The selection criteria for sdBs are based on the stellar mass at He ignition, which has to be below 0. And the absence of a hydrogen burning shell. In our models, there is a clear mass gap between the sdB models, and the models that ignite He on the RGB.
By increasing the wind mass loss, the radius of the donor can be kept small in comparison to its Roche-lobe, thus the tidal forces that circularise the system are weaker. This difference between the mass lost at periastron and apastron can increase the orbital eccentricity, depending on the mass ratio of the system and the fraction of the wind mass loss that is accreted by the companion.
The strength of the eccentricity pumping will depend on the mass-loss rate, and the fraction that is accreted by the companion. Mass lost to infinity increases the eccentricity. Mass that is accreted, drives a change in eccentricity that is positive only if the donor mass is lower than the accretor mass. See Appendix A. The initial period is days, and the initial donor and companion masses are 1.
For discussion see Sect. In Fig. As it can be seen, this method can result in a significant final eccentricity of the orbit after the RGB evolution of the primary, if the enhancement parameter is sufficiently large. For the system shown in Fig.
When comparing all calculated models, we find that the amount of mass that the donor needs to lose to maintain an eccentric orbit is so high that the final core mass is too low for He ignition. The donor star will then end its life as a cooling He-WD on an eccentric orbit. To reach a final donor mass high enough to ignite He, the mass loss has to be lower so that the system completely circularises on the RGB. Therefore, the sdB binaries formed in this channel are all circularised.
The final mass of the donor changes with the initial parameters of the system, like orbital period, donor and accretor mass, and wind accretion fraction, but none of the combinations of initial binary parameters resulted in an eccentric system containing an sdB companion. We conclude that in the parameter regime considered here, the progenitors of sdBs do not evolve with a strong enhanced mass loss on the RGB.
To form an sdB by tidally-enhanced wind mass-loss, the enhancement of the wind needs to be small, so essentially no eccentricity pumping occurs.
Furthermore, if there is a significant enhanced wind mass-loss, the sdB progenitor will lose its hydrogen envelope in a stellar wind, and will not undergo RLOF during its later evolution.
If this mass loss happens on a slightly eccentric orbit, the mass-loss rate will not be constant over the orbit, and the mass loss can have an eccentricity-pumping effect. The strength of the eccentricity pumping will depend on the mass-loss rate, and the fraction that is accreted by the companion, in the same way as in the tidally-enhanced-mass-loss mechanism. In this section we will describe behaviour of the mass lost from the donor star in our study of phase-dependent RLOF.
This mass can be accreted by the companion star, or is lost from the system to infinity. A detailed description of that mechanism is given in Appendix A.
The default system has an initial period of days, initial eccentricity of 0. The companion star will be on the MS during the relevant part of the evolution, thus the radiative dissipation mechanism for the tidal energy is assumed.
The sdB progenitor will be on the RGB when the tidal forces are strongest, and the dissipation mechanism is that of a convective star. Dissipation becomes radiative when the envelope is completely lost. The accretion fraction onto the companion star is zero in this default model.
These parameters are summarised in Table 2. Only the parameters mentioned in the text are changed. As it is not known when exactly a common envelope would start to form we assume that RLOF is stable in our models. The latter limit is also imposed as our physical model breaks down at high Roche-lobe overfilling. However, Nelemans et al. Furthermore, based on BPS studies, Clausen et al.
Table 2 Standard parameters of the binary models. The evolution of the default model is plotted in Fig. These events indicate two phases in the period-eccentricity evolution of the binary. The time scale between these phases differs on the figure, and within one phase the time scale is linear. Underneath the figure the duration of each phase is given. The eccentricity starts increasing, and reaches a value of 0.
During the mass-loss phase the period increases as well, from to days. This phase of strong mass loss only takes years. During this phase the sdB progenitor is contracting, thus the tidal forces are weakening. The eccentricity diminishes from 0. For this system, the eccentricity increased by roughly a factor The time scale differs between the phases, but is linear withing each phase.
The duration of each phase is shown under the figure. Panel A the change in angular momentum. Panel C the tidal forces blue and eccentricity pumping due to mass loss red. Panel D the orbital eccentricity. Panel E the orbital period. The effect of the initial period and companion mass is plotted in Fig. With increasing initial period, the final period of the sdB binary will increase as well, while at the same time the eccentricity decreases.
A similar effect is visible when increasing the companion mass. The closer the companion mass is to the donor mass of 1. We first discuss the effect of the initial binary parameters, as they clearly illustrate the effect of the orbital period on the eccentricity pumping force that is also essential in the discussion of the process dependent parameters.
Panel A the Roche-lobe overfilling factor. Panel C the tidal forces in log s Panel D the eccentricity pumping forces in log s Panel E net change in eccentricity per second. Panel F mass ratio. Panel G the eccentricity. Panel H the orbital period in days. To explain these effects, the evolution of several parameters for models with different companion masses of 0. These parameters are plotted in function of the donor mass instead of time, so that the different models can be more easily compared.
The connection between eccentricity and orbital period is found in the change in mass-loss rate during RLOF. This lower overfilling of the Roche lobe leads to a lower mass-loss rate and a shorter total time during which mass is lost at the maximum rate. For the models described here, the maximum mass-loss rate is the same, but the time during which it is sustained differs from roughly 25 years for the 0.
This diminishes the total eccentricity pumping force, even though the time during which it overpowers the tidal forces is longer for the 1. Due to the lower Roche-lobe overfilling, the tidal forces will also decrease with increasing orbital period, with a factor 10 difference between the 0.
Even so, there is a net effect of lower eccentricity enhancement for the model with the heaviest companion. For a system with a specified donor and accretor mass, the effect of the initial period on the final mass is important.
If the orbital period is too short, the mass loss will be too strong, and the donor star will lose too much mass to ignite helium, ending up on the He-WD cooling track. The lower limit on helium ignition in MESA is around 0. If the initial orbital period is too high, the mass-loss rate during RLOF will be too low, and the donor will ignite helium on the RGB and hence no sdB is created. The system with the lowest orbital period panel A ends up with a donor star mass of 0.
The system with the days initial period panel B ends with a donor star of 0. With an even higher period of days, the final donor mass is 0. If the period is increased further, to days, the donor will be too massive, and will ignite helium on the RGB when it still has a total mass of 0.
The sum of the mass loss fractions is unity for each model. See Sects. The different ways to lose mass to infinity will change the final orbital parameters of the system. These parameters mainly influence the amount of angular momentum that is removed from the system with the lost mass. The effect of these mass-loss fractions and the location of the outer Lagrange point is shown in Figs.
When most mass is lost from around the donor the resulting period will be lower than when the mass is lost from around the companion star.
This is easily explained by Eq. The mass that is lost from around the donor star will thus carry more angular momentum with it, which will result in a shorter orbital period. This change in period will influence the change in eccentricity by altering the mass-loss rates during RLOF, similar as was explained in Sect.
From a certain threshold period that depends on the initial and mass-loss parameters, the eccentricity pumping is smaller than the tidal forces, and the orbits stay circularised. The effect of these two mass-loss parameters on the period is large, of the order of several hundred days. By changing from most mass lost around the donor to most mass lost around the companion, the final period can double.
The two models on Fig. We note that our description breaks down at high Roche-lobe-overfilling values, and the models in Fig. The exact threshold value depends on the other parameters. By increasing the accreted fraction, the eccentricity increases while the orbital period stays more or less constant. By accreting a certain fraction of the mass loss, the size of the Roche-lobes will differ between the models.
Higher accretion leads to slightly higher Roche-lobe overfilling, and a slightly higher eccentricity pumping. The period during which the eccentricity-pumping forces are stronger than the tidal forces also increases with increasing accretion rates. By increasing the accretion, the final eccentricity can almost be doubled, while the period remains constant. The models with phase-dependent RLOF can indeed explain a certain part of the period-eccentricity diagram, but have problems in the high-period high-eccentricity range, and cannot reproduce the circular systems.
Circumbinary disks could potentially explain the high-period, high-eccentricity systems as they add extra eccentricity-pumping forces on top of those from phase-dependent RLOF. Circumbinary disks can form around binaries during the RLOF phase, if part of the mass can leave the system through the outer Lagrange points and form a Keplerian disk around the binary. The effect of the CB disk-binary resonances on the orbital parameters has been the subject of many studies.
The effect of the disk on the binary separation is given by: 3 where J D is the angular momentum of the disk and J B the orbital angular momentum of the binary.
The derivation and implementation of these equations is given in Appendix A. In our model, the disk is formed by matter lost from the binary through the outer Lagrange point, and the disk itself loses mass at a rate determined by its life time. The mass in the disk is not constant, and only the maximum disk mass is a defined input parameter. The inner radius of the disk is determined based on SPH simulations, and the dust-condensation radius of the binary see Eqs.
Based on observations and the assumed surface density behaviour, the outer radius of the disk is fixed at AU see also Appendix A. Maximum mass, life time, viscosity and distribution, are input parameters in the model. Based on observations of post-AGB disks Gielen et al. Disk life times after the end of RLOF range from 10 4 to 10 5 years.
Four different events are indicated on the x -axes. The time scale differs between the phases, but is linear within each phase. Panel B the change in angular momentum due to mass loss red dashed line and the CB disk — binary interaction green dashed dotted line.
Panel C the tidal forces blue full line , eccentricity pumping due to mass loss red dashed line and eccentricity pumping through CB disk — binary interactions green dashed dotted line. Panel D the eccentricity. The model that is shown has the same parameters as the model shown in Fig.
Four different events are indicated on the figure. These events define three different phases in the period-eccentricity evolution of the binary. On the figure, the time of each phase is linear, but the time scales between phases differ. The duration of each phase is plotted under the figure. The mass in the disk continues to grow while the donor star continues to expand, eventually filling and overfilling its Roche-lobe. The tidal forces continue to increase as well, and when the CB disk reaches its maximum mass after roughly years, the tidal forces again overtake the disk-binary interaction.
By this time the eccentricity of the system reached 0. The change in eccentricity is solely due to the disk-binary interactions, but the loss of angular momentum from the binary is caused mainly by the mass loss. Due to phase-dependent mass loss, the eccentricity continues to increase. What happens in this phase is very similar to what is shown in Fig. However, because the binary has a higher eccentricity than that in the model without a disk, the eccentricity pumping due to mass loss is stronger.
This leads to an eccentricity of 0. The orbital period first decreases and then increases again, reaching days. This whole phase lasts only years. The CB disk-binary interactions continue to increase the eccentricity, while at the same time angular momentum is transported from the binary to the CB disk, thus decreasing the orbital period again.
The effect of the CB disk-binary interactions diminishes due to a decrease in disk mass while simultaneously, the resonances that are responsible for the eccentricity pumping become less effective at higher eccentricities. When the disk is completely dissipated, the binary has an eccentricity of 0. However, no models we have tried reach this eccentricity limit. There are four parameters in the CB disk model that can influence the CB disk-binary interactions: the maximum mass in the disk, the life time of the disk, the viscosity parameter and the assumed distribution of the mass with radius.
The effect of these five parameters is shown in Fig. These life times can also be interpreted as the evolution in the period-eccentricity diagram after the end of RLOF. Based on the equations that govern the CB disk-binary interaction given in Sect.
By increasing the maximum disk mass as shown in panel A, the final eccentricity will be higher, while the final period will be slightly lower. The mass-loss rates during RLOF are high enough that there is very little difference in time to fill a disk of 0. An increased disk mass will lead to a higher angular momentum of the disk, and, according to Eq.
A higher viscosity will lead to a higher final eccentricity and lower period. The effect of the mass distribution in the disk is plotted in panel C.
The mass distribution will determine the mass fraction that resides close to the inner rim of the disk, where it has the strongest effect on the binary. Normally a radial mass distribution of r -1 is chosen, which takes into account that the disk is not just flat, but also has a specific thickness that varies with the radius. A distribution of r -2 assumes that the thickness of the disk is more constant with radius.
There may or may not be a cause-effect relationship between the astronomical phenomenon and glacial cycles. Nevertheless, it is being rigorously researched. It is similar to carbon dating used to determine the age of fossils.
The oldest data was 1. Previous analyses, looking for such a relationship had been unable to find a statistically sound argument for that. Lisiecki used a different type of analysis on her set of data to arrive at a significant correlation. She was able to show a link between occurrence of the largest glacial cycles and weakest changes in orbital eccentricity.
The negative correlation between the two is unusual and tricky to explain in a cause-effect relationship. Hence, Lisiecki theorizes a complex interplay of various factors including eccentricity, obliquity and the climate system in its entirety.
Note that there is some disagreement between the maximum and minimum values of our approximate method and the brute force calculation due to the fact that the system has insufficient time to reach the theoretical long-term extrema due to the rapid orbital decay. Although the presence of the secular resonance can help to speed up tidal evolution of proto-USP systems, it is not a necessary condition to form USPs.
In the next section, we discuss the conditions under which USPs may form in three-planet systems. Similar to Fig. The two dips in the solid curves correspond to the two resonant mode crossings discussed in Section 4.
Note that some values of a 2 may result in dynamically unstable systems. In general, for three-planet systems, the AMD constraint is more stringent than the decay time constraint. To illustrate this, we show the final value of a 1,f reached after 10 Gyr of evolution as a function of a 2 in Fig. The dashed curves in Fig. One would expect USPs to be systematically lower in mass, as lower mass inner planets are more likely to meet the AMD constraint see equation At the same time, we expect the external companions of USPs to have systematically larger masses, although giant planet companions are not required.
The observational implications are explored in more detail in Section 6 , where we develop a population model for USP generation. The final value of a 1,f after 10 Gyr of evolution in a three-planet system solid curves and its theoretical minimum dictated by the AMD constraint dashed curves, equation 48 , plotted as a function of a 2. Regions where the solid curves lie on top of the dashed curves indicate the system is AMD-constrained, while regions where the solid curve is well separated from the dashed curve correspond to tidal decay time-constrained systems.
In this section, we have explored USP formation from three-planet systems. At first glance, there is a tension between USP generation from multiplanet systems and the fact that observed USPs have a dearth of exterior transiting companions compared with their non-USP counterparts. This prima facie contradiction can be rectified when we consider the mutual inclination evolution of USP-forming systems, in Section 5.
We are interested in the inclination evolution of the proto-USP system because the evolution of the mutual inclination of planets determine the extent to which USPs will transit simultaneously with their companions, a quantity that can be observationally constrained see Section 6. In Section 5. To illustrate the possibility of resonance, in Fig. Note that in the examples shown in Fig. To illustrate this, in Fig.
Same as Fig. Similar to the right-hand panels of Fig. In real systems, the stellar rotation period increases over time, thus the importance of the spin-orbit coupling depends on the time-scale of the proto-USP orbital decay: if the orbital decay occurs well within a Gyr, then spin-orbit coupling can be important.
Otherwise, the star would have already spun down by the time the final USP semimajor axis is reached, and the effect of spin-orbit coupling is small. We synthesize the results of Sections 2 — 5 by performing a population synthesis calculation of USPs generated through the low- e migration mechanism. Given the inherent uncertainties in various population statistics of both USPs and larger period planets , the purpose of this study is not to accurately reproduce all the observed population of USPs.
Instead, our goal is to illustrate the statistical trends that would be expected when USPs are generated by low- e migration. Using the above stability criterion, we find that systems can indeed become potentially dynamically unstable before forming USPs. We evolve our systems for 10 Gyr. We found that our initial population of planet systems indeed formed USPs during its evolution, with statistical properties similar to the observed population.
The USP population show substantial statistical differences with the longer period planets. We summarize their main properties below. Note that in our simulations, the final distribution for P 1 is not the same as the distribution for P of all the planets , since in some cases P 2 can also be in the range [1, 8] d, although this mixing does not affect our conclusions.
Inner planet mass: Less massive inner planets are more likely to become USPs. Systems with 0. Outer planet masses: Conversely, we find that USP production favours systems with more massive outer planets, although this is a weak effect.
The average mass for the exterior planets across all samples is Initial eccentricities: We find that USP generation is strongly dependent on the initial eccentricities, with the fraction of systems producing USPs roughly doubling for every 0.
We show the dependence of the final period distribution for various initial eccentricities in Fig. Initial inner period cut-off P min : Our results can be used to constrain the minimum period P min for the initial planet population.
Mutual inclinations: Observationally, USPs show substantially larger mutual inclinations with their closest neighbours compared with typical Kepler multis Dai et al. We find that our low- e formation mechanism for USPs naturally generates larger mutual inclinations between the inner planets.
Using this criterion, we find that The normalization of the black lines is chosen so that the total probability density integrates to unity. The solid bars show the initial period distribution for the four values of P min while the lines show the final distribution. PDF of the final RMS mutual inclination between the two inner planets after 10 Gyr of integrations for all systems in our population synthesis. The blue curve is the initial period ratio, while the green and blue curves are the final period ratios for systems that resulted in USPs and no USPs, respectively.
In this section, we evaluate this proposed formation mechanism in light of the observations of USPs and their population statistics. We then discuss some specific USP sources, potential uncertainties and future extensions to this work. As discussed in Section 1 , USPs have a number of distinct properties compared to the bulk of longer period super-Earth systems see Winn et al.
Our study shows that our low- e migration scenario produce USPs with the observed properties under a variety of initial conditions see Section 6. USPs are preferentially formed from smaller terrestrial planets with more eccentric external companions. Note that this combination of parameters is not the only one that can produce the observed P 1 distribution; there is a hyper-surface of possible initial system parameters that can fit the observations.
Even more encouragingly, this formation mechanism naturally produces higher mutual inclinations between USPs and their closest companions, a trend which has been observed by empirical studies. These two mechanisms require different initial conditions and produce USPs with distinct final configurations. In the Petrovich et al. In contrast, our low- e migration mechanism requires a proto-USP with a 1, 0 between 0. Observations e.
In addition, while Petrovich et al. In the absence of strong spin-orbit coupling, Petrovich et al. In this scenario, the fact that the hypothetical Keplere would fail to transit is compatible with observations. The observed population of USPs have radii that are mostly within the range 1. One common explanation for this observation is the scenario that USPs were initially mini-Neptunes that have had their envelopes stripped due to photoevaporation e. Winn et al. This picture may be incompatible with our model, and an alternative explanation might be preferred.
Another factor that can potentially explain the lack of larger radius USPs is the dichotomy in tidal Q 1 between rocky planets and those with more extended gaseous envelopes. In carrying out this work, we made several simplifications, which may cause additional uncertainties; we discuss them below.
Effects of mean-motion resonance: One source of uncertainty is the role of MMR in modulating the secular interactions between planets. In our population synthesis model, we considered planet systems with semimajor axes ratios 1. A careful study of the effect of MMRs on the secular interactions of multiplanet systems is beyond the scope of this work. MMRs can excite the eccentricities of the planets, independent of secular interactions. One example is Kepler, a system containing an USP accompanied by five external planets.
MacDonald et al. The resulting librations may provide the entire system with an additional source of AMD that ameliorates the effect of tidal dissipation. MMRs could bring about unexpected and interesting interactions in proto-USP systems and deserves to be the subject of further study.
In this linear regime, the eccentricity and mutual inclination evolution of the planet orbits are decoupled. Since the collisional time-scale is much shorter than the eccentricity damping and orbital decay time-scale, once two planets cross orbits, they will quickly undergo a physical collision, which can potentially inhibit USP formation.
The extent to which these dynamical instabilities occur requires investigations using numerical N -body simulations and is outside the scope of this work. Effect of additional planets : In this work, we have limited our attention to USP formation in systems with two or three planets. What happens when additional planets are present? Our framework for three-planet proto-USP systems can be easily generalized to systems with more than three planets.
In general, the generation of USPs is constrained by the dual criteria that the system must have sufficient AMD equation 46 , and the forced eccentricity on the inner planet must be sufficiently large equation The presence of additional exterior planets only help to overcome this constraint and bolster the chances of USP generation, since having more outer planets will increase the total reservoir of AMD to maintain the tidal decay of the inner planet.
Moreover, the presence of additional planets and thereby eigenmodes increases the likelihood of hitting one of eccentricity secular resonances that can speed up the tidal orbital decay time-scale.
Tidal dissipation in the host star further enhances this orbital decay when the inner planet reaches a sufficiently small period. We find that this low- e mechanism naturally produce USPs from the large population of Kepler multis, and can explain most of the observed population properties of USPs. USP formation is governed by two criteria Section 3.
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