The BAGLE Python Package for Bayesian Analysis of Gravitational Lensing Events

TL;DR

It is shown that Roman GBTDS will detect significant microlensing parallax signals for events that are 2x shorter in duration than from ground-based surveys, enabling confident identification of isolated stellar-mass black holes that can be modeled both astrometrically and photometrically with BAGLE for precise mass determinations.

Abstract

We present the open-source Python package, BAGLE (Bayesian Analysis of Gravitational Lensing Events), which enables modeling and joint fitting of photometric and astrometric data sets. We describe the model parameterizations and present the equations for microlensing events containing either a point-source, point-lens or a finite-source, point-lens geometry both with and without microlensing parallax due to the motion of the Earth or a satellite around the Sun. Conversions between different coordinate reference frames are also derived. We compare our model light curves to those from other papers and microlens modeling software, finding good agreement, although with some differences in finite-source models at a ~1% level detectable with upcoming observations from space-based facilities. We also use BAGLE to demonstrate the impact of changing lens mass, lens distance, and blended source flux fraction on photometric lightcurves and astrometric trajectories in preparation for upcoming Gaia data releases and the launch of the Nancy Grace Roman Space Telescope and its Galactic Bulge Time Domain Survey (GBTDS). In particular, we show that Roman GBTDS will detect significant microlensing parallax signals for events that are 2x shorter in duration than from ground-based surveys. Additionally, long-duration events with durations of $\t_{E,\odot} >$ 100 days will yield microlensing parallax uncertainties of $σ_{π_E} <$ 0.01 with Roman, enabling confident identification of isolated stellar-mass black holes that can be modeled both astrometrically and photometrically with BAGLE for precise mass determinations. BAGLE is an open-source code and community development is encouraged.

Figures (21)

• Figure 1: The BAGLE python package includes microlensing event models, data, model fitters, and numerous utility functions.
• Figure 2: The basic geometry of a point-source, point-lens microlensing event, including parallax, toward the Galactic bulge as seen from Earth. The lens ( black) and the unlensed source ( orange) move in small parallax ellipses aligned with the ecliptic as seen from the geocentric perspective when at a distance of 4 kpc and 8 kpc, respectively. The compass rose for equatorial ( purple) and galactic ( green) coordinate systems are also shown.
• Figure 3: A BAGLE model for point-source, point-lens (PSPL) with parallax with a dark lens of $M_L$ = 1 $M_\odot$, $\mu_{\mathrm{rel},\sun}$ = 4 mas/yr, $\beta$ = 0.5 mas, $d_L$ = 3 kpc, $d_S$ = 8 kpc, and observed from Earth towards the bulge. The top panel shows the position of the lens and the unlensed source. The middle panel shows the two sets of lensed source images. The bottom panel shows the flux-weighted centroid as would be observed on sky. The circle illustrates the Einstein radius ($\theta_E =$ 1.3 mas) centered on the lens position at time $t_0 =$ 57200 MJD.
• Figure 4: The on-sky photometry ( top) and astrometry ( middle) for the PSPL event from Figure \ref{['fig:lens_geometry']}. The observed centroid position of the lensed source is shown as a solid line in the astrometric panels. Color indicates time as shown in the top panel horizontal axis. The astrometry in different reference frames with the proper motion (PM) removed ( bottom left) or PM and parallax removed ( bottom right) show the astrometric microlensing signal traces an ellipse. The complete set of parameters is $M_L$ = 1 M$_\odot$, $\beta$ = 0.5 mas, $X_{S,0}$ = [0.00, 0.50] mas, $\mu_S$ = [10, 0] mas/yr, $\mu_L$ = [0, 0] mas/yr, $d_L$ = 3 kpc, $d_S$ = 8 kpc, $b_{SFF}$ = 1, $mag_S$ = 19 mag, $t_0$ = 57200 MJD, $\alpha$ = 262.5 deg, $\delta$ = -30 deg.
• Figure 5: A microlensing event as viewed from heliocentric ( orange) and true geocentric ( purple) perspectives. We note that the heliocentric frame is rectilinear while the true geocentric frame is non-inertial. The separation, $|\boldsymbol{u}_{\sun}(t)|$ ( top), and amplification, $A(t)$ ( bottom) illustrate the differences between $\boldsymbol{u}_{\boldsymbol{0},\sun}$ and $\boldsymbol{u}_{\boldsymbol{0},\earth_r}$ and $t_{0,\sun}$ and $t_{0,\earth_r}$. The complete list of heliocentric parameters for this event are $\alpha_L=262.5\degr$, $\delta_L=-30\degr$, $t_{0,\sun} = 60000$, $M_L = 0.1 M_\odot$, $\beta = 0.1$ mas, $\boldsymbol{X}_{\boldsymbol{S,0},\sun} = [0.0, 0.1]$ mas, $\boldsymbol{\mu}_{L,\odot} = [0, 0]$ mas/yr, $\boldsymbol{\mu}_{S,\odot} = [3, 0]$ mas/yr, $d_L = 3$ kpc, $d_S = 8$ kpc, $b_{sff} = 1.0$.