Caption: Understanding Lensing Mass and Velocity in Particle Astrophysics

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The discussion focuses on understanding the geometric relationships in lensing mass and velocity in particle astrophysics, specifically referencing equations from D H Perkins' textbook. The key point is the use of right triangles to relate distances and angles, with LS' and AS' being derived from small angle approximations in a large distance context. Participants highlight that for small angles, the sine of the angle approximates the angle itself in radians, which simplifies the calculations. The Pythagorean theorem is also noted as essential for connecting these geometric relationships. Overall, the thread emphasizes the importance of geometry and trigonometry in astrophysical lensing scenarios.
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Homework Statement



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

keuc5h.png
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..
 
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Small angles in radian measure approximate their sin() value and that maybe that's what's going on here.
 
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.
 
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