BinaryTrajectories Class Reference

Class to compute post-Newtonian trajectories. More...

#include <BinaryTrajectories.hpp>

Public Member Functions
	BinaryTrajectories (double initial_separation, const std::array< double, 3 > &center_of_mass_velocity=std::array< double, 3 >{{0.0, 0.0, 0.0}}, bool newtonian=false)
	BinaryTrajectories (BinaryTrajectories &&)=default
BinaryTrajectories &	operator= (BinaryTrajectories &&)=default
	BinaryTrajectories (const BinaryTrajectories &)=default
BinaryTrajectories &	operator= (const BinaryTrajectories &)=default
template<typename DataType >
DataType	separation (const DataType &time) const
	Gives separation as function of time.
template<typename DataType >
DataType	orbital_frequency (const DataType &time) const
	Gives orbital frequency \(f\) as a function of time calculated from Kepler's third law \(f^2\propto\frac{1}{a^3}\) where \(a\) is calculated from separation.
template<typename DataType >
DataType	angular_velocity (const DataType &time) const
	Gives the angular velocity of the objects as a function of time. Calculated by \(\omega(t)=\frac{d\theta(t)}{dt}\) where \(\theta(t)=f(t)t\). More...
template<typename DataType >
std::array< tnsr::I< DataType, 3 >, 2 >	positions (const DataType &time) const
	Gives the positions of the two objects as a function of time.
template<typename DataType >
std::array< tnsr::I< DataType, 3 >, 2 >	positions_no_expansion (const DataType &time) const
	Same as positions, except the separation remains constant (equal to the initial separation). More...

Detailed Description

Class to compute post-Newtonian trajectories.

Details

Computes the leading post-Newtonian trajectories \(x_1^i(t)\) and \(x_2^i(t)\) for an equal-mass binary with a total mass of 1. Currently, this class implements the leading-order terms of the integral of Eq. (226) and the square root of Eq. (228) of [24] :

\begin{align} r(t) &= \left(r_0^4 - \frac{64}{5}t\right)^{1/4}, \\ \Omega(t) &= r^{-3/2}(t). \end{align}

In terms of these functions, the positions of objects 1 and 2 are

\begin{align} x_1(t) &= -\frac{r(t)}{2}\cos\left[\Omega(t) t\right] + v_x t, \\ y_1(t) &= -\frac{r(t)}{2}\sin\left[\Omega(t) t\right], + v_y t\\ x_2(t) &= \frac{r(t)}{2}\cos\left[\Omega(t) t\right], + v_x t\\ y_2(t) &= \frac{r(t)}{2}\sin\left[\Omega(t) t\right], + v_y t\\ z_1(t) &= z_2(t) = v_z t. \end{align}

These trajectories are useful for, e.g., testing a horizon-tracking control system.

Note

The trajectories could be generalized to higher post-Newtonian order if needed.

Note

If the newtonian argument is true, then this will just give Kepler's third law

Member Function Documentation

angular_velocity()

template<typename DataType >

DataType BinaryTrajectories::angular_velocity

(

const DataType &

time

)

const

Gives the angular velocity of the objects as a function of time. Calculated by \(\omega(t)=\frac{d\theta(t)}{dt}\) where \(\theta(t)=f(t)t\).

Note

For the newtonian case, orbital_frequency and angular_velocity give the same result because the orbital frequency is independent of time.

positions_no_expansion()

template<typename DataType >

std::array< tnsr::I< DataType, 3 >, 2 > BinaryTrajectories::positions_no_expansion

(

const DataType &

time

)

const

Same as positions, except the separation remains constant (equal to the initial separation).

Note

This is useful for testing the rotation control system by itself, because we want the frequency to vary, but the separation to remain the same. This way, we don't have to worry about expansion effects.

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The documentation for this class was generated from the following file:

• tests/Unit/Helpers/PointwiseFunctions/PostNewtonian/BinaryTrajectories.hpp