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: