Missing factor in focusing factor & errors in analytic path length / travel time equations

[sjoerd-bouma]

While writing this up for my thesis (draft attached), I found

1. mistakes in the analytic formulas for the path length and travel time: Original: Corrected: These were checked against numerical integration: check_new_analytic_traveltimes.pdf check_new_analytic_pathlength.pdf and observed to agree within the 1e-4 numerical integration precision. Note that the previous formulas
   • used a separate ad-hoc formula for the deep ice
   • regularly raised ArithmeticErrors and fell back on numerical integration I see no reason why any issues should have arisen in these (clearly) finite integrals in the first place, but someone else can plug the equations into WolframAlpha again if they fancy.

[sjoerd-bouma]

2. The focusing factor was missing a factor in the $\phi$ direction. Previously, it assumed the focusing only takes place in the $\theta$ direction. But focusing also takes place in the $\phi$-direction, where the angular distance changes as $\sin \theta d\phi$. A straightforward example to show that the previous version was wrong is the ground-bounce experiment (vertical propagation) - previously, this would give a focusing factor of $n{ice}/n{surface}$ instead of $n{ice}^2/n{surface}^2$. However, for most neutrino trajectories the $\phi$-factor is very close to 1 as $n\approx n_{ice}$ for most of the ray path. Derivation in pdf attached $\phi$-factor only:

[sjoerd-bouma]

3. Other minor corrections: For some reason, a separate expression was derived for $dy/dz$ and used in the code that looks really complicated: I suggest to replace this with the much simpler expression derived already in C.12 These were numerically verified to give the same result.

There is also a 'typo' in equation 10: which is however correct in the appendix (C.19) and in the code (i.e. the first $\sqrt{...}$ factor in (10) should be $1/\sqrt{...}$ instead)

[sjoerd-bouma]

I'm attaching a draft of the appendix of my thesis in which the focusing factor is derived. focusing_factor_derivation.pdf

The above issues have been fixed in PR #649

[sjoerd-bouma]

As requested by @cg-laser , here are the checks on the impact of the new focusing factor (including the $\phi$-term) for a deep (100 m ) and a shallow (3 m) antenna in Greenland ice focusing_factor_comparison.pdf

As expected, for the most part, this results in a slight decrease in the focusing factor (of up to 10%, though the 90th percentile is < 1 %).