Point-Source Point-Lens (PSPL) Models

PSPL models involve a single lens moving in front of a single star on the plane of the sky. The majority of PSPL models in BAGLE have parameterizations with input parameters in Solar System barycentric coordinates, unless otherwise specified. The Solar System barycentric (SSB) coordinates are most useful in when modeling astrometry and photometry jointly, which we will build up to.

Photometry-Only

The simplest form of microlens model is one used for generating a photometric light curve only. In these events, the absolute position of the lens and source on the sky are not known. Instead, only the relative separation is known and there is some information about the relative orientation of their separation and relative proper motion vectors on the sky at closest approach.

For classes with photometry only, input parameters typically include

Parameter

Units

Description

\(t_0\)

MJD (day)

Time of closest approach in SSB coordinates.

\(u_0\)

\(\theta_E\)

Closest approach distance.

\(t_E\)

(day)

Einstein crossing time.

\(\pi_{E,E}\)

Microlensing parallax in the East direction.

\(\pi_{E,N}\)

Microlensing parallax in the North direction.

\(b_{sff}\)

Blending source flux fraction [0-1].

\(m_{src}\)

(mag)

Source magnitude.

For microlensing events where parallax should be modeled, the following additional parameters are needed. Note these parameters are typically fixed and won’t need to be varied when we fit data with these models.

Parameter

Units

Description

\(\alpha_L\)

(deg)

R.A. (for parallax)

\(\delta_L\)

(deg)

Dec. (for parallax)

obsLocation

str

earth or satellite name (for parallax)

Photometry and Astrometry

BAGLE’s strength is jointly modeling or fitting photometric and astrometric data sets. In these events, the absolution position of the lens and source on the sky are known and controlled by additional parameters in the model. All orientations of position and velocity vectors in these models are with respect to North and East as defined by the Earth’s equator, even if models are SSB or observed from some other satellite (e.g. Roman).

Parameterizations for photometric+astrometric models are much more varied. First, we can start with an expansion approach where we start with photometric parameters and add more parameters to describe the astrometry. An example of this parameterization is PSPL_PhotAstrom_Par_Param2 with the following parameters:

Parameter

Units

Description

\(t_0\)

MJD (day)

Time of closest approach in SSB coordinates.

\(u_0\)

\(\theta_E\)

Closest approach distance.

\(t_E\)

(day)

Einstein crossing time.

\(\theta_E\)

(mas)

Einstein radius.

\(\pi_S\)

(mas)

Parallax of the source.

\(\pi_{E,E}\)

Microlensing parallax in the East direction.

\(\pi_{E,N}\)

Microlensing parallax in the North direction.

\(X_{S_0,E}\)

(arcsec)

R.A. source position on sky at \(t = t_0\).

\(X_{S_0,N}\)

(arcsec)

Dec. source position on sky at \(t = t_0\).

\(\mu_{S,E}\)

(mas/yr)

Source proper motion in R.A. direction.

\(\mu_{S,N}\)

(mas/yr)

Source proper motion in Dec. direction.

\(b_{sff}\)

Blending source flux fraction [0-1].

\(m_{src}\)

(mag)

Source magnitude.

along with the fixed parameters:

Parameter

Units

Description

\(\alpha_L\)

(deg)

R.A. (for parallax)

\(\delta_L\)

(deg)

Dec. (for parallax)

obsLocation

str

earth or satellite name (for parallax)

Note that the \(X_{S_0}\) source positions are in an arbitrary reference frame and are designed for relative astrometric measurements in a tangential plane projection (i.e. small proper motions, not large proper motions where spherical coordinates are important and the tangential plane projection is no longer valid).

Alternatively, parameterizations can be expressed entirely in physical quantities as is the case for the PSPL_PhotAstrom_Par_Param1 model class with the following parameters:

Parameter

Units

Description

\(m_L\)

(\(M_\odot\))

Lens mass.“

\(t_0\)

MJD (day)

Time of closest approach in SSB coordinates.

\(\beta\)

(mas)

Closest approach distance, projected on sky.

\(d_L\)

(pc)

Distance to the lens.

\(d_L/d_S\)

Ratio of lens distance to source distance.

\(X_{S_0,E}\)

(arcsec)

R.A. source position on sky at t = t0.

\(X_{S_0,N}\)

(arcsec)

Dec. source position on sky at t = t0.

\(\mu_{L,E}\)

(mas/yr)

Lens proper motion in R.A. direction.

\(\mu_{L,N}\)

(mas/yr)

Lens proper motion in Dec. direction.

\(\mu_{S,E}\)

(mas/yr)

Source proper motion in R.A. direction.

\(\mu_{S,N}\)

(mas/yr)

Source proper motion in Dec. direction.

\(b_{sff}\)

Blending source flux fraction [0-1].

\(m_{src}\)

(mag)

Source magnitude.

\(u_0\) Orientation Conventions

In both photometric and astrometric models, the \(u_0\) paramater quantifies both the amplitude of the \(\vec{u}_0\) and has a \(\pm\) sign convention used to break the degeneracy between the lens passing to the East or the West of the source. FILL IN MORE. GOULD REFERENCE.

Gaussian Process Noise

We occasionally need models that have underlying noise in them as well (often in excess of the noise from measurements alone). This is particular useful to model events where the source star is stochastically variable from spots or activity. But this noise model can also capture systematic red noise from atmospheric and instrumentation sources.

We utilize a Gaussian Process to model this additional noise using the celerite package. The default GP kernel includes two temporally correlated noise terms: a Matern-3/2 and a damped simple harmonic oscillator. We also include a white-noise jitter term. The GP classes require additional parameters to specify the noise as shown in the table below.

Parameter

Units

Description

\(\log \sigma_{GP}\)

\(\log \rho_{GP}\)

\(\log S_0\)

\(log \omega_{0,GP}\)

PSPL Developer Classes