Multiple imaging

Let’s explore simulation of signals generated by neutron stars sufficiently compact to generate multiple images of a subset or even all of the surface. En route we demonstrate correctness for an oblate surface embedded in an ambient Schwarzschild spacetime by calling four integrators whose algorithms are (vastly) different. In particular integrators that directly discretise the surface with a regular mesh will be compared to an integrator that discretises a moderately distant image-plane; integrators that employs these distinct discretisation schemes in the computational domain are effectively orthogonal and provide strong internal validation of the ray-tracing backend used for statistical computation in practice. We will also replace numerical primary-image ray solutions (lensing integral computed via numerical quadrature and inversion by spline interpolation) with a high-order lensing integral expansion via linking to the rayXpanda package.

Ultimately, we conclude that the inclusion of secondary images is important in practice, whilst the inclusion of tertiary images is unimportant in practice, and the inclusion of quaternary order and higher is entirely unnecessary as far as we can forsee.

Note that if you wish to execute this notebook as is, it is costly because we use rather high-resolution calculations to probe convergence of different integrators. You can also lower the integrator resolution settings and change the model parameters including the stellar spin frequency, the spacetime properties, and the source-receiver configuration.

The unexecuted notebook for this tutorial may be found in a GitHub repository:

git clone https://github.com/ThomasEdwardRiley/xpsi_workshop.git </path/to/clone>

cd </path/to/clone>/tutorials/v0.6/

You can use the default atmosphere extension modules xpsi/surface_radiation_field/archive/hot/blackbody.pyx and xpsi/surface_radiation_field/archive/local_variables/uniform.pyx. To run this tutorial, you should therefore be able to simply use the default xpsi/surface_radiation_field/hot.pyx extension and xpsi/surface_radiation_field/local_variables.pyx that are automatically compiled when X-PSI is installed.

The first part of this tutorial largely follows the Global surface emission tutorial. We use a surface radiation field that is globally uniform to compare all integrators. We use an effectively spherical star by entering the limit of zero spin.

[1]:
%matplotlib inline

from __future__ import print_function, division

import os
import numpy as np
import math

from matplotlib import pyplot as plt
from matplotlib import rcParams
from matplotlib.ticker import MultipleLocator, AutoLocator, AutoMinorLocator
from matplotlib import gridspec
from matplotlib import cm

from IPython.display import Image

import xpsi
from xpsi import Parameter

try:
    xpsi.cellmesh
except AttributeError:
    pass
else:
    # use the override switch to deactivate external library for now
    # this means that lensing integral solutions will be fully numerical
    xpsi.cellmesh.__deactivate_rayXpanda__ = True

from xpsi.global_imports import _c, _G, _dpr, gravradius, _csq, _km, _2pi
/=============================================\
| X-PSI: X-ray Pulse Simulation and Inference |
|---------------------------------------------|
|                Version: 0.6.0               |
|---------------------------------------------|
|  https://thomasedwardriley.github.io/xpsi/  |
\=============================================/

Imported GetDist version: 0.3.1
Imported nestcheck version: 0.2.0

First we need to do some setup of the ambient spacetime and the surface embedded in it that the photosphere exists on.

[2]:
bounds = dict(frequency = (0.1, 600.0),
              distance = (0.1, 1.0),                     # (Earth) distance
              mass = (1.0, 3.0),                         # mass
              radius = (3.0 * gravradius(1.0), 16.0),    # equatorial radius
              cos_inclination = (0.0, 1.0))              # (Earth) inclination to rotation axis

spacetime = xpsi.Spacetime(bounds=bounds, values=dict(frequency=0.1)) # spherical star
Creating parameter:
    > Named "frequency" with bounds [1.000e-01, 6.000e+02] and initial value 1.000e-01.
    > Spin frequency [Hz].


Creating parameter:
    > Named "mass" with bounds [1.000e+00, 3.000e+00].
    > Gravitational mass [solar masses].


Creating parameter:
    > Named "radius" with bounds [4.430e+00, 1.600e+01].
    > Coordinate equatorial radius [km].


Creating parameter:
    > Named "distance" with bounds [1.000e-01, 1.000e+00].
    > Earth distance [kpc].


Creating parameter:
    > Named "cos_inclination" with bounds [0.000e+00, 1.000e+00].
    > Cosine of Earth inclination to rotation axis.


Default (phase-invariant)

First we invoke a globally uniform temperature field. There is no azimuthal dependence, meaning that the signal generated by the star is time-invariant. We are in need of an object that embeds a globally discretised surface into the ambient spacetime and exposes methods for integration over solid angle on our sky.

[3]:
bounds = dict(temperature = (None, None))

everywhere = xpsi.Everywhere(time_invariant=True,
                             bounds=bounds,
                             values={}, # no fixed/derived variables
                             sqrt_num_cells=1024, # in colatitude, and in azimuth separately
                             num_rays=1024, # O(1000) useful for secondary and tertiary images
                             num_leaves=1024, # specify leaves if time-dependent
                             num_phases=100, # of the output signal
                             image_order_limit=None) # as many images as possible within resolution
Creating parameter:
    > Named "temperature" with bounds [3.000e+00, 7.000e+00].
    > log10(effective temperature [K] everywhere).


We are free to subclass Everywhere and implement custom functionality beyond the simple default above. The argument specifying the number of rays has the familiar meaning. The argument for the number of cells is now used to discretise the surface in azimuth and colatitude with respect to the stellar rotation axis, as was the case for the Elsewhere module. The new argument time_invariant declares whether or not the surface radiation field is dependent on azimuth; if it is independent of azimuth, a faster integrator is called.

Now we need an instance of Photosphere that we can feed our everywhere object to. If we are not imaging the photosphere or we are satisfied with the default behaviour we do not need to subclass Photosphere.

[4]:
class derive(xpsi.Derive):
    def __init__(self):
        pass
    def __call__(self, boundto, caller = None):
        global spacetime
        return spacetime['frequency']
[5]:
photosphere = xpsi.Photosphere(hot = None, elsewhere = None, everywhere = everywhere,
                               values=dict(mode_frequency = derive()))
Creating parameter:
    > Named "mode_frequency" that is derived from ulterior variables.
    > Coordinate frequency of the mode of radiative asymmetry in the
photosphere that is assumed to generate the pulsed signal [Hz].


[6]:
star = xpsi.Star(spacetime = spacetime, photospheres = photosphere)

Let’s check the vector of parameter values in the Star instance and the other objects it encapsulates references to.

[7]:
star
[7]:
Free parameters
---------------
frequency: Spin frequency [Hz].
mass: Gravitational mass [solar masses].
radius: Coordinate equatorial radius [km].
distance: Earth distance [kpc].
cos_inclination: Cosine of Earth inclination to rotation axis.
temperature: log10(effective temperature [K] everywhere).

We assign parameter values and update the star as follows:

[8]:
# (Earth) distance
star['distance'] = 0.33
# gravitational mass
star['mass'] = 2.7088795
# coordinate equatorial radius
star['radius'] = 12.0
# (Earth) inclination to rotation axis
star['cos_inclination'] = math.cos(1.0)
# isotropic blackbody temperature
star['temperature'] = 6.3

star.update()

The temperature is chosen to be something appropriately hot for the energy range, so that the specific flux at the lowest energies is not dominated by bright stellar limb, which will be the case for targets in X-ray telescope wavebands. More on this later.

[9]:
spacetime.R_r_s # rather compact! effectively @ the Schwarzschild photon sphere
[9]:
1.5000001812176318

Let’s see what the maximum deflection is calculated to be conditional on the resolution:

[10]:
everywhere._maxDeflection[0]/xpsi.global_imports._pi
[10]:
5.44436523851195

A true maximum deflection of this value, for a spherical star, would mean a subset of the stellar surface would be imaged six times. This subset is defined by having a principal deflection of \(\psi\gtrsim(1-0.4444)\pi\), meaning a spherical cap of this angular radius centred on the direction between Earth and the centre of the star. The subset of the surface with principal deflection of \(\psi\lesssim(1-0.4444)\pi\), and by definition less than \(\pi\), is only imaged five times.

Let’s compute the incident specific flux signal, up to some constant coefficient.

[11]:
energies = np.logspace(-2.0, np.log10(3.0), 100, base=10.0)
[12]:
photosphere.integrate(energies=energies, threads=4) # calls to xpsi/cellmesh/integrator_for_time_invariance.pyx

The signal is time-invariant and therefore we need to copy the spectrum to a sequence of matrix columns to get the desired energy-phase signal matrix:

[13]:
temp_int1 = np.tile(photosphere.signal[0][0], (1,len(everywhere.phases_in_cycles)))

We need a helper function to plot the signal, normalised to the maximum specific flux:

[2]:
from xpsi.tools import phase_interpolator

def plot_2D_pulse(z, x, shift, y, ylabel,
                  num_rotations=5.0, res=5000, normalise=True,
                  cm=cm.viridis, vmin=None, vmax=None, label=None,
                  combine=False, error=False):
    """ Helper function to plot a phase-energy pulse.

    :param array-like z:
        A pair of *ndarray[m,n]* objects representing the signal at
        *n* phases and *m* values of an energy variable.

    :param ndarray[n] x: Phases the signal is resolved at.

    :param tuple shift: Hot region phase parameters.

    :param ndarray[m] x: Energy values the signal is resolved at.

    """

    fig = plt.figure(figsize = (12,6))

    gs = gridspec.GridSpec(1, 2, width_ratios=[50,1], wspace=0.025)
    ax = plt.subplot(gs[0])
    ax_cb = plt.subplot(gs[1])

    new_phases = np.linspace(0.0, num_rotations, res)

    if combine:
        joint = phase_interpolator(x, x, np.ascontiguousarray(z[0]), shift[0])
        joint += phase_interpolator(x, x, np.ascontiguousarray(z[1]), shift[1])

    if error:
        _temp = phase_interpolator(new_phases,
                                      x,
                                      np.ascontiguousarray(z[0]), shift[0])

        interpolated = _temp - phase_interpolator(new_phases,
                                           x,
                                           np.ascontiguousarray(z[1]), shift[1])
        interpolated /= _temp/100.0
    else:
        if not isinstance(z, (list, tuple)):
            interpolated = phase_interpolator(new_phases,
                                          x,
                                          np.ascontiguousarray(z), shift)
        else:
            interpolated = phase_interpolator(new_phases,
                                          x,
                                          np.ascontiguousarray(z[0]), shift[0])
            if len(z) > 1:
                interpolated += phase_interpolator(new_phases,
                                                   x,
                                                   np.ascontiguousarray(z[1]), shift[1])

    if normalise:
        interpolated /= np.max(interpolated)

    if vmin is None:
        vmin = np.min(interpolated)

    if vmax is None:
        vmax = np.max(interpolated)

    profile = ax.pcolormesh(new_phases,
                             y,
                             interpolated,
                             cmap = cm,
                             vmin = vmin,
                             vmax = vmax,
                             linewidth = 0,
                             rasterized = True)

    profile.set_edgecolor('face')

    ax.set_xlim([0.0, num_rotations])
    ax.set_yscale('log')
    ax.set_ylabel(ylabel)
    ax.set_xlabel(r'Phase (cycles)')
    veneer((0.1, 0.5), (None,None), ax)

    cb = plt.colorbar(profile,
                      cax = ax_cb,
                      ticks = AutoLocator())

    cb.set_label(label=label or r'Signal (normalised by maximum)', labelpad=25)
    cb.solids.set_edgecolor('face')

    cb.outline.set_linewidth(1.0)

    try:
        return joint
    except NameError:
        pass

rcParams['text.usetex'] = False
rcParams['font.size'] = 18.0

def veneer(x, y, axes, lw=1.0, length=8):
    """ Make the plots a little more aesthetically pleasing. """
    if x is not None:
        if x[1] is not None:
            axes.xaxis.set_major_locator(MultipleLocator(x[1]))
        if x[0] is not None:
            axes.xaxis.set_minor_locator(MultipleLocator(x[0]))
    else:
        axes.xaxis.set_major_locator(AutoLocator())
        axes.xaxis.set_minor_locator(AutoMinorLocator())

    if y is not None:
        if y[1] is not None:
            axes.yaxis.set_major_locator(MultipleLocator(y[1]))
        if y[0] is not None:
            axes.yaxis.set_minor_locator(MultipleLocator(y[0]))
    else:
        axes.yaxis.set_major_locator(AutoLocator())
        axes.yaxis.set_minor_locator(AutoMinorLocator())

    axes.tick_params(which='major', colors='black', length=length, width=lw)
    axes.tick_params(which='minor', colors='black', length=int(length/2), width=lw)
    plt.setp(axes.spines.values(), linewidth=lw, color='black')
[15]:
plot_2D_pulse(temp_int1,
              x=everywhere.phases_in_cycles,
              shift=np.array([0.0]),
              y=energies,
              num_rotations=1.0,
              ylabel=r'Energy (keV)')
_images/multiple_imaging_34_0.png

If we declare the signal as time-dependent, a different integrator is called:

[16]:
everywhere.time_invariant = False # switches to xpsi/cellmesh/integrator.pyx
[17]:
everywhere._integrator_toggle = False
[18]:
photosphere.integrate(energies=energies, threads=4)
[19]:
temp_int2 = photosphere.signal[0][0].copy()
[20]:
plot_2D_pulse((temp_int2, temp_int1),
              x=everywhere.phases_in_cycles,
              shift=(0.0, 0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              vmin=-0.001,
              vmax=0.001,
              num_rotations=1.0,
              error=True,
              normalise = False)
_images/multiple_imaging_40_0.png
[21]:
everywhere._integrator_toggle = True # switch to final integrator that discretises surface, xpsi/cellmesh/integrator_for_azimuthal_invariance.pyx, for testing
[22]:
photosphere.integrate(energies=energies, threads=4)
[23]:
temp_int3 = photosphere.signal[0][0].copy()
[24]:
plot_2D_pulse((temp_int3, temp_int2),
              x=everywhere.phases_in_cycles,
              shift=(0.0,0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              vmin=-0.000001,
              vmax=0.000001,
              num_rotations=1.0,
              error=True,
              normalise = False)
_images/multiple_imaging_44_0.png

We can also call a fourth integrator. This integrator is more general purpose, and thus inexorably more expensive to call. First we need to force the spacetime to be static (otherwise univeral relations are invoked based on the stellar spin frequency as set above):

[25]:
spacetime.a = 0.0 # spacetime spin parameter (~angular momentum)
spacetime.q = 0.0 # spacetime mass quadrupole moment

Now we call the integrator. The integrator discretises a distant image plane instead of the stellar surface. The image of the star is spatially resolved on the image plane. The integrator yields four-dimensional information about the signal. We trace a set of rays from the image plane to the star; the set is roughly equal in cardinality to the number of cells that discretise the surface above. Note that when this extension module is called, some output for diagnostics is directed to the terminal in which you launched this Jupyter notebook.

[26]:
photosphere.image(reimage = True,
                  reuse_ray_map = False,
                  energies = energies,
                  phases = everywhere.phases_in_cycles * _2pi,
                  sqrt_num_rays = 1024,
                  # squeeze rays towards limb where higher-order images are
                  image_plane_radial_increment_power=0.2,
                  threads = 4,          # OpenMP
                  max_steps = 100000,   # max number of steps per ray
                  epsrel_ray = 1.0e-12) # ray relative tolerance
Imaging the star...
Commencing ray tracing and imaging...
Ray tracing complete.
Ray set cached.
Phase-resolved specific flux integration complete.
Star imaged.

We now compare the signal to those computed above. The phase-energy resolved specific flux signal (integrated over sky solid angle) can be accessed through the images property of the photosphere object. The elements of this property also contain image plane coordinates, stellar surface coordinates, and quantities such as the specific photon intensity as a function of phase, energy, and sky direction (image plane coordinates). Note that the units of the specific flux signal are photons/cm\(^{2}\)/s/keV because it has already been scaled by the square of the distance. The signals generated by the integrators above have not been scaled by the square of the distance (an implementation specific detail that is susceptible to change in the future).

[27]:
plot_2D_pulse((photosphere.images[0]*spacetime.d_sq, temp_int1),
              x=everywhere.phases_in_cycles,
              shift=(0.0, 0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              vmin=-0.1,
              vmax= 0.1,
              error=True,
              num_rotations=1.0,
              normalise=False)
_images/multiple_imaging_50_0.png

Let’s make a comparison of the integrators in one plot:

[28]:
fig = plt.figure(figsize=(10,10))

ax = fig.add_subplot(211)

MAX = np.max(temp_int1[:,0])

ax.plot(energies,
         temp_int1[:,0]/MAX,
         'k-')

ax.plot(energies,
         temp_int2[:,0]/MAX,
         'k--')

ax.plot(energies,
         temp_int3[:,0]/MAX,
         'k:')

ax.plot(energies,
         photosphere.images[0][:,0]*spacetime.d_sq/MAX,
         'k-.')

ax.set_xscale('log')
ax.set_ylabel(r'Normalised photons/cm$^{2}$/s/keV')
ax.set_yscale('log')
ax.xaxis.set_ticklabels([])
veneer((None, None), (None, None), ax)

ax = fig.add_subplot(212)

ax.plot(energies,
         100.0*np.abs(temp_int2[:,0]/temp_int1[:,0] - 1.0),
         'k--')

ax.plot(energies,
         100.0*np.abs(temp_int3[:,0]/temp_int1[:,0] - 1.0),
         'k:')

ax.plot(energies,
         100.0*np.abs(photosphere.images[0][:,0]*spacetime.d_sq/temp_int1[:,0] - 1.0),
         'k-.')

ax.set_xscale('log')
ax.set_xlabel('Energy [keV]')

ax.set_yscale('log')
ax.set_ylabel('Abs. % frac. diff.')

veneer((None, None), (None, None), ax)

plt.subplots_adjust(hspace=0.1)
_images/multiple_imaging_52_0.png

The differences can be reduced, within the scope of a given algorthm, by defining higher resolution integration settings. The integration algorithms are so distinct that this consistency validates the tools internally; verification against external packages would nevertheless permit stronger guarantees of robustness. The simple package rayXpanda offered a weak validation (i.e., for primary images only) of the Schwarzschild ray integration routines called by the surface-discretisation signal integrators. In fact, let’s now link rayXpanda into X-PSI to make a final validating cross-check of the primary image computation: X-PSI calculating the lensing integral \(\psi(\alpha;u)\) and the convergence \(\sim\partial\cos\alpha/\partial\cos\psi\) via numerical quadrature and inversion by spline interpolation, and rayXpanda calculating a high-order integral expansion to simultaneously evaluate \(\alpha(\psi;u)\) and the convergence. Here the deflection \(\psi\) depends on the ray angle to the radial direction (in a local Eulerian frame) and on the compactness coordinate \(u=r_{\rm s}/R(\theta;M,\Omega,R_{\rm eq})\).

Linking in rayXpanda

If you do not have rayXpanda installed, the surface-discretisation integrators will simply revert to the default behaviour already shown above.

If rayXpanda is linked in, then it replaces the primary image lensing calculation up to some deflection limit. There are however two behaviours to be of that we demonstrate below. If the deflection limit is too low high but below \(\pi\) radians (by definition), then the expansion truncation error manifests. And non-intuitively, if the deflection limit is too low, then due to the design of the algorithm to support inclusion of higher-order images, the numerical solution will become inaccurate for small deflections. If rayXpanda is not linked in, a different numerical scheme is substituted in that handles primary images that are deflected by \(\psi<\pi/2\) radians; the rayXpanda expansion is highly consistent with this numerical (quadrature + interpolation) solution as shown below.

[29]:
%%capture

try:
    xpsi.cellmesh
except AttributeError:
    pass
else:
    # use the override switch to deactivate external library for now
    # this means that lensing integral solutions will be fully numerical
    xpsi.cellmesh.__deactivate_rayXpanda__ = False
    everywhere.time_invariant = True # calls to xpsi/cellmesh/integrator_for_time_invariance.pyx
    everywhere.image_order_limit = 0 # in case it changed anywhere, e.g., cells executed out of order
    limits = np.linspace(0.01, 0.99*math.pi, 100)
    def helper(lim):
        xpsi.set_rayXpanda_deflection_limit(lim)
        photosphere.integrate(energies=energies, threads=4)
        return np.max(100.0*np.abs(photosphere.signal[0][0][:,0]/temp_int1[:,0] - 1.0))

    errors = []
    for lim in limits:
        errors.append(helper(lim))

If we do not capture the output, the warning below will be printed a number of times:

Warning: invoking rayXpanda for a signal integration over a subdomain of the stellar image.

Warning: the larger the primary image subdomain chosen for rayXpanda calls,

Warning: the larger the rayXpanda expansion truncation error.

Warning: you can control this by setting the rayXpanda deflection limit manually.

Warning: please use the top-level function xpsi.set_rayXpanda_deflection_limit(float).

Warning: please refer to the documentation at https://thomasedwardriley.github.io/rayXpanda/theory

[30]:
try:
    xpsi.cellmesh
except AttributeError:
    pass
else:
    fig = plt.figure(figsize=(10,10))
    ax = fig.add_subplot(111)

    ax.plot(limits/xpsi.global_imports._pi, np.array(errors)/100.0, 'k-')

    ax.set_xlabel('rayXpanda primary image limit [$\pi$ radians]')

    ax.set_yscale('log')
    ax.set_ylabel('Max. abs. frac. diff.')

    veneer((0.05, 0.2), (None, None), ax)
_images/multiple_imaging_60_0.png

The default rayXpanda deflection limit, if the library is linked in, is also \(\psi<\pi/2\) radians. If you are interested in a straightforward comparison of the high-order rayXpanda expansion and the default numerical primary image scheme for \(\psi<\pi/2\), please refer to the rayXpanda documentation for demonstrative plots.

Decreasing the image order limit

For integrators that directly discretise the stellar surface with a regular mesh we can straightforwardly control the limiting image order to try to sum up to. If the image order is conservative, the integrators automatically detect invisibility and truncate. Note that the sum is executed from low- to high-order, which generally means adding small numbers to much larger numbers, and thus in principle accuracy loss, but the number magnitude decays very rapidly with order so the contribution from combining higher-order images first shouldn’t make an importance difference. If higher-order images are visible, they are simply missed because of enforced truncation. It follows that because the flux contribution from higher-order images (even at fixed phase and energy) decays with order, setting a limit of three images or even two images is going to be entirely sufficient in practice. On the other hand, not setting a limit won’t affect integration time in an important way for less compact stars, but if prior support extends to very compact stars, the computation time increases by a factor of a few, until the resolution limit is reached.

Choosing a limit of two or three, and allowing the prior support to encompass configurations with polar radius greater than the Schwarzschild photon sphere radius should be sufficient. Then, if of interest, incrementing the order limit during post-processing to execute sensitivity analysis via importance sampling should allow one to demonstrate insensitivity to higher-order image inclusion; this is only relevant if the samples are from a posterior mode at high compactness where part of the surface is doubly or multiply-imaged.

[31]:
everywhere.time_invariant = True # calls to xpsi/cellmesh/integrator_for_time_invariance.pyx
[32]:
def helper(order):
    everywhere.image_order_limit = order
    photosphere.integrate(energies=energies, threads=4)
    return photosphere.signal[0][0].copy()
[33]:
xpsi.cellmesh.__deactivate_rayXpanda__ = True
[34]:
temp_int1_rayX_1 = helper(1) # primary image only
temp_int1_rayX_2 = helper(2) # up to secondary images
temp_int1_rayX_3 = helper(3) # up to tertiary images
temp_int1_rayX_4 = helper(4) # and so on...
temp_int1_rayX_5 = helper(5) # we will compare to the adaptive computation above
[36]:
fig = plt.figure(figsize=(10,5))
ax = fig.add_subplot(111)

ax.plot(energies,
     100.0*(-temp_int1_rayX_1[:,0]/temp_int1[:,0] + 1.0),
     'k-')

ax.plot(energies,
         100.0*(-temp_int1_rayX_2[:,0]/temp_int1[:,0] + 1.0),
         'k-.')

ax.plot(energies,
         100.0*(-temp_int1_rayX_3[:,0]/temp_int1[:,0] + 1.0),
         'k--')

ax.plot(energies,
         100.0*(-temp_int1_rayX_4[:,0]/temp_int1[:,0] + 1.0),
         'k:')

ax.plot(energies,
         100.0*(-temp_int1_rayX_5[:,0]/temp_int1[:,0] + 1.0),
         'r:')

ax.plot(energies,
         100.0*np.abs(photosphere.images[0][:,0]*spacetime.d_sq/temp_int1[:,0] - 1.0),
         'b-.')

ax.set_xscale('log')
ax.set_xlabel('Energy [keV]')

ax.set_yscale('log')
ax.set_ylabel('Abs. % frac. diff.')

veneer((None, None), (None, None), ax)
_images/multiple_imaging_68_0.png

For this compact star only the secondary images are demonstrably important, with a massive increase in accuracy. We also show again the level of validation achieved via image-plane calculation in blue. To further demonstrate either the convergence of the image-plane and surface discretisation algorithms, and thus determine if the algorithms need to be improved to be accurate for image orders beyond two, we’d need to tune up the resolution settings. In particular we sacrificed primary image resolution when squeezing the image-plane rays towards the stellar limb, given that we were also in pursuit of visualisation the higher-order image squeezing. Resolution settings that target flux accuracy will improve the consistency.

Multiple images

Let’s use the cached ray map to try to visualise the multiple imaging:

[37]:
fig = plt.figure(figsize=(10,10))

ref = photosphere.images[3].copy()
ref[ref > math.pi/2.0] *= -1.0
plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)
# rays that scatter have negative constant values <-100
# for map quantities such as the colatitude

ax = plt.gca()
ax.set_xlim([-1.025,1.025])
ax.set_ylim([-1.025,1.025])
veneer((0.05,0.25), (0.05,0.25), ax)
_images/multiple_imaging_72_0.png

The artefact near the origin along the equator is because we squeezed the image-plane polar ray set towards the image boundary, leaving fewer rays around the origin for accurate contour computation (via Delaunay triangulation).

[38]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([0.95, 1.0])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_74_0.png
[39]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([-1.0,-0.95])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_75_0.png

Let’s zoom in again to see the insane squeezing of just the third image of the southern pole:

[40]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.025,0.025])
ax.set_ylim([-1.0,-0.9995])
_ = veneer((0.0025,0.01), (0.000025,0.0001), ax)
_images/multiple_imaging_77_0.png

We’d evidently need more rays and potentially more accurate ray tracing (in particular surface detection for near-grazing rays) to resolve even these third-order images if they contributed to the flux in an important way. Note that in theory the image of this almost exactly spherical star should extend to coordinates \(y=\pm1\) and \(x=\pm1\) almost exactly (for an oblate star and the definition of the coordinate units, the maximum extent of the star is \(x=\pm1\), with the actual extent being a function of both viewing angle and the surface oblateness and compactness). We continue discussion of performance below.

Let’s first plot the azimuth map to distinguish images:

[41]:
fig = plt.figure(figsize=(10,10))

phi = np.copy(photosphere.images[4][...])

for i in range(len(phi)):

    if phi[i] > -100.0:
        if phi[i] > math.pi:
            while phi[i] > math.pi:
                phi[i] -= 2.0 * math.pi
        elif phi[i] < -math.pi:
            while phi[i] < -math.pi:
                phi[i] += 2.0 * math.pi

phi_lvls = np.linspace(np.min(phi[phi > -100.0]),
                       np.max(phi[phi > -100.0]), 100)

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                phi,
                levels=phi_lvls,
                cmap=cm.magma_r)
# rays that scatter have negative constant values <-100
# for map quantities such as the azimuth

ax = plt.gca()
ax.set_xlim([-1.025,1.025])
ax.set_ylim([-1.025,1.025])
veneer((0.05,0.25), (0.05,0.25), ax)
_images/multiple_imaging_80_0.png
[42]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                phi,
                levels=phi_lvls,
                cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([0.95, 1.0])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_81_0.png
[43]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                phi,
                levels=phi_lvls,
                cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([-1.0,-0.95])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_82_0.png

Millisecond pulsar spin (phase-invariant)

Let’s spin the star up to include two additional effects: relativistic beaming of the surface emission and rotational deformation (oblateness). The former means that for an observer substantially inclined to the rotational axis, the flux at (gravitationally blueshifted) energies beyond a thermal knee in the photosphere specific intensity spectrum becomes dominated by images at the edge of the stellar image on our sky (i.e., at the stellar limb). An interesting effect to note is that relatvistic beaming brightens odd-order images and dims even-order images, or vice versa dependending on whether the radiating neighbourhood of the surface is receeding or approaching based on the primary image. That means for instance, that if the primary is dimmed due to beaming, and the secondary is brightened, the tertiary image will be dimmed even more than the primary due to beaming which compounds the solid angle decay. This is one way in which a secondary image can in principle contribute at least as much flux as a primary image at some rotational phases and viewing configurations, whilst the quaternary image contributes a similar flux to the tertiary image, and so on relating adjacent odd and even order images; the tertiary and higher-order images are would be unimportant though. On the other hand, if the primary image is brighened, the secondary is dimmed and the tertiary is brightened. In this case, a tertiary image can contribute more flux than a secondary image at select phases and viewing configurations, but will itself generally be entirely dominated by the primary. The relationships between unimportant higher-order adjacent even and odd images is similar to the above description.

Note that another way to weight the flux integral towards the stellar limb is to have temperature inversion in the atmosphere, leading to bright near-tangential emission, but the effect is not as a strong as tuning up to millisecond spin periods and higher.

Only at energies well beyond a thermal knee in the surface emission will the relativistic beaming mean that the flux is dominated by the brightness at the stellar limb (when there is somewhat uniform, global surface emission as assumed here), where the high-order images will be squeezed. The brightness of near-tangential emission after beaming partially counteracts the solid angle decay with image order, in such a case, but will not realistically ever mean high-order images contribute more flux at some energy. However, X-ray telescopes will generally be sensitive to softer emission where there are far more counts, meaning relative insensitivity of a well-performing likelihood function jointly over the instrument waveband to flux integration error in the vicinity of the stellar limb.

Realistically, if a surface does not radiate somewhat unformly in the instrument waveband, leading to pulsed emission, effectively the same reasoning applies to the importance of accurately calculating the flux at phases where the flux is dominated by the signal from the stellar limb where the limb is constituted by only primary images, simply because of geometric projected effects. This is without or without additional strong relativistic beaming effects leading to flux (at energies beyond a thermal knee in the surface emission spectrum) being dominated by the contribution from the relevant radiating regions when they reach the stellar limb on the approaching side of the star.

Oblateness has the effect of increasing the maximum deflection that can be attained by a point on the surface, relative to an unperturbed spherical background solution. Some points, such as the locus in the vicinity of the equator only lower in maximum deflection. In the vicinities of the poles, points gain in maximum deflection. In these cases the surface normal remains effectively radial. For intermediate points between a pole and the equator, the maximum deflection generally increases due to tilt even if the compactness coordinate decreases, but also due to tilt, some images (either even or odd beyond some order) are also lost. This is because the maximum deflection depends on the ray direction aside from the angle subtended to the radial direction.

Note that for very compact stars with polar radius near the Schwarzschild photon sphere, higher order images need to graze closer and closer to the photon sphere to complete revolutions before scattering. If the surface is slightly oblate, then an opaque surface quickly truncates higher order images. Alas, for an oblate surface in this case to be valid in the ambient Schwarzschild spacetime, it would have to be effectively massless; it requires substantial rotation speed for such soft equations of state that lead to high compactness, which thus nullifies the Schwarzschild ambient spacetime and photon sphere. The issue of rotational deformation leading to a (highly compact) photosphere which lies within it’s own null future due to high (outgoing) ray curvature is somewhat of open problem to our knowledge but it is thought that the source matter curvature will always be too high to intercept outgoing rays that would otherwise scatter in the ambient spacetime in the absence of the opaque surface (private communication with the NICER collaboration light-curve SWG, Joonas Nättilä, and Juri Poutanen, July 2016, Seattle). Beyond this reasoning, it is not considered even mildly important for realistic experiements because the flux contribution from higher order images is so small and entirely dominated by other modelling inaccuracies.

Note that in principle, for a maximally compact toy equation of state (see, e.g., Lattimer 2015 and references therein), non-rotating scale-free solutions exist within the Schwarzschild photon sphere, but rather close to it, at \(R\approx1.41r_{\rm s}\).

[44]:
spacetime['frequency'] = 205.0 # ~J0030+0451 spin

We’ll reset the mass so that the oblate surface lies entirely, and barely just outside ambient Schwarzschild photon sphere:

[45]:
star['mass'] = 2.695 # choose mass so that surface lies entirely, and barely just outside Schwarzschild photon sphere
[46]:
spacetime.R_r_s
[46]:
1.5077253212974875
[47]:
star.update()
[48]:
everywhere.time_invariant = True
[49]:
photosphere.integrate(energies=energies, threads=4)
[50]:
rtemp_int1 = np.tile(photosphere.signal[0][0], (1,len(everywhere.phases_in_cycles)))
[51]:
plot_2D_pulse(rtemp_int1,
              x=everywhere.phases_in_cycles,
              shift=np.array([0.0]),
              y=energies,
              num_rotations=1.0,
              ylabel=r'Energy (keV)')
_images/multiple_imaging_96_0.png
[77]:
everywhere.time_invariant = False # switches to xpsi/cellmesh/integrator.pyx
# a few cells from this one onwards were executed again, and thus their execution numbers
# are out of order, because this switch was forgotten at first
everywhere._integrator_toggle = False
[78]:
photosphere.integrate(energies=energies, threads=4)
[79]:
rtemp_int2 = photosphere.signal[0][0].copy()
[83]:
plot_2D_pulse((rtemp_int2, rtemp_int1),
              x=everywhere.phases_in_cycles,
              shift=(0.0, 0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              vmin=-0.05,
              vmax=0.05,
              num_rotations=1.0,
              error=True,
              normalise = False)
_images/multiple_imaging_100_0.png
[56]:
everywhere._integrator_toggle = True # switch to final integrator that discretises surface, xpsi/cellmesh/integrator_for_azimuthal_invariance.pyx, for testing
[57]:
photosphere.integrate(energies=energies, threads=4)
[58]:
rtemp_int3 = photosphere.signal[0][0].copy()
[59]:
plot_2D_pulse((rtemp_int3, rtemp_int1),
              x=everywhere.phases_in_cycles,
              shift=(0.0, 0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              vmin=-0.01,
              vmax=0.01,
              num_rotations=1.0,
              error=True,
              normalise = False)
_images/multiple_imaging_104_0.png
[84]:
plot_2D_pulse((rtemp_int3, rtemp_int2),
              x=everywhere.phases_in_cycles,
              shift=(0.0, 0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              vmin=-0.05,
              vmax=0.05,
              num_rotations=1.0,
              error=True,
              normalise = False)
_images/multiple_imaging_105_0.png
[61]:
spacetime.a = 0.0 # spacetime spin parameter (~angular momentum)
spacetime.q = 0.0 # spacetime mass quadrupole moment
[62]:
photosphere.image(reimage = True,
                  reuse_ray_map = False,
                  energies = energies,
                  phases = everywhere.phases_in_cycles * _2pi,
                  sqrt_num_rays = 1024,
                  # squeeze rays towards limb where higher-order images are
                  image_plane_radial_increment_power=0.33,
                  threads = 4,          # OpenMP
                  max_steps = 100000,   # max number of steps per ray
                  epsrel_ray = 1.0e-12) # ray relative tolerance
Imaging the star...
Commencing ray tracing and imaging...
Ray tracing complete.
Ray set cached.
Phase-resolved specific flux integration complete.
Star imaged.
[63]:
plot_2D_pulse((photosphere.images[0]*spacetime.d_sq, rtemp_int1),
              x=everywhere.phases_in_cycles,
              shift=(0.0, 0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              vmin=-0.1,
              vmax= 0.1,
              error=True,
              num_rotations=1.0,
              normalise = False)
_images/multiple_imaging_108_0.png
[64]:
fig = plt.figure(figsize=(10,10))

ref = photosphere.images[3].copy()
ref[ref > math.pi/2.0] *= -1.0
plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)
# rays that scatter have negative constant values <-100
# for map quantities such as the colatitude

ax = plt.gca()
ax.set_xlim([-1.025,1.025])
ax.set_ylim([-1.025,1.025])
veneer((0.05,0.25), (0.05,0.25), ax)
_images/multiple_imaging_109_0.png
[65]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([0.95, 1.0])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_110_0.png
[66]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([-1.0,-0.95])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_111_0.png
[67]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                ref,
                levels=np.linspace(np.min(ref[ref > -100.0]),
                                   np.max(ref[ref > -100.0]),
                                   200),
               cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.025,0.025])
ax.set_ylim([-1.0,-0.9995])
_ = veneer((0.0025,0.01), (0.000025,0.0001), ax)
_images/multiple_imaging_112_0.png

Notice that due to finite oblateness the resolved limb is slightly further from \(y=-1\) that is the case for the effectively spherical star above. For the \(x\) coordinate, the proximity to \(x±1\) depends on the viewing angle and surface oblateness and compactness (besides the ray resolution).

[68]:
fig = plt.figure(figsize=(10,10))

phi = np.copy(photosphere.images[4][...])

for i in range(len(phi)):

    if phi[i] > -100.0:
        if phi[i] > math.pi:
            while phi[i] > math.pi:
                phi[i] -= 2.0 * math.pi
        elif phi[i] < -math.pi:
            while phi[i] < -math.pi:
                phi[i] += 2.0 * math.pi

phi_lvls = np.linspace(np.min(phi[phi > -100.0]),
                       np.max(phi[phi > -100.0]), 100)

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                phi,
                levels=phi_lvls,
                cmap=cm.magma_r)
# rays that scatter have negative constant values <-100
# for map quantities such as the azimuth

ax = plt.gca()
ax.set_xlim([-1.025,1.025])
ax.set_ylim([-1.025,1.025])
veneer((0.05,0.25), (0.05,0.25), ax)
_images/multiple_imaging_114_0.png
[69]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                phi,
                levels=phi_lvls,
                cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([0.95, 1.0])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_115_0.png
[70]:
fig = plt.figure(figsize=(10,10))

plt.tricontourf(photosphere.images[1],
                photosphere.images[2],
                phi,
                levels=phi_lvls,
                cmap=cm.magma_r)

ax = plt.gca()
ax.set_xlim([-0.4,0.4])
ax.set_ylim([-1.0,-0.95])
veneer((0.025,0.1), (0.005,0.025), ax)
_images/multiple_imaging_116_0.png

Custom (phase-dependent)

Let’s now play with localised hot regions so that secondary images in particular can be visualised directly in intensity sky maps.

You now need to reinstalled the package after replacing the contents of xpsi/surface_radiation_field/local_variables.pyx with the exact contents of xpsi/surface_radiation_field/archive/two_spots.pyx. The local_variables.pyx extension module must transform some set of global variables into local variables at the spacetime event defined by the intersection of a ray with the stellar surface. A vector of local variables is then passed to the xpsi/surface_radiation_field/hot.pyx module for evaluation of the specific intensity of radiation, w.r.t a local comoving surface frame, that after Lorentz transformation is transported along the ray to the image plane.

Once you can reinstalled the package, you need to restart the IPython kernel and then execute code cells [1] and [14] above.

[3]:
bounds = dict(frequency = (0.1, 600.0),
              distance = (0.1, 1.0),                     # (Earth) distance
              mass = (1.0, 3.0),                         # mass
              radius = (3.0 * gravradius(1.0), 16.0),    # equatorial radius
              cos_inclination = (0.0, 1.0))              # (Earth) inclination to rotation axis

spacetime = xpsi.Spacetime(bounds=bounds, values=dict(frequency=205.0)) # ~J0030 spin
Creating parameter:
    > Named "frequency" with bounds [1.000e-01, 6.000e+02] and initial value 2.050e+02.
    > Spin frequency [Hz].


Creating parameter:
    > Named "mass" with bounds [1.000e+00, 3.000e+00].
    > Gravitational mass [solar masses].


Creating parameter:
    > Named "radius" with bounds [4.430e+00, 1.600e+01].
    > Coordinate equatorial radius [km].


Creating parameter:
    > Named "distance" with bounds [1.000e-01, 1.000e+00].
    > Earth distance [kpc].


Creating parameter:
    > Named "cos_inclination" with bounds [0.000e+00, 1.000e+00].
    > Cosine of Earth inclination to rotation axis.


[4]:
bounds = dict(super_colatitude = (0.001, math.pi - 0.001),
              super_radius = (0.001, math.pi/2.0 - 0.001),
              phase_shift = (-0.25, 0.75),
              super_temperature = (5.1, 6.8))

primary = xpsi.HotRegion(bounds=bounds,
                            values={},
                            symmetry=True,
                            omit=False,
                            cede=False,
                            concentric=False,
                            sqrt_num_cells=512,
                            min_sqrt_num_cells=10,
                            max_sqrt_num_cells=1024,
                            num_leaves=1024,
                            num_phases=100,
                            num_rays=1024,
                            is_secondary=False,
                            prefix='p')

bounds = dict(super_colatitude = (0.001, math.pi - 0.001),
              super_radius = (0.001, math.pi/2.0 - 0.001),
              phase_shift = (-0.25, 0.75),
              super_temperature = (5.1, 6.8))

secondary = xpsi.HotRegion(bounds=bounds,
                            values={},
                            symmetry=True,
                            omit=False,
                            cede=False,
                            concentric=False,
                            sqrt_num_cells=512,
                            min_sqrt_num_cells=10,
                            max_sqrt_num_cells=1024,
                            num_leaves=1024,
                            num_phases=100,
                            num_rays=1024,
                            is_antiphased=True,
                            prefix='s')

from xpsi import HotRegions

hot = HotRegions((primary, secondary))
Creating parameter:
    > Named "super_colatitude" with bounds [1.000e-03, 3.141e+00].
    > The colatitude of the centre of the superseding region [radians].


Creating parameter:
    > Named "super_radius" with bounds [1.000e-03, 1.570e+00].
    > The angular radius of the (circular) superseding region [radians].


Creating parameter:
    > Named "phase_shift" with bounds [-2.500e-01, 7.500e-01].
    > The phase of the hot region, a periodic parameter [cycles].


Creating parameter:
    > Named "super_temperature" with bounds [5.100e+00, 6.800e+00].
    > log10(superseding region effective temperature [K]).


Creating parameter:
    > Named "super_colatitude" with bounds [1.000e-03, 3.141e+00].
    > The colatitude of the centre of the superseding region [radians].


Creating parameter:
    > Named "super_radius" with bounds [1.000e-03, 1.570e+00].
    > The angular radius of the (circular) superseding region [radians].


Creating parameter:
    > Named "phase_shift" with bounds [-2.500e-01, 7.500e-01].
    > The phase of the hot region, a periodic parameter [cycles].


Creating parameter:
    > Named "super_temperature" with bounds [5.100e+00, 6.800e+00].
    > log10(superseding region effective temperature [K]).


We want to image the photosphere and the default behaviour of Photosphere is insufficient. We therefore subclass Photosphere to provide a custom implementation of a higher-complexity radiation field. The customisation is actually very simple: we must make a property return a vector of global variable values that are relayed to a compiled extension module xpsi.surface_radiation_field.local_variables by the image-plane discretisation integrator. Thus, the bulk of the customisation must be written in xpsi/surface_radiation_field/local_variables.pyx for compilation; this has already been handled for this tutorial (see instructions above).

[5]:
class CustomPhotosphere(xpsi.Photosphere):
    """ Implement custom global variables property. """

    @property
    def global_variables(self):
        """ This method is needed if we also want to invoke the image-plane signal simulator.

        The extension module compiled is surface_radiation_field/archive/local_variables/two_spots.pyx,
        which replaces the contents of surface_radiation_field/local_variables.pyx.

        """
        return np.array([self['p__super_colatitude'],
                          self['p__phase_shift'] * 2.0 * math.pi,
                          self['p__super_radius'],
                          self['p__super_temperature'],
                          self['s__super_colatitude'],
                          (self['s__phase_shift'] + 0.5) * 2.0 * math.pi,
                          self['s__super_radius'],
                          self['s__super_temperature']])
[6]:
class derive(xpsi.Derive):
    def __init__(self):
        pass
    def __call__(self, boundto, caller = None):
        global spacetime
        return spacetime['frequency']
[7]:
photosphere = CustomPhotosphere(hot = hot,
                                elsewhere = None,
                                everywhere = None,
                                values=dict(mode_frequency = derive()))
Creating parameter:
    > Named "mode_frequency" that is derived from ulterior variables.
    > Coordinate frequency of the mode of radiative asymmetry in the
photosphere that is assumed to generate the pulsed signal [Hz].


[8]:
star = xpsi.Star(spacetime = spacetime, photospheres = photosphere)
[9]:
# (Earth) distance
star['distance'] = 0.33
# gravitational mass
star['mass'] = 2.695
# coordinate equatorial radius
star['radius'] = 12.0
# (Earth) inclination to rotation axis
star['cos_inclination'] = 0.0

star['p__phase_shift'] = 0.0
# isotropic blackbody temperature
star['p__super_temperature'] = 6.3
star['p__super_colatitude'] = 1.0
star['p__super_radius'] = 0.2

star['s__phase_shift'] = 0.0
# isotropic blackbody temperature
star['s__super_temperature'] = 6.3
star['s__super_colatitude'] = math.pi - 1.0
star['s__super_radius'] = 0.2
[10]:
star.update()
[11]:
energies = np.logspace(-2.0, np.log10(3.0), 100, base=10.0)
[12]:
photosphere.integrate(energies=energies, threads=4)
[13]:
joint = plot_2D_pulse((photosphere.signal[0][0], photosphere.signal[1][0]),
                          x=hot.phases_in_cycles[0],
                          shift=( hot['p__phase_shift'], hot['s__phase_shift'] ),
                          y=energies,
                          ylabel=r'Energy (keV)',
                          combine=True)
_images/multiple_imaging_131_0.png
[14]:
primary.symmetry = False
secondary.symmetry = False
[15]:
photosphere.integrate(energies=energies, threads=4)
[16]:
_joint = plot_2D_pulse((photosphere.signal[0][0], photosphere.signal[1][0]),
                      x=hot.phases_in_cycles[0],
                      shift=( hot['p__phase_shift'], hot['s__phase_shift'] ),
                      y=energies,
                      ylabel=r'Energy (keV)',
                      combine=True)
_images/multiple_imaging_134_0.png
[19]:
plot_2D_pulse((joint, _joint),
              x=hot.phases_in_cycles[0],
              shift=(0.0, 0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              num_rotations=1.0,
              vmin=-0.000001,
              vmax= 0.000001,
              error=True,
              normalise = False)
_images/multiple_imaging_135_0.png
[20]:
spacetime.a = 0.0 # spacetime spin parameter (~angular momentum)
spacetime.q = 0.0 # spacetime mass quadrupole moment

Now we call the image-plane integrator. The image-plane ray mesh is distributed globally over the stellar image, and we therefore for cross-validation, we need a mesh constituted by a larger number of elements than the moving meshes that discretise just the surface hot regions and near vicinities.

[21]:
photosphere.image(reimage = True,
                  reuse_ray_map = False,
                  energies = energies,
                  phases = hot.phases_in_cycles[0] * _2pi,
                  sqrt_num_rays = 4096, # ~16 million rays to try to converge
                  threads = 4,          # OpenMP
                  max_steps = 100000,   # max number of steps per ray
                  epsrel_ray = 1.0e-12) # ray relative tolerance
Imaging the star...
Commencing ray tracing and imaging...
Ray tracing complete.
Ray set cached.
Phase-resolved specific flux integration complete.
Star imaged.

Let’s examine the phase-energy resolved photon specific flux signal and compare it to the signal computed by via surface discretisation:

[22]:
plot_2D_pulse(photosphere.images[0]*spacetime.d_sq,
                  x=hot.phases_in_cycles[0],
                  shift=0.0,
                  y=energies,
                  ylabel=r'Energy (keV)')
_images/multiple_imaging_140_0.png
[24]:
plot_2D_pulse((photosphere.images[0]*spacetime.d_sq, joint),
              x=hot.phases_in_cycles[0],
              shift=(0.0,0.0),
              y=energies,
              ylabel=r'Energy (keV)',
              cm=cm.RdBu,
              label=r"Pct. frac. diff.",
              num_rotations=1.0,
              vmin=-0.1,
              vmax= 0.1,
              error=True,
              normalise = False)
_images/multiple_imaging_141_0.png
[25]:
sky_map_kwargs = {'panel_indices': (0,1,2,3,4,5),
                  'num_levels': 100, # in intensity field rendering
                  'colormap': cm.Purples_r,
                  'phase_average': False,
                  'annotate_energies': True,  # background from the surface and behind the star
                  'energy_annotation_format': '[%.2f keV]',
                  'annotate_location': (0.025,0.025)}

# you can install ffmpeg with conda in order to animate
animate_kwargs = {'cycles': 4, 'fps': 32}
[26]:
photosphere.image(reimage = True,
                  reuse_ray_map = False,
                  cache_intensities = 1.0, # cache size limit in GBs
                  energies = np.array([0.01,0.1,0.5,1.0,2.0,5.0]),
                  phases = np.linspace(0.0, 1.0, 96) * _2pi,
                  sqrt_num_rays = 512, # do not need as many to see the important imagess
                  threads = 4,
                  max_steps = 100000,
                  epsrel_ray = 1.0e-12,
                  plot_sky_maps = True, # activate if you want to plot frames
                  sky_map_kwargs = sky_map_kwargs,
                  animate_sky_maps = False, # activate if you want to animate
                  free_memory = False, # activate if memory is a concern, to delete ray-map/intensity caches
                  animate_kwargs = animate_kwargs)
Imaging the star...
Commencing ray tracing and imaging...
Ray tracing complete.
Ray set cached.
Intensity caching complete.
Plotting intensity sky maps...
Normalising each sky map panel separately...
Normalised sky map panels separately.
Rendering images for phase numbers [1, 10]...
Rendering images for phase numbers (10, 20]...
Rendering images for phase numbers (20, 30]...
Rendering images for phase numbers (30, 40]...
Rendering images for phase numbers (40, 50]...
Rendering images for phase numbers (50, 60]...
Rendering images for phase numbers (60, 70]...
Rendering images for phase numbers (70, 80]...
Rendering images for phase numbers (80, 90]...
Rendering images for phase numbers (90, 100]...
Intensity sky maps plotted.
Animating intensity sky maps...
Writing to disk: ./images/skymap_animated.mp4...
Intensity sky maps animated.
Star imaged.

If you are executing this notebook, you can view the video file:

[33]:
%%HTML
<div align="middle">
<video width="100%" controls loop>
    <source src="images/skymap_animated.mp4" type="video/mp4">
</video></div>
[28]:
!mkdir images/frames_multiple_imaging
!mv images/*.png images/frames_multiple_imaging/.

Here is a frame for the purpose of the documentation notebook. Each panel displays the photon specific intensity field, on the sky, at a given energy; energy increases from top-left to bottom-right. The intensity field in each panel is normalised over sky direction and phase, for each energy in the sequence.

[29]:
Image("./images/frames_multiple_imaging/skymap_50.png")
[29]:
_images/multiple_imaging_148_0.png

Finally, let’s phase-average the intensity sky maps:

[30]:
sky_map_kwargs['phase_average'] = True
# only one set of panels so why not choose higher res.?
sky_map_kwargs['num_levels'] = 500

photosphere.image(reimage = False, # because we decided not to free_memory earlier
                  energies = np.array([0.01,0.1,0.5,1.0,2.0,5.0]),
                  phases = np.linspace(0.0, 1.0, 96) * _2pi,
                  plot_sky_maps = True,
                  sky_map_kwargs = sky_map_kwargs)
Imaging the star...
Plotting intensity sky maps...
Averaging (specific) intensity over rotational phase...
Averaged (specific) intensity over rotational phase.
Normalising each sky map panel separately...
Normalised sky map panels separately.
Rendering phase-averaged images...
Intensity sky maps plotted.
Star imaged.
[31]:
!mv images/skymap_0.png images/frames_multiple_imaging/skymap_multiple_imaging_phase_averaged.png
[32]:
Image("images/frames_multiple_imaging/skymap_multiple_imaging_phase_averaged.png")
[32]:
_images/multiple_imaging_152_0.png