Point Mass

Point Mass

Usage

point  
  -0.03     #x-coord  (= x centroid)  
  -0.04     #y-coord  (= y centroid)  
   0.10     #theta_e2  (= $\theta_E^2$, strength of the point mass, log option available)  

The deflection angle for a point mass at the origin is defined as:

$\alpha_1 = \frac{\theta_E^2}{r^2} x_1 \quad \text{and} \quad \alpha_2 = \frac{\theta_E^2}{r^2} x_2$

where:

$r^2 = x_1^2 + x_2^2$

Here, $\theta_E^2$ is the strength of the point mass, related to the physical mass $M$ via:

$\theta_E^2 = \frac{4GM}{c^2 D_d}$

The factor $\left( \frac{D_{ds}}{D_s} \right)$ is incorporated separately in the config to convert this to a scaled deflection angle.

Calculating Mass $M$

To compute the mass $M$ from a given $\theta_E^2$, use:

$\theta_E^2 \cdot \frac{D_{ds}}{D_s} = \frac{4GM}{c^2} \cdot \left( \frac{D_{ds}}{D_s D_d} \right)$

Thus, with known distances from cosmology, one can back out the physical point mass $M$.

Einstein Radius

• $\theta_E$ is the Einstein radius of the point mass assuming a source at $z = \infty$.
• To get the Einstein radius for a source at finite redshift:

$\theta_{E,\text{eff}}^2 = \theta_E^2 \cdot \frac{D_{ds}}{D_s}$

Notes

• The third parameter theta_e2 (i.e., $\theta_E^2$) supports the log: option.
• When using log: (log base 10):

$\theta_E^2 = 10^{\text{theta\_e2\_gleeconfig\_value}}$

• Units: $\theta_E$ (and thus $\theta_E^2$) should be in arcsec, to match the unit convention of other lens models in GLEE.
• log: option is avialable for #theta_e2