Point Source

Point-source microlensing utilities (JAX).

This module provides JAX-friendly implementations of core operations for point-source gravitational microlensing by up to three point-mass lenses.

Functions

  • lens_eq: complex lens equation mapping image-plane coordinates z to source-plane coordinates w.

  • lens_eq_det_jac: determinant of the Jacobian of the lens mapping, used to compute magnification through |det J|^{-1}.

  • critical_and_caustic_curves: critical curves in the image plane and their mapped caustics in the source plane.

  • _images_point_source: image positions for given source position(s) by solving the corresponding complex polynomial.

  • _images_point_source_sequential: helper that tracks images across a sequence of sources by reusing previous roots as initialization.

  • mag_point_source: total point-source magnification (sum over images).

All functions are compatible with jax.jit and support batched evaluation where appropriate. Complex numbers encode 2D coordinates using the x + i y convention.

Coordinate conventions

  • Binary/triple lens configurations use the mid-point coordinate system for coefficient construction. mag_point_source shifts input source-plane coordinates to the center of mass when required and handles the inverse shift internally.

References

  • Bartolic, F. caustics (MIT License) — original inspiration and some coefficient-generation routines, modified here for JAX and extended triple lens support.

microjax.point_source.critical_and_caustic_curves(npts: int = 200, nlenses: int = 2, **params)

Compute critical curves and mapped caustics.

Parameters

  • npts (int, optional) – Number of sampling points on the unit circle used to construct the critical-curve polynomial. Defaults to 200.

  • nlenses (int, optional) – Number of point-mass lenses (1, 2, or 3). Defaults to 2.

  • params (dict) –

    Lens configuration parameters:

    • nlenses = 1: no additional parameters.

    • nlenses = 2: s (separation) and q (mass ratio m2/m1). Internally we set a = s / 2 and e1 = q / (1 + q).

    • nlenses = 3: s, q, q3 (third mass ratio), r3 (radius), psi (angle). Internally a = s / 2, e1 = q / (1 + q + q3), e2 = 1 / (1 + q + q3).

Returns

(z_cr, z_ca) where z_cr contains critical curves and z_ca the mapped caustics. Both arrays have shape (N_branches, npts).

Return type

tuple[jax.Array, jax.Array]

Notes

  • For nlenses = 1 the critical curve is the unit circle and the caustic collapses to the origin.

  • Output is shifted from mid-point to center of mass for consistency with the rest of the library.

microjax.point_source.lens_eq(z: Array, nlenses: int = 2, **params) → Array

Lens equation mapping image-plane z to source-plane w.

Parameters

  • z (jax.Array) – Complex scalar or array of image-plane coordinates.

  • nlenses (int, optional) – Number of point-mass lenses (1, 2, or 3). Defaults to 2.

  • params (dict) – Lens configuration parameters. For nlenses = 2 expect a (half separation) and e1 (mass fraction at +a). For nlenses = 3 expect a, r3, psi, e1 and e2.

Returns

Source-plane coordinates with the same shape as z.

Return type

jax.Array

Notes

All arithmetic is performed in complex form; gradients propagate through JAX as expected.

microjax.point_source.lens_eq_det_jac(z: Array, nlenses: int = 2, **params) → Array

Determinant of the Jacobian of the lens mapping at z.

Parameters

  • z (jax.Array) – Complex scalar or array of image-plane coordinates.

  • nlenses (int, optional) – Number of point-mass lenses (1, 2, or 3). Defaults to 2.

  • params (dict) – Lens configuration parameters matching those accepted by lens_eq().

Returns

Real array with the same shape as z storing det J(z).

Return type

jax.Array

Notes

Point-source magnification is |det J|^{-1} for each image.

microjax.point_source.mag_point_source(w, nlenses=2, **params)

Total point-source magnification for 1–3 lens configurations.

Parameters

  • w (jax.Array) – Complex scalar or array of source-plane coordinates.

  • nlenses (int, optional) – Number of point-mass lenses (1, 2, or 3). Defaults to 2.

  • params (dict) –

    Lens parameters depend on nlenses:

    • nlenses = 1: no additional parameters required.

    • nlenses = 2: s (separation) and q (mass ratio m2/m1). Internally a = s / 2 and e1 = q / (1 + q); the source is shifted to the center of mass for the polynomial construction.

    • nlenses = 3: s, q, q3, r3 and psi. Internally a = s / 2, e1 = q / (1 + q + q3), e2 = 1 / (1 + q + q3), and the same center-of-mass shift is applied.

Returns

Real-valued magnification with the same shape as w.

Return type

jax.Array

Notes

Magnification is computed as the sum of |det J|^{-1} over valid image branches returned by _images_point_source.