S14TimeDilation

This experiment seeks to determine the effects of time dilation through the study of muon detection rates at varying atmospheric depths. Balloon borne Geiger tube systems logged muon detections from a depth of 1030g/cm² to 10g/cm² aboard a high-altitude balloon flight. The resulting flux counts were exponentially fit to extract an average muon decay lifetime of:

. This value is 0.2σ off of the globally accepted value of

. The gamma factor necessary to obtain this value was 11.19, which implied an average measured muon velocity of 0.996c.

I. Introduction

Muons have a mean lifetime of 2.2μs[1], which means cosmic ray muons, with a peak creation at 15km, would need to travel above the speed of light to be observed in the quantities seen at ground level [2]. Time dilation extends their apparent lifetimes, allowing for muon penetration to high atmospheric depths [3]. This experiment uses balloon-borne Geiger tubes which travel from ground level (1030g/cm²) to above the peak muon creation range of 170 g/cm². Geiger tubes will measure flux rates over the range of atmospheric depths, and the results will be analyzed to determine muon decay lifetimes. Energy loss due to atmosphere will be accounted for as will the propagation of non-muon particle species [4]. This will allow extraction of the gamma factor for time dilation.

II. Theory

Muons are created in the upper atmosphere when protons from cosmic rays collide with gas nuclei. Protons comprise about 90% of these cosmic rays and when the protons collide, the amount of energy released amounts in a cascade of particles heading down towards sea level. Many of the produced particles are lighter charged mesons, called muons, which are made via pion decay governed by the following decay formulas [5]:

Muon flux, or count rate N , is described by the differential decay relation shown in Equation 1.

(1)

These particles travel at relativistic speeds, so the increased lifetime (Tau) is defined as in Equation 2.

(2)

is the Lorentz factor defined by, which is the muon speed with respect to the speed of light. This implies a length contracted penetration distance :

(3)

While decaying through the atmosphere, energy is lost to the atmospheric medium at a rate established by the Bethe formula for air (or N₂) shown in Equation 4 [5].

(4)

This loss of energy to the atmosphere reduces velocity, ranging out the lower energy muons. The average energy of muons at creation is approximately 6GeV and this drops to 2GeV at sea level due to the described energy loss. The spectrum of muons at sea level for two different incident angles is shown in Figure 1a. A Bethe plot for muons is shown in Figure 1b. Notice that the energy loss does not fluctuate noticeably over the range of muon energies.

Figure 1a: Muon energy spectrum at sea level for incident angles of 0° and 75° [3]

Figure 1b: Bethe function for range of cosmic ray particle energies, notice the function is essentially flat for any medium in the range of muon energies studied [5]

Atmospheric depth is a theoretical function of medium density (X) and was obtained with measurements of pressure (P) in units of milliBars using Equation 5 [6]. X=1.02P(5)

A plot of atmospheric depth versus altitude is shown in Figure 2. This plot demonstrates equation 6, which gives an empirical relation of depth to altitude z .

(6)

The two primed variables have values of z'=8.33 km and X-=1030 g/cm², and are obtained via the fit obtained for the depth versus altitude data.

Figure 2: Measured atmospheric depth versus altitude. The fit of the data is the function

(original figure)

During the progression through depths from X_0 to X, high energy muons remain relatively uninfluenced by the atmospheric medium, while the lower energy muons range out. The energy threshold of muons which will range out over this depth range is approximately:

. This average loss of energy was then related to the flux according to the power law relation in Equation 7 [7].

(7)

With these two factors, the decay and ranging out of muons, a differential flux model can be constructed. Not accounting for the time dilation effect on the decay times, Equation 8 describes an un-relativistic prediction model of differential flux with respect to atmospheric depth.

(8)

The k' in Equation 8 is the absorption of two constants, the scaling constant in the muon energy spectrum and the average energy loss to the atmosphere. The second term then describes the ranging out of muons in the upper atmosphere. The first term is obtained using simple kinematic adjustments from time to altitude using Equation 5, which gives us the necessary conversion of the differentials:

.

Accounting for the effect of relativistic motion, using Equations 3 and 4 with the Lorentz factor rewritten as , where is the average initial energy of the muons and

. The differential flux with respect to atmospheric depth, using this established Lorentz factor, then became:

(9)

Approximating, using the known values of k and α[3], and then using separation of variables to solve, yielded the exponential function shown in Equation 10 describing the flux of muons as a function of atmospheric depth.:

(10)

Equation 10 was then used to fit the experimental results obtained in order to prove the existence of time dilation effects on the lifetimes of cosmic-ray muons.

III. Experimental Setup

Apparatus consisted of a Digilent Nexus Field-Programmable Array (FPGA), an internal and an external PModTMP Temperature Sensor, a PModSF Memory Chip, an MXP-Lab constructed pressure sensor, two vertically stacked RM-60 Geiger Tubes, and 4 AA batteries as a power source. This setup was duplicated for redundancy and is pictured in Figure 3. It was secured in a 1” insulating foam case and wrapped in conductive adhesive to prevent interference. Apparatus was then attached to a helium filled, high-altitude, polyurethane balloon via four 40lb test lines. Total weight of 1.84kg was within Federal Aviation Administration limit of 2.7kg.

Figure 3. Configuration of the components inside protective foam is shown above, this setup was duplicated for each of the two Nexus boards (original figure)

Full incidence detection intensity is given by given by I = cos²Θ [3]. Stacked Geiger tubes reduce angle of incidence. Coincidence counts logged to PmodSF Memory Chip.

IV. Results

After data extraction, it was seen that on the first flight data failed to compile after 3 minutes. On the second flight, it was seen that data from both sides was collected during the entire balloon flight and recovery. The data form flight two is shown in Figures 4a,b,c., binned in 1-minute intervals.

Figure 4a. Left Board Counts vs. Atmospheric Depth(original figure)

Figure 4b. Right Board Counts vs. Atmospheric Depth(original figure)

Figure 4c. Total Summed Counts vs. Atmospheric Depth(original figure)

This shows a preliminary match to expected results as given by Rossi [9]. Coincidence counts were then similarly binned and are shown in Figure 5a,b,c. Coincidence counts were below expectations and showed significant deviation from left to right boards. The source of this remains indeterminate and led to exclusion of coincidence results.

Figure 5a. Left Board Coincidence Counts vs. Atmospheric Depth (original figure)

Figure 5b. Right Board Coincidence Counts vs. Atmospheric Depth (original figure)

Figure 5c. Summed Total Coincidence Counts vsAtmospheric Depth (original figure)

V. Analysis

After initial binning, Origins 8.6 Software was used for fit analysis. Vertical error bars reflect Poisson counting statistics, as variation in counts was larger by orders of magnitude than component error. Horizontal error bars, included in Figure 6 below but too small to be observed, arose from the Pressure sensor. Using one minute interval averaging, with the error at each measurement, the final error was given by Equation 11.

(11)

Figure 6. Truncated Total Binned and Summed Counts vs. Atmospheric Depth, here the fit line and equation are indicated in red

Above peak muon creation of 170 , the data was truncated as the pattern of particle creation does not follow decay relation [10]. Data was additionally truncated to remove excess time spent during ground level measurements. This achieved a Chi Squared of 2.826. This value is within acceptable range, as estimations were used in this experiment. It also deemed likely that the inability to distinguish particle types had significant effect on results.

Using averaged muon velocities, the fit was then used with Equation 11. Observable muons not likely to have ranged out in the high-atmosphere travel with velocities between 0.994c to 0.998c, or average velocity <v> = 0.996c This revealed a decay time of

. This value was impossible to achieve without use of the relativistic Lorentz factor. This corresponding Lorentz factor found through quadratic reduction was 11.19±.97. Error on decay lifetime was small, but likely far underestimated due to averaging processes. Error on Lorentz factor comes from quadratic solution parameters. Although significant averaging was used, this verifies the established theoretical prediction requiring relativistic Lorentz factor to account for observed muon flux.

VI. Conclusions

Utilizing derived averages to obtain a fit for muon detection rates, the resulting χ²_RED=2.86. The time dilation adjusted decays were found to be τ₀=2.19±.04x10^-6s. This is 2.2σ away from the mean decay lifetime of accepted τ₀=2.2x10^-6s [3]. The time dilation factor necessary for the derived decay times was found to be optimized at 11.2±.97. This varies form accepted value of 9.02 by 2.3σ. Despite goodness of fit and similarities to previous results [10], this experiment was unable to quantifiably verify time dilation as the only mechanism for muon penetration. Further study with models utilizing non-averaged values and incorporating sub-peak muon production methods is necessary to achieve confidence.

VII. Acknowledgements

The author wishes to thank lab partner Rob Carlon, lab supervisor Kurt Wick, advisor Greg Pawloski, and ballooning expert James P. Flaten.

VIII. References

[1] Hess, Victor “The Electrical Conductivity of the Atmosphere and Its Causes.” Constable & Company. 1928.

[2] Rossi, Bruno, Norman Hilberry, and J. Barton Hoag. "The Variation of the Hard Component of Cosmic Rays with Height and the Disintegration of Mesotrons." Physical Review 57.6 (1940): 461-69. Print.

[3] J.J. Beatty, Matthews, and S.P. Wakel. “Cosmic Rays.” Bartol Research Inst., Univ. of Delaware (2009). Print.

[4] Obrien, Keith. “Atmospheric Cosmic Rays and Solar Energetic Particles at Aircraft Altitudes” Northern Arizona University (1995). Print.

[5] Coulon, P. “Interactions of Particle Matter” Heidelberg, Germany (2010). Web.

[6] Mizuno, T. “Cosmic-Ray Background Flux Model Based On Gamma-Ray Large Area Space Telescope Balloon Flight Engineering”Astrophysical Journal (2004). Web. http://www-heaf.hepl.hiroshima-u.ac.jp/~mizuno/GLAST/Balloon/BalloonInfo/Info.html

[7] Obrien, Keith. “Atmospheric Cosmic Rays and Solar Energetic Particles at Aircraft Altitudes” Northern Arizona University (1995). Print

[8] M. Aglietta. “Muon ‘Depth – Intensity’ Relation Measured by LVD Underground Experiment and Cosmic-Ray Muon Spectrum at Sea Level” University of Bologna (1998). Print.

[9] Rossi, Bruno. “Interpretation of Cosmic-Ray Phenomena” Rev. Mod. Phys. 20, 537 (1948). Print.

[10] Easwar, Nalini & MacIntire, Douglas A. “Study of the effect of relativistic time dilation on cosmic ray muon flux” Smith College (1990). Print.

-- Main.corga005 - 17 May 2014