🪐Exoplanetary Science Unit 7 – Exoplanetary Orbital Dynamics & Stability

Key Concepts and Definitions

• Exoplanets are planets that orbit stars other than our Sun located outside of our solar system
• Orbital dynamics is the study of the motion of objects in gravitational fields, such as planets orbiting stars
• Kepler's laws of planetary motion describe the orbits of planets around the Sun and can be applied to exoplanets
  • First law states that planets orbit in ellipses with the star at one focus
  • Second law states that a line connecting a planet and star sweeps out equal areas in equal time intervals
  • Third law relates the orbital period to the semi-major axis of the orbit: $P^2 = a^3$, where $P$ is the orbital period and $a$ is the semi-major axis
• Orbital elements are a set of parameters that uniquely define an orbit, including semi-major axis, eccentricity, inclination, longitude of ascending node, argument of periapsis, and true anomaly
• Orbital stability refers to the long-term behavior of an orbit and whether it remains bounded and predictable over time
• Mean motion resonance occurs when the orbital periods of two bodies are in a simple integer ratio (2:1, 3:2)
• Chaos theory studies systems that are highly sensitive to initial conditions, where small perturbations can lead to drastically different outcomes over time

Orbital Elements and Parameters

• Semi-major axis ($a$) is half the longest diameter of an elliptical orbit and determines the size of the orbit
• Eccentricity ($e$) describes the shape of the orbit, with $e=0$ being a perfect circle and $0<e<1$ being an ellipse
• Inclination ($i$) is the angle between the orbital plane and a reference plane (usually the plane of the sky)
• Longitude of ascending node ($\Omega$) is the angle from a reference direction to the point where the orbit crosses the reference plane from below to above
• Argument of periapsis ($\omega$) is the angle from the ascending node to the point of closest approach to the star (periapsis)
• True anomaly ($\nu$) is the angle between the periapsis and the current position of the planet in its orbit
• Mean anomaly ($M$) is a fictitious angle that varies linearly with time, used to calculate the position of a planet in its orbit
• Orbital period ($P$) is the time it takes for a planet to complete one full orbit around its star

Types of Exoplanetary Orbits

• Circular orbits have an eccentricity of zero and maintain a constant distance between the planet and star
• Elliptical orbits have an eccentricity between 0 and 1, with the star located at one focus of the ellipse
  • Planets in elliptical orbits experience variations in distance and velocity throughout their orbit
  • Most exoplanets discovered to date have elliptical orbits
• Parabolic orbits have an eccentricity equal to 1 and are not closed, with the planet escaping the gravitational influence of the star
• Hyperbolic orbits have an eccentricity greater than 1 and are also not closed, with the planet approaching the star from infinity and then receding back to infinity
• Retrograde orbits have an inclination greater than 90°, meaning the planet orbits in the opposite direction compared to the rotation of the star
• Polar orbits have an inclination near 90°, with the planet's orbit perpendicular to the star's equatorial plane
• Kozai-Lidov mechanism can cause periodic oscillations in eccentricity and inclination for planets in highly inclined orbits around binary star systems

Factors Affecting Orbital Stability

• Mass ratio between the planet and star influences the stability of the orbit, with more massive planets having a greater effect on the star's motion
• Orbital resonances can stabilize or destabilize orbits, depending on the specific integer ratio and the eccentricities of the orbits involved
  • Mean motion resonances (2:1, 3:2) can prevent close encounters and stabilize orbits
  • Secular resonances involve long-term gravitational interactions and can lead to chaotic behavior
• Tidal forces between the planet and star can cause orbital decay and circularization over long timescales
  • Tidal heating can also affect the internal structure and habitability of exoplanets
• Gravitational interactions with other planets in the system can perturb orbits and lead to instabilities
  • Closely-spaced planets are more likely to experience strong gravitational interactions
• Stellar companions in binary or multiple star systems can dynamically influence planetary orbits through the Kozai-Lidov mechanism or other secular effects
• Galactic tides and close encounters with passing stars can perturb planetary orbits over very long timescales

Detection Methods and Orbital Determination

• Radial velocity method measures the wobble of a star caused by the gravitational pull of an orbiting planet
  • Variations in the star's radial velocity can reveal the presence of a planet and constrain its orbital elements
  • Sensitive to planets with short orbital periods and high masses
• Transit method detects the periodic dimming of a star's light as a planet passes in front of it from our perspective
  • Provides information on the planet's radius, orbital period, and inclination
  • Kepler mission used the transit method to discover thousands of exoplanets
• Direct imaging captures light from the planet itself, allowing for the determination of its orbit and atmospheric properties
  • Requires high-contrast imaging techniques to separate the planet's light from the glare of its host star
  • Sensitive to massive planets on wide orbits
• Astrometry measures the tiny back-and-forth motion of a star on the sky caused by an orbiting planet
  • Gaia mission is expected to discover thousands of exoplanets using astrometry
• Microlensing detects the temporary brightening of a background star due to the gravitational lensing effect of a foreground star and planet
  • Provides a snapshot of the planet's orbit at the time of the lensing event
  • Sensitive to planets at large orbital distances from their host stars

Multi-Planet Systems and Resonances

• Many exoplanetary systems contain multiple planets, with some hosting up to eight or more known planets
• Orbital resonances are common in multi-planet systems, with planets often locked in mean motion resonances (2:1, 3:2)
  • Trappist-1 system contains seven Earth-sized planets, with several in a chain of near-resonant orbits
  • Resonances can stabilize orbits by preventing close encounters between planets
• Secular resonances involve long-term gravitational interactions between planets and can lead to chaotic orbital evolution
  • Kozai-Lidov mechanism is a type of secular resonance that can cause oscillations in eccentricity and inclination
• Laplace resonance is a three-body resonance where the orbital periods of three planets are in a 1:2:4 ratio
  • Galilean moons of Jupiter (Io, Europa, Ganymede) are in a Laplace resonance
• Stability of multi-planet systems depends on factors such as mass ratios, orbital spacings, and resonances
  • Closely-spaced systems with high mass ratios are more likely to be unstable over long timescales
• Planetary migration can lead to the capture of planets into resonant orbits
  • Grand Tack model proposes that Jupiter and Saturn migrated inward and then outward, shaping the architecture of the inner solar system

Long-Term Stability and Chaos Theory

• Long-term stability of planetary orbits is governed by the interplay of gravitational interactions and resonances
• Chaos theory studies systems that are highly sensitive to initial conditions, where small perturbations can lead to drastically different outcomes
  • Lyapunov exponent quantifies the rate of divergence of nearby orbits and is a measure of chaos
  • Positive Lyapunov exponents indicate chaotic behavior, while negative values indicate stability
• Kolmogorov-Arnold-Moser (KAM) theorem states that stable orbits can exist in chaotic systems under certain conditions
  • KAM tori are stable regions in phase space where orbits remain quasi-periodic and bounded
• Resonance overlap criterion predicts the onset of chaos when the separation between neighboring resonances is smaller than the sum of their widths
• Stability timescales for planetary systems can range from millions to billions of years, depending on the specific configuration and perturbations
• Numerical simulations (N-body) are used to study the long-term evolution and stability of exoplanetary systems
  • Symplectic integrators are efficient numerical methods that preserve the Hamiltonian structure of the system
• Hill stability criterion determines whether a planet's orbit is stable against perturbations from a more massive companion
  • Depends on the mass ratio and separation between the planets

Applications and Case Studies

• Habitability of exoplanets is influenced by orbital dynamics, as stable orbits within the habitable zone are necessary for liquid water and life
  • Proxima Centauri b is an Earth-sized planet in the habitable zone of our nearest stellar neighbor
  • Trappist-1 system contains three planets in the habitable zone, but their orbits may be influenced by tidal heating and resonances
• Hot Jupiters are gas giant planets orbiting very close to their host stars, with orbital periods of a few days
  • 51 Pegasi b was the first hot Jupiter discovered and challenged theories of planet formation and migration
  • Tidal interactions can cause orbital decay and circularization of hot Jupiters
• Debris disks around young stars provide evidence of planet formation and orbital evolution
  • Beta Pictoris system contains a warped debris disk and a directly imaged giant planet, suggesting ongoing dynamical interactions
• Protoplanetary disks are the birthplaces of planets, and their structure and evolution are influenced by gravitational instabilities and planet-disk interactions
  • ALMA observations have revealed gaps and rings in protoplanetary disks, indicating the presence of forming planets
• Planetary system architecture and orbital dynamics can provide clues about the formation and evolution of exoplanets
  • Solar system's terrestrial planets may have formed from a narrow annulus of material, while the giant planets likely migrated to their current orbits
  • Kepler-11 system contains six planets with tightly-packed, coplanar orbits, suggesting a quiescent formation environment