3) * M₁₂^(2/3)

So when we take the ratio (1)/(2):

10 / 15 = [ (π G / c) * 2 / 3^(1/3) * M₁₁^(2/3) ] / [ (2π G / c) * (3/5) / (5/2)^(1/3) * M₁₂^(2/3) ]

Simplify:

2/3 = [ (2 / 3^(1/3)) / ( (6/5) / (5/2)^(1/3) ) ) ] * (M₁₁^(2/3) / M₁₂^(2/3))

Wait, let's compute the constants:

Denominator term: (3/5) / (5/2)^(1/3) = (3/5) * (2/5)^(1/3)

Wait, (5/2)^(1/3) = (5)^(1/3) / (2)^(1/3), so 1 / (5/2)^(1/3) = (2/5)^(1/3)

So (3/5) * (2/5)^(1/3) = 3/(5) * (2)^(1/3)/5^(1/3) = 3/(5) * (2/5)^(1/3)

But this seems complicated. Let me compute the numerical values step by step.

First, compute 3^(1/3) ≈ 1.44225

(5/2)^(1/3) ≈ (2.5)^(1/3) ≈ 1.3572

So denominator term:

(3/5) / (5/2)^(1/3) ≈ (0.6) / 1.3572 ≈ 0.6 / 1.3572 ≈ 0.442

Numerator term: 2 / 3^(1/3) ≈ 2 / 1.44225 ≈ 1.386

So the ratio of numerator to denominator is 1.386 / 0.442 ≈ 3.137

So now, the equation becomes:

2/3 = 3.137 * (M₁₁^(2/3) / M₁₂^(2/3))

Therefore:

M₁₁^(2/3) / M₁₂^(2/3) = (2/3) / 3.137 ≈ 0.6667 / 3.137 ≈ 0.2123

Take both sides to the power of 3/2:

(M₁₁ / M₁₂) = (0.2123)^(3/2) ≈ (0.2123)^1.5 ≈ sqrt(0.2123^3) ≈ sqrt(0.0095) ≈ 0.0975. Wait, that can't be right. Wait, 0.2123^1.5 = 0.2123 * sqrt(0.2123) ≈ 0.2123 * 0.4607 ≈ 0.0978. So M₁₁ / M₁₂ ≈ 0.0978. Therefore, M₁₁ ≈ 0.0978 M₁₂.

But that's conflicting with the previous calculation. Wait, perhaps I made a mistake in the ratio. Let me re-express the equation.

From the ratio:

2/3 = [ (2 / 3^(1/3)) / ( (3/5) / (5/2)^(1/3) ) ) ] * (M₁₁^(2/3) / M₁₂^(2/3))

Let me compute the constant factor:

First, compute (2 / 3^(1/3)) divided by ( (3/5) / (5/2)^(1/3) )

= [2 / 3^(1/3)] * [ (5/2)^(1/3) / (3/5) ]

= (2 * 5/2)^(1/3) / 3^(1/3) * 5/3

= (5)^(1/3) / 3^(1/3) * 5/3

= (5/3)^(1/3) * 5/3

= (5/3)^(1/3 + 1) = (5/3)^(4/3)

So the constant factor is (5/3)^(4/3)

Therefore, the equation becomes:

2/3 = (5/3)^(4/3) * (M₁₁/M₁₂)^(2/3)

Therefore:

(M₁₁/M₁₂)^(2/3) = (2/3) / (5/3)^(4/3) = (2/3) * (3/5)^(4/3) = 2 * (3/5)^(4/3) / 3 = (2/3) * (3/5)^(4/3)

Compute (3/5)^(4/3) = (3^4 / 5^4)^(1/3) = (81/625)^(1/3) ≈ (0.1296)^(1/3) ≈ 0.506

So:

(2/3) * 0.506 ≈ (0.6667) * 0.506 ≈ 0.337

Therefore:

(M₁₁/M₁₂)^(2/3) ≈ 0.337

Take both sides to the 3/2 power:

M₁₁/M₁₂ ≈ (0.337)^(3/2) ≈ sqrt(0.337^3) ≈ sqrt(0.0382) ≈ 0.195

So M₁₁ ≈ 0.195 M₁₂

Then, the total masses:

M_total₁ = 3 M₁₁ ≈ 3 * 0.195 M₁₂ ≈ 0.585 M₁₂

M_total₂ = 2.5 M₁₂

So the ratio M_total₁ / M_total₂ ≈ 0.585 / 2.5 ≈ 0.234

Which is approximately 0.23, closest to option D, which is ~0.7. But wait, 0.23 is even smaller. Hmm, this suggests that the total mass of system_1 is about 0.23 times that of system_2, which would mean system_1 is less massive than system_2. But according to the options, the closest is D (~0.7), which is still higher than 0.23. Wait, perhaps I made a mistake in the calculation.

Wait, let's recompute (5/3)^(4/3):

(5/3)^(1/3) ≈ 1.710, so (5/3)^(4/3) = (5/3)^(1) * (5/3)^(1/3) ≈ (1.6667) * 1.710 ≈ 2.85

So (5/3)^(4/3) ≈ 2.85

Then:

(M₁₁/M₁₂)^(2/3) = (2/3) / 2.85 ≈ 0.6667 / 2.85 ≈ 0.233

So (M₁₁/M₁₂) = (0.233)^(3/2) ≈ sqrt(0.233^3) ≈ sqrt(0.0126) ≈ 0.112

Therefore, M₁₁ ≈ 0.112 M₁₂

Total mass system_1: 3 M₁₁ ≈ 0.336 M₁₂

Total mass system_2: 2.5 M₁₂

Ratio: 0.336 / 2.5 ≈ 0.1344, which is ~0.13. Still much lower than the options. This suggests a mistake in the approach.

Wait, perhaps there's a different approach. Let's think differently.

The mass ratio M₂/M₁ is equal to K₁/K₂. So M₂/M₁ = K₁/K₂.

But the total mass is M₁ + M₂ = M₁ + (K₁/K₂) M₁ = M₁ (1 + K₁/K₂) = M₁ (K₂ + K₁)/K₂.

Similarly, for the second system, M₁₂ (1 + K₂₂/K₁₂) = M₁₂ (K₁₂ + K₂₂)/K₁₂.

But we need to relate M₁ and M₁₂. However, the periods are given, so perhaps we can use Kepler's third law to relate the separation a to the period P, and then relate the masses through the RV amplitudes.

Kepler's third law: P² = (4π²/G(M₁ + M₂)) * a³

But for each system, the RV amplitude is related to the separation. The semi-major axis of the stars' orbits around the center of mass is a₁ = (M₂/(M₁ + M₂)) a, and a₂ = (M₁/(M₁ + M₂)) a, where a is the total separation between the stars. So the RV semi-amplitude K₁ = (2π G / (P c)) * (M₂ sin i) / (M₁ + M₂)^(1/3)

But since sin i = 1, K₁ = (2π G / (P c)) * M₂ / (M_total)^(1/3)

Similarly, K₂ = (2π G / (P c)) * M₁ / (M_total)^(1/3)

Wait, for system_1:

K₁₁ = (2π G / (P₁ c)) * M₂₁ / (M_total₁)^(1/3)

Similarly, for system_2:

K₂₁ = (2π G / (P₂ c)) * M₂₂ / (M_total₂)^(1/3)

But we need to relate M_total₁ and M_total₂. Let's express M_total₁ and M_total₂ in terms of their respective K's.

For system_1:

M_total₁ = (M₂₁ + M₁₁) = M₂₁ + (M₂₁ K₁₁ / K₁₂) ) = M₂₁ (1 + K₁₁/K₁₂)

But K₁₁/K₁₂ = (10)/(15) = 2/3. So M_total₁ = M₂₁ (1 + 2/3) = (5/3) M₂₁

Similarly, M_total₂ = M₂₂ (1 + K₂₂/K₂₁) = M₂₂ (1 + 15/10) = (5/2) M₂₂

Now, from system_1:

K₁₁ = (2π G / (P₁ c)) * M₂₁ / (M_total₁)^(1/3)

Substitute M_total₁ = (5/3) M₂₁:

K₁₁ = (2π G / (2 c)) * M₂₁ / ((5/3) M₂₁)^(1/3)

= (π G / c) * M₂₁ / ( (5/3)^(1/3) M₂₁^(1/3) )

= (π G / c) * M₂₁^(2/3) / ( (5/3)^(1/3) )

Similarly, for system_2:

K₂₁ = (2π G / (1 c)) * M₂₂ / (M_total₂)^(1/3)

= (2π G / c) * M₂₂ / ( (5/2) M₂₂ )^(1/3)

= (2π G / c) * M₂₂ / ( (5/2)^(1/3) M₂₂^(1/3) )

= (2π G / c) * M₂₂^(2/3) / ( (5/2)^(1/3) )

Now, divide system_1 equation by system_2 equation:

K₁₁ / K₂₁ = [ (π G / c) * M₂₁^(2/3) / (5/3)^(1/3) ] / [ (2π G / c) * M₂₂^(2/3) / (5/2)^(1/3) ) ]

Simplify:

(10 / 15) = [ (π G / c) / (2π G / c) ) * (M₂₁^(2/3) / M₂₂^(2/3)) * ( (5/2)^(1/3) / (5/3)^(1/3) ) ]

Simplify constants:

(10/15) = (1/2) * (M₂₁/M₂₂)^(2/3) * ( (5/2)^(1/3) / (5/3)^(1/3) )

Simplify the 5 terms:

(5/2)^(1/3) / (5/3)^(1/3) = (5^(1/3)/2^(1/3)) / (5^(1/3)/3^(1/3)) ) = (3^(1/3)/2^(1/3)) ) = (3/2)^(1/3)

So:

1/2 = (1/2) * (M₂₁/M₂₂)^(2/3) * (3/2)^(1/3)

Multiply both sides by 2:

1 = (M₂₁/M₂₂)^(2/3) * (3/2)^(1/3)

Raise both sides to the power of 3/2:

(M₂₁/M₂₂) = (3/2)^( (1/3) * (3/2) ) = (3/2)^(1/2) = sqrt(3/2) ≈ 1.2247

So M₂₁ ≈ 1.2247 M₂₂

Therefore, the total mass of system_1 is (5/3) M₂₁ ≈ (5/3)*1.2247 M₂₂ ≈ 2.041 M₂₂

Total mass of system_2 is (5/2) M₂₂ ≈ 2.5 M₂₂

So the ratio M₁_total / M₂_total ≈ 2.041 / 2.5 ≈ 0.816, which is approximately 0.82. This is close to option B (~1.2) but not exactly. However, none of the options are exactly 0.82. The closest option is B (~1.2), but that's higher. Alternatively, perhaps I made a calculation error.

Wait, let's recompute the exponents. From the earlier equation:

(M₂₁/M₂₂)^(2/3) * (3/2)^(1/3) = 1

Let me solve for (M₂₁/M₂₂):

Let x = M₂₁/M₂₂

Then, x^(2/3) = 1 / (3/2)^(1/3) = (2/3)^(1/3)

Raise both sides to the power of 3/2:

x = (2/3)^( (1/3) * (3/2) ) = (2/3)^(1/2) ≈ 0.8165

So M₂₁ ≈ 0.8165 M₂₂

Then, M_total₁ = (5/3) * 0.8165 M₂₂ ≈ (1.6667) * 0.8165 ≈ 1.361 M₂₂

M_total₂ = (5/2) M₂₂ = 2.5 M₂₂

So ratio M_total₁ / M_total₂ ≈ 1.361 / 2.5 ≈ 0.544, which is ~0.544, closer to option C (~0.6). Hmm, this is conflicting with the previous result. I must have made a mistake in the calculation.

Wait, let's clarify. The ratio M_total₁ / M_total₂ is (5/3 * M₂₁) / (5/2 * M₂₂) ) = (5/3) / (5/2) * (M₂₁ / M₂₂) ) = (2/3) * (M₂₁ / M₂₂)

From earlier, we found that (M₂₁ / M₂₂) = (3/2)^(1/2) ≈ 1.2247

So:

M_total₁ / M_total₂ = (2/3) * 1.2247 ≈ 0.5498 ≈ 0.55

Which is approximately 0.55, closest to option C (~0.6). But the options don't include ~0.55. However, option B is ~1.2, which is the inverse of 0.83. Wait, maybe the question is asking by what factor system_1 is more massive than system_2, so the reciprocal.

Wait, the problem states: "by what factor is system_1 more massive than system_2?" So if system_1 is 0.55 times as massive as system_2, then system_2 is 1/0.55 ≈ 1.8 times more massive. But the options don't have that. Alternatively, maybe I messed up the ratio.

Wait, let's re-express. If M_total₁ = 0.55 M_total₂, then system_1 is 0.55 times as massive as system_2. But the question asks by what factor system_1 is more massive than system_2. So the factor is M_total₁ / M_total₂ ≈ 0.55, which is approximately 0.55. However, none of the options match. The closest is option C (~0.6). So perhaps the answer is C.

But according to the previous calculation, when we correctly solve for the ratio (M₂₁/M₂₂) = (3/2)^(1/2) ≈ 1.2247, then M_total₁ = (5/3) * 1.2247 ≈ 2.041, and M_total₂ = 2.5, so the ratio is 2.041 / 2.5 ≈ 0.816, which is ~0.82. This is approximately 0.82, which is closest to option B (~1.2) if we consider rounding errors, but 0.82 is closer to 0.8, but there's no option for that. Alternatively, perhaps the answer is D (~0.7), but 0.82 is closer to 0.8. Wait, the options are:

A. ~0.4

B. ~1.2

C. ~0.6

D. ~0.7

0.82 is closer to 0.8, which is not an option, but perhaps the intended answer is B (~1.2), as it's the closest to the inverse. Alternatively, maybe there's an error in the calculation.

Wait, let's check the initial ratio again. From system_1:

M₂₁ = (M₂₁ / M₁₁) * M₁₁ = (K₁/K₂) * M₁₁ = (10/15) M₁₁ = (2/3) M₁₁

So M_total₁ = M₁₁ + M₂₁ = M₁₁ + (2/3) M₁₁ = (5/3) M₁₁

Similarly, M_total₂ = (5/2) M₂₂

From the RV amplitudes and the periods, we derived that (M₂₁ / M₂₂) = sqrt(3/2) ≈ 1.2247

So M_total₁ = (5/3) * 1.2247 ≈ 2.041, M_total₂ = 2.5

Ratio: 2.041 / 2.5 ≈ 0.816

This is approximately 0.82, so the factor is ~0.82. But the options don't have that. The closest is option B (~1.2), which is the inverse. Alternatively, perhaps the problem intended to ask for the inverse ratio, i.e., system_2 is more massive than system_1. But the question says "by what factor is system_1 more massive than system_2?"

Alternatively, perhaps the error comes from the assumption of the same separation. But the periods are different, so the separation must be different.

Wait, another approach: since the period is related to the separation (a) via Kepler's law, and the mass ratio affects the separation, perhaps we can relate the two systems.

For system_1:

P₁² = (4π² / G(M₁ + M₂)) * a₁³

For system_2:

P₂² = (4π² / G(M₁' + M₂')) * a₂³

But without knowing the actual separations, it's hard to relate them directly. However, if we assume that the stars in each system have the same radius, then the RV semi-amplitude K is proportional to (M₂ / a) * (1 + M₂/M₁)^(1/3) * sin i. But this seems complicated.

Alternatively, since the problem gives the RV semi-amplitudes and the periods, and asks for the mass ratio, perhaps the initial approach of using the RV ratio to find the mass ratio for each system and then combining them with Kepler's law to relate the total masses is the way to go.

Earlier, when we found that M_total₁ ≈ 0.55 M_total₂, but the options don't have that. However, if we consider that the mass ratio M₁_total / M₂_total ≈ 0.55, then system_2 is approximately 1.8 times more massive, but that's not an option. Alternatively, maybe there's a miscalculation in the exponents.

Alternatively, perhaps the question is to find the ratio of the total mass of system_1 to system_2, which is approximately 0.55, so closest to option C (~0.6). Alternatively, if the question had asked for the inverse, but no.

Alternatively, maybe the answer is D (~0.7), as it's the closest to 0.82 when rounded. But 0.82 is closer to 0.8, which isn't an option. However, perhaps due to rounding in the problem's numbers, the intended answer is D (~0.7).

Alternatively, perhaps the problem expects us to ignore the periods and only use the RV amplitudes to find the mass ratios.

If we ignore the periods and use only the RV semi-amplitudes:

For system_1: M₂/M₁ = K₁₁/K₁₂ = 10/15 = 2/3

Total mass: M₁ + M₂ = M₁ + 2/3 M₁ = (5/3) M₁

Similarly, system_2: M₂/M₁ = 15/10 = 3/2

Total mass: M₁ + 1.5 M₁ = 2.5 M₁

Thus, ratio of total masses: (5/3 M₁) / (2.5 M₁) = (5/3)/2.5 = (5/3)/(5/2) )= (5/3)*(2/5) = 2/3 ≈ 0.6667, which is approximately 0.67, closest to option C (~0.6). However, this ignores the periods, which are given as 2 and 1 years. Therefore, this approach is incorrect, as the periods must be considered.

Therefore, despite the confusion, the correct approach using Kepler's law and the given periods leads to a ratio of approximately 0.82, which is closest to option B (~1.2) if we consider the inverse. However, since the question asks for the factor by which system_1 is more massive than system_2, and we found that system_1 is less massive (0.82 times), but none of the options match exactly, perhaps the intended answer is option D (~0.7), as it's the closest to 0.82 when rounded down. Alternatively, there might be a mistake in the problem's options.

Alternatively, perhaps the problem is designed to have the answer as D (~0.7). Given the calculations, the closest option is D.
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