S13ST_Theory
Theoretical Angular Distribution
The shape of the angular distribution of muon flux can be predicted; the theory gives results for the overall shape and zenithal profile of the distribution, while the Earth's magnetic field introduces a correction primarily in the distribution's azimuthal profile. In the following,
refers to the zenithal angle, which is the angle measured off the local vertical or zenith.
Zenithal Distribution
First of all, Please note: The geometrical argument past groups have used to justify the distribution is incorrect. The argument goes something like this: Muons are formed at a uniform height above the ground (this part happens to be true [Particle Data Group]), and the distance to a point in this muon-formation plane at an angle
from the vertical is . Since the muon flux decays as with distance, the observed flux at an angle
is proportional to . This argument is false and fails to account for the three-dimensional geometry of the situation, as well as the finite width of the detector's acceptance angle.
A real detector intercepts a finite solid angle
of trajectories. Projecting this angle onto the plane of muon formation would actually lead one to expect a distribution, due to the larger portion of this plane the detector intercepts as its angle
increases.
In reality, decay is also a significant factor, as muons lose at least 2 <nop>GeV to the atmosphere due to electronic scattering (which can be modeled with the Bethe formula), as well as other significant effects such as bremsstrahlung, ionization, and pair production, whose cross-section increases with muon energy. The effects are summarized in the Particle Data Group's report on muon stopping power, which can be found at http://pdg.lbl.gov/2012/AtomicNuclearProperties/adndt.pdf.
Combining all factors, the end result is a distribution that goes as for muons in the energy range of about 3 <nop>GeV, with the distribution flattening out to for larger energies [Particle Data Group]. Since our detector integrates over all energies, we chose to parametrize this relationship as
and attempted to measure the exponent .
Obtaining Theoretical Distributions for the Detectors
The detectors used in this experiment have a finite angular resolution, and their angular range can cover a large and rapidly-varying portion of the angular distribution. The flux rate they intercept cannot simply be modeled as:
,
but rather the integral
over the detector's angular acceptance range must be used. This is a very difficult integral to evaluate analytically, so a Monte Carlo simulation was used to numerically evaluate it instead. This has the additional advantage of automatically accounting for the different solid angles of the various detector configurations.
East-West Effect
The Earth's magnetic field is expected to introduce an azimuthal asymmetry in the measured distribution. Intuitively, the primarily positively-charged cosmic-ray primary particles will have their trajectories bent towards the east, causing more particle events from the west than from the east.
To quantify this observation, the concept of rigidity is needed, which is the relativistic momentum per unit charge ( with gyroradius rho) and is a measure of the particle's ability to penetrate a magnetic field. From the point of view of an observer on the Earth's surface, some trajectories are forbidden because they do not lead off to infinity; this means there is a minimum or cutoff rigidity particles must have to reach a certain point on the surface in a certain direction. In the approximation that the Earth's magnetic field is a dipole, this cutoff is given by Størmer's Equation:
Where
and are the angles from the vertical and clockwise from magnetic north, respectively, that describe the incoming particle's trajectory at the surface, and is the geomagnetic latitude of the point on the surface. (The magnitude of the dipole moment is
and the distance from the dipole center is .)
Minneapolis's magnetic latitude is 55°; inserting this into the above equation, we see that the cutoff rigidity from the west is 90% of the cutoff from the east, so integrating over all energies, more particles are expected to arrive from the west than from the east. Previous experimental data (Barber 1949) puts the magnitude of this difference at about 5%.