The possibility to construct a unit sphere of access that describes the cosmic radiation allowed to an Earth-orbiting spacecraft is discussed. It is found that it is possible to model the occluded portion of the cosmic ray sphere of access as a circular projection with a diameter bounded by the satellite-Earth horizon. Maintaining tangency at the eastern edge of the spacecraft-Earth horizon, this optically occluded area is projected downward by an angle beta which is a function of the magnetic field inclination and cosmic ray arrival direction. This projected plane, corresponding to the forbidden area of cosmic ray access, is bounded by the spacecraft-Earth horizon in easterly directions, and is rotated around the vertical axis by an angle alpha from the eastern direction, where the angle alpha is a function of the offset dipole latitude of the spacecraft

Humble, J. E.

Shea, M. A.

Smart, D. F.

English

NASA Technical Reports Server

34.0 SH 6.1-14
EMPIRICAL MODELFOR THE EARTH'S COSMICRAY SHADOWAT 400 KM:
PROHIBITED COSMICRAY ACCESS
J. E. Humble
Department of Physics, University of Tasmania
Hobart, Tasmania, 7001, Australia
D. F. Smart and M. A. Shea
Air Force Geophysics Laboratory
Hanscom AFB, Bedford, MA., 01731, U.S.A.
ABSTRACT
It is possible to construct a unit sphere of access that de-
scribes the cosmic radiation allowed to an earth-orbiting
spacecraft. In the upper hemisphere of the allowed portion
of the cosmic ray sphere of access, the cosmic ray cutoffs
can be ordered by application of Stormer theory in offset
dipole coordinates. In the downward hemisphere, in westerly
directions, the cosmic radiation is allowed at large zenith
angles well below the spacecraft-earth horizon, with particles
being able to reach satellites at 400 km altitude from large
zenith angles of _40 ° at azimuthal directions equatorward
of west in both the northern and southern hemispheres. We
have found it is possible to model the occluded portion of the
cosmic ray sphere of access as a circular projection having a
diameter bounded by the satellite-earth horizon. Maintaining
tangency at the eastern edge of the spacecraft-earth horizon,
this optically occluded area is projected downward (toward the
earth) by an angle _ which is a function of the magnetic field
inclination and cosmic ray arrival direction. This projected
plane, corresponding to the forbidden area of cosmic ray access,
is bounded by the spacecraft-earth horizon in easterly direct-
ions, and is rotated around the vertical axis by an angle
from the eastern direction, where the angle _ is a function of
the offset dipole latitude of the spacecraft.
1. Introduction. It is of considerable interest to evaluate the primary
cosmic ray flux which is able to reach a satellite in earth orbit from
any specified direction of arrival. The evaluation requires a knowledge
of the geomagnetic cutoffs for all four pi steradins of possible arrival °
directions at all points along the spacecraft orbit. In practice, in
order to keep the computational problem within manageable (and economic)
bounds, cutoffs are generally calculated for a selected set of directions
at each of a chosen set of representative directions along the orbit
(Humble et al., 1979). It is then necessary to interpolate cutoffs for
directions intermediate to those for which explicit calculations have
been performed for a specific location, and for locations intermediate
to those for which calculations have been undertaken for particular
directions of particle arrival. For the upper hemisphere of arrival
directions Smart and Shea (1977) showed that the cutoffs are reasonably
https://ntrs.nasa.gov/search.jsp?R=19850026774 2020-03-20T16:51:13+00:00Z
SH 6.1-14
341
well ordered, for some locations, by Stormer theory. Humble et al.
(1979) pointed out that at least some of the exceptions were due to the
intervention of the 'shadow cone'.
Recent interest has focussed on particles arriving at the spacecraft
from the lower, earthward facing, hemisphere. We have previously report-
ed finding that primary cosmic rays are able to reach a satellite orbit-
ing at quite low altitudes from zenith angles considerably larger than
that of the local earth horizon. For a spacecraft at 400 km these angles
are at least 140° at all latitudes we have investigated, and reach 150°
at one location (Humble et al., 1983; Humble, 1983). We report here
results from some further calculations of this type, and then describe a
possible method for modelling the bounds of that portion of the unit
sphere of access to which primary particles of any energy are unable to
gain access because of the presence of the solid earth.
2. Method. A trajectory tracing technique has been devised to search
rigidity/zenith-angle space for accessible arrival directions (Humble et
al., 1983). The search proceeds, for a given location and azimuth, in
the directions of increasing zenith angle and decreasing rigidity, until
no further accessible directions of arrival can be found. A number of
searches has been performed using this technique. Each was started at
a zenith angle of I00 ° which is above the local satellite-earth horizon
at all altitudes considered (Humble, 1984). It was found that, for ze-
nith angles greater than that of the satellite-earth horizon, primary
particles are only able to reach the satellite from generally westerly
directions. Such particles experience a V x B force having a positive
radial component in the final stages of t_eir-approach to the satellite.
Their trajectories consequently have positive upwards curvature, and the
local zenith angle of arrival can be greater than that of the satellite-
earth horizon. The range of accessible azimuths increases with altitude,
as would be expected.
3. Results at 400 km. A considerable number of trajectories have been
_alculated for the 400 km altitude. In analyzing these results we have
found that it is possible to construct a unit sphere of access which de-
, scribes the access of primary cosmic ray particles to any given detector.
In the relatively simple case of a ground-level location the upper half
of this sphere is totally allowed and the lower hemisphere is totally
forbidden, due to the presence of tlre earth. For a satellite in earth
" orbit the situation is more complicated. All of the upper hemisphere is
allowed to cosmic rays of some energy, as is part of the downward hemi-
sphere. The latter may be divided into two parts. One is essentially a
continuation of the allowed upper hemisphere - an annulus of allowed di-
rections at zenith angles ze such that 90° _ ze < Zh, where zh is the
zenith angle of the horizon seen by the spacecraft. The second region
lies at ze > zh, opens to the west, and is due to the V x B effect dis-
cussed in tile previous section. The remainder of the _phc_e of access
represents forbidden directions of arrival. We have found that it is
possible to model the forbidden, or occluded, portion of the unit sphere
of access, in an empirical manner and to varying degrees of accuracy, by
the following technique.
SH 6.1-14
342
Consider the equatorial plane of the unit sphere surrounding the
spacecraft. Project the sphere onto a plane calibrated as a polar plot
in zenith (as radius) and azimuth. Rotate the disc corresponding to the
equatorial plane of the sphere through an angle _ about a magnetic north
south axis, until its projection, now an ellipse, on the polar plot has
a minor axis which matches the distance between the largest and smallest
values of zenith indicated. The rotated plane is then relocated, and sub-
sequently rotated about a vertical axis, such that it is tangent to the
satellite-earth horizon at the inner surface of the unit sphere at an ang-
le _ from the easterly direction. In our present model _ is equal to the
offset dipole latitude of the spacecraft. Figure 1 shows examples of the
fit between the model curve and the points calculated for particular dir-
ections of arrival. The rms difference is about 3°; however, the great-
est differences occur at positions where there is the largest deviation
between the invariant latitude and the offset dipole latitude contours.
4. Results for Higher Altitudes. In general, it has been found that
the largest accessible zenith angle for a given latitude, longitude, and
azimuth, would also increase with increasing altitude. More detailed re-
sults are given by Humble et al. (1983) and Humble (1983). We have not
yet applied the empirical model described in this paper to altitudes of
1000 km and above. Preliminary calculations have, however, been made.
As the altitude increases above 1250 km a range of large zenith angles
in generally easterly directions begins to become accessible, while smal-
ler (but still below-horizon) zenith angles in the same azimuthal direct-
ions remain inaccessible. The higher altitude permits particles approach-
ing the general region of the sate!lite at low altitudes from the west to
pass above the top of the atmospher_ more than one gyro-radius beneath
the satellite. Such particles will arrive at the satellite from easterly
directions at large zenith angles. The following table summarizes the
results which have been obtained.
Altitude km Maximum Zenith Angle
400 150°
600 156°
800 166°
1000 172°
1250 180°
Note that a zenith of 180° for an altitude of 1250 km means that primary
cosmic ray nuclei are able to reach a satellite at that altitude from the *
nadir direction, directly underneath it, for at least one location some-
where on the orbit. This does not, however, mean that all possible di-
rections of arrival are accessible at such a location. Easterly arrival ,
directions at below-horizon zeniths are still forbidden.
References.
Humble, J.E: Proc. Astron.Soc.Australia, 5, 265-267, 1983.
Humble, J.E., D.F.Smart, and M.A.Shea: 16-th ICRC, Kyoto, Conference
_, 4, 303-308, 1979.
Humble, J.E., D.F.Smart, and M.A.Shea: 18th ICRC, Bangalore, Conference
Papers, 442-445, 1983.
Humble, J.E., AFGL-TR-84-0258, 1984.
Smart, D.F., and M.A.Shea: 15th ICRC, Plovdiv, Conference Papers, II,
256-261, 1977.
SH 6.1-14
343
LT LT
W.ST I _T
ECLAT. _._ SOUTH IB]LRT. _._ S_UTH ElaLRI'. 4.1Z
0 NORTH _T 0 NORTH L_ 0 NORTH
_T
_ _T
" ED£RT'_. n 80UTH EI)LqT_. 71 SOUTH £D..RT_. R 8QUTH
Figure 1. Polar projections of the downward hemisphere beneath an orbit-
ing satellite. The occluded area beneath a satellite to which access is
- completely forbidden to galactic cosmic radiation of a_ ener_ is the
area enclosed _ the large dots in each projection. In this projection
the observer is at the center of the spacecraft. This projection is
looking downward from the center of the spacecraft; concentric rings
represent 15 degree projections from the spacecraft equator to the nadi,-.
Zero degrees is upward from the top of the spacecraft, 90 degrees is the
spacecraft equator, and 180 degrees is in the nadir direction. At the
400 km altitude the local satellite-earth horizon is 109.8 ° from the
zenith. The large open dots in each panel indicate where the maximum
accessible zenith directions were determined _ the cosmic r_ trajecto_-
tracing method. The line connected with small solid squares indicates
the result of applying the model described in this paper.


Empirical model for the Earth's cosmic ray shadow at 400 KM: Prohibited cosmic ray access

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