PX158 - G3 - gravitational lensing

general relativity predicts that light will bend in a gravitational field
the angle of deflection due to gravitational lensing, $α$ , by a mass, $M :$

α = \frac{4 G M}{c^{2} r} $ $ n i m p - t h e a n g l e o f d e f l e c t i o n f o r l i g h t f r o m a s t a r n e a r t h e l i m b o f t h e s u n : $ $ Δ α = {1.73}^{^{″}}

in $1919$ , measurements were made that helped prove general relativity

einstien ring radius:

θ_{E} = \sqrt{\frac{4 G M}{c^{2} D}}

where, $M =$ mass of gravitational lens object, $D =$ observer-lens distance $=$ lens-source distance

note: need perfect alignment
micro-lensing: $M \sim M_{⊙}, D \sim k p c ⟹ θ_{E} \sim m i l l i - a r c s e c o n d s$
not spatially resolvable, but magnification can be seen in brightness
galaxy-galaxy gravitational lensing: $M \sim M_{g a l a x y}, D \sim M p c ⟹ θ_{E} \sim a r c s e c o n d s$
$M \sim M_{g a l a c y c l u s t e r}, D \sim \frac{1}{2} D_{o b s e r v a b l e u n i v e r s e} ⟹ θ \sim a r c m i n u t e s$