Caption: Understanding Lensing Mass and Velocity in Particle Astrophysics

[FatPhysicsBoy]

Homework Statement

I just came across a couple of expressions in a textbook I don't particularly understand.

Caption: "A point lensing mass L moving with velocity v perpendicular to the line of sight. O is the observer and S' is the projected position of the source in the plane of the lens.

The textbook is D H Perkins - Particle Astrophysics 2nd edition, Pg 163.

Homework Equations

An excerpt from the textbook is ".. the right-angled triangle AS'L gives us $LS'^{2} = AS'^{2} + AL^{2}$, where $LS' = D_{L}\theta_{s}$, $AS' = D_{L}\theta_{s}(min)$ .."

The Attempt at a Solution

Tried to refresh geometry/trig, looked at sine and cosine rules and different combinations of lines and angles. I still don't understand the last two equations, how does multiplying by the angle give you LS' and AS'? Looks simple but why you can do it escapes me..

 

[jedishrfu]

Small angles in radian measure approximate their sin() value and that maybe that's what's going on here.

 

[gneill]

In an Astrophysical context, $D_L$ is probably assumed to be very large with respect to other dimensions, making the angles small (as stated by @jedishrfu) so that the sides AS' and LS' are essentially equal to the arc lengths subtended by $D_L$ swept out by those angles. The rest is Pythagoras.