Orbit and phasespace position objects in more detail¶
We’ll assume the following imports have already been executed:
>>> import astropy.units as u
>>> import numpy as np
>>> import gala.potential as gp
>>> import gala.dynamics as gd
>>> from astropy.coordinates import (CylindricalRepresentation,
... CylindricalDifferential)
>>> from gala.units import galactic
>>> np.random.seed(42)
We will also set the default Astropy Galactocentric frame parameters to the values adopted in Astropy v4.0:
>>> import astropy.coordinates as coord
>>> _ = coord.galactocentric_frame_defaults.set('v4.0')
Introduction¶
The astropy.units
subpackage is excellent for working with numbers and
associated units, but dynamical quantities often contain many quantities with
mixed units. An example is a position in phasespace, which may contain some
quantities with length units and some quantities with velocity or momentum
units. The PhaseSpacePosition
and Orbit
classes are designed to work with these data
structures and provide a consistent API for visualizing and computing further
dynamical quantities. Click these shortcuts to jump to a section below, or start
reading below:
Phasespace positions¶
The PhaseSpacePosition
class provides an interface for representing full phasespace
positions–coordinate positions and momenta (velocities). This class is useful
as a container for initial conditions and for transforming phasespace positions
to new coordinate representations or reference frames.
The easiest way to create a PhaseSpacePosition
object is to pass in a pair of
Quantity
objects that represent the Cartesian position and
velocity vectors:
>>> gd.PhaseSpacePosition(pos=[4.,8.,15.]*u.kpc,
... vel=[150.,50.,15.]*u.km/u.s)
<PhaseSpacePosition cartesian, dim=3, shape=()>
By default, passing in Quantity
’s are interpreted as Cartesian
coordinates and velocities. This works with arrays of positions and velocities
as well:
>>> x = np.arange(24).reshape(3,8)
>>> v = np.arange(24).reshape(3,8)
>>> w = gd.PhaseSpacePosition(pos=x * u.kpc,
... vel=v * u.km/u.s)
>>> w
<PhaseSpacePosition cartesian, dim=3, shape=(8,)>
This is interpreted as 8, 6dimensional phasespace positions.
The class internally stores the positions and velocities as
BaseRepresentation
and
BaseDifferential
subclasses; in this case,
CartesianRepresentation
and
CartesianDifferential
:
>>> w.pos
<CartesianRepresentation (x, y, z) in kpc
[(0., 8., 16.), (1., 9., 17.), (2., 10., 18.), (3., 11., 19.),
(4., 12., 20.), (5., 13., 21.), (6., 14., 22.), (7., 15., 23.)]>
>>> w.vel
<CartesianDifferential (d_x, d_y, d_z) in km / s
[(0., 8., 16.), (1., 9., 17.), (2., 10., 18.), (3., 11., 19.),
(4., 12., 20.), (5., 13., 21.), (6., 14., 22.), (7., 15., 23.)]>
All of the components of these classes are mapped to attributes of the
phasespace position class for convenience, but with more userfriendly names.
These mappings are defined in the class definition of
PhaseSpacePosition
. For example, to access the x
component
of the position and the v_x
component of the velocity:
>>> w.x, w.v_x
(<Quantity [0.,1.,2.,3.,4.,5.,6.,7.] kpc>, <Quantity [0.,1.,2.,3.,4.,5.,6.,7.] km / s>)
The default representation is Cartesian, but the class can also be instantiated
with representation objects instead of Quantity
’s – this is
useful for creating PhaseSpacePosition
or Orbit
instances from nonCartesian
representations of the position and velocity:
>>> pos = CylindricalRepresentation(rho=np.linspace(1., 4, 4) * u.kpc,
... phi=np.linspace(0, np.pi, 4) * u.rad,
... z=np.linspace(1, 1., 4) * u.kpc)
>>> vel = CylindricalDifferential(d_rho=np.linspace(100, 150, 4) * u.km/u.s,
... d_phi=np.linspace(1, 1, 4) * u.rad/u.Myr,
... d_z=np.linspace(15, 15., 4) * u.km/u.s)
>>> w = gd.PhaseSpacePosition(pos=pos, vel=vel)
>>> w
<PhaseSpacePosition cylindrical, dim=3, shape=(4,)>
>>> w.rho
<Quantity [1., 2., 3., 4.] kpc>
We can easily transform the full phasespace vector to new representations or
coordinate frames. These transformations use the astropy.coordinates
representations framework:
>>> cart = w.represent_as('cartesian')
>>> cart.x
<Quantity [ 1. , 1. , 1.5, 4. ] kpc>
>>> sph = w.represent_as('spherical')
>>> sph.distance
<Distance [1.41421356, 2.02758751, 3.01846171, 4.12310563] kpc>
There is also support for transforming the positions and velocities (assumed to
be in a Galactocentric
frame) to any of the other
coordinate frames. For example, to transform to
Galactic
coordinates:
>>> from astropy.coordinates import Galactic
>>> gal_c = w.to_coord_frame(Galactic)
>>> gal_c
<Galactic Coordinate: (l, b, distance) in (deg, deg, kpc)
[(4.40971301e05, 6.23850462, 9.17891228),
(1.07501936e+01, 2.04017409, 9.29170644),
(2.14246214e+01, 2.65220588, 7.12026744),
(7.35169893e05, 13.50991169, 4.23668468)]
(pm_l_cosb, pm_b, radial_velocity) in (mas / yr, mas / yr, km / s)
[( 28.11596908, 0.297625 , 89.093095 ),
( 13.077309 , 0.15891073, 511.60269726),
( 7.04751509, 1.33976418, 1087.52574084),
(206.97042166, 2.22471526, 156.82064814)]>
We can easily plot projections of the phasespace positions using the
plot
method:
>>> np.random.seed(42)
>>> x = np.random.uniform(10, 10, size=(3,128))
>>> v = np.random.uniform(200, 200, size=(3,128))
>>> w = gd.PhaseSpacePosition(pos=x * u.kpc,
... vel=v * u.km/u.s)
>>> fig = w.plot()
(Source code, png)
This is a thin wrapper around the plot_projections
function and any keyword arguments are passed through to that function:
>>> fig = w.plot(components=['x', 'v_z'], color='r',
... facecolor='none', marker='o', s=20, alpha=0.5)
(Source code, png)
Phasespace position API¶
Classes¶

Represents phasespace positions, i.e. positions and conjugate momenta (velocities). 
Class Inheritance Diagram¶
Orbits¶
The Orbit
class inherits much of the functionality from PhaseSpacePosition
(described above)
and adds some additional features that are useful for timeseries orbits.
An Orbit
instance is initialized like the PhaseSpacePosition
–with arrays of positions and
velocities– but usually also requires specifying a time array as well. Also,
the extra axes in these arrays hold special meaning for the Orbit
class. The
position and velocity arrays passed to PhaseSpacePosition
can have arbitrary numbers of
dimensions as long as the 0th axis specifies the dimensionality. For the Orbit
class, the 0th axis remains the axis of dimensionality, but the 1st axis now is
always assumed to be the time axis. For example, an input position with shape
(2,128)
to a PhaseSpacePosition
represents 128 independent 2D positions, but to a Orbit
it represents a single orbit’s positions at 128 times:
>>> t = np.linspace(0, 100, 128) * u.Myr
>>> Om = 1E1 * u.rad / u.Myr
>>> pos = np.vstack((5*np.cos(Om*t), np.sin(Om*t))).value * u.kpc
>>> vel = np.vstack((5*np.sin(Om*t), np.cos(Om*t))).value * u.kpc/u.Myr
>>> orbit = gd.Orbit(pos=pos, vel=vel)
>>> orbit
<Orbit ndcartesian, dim=2, shape=(128,)>
To create a single object that contains multiple orbits, the input position
object should have 3 axes. The last axis (axis=2
) specifies the number of
orbits. So, an input position with shape (2,128,16)
would represent 16, 2D
orbits, each with the same 128 times:
>>> t = np.linspace(0, 100, 128) * u.Myr
>>> Om = np.random.uniform(size=16) * u.rad / u.Myr
>>> angle = Om[None] * t[:,None]
>>> pos = np.stack((5*np.cos(angle), np.sin(angle))).value * u.kpc
>>> vel = np.stack((5*np.sin(angle), np.cos(angle))).value * u.kpc/u.Myr
>>> orbit = gd.Orbit(pos=pos, vel=vel)
>>> orbit
<Orbit ndcartesian, dim=2, shape=(128, 16)>
To make full use of the orbit functionality, you must also pass in an array with
the time values and an instance of a PotentialBase
subclass
that represents the potential that the orbit was integrated in:
>>> pot = gp.PlummerPotential(m=1E10, b=1., units=galactic)
>>> orbit = gd.Orbit(pos=pos*u.kpc, vel=vel*u.km/u.s,
... t=t*u.Myr, potential=pot)
(note, in this case pos
and vel
were not generated from integrating
an orbit in the potential pot
!). However, most of the time you won’t need to
create Orbit
objects from scratch! They are returned from any of the numerical
intergration routines provided in gala
. For example, they are returned by the
integrate_orbit
method of potential objects and
will automatically contain the time
array and potential
object. For
example:
>>> pot = gp.PlummerPotential(m=1E10 * u.Msun, b=1. * u.kpc, units=galactic)
>>> w0 = gd.PhaseSpacePosition(pos=[10.,0,0] * u.kpc,
... vel=[0.,75,0] * u.km/u.s)
>>> orbit = gp.Hamiltonian(pot).integrate_orbit(w0, dt=1., n_steps=5000)
>>> orbit
<Orbit cartesian, dim=3, shape=(5001,)>
>>> orbit.t
<Quantity [0.000e+00, 1.000e+00, 2.000e+00, ..., 4.998e+03, 4.999e+03,
5.000e+03] Myr>
>>> orbit.potential
<PlummerPotential: m=1.00e+10, b=1.00 (kpc,Myr,solMass,rad)>
Just like for PhaseSpacePosition
, we can quickly visualize an orbit using the
plot
method:
>>> fig = orbit.plot()
(Source code, png)
Again, this is a thin wrapper around the plot_projections
function and any keyword arguments are passed through to that function:
>>> fig = orbit.plot(linewidth=4., alpha=0.5, color='r')
(Source code, png)
We can also quickly compute quantities like the angular momentum, and estimates for the pericenter, apocenter, eccentricity of the orbit. Estimates for the latter few get better with smaller timesteps:
>>> orbit = gp.Hamiltonian(pot).integrate_orbit(w0, dt=0.1, n_steps=100000)
>>> np.mean(orbit.angular_momentum(), axis=1)
<Quantity [0. ,0. ,0.76703412] kpc2 / Myr>
>>> orbit.eccentricity()
<Quantity 0.31951765618193967>
>>> orbit.pericenter()
<Quantity 10.00000005952518 kpc>
>>> orbit.apocenter()
<Quantity 19.390916871970223 kpc>
More information¶
Internally, both of the above classes rely on the Astropy representation
transformation framework (i.e. the subclasses of
BaseRepresentation
and
BaseDifferential
). However, at present these classes only
support 3D positions and differentials (velocities). The PhaseSpacePosition
and Orbit
classes
both support arbitrary numbers of dimensions and, when relevant, rely on custom
subclasses of the representation classes to handle such cases. See the
Ndimensional representation classes page for more information about these classes.