SingleLenses.md

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Single lenses

The lens equation for a single lens is

$$ u=x-\frac{1}{x}$$

For a given lens-source angular separation $u$ there are two images $x$ solving this equation. We note that all angular distances are in units of the Einstein angle $\theta_E$.

Point-Source-Point-Lens

If the source is point-like, the magnification is given by the famous Paczynski formula

$$ \mu = \frac{u^2+2}{u\sqrt{u^2+4}}$$

In VBMicrolensing, this formula is obtained through the function PSPLMag as follows:

VBM = VBMicrolensing.VBMicrolensing()

u=0.1;  # Source-lens separation in Einstein radii
Mag = VBM.PSPLMag(u)
print(f"PSPL Magnification = {Mag}") # Output should be 10.037...

Extended-Source-Point-Lens

For extended sources, the magnification depends on $\rho$, the source radius normalized to the Einstein angle, and can be calculated through elliptic integrals. In order to make VBMicrolensing as fast as possible, we provide pre-calculated tables in the file "ESPL.tbl". This file is automatically loaded upon the first calculation involving Extended-Source-Point-Lenses (ESPL).

VBM = VBMicrolensing.VBMicrolensing()

u = 0.1 # Source-lens separation in Einstein radii
rho = 0.01 # Source radius in units of the Einstein angle

Mag = VBM.ESPLMag2(u, rho) # Call to the ESPLMag2 function with these parameters
print(f"\nMagnification of Extended-source-point-lens = {Mag}\n")  # Output should be 10.050.....

The current range for the pre-calculated table is $10^{-4} \leq \rho \leq 10^{+2}$. Sources smaller than the minimum are considered equal to the minimum. Sources larger than the maximum generate an error message.

By default, VBMicrolensing works with uniform sources. We will introduce Limb Darkening in a later section: arbitrary Limb Darkening laws can be implemented in VBMicrolensing.

Astrometry

For a Point-Source, in the reference frame in which the lens is in the origin, the centroid of the images is simply

$$ \bar x = \frac{u}{u^2+2} + u$$

If you need astrometry calculations together with magnification, you have to turn astrometry on by VBM.astrometry = True and read the results in VBM.astrox1. This works in the same way for PSPLMag and ESPLMag2.

VBM = VBMicrolensing.VBMicrolensing()

u = 0.1 # Source-lens separation in Einstein radii
rho = 0.01 # Source radius in units of the Einstein angle

VBM.astrometry = True # We want astrometry

Mag = VBM.ESPLMag2(u, rho) # Call to the ESPLMag2 function with these parameters
print(f"Magnification of Extended-source-point-lens = {Mag}\n")  # Output should be 10.050.....
print(f"Centroid shift = {VBM.astrox1 - u}\n")  # Output should be 0.0493.....

Note that VBM.astrox1 reports the centroid position with respect to the lens. The centroid position with respect to the source is VBM.astrox1 - u.

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