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dcyphr | Predicting the long-term stability of compact multi-planet systems

Abstract

The researchers have developed a machine learning model called the Stability of Planetary Orbital Configurations Klassifier (SPOCK). This model can characterize thousands of planetary systems at once. SPOCK is much quicker and more efficient than former models. The former models based predictions off of angular momentum deficit and chaos. Researchers input the eccentricity of planet orbits, radial velocity, and transit duration into SPOCK. Then, SPOCK can predict how stable the planetary system is. Any unstable combinations can then be rejected since they cannot occur. Thus, researchers can precisely characterize planetary systems.

Aims

The researchers provide a new model, SPOCK, which precisely characterizes exoplanet systems.

Introduction

It is difficult to predict how stable a planetary system is due to resonance. Unstable systems can collapse, or planets may collide into one another. Orbital resonance is the gravitational effect that planets have on one another. This effect may affect the planets’ orbit. Thus, the researchers focus on planets that are far from the host star. Past researchers have also relied on integrations from the numerical (N)-body model to rule out various planetary system characteristics. The N-body problem is that it is difficult to predict the gravitational effects that planets have on one another. As more dimensions or planets are added into the calculations, it is impossible to compute models.


Two planet systems can be precisely predicted. These models depend on mean motion resonances (MMRs), which are a type of orbital resonance. The orbits of the planets can only be separated by a certain distance as determined by the planets’ masses. But, for systems with three or more planets, the instability of the system is difficult to calculate. There seem to be other factors playing a huge role besides MMRs. Thus, many researchers disagree on predictions of these unstable systems. Every researcher seems to make a different simplification of the model.


SPOCK can characterize systems with three or more planets accurately. It can make computations for systems with over 109 orbits in time. SPOCK functions 100,000 times faster than previous models.


There have been four main previous models to predict stability. These include N-body calculations, Hill radii, angular momentum deficit, and the Mean Exponential Growth Factor of Nearby Orbits (MEGNO). MEGNO predicts chaos. All four models are useful, but lack some information.


The authors define instability as if a planet does not follow a path as expected by gravity. The researchers categorize systems with greater than 109 orbits as stable. Systems less than 109 orbits are not stable. This simplifies calculations while reasonably including valid models. They ran ~100,000 configurations to train SPOCK.


How SPOCK works is that it analyzes the first 104 orbits of a particular planetary system. The researchers provide the initial measurements. SPOCK can provide ten different predictions, such as chaos and values based on MMRs (Table 1). The data is plotted in a ten dimensional space. Afterwards, an algorithm called the XGBoost model predicts the stability of these characteristics. SPOCK then outputs the probability that the system is stable after 109 orbits. SPOCK can conduct this computation in 0.5 seconds. Using traditional integration methods would take 10 hours (Figure 1). SPOCK can also predict systems that normally have unclear predictions due to resonance.

Results

SPOCK outperforms all of the other models. The N-body model would be the perfect scenario if it could be solved. The researchers calculate the accuracy of the different models. They look at the true positive rate, which is the percentage of systems correctly classified as stable. They also look at a false positive rate, which is the percentage of systems that are incorrectly identified as stable. In a scenario where a false positive rate of 10% is acceptable, SPOCK has a true positive rate of 85%. However, the MEGNO, AMD, and Hill models all have a true positive rate of less than 50% (Figure 3). The reason why SPOCK does so well is because it combines both chaos and MMR data.


The researchers inputted a different random data set into SPOCK. The data set is representative of what real exoplanet data would look like. They wanted to ensure that SPOCK could make predictions beyond the data set that SPOCK used to learn how to make predictions. They find that SPOCK was accurate in making predictions with the new data set. It had a 94.2% true positive rate at the fixed 10% false positive rate (Figure 4). The predictions also suggest that MMRs are the primary factor that results in instability.


SPOCK can make further generalizations in systems with multiple planets. The researchers trained SPOCK with a three planet system. But, SPOCK was still reliable and can adjust its calculations for stability with five planets. SPOCK had a true positive rate of 94% and a false positive rate of 6% (Figure 5).


The researchers plot different possible eccentricities and polar angles for Kepler 431, a three-planet system. SPOCK matched the N-body predictions 96% of the time (Figure 6). Stability is important to measure because it can provide constraints for certain planet system characteristics.


Limitations

Some planet systems are restricted by certain transit-timing variations (TTVs). TTVs are differences in how long a planet takes to orbit due to gravitational effects from another planet. In these systems with certain constraints, SPOCK failed to make accurate predictions (Figure 7). The reason for this failure is that SPOCK only looks at 10 characteristics well rather than the original 60 the researchers had designed.

Methods

The researchers tested a variety of parameters for the planets. They included different masses and orbit eccentricity. The researchers also calculated various integrations necessary for the N-body model. 

Conclusion

SPOCK is a model that uses machine learning to make accurate predictions about the stability of planetary systems. SPOCK is extremely quick and works better than the current models. SPOCK will hopefully advance machine learning. The SPOCK training and data is open-source. It can be found at this link: https://github.com/dtamayo/spock