Monte Carlo Simulations of Timing Structure: Analysis

3. Analytical Analysis

We begin the derivation of the expected shape of the light arrival times by considering anisotropically emitting point source moving vertically in a straight line at speed c from zenith to theobservatory altitude where it impacts the ground. We follow this with a generalization to inclinedemission tracks. It is expected that the timing properties of showers, which can bethought to consistof many emission tracks, will follow from the properties of these single tracks.

3.1 Zenith Emission Tracks

The amount of time it takes a visible photon moving in a direction that is at an angle Theta tothevertical, traveling from an emission altitude Z to the observatory altitude Z_obs is

(1)

where Eta(z) is related to the atmospheric index of refraction n(z)=(1+Eta(z)) and is relatedto the atmospheric density

where Eta₀ and Rho₀ are the values at sea level. We are interested in thetiming as a function of the distance x of the impact from the origin which is defined as the pointwhere the original emitting particle will hit. We define d as (Z-Z_obs)as the vertical distance the photon travels. We then have

and since are interested in the timing of the wavefront relative to the time of impact of the v=cemitting particle we subtract the time the emitting particle will take to reach altitudeZ_obs.The last integral in Equation 1 is the amount of matter(gm/cm²) a photon traverses going vertically from altitude Z to altitudeZ_obs and we can define g(Z) to be the depth in gm/cm²at altitude Z. T becomes:

(2)

where we have made allowance for the possibility of the emitting particle moving in a non-verticaldirection with a direction Theta_i. If we define

	(3)
and
	(4)

then Equation 2 can be rearranged as:

(5)

This is the equation of a hyperbola. We show in Figure 1 thetiming structure for various emissionaltitudes for a v=c vertical (Theta_i=0) emitting particle.

3.2 Inclined Emission Tracks

For our model of an inclined shower we consider a v=c particle inclined from zenith by angleTheta_i. The timing structure is made up of point sources along this emissiontrack which again give hyperbolas as specified in Equation 5. Now however the horizontallocations of the emission points are displaced by

as the vertical altitude d of the emission points change. In Equation 5 x is replaced byx-x_step.Figure 2 shows the arrival timestructures from different altitudes for an emission track inclined by 30 deg. from zenith. The origin(x=0) has been designated as the impact point of the emission track. To can get a clearer idea of thetiming structure we make a timing correction,

which is designed to make a plane wave moving at speed v=c down the emission track appear to'simultaneously' impact the ground at all points x. If we apply this correction to Figure 2 we get Figure 3. If we make thesechanges to Equation 5 to derive what the equations are that govern Figure3 we get

(6)

It can be shown that the cross term T^. x can be eliminated by a rotation of axis in thex-t plane. This results again in an equation for a hyperbola but with the symmetry axis slightlyrotated in the x-t plane. This rotation can just barely be seen in Figure3.

We now look at the shape of the wavefront at points transverse to the direction of inclination.i.e. along the y axis. It can be seen that again from a particular altitude d the emission point is offsetby x_step. The timing curve from this point is a hyperbola which is symmetricin rotation about its z axis. The transverse y axis going through the origin slices up through thisoffset form. It can be shown then that y axis timing is equivalent to the timing at points

Substituting this y for x in Equation 5 gives after rearrangement:

(7)

which is again a hyperbola. We show in Figure 4 the transverse(y-axis) timing structures for aemission track inclined 30 deg from the vertical along the x axis.

3.3 Discussion

We can make several observations about the results shown in Figures1, 3, and 4. First, forinclined emission tracks the timing structure is independent of inclination. All inclinations showevidence of timing focusing. For zenith emission tracks this focusing occurs in an annulus whichhas a inner radius of 80 m and a width of about 80 m. The timing width of the light pulse at thecenter of the annulus (radius=120 m) is less then .5 ns. From Figures 3and 4 we see that for aninclined emission track the annulus now has the shape an ellipse. The long axis of the ellipse has aninner radius that goes like 80 m/cos(Theta_i). The inner radius of the transverseaxis has grown to 100 m. Across the annulus we can also see that the envelope of the timing has aconical behavior spatially.

A second observation is that the timing structure at the center of a shower is a goodmeasurement of the altitude of emission distribution. From Equation 2 we can see that at x, y=0 i.e.at Theta=0, the timing structure as a function of altitude of emission d for azenith emission track, Theta_i=0, is:

(8)

Thus Figure 1 and Equation 8 indicate that at the core of a shower theshape of thetiming pulse reflects the structure of the altitude of emission distribution of the shower. The timingpulse shape can be used to determine an altitude of shower-maximum which is an important quantityto know when trying to determine the energy of a shower. To calibrate the timing pulse the timing ofthe annulus can be used. The timing there is very sharp and from Figure 1 appears for zenithshowers to be at t=5.5 ns.