Case Study: Extending the Tractor to do Strong Gravitational Lensing

This tutorial (really more of a case study) goes through extending the Tractor for a new kind of astronomical source. Hi, Phil!

We want to create a new kind of Source object: a strongly gravitationally lensed quasar. The lens will have a visible component: a DevGalaxy galaxy profile, as well as a dark component that defines its mass. The lens will produce 2 to 4 images of the quasar.

The lens model produces only approximate estimates of the brightness of the multiple quasar images, so we will need a “fudge factor” for the magnitudes predicted by the lens model.

We want to create a class to hold our Lensed Quasar:

from tractor import MultiParams
from tractor.sdss_galaxy import DevGalaxy

class LensedQuasar(MultiParams):
    def getNamedParams():
        return dict(light=0, mass=1, quasar=2, magfudge=3)

We chose to make it inherit from MultiParams because we want to think of it as being composed of the light (visible component of the lens that determines its appearance), mass (the dark mass of the lens that determines its lensing behavior), and the quasar being lensed. We also have magfudge, our fudge-factor for the quasar’s brightness at each image.

We want our LensedQuasar to be a Source that can be manipulated by the Tractor, though. We therefore have to implement the interface described by Source; we have to make our LensedQuasar quack like a Source duck.

To do this, we add the methods defined in Source to our LensedQuasar:

from tractor import PointSource

class LensedQuasar(MultiParams):
    # ... as before ...

    def getModelPatch(self, img):
        # We start by rendering the visible lens galaxy.
        patch = self.light.getModelPatch(img)

        # We will use the lens model to predict the quasar's image positions.
        positions,mags = self.mass.getLensedImages(self.light.position, self.quasar)
        # 'positions' should be a list of RaDecPos objects
        # 'mags' should be a list of Mags objects

        for pos,mag,fudge in zip(positions, mags, self.magfudge):
            # For each image of the quasar, we will create a PointSource
            ps = PointSource(pos, mag + fudge)
            # ... and add it to the patch.
            patch += ps.getModelPatch(img)

        return patch

    def getParamDerivatives(img):

In the getModelPatch method, we have to return a Patch object: a synthetic rendering of our LensedQuasar as it would appear in the given Image. We will do that by combining the appearance of self.light – the visible component of the lens – with the multiple images of the quasar whose positions and brightnesses are estimated by the lensing model, self.mass.

Now, what is the mass going to look like? It is going to have parameters that we want the Tractor to be able to optimize, so it has to be a Params. Actually, as you might have guessed, it just has to quack like a Params. Since our mass is just going to have a few parameters, we could inherit from ParamList:

from tractor import ParamList

class LensingMass(ParamList):

    def getNamedParams():
        return dict(mass=0, radius=1)

    def getStepSizes(self):
        '''We're using units of solar masses and arcsec'''
        return [1e12, 0.1]

    def getLensedImages(self, mypos, quasar):

The getLensedImages function is the one we’re going to call from LensedQuasar.getModelPatch() to predict the lensed image properties.

Let’s fill in the blanks and get the code to run. To create a LensedQuasar object, we’ll have to create its components. We will mock up the Quasar and MagFudge classes. Currently Quasar doesn’t even have any parameters, and that’s ok:

from tractor import RaDecPos, Mags
from tractor.sdss_galaxy import GalaxyShape

class Quasar(ParamList):

class MagFudge(ParamList):

if __name__ == '__main__':
    # Create properties of the lensing galaxy:
    pos = RaDecPos(234.5, 17.9)
    bright = Mags(r=17.4, g=18.9, order=['g','r'])
    # GalaxyShape( re [arcsec], ab ratio, phi [deg] )
    shape = GalaxyShape(2., 0.5, 48.)
    light = DevGalaxy(pos, bright, shape)

    mass = LensingMass(1e14, 0.1)

    quasar = Quasar()

    # Four parameters for up to four images.
    fudge = MagFudge(0., 0., 0., 0.)

    # Create a LensedQuasar object from its components.
    lq = LensedQuasar(light, mass, quasar, fudge)

    print 'LensedQuasar params:'
    for nm,val in zip(lq.getParamNames(), lq.getParams()):
        print '  ', nm, '=', val

and this will print:

LensedQuasar params:
   light.pos.ra = 234.5
   light.pos.dec = 17.9
   light.brightness.g = 18.9
   light.brightness.r = 17.4 = 2.0
   light.shape.ab = 0.5
   light.shape.phi = 48.0
   mass.mass = 1e+14
   mass.radius = 0.1
   magfudge.param0 = 0.0
   magfudge.param1 = 0.0
   magfudge.param2 = 0.0
   magfudge.param3 = 0.0