# Alpha Centauri: Disc Dynamics, Planet Stability, Detectability
## 3 Stability of Planets in Alpha Centauri
### Stability for Inner Solar System Analogues in Alpha Centauri  
In this section, we study the stability of Solar System planetary analogues orbiting around stars A, B, and C in Alpha Centauri. To set the planetary masses as a function of the distance from the host star, we use an empirical relation based on a second-order polynomial equation. This relation is calibrated based on the inner Solar System's masses and has the following expression:  
\[m(r)=-2.8664\cdot r^{2}+5.5251\cdot r-1.6671, \tag{1}\]  
Figure 5: Stability maps based on MEGNO values for test particles over a 10,000-year period. Dynamically stable regions are coloured in blue.  
Figure 6: Stability maps based on MEGNO values for 1 Earth-mass planet over a 10,000-year period. Dynamically stable regions are coloured in blue.  
where \(m\) is the planet mass in Earth masses, and \(r\in[0.35,1.9]\) represents the distance from the host star in au (see Figure 7). This oversimplified function offers a useful method for estimating the mass of hypothetical Solar System-like planets in the Alpha Centauri system. The polynomial function is a mere fitting of four data points, thereby corresponding to the inner rocky planets of the Solar System: Mercury, Venus, Earth, and Mars. Consequently, this model does not possess a robust physical justification and should be utilised with caution.  
Figure 8 illustrates the MEGNO map on the \(a\)-\(e\) plane for this kind of planets for semi-major axes ranging between 0.3 and 1.5 au. For the considered timescale (10,000 yr), these celestial bodies exhibited perfect regularity in the region under study. Therefore, stars A and B could host inner Solar System analogues on stable orbits. It is worth reminding that the stability maps presented here were done for four bodies only: three stars and a planet. The initial conditions for the planet's semimajor axis and eccentricity varies within specified values, arranged in a 100 \(\times\) 100 grid. In essence, to construct each map, 10,000 integrations of the system were required, thus spanning a duration of 10,000 years. The stability analysis of the multiplanetary systems in Alpha Cen using the MEGNO indicator is left for future work. The implications of these results are discussed further in Section 5.  
Figure 8: Stability maps based on MEGNO values for the inner Solar System analogues over a 10,000-year period. Dynamically stable regions are coloured in blue.  
Figure 7: Estimated vs. real mass of rocky planets as a function of distance.  
Finally, we examined the stability of a system with a single circumstellar planet around each star, thus focusing on the planetary mass and its semimajor axis. The map in Figure 9 displays MEGNO results for 10,000-year-long simulations of planets on circular orbits (using a 20 \(\times\) 20 grid). Dark and copper areas correspond to unstable and stable regions, respectively. Our findings suggest that there is no maximum mass limit in the region spanning approximately from 0.5 to 2.5 au, thereby indicating that planets of varied masses should be stable in this area of the parameter space.