Reframing Microlensing: Blog 11, Worksheet 4.1, Problem 3

When speaking about microlensing, it is often easier to refer to angular quantities in units of θE. Let’s define u = β/θE and y = θ/θE.

Show that the lens equation can be written as: \[ u = y - y^{-1} \]
A)  Since we know \[ \Theta_E = \left( \frac{4GM_L \pi_{rel} }{ c^2 } \right)^{ \frac{1}{2}} \] and \[ \beta =  \Theta_E - \frac{4GM_L \pi_{rel}}{\Theta c^2} \] we can find \[u =  \frac{ \beta}{\Theta_E} = \frac{\Theta_E - \frac{4GM_L \pi_{rel}}{\Theta c^2} }{\left( \frac{4GM_L \pi_{rel}}{ c^2} \right)^{ \frac{1}{2}} } \] And \[ y = \frac{ \Theta}{\Theta_E} = \frac{\Theta}{ \left(  \frac{4GM_L \pi_{rel} }{ c^2} \right)^{ \frac{1}{2}} } \] Rearranging that gives: \[ u = \frac{ \Theta}{\Theta_E} - \frac{\frac{ \Theta_E^2}{\Theta}}{ \Theta_E} = \frac{ \Theta}{\Theta_E} - \frac{ \Theta_E}{\Theta}= y - y^{-1} \]  \[ u = y - y^{-1} \]

Solve for the roots of y(u) in terms of u. These equations prescribe the angular position of the images as a function of the (mis)alignment between the source and lens. For the situation given in Question 2(f) and a lens-source angular separation of 100 µas (micro-arcseconds), indicate the positions of the images in a drawing.
B)  \[ u = y - y^{-1} \] Multiplying everything by y gives:  \[ yu = y^2 - 1 \] \[ y^2 - uy -1 = 0\] Plugging this into a quadratic equation gives: \[ y = \frac{ u \pm \sqrt{ u^2 + 4} }{2} \] The fact that we have two possible answers is consitent with the two images often seen in gravitational lensing.

Source

In the case of 2(f), we already have \(\beta\) and \( \Theta_E\) so we can rearrange the equation to get:  \[ \frac{ \Theta}{\Theta_E}  = \frac{ \frac{ \beta}{\Theta_E} \pm \sqrt{ \left( \frac{ \beta}{\Theta_E} \right)^2 + 4} }{2} \]  \[ \frac{ \Theta}{1 \times 10^{-6}}  = \frac{ \frac{ 1 \times 10^{-4}}{1 \times 10^{-6}} \pm \sqrt{ \left( \frac{1 \times 10^{-4}}{1 \times 10^{-6}} \right)^2 + 4} }{2} \] \[ \Theta_E = 10^{-4} , -10^{-10} \: arcseconds \] This would look something like this, though this is extremely exaggerated: