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Implementation
==============

In order to facilitated modifying or extending the code we provide detailed
implementation documentation.

The C++ code can be split in five parts:

 * [Stellar system](\ref StellarSystem_group):
   describing the star and the planet, including the dependence of
   non-constant quantities on the system age and other parameters.

 * [Orbit solver](\ref OrbitSolver_group): implements the steps necessary to
   calculates the orbital evolution and provides an interface for specifying
   the problem.

 * [Utilities](\ref Utilities_group): general utilities useful when defining
   the stellar system or the orbit solver.

 * [Unit tests](\ref UnitTests_group): various tests defined to make sure the
   other parts do what they claim to do.

\defgroup StellarSystem_group Stellar System
\brief Star-planet system for which the orbital evolution will be calculated.

Describes the star and the planet, including:

 * [Star](\ref Star)

   * Stellar properties (e.g. radius, moments of inertia, ...) evolving with
     age.

   * Frequency dependent tidal dissipation efficiency

   * Coupled rotational evolution of a radiative core and a convective
     envelope (for low mass stars only).

   * Loss of angular momentum due to wind

   * Having initial surface rotation locked to a proto-planetary disk

 * [Planet](\ref Planet)

   * Various orbital properties

   * Rate of tidal decay and its derivatives with respect to age and system
     or orbital parameters, as well as the torque exerted on the star due to
     the tidal dissipation.

\defgroup OrbitSolver_group Orbit Solver
\brief Calculates the orbital evolution under tides.

\section evolution Orbital Evolution

The general evolution equations that govern the tidal evolution of a planet
around a star in a circular orbit aligned with the stellar rotation are given
by:

\f{eqnarray}{
    \frac{da}{dt}&=&\mathrm{sign}(\omega_\mathrm{conv}-\omega_\mathrm{orb})
    \frac{9}{2}\sqrt{\frac{G}{aM_*}}\left( \frac{R_*}{a}
    \right)^5\frac{m_p}{Q_*}
\\
    \left(\frac{dL}{dt}\right)_\mathrm{tide}&=&-\frac{1}{2}m_p
    M_*\sqrt{\frac{G}{a(M_*+m_p)}}\frac{da}{dt}
\\
    \left(\frac{dL}{dt}\right)_\mathrm{wind}&=&
        -K\omega_\mathrm{conv}
        \min(\omega_\mathrm{conv}, \omega_\mathrm{sat})^2
    \left(
        \frac{R_*}{R_\odot}\right)^{1/2} \left( \frac{M_*}{M_\odot}
    \right)^{-1/2}
\\
    \frac{dL_\mathrm{conv}}{dt}&=&\frac{\Delta L}{\tau_c} - \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt} +
    \left(\frac{dL}{dt}\right)_\mathrm{wind} +
    \left(\frac{dL}{dt}\right)_\mathrm{tide}
\\
    \frac{dL_\mathrm{rad}}{dt}&=&-\frac{\Delta L}{\tau_c} + \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt}
\\
    \Delta
    L&=&\frac{I_\mathrm{conv}L_\mathrm{rad}-
        I_\mathrm{rad}L_\mathrm{conv}}{I_\mathrm{conv}+I_\mathrm{rad}}
\f}

Where:
    * \f$M_*\f$ is the mass of the star

    * \f$R_*\f$ is the radius of the star

    * \f$m_p\f$ is the mass of the planet

    * \f$Q_*\f$ is the tidal quality factor of the star, which is possibly
      dependent on the tidal frequency
      (\f$\omega_\mathrm{conv}-\omega_\mathrm{orb}\f$)

    * \f$\mathrm{sign}(\omega_\mathrm{conv}-\omega_\mathrm{orb})\f$ takes the
      value 1 when the stellar convective zone is spinning faster than the
      planet and -1 when the reverse is true

    * \f$K\f$ is the proportionality constant, parametrizing the strength of
      the magnetic wind of the star.

    * \f$\omega_\mathrm{sat}\f$ is the wind saturation frequency

    * \f$I_\mathrm{conv}\f$ is the moment of inertia of the stellar
      convective zone

    * \f$L_\mathrm{conv}\f$ is the angular momentum of the stellar convective
      zone

    * \f$I_\mathrm{rad}\f$ is the moment of inertia of the stellar radiative
      core

    * \f$L_\mathrm{rad}\f$ is the angular momentum of the stellar radiative
      core

    * \f$\tau_c\f$ is the stellar core-envelope coupling timescale

    * \f$M_\mathrm{rad}\f$ is the mass of the stellar radiative core

    * \f$R_\mathrm{rad}\f$ is the radius of the radiative-convective boundary
      in the star

    * \f$\omega_\mathrm{conv}\equiv L_\mathrm{conv}/I_\mathrm{conv}\f$ is the
      angular frequency of the stellar convective zone.

The following default values for the stellar rotation properties reproduce
the present rotation rate of the Sun at the present age:
    * \f$K=0.35\ \frac{M_{\odot}R_{\odot}^2
      \mathrm{day}^2}{\mathrm{rad}^2\mathrm{Gyr}}\f$
    * \f$\omega_\mathrm{sat}=1.84\ \mathrm{rad}/\mathrm{day}\f$
    * \f$\tau_c=5\ \mathrm{Myr}\f$

\section stop_cond Stopping Conditions

Because Equations 1 and 3 have discontinuities (when
\f$\omega_\mathrm{conv}=\omega_\mathrm{orb}\f$ for equation 1 and when
\f$\omega_\mathrm{conv}=\omega_\mathrm{sat}\f$), it is beneficial to detect
when the evolution goes through these discontinuities and ensure that it does
not simply jump over such points, but lands on them (to some precision of
course). Such special treatment of allows the evolution to be calculated both
more accurately and more efficientlfy.

To see this, consider the discontinuity in Equation 1. Because the sign of
the tidal evolution changes when the spin of the star goes through
synchroneity with the orbit, it is possible to lock the system in a state
where the two quantities are locked to each other. If we simply let the ODE
solver handle this for us, it would result in tiny steps being taken,
oscillating between super- and sub-synchronous rotation. If we go through the
effort of detecting this and stopping the evolution precisely at the point
where synchronous rotation is achieved, we can switch to a different system
of differential equations that explicitly imposes the spin-orbit lock,
getting rid of the oscillatory behavior in the evolution of the semimajor
axis.

The spin-orbit lock may not persist indefinitely. The orbit continues to
evolve since the system is losing angular momentum due to the stellar wind.
Because of this, there may come a point when the tidal dissipation in the
star cannot drain sufficient amount of angular momentum from the orbit to
compensate for the wind loss and the extra spin up required of the star in
order to match the shorter orbital period, at which point the evolution has
to revert back to the non-locked equations.

Next, consider the discontinuity in Equation 3. Because in this case the rate
of evolution of the convective angular momentum is not discontinuous, but its
derivative is, the ODE solver can blindly jump over the
\f$\omega_\mathrm{conv}=\omega_\mathrm{sat}\f$ point resulting in the change
from saturated to non-saturated wind (or vice-versa) happening later than it
should. If on the other hand, we detect this and force the solver to land
precisely on the critical point, the calculated evolution will be more
precise.

In addition to the above discontinuities, we have several others, which are
due to the fact that we may want to include parts of the evolution of the
system before and after the planet is present.

In the present version of the code, we would like to start the evolution when
the protoplanetary disk is still present, assuming that the stellar surface
rotation is locked to the rotation rate of the inner edge of the disk. Then
at some specified age, the disk is removed (releasing the surface rotation
rate of the star from the lock) and replaced with a planet in a circular
orbit, whose evolution we then follow to the point when either the star moves
off the main-sequenc or the orbit shrinks so much that the planet is tidally
destroyed or engulfed by the star. If the planet dies before the star, the
angular momentum of the planetary orbit at the moment of death is added to
the stellar convective zone and the subseqent rotational evolution of the
star is followed until the end of its main sequence phase.

Finally, often various applications for which calculating the planet-star
orbital evolution is necessary require detecting when some special conditions
in the evolution are met.

In order to accomplish stopping the evolution when some condition is met,
StoppingCondition objects have been introduced. Each Stopping Condition
should be a quantity that varies smoothly with the evolution (continuous and
continuously differentiable up to at least third order) and is zero exactly
when the condition which should stop the evolution is met (see for example
SynchronizedCondition, BreakLockCondition, PlanetDeathCondition).

It is possible for users to define external stopping conditions. This is
accomplished by editing the files ExternalStoppingConditions.h and
ExternalStoppingConditions.cpp. The user needs to inherit a class from
the #ExternalStoppingCondition class and define operator() for it, which
should return a valarray of real values which vary smoothly with age and the
orbital parameters and one of which is zero exactly for the evolution states
that the evolution needs to meet precisely. If it is known, operator() should
also overwrite an input derivatives valarray with the rate at which the
stopping condition values change with age (per Gyr). For a more detailed
description of the input arguments see #StoppingCondition::operator()().
**Recompliing the code is necessary after defining or changing external
stopping conditions!**

\section modes Evolution Modes

As discussed above, discontinuities in the evolution require switching
between different systems of differential equations when some
StoppingCondition is encountered. The system of differential equations to use
at any given time is determined by an evolution mode (#EvolModeType) and the
wind saturation state (#WindSaturationState).

The wind saturation state only affects how the following quantity is
calculated:
\f[
    \left(\frac{dL_\mathrm{conv}}{dt}\right)_\mathrm{wind}=\left\{
    \begin{array}{l@{\quad\mathrm{if}\quad}l}
        -K\omega_\mathrm{conv}^3
        \left(
            \frac{R_*}{R_\odot}\right)^{1/2} \left( \frac{M_*}{M_\odot}
        \right)^{-1/2} & \mathrm{NOT\_SATURATED}\\
        -K\omega_\mathrm{conv}
            \min(\omega_\mathrm{conv}, \omega_\mathrm{sat})^2
        \left(
            \frac{R_*}{R_\odot}\right)^{1/2} \left( \frac{M_*}{M_\odot}
        \right)^{-1/2} & \mathrm{UNKNOWN}\\
        -K\omega_\mathrm{conv} \omega_\mathrm{sat}^2
        \left(
            \frac{R_*}{R_\odot}\right)^{1/2} \left( \frac{M_*}{M_\odot}
        \right)^{-1/2} & \mathrm{SATURATED}
    \end{array}
    \right.
\f]

Depending on the evolution mode, the evolution is calculated using different
sets of  variables and equations governing their evolution.

\subsection LOCKED_TO_DISK

This is the evolution mode for a system for which the protoplanetary disk is
still present. In this case, the spin of the surface convective zone is held
at some prescribed constant value \f$\omega_\mathrm{disk}\f$, representing
the orbital frequency of the inner edge of the protoplanetary disk, to which
the stellar surface rotation is locked.

In this case, the only evolution variable is \f$L_\mathrm{rad}\f$ and the
equation governing its evolution is:
\f{eqnarray}{
    \frac{dL_\mathrm{rad}}{dt}&=&-\frac{\Delta L}{\tau_c} + \frac{2}{3}
        R_\mathrm{rad}^2 \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt}
        \Delta
\\
    L&=&\frac{I_\mathrm{conv}}{I_\mathrm{conv}+I_\mathrm{rad}}
        \left(L_\mathrm{rad}-I_\mathrm{rad}\omega_\mathrm{disk}\right)
\f}

The evolution will switch out of this mode at a prescribed disk-dissipation
age. The subsequent evolution mode is FAST_PLANET, LOCKED_TO_PLANET or
SLOW_PLANET depending on the initial semimajor axis at which the planet
appears.

\subsection FAST_PLANET

This is the evolution mode for a system in which the orbital period is
shorter than the spin period of the stellar surface convective zone. In this
case the evolution variables are: \f$a^{6.5}\f$, \f$L_\mathrm{conv}\f$,
\f$L_\mathrm{rad}\f$, and the equations for their evolution are:

\f{eqnarray*}{
    \frac{da^{6.5}}{dt}&=&
        -\frac{9}{13}\sqrt{\frac{G}{M_*}}R_*^5\frac{m_p}{Q_*}
\\
    \left(\frac{dL_\mathrm{conv}}{dt}\right)_\mathrm{tide}&=&-\frac{1}{2}m_p
    M_*\sqrt{\frac{G}{a(M_*+m_p)}}\frac{da}{dt}
\\
    \frac{dL_\mathrm{conv}}{dt}&=&\frac{\Delta L}{\tau_c} - \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt} +
    \left(\frac{dL}{dt}\right)_\mathrm{wind} +
    \left(\frac{dL}{dt}\right)_\mathrm{tide}
\\
    \frac{dL_\mathrm{rad}}{dt}&=&-\frac{\Delta L}{\tau_c} + \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt}
\\
    \Delta L&=&\frac{I_\mathrm{conv}L_\mathrm{rad}-
        I_\mathrm{rad}L_\mathrm{conv}}{I_\mathrm{conv}+I_\mathrm{rad}}
\f}

The reason for using \f$a^{6.5}\f$ instead of \f$a\f$ as the evolution
variable is evident from the first equation above. The rate at which
\f$a^{6.5}\f$ evolves is independent of \f$a\f$. In fact, for a constant
\f$Q_*\f$ it only changes due to \f$R_*\f$ evolving. This allows the ODE
solver to take much larger steps when the orbit has shrunk than would
otherwise be possible.

This evolution mode can end in one of two ways:
    * The orbit shrinks sufficiently to either be inside the star or inside
      the critical size when the planet gets destroyed by the tides from the
      star. In which case the subsequent evolution mode is NO_PLANET.
    * The spin period of the stellar surface convective zone matches the
      orbital period. In which case the subsequent evolution mode is either
      LOCKED_TO_PLANET or SLOW_PLANET, depending on whether the transfer of
      angular momentum due to tides is sufficient to keep the lock.

\subsection LOCKED_TO_PLANET

This is the evolution mode for a system in which the surface rotation of the
star is held locked to the orbit by the dissipation of the stellar tides. The
evolution variables are \f$a\f$ and \f$L_\mathrm{rad}\f$ and the equations:
\f{eqnarray*}{
    \frac{da}{dt}&=&2\frac{T - \dot{I}_\mathrm{conv}/a}
        {M_*m_p/(M_*+m_p) - 3I_\mathrm{conv}/a^2}
\\
    \left(\frac{dL}{dt}\right)_\mathrm{tide}&=&-\frac{1}{2}m_p
    M_*\sqrt{\frac{G}{a(M_*+m_p)}}\frac{da}{dt}
\\
    \left(\frac{dL}{dt}\right)_\mathrm{wind}&=&
        -K\omega_\mathrm{conv}
        \min(\omega_\mathrm{conv}, \omega_\mathrm{sat})^2
    \left(
        \frac{R_*}{R_\odot}\right)^{1/2} \left( \frac{M_*}{M_\odot}
    \right)^{-1/2}
\\
    \left(\frac{dL}{dt}\right)_\mathrm{coup}&=&
        \frac{\Delta L}{\tau_c} - \frac{2}{3}
        R_\mathrm{rad}^2
        \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt}
\\
    T&=&\frac{a}{(M_*+m_p)G}\left[\left(\frac{dL}{dt}\right)_\mathrm{coup} +
        \left(\frac{dL}{dt}\right)_\mathrm{wind}\right]
\\
    \frac{dL_\mathrm{conv}}{dt}&=&
    \left(\frac{dL}{dt}\right)_\mathrm{coup} +
    \left(\frac{dL}{dt}\right)_\mathrm{wind} +
    \left(\frac{dL}{dt}\right)_\mathrm{tide}
\\
    \frac{dL_\mathrm{rad}}{dt}&=&-\frac{\Delta L}{\tau_c} + \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt}
\\
    \Delta
    L&=&\frac{I_\mathrm{conv}L_\mathrm{rad}-
        I_\mathrm{rad}L_\mathrm{conv}}{I_\mathrm{conv}+I_\mathrm{rad}}
\f}

This evolution mode can end either by the planet dying or by the rate of
transfer of angular momentum from the orbit to the star falling below what is
required to keep the lock. In the first case, the subsequent evolution mode
is NO_PLANET and in the other case it is either FAST_PLANET or SLOW_PLANET.

\subsection SLOW_PLANET

This is the evolution mode for a system in which the orbital period is
longer than the spin period of the stellar surface convective zone. In this
case the evolution variables are: \f$a^{6.5}\f$, \f$L_\mathrm{conv}\f$,
\f$L_\mathrm{rad}\f$, and the equations for their evolution are:

\f{eqnarray*}{
    \frac{da^{6.5}}{dt}&=&
        \frac{9}{13}\sqrt{\frac{G}{M_*}}R_*^5\frac{m_p}{Q_*}
\\
    \left(\frac{dL_\mathrm{conv}}{dt}\right)_\mathrm{tide}&=&-\frac{1}{2}m_p
    M_*\sqrt{\frac{G}{a(M_*+m_p)}}\frac{da}{dt}
\\
    \frac{dL_\mathrm{conv}}{dt}&=&\frac{\Delta L}{\tau_c} - \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt} +
    \left(\frac{dL}{dt}\right)_\mathrm{wind} +
    \left(\frac{dL}{dt}\right)_\mathrm{tide}
\\
    \frac{dL_\mathrm{rad}}{dt}&=&-\frac{\Delta L}{\tau_c} + \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt}
\\
    \Delta L&=&\frac{I_\mathrm{conv}L_\mathrm{rad}-
        I_\mathrm{rad}L_\mathrm{conv}}{I_\mathrm{conv}+I_\mathrm{rad}}
\f}

This evolution mode can end only if The spin period of the stellar surface
convective zone matches the orbital period. In which case the subsequent
evolution mode is either LOCKED_TO_PLANET or FAST_PLANET, depending on
whether the transfer of angular momentum due to tides is sufficient to keep
the lock.

\subsection NO_PLANET

This is the evolution mode for a star without a planet in orbit and no
proto-planetary disk. Usually this state is reached after the planet dies.
The evolution variables are \f$L_\mathrm{conv}\f$ and \f$L_\mathrm{rad}\f$,
and their evolution is given by:
\f{eqnarray*}{
    \frac{dL_\mathrm{conv}}{dt}&=&\frac{\Delta L}{\tau_c} - \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt} +
    \left(\frac{dL}{dt}\right)_\mathrm{wind}
\\
    \frac{dL_\mathrm{rad}}{dt}&=&-\frac{\Delta L}{\tau_c} + \frac{2}{3}
    R_\mathrm{rad}^2
    \omega_\mathrm{conv} \frac{dM_\mathrm{rad}}{dt}
\\
    \Delta
    L&=&\frac{I_\mathrm{conv}L_\mathrm{rad}-
        I_\mathrm{rad}L_\mathrm{conv}}{I_\mathrm{conv}+I_\mathrm{rad}}
\f}

This evolution mode persists until the end of the star's lifetime.

\defgroup Utilities_group Utilities
\brief General utilities useful when defining the stellar system or
the orbit solver.

\defgroup UnitTests_group Unit Tests
\brief A collection of tests to verify the rest of the code.