• It can be cast in terms of the relative motion and total gravitational mass
• motion is in a plane

The two-body problem for bodies at a mass ratio . ↗ In the inertial frame both bodies orbit around the center of mass (cross). ↘ relative motion with one of the bodies put at the center. This representation is far more common (and useful) when . However, motion is not in an inertial coordinate frame. The orbit is elliptical, here with an eccentricity of e=0.7.

• It can be cast in terms of the relative motion and total gravitational mass
• motion is in a plane
• conserved quantities:

  Energy
  Angular momentum
  Eccentricity vector

• the orbit

  which describes an ellipse with semi-major axis and semi-minor axis .

• the orbit

  which describes an ellipse with semi-major axis and semi-minor axis .

• energy, angular momentum in terms of orbital elements:
• the orbital period

  where n is the mean motion. This is Kepler's law (not to be confused with Kepler's equation [see below], which describes how is related to time ).

Three anomalies

• Mean anomaly (time),

  where t is time, n the mean motion, and t₀ the time of pericenter passage

• True anomaly

  indicates the position of the particle in the orbit

• Eccentric anomaly E

  another indicator for the position of the orbit (see figure) the eccentric and true anomalies are geometrically related:

Kepler's equation relates the eccentric and mean anomalies

Hence, while we can obtain the mean anomaly (or time) as function of , i.e., , no analytic solutions exist for the reverse problem, . This amounts to solving Kepler's equation — a transcendental equation that must be solved numerically.

There are 6 orbital elements:

1. semi-major axis a
2. eccentricity e
3. argument of pericenter ω.

   this is the angle with a reference direction. In the planar case the reference is fixed (e.g., the x-axis); in the 3D case, it is the angle with respect to the ascending node Ω.

   Sometimes, this angle is replaced by the longitude of pericenter

4. true anomaly

   the position of the particle w.r.t. pericenter. Equivalently, the true longitude Another option is to indicate the positon by the mean anomaly M or by the mean longitude which is not a geometric angle.

for 2D problems, these 4 elements suffice.

There are 6 orbital elements:

1. semi-major axis a
2. eccentricity e
3. argument of pericenter ω.
4. true anomaly
5. inclination i: the angle of the orbit w.r.t. the reference plane.
6. longitude of the ascending node (Ω in the figure →). The angle to the line in the reference plane where the particle intersects the reference planet in the ascending direction

In the guiding center approximation the orbit solution, as function of mean anomaly M reads, to first order in eccentricity:

This is a superposition of:

1. uniform circular motion (M is linear with time); and
2. uniform elliptical motion with the ellipse having a 2:1 aspect ratio.

Note. The 2:1 aspect ratio and the epicycle frequency are valid for a Keplerian potential. This can be generalized to arbitrary potentials.

Note. Historically, pre-Copernican Europeans used circular epicyles to describe the motion of planets under the Ptolemaic model (Earth at center). They did not stop with one epicycle!

↗ Guiding center approximation for a particle in an orbit.
→ motion with respect to the guiding center.

• conduct analysis in comoving, non-inertial frame where bodies 1 and 2 are at rest and:
  • (test particle)
  • the binary is in a circular orbit resulting in the e.o.m.:

    the first term on the RHS is the Coriolis force, the second the centrifugal force. We insert the mean motion of the binary n for ω.

• one integral of motion, the Jacobean energy:

  where and are the two-body potentials

• The e.o.m. is then

• Equipotential curves

  in the rotating frame. Applications range from:
  — stellar binaries evolution
  — dynamics of asteroids

• The J value corresponding to these zero velocity curves determine the regions that are (potentially) accessible to the particle.

• Equipotential curves

  in the rotating frame. Applications range from:
  — stellar binaries evolution
  — dynamics of asteroids

• The J value corresponding to these zero velocity curves determine the regions that are (potentially) accessible to the particle.

• Equipotential curves

  in the rotating frame. Applications range from:
  — stellar binaries evolution
  — dynamics of asteroids

• The J value corresponding to these zero velocity curves determine the regions that are (potentially) accessible to the particle.

• Equipotential curves

  in the rotating frame. Applications range from:
  — stellar binaries evolution
  — dynamics of asteroids

• Stationary Lagrangian points where the potential vanishes

  5 in total:

  • L1, L2, and L3 along the binary axis, the colinear points
  • L4, and L5, the triangular points (r₁ = r₂ = d)

• Equipotential curves

  in the rotating frame. Applications range from:
  — stellar binaries evolution
  — dynamics of asteroids

• Stationary Lagrangian points where the potential vanishes

  5 in total:

  • L1, L2, and L3 along the binary axis, the colinear points
  • L4, and L5, the triangular points (r₁ = r₂ = d)
• Two types of coorbital orbits
  • Trojans orbit either L4 or L5
  • Horseshoes orbit both L4 and L5

Stellar binaries are a classic application of the three-body problem

1. One of the stars in the binary expands...

   e.g., during the RGB phase. This does not necessarily have to be the most massive star

2. The star gradually fill its Roche lobe...

   If one star has (almost) fills its Roche lobe, the binary is referred to as semi-detached

3. Once it has reached the L1 point, material flows to the secondary

   Possible (but not necessarily) it will also fill the Roche Lobe of the secondary.
   Accretion onto the secondary can substantially change its evolution

4. Upon, further expansion, the Roche lobes merge

   This is the common envelope phase. The system has now become a contact binary

Stellar binaries are a classic application of the three-body problem

1. One of the stars in the binary expands...

   e.g., during the RGB phase. This does not necessarily have to be the most massive star

2. The star gradually fill its Roche lobe...

   If one star has (almost) fills its Roche lobe, the binary is referred to as semi-detached

3. Once it has reached the L1 point, material flows to the secondary

   Possible (but not necessarily) it will also fill the Roche Lobe of the secondary.
   Accretion onto the secondary can substantially change its evolution

4. Upon, further expansion, the Roche lobes merge

   This is the common envelope phase. The system has now become a contact binary

Stellar binaries are a classic application of the three-body problem

1. One of the stars in the binary expands...

   e.g., during the RGB phase. This does not necessarily have to be the most massive star

2. The star gradually fill its Roche lobe...

   If one star has (almost) fills its Roche lobe, the binary is referred to as semi-detached

3. Once it has reached the L1 point, material flows to the secondary

   Possible (but not necessarily) it will also fill the Roche Lobe of the secondary.
   Accretion onto the secondary can substantially change its evolution

4. Upon, further expansion, the Roche lobes merge

   This is the common envelope phase. The system has now become a contact binary

Stellar binaries are a classic application of the three-body problem

1. One of the stars in the binary expands...

   e.g., during the RGB phase. This does not necessarily have to be the most massive star

2. The star gradually fill its Roche lobe...

   If one star has (almost) fills its Roche lobe, the binary is referred to as semi-detached

3. Once it has reached the L1 point, material flows to the secondary

   Possible (but not necessarily) it will also fill the Roche Lobe of the secondary.
   Accretion onto the secondary can substantially change its evolution

4. Upon, further expansion, the Roche lobes merge

   This is the common envelope phase. The system has now become a contact binary

Consequences Roche Lobe overflow

• Accretion. Angular momentum conservation results in an accretion disk
• Energy release. The potential at L₁ is much higher than at the surface of the secondary
• Nova

  hydrogen spills over and is ignited on the surface of the White Dwarf

• Type 1a supernova

  upon reaching the Chandrasekhar mass, the Carbon in the White Dwarf's core will explosively ignite

In Hill's approximation a new coordinate system is erected centered on the secondary and co-moving with the mean motion n of the system.

• In Hill's approximation, we put the origin at the secondary;
• The L1 and L2 points lie at a distance
  which is known as the Hill radius.

  Interpretation. At distances we can ignore m₁'s (the stellar) gravity; at distances the secondary's gravity amounts to a small perturbation (small angle scattering). At distances both bodies contribute equally, giving rise to complex 3-body dynamics.

• The L1 and L2 points lie at a distance
  which is known as the Hill radius.

  Interpretation. At distances we can ignore m₁'s (the stellar) gravity; at distances the secondary's gravity amounts to a small perturbation (small angle scattering). At distances both bodies contribute equally, giving rise to complex 3-body dynamics.

• The e.o.m. are, in terms of Cartesian coordinates x, y :

→ sample integrations of trajectories in Hill's approximation. Particles initially have 0 eccentricity. Grey: particles do not enter the Hill sphere; orange: particles enter and exit the Hill sphere; red: particles that accrete onto the planet Encounters that enter the Hill sphere will experience strong scatterings.

• The Roche limit refers to the point where the secondary overflow its Roche lobe; it is tidally destructed when or at a distance

  Based on the Hill radius expression, the prefactor should have been 3^1/3. But the point-mass assumptions underlying these expressions are no longer be valid at the point where bodies are tidally torn apart. The expression (first derived by Eduard Roche) accounted for these effects

  Note in case of planet satellites, we are represented with a hierarchy: star—planet—satellite. We say that the satellite orbits within the planet's Hill sphere, and that they are destroyed once they pertrude inside the planet's Roche limit.

• Therefore, each body has:
  • a Roche limit, determined by the density ratio with a (smaller) secondary body
  • a Hill sphere, determined by the mass ratio and the distance to a larger (primary) body

• Evaluate the effective potential at :

  where is the tidal potential and is the Legendre polynomial.

• The tidal acceleration on the planet's surface is

  where the components are the Cartesian x and y. Tides on a planet are created due to a difference in the gravitational force of the perturber over its surface.

• The (half-) amplitude of the tidal bulge is estimated as

we consider tides raised on the planet, by the satellite, in the equatorial plane

Tides are raised because the gravitational force from the companion is differently distributed across the planet's surface. The tidal force is the difference between Fs and the average force

Q: What are the amplitudes of the tides on the Earth, induced by the Moon and the Sun, respectively?

• Evaluate the effective potential at :

  where is the tidal potential and is the Legendre polynomial.

• The tidal acceleration on the planet's surface is

  where the components are the Cartesian x and y. Tides on a planet are created due to a difference in the gravitational force of the perturber over its surface.

• The (half-) amplitude of the tidal bulge is estimated as

  for the Earth this gives a tide (full amplitude) of 71cm (Moon) and 33cm (Sun)?

• Due to friction, the tidal bulge is mis-aligned with the planet by an angle the lag angle
• The tidal quality factor Q expresses the rate at which the energy contained in the tidal bulge dissipates

  It's defined as

  where is the tidal frequency — for a diurnal tide (Earth-Moon system) — and is the maximum energy stored in the tidal deformation. This expression arises, because it is reasonable to expect that the rate of energy dissipation is proportional to tidal frequency and .

  • If there is time delay between the tidal forcing and the tidal response, it is straightforward to show that .
  • In the case of an diurnal tide, a lag of between forcing & response corresponds to an angle of . Hence, .

• Due to friction, the tidal bulge is mis-aligned with the planet by an angle the lag angle
• The tidal quality factor Q expresses the rate at which the energy contained in the tidal bulge dissipates

• The (tilted!) tidal bulge results in a back-reaction force on the planet, which:

• dissipates energy at rate :

  This heats the planets (tidal heating) at the expense of the total spin and orbital angular momentum

• produce a torque on the satellite
  • For positive ε (a tidal lag), Γ is positive, the satellite gains orbital angular momentum, whereas the planet spins down. Angular momentum is transferred from the planet (spin) to the satellite (orbit) .
  • Vice-versa for negative ε. As a result the system will move towards co-rotation where the orbit's mean motion equals the spin rotation rate:

• The tidal bulge generates a quadrupole potential

  This can be understood as follows. The excess/deficit mass in the tidal bulge is . The quadrupole potential generated by will be be proportial to , and where is aligned with the bulge. Finally, k_2p — the Love number — encapsulates all our uncertainty regarding the tidal response. For example, rigid planets will have small .

  In expressions like these, the tidal lag parameter ε is usually replaced by the tidal quality factor Q_p.

• The net energy (loss) is .

  (for positive Γ_T) the satellite gains energy while the planet loses energy

Q: What happens to this energy?

• The tidal torque elevates the orbit of the satellite while spinning down the planet

  where is the moment of inertia of the planet its spin, m_μ the reduced mass, the mean motion and the semi-major axes of the orbit.

• Energy is dissipated in the planetary body

• Key timescales

  where is the moment of inertia of the planet and signs are consistent with a positive torque (orbit expands and planet spins down).

The equations

also hold when there is no (net) torque ( )!

Note.: In discussion below (↓), and (↗) I have interchanged the satellite/planet ( ) as it is more usual that a massive body (planet/star) induces tides in a smaller body (satellite/planet).

When the satellite is tidally locked ( ), there is no (fixed/net) tidal lag: and does not change. However, tides still operate, e.g., when the satellite is inclined or is in an eccentric orbit. This is because tides dissipate energy in the interior of the planet ( ), resulting in energy being extracted from the orbit and circularizing it (due to angular momentum conservation).

↑ Tidal forcing due to a satellite on an eccentric orbit, in a frame where the guiding center (open circle) is at rest. As the planet is tidally locked, there is no rotation in this frame (black arrow is fixed). The tidal bulge follows (but lags) the position of the satellite. Energy is dissipated due to the radial and azimuthal flexing of the bulge. The tidal deformation and lag angle are greatly exaggerated for illustrative purposes.

The Galilean satellite system is an example of three-body resonance (Laplace resonance)

The period ratios of the planets obey a conmensurability, where the orbital period ratio is a rational number:

• same for the second and third planet
• o is called the resonance order

Q: Shown are planets at/near a 3:2 mean motion resonance. Inner body is on a circular orbit, outer on an eccentric. Which are in resonance?

• Consider a first order j:j+1 resonance between
  1. an inner planet on a circular orbit
  2. an exterior planet on an eccentric orbit
• Let the resonance angle be defined

  Hence, the resonance angle corresponds to the mean anomaly M₂ at conjuction. When is fixed, conjunction always take place at the same point in the orbit.

• Generally, the condition for resonance is that librates around a fixed value

  That is, is restricted to a certain range. To librate contrasts to circulate — in that case takes on all values between 0 and 360 degrees.

• At conjunction, the inner planet 1 overtakes the outer planet 2

→ inner planet on circular orbit; outer on eccentric orbit with conjunctions just past pericenter. Positions at equal times are connected by dots.

• At conjunction, the inner planet 1 overtakes the outer planet 2

  for an eccentric outer planet, stable conjunctions take place at apocenter.

• If the conjuction is not precisely at pericenter (apocenter) the encounter is asymmetric:
  • the distance pre- and post encounter;
  • the relative motion pre- and post encounter;
• The asymmetry results in a net torque on planet 2, which changes its angular momentum

  where F_θ is the azimuthal component and Δl the change in angular momentum of planet 2.

→ inner planet on circular orbit; outer on eccentric orbit with conjunctions just past pericenter. Positions at equal times are connected by dots. Before the conjunction 2 experiences a gravitational force (from 1) opposite to its direction of motion. After the encounter, planet 1 pulls planet 2 along. Only the azimuthal component of the direct force is shown with red arrows. In this setup the resonance would move to apocenter.

The resonance forcing equations for a first order (j+1:j) resonance read:

Above expression valid for a j+1:j resonance, inner perturber on circular orbit, where , is the mean motion of the inner perturber and ( ) refer to the outer body on an eccentric orbit.

Notes:

1. For the perturber on an outer, circular orbit, re-label 1→2. Furthermore, and . See Table for values of .
2. Expressions are valid to first order in eccentricity only.
3. has been substituted for ω through the resonance angle and has been defined as a "distance" from resonance.
4. These expression only hold water if librates (domain of is restricted).
5. If does not librate, there is no resonance forcing. But, to second order in eccentricity (expressions proportional to e²), eccentricity can still be forced in the non-resonant (secular) way.

valid for an inner perturber on a circular orbit (left column; with δ₁=1) or an outer perturber on circular orbit (right column; with δ₂=0)

Several applications and insights follow from these equations

• the timescale over which oscillates — the libration time — and the libration width
• the condition for trapping into resonance.

Another, complementary way to study these problems are direct N-body integrations, which numerically solve the equation of motion.

Several applications and insights follow from these equations

• the timescale over which oscillates — the libration time — and the libration width
• the condition for trapping into resonance.

Another, complementary way to study these problems are direct N-body integrations, which numerically solve the equation of motion.

Several applications and insights follow from these equations

• the timescale over which oscillates — the libration time — and the libration width
• the condition for trapping into resonance.

Another, complementary way to study these problems are direct N-body integrations, which numerically solve the equation of motion.

Do planets get trapped?

• certain systems, like TRAPPIST-1, are characterized by a many-planet resonance chain

  in case of TRAPPIST-1, its seven planets are located near 8:5, 5:3, 3:2, 3:2, 4:3, and 3:2.

• statistically, there is an excess of the planet frequency at resonance locations

  certainly not every planet pair is now in resonance, e.g., solar system

Do planets get trapped?

• certain systems, like TRAPPIST-1, are characterized by a many-planet resonance chain

  in case of TRAPPIST-1, its seven planets are located near 8:5, 5:3, 3:2, 3:2, 4:3, and 3:2.

• statistically, there is an excess of the planet frequency at resonance locations

  certainly not every planet pair is now in resonance, e.g., solar system