Earth-size planet formation in the habitable zone of circumbinary stars

The sum of the masses of the binary pair equals approximately 0.9 $M_{\odot}$, so it is the least massive pair in our simulations. Thus, it is the system that has the narrowest and closest HZ to the system’s barycenter, causing much disturbance in the particle disk by the motion of the stars. Another factor that increases disk disturbance is the semi-major axis of the stars, which is the second largest of our test cases, equal to 0.224 $au$. Also, the system planet is inside of HZ. These ingredients cause all particles to be ejected within a few thousand years. Only some coorbitant particles to the host planet survived in the simulation, see Figure 2. Which indicates that the habitable region of the system is unstable. This result shows us that forming a planet in HZ is difficult, but this does not exclude the possibility of any body orbiting these regions. Since this system has unstable HZ, we do not simulate the formation of planets in this case.

Although this system is unstable throughout its HZ, it is possible to note that the system has bodies coorbital to the host planet, see Figure 2. This is an important result mainly for two reasons; (1) demonstrates the condition of binary systems having bodies coorbital to the host planets; (2) As in this case the planet is within the HZ, consequently coorbital bodies will also be. Thus, even if the HZ is dynamically unstable, the system can still have bodies within the HZ sharing the orbit with the host planet.

The system’s HZ is away from the system center of mass. This makes the disk not very disturbed by the stars. The dominant disk disturbance comes from the confirmed planet. The planet is positioned at the beginning of HZ and consequently on the particle disk, see Figure 2. This causes further disturbance in the early part of the disk. Even with the planet’s presence, HZ is practically entirely stable. Therefore, the system presents stability conditions to study the formation of planets in HZ.

The Kepler-35 system is very similar with Kepler-34. The system also has the planet very close to the interior of the habitable region, which makes the planet the main disturbing agent of the disk. Although close, the planet never orbits within to HZ as in the previous case, so what makes it completely stable and the particles practically did not vary their positions. This can be checked in Figure 3, where snapshots of the evolution of the semimajor axis and eccentricity in the Kepler-35 system are shown. Only the particles closest to the planet were ejected giving us a large region for the study of planetary formation.

This system is the most stable of all studied in this work, as can be seen in Figure 3. This feature comes from the great distance that HZ is from the planet and consequently from the stars. In addition, the planet is the least massive of all eight selected. with reasonable mass of 0.016 Jupiter masses. Throughout the simulation, no particles were ejected, and as we can see from Figure 3, only the early disk particles had a slight increase in eccentricity. Thus, the system has a large region for the study of planetary formation within its HZ.

The Kepler-47 system has already been the subject of a stability study in some works before the third planet was confirmed. In Kratter & Shannon (2014), the system was used as a case study to verify that the system is dynamically packed with the three planets. In Hinse et al. (2015), a stability study was performed using MEGNO technique to predict the third planet.

In our work, with the confirmation of the three planets, we used the three to test the dynamic stability of the HZ. Thus, this case is the only multiplanetary system present in our stability test. The most recent confirmation is from the planet Kepler-47d (Orosz et al., 2019). One of the planets is orbiting within the HZ, which causes great instability in that region. We can check in Figures 2 and 4 that almost all particles were ejected at 1 $Myr$, with only a few particles remaining on the outside of the HZ. Thus, we can conclude that HZ is unstable to its full extent and does not provide conditions to study the ability of this system to form a planet within HZ.

Together with the Kepler-16 system, this case is the most unstable HZ of our simulations. In addition to the instability present in both cases, one of the host planets also has a coorbital body to it, see Figure 2.

This system has a very narrow HZ and a host planet on its inner edge. In this case, the particles that are in the innermost region of the disk are ejected because they are very close to the planet. Although many particles have been ejected and collided with the planet, a reasonable fraction of the HZ of the Kepler-413 system has stability.

The planet of the system orbits within to HZ, which causes a lot of ejections and collisions of the particles. We can see in Figure 2 two regions, one at the beginning and one at the end of HZ, that have stability, the largest being the outermost region. So even though it has no stability to its full extent, the system has parts that are stable.

Besides the system has a stable part of the HZ, the same result of the Kepler-16 system can be noticed in this case, see Figure 2. The system has bodies coorbital to the host planet as well.

The Kepler-1647 system has the largest exoplanet in binary systems with approximately 1.5 Jupiter masses. In addition to the planet, stars also have the largest mass sum in our work, which makes HZ farther and wider. The planet is in the middle of HZ, causing many particles to be ejected. In Figure 2 we can see three subregions within the HZ. An inner region, an coorbital region and another outer region in relation to the planet. In this case, because it is a peculiar and complex system, we will study the formation of terrestrial planets within these three regions in a future work.

With the stable regions found in the above systems, we can now study the ability to form terrestrial planets in these regions. Some systems, as in the case of Kepler-16, 47 and 1647, do not have a stable region sufficient for this study. Others, such as the Kepler-413 and 453 systems, do not have completely stable HZ, but we can study at least some parts of these regions. The red dashed lines in Figure 2 are the inner and outer limits where we distribute the embryos and planetesimals to explore the terrestrial planet formation.

The Kepler-34 system has two stars with masses close to the Sun (see parameters in Table 3), consequently belong to spectral class G with age of 5-6 Gyr. They complete an orbital period in about 28 days (Welsh et al., 2012). This system has one giant planet with $\approx$ 22% of the mass of Jupiter, 76% of the radius of Jupiter and is known as Kepler-34b. The Kepler-34 system has its planet very close to the disk of planetesimals and embryos. This makes the planet the main disturber. This causes a lot of disk mass to be lost as early as the first thousand of years of integration, especially in the early part of the disk. At $x=1.5$, 24% of the initial mass was lost, while in the case of $x=2.5$ 36% of the mass was lost at the same length of time. The percentage is higher in the case of $x=2.5$, because in this case the most massive embryos are at the inner part of the disk, the most disturbed region. Thus being ejected or collided with the planet. With both mass scale parameters, protoplanets are formed within the HZ in all simulations. The closest to a terrestrial mass appears in the case Kepler-34-1 (with $x=1.5$), where a planet with a mass of about 0.6 $M_{\oplus}$ is formed in the simulation, see Figure 6. In all others, with both values of $x$, only bodies with masses < 0.4 $M_{\oplus}$ are formed, see Figures 6 and 7. Therefore, from our initial conditions, this system can form a terrestrial planet with up to 0.6 Earth mass within HZ.

This system also has stars with mass close to that of the Sun, and with this also belonging to the spectral class G. These stars have a 0.176 $au$ semi-major axis and an orbital period of 21 days with an age of 8-12 Gyr (Welsh et al., 2012). The host planet of the system, known as Kepler-35b, has 13% of the mass and 73% of the radius of Jupiter (Welsh et al., 2012), it is $\approx$ 0.3 $au$ away from the beginning of the disk. In the stability test within the HZ, we saw that the test particles hardly changed in their semi-major axis and eccentricities and thus little mass is lost.

In the case of $x=1.5$, at least one planet with mass > 0.7 $M_{\oplus}$ has been formed, see Figure 6. Analyzing only the bodies that were formed in the simulation inside the HZ, we have that in the Kepler-35-1 simulation, two bodies with masses of 0.6 and 0.85 $M_{\oplus}$ are formed. The Kepler-35-2 simulation is the case in which less massive bodies are formed, which are approximately 0.6 and 0.3 $M_{\oplus}$. In Kepler-35-3 the largest planet is formed, with a mass equal to 1.2 $M_{\oplus}$. The Kepler-35-4 and 5 simulations have very similar results. Where in both cases planets with about 0.85 Earth mass were formed in the simulation and with similar semi-major axis. Figure 8 shows snapshots in time of the dynamic evolution of the semi-major axis and the eccentricity of embryos and planetesimals of the Kepler-35-5 simulation.

In the case where we have $x=2.5$, we also found that in all simulations, at least one planet with mass greater than 0.6 $M_{\oplus}$ is formed inside the HZ, see Figure 7. Looking at these planets, we have that in the Kepler-35-1 case, two planets with masses of 0.44 and 0.78 $M_{\oplus}$ are formed. In the Kepler-35-2, the largest of them is formed, with mass equal to $1.33M_{\oplus}$. In the Kepler-35-3 simulation, two planets with very similar masses were formed (0.65 $M_{\oplus}$). The Kepler-35-4 simulation has an Earth size with a mass of 1.04 $M_{\oplus}$, Figure 9 shows snapshots in time of the dynamical evolution of the semi-major axis and eccentricity of embryos and planetesimals. Finally, in the case Kepler-35-5, two planets are formed, the largest with a mass of 0.75 $M_{\oplus}$.

The results shown above showed that with both values of $x$ (1.5 and 2.5) terrestrial planets were formed within the HZ. Therefore, we can conclude that this system has a great potential to house a habitable Earth type planet.

The binary star system Kepler-38, has two stars with masses of 95% and 25% solar masses. The brighter star is spectral class G while the secondary has spectral class M. The stars have semi-major axis 0.147 $au$ and complete an eccentric orbit around a common center of mass every 18.8 days (Orosz et al., 2012b). The system has a planet called Kepler-38b, known as a Neptune-sized type. Since it has not yet noticeably perturbed the stellar orbits, the planet does not have exact mass, being it an upper limit of 122 Earth masses, being its reasonable mass equal to 21 Earth masses¹¹1 This mass is found assuming that the planet follows the empirical mass-radius relationship of $M_{b}=(R_{b}/R_{\oplus})^{2.06}M_{\oplus}$ (Lissauer et al., 2011). Where $M_{b}$ and $R_{b}$ are respectively the mass and the radius of the planet., which was used in our work as can be seen in Table 2. The system has the most stable HZ. The planet is 0.7 $au$ away from the disk, in addition to being a planet with mass 0.016 $M_{j}$, that is, the smallest of the planets studied in our work. With this, the planet causes little disturbance to the disk. At $x=1.5$, the first ejection occurred at 7 million years and at $x=2.5$ in only 10 million years. In addition, this small disturbance causes collisions to occur in a longer timescale and consequently the formation of planets is slower.

For $x=1.5$, Figure 6 shows that planets with Earth-like mass are formed in all simulations within the HZ. The Kepler-38-1 simulation is the one containing the smallest planet formed, with a mass of 0.74 $M_{\oplus}$. The Kepler-38-2, 3 and 4 cases have the largest ones, with masses of 1.23, 1.14 and 1.27 $M_{\oplus}$, respectively. In addition to these three simulations having planets with similar masses, the positions in which they were formed are also similar. The Kepler-38-5 case has the planet with the largest semi-major axis, being 2.5 $au$ with mass approximately equal to 1.01 $M_{\oplus}$.

In the cases for $x=2.5$, all simulations have at least one planet inside HZ, see Figure 7. Regarding these planets, in the Kepler-38-1 simulation, a planet with mass 1.1 $M_{\oplus}$ was formed. In the case Kepler-38-2, one planet with mass equal to 1.26 $M_{\oplus}$ is formed in the same position as the previous case. The final configuration of the other three cases is quite similar. In these cases, three planets were formed outside the HZ and three others inside. The semi-major axis of HZ’s three external are close to 1 $au$ with masses around 0.65 $M\oplus$ and the three interns have semi-major axis close to 2 $au$ with masses ranging from 0.4 to 0.65 $M_{\oplus}$.

From the results found, we can conclude that the system has conditions of forming and housing a habitable Earth type planet. With less massive disks on the inside (with $x=1.5$), we show that in almost all cases (except simulation 1) Earth-like planets form inside the HZ. In the cases where the simulations had more massive disks inside, only the simulations 1 and 2 Earth-type planets inside the HZ were formed.

In this system, the two stars Kepler-413A and Kepler-413B have masses of 82% and 54% of the mass of the Sun. The brightest is a K dwarf spectral type while its companion a M dwarf, which results in the thinnest HZ of the systems present in our simulations. In this system, we have the second least massive host planet of the systems treated in our work. However, unlike the Kepler-38 system that has a planet that is $\approx$ 0.6 $au$ from the disk, and therefore, resulting in a disk in which the collisions happen slowly, in this case, the planet is at the inner boundary of HZ and very close to the disk ($\approx$ 0.05 $au$), causing great disturbance. Much of the mass present in the innermost region of the disk is ejected as early as the first instants of the simulation. In the simulations where $x=1.5$, 24% of the total disk mass was lost in 0.5 $Myr$, while for $x=2.5$, 38% of the total mass was lost in the same length of time.

In the case of $x=1.5$, where the outermost part of the disk are conserved, thus forming more massive planets, see Figure 6. In the Kepler-413-1 simulation the largest one is formed, with mass close to 1.4 $M_{\oplus}$, but outside of HZ. In simulation 2 of this system, a planet with 0.6 Earth mass is formed, also outside of HZ. In simulation 3, a planet with 1.1 $M_{\oplus}$ is formed inside the HZ very close to the outer boundary. In the following two cases, Kepler-413-4 and 5, a planet was formed in each case with masses of 0.6 and 1 $M_{\oplus}$, respectively.

In simulations with $x=2.5$, due to the factors mentioned above, less massive planets were formed, see Figure 7. In simulation 1, one planet with 0.6 Earth mass forms far from HZ. In simulations 2 and 3, two planets with similar masses, approximately 0.75 $M_{\oplus}$, are formed outside HZ in similar regions. In cases 4 and 5, two planets also with similar masses, approximately 0.6 $M_{\oplus}$, are formed within the HZ.

Thus, we can conclude that the system is capable of housing habitable Earth-like planets within HZ in some cases. These cases occur when the system has a less massive embryo and planetesimal disk in its internal region ($x=1.5$).

The system Kepler-453 has two stars, Kepler-453A and Kepler-453B. The brightest is similar to our Sun, containing 94% of its mass, while the smaller star, Kepler-453B, has about 20% of its mass. Stars orbit one another every 27.3 days (Welsh et al., 2015). This system has a giant planet that is within the HZ of the system with $\approx$ 0.8 mass of Jupiter. It is important to remember that not every planet within the HZ of a system can harbor life since the boundaries of these regions are calculated on the basis of the flow of energy received at the top of Earth’s atmosphere. In calculating the stableness of the habitable region, we discovered a small band on which it was possible to distribute a disk. Thus, the beginning of the disk of material is approximately 0.85 $au$, being that the planet has semi-major axis equal to 0.790 $au$. This way the disk is very affected by the presence of the planet so close. As in previous cases, in simulations with $x=1.5$, on average less mass was ejected. In this case 10% of the disk mass lost to 0.5 $Myr$, and 16% mass lost in the same time interval to $x=2.5$. Although the planet is extremely close to the disk, and even within the HZ, due to its low eccentricity (0.036), not as much mass is ejected as in the case of the Kepler-34 system. Thus, planets with masses close to Earth‘s mass (0.6 - 0.9 M_⊕) are formed in all simulations, but all outside of HZ, see Figures 6 and 7.