Triangles inside a circle to represent raypaths inside an ideal Earth

[Remixex]

Homework Statement

Hello everyone, I am having some problems with what the professor calls basic geometry....

Imagine we have our planet Earth divided into two homogeneous segments. Mantle and core. a ray takes off with an angle $\beta$ and travels straight through the mantle a linear distance $a$ which can also be expressed as a surface distance of 110 degrees (to go from degrees to arc distance just multiply by 111.2).



A second ray takes off with angle $\alpha$, smaller than $\beta$ and enters the core with an incident angle $\theta_1$ and refracts with an outgoing angle $\theta_2$. Travels through the core, refracts again through the mantle and reaches the surface, creating what's known as a "shadow zone" (P wave shadow zone is between 110 and 140 degrees, but $\alpha$ doesn't necessarily reach exactly at 140 degrees, it's arbitrary)



What are the angles of incidence and refraction? (This will yield the velocity at the core)

Relevant Equations

Law of sines, cosines and Snell's law

I have managed to get some of the required distances and angles. I have the distance $a$, the velocity inside the mantle, the total radius of the Earth $R_t$ as well as mantle and core radii. I have also figured out the angle of incidence, however I cannot get the refracted angle with the available information. I have drawn and created a couple of more distances and angles I have been able to figure out, such as $\epsilon$ which contains the refracted angle, $\epsilon=\delta+\theta_1$ however I do not know how to get $\delta$...
Any help or direction would be appreciated. I have reached several times at solutions that turn out to be wrong, and I have started over several times. A high resolution version of the image can be found in imgur : (imgur.com)/a/Y0rvkg9

 

[PeroK]

What is the precise question you are trying to answer?

 

[Remixex]

What is the precise question you are trying to answer?

What are the angles of incidence and refraction essentially. What is $\theta_2$ ? In terms of known variables

 

[PeroK]

What are the angles of incidence and refraction essentially. What is $\theta_2$ ? In terms of known variables

That's not a precise question!

If you went up to a physicist or mathematician in the street and asked him or her that and only that, how could they answer?

Excuse me, sir or madam: "What are the angles of incidence and refraction essentially. What is $\theta_2$ ? In terms of known variables?"

 

[Remixex]

I will rephrase then, what is the velocity of the P wave through the core?
I apologize for being imprecise.

That's not a precise question!

If you went up to a physicist or mathematician in the street and asked him or her that and only that, how could they answer?

Excuse me, sir or madam: "What are the angles of incidence and refraction essentially. What is $\theta_2$ ? In terms of known variables?"

 

[PeroK]

I will rephrase then, what is the velocity of the P wave through the core?

According to Google it's about $11.3 \ km/s$.

 

[Remixex]

I will post this in a more precise way on the relevant forum thread

According to Google it's about $11.3 \ km/s$.

Thanks for your help

 

[PeroK]

I will post this in a more precise way on the relevant forum thread Thanks for your help

My guess is you're trying to determine the size of the shadow zone?

 

[Remixex]

My guess is you're trying to determine the size of the shadow zone?

I posted the question again at https://www.physicsforums.com/threa...mic-waves-through-an-idealized-earth.1012000/. Hopefully it will show?

 

[berkeman]

I posted the question again at https://www.physicsforums.com/threa...mic-waves-through-an-idealized-earth.1012000/. Hopefully it will show?

In general it is not good to post new versions of the same question. I'll lock this thread for now -- please continue the discussion in the new thread.