All basic magnification functions illustrated in Single Lenses and Binary Lenses allow the calculation of the centroid of the images in a frame centered on the lens in Einstein units.
However, in order to exploit this information and calculate the centroid positions in the sky during a microlensing event, we need some additional information:
1 - The source heliocentric proper motion in declination and right ascension in mas/yr.
2 - The source parallax in mas.
3 - The Einstein angle in mas.
The lens parallax and proper motion components are obtained from the source parallax and proper motion components once the microlensing parallax components
VBMicrolensing contains the following astro-photometric functions:
-
PSPLAstroLightCurve(parameters, times), extending PSPLLightCurveParallax
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_list, y2_list] -
ESPLAstroLightCurve(parameters, times), extending ESPLLightCurveParallax
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_list, y2_list] -
BinaryAstroLightCurve(parameters, times), extending BinaryLightCurveParallax
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_list, y2_list] -
BinaryAstroLightCurveOrbital(parameters, times), extending BinaryLightCurveOrbital
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_list, y2_list, separations_list] -
BinaryAstroLightCurveKepler(parameters, times), extending BinaryLightCurveKepler
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_list, y2_list, separations_list] -
BinSourceAstroLightCurveXallarap(parameters, times), extending BinSourceExtLightCurveXallarap
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_s1_list, y2_s1_list, y1_s2_list, y2_s2_list] -
BinSourceBinLensAstroLightCurve(parameters, times), extending BinSourceBinLensLightCurve
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_s1_list, y2_s1_list, y1_s2_list, y2_s2_list, separations_list] -
TripleAstroLightCurve(parameters, times), extending TripleLightCurveParallax
returns[magnifications, source_centroid_dec, source_centroid_ra, lens_centroid_dec, lens_centroid_ra, y1_list, y2_list]
Similarly to their corresponding original functions, these new functions take a parameters list and a list of observation times as arguments. The output contains a list of magnifications calculated at the epochs in times, centroid positions in (dec,ra) for source and lens, and source positions in the lens reference frame.
Here is a full example with the PSPLAstroLightCurve. For the other functions we just have to change the standard parameters accordingly, as explained in the corresponding sections Light Curves, Parallax, Orbital Motion, Binary Sources.
import VBMicrolensing
VBM = VBMicrolensing.VBMicrolensing()
import numpy as np
import math
import matplotlib.pyplot as plt
# Standard parameters for PSPL with parallax
t0 = 5034.0
tE = 27.0
u0 = 0.1
paiN = -0.1
paiE = +0.2
# Additional parameters required for centroid trajectory
muS_Dec = -3.597 # Source proper motion (Dec) in mas/yr
muS_RA = -2.263 # Source proper motion (RA) in mas/yr
paiS = 0.12 # Source parallax in mas
thetaE = 5.15 # Einstein angle in mas
VBM.SetObjectCoordinates("17:51:40.2082 -29:53:26.502"); # Coordinates of the microlensing event
# Here we fill the array of parameters
pr = [u0,math.log(tE),t0, paiN,paiE, # Standar light curve parameters for PSPL including parallax
muS_Dec,muS_RA, paiS, thetaE] # Additional parameters required for centroid trajectory
t = np.linspace(-3*365.25+t0,3*365.25+t0,10000) # Array of observation epochs
results = VBM.PSPLAstroLightCurve(pr,t)
magnifications = results[0]
plt.plot(t,magnifications) # Here we plot the light curve as usual
Now, let us plot the centroids for the magnified source and for the lens
sourcecentroid = [results[1],results[2]]
lenscentroid = [results[3],results[4]]
fig, ax = plt.subplots(figsize=(5,5))
ax.plot(sourcecentroid[1],sourcecentroid[0])
ax.plot(lenscentroid[1],lenscentroid[0])
ran = 10
ax.set_ylim(-ran,ran)
ax.set_xlim(ran,-ran)
ax.set_xlabel('dRA (mas)')
ax.set_ylabel('dDec (mas)')
In this figure the source centroid is in blue and the lens centroid is in yellow. We can appreciate the wiggles due to annual parallax (the time span is six years in this figure). The source is deflected at the time of the microlensing event.
We can subtract the average source proper motion to check the astrometric deflection
fig, ax = plt.subplots(figsize=(5,5))
ax.plot(np.array(sourcecentroid[1]) - muS_RA*(t - t0)/365.25,np.array(sourcecentroid[0]) - muS_Dec*(t - t0)/365.25)
ran = 3
ax.set_ylim(-ran,ran)
ax.set_xlim(ran,-ran)
Here we see the astrometric deflection along with the parallax wiggles, which have not been subtracted.
Finally, we may combine lens and source centroid if we know the blending ratio
fig, ax = plt.subplots(figsize=(5,5))
g = 0.
combinedcentroid = VBM.CombineCentroids(results,g)
ax.plot(combinedcentroid[1],combinedcentroid[0],'b')
g = 0.1
combinedcentroid = VBM.CombineCentroids(results,g)
ax.plot(combinedcentroid[1],combinedcentroid[0],'g')
g = 1
combinedcentroid = VBM.CombineCentroids(results,g)
ax.plot(combinedcentroid[1],combinedcentroid[0],'y')
ran = 10
ax.set_ylim(-ran,ran)
ax.set_xlim(ran,-ran)
ax.set_xlabel('dRA (mas)')
ax.set_ylabel('dDec (mas)')
In this figure the blue line is the centroid trajectory for zero blending (basically the centroid for the magnified source), the green line is for blending = 0.1 and the yellow line for blending = 1. By increasing the blending, the lens dominates the centroid more and more.
The lens centroid for binary lenses is not the center of mass but the center of light, which depends on the flux ratio, if both components are luminous. The flux ratio between the two lenses is determined using the mass ratio parameter VBM.lens_mass_luminosity_exponent. The default value is 4. This works also for planetary systems, since the flux ratio would be negligible. If we want to force the secondary to be dark, then we may set VBM.turn_off_secondary_lens = True. In this way, all the lens flux will come from the primary.
For binary sources, the centroid is determined using the flux ratio as already explained in Binary Sources.