Gravitational Lensing and Angular Diameter Distances

[maxhersch]

Homework Statement

Given this diagram, the problem is to find an expression for β/Θ_E in terms of X/Θ_E and Y/Θ_E.

Homework Equations

β = Θ – α(Θ)
D_sβ = D_sΘ – D_lsα'(Θ)

The Attempt at a Solution

I really only need help starting this problem. In my textbook and every document I can find online about gravitational lensing it says that the distance from the lens to the source is not simply the difference between the distance from the source to the observer and the distance from the lens to the observer. They say that this is because these distances are actually angular diameter distances and that we will learn about them later on.

What I would like to know is if you are given the distance to the source and the distance to the lens, how do you get the distance between the source and lens? The textbook seems to tell you how not to calculate it, but then they don't explain how to calculate it until 5 chapters later. Meanwhile in my homework I have this problem along with another problems that asks me to use distance to source and lens to find the Einstein radius of the lens, which also requires a value for β. Maybe I am missing something obvious but any help would really be appreciated.

 

[anathema]

Thank you for your question. I understand your frustration with not being able to find a clear explanation for this problem. I will do my best to help you get started on finding an expression for β/ΘE in terms of X/ΘE and Y/ΘE.

First, let's define some variables:
- Θ is the angular diameter distance from the observer to the source
- α(Θ) is the deflection angle of light passing near a massive object, such as a galaxy or a star
- β is the apparent angular position of the source as seen by the observer
- Ds is the angular diameter distance from the lens to the source
- Dls is the angular diameter distance from the lens to the observer

Now, we can use the equations you provided to start finding an expression for β/ΘE. We know that β = Θ – α(Θ), so we can substitute this into the second equation: Dsβ = DsΘ – Dlsα'(Θ).

Next, we can use the small angle approximation, which states that for small angles, the sine of the angle is approximately equal to the angle itself. This is useful in gravitational lensing because the deflection angle, α(Θ), is usually very small. So, we can rewrite the equation as: Dsβ = DsΘ – Dlsα(Θ).

Now, we can rearrange this equation to solve for β/ΘE: β/ΘE = (DsΘ – Dlsα(Θ)) / (DsΘE)

Finally, we can use the definition of the angular diameter distance, ΘE = Ds / Dls, to substitute for DsΘE in the equation: β/ΘE = (DsΘ – Dlsα(Θ)) / (ΘE * Dls).

I hope this helps you get started on finding an expression for β/ΘE. Remember to always check your units and make sure they cancel out to give you the correct units for the final answer. Good luck with your homework!