Taylor adaptive propagators#
Taylor adaptive integrators are offered in pykep wrapping some of the functionalities of Heyoka [BI21] python package. Their variational version is also offered (at order one) as to be able to produce stms and, where needed, more. Higher order variational equations can also be obtained directly using the available dynamics and using Heyoka [BI21] syntax.
Stark#
- pykep.ta.get_stark(tol)#
Returns a Taylor adaptive propagator (Heyoka) for the Stark problem retreiving one from a global cache and making a copy.
In pykep, abusing a term well established in electrodynamics, this is the initial value problem of a fixed inertial thrust mass-varying spacecraft orbiting a main body.
If the requested propagator was never created this will create it, else it will return the one from the global cache, thus avoiding jitting.
The dynamics is that returned by
stark_dyn().- Args:
tol (
float): the tolerance of the Taylor adaptive propagator.- Returns:
hy::taylor_adaptive: The Taylor adaptive propagator.- Examples:
>>> import pykep as pk >>> ta = pk.ta.get_stark(tol = 1e-16) >>> ta.time = 0. >>> ta.state[:] = [1.,0.,0.,0.,1.,0.,1.] >>> mu = 1. >>> veff = 1. >>> thrust = [0., 0., 0.] >>> tof = 1. >>> ta.pars[:] = [mu, veff] + thrust >>> ta.propagate_until(tof)
- pykep.ta.get_stark_var(tol)#
Returns a (order 1) variational Taylor adaptive propagator (Heyoka) for the Stark problem retreiving one from a global cache and making a copy.
In pykep, abusing a term well established in electrodynamics, this is the initial value problem of a fixed inertial thrust mass-varying spacecraft orbiting a main body.
The dynamics is that returned by
stark_dyn(): and also used inget_stark()- Args:
tol (
float): the tolerance of the Taylor adaptive propagator.- Returns:
hy::taylor_adaptive: The Taylor adaptive propagator.- Examples:
>>> import pykep as pk >>> ta = pk.ta.get_stark_var(tol = 1e-16) >>> ta.time = 0. >>> ta.state[:] = [1.,0.,0.,0.,1.,0.,1.] >>> mu = 1. >>> veff = 1. >>> thrust = [0., 0., 0.] >>> tof = 1. >>> ta.pars[:5] = [mu, veff] + thrust >>> ta.propagate_until(tof)
- pykep.ta.stark_dyn()#
The dynamics of the Stark problem.
In pykep, abusing a term well established in electrodynamics, this is the initial value problem of a fixed inertial thrust mass-varying spacecraft orbiting a main body.
\[\begin{split}\left\{ \begin{array}{l} \dot{\mathbf r} = \mathbf v \\ \dot{\mathbf v} = -\frac{\mu}{r^3} \mathbf r + \frac{\mathbf T}{m} \\ \dot m = - \frac{|\mathbf T|}{I_{sp} g_0} \end{array}\right.\end{split}\]where \(\mu, v_{eff} = I_{sp}g_0\) and \(\mathbf T = [T_x, T_y, T_z]\) are parameters.
Circular Restricted Three Body Problem#
- pykep.ta.get_cr3bp(tol)#
Returns a Taylor adaptive propagator (Heyoka) for the CR3BP problem retreiving one from a global cache and making a copy.
If the requested propagator was never created, a call to this function will trigger its creation, else it will return the one from a global cache, thus avoiding jitting.
In pykep, the CR3BP is defined in Cartesian coordinates (thus it is not symplectic as not in a Hamiltonian form).
The specific dynamics used is that returned by
cr3bp_dyn().- Args:
tol (
float): the tolerance of the Taylor adaptive propagator.- Returns:
hy::taylor_adaptive: The Taylor adaptive propagator.- Examples:
>>> import pykep as pk >>> ta = pk.ta.get_cr3bp(tol = 1e-16) >>> ta.time = 0. >>> ta.state[:] = [1.01238082345234, -0.0423523523454, 0.22634376321, -0.1232623614, 0.123462698209365, 0.123667064622] >>> mu = 0.01215058560962404 >>> tof = 5.7856656782589234 >>> ta.pars[0] = mu >>> ta.propagate_until(tof)
- pykep.ta.get_cr3bp_var(tol)#
Returns a (order 1) variational Taylor adaptive propagator (Heyoka) for the CR3BP problem retreiving one from a global cache and making a copy.
If the requested propagator was never created, a call to this function will trigger its creation, else it will return the one from a global cache, thus avoiding jitting.
In pykep, the CR3BP is defined in Cartesian coordinates (thus it is not symplectic as not in a Hamiltonian form).
The specific dynamics used is that returned by
cr3bp_dyn().- Args:
tol (
float): the tolerance of the Taylor adaptive propagator.- Returns:
hy::taylor_adaptive: The Taylor adaptive propagator.- Examples:
>>> import pykep as pk >>> ta = pk.ta.get_cr3bp_var(tol = 1e-16) >>> ta.time = 0. >>> ta.state[:] = [1.01238082345234, -0.0423523523454, 0.22634376321, -0.1232623614, 0.123462698209365, 0.123667064622] >>> mu = 0.01215058560962404 >>> tof = 5.7856656782589234 >>> ta.pars[0] = mu >>> ta.propagate_until(tof)
- pykep.ta.cr3bp_dyn()#
The dynamics of the Circular Restricted Three Body Problem (CR3BP).
In pykep, the CR3BP is defined in Cartesian coordinates (thus it is not symplectic as not in a Hamiltonian form).
The parameter \(\mu\) is defined as \(\frac{m_2}{m_1+m_2}\) where \(m_2\) is the mass of the secondary body (i.e. placed on the positive x axis).
The equations are non-dimensional with units \(L = r_{12}\) (distance between the primaries), \(M = m_1 + m_2\) (total system mass) and \(T = \sqrt{\frac{r_{12}^3}{m_1+m_2}}\) (period of rotation of the primaries).
\[\begin{split}\left\{ \begin{array}{l} \dot{\mathbf r} = \mathbf v \\ \dot v_x = 2v_y + x - (1 - \mu) \frac{x + \mu}{r_1^3} - \mu \frac{x + \mu - 1}{r_2^3} \\ \dot v_y = -2 v_x + y - (1 - \mu) \frac{y}{r_1^3} - \mu \frac{y}{r_2^3} \\ \dot v_z = -(1 - \mu) \frac{z}{r_1^3} - \mu \frac{z}{r_2^3} \end{array}\right.\end{split}\]where \(\mu\) is the only parameter.