Ratio of the Earth's core radius to total radius via seismic turning point

[peeballs]

Homework Statement

S-waves are observed to an epicentral distance of about 103 degrees on the Earth (but no further because of the presence of the fluid core). Assuming that the S-wave velocity is constant throughout the mantle (rays are straight lines), compute the ratio of the core radius to the total radius of the Earth (You can ignore the crust).

Relevant Equations

B = sqrt(mu/rho), X(p) = the integral from 0 to turning point of p/(sqrt(u^2-p^2))dz

I've attached a figure I've made; I know I'm to assume the Earth is a perfect sphere in this case. Assuming the 103 degrees is measured as latitude, I've calculated the distance in kilometers (Xp in the second equation above) to be 1.1453e4 km. I know I need u = p at the turning point, but not where to go from here. I'm not convinced this figure is correct, but I'm doing my best.

 

[haruspex]

Homework Statement:: S-waves are observed to an epicentral distance of about 103 degrees on the Earth (but no further because of the presence of the fluid core). Assuming that the S-wave velocity is constant throughout the mantle (rays are straight lines), compute the ratio of the core radius to the total radius of the Earth (You can ignore the crust).
Relevant Equations:: B = sqrt(mu/rho), X(p) = the integral from 0 to turning point of p/(sqrt(u^2-p^2))dz

I've attached a figure I've made; I know I'm to assume the Earth is a perfect sphere in this case. Assuming the 103 degrees is measured as latitude, I've calculated the distance in kilometers (Xp in the second equation above) to be 1.1453e4 km. I know I need u = p at the turning point, but not where to go from here. I'm not convinced this figure is correct, but I'm doing my best.

The top of the upper diagram is cut off, and I do not understand the lower diagram. The angle marked as 103° is clearly less than 90, and there is no basis given for the wider angle where there seems to be a reflection off the core.
From what I can see of the upper diagram, it looks more like what I expected to see, but I've no expertise on this subject, so I could be wrong.