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Module airball.analytic

Analytic functions for comparison to n-body results to assist in verification of n-body results.

The following documentation was automatically generated from the docstrings.

Energy Changes

airball.analytic.binary_energy(sim, particle_index=1)

Calculate the energy of a binary system, \(-(G M m)/(2 a)\).

Parameters:

Name Type Description Default
sim Simulation

The simulation with two bodies, a central star and a planet.

 required
particle_index int

The index of the particle in the simulation to calculate the energy for. Default is 1.

 1

Returns:

Name Type Description
energy Quantity

The energy of the binary system.
Example 
import rebound
import airball

sim = rebound.Simulation()
sim.add(m=1.0)
sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
energy = airball.analytic.binary_energy(sim)
Source code in src/airball/analytic.py 
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def binary_energy(sim, particle_index=1):
    """
    Calculate the energy of a binary system, $-(G M m)/(2 a)$.

    Args:
        sim (Simulation): The simulation with two bodies, a central star and a planet.
        particle_index (int, optional): The index of the particle in the simulation to calculate the energy for. Default is 1.

    Returns:
        energy (Quantity): The energy of the binary system.

    Example:
        ```python
        import rebound
        import airball

        sim = rebound.Simulation()
        sim.add(m=1.0)
        sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
        energy = airball.analytic.binary_energy(sim)
        ```
    """
    index = int(particle_index)
    unit_set = _tools.rebound_units(sim)
    G = sim.G * unit_set.length**3 / unit_set.mass / unit_set.time**2
    p = sim.particles
    if p[index].m == 0:
        message = "Planet has zero mass therefore the binary energy is zero. This may cause divide by zero error when calculating the relative change in energy."
        _warnings.warn(message, RuntimeWarning)
    return (-G * p[0].m * unit_set.mass * p[index].m * unit_set.mass / (2.0 * p[index].a * unit_set["length"])).decompose(
        list(unit_set.values())
    )

airball.analytic.energy_change_adiabatic_estimate(sim, star, averaged=False, particle_index=1)

An analytical estimate for the change in energy of a binary system due to a stellar flyby.

This function is based on the conclusions of Roy & Haddow (2003) and Heggie (2006). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

Parameters:

Name Type Description Default
sim Simulation

The simulation to calculate the energy change for.

 required
star Star or Stars

The star or stars that are flying by the binary system.

 required
averaged bool

Whether to return the averaged energy change. Default is False.

 False
particle_index int

The index of the particle in the simulation to calculate the energy change for. Default is 1.

 1

Returns:

Name Type Description
result Quantity

The estimated change in energy.
Example 
import rebound
import airball

sim = rebound.Simulation()
sim.add(m=1.0)
sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
star = airball.Star(m=1.0, b=500, v=5)
energy_change = airball.energy_change_adiabatic_estimate(sim, star)
Source code in src/airball/analytic.py 
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def energy_change_adiabatic_estimate(sim, star, averaged=False, particle_index=1):
    """
    An analytical estimate for the change in energy of a binary system due to a stellar flyby.

    This function is based on the conclusions of [Roy & Haddow (2003)](https://ui.adsabs.harvard.edu/abs/2003CeMDA..87..411R/abstract) and [Heggie (2006)](https://ui.adsabs.harvard.edu/abs/2006fbp..book...20H/abstract). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

    Args:
        sim (Simulation): The simulation to calculate the energy change for.
        star (Star or Stars): The star or stars that are flying by the binary system.
        averaged (bool, optional): Whether to return the averaged energy change. Default is False.
        particle_index (int, optional): The index of the particle in the simulation to calculate the energy change for. Default is 1.

    Returns:
        result (Quantity): The estimated change in energy.

    Example:
        ```python
        import rebound
        import airball

        sim = rebound.Simulation()
        sim.add(m=1.0)
        sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
        star = airball.Star(m=1.0, b=500, v=5)
        energy_change = airball.energy_change_adiabatic_estimate(sim, star)
        ```
    """
    index = int(particle_index)
    units = _tools.rebound_units(sim)
    t0 = 0 * units.time
    G = sim.G * units.length**3 / units.mass / units.time**2

    sim = sim.copy()
    _rotate_into_plane(sim, plane=index)
    # add_star_to_sim(sim, star, hash='flybystar', rmax=0) # Initialize Star at perihelion
    sim.move_to_hel()

    p = sim.particles
    m1, m2, m3 = (
        p[0].m * units.mass,
        p[index].m * units.mass,
        star.mass,
    )  # redefine the masses for convenience
    M12 = m1 + m2  # total mass of the binary system
    M123 = m1 + m2 + m3  # total mass of all the objects involved

    mu = G * (_tools.system_mass(sim) * units.mass + m3)
    es = _tools.calculate_eccentricity(sim, star)

    a, e = (
        p[index].a * units.length,
        p[index].e,
    )  # redefine the orbital elements of the planet for convenience
    # Case: Non-circular Binary

    n = _np.sqrt(G * M12 / a**3)  # compute the mean motion of the planet

    w, W, i = (
        star.omega,
        star.Omega,
        star.inc,
    )  # redefine the orientation elements of the flyby star for convenience
    V = star.v
    # GM123 = G*M123
    q = (-mu + _np.sqrt(mu**2.0 + star.b**2.0 * V**4.0)) / V**2.0  # compute the periapsis of the flyby star

    # Calculate the following convenient functions of the planet's eccentricity and Bessel functions of the first kind of order n.
    e1 = _jv(-1, e) - 2 * e * _j0(e) + 2 * e * _jv(2, 0) - _jv(3, e)
    e2 = _jv(-1, e) - _jv(3, e)
    e4 = _jv(-1, e) - e * _j0(e) - e * _jv(2, e) + _jv(3, e)

    # Calculate a convenient function of the planet's semi-major axis and the flyby star's periapsis.
    k = _np.sqrt((2 * M12 * q**3) / (M123 * a**3))

    # Calculate convenient functions of the flyby star's eccentricity.
    f1 = ((es + 1.0) ** (0.75)) / ((2.0 ** (0.75)) * (es * es))
    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        f2 = (3.0 / (2.0 * _np.sqrt(2.0))) * (_np.sqrt((es * es) - 1.0) - _np.arccos(1.0 / es)) / ((es - 1.0) ** (1.5))

    # Compute the prefactor and terms of the calculation done by Roy & Haddow (2003)
    prefactor = (
        (-_np.sqrt(_np.pi) / 8.0)
        * ((G * m1 * m2 * m3) / M12)
        * ((a * a) / (q * q * q))
        * f1
        * k ** (2.5)
        * _np.exp((-2.0 * k / 3.0) * f2)
    )
    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        term1 = e1 * (
            _np.sin(2.0 * w + n * t0) * _np.cos(2.0 * i - 1.0)
            - _np.sin(2.0 * w + n * t0) * _np.cos(2.0 * i) * _np.cos(2.0 * W)
            - 3.0 * _np.sin(n * t0 + 2.0 * w) * _np.cos(2.0 * W)
            - 4.0 * _np.sin(2.0 * W) * _np.cos(2.0 * w + n * t0) * _np.cos(i)
        )
        term2 = (
            e2
            * (1.0 - e * e)
            * (
                _np.sin(2.0 * w + n * t0) * (1.0 - _np.cos(2.0 * i))
                - _np.sin(2.0 * w + n * t0) * _np.cos(2.0 * i) * _np.cos(2.0 * W)
                - 3.0 * _np.sin(n * t0 + 2.0 * w) * _np.cos(2.0 * W)
                - 4.0 * _np.cos(n * t0 + 2.0 * w) * _np.sin(2.0 * W) * _np.cos(i)
            )
        )
        term3 = (
            e4
            * _np.sqrt(1.0 - e * e)
            * (
                -2.0 * _np.cos(2.0 * i) * _np.cos(2.0 * w + n * t0) * _np.sin(2.0 * W)
                - 6.0 * _np.cos(2.0 * w + n * t0) * _np.sin(2.0 * W)
                - 8.0 * _np.cos(2.0 * W) * _np.sin(2.0 * w + n * t0) * _np.cos(i)
            )
        )

    noncircular_result = None
    if averaged:
        noncircular_result = (prefactor * (e1 + e2 * (1 - e * e) + 2 * e4 * _np.sqrt(1 - e * e))).decompose(
            list(units.values())
        )
    else:
        noncircular_result = (prefactor * (term1 + term2 + term3)).decompose(list(units.values()))

    # Case: Circular Binary

    # Compute the prefactor and terms of the calculation done by Roy & Haddow (2003)
    prefactor = (
        (-_np.sqrt(_np.pi) / 8.0)
        * ((G * m1 * m2 * m3) / M12)
        * ((a * a * a) / (q * q * q * q))
        * f1
        * k ** (3.5)
        * _np.exp((-2.0 * k / 3.0) * f2)
        * (m2 / M12 - m1 / M12)
    )
    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        term1 = (1.0 + _np.cos(i)) * _np.sin(i) ** 2.0
        term2 = ((_np.cos(w) ** 3.0) - 3.0 * (_np.sin(w) ** 2.0) * _np.cos(w)) * _np.sin(n * t0)
        term3 = (3.0 * (_np.cos(w) ** 2.0) * _np.sin(w) - (_np.sin(w) ** 3.0)) * _np.cos(n * t0)

    circular_result = None
    if averaged:
        circular_result = (prefactor / _np.pi).decompose(list(units.values()))
    else:
        circular_result = (prefactor * term1 * (term2 + term3)).decompose(list(units.values()))

    return circular_result + noncircular_result

airball.analytic.energy_change_close_encounter_estimate(sim, star, particle_index=1)

An analytical estimate for the change in energy of a binary system due to a flyby star. From Equation (4.7) of Heggie (1975). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

Parameters:

Name Type Description Default
sim Simulation

The simulation with two bodies, a central star and a planet.

 required
star Star

The star used to estimate the change in energy of a binary due to the flyby star

 required
particle_index int

The index of the particle in the simulation to calculate the energy for. Default is 1.

 1

Returns:

Name Type Description
result Quantity

The change in energy of the binary system.
Example 
import rebound
from airball import Star

sim = rebound.Simulation()
sim.add(m=1.0)
sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
star = Star(m=1.0, b=100, v=5)
energy_change = airball.analytic.energy_change_close_encounter_estimate(sim, star)
Source code in src/airball/analytic.py 
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def energy_change_close_encounter_estimate(sim, star, particle_index=1):
    """
    An analytical estimate for the change in energy of a binary system due to a flyby star. From Equation (4.7) of [Heggie (1975)](https://ui.adsabs.harvard.edu/abs/1975MNRAS.173..729H/abstract). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

    Args:
        sim (Simulation): The simulation with two bodies, a central star and a planet.
        star (Star): The star used to estimate the change in energy of a binary due to the flyby star
        particle_index (int, optional): The index of the particle in the simulation to calculate the energy for. Default is 1.

    Returns:
        result (Quantity): The change in energy of the binary system.

    Example:
        ```python
        import rebound
        from airball import Star

        sim = rebound.Simulation()
        sim.add(m=1.0)
        sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
        star = Star(m=1.0, b=100, v=5)
        energy_change = airball.analytic.energy_change_close_encounter_estimate(sim, star)
        ```
    """
    index = int(particle_index)
    units = _tools.rebound_units(sim)

    c = _tools.cartesian_elements(sim, star, rmax=0)
    vx, vy, vz = c["vx"], c["vy"], c["vz"]
    dat = _np.array([c["x"].value, c["y"].value, c["z"].value]).T << units.length
    x, y, z = _tools.unit_vector(dat).T
    G = _c.G.decompose(units.UNIT_SYSTEM)  # Newton's Gravitational constant

    m1, m2 = (
        sim.particles[0].m * units.mass,
        sim.particles[index].m * units.mass,
    )  # redefine the masses for convenience
    m3 = star.m
    M12 = m1 + m2  # total mass of the binary system
    M23 = m2 + m3  # total mass of the second and third bodies

    V = _np.sqrt(vx * vx + vy * vy + vz * vz)  # compute the velocity of the flyby star at perihelion

    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        cosϕ = 1.0 / _np.sqrt(1.0 + (((star.b**2.0) * (V**4.0)) / ((G * M23) ** 2.0)).decompose())

    prefactor = (-2.0 * m1 * m2 * m3) / (M12 * M23) * V * cosϕ
    t1 = -(x * vx + y * vy + z * vz)
    t2 = (m3 * V * cosϕ) / M23
    return (prefactor * (t1 + t2)).decompose(units.values())

airball.analytic.relative_energy_change(sim, star, averaged=False, particle_index=1)

A convenience function for computing the analytical estimate for the relative change in energy of a binary system due to a flyby star. Combines energy_change_adiabatic_estimate and binary_energy functions.

Parameters:

Name Type Description Default
sim Simulation

The simulation with two bodies, a central star and a planet.

 required
star Star or Stars

The star or stars used to estimate the change in energy of a binary due to the flyby star

 required
averaged bool

Boolean indicator for averaging over the incident angles of the flyby star. Default is False.

 False
particle_index int

The index of the particle in the simulation to calculate the energy for. Default is 1.

 1

Returns:

Name Type Description
energy_change Quantity

The relative change in energy of the binary system.
Example 
import rebound
import airball

sim = rebound.Simulation()
sim.add(m=1.0)
sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
star = airball.Star(m=1.0, b=500, v=5)
energy_change = airball.analytic.relative_energy_change(sim, star)
Source code in src/airball/analytic.py 
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def relative_energy_change(sim, star, averaged=False, particle_index=1):
    """
     A convenience function for computing the analytical estimate for the relative change in energy of a binary system due to a flyby star. Combines [`energy_change_adiabatic_estimate`][airball.analytic.energy_change_adiabatic_estimate] and [`binary_energy`][airball.analytic.binary_energy] functions.

    Args:
        sim (Simulation): The simulation with two bodies, a central star and a planet.
        star (Star or Stars): The star or stars used to estimate the change in energy of a binary due to the flyby star
        averaged (bool, optional): Boolean indicator for averaging over the incident angles of the flyby star. Default is False.
        particle_index (int, optional): The index of the particle in the simulation to calculate the energy for. Default is 1.

    Returns:
        energy_change (Quantity): The relative change in energy of the binary system.

    Example:
        ```python
        import rebound
        import airball

        sim = rebound.Simulation()
        sim.add(m=1.0)
        sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
        star = airball.Star(m=1.0, b=500, v=5)
        energy_change = airball.analytic.relative_energy_change(sim, star)
        ```

    """
    unit_set = _tools.rebound_units(sim)
    qs = star.q(sim) / (sim.particles[particle_index].a * unit_set.length)
    q_crit = 2 ** (2 / 3)
    if star.N == 1:
        if qs < q_crit:
            return energy_change_close_encounter_estimate(sim, star, particle_index=particle_index) / binary_energy(
                sim, particle_index=particle_index
            )
        else:
            return energy_change_adiabatic_estimate(
                sim, star, averaged=averaged, particle_index=particle_index
            ) / binary_energy(sim, particle_index=particle_index)
    else:
        de_close = energy_change_close_encounter_estimate(sim, star, particle_index=particle_index) / binary_energy(
            sim, particle_index=particle_index
        )
        de = energy_change_adiabatic_estimate(sim, star, averaged=averaged, particle_index=particle_index) / binary_energy(
            sim, particle_index=particle_index
        )
        close_inds = qs < q_crit
        far_inds = ~close_inds
        results = _np.zeros(star.N)
        results[close_inds] = de_close[close_inds]
        results[far_inds] = de[far_inds]
        return results

airball.analytic.parallel_relative_energy_change(sims, stars, averaged=False, particle_index=1)

A convenience function for computing the analytical estimate for the relative change in energy of a binary system due to a flyby star. Combines energy_change_adiabatic_estimate and binary_energy functions parallelized over the input parameters.

Parameters:

Name Type Description Default
sims list of Simulations

The simulation with two bodies, a central star and a planet.

 required
stars Stars

The stars used to estimate the change in energy of a binary due to the flyby star

 required
averaged bool

Boolean indicator for averaging over the incident angles of the flyby star. Default is False.

 False
particle_index int

The index of the particle in the simulation to calculate the energy for. Default is 1.

 1

Returns:

Name Type Description
result list of Quantities

The relative change in energy of the binary systems.
Example 
import rebound
import airball


def setup():
    sim = rebound.Simulation()
    sim.add(m=1.0)
    sim.add(m=5e-5, a=30.0, e=0.1, f="uniform")
    return sim


stars = airball.Star(m=1.0, b=500, v=5, size=10)
sims = [setup() for _ in range(stars.N)]
energy_change = airball.analytic.parallel_relative_energy_change(sims, stars)
Source code in src/airball/analytic.py 
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def parallel_relative_energy_change(sims, stars, averaged=False, particle_index=1):
    """
    A convenience function for computing the analytical estimate for the relative change in energy of a binary system due to a flyby star. Combines [`energy_change_adiabatic_estimate`][airball.analytic.energy_change_adiabatic_estimate] and [`binary_energy`][airball.analytic.binary_energy] functions parallelized over the input parameters.

    Args:
        sims (list of Simulations): The simulation with two bodies, a central star and a planet.
        stars (Stars): The stars used to estimate the change in energy of a binary due to the flyby star
        averaged (bool, optional): Boolean indicator for averaging over the incident angles of the flyby star. Default is False.
        particle_index (int, optional): The index of the particle in the simulation to calculate the energy for. Default is 1.

    Returns:
        result (list of Quantities): The relative change in energy of the binary systems.

    Example:
        ```python
        import rebound
        import airball


        def setup():
            sim = rebound.Simulation()
            sim.add(m=1.0)
            sim.add(m=5e-5, a=30.0, e=0.1, f="uniform")
            return sim


        stars = airball.Star(m=1.0, b=500, v=5, size=10)
        sims = [setup() for _ in range(stars.N)]
        energy_change = airball.analytic.parallel_relative_energy_change(sims, stars)
        ```
    """
    unit_set = _tools.rebound_units(sims[0])
    qs = stars.q(sims[0])
    de = _np.array(
        _joblib.Parallel(n_jobs=-1)(
            _joblib.delayed(relative_energy_change)(
                sim=sims[i],
                star=stars[i],
                averaged=averaged,
                particle_index=particle_index,
            )
            for i in range(stars.N)
        )
    )
    de_close = _np.array(
        _joblib.Parallel(n_jobs=-1)(
            _joblib.delayed(relative_energy_change_close_encounter_estimate)(
                sim=sims[i], star=stars[i], particle_index=particle_index
            )
            for i in range(stars.N)
        )
    )
    close_inds = qs / (sims[0].particles[particle_index].a * unit_set.length) < 2 ** (2 / 3)
    far_inds = ~close_inds
    results = _np.zeros(stars.N)
    results[close_inds] = de_close[close_inds]
    results[far_inds] = de[far_inds]
    return results

Eccentricity Changes

airball.analytic.eccentricity_change_adiabatic_estimate(sim, star, averaged=False, particle_index=1, rmax=100000.0 * _u.au)

An analytical estimate for the change in eccentricity of a binary system due to a flyby star. From Equations (7, 9, & 12) of Heggie & Rasio (1996). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

Parameters:

Name Type Description Default
sim Simulation

The simulation with two bodies, a central star and a planet.

 required
star Star

The star used to estimate the change in energy of a binary due to the flyby star

 required
averaged bool

Boolean indicator for averaging over the incident angles of the flyby star. Default is False.

 False
particle_index int

The index of the particle in the simulation to calculate the energy for. Default is 1.

 1
rmax float or Quantity

Needed to determine the time when the star will be at perihelion. Default is \(10^5\) AU.

 100000.0 * au

Returns:

Name Type Description
result Quantity

The change in eccentricity of the binary system.
Example 
import rebound
import airball

sim = rebound.Simulation()
sim.add(m=1.0)
sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
star = airball.Star(m=1.0, b=500, v=5)
eccentricity_change = airball.analytic.eccentricity_change_adiabatic_estimate(sim, star)
Source code in src/airball/analytic.py 
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def eccentricity_change_adiabatic_estimate(sim, star, averaged=False, particle_index=1, rmax=1e5 * _u.au):
    """
    An analytical estimate for the change in eccentricity of a binary system due to a flyby star. From Equations (7, 9, & 12) of Heggie & Rasio [(1996)](https://ui.adsabs.harvard.edu/abs/1996MNRAS.282.1064H/abstract). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

    Args:
        sim (Simulation): The simulation with two bodies, a central star and a planet.
        star (Star): The star used to estimate the change in energy of a binary due to the flyby star
        averaged (bool, optional): Boolean indicator for averaging over the incident angles of the flyby star. Default is False.
        particle_index (int, optional): The index of the particle in the simulation to calculate the energy for. Default is 1.
        rmax (float or Quantity, optional): Needed to determine the time when the star will be at perihelion. Default is $10^5$ AU.

    Returns:
        result (Quantity): The change in eccentricity of the binary system.

    Example:
        ```python
        import rebound
        import airball

        sim = rebound.Simulation()
        sim.add(m=1.0)
        sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
        star = airball.Star(m=1.0, b=500, v=5)
        eccentricity_change = airball.analytic.eccentricity_change_adiabatic_estimate(sim, star)
        ```
    """

    index = int(particle_index)

    unit_set = _tools.rebound_units(sim)

    t0 = sim.t * unit_set.time
    G = sim.G * unit_set.length**3 / unit_set.mass / unit_set.time**2

    p = sim.particles
    m1, m2, m3 = (
        p[0].m * unit_set.mass,
        p[index].m * unit_set.mass,
        star.mass,
    )  # redefine the masses for convenience
    M12 = m1 + m2  # total mass of the binary system
    M123 = m1 + m2 + m3  # total mass of all the objects involved

    mu = G * (_tools.system_mass(sim) * unit_set.mass + m3)
    es = _tools.vinf_and_b_to_e(mu=mu, star_b=star.b, star_v=star.v)

    a, e = (
        p[index].a * unit_set.length,
        p[index].e,
    )  # redefine the orbital elements of the planet for convenience
    # n = _np.sqrt(G*M12/a**3) # compute the mean motion of the planet

    w, W, i = (
        star.omega,
        star.Omega,
        star.inc,
    )  # redefine the orientation elements of the flyby star for convenience

    star_params = _tools.hyperbolic_elements(sim, star, rmax)
    tperi = star_params["T"]
    Mperi = p[index].M + (p[index].n / unit_set.time) * (
        tperi - t0
    )  # get the Mean anomaly when the flyby star is at perihelion
    f = _tools.M_to_f(p[index].e, Mperi.value) << _u.rad  # get the true anomaly when the flyby star is at perihelion
    Wp = W + f
    V = star.v
    q = (-mu + _np.sqrt(mu**2.0 + star.b**2.0 * V**4.0)) / V**2.0  # compute the periapsis of the flyby star

    # Case: Non-Circular Binary

    prefactor = (
        (-15.0 / 4.0) * m3 / _np.sqrt(M12 * M123) * ((a / q) ** 1.5) * ((e * _np.sqrt(1.0 - e * e)) / ((1.0 + es) ** 1.5))
    )
    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        t1 = (_np.sin(i) * _np.sin(i) * _np.sin(2.0 * W) * (_np.arccos(-1.0 / es) + _np.sqrt(es * es - 1.0))).value
        t2 = ((1.0 / 3.0) * (1.0 + _np.cos(i) * _np.cos(i)) * _np.cos(2.0 * w) * _np.sin(2.0 * W)).value
        t3 = (2.0 * _np.cos(i) * _np.sin(2.0 * w) * _np.cos(2.0 * W) * ((es * es - 1.0) ** 1.5) / (es * es)).value

    if averaged:
        noncircular_result = (
            prefactor
            * (
                (_np.arccos(-1.0 / es).value + _np.sqrt(es * es - 1.0))
                + (2.0 / 3.0)
                + (2.0 * ((es * es - 1.0) ** 1.5) / (es * es))
            )
        ).decompose(list(unit_set.values()))
    else:
        noncircular_result = (prefactor * (t1 + t2 + t3)).decompose(list(unit_set.values()))

    # Case: Circular Binary

    prefactor = (
        (15.0 / 8.0)
        * (m3 * _np.abs(m1 - m2) / (M12 * M12))
        * _np.sqrt(M12 / M123)
        * ((a / q) ** 2.5)
        * 1.0
        / (es * es * es * ((1.0 + es) ** 2.5))
    )

    def f1(e):
        with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
            return ((e**4.0) * _np.arccos(-1.0 / e) + (-2.0 * 9.0 * e * e + 8.0 * e**4.0) * _np.sqrt(e * e - 1.0) / 15.0).to(
                _u.dimensionless_unscaled
            )

    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        t1 = (_np.cos(i) ** 2.0 * _np.sin(w) ** 2.0) * (
            f1(es) * (1.0 - 3.75 * _np.sin(i) ** 2.0)
            + (2.0 / 15.0) * (es * es - 1.0) ** 2.5 * (1.0 - 5.0 * _np.sin(w) ** 2.0 * _np.sin(i) ** 2.0)
        ) ** 2.0
        t2 = (_np.cos(w) ** 2.0) * (
            f1(es) * (1.0 - 1.25 * _np.sin(i) ** 2.0)
            + (2.0 / 15.0) * (es * es - 1.0) ** 2.5 * (1.0 - 5.0 * _np.sin(w) ** 2.0 * _np.sin(i) ** 2.0)
        ) ** 2.0

    # (187 (-1 + e^2)^5)/14400 + (199 f1[e] (4 (-1 + e^2)^(5/2) + 15 f1[e]))/7680
    if averaged:
        with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
            circular_result = prefactor * _np.sqrt(
                ((187.0 / 14400.0) * (es * es - 1.0) ** 5.0)
                + ((199.0 * f1(es)) * (4.0 * (es * es - 1.0) ** 2.5 + 15.0 * f1(es))) / 7680.0
            )
    else:
        circular_result = prefactor * _np.sqrt(t1 + t2)

    # Case: Exponential

    prefactor = (
        (3.0 * _np.sqrt(2.0 * _np.pi))
        * (m3 * (M12**0.25) / (M123**1.25))
        * ((q / a) ** 0.75)
        * (((es + 1.0) ** 0.75) / (es * es))
    )
    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        i2 = i / 2.0
        exponential = (
            -_np.sqrt(M12 / M123) * ((q / a) ** 1.5) * ((_np.sqrt(es * es - 1.0) - _np.arccos(1.0 / es)) / ((es - 1.0) ** 1.5))
        )
        angles = (_np.cos(i2) ** 2.0) * _np.sqrt(
            (_np.cos(i2) ** 4.0)
            + ((4.0 / 9.0) * _np.sin(i2) ** 4.0)
            + ((4.0 / 3.0) * (_np.cos(i2) ** 2.0) * (_np.sin(i2) ** 2.0) * _np.cos(4.0 * w + 2.0 * Wp))
        )

    if averaged:
        exponential_result = prefactor * _np.exp(exponential)
    else:
        exponential_result = prefactor * _np.exp(exponential) * angles

    # Add the ABS together, deliberately choose a norm.
    return (
        _np.nan_to_num(circular_result) + _np.nan_to_num(noncircular_result) + _np.nan_to_num(exponential_result)
    ).decompose()

airball.analytic.eccentricity_change_impulsive_estimate(sim, star, particle_index=1)

An analytical estimate for the change in eccentricity of an eccentric binary system due to a flyby star. From Equation (28) of Heggie & Rasio (1996). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

The base assumptions are that the flyby is coplanar, \(q_\star \gg a\), and \(v_\star \gg v_\mathrm{planet}\). It may work outside of this regime, but it is not guaranteed.

TODO

Make sure that the time of perihelion is calculated correctly and that the phase of the planet is correct.

Parameters:

Name Type Description Default
sim Simulation

The simulation with two bodies, a central star and a planet.

 required
star Star

The star used to estimate the change in energy of a binary due to the flyby star

 required
particle_index int

The index of the particle in the simulation to calculate the energy for. Default is 1.

 1

Returns:

Name Type Description
result Quantity

The change in eccentricity of the binary system.
Example 
import rebound
import airball

sim = rebound.Simulation()
sim.add(m=1.0)
sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
star = airball.Star(m=1.0, b=500, v=5)
eccentricity_change = airball.analytic.eccentricity_change_impulsive_estimate(sim, star)
Source code in src/airball/analytic.py 
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def eccentricity_change_impulsive_estimate(sim, star, particle_index=1):
    """
    An analytical estimate for the change in eccentricity of an eccentric binary system due to a flyby star. From Equation (28) of [Heggie & Rasio (1996)](https://ui.adsabs.harvard.edu/abs/1996MNRAS.282.1064H/abstract). The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

    The base assumptions are that the flyby is coplanar, $q_\\star \\gg a$, and $v_\\star \\gg v_\\mathrm{planet}$. It may work outside of this regime, but it is not guaranteed.

    TODO:
        Make sure that the time of perihelion is calculated correctly and that the phase of the planet is correct.

    Args:
        sim (Simulation): The simulation with two bodies, a central star and a planet.
        star (Star): The star used to estimate the change in energy of a binary due to the flyby star
        particle_index (int, optional): The index of the particle in the simulation to calculate the energy for. Default is 1.

    Returns:
        result (Quantity): The change in eccentricity of the binary system.

    Example:
        ```python
        import rebound
        import airball

        sim = rebound.Simulation()
        sim.add(m=1.0)
        sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
        star = airball.Star(m=1.0, b=500, v=5)
        eccentricity_change = airball.analytic.eccentricity_change_impulsive_estimate(sim, star)
        ```
    """

    index = int(particle_index)
    units = _tools.rebound_units(sim)
    G = sim.G * units.length**3 / units.mass / units.time**2

    p = sim.particles
    m1, m2, m3 = (
        p[0].m * units.mass,
        p[index].m * units.mass,
        star.mass,
    )  # redefine the masses for convenience
    M12 = m1 + m2  # total mass of the binary system

    a = p[index].a * units.length  # redefine the orbital elements of the planet for convenience
    V = star.v
    q = star.q(sim)  # compute the periapsis of the flyby star
    c = _tools.cartesian_elements(sim, star, rmax=0)  # Get the heliocentric coordinates of the flyby star at periapsis
    dat = _np.array([c["x"].value, c["y"].value, c["z"].value]).T << units.length
    theta = _tools.angle_between(p[index].xyz, dat)

    # print(f"{G=}, {M12=}, {m3=}, {V=}, {a=}, {q=}, {theta=}")
    return (
        (2.0 * _np.sqrt(G / M12) * (m3 / V) * _np.sqrt(a * a * a) / (q * q))
        * _np.abs(_np.cos(theta))
        * _np.sqrt((_np.cos(theta) ** 2.0) + (4.0 * _np.sin(theta) ** 2.0))
    ).decompose()

airball.analytic.parallel_eccentricity_change_adiabatic_estimate(sims, stars, averaged=False, particle_index=1, rmax=100000.0 * _u.au)

An analytical estimate for the changes in eccentricity of binary systems due to a flyby stars.

Parameters:

Name Type Description Default
sims list of Simulations

the simulation with two bodies, a central star and a planet.

 required
stars Stars

the star or stars used to estimate the change in energy of a binary due to the flyby star

 required
averaged bool

boolean indicator for averaging over the incident angles of the flyby star. Default is False.

 False
particle_index int

the index of the particle in the simulation to calculate the energy for. Default is 1.

 1
rmax float or Quantity

needed to determine the time when the star will be at perihelion. Default is \(10^5\) AU.

 100000.0 * au

Returns:

Name Type Description
results list of Quantities

The changes in eccentricity of the binary systems.
Example 
import rebound
import airball


def setup():
    sim = rebound.Simulation()
    sim.add(m=1.0)
    sim.add(m=5e-5, a=30.0, e=0.1, f="uniform")
    return sim


stars = airball.Star(m=1.0, b=500, v=5, size=10)
sims = [setup() for _ in range(stars.N)]
eccentricity_changes = airball.analytic.parallel_eccentricity_change_adiabatic_estimate(sims, stars)
Source code in src/airball/analytic.py 
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def parallel_eccentricity_change_adiabatic_estimate(sims, stars, averaged=False, particle_index=1, rmax=1e5 * _u.au):
    """
    An analytical estimate for the changes in eccentricity of binary systems due to a flyby stars.

    Args:
        sims (list of Simulations): the simulation with two bodies, a central star and a planet.
        stars (Stars): the star or stars used to estimate the change in energy of a binary due to the flyby star
        averaged (bool, optional): boolean indicator for averaging over the incident angles of the flyby star. Default is False.
        particle_index (int, optional): the index of the particle in the simulation to calculate the energy for. Default is 1.
        rmax (float or Quantity, optional): needed to determine the time when the star will be at perihelion. Default is $10^5$ AU.

    Returns:
        results (list of Quantities): The changes in eccentricity of the binary systems.

    Example:
        ```python
        import rebound
        import airball


        def setup():
            sim = rebound.Simulation()
            sim.add(m=1.0)
            sim.add(m=5e-5, a=30.0, e=0.1, f="uniform")
            return sim


        stars = airball.Star(m=1.0, b=500, v=5, size=10)
        sims = [setup() for _ in range(stars.N)]
        eccentricity_changes = airball.analytic.parallel_eccentricity_change_adiabatic_estimate(sims, stars)
        ```
    """
    return _joblib.Parallel(n_jobs=-1)(
        _joblib.delayed(eccentricity_change_adiabatic_estimate)(
            sim=sims[i],
            star=stars[i],
            averaged=averaged,
            particle_index=particle_index,
            rmax=rmax,
        )
        for i in range(stars.N)
    )

Inclination Changes

airball.analytic.inclination_change_adiabatic_estimate(sim, star, averaged=False, particle_index=1)

An analytical estimate for the sine of the change in inclination of an eccentric binary system due to a flyby star, \(\sin(\Delta i)\). This function is based on Equation (A5) of Malmberg, Davies, & Heggie (2011), with the addition of a factor of \((1 + e_\star)^{-1/2}\) to account for hyperbolic flybys. The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

Parameters:

Name Type Description Default
sim Simulation

the simulation with two bodies, a central star and a planet.

 required
star Star or Stars

The star or stars that are flying by the binary system.

 required
averaged bool

Whether to return the inclination change averaged over the incident angles. Default is False.

 False
particle_index int

the index of the particle in the simulation to calculate the inclination change for. Default is 1.

 1

Returns:

Name Type Description
result Quantity

Sine of the estimated change in the inclination, \(\sin(\Delta i)\).
Example 
import rebound
import airball

sim = rebound.Simulation()
sim.add(m=1.0)
sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
star = airball.Star(m=1.0, b=500, v=5)
inclination_change = airball.analytic.inclination_change_adiabatic_estimate(sim, star)
Source code in src/airball/analytic.py 
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def inclination_change_adiabatic_estimate(sim, star, averaged=False, particle_index=1):
    """
    An analytical estimate for the sine of the change in inclination of an eccentric binary system due to a flyby star, $\\sin(\\Delta i)$. This function is based on Equation (A5) of [Malmberg, Davies, & Heggie (2011)](https://ui.adsabs.harvard.edu/abs/2011MNRAS.411..859M/abstract), with the addition of a factor of $(1 + e_\\star)^{-1/2}$ to account for hyperbolic flybys. The orbital element angles of the flyby star are determined with respect to the plane defined by the binary orbit.

    Args:
        sim (Simulation): the simulation with two bodies, a central star and a planet.
        star (Star or Stars): The star or stars that are flying by the binary system.
        averaged (bool, optional): Whether to return the inclination change averaged over the incident angles. Default is False.
        particle_index (int, optional): the index of the particle in the simulation to calculate the inclination change for. Default is 1.

    Returns:
        result (Quantity): Sine of the estimated change in the inclination, $\\sin(\\Delta i)$.

    Example:
        ```python
        import rebound
        import airball

        sim = rebound.Simulation()
        sim.add(m=1.0)
        sim.add(m=5e-5, a=30.0, e=0.1, inc=0.1)
        star = airball.Star(m=1.0, b=500, v=5)
        inclination_change = airball.analytic.inclination_change_adiabatic_estimate(sim, star)
        ```
    """

    index = int(particle_index)

    unit_set = _tools.rebound_units(sim)
    # t0 = sim.t * unit_set["time"]
    G = sim.G * unit_set["length"] ** 3 / unit_set["mass"] / unit_set["time"] ** 2

    p = sim.particles
    m1, m2, m3 = (
        p[0].m * unit_set["mass"],
        p[index].m * unit_set["mass"],
        star.mass,
    )  # redefine the masses for convenience
    M12 = m1 + m2  # total mass of the binary system
    M123 = m1 + m2 + m3  # total mass of all the objects involved

    mu = G * (_tools.system_mass(sim) * unit_set["mass"] + m3)
    es = _tools.calculate_eccentricity(sim, star)

    # redefine the orbital elements of the planet for convenience
    a, e = p[index].a * unit_set["length"], p[index].e

    # redefine the orientation elements of the flyby star for convenience
    W, i = star.Omega, star.inc

    # compute the periapsis of the flyby star
    V = star.v
    q = (-mu + _np.sqrt(mu**2.0 + star.b**2.0 * V**4.0)) / V**2.0

    # Case: Non-Circular Binary

    prefactor = (3.0 * _np.pi * m3 / 8.0) * _np.sqrt(2.0 / (M123 * M12 * (1.0 - e * e) * (1.0 + es))) * (a / q) ** 1.5
    with _u.set_enabled_equivalencies(_u.dimensionless_angles()):
        angles = (
            _np.sin(i)
            * _np.cos(i)
            * _np.sqrt((1.0 + 3.0 * e * e) ** 2.0 * _np.sin(W) ** 2.0 + (1.0 + e * e) ** 2.0 * _np.cos(W) ** 2.0)
        ).value

    if averaged:
        noncircular_result = (prefactor).decompose(list(unit_set.values()))
    else:
        noncircular_result = (prefactor * angles).decompose(list(unit_set.values()))

    return noncircular_result